understanding mathematical problem solving

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

• The problem has important, useful mathematics embedded in it.
• The problem requires high-level thinking and problem solving.
• The problem contributes to the conceptual development of students.
• The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
• The problem can be approached by students in multiple ways using different solution strategies.
• The problem has various solutions or allows different decisions or positions to be taken and defended.
• The problem encourages student engagement and discourse.
• The problem connects to other important mathematical ideas.
• The problem promotes the skillful use of mathematics.
• The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

• It must begin where the students are mathematically.
• The feature of the problem must be the mathematics that students are to learn.
• It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
• What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
• Can the activity accomplish your learning objective/goals?

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

• Allows students to show what they can do, not what they can’t.
• Provides differentiation to all students.
• Promotes a positive classroom environment.
• Advances a growth mindset in students
• Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

• YouCubed – under grades choose Low Floor High Ceiling
• NRICH Creating a Low Threshold High Ceiling Classroom
• Inside Mathematics Problems of the Month

Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

• Dan Meyer’s Three-Act Math Tasks
• Graham Fletcher3-Act Tasks ]
• Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

• The teacher presents a problem for students to solve mentally.
• Provide adequate “ wait time .”
• The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
• For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
• Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

• “Everyone else understands and I don’t. I can’t do this!”
• Students may just give up and surrender the mathematics to their classmates.
• Students may shut down.

Instead, you and your students could say the following:

• “I think I can do this.”
• “I have an idea I want to try.”
• “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

• Provide your students a bridge between the concrete and abstract
• Serve as models that support students’ thinking
• Provide another representation
• Support student engagement
• Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Problem Solving in Mathematics Education

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• First Online: 28 June 2016

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• Peter Liljedahl 6 ,
• Manuel Santos-Trigo 7 ,
• Uldarico Malaspina 8 &
• Regina Bruder 9  

Part of the book series: ICME-13 Topical Surveys ((ICME13TS))

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Problem solving in mathematics education has been a prominent research field that aims at understanding and relating the processes involved in solving problems to students’ development of mathematical knowledge and problem solving competencies. The accumulated knowledge and field developments include conceptual frameworks to characterize learners’ success in problem solving activities, cognitive, metacognitive, social and affective analysis, curriculum proposals, and ways to foster problem solving approaches. In the survey, four interrelated areas are reviewed: (i) the relevance of heuristics in problem solving approaches—why are they important and what research tells us about their use? (ii) the need to characterize and foster creative problem solving approaches—what type of heuristics helps learners think of and practice creative solutions? (iii) the importance for learners to formulate and pursue their own problems; and (iv) the role played by the use of both multiple purpose and ad hoc mathematical action types of technologies in problem solving activities—what ways of reasoning do learners construct when they rely on the use of digital technologies and how technology and technology approaches can be reconciled?

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• Mathematical Problem
• Prospective Teacher
• Creative Process
• Digital Technology
• Mathematical Task

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Mathematical problem solving has long been seen as an important aspect of mathematics, the teaching of mathematics, and the learning of mathematics. It has infused mathematics curricula around the world with calls for the teaching of problem solving as well as the teaching of mathematics through problem solving. And as such, it has been of interest to mathematics education researchers for as long as our field has existed. More relevant, mathematical problem solving has played a part in every ICME conference, from 1969 until the forthcoming meeting in Hamburg, wherein mathematical problem solving will reside most centrally within the work of Topic Study 19: Problem Solving in Mathematics Education. This booklet is being published on the occasion of this Topic Study Group.

To this end, we have assembled four summaries looking at four distinct, yet inter-related, dimensions of mathematical problem solving. The first summary, by Regina Bruder, is a nuanced look at heuristics for problem solving. This notion of heuristics is carried into Peter Liljedahl’s summary, which looks specifically at a progression of heuristics leading towards more and more creative aspects of problem solving. This is followed by Luz Manuel Santos Trigo’s summary introducing us to problem solving in and with digital technologies. The last summary, by Uldarico Malaspina Jurado, documents the rise of problem posing within the field of mathematics education in general and the problem solving literature in particular.

Each of these summaries references in some critical and central fashion the works of George Pólya or Alan Schoenfeld. To the initiated researchers, this is no surprise. The seminal work of these researchers lie at the roots of mathematical problem solving. What is interesting, though, is the diverse ways in which each of the four aforementioned contributions draw on, and position, these works so as to fit into the larger scheme of their respective summaries. This speaks to not only the depth and breadth of these influential works, but also the diversity with which they can be interpreted and utilized in extending our thinking about problem solving.

Taken together, what follows is a topical survey of ideas representing the diversity of views and tensions inherent in a field of research that is both a means to an end and an end onto itself and is unanimously seen as central to the activities of mathematics.

1 Survey on the State-of-the-Art

1.1 role of heuristics for problem solving—regina bruder.

The origin of the word heuristic dates back to the time of Archimedes and is said to have come out of one of the famous stories told about this great mathematician and inventor. The King of Syracuse asked Archimedes to check whether his new wreath was really made of pure gold. Archimedes struggled with this task and it was not until he was at the bathhouse that he came up with the solution. As he entered the tub he noticed that he had displaced a certain amount of water. Brilliant as he was, he transferred this insight to the issue with the wreath and knew he had solved the problem. According to the legend, he jumped out of the tub and ran from the bathhouse naked screaming, “Eureka, eureka!”. Eureka and heuristic have the same root in the ancient Greek language and so it has been claimed that this is how the academic discipline of “heuristics” dealing with effective approaches to problem solving (so-called heurisms) was given its name. Pólya ( 1964 ) describes this discipline as follows:

Heuristics deals with solving tasks. Its specific goals include highlighting in general terms the reasons for selecting those moments in a problem the examination of which could help us find a solution. (p. 5)

This discipline has grown, in part, from examining the approaches to certain problems more in detail and comparing them with each other in order to abstract similarities in approach, or so-called heurisms. Pólya ( 1949 ), but also, inter alia, Engel ( 1998 ), König ( 1984 ) and Sewerin ( 1979 ) have formulated such heurisms for mathematical problem tasks. The problem tasks examined by the authors mentioned are predominantly found in the area of talent programmes, that is, they often go back to mathematics competitions.

In 1983 Zimmermann provided an overview of heuristic approaches and tools in American literature which also offered suggestions for mathematics classes. In the German-speaking countries, an approach has established itself, going back to Sewerin ( 1979 ) and König ( 1984 ), which divides school-relevant heuristic procedures into heuristic tools, strategies and principles, see also Bruder and Collet ( 2011 ).

Below is a review of the conceptual background of heuristics, followed by a description of the effect mechanisms of heurisms in problem-solving processes.

1.1.1 Research Review on the Promotion of Problem Solving

In the 20th century, there has been an advancement of research on mathematical problem solving and findings about possibilities to promote problem solving with varying priorities (c.f. Pehkonen 1991 ). Based on a model by Pólya ( 1949 ), in a first phase of research on problem solving, particularly in the 1960s and the 1970s, a series of studies on problem-solving processes placing emphasis on the importance of heuristic strategies (heurisms) in problem solving has been carried out. It was assumed that teaching and learning heuristic strategies, principles and tools would provide students with an orientation in problem situations and that this could thus improve students’ problem-solving abilities (c.f. for instance, Schoenfeld 1979 ). This approach, mostly researched within the scope of talent programmes for problem solving, was rather successful (c.f. for instance, Sewerin 1979 ). In the 1980s, requests for promotional opportunities in everyday teaching were given more and more consideration: “ problem solving must be the focus of school mathematics in the 1980s ” (NCTM 1980 ). For the teaching and learning of problem solving in regular mathematics classes, the current view according to which cognitive, heuristic aspects were paramount, was expanded by certain student-specific aspects, such as attitudes, emotions and self-regulated behaviour (c.f. Kretschmer 1983 ; Schoenfeld 1985 , 1987 , 1992 ). Kilpatrick ( 1985 ) divided the promotional approaches described in the literature into five methods which can also be combined with each other.

Osmosis : action-oriented and implicit imparting of problem-solving techniques in a beneficial learning environment

Memorisation : formation of special techniques for particular types of problem and of the relevant questioning when problem solving

Imitation : acquisition of problem-solving abilities through imitation of an expert

Cooperation : cooperative learning of problem-solving abilities in small groups

Reflection : problem-solving abilities are acquired in an action-oriented manner and through reflection on approaches to problem solving.

Kilpatrick ( 1985 ) views as success when heuristic approaches are explained to students, clarified by means of examples and trained through the presentation of problems. The need of making students aware of heuristic approaches is by now largely accepted in didactic discussions. Differences in varying approaches to promoting problem-solving abilities rather refer to deciding which problem-solving strategies or heuristics are to imparted to students and in which way, and not whether these should be imparted at all or not.

1.1.2 Heurisms as an Expression of Mental Agility

The activity theory, particularly in its advancement by Lompscher ( 1975 , 1985 ), offers a well-suited and manageable model to describe learning activities and differences between learners with regard to processes and outcomes in problem solving (c.f. Perels et al. 2005 ). Mental activity starts with a goal and the motive of a person to perform such activity. Lompscher divides actual mental activity into content and process. Whilst the content in mathematical problem-solving consists of certain concepts, connections and procedures, the process describes the psychological processes that occur when solving a problem. This course of action is described in Lompscher by various qualities, such as systematic planning, independence, accuracy, activity and agility. Along with differences in motivation and the availability of expertise, it appears that intuitive problem solvers possess a particularly high mental agility, at least with regard to certain contents areas.

According to Lompscher, “flexibility of thought” expresses itself

… by the capacity to change more or less easily from one aspect of viewing to another one or to embed one circumstance or component into different correlations, to understand the relativity of circumstances and statements. It allows to reverse relations, to more or less easily or quickly attune to new conditions of mental activity or to simultaneously mind several objects or aspects of a given activity (Lompscher 1975 , p. 36).

These typical manifestations of mental agility can be focused on in problem solving by mathematical means and can be related to the heurisms known from the analyses of approaches by Pólya et al. (c.f. also Bruder 2000 ):

Reduction : Successful problem solvers will intuitively reduce a problem to its essentials in a sensible manner. To achieve such abstraction, they often use visualisation and structuring aids, such as informative figures, tables, solution graphs or even terms. These heuristic tools are also very well suited to document in retrospect the approach adopted by the intuitive problem solvers in a way that is comprehensible for all.

Reversibility : Successful problem solvers are able to reverse trains of thought or reproduce these in reverse. They will do this in appropriate situations automatically, for instance, when looking for a key they have mislaid. A corresponding general heuristic strategy is working in reverse.

Minding of aspects : Successful problem solvers will mind several aspects of a given problem at the same time or easily recognise any dependence on things and vary them in a targeted manner. Sometimes, this is also a matter of removing barriers in favour of an idea that appears to be sustainable, that is, by simply “hanging on” to a certain train of thought even against resistance. Corresponding heurisms are, for instance, the principle of invariance, the principle of symmetry (Engel 1998 ), the breaking down or complementing of geometric figures to calculate surface areas, or certain terms used in binomial formulas.

Change of aspects : Successful problem solvers will possibly change their assumptions, criteria or aspects minded in order to find a solution. Various aspects of a given problem will be considered intuitively or the problem be viewed from a different perspective, which will prevent “getting stuck” and allow for new insights and approaches. For instance, many elementary geometric propositions can also be proved in an elegant vectorial manner.

Transferring : Successful problem solvers will be able more easily than others to transfer a well-known procedure to another, sometimes even very different context. They recognise more easily the “framework” or pattern of a given task. Here, this is about own constructions of analogies and continual tracing back from the unknown to the known.

Intuitive, that is, untrained good problem solvers, are, however, often unable to access these flexibility qualities consciously. This is why they are also often unable to explain how they actually solved a given problem.

To be able to solve problems successfully, a certain mental agility is thus required. If this is less well pronounced in a certain area, learning how to solve problems means compensating by acquiring heurisms. In this case, insufficient mental agility is partly “offset” through the application of knowledge acquired by means of heurisms. Mathematical problem-solving competences are thus acquired through the promotion of manifestations of mental agility (reduction, reversibility, minding of aspects and change of aspects). This can be achieved by designing sub-actions of problem solving in connection with a (temporarily) conscious application of suitable heurisms. Empirical evidence for the success of the active principle of heurisms has been provided by Collet ( 2009 ).

Against such background, learning how to solve problems can be established as a long-term teaching and learning process which basically encompasses four phases (Bruder and Collet 2011 ):

Intuitive familiarisation with heuristic methods and techniques.

Making aware of special heurisms by means of prominent examples (explicit strategy acquisition).

Short conscious practice phase to use the newly acquired heurisms with differentiated task difficulties.

Expanding the context of the strategies applied.

In the first phase, students are familiarised with heurisms intuitively by means of targeted aid impulses and questions (what helped us solve this problem?) which in the following phase are substantiated on the basis of model tasks, are given names and are thus made aware of their existence. The third phase serves the purpose of a certain familiarisation with the new heurisms and the experience of competence through individualised practising at different requirement levels, including in the form of homework over longer periods. A fourth and delayed fourth phase aims at more flexibility through the transfer to other contents and contexts and the increasingly intuitive use of the newly acquired heurisms, so that students can enrich their own problem-solving models in a gradual manner. The second and third phases build upon each other in close chronological order, whilst the first phase should be used in class at all times.

All heurisms can basically be described in an action-oriented manner by means of asking the right questions. The way of asking questions can thus also establish a certain kind of personal relation. Even if the teacher presents and suggests the line of basic questions with a prototypical wording each time, students should always be given the opportunity to find “their” wording for the respective heurism and take a note of it for themselves. A possible key question for the use of a heuristic tool would be: How to illustrate and structure the problem or how to present it in a different way?

Unfortunately, for many students, applying heuristic approaches to problem solving will not ensue automatically but will require appropriate early and long-term promoting. The results of current studies, where promotion approaches to problem solving are connected with self-regulation and metacognitive aspects, demonstrate certain positive effects of such combination on students. This field of research includes, for instance, studies by Lester et al. ( 1989 ), Verschaffel et al. ( 1999 ), the studies on teaching method IMPROVE by Mevarech and Kramarski ( 1997 , 2003 ) and also the evaluation of a teaching concept on learning how to solve problems by the gradual conscious acquisition of heurisms by Collet and Bruder ( 2008 ).

1.2 Creative Problem Solving—Peter Liljedahl

There is a tension between the aforementioned story of Archimedes and the heuristics presented in the previous section. Archimedes, when submersing himself in the tub and suddenly seeing the solution to his problem, wasn’t relying on osmosis, memorisation, imitation, cooperation, or reflection (Kilpatrick 1985 ). He wasn’t drawing on reduction, reversibility, minding of aspects, change of aspect, or transfer (Bruder 2000 ). Archimedes was stuck and it was only, in fact, through insight and sudden illumination that he managed to solve his problem. In short, Archimedes was faced with a problem that the aforementioned heuristics, and their kind, would not help him to solve.

According to some, such a scenario is the definition of a problem. For example, Resnick and Glaser ( 1976 ) define a problem as being something that you do not have the experience to solve. Mathematicians, in general, agree with this (Liljedahl 2008 ).

Any problem in which you can see how to attack it by deliberate effort, is a routine problem, and cannot be an important discover. You must try and fail by deliberate efforts, and then rely on a sudden inspiration or intuition or if you prefer to call it luck. (Dan Kleitman, participant cited in Liljedahl 2008 , p. 19).

Problems, then, are tasks that cannot be solved by direct effort and will require some creative insight to solve (Liljedahl 2008 ; Mason et al. 1982 ; Pólya 1965 ).

1.2.1 A History of Creativity in Mathematics Education

In 1902, the first half of what eventually came to be a 30 question survey was published in the pages of L’Enseignement Mathématique , the journal of the French Mathematical Society. The authors, Édouard Claparède and Théodore Flournoy, were two Swiss psychologists who were deeply interested in the topics of mathematical discovery, creativity and invention. Their hope was that a widespread appeal to mathematicians at large would incite enough responses for them to begin to formulate some theories about this topic. The first half of the survey centered on the reasons for becoming a mathematician (family history, educational influences, social environment, etc.), attitudes about everyday life, and hobbies. This was eventually followed, in 1904, by the publication of the second half of the survey pertaining, in particular, to mental images during periods of creative work. The responses were sorted according to nationality and published in 1908.

During this same period Henri Poincaré (1854–1912), one of the most noteworthy mathematicians of the time, had already laid much of the groundwork for his own pursuit of this same topic and in 1908 gave a presentation to the French Psychological Society in Paris entitled L’Invention mathématique —often mistranslated to Mathematical Creativity Footnote 1 (c.f. Poincaré 1952 ). At the time of the presentation Poincaré stated that he was aware of Claparède and Flournoy’s work, as well as their results, but expressed that they would only confirm his own findings. Poincaré’s presentation, as well as the essay it spawned, stands to this day as one of the most insightful, and thorough treatments of the topic of mathematical discovery, creativity, and invention.

Just at this time, I left Caen, where I was living, to go on a geological excursion under the auspices of the School of Mines. The incident of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuschian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had the time, as, upon taking my seat in the omnibus, I went on with the conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the results at my leisure. (Poincaré 1952 , p. 53)

So powerful was his presentation, and so deep were his insights into his acts of invention and discovery that it could be said that he not so much described the characteristics of mathematical creativity, as defined them. From that point forth mathematical creativity, or even creativity in general, has not been discussed seriously without mention of Poincaré’s name.

Inspired by this presentation, Jacques Hadamard (1865–1963), a contemporary and a friend of Poincaré’s, began his own empirical investigation into this fascinating phenomenon. Hadamard had been critical of Claparède and Flournoy’s work in that they had not adequately treated the topic on two fronts. As exhaustive as the survey appeared to be, Hadamard felt that it failed to ask some key questions—the most important of which was with regard to the reason for failures in the creation of mathematics. This seemingly innocuous oversight, however, led directly to his second and “most important criticism” (Hadamard 1945 ). He felt that only “first-rate men would dare to speak of” (p. 10) such failures. So, inspired by his good friend Poincaré’s treatment of the subject Hadamard retooled the survey and gave it to friends of his for consideration—mathematicians such as Henri Poincaré and Albert Einstein, whose prominence were beyond reproach. Ironically, the new survey did not contain any questions that explicitly dealt with failure. In 1943 Hadamard gave a series of lectures on mathematical invention at the École Libre des Hautes Études in New York City. These talks were subsequently published as The Psychology of Mathematical Invention in the Mathematical Field (Hadameard 1945 ).

Hadamard’s classic work treats the subject of invention at the crossroads of mathematics and psychology. It provides not only an entertaining look at the eccentric nature of mathematicians and their rituals, but also outlines the beliefs of mid twentieth-century mathematicians about the means by which they arrive at new mathematics. It is an extensive exploration and extended argument for the existence of unconscious mental processes. In essence, Hadamard took the ideas that Poincaré had posed and, borrowing a conceptual framework for the characterization of the creative process from the Gestaltists of the time (Wallas 1926 ), turned them into a stage theory. This theory still stands as the most viable and reasonable description of the process of mathematical creativity.

1.2.2 Defining Mathematical Creativity

The phenomena of mathematical creativity, although marked by sudden illumination, actually consist of four separate stages stretched out over time, of which illumination is but one stage. These stages are initiation, incubation, illumination, and verification (Hadamard 1945 ). The first of these stages, the initiation phase, consists of deliberate and conscious work. This would constitute a person’s voluntary, and seemingly fruitless, engagement with a problem and be characterized by an attempt to solve the problem by trolling through a repertoire of past experiences. This is an important part of the inventive process because it creates the tension of unresolved effort that sets up the conditions necessary for the ensuing emotional release at the moment of illumination (Hadamard 1945 ; Poincaré 1952 ).

Following the initiation stage the solver, unable to come up with a solution stops working on the problem at a conscious level and begins to work on it at an unconscious level (Hadamard 1945 ; Poincaré 1952 ). This is referred to as the incubation stage of the inventive process and can last anywhere from several minutes to several years. After the period of incubation a rapid coming to mind of a solution, referred to as illumination , may occur. This is accompanied by a feeling of certainty and positive emotions (Poincaré 1952 ). Although the processes of incubation and illumination are shrouded behind the veil of the unconscious there are a number of things that can be deduced about them. First and foremost is the fact that unconscious work does, indeed, occur. Poincaré ( 1952 ), as well as Hadamard ( 1945 ), use the very real experience of illumination, a phenomenon that cannot be denied, as evidence of unconscious work, the fruits of which appear in the flash of illumination. No other theory seems viable in explaining the sudden appearance of solution during a walk, a shower, a conversation, upon waking, or at the instance of turning the conscious mind back to the problem after a period of rest (Poincaré 1952 ). Also deducible is that unconscious work is inextricably linked to the conscious and intentional effort that precedes it.

There is another remark to be made about the conditions of this unconscious work: it is possible, and of a certainty it is only fruitful, if it is on the one hand preceded and on the other hand followed by a period of conscious work. These sudden inspirations never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come … (Poincaré 1952 , p. 56)

Hence, the fruitless efforts of the initiation phase are only seemingly so. They not only set up the aforementioned tension responsible for the emotional release at the time of illumination, but also create the conditions necessary for the process to enter into the incubation phase.

Illumination is the manifestation of a bridging that occurs between the unconscious mind and the conscious mind (Poincaré 1952 ), a coming to (conscious) mind of an idea or solution. What brings the idea forward to consciousness is unclear, however. There are theories of the aesthetic qualities of the idea, effective surprise/shock of recognition, fluency of processing, or breaking functional fixedness. For reasons of brevity I will only expand on the first of these.

Poincaré proposed that ideas that were stimulated during initiation remained stimulated during incubation. However, freed from the constraints of conscious thought and deliberate calculation, these ideas would begin to come together in rapid and random unions so that “their mutual impacts may produce new combinations” (Poincaré 1952 ). These new combinations, or ideas, would then be evaluated for viability using an aesthetic sieve, which allows through to the conscious mind only the “right combinations” (Poincaré 1952 ). It is important to note, however, that good or aesthetic does not necessarily mean correct. Correctness is evaluated during the verification stage.

The purpose of verification is not only to check for correctness. It is also a method by which the solver re-engages with the problem at the level of details. That is, during the unconscious work the problem is engaged with at the level of ideas and concepts. During verification the solver can examine these ideas in closer details. Poincaré succinctly describes both of these purposes.

As for the calculations, themselves, they must be made in the second period of conscious work, that which follows the inspiration, that in which one verifies the results of this inspiration and deduces their consequences. (Poincaré 1952 , p. 62)

Aside from presenting this aforementioned theory on invention, Hadamard also engaged in a far-reaching discussion on a number of interesting, and sometimes quirky, aspects of invention and discovery that he had culled from the results of his empirical study, as well as from pertinent literature. This discussion was nicely summarized by Newman ( 2000 ) in his commentary on the elusiveness of invention.

The celebrated phrenologist Gall said mathematical ability showed itself in a bump on the head, the location of which he specified. The psychologist Souriau, we are told, maintained that invention occurs by “pure chance”, a valuable theory. It is often suggested that creative ideas are conjured up in “mathematical dreams”, but this attractive hypothesis has not been verified. Hadamard reports that mathematicians were asked whether “noises” or “meteorological circumstances” helped or hindered research [..] Claude Bernard, the great physiologist, said that in order to invent “one must think aside”. Hadamard says this is a profound insight; he also considers whether scientific invention may perhaps be improved by standing or sitting or by taking two baths in a row. Helmholtz and Poincaré worked sitting at a table; Hadamard’s practice is to pace the room (“Legs are the wheels of thought”, said Emile Angier); the chemist J. Teeple was the two-bath man. (p. 2039)

1.2.3 Discourses on Creativity

Creativity is a term that can be used both loosely and precisely. That is, while there exists a common usage of the term there also exists a tradition of academic discourse on the subject. A common usage of creative refers to a process or a person whose products are original, novel, unusual, or even abnormal (Csíkszentmihályi 1996 ). In such a usage, creativity is assessed on the basis of the external and observable products of the process, the process by which the product comes to be, or on the character traits of the person doing the ‘creating’. Each of these usages—product, process, person—is the roots of the discourses (Liljedahl and Allan 2014 ) that I summarize here, the first of which concerns products.

Consider a mother who states that her daughter is creative because she drew an original picture. The basis of such a statement can lie either in the fact that the picture is unlike any the mother has ever seen or unlike any her daughter has ever drawn before. This mother is assessing creativity on the basis of what her daughter has produced. However, the standards that form the basis of her assessment are neither consistent nor stringent. There does not exist a universal agreement as to what she is comparing the picture to (pictures by other children or other pictures by the same child). Likewise, there is no standard by which the actual quality of the picture is measured. The academic discourse that concerns assessment of products, on the other hand, is both consistent and stringent (Csíkszentmihályi 1996 ). This discourse concerns itself more with a fifth, and as yet unmentioned, stage of the creative process; elaboration . Elaboration is where inspiration becomes perspiration (Csíkszentmihályi 1996 ). It is the act of turning a good idea into a finished product, and the finished product is ultimately what determines the creativity of the process that spawned it—that is, it cannot be a creative process if nothing is created. In particular, this discourse demands that the product be assessed against other products within its field, by the members of that field, to determine if it is original AND useful (Csíkszentmihályi 1996 ; Bailin 1994 ). If it is, then the product is deemed to be creative. Note that such a use of assessment of end product pays very little attention to the actual process that brings this product forth.

The second discourse concerns the creative process. The literature pertaining to this can be separated into two categories—a prescriptive discussion of the creativity process and a descriptive discussion of the creativity process. Although both of these discussions have their roots in the four stages that Wallas ( 1926 ) proposed makes up the creative process, they make use of these stages in very different ways. The prescriptive discussion of the creative process is primarily focused on the first of the four stages, initiation , and is best summarized as a cause - and - effect discussion of creativity, where the thinking processes during the initiation stage are the cause and the creative outcome are the effects (Ghiselin 1952 ). Some of the literature claims that the seeds of creativity lie in being able to think about a problem or situation analogically. Other literature claims that utilizing specific thinking tools such as imagination, empathy, and embodiment will lead to creative products. In all of these cases, the underlying theory is that the eventual presentation of a creative idea will be precipitated by the conscious and deliberate efforts during the initiation stage. On the other hand, the literature pertaining to a descriptive discussion of the creative process is inclusive of all four stages (Kneller 1965 ; Koestler 1964 ). For example, Csíkszentmihályi ( 1996 ), in his work on flow attends to each of the stages, with much attention paid to the fluid area between conscious and unconscious work, or initiation and incubation. His claim is that the creative process is intimately connected to the enjoyment that exists during times of sincere and consuming engagement with a situation, the conditions of which he describes in great detail.

The third, and final, discourse on creativity pertains to the person. This discourse is space dominated by two distinct characteristics, habit and genius. Habit has to do with the personal habits as well as the habits of mind of people that have been deemed to be creative. However, creative people are most easily identified through their reputation for genius. Consequently, this discourse is often dominated by the analyses of the habits of geniuses as is seen in the work of Ghiselin ( 1952 ), Koestler ( 1964 ), and Kneller ( 1965 ) who draw on historical personalities such as Albert Einstein, Henri Poincaré, Vincent Van Gogh, D.H. Lawrence, Samuel Taylor Coleridge, Igor Stravinsky, and Wolfgang Amadeus Mozart to name a few. The result of this sort of treatment is that creative acts are viewed as rare mental feats, which are produced by extraordinary individuals who use extraordinary thought processes.

These different discourses on creativity can be summed up in a tension between absolutist and relativist perspectives on creativity (Liljedahl and Sriraman 2006 ). An absolutist perspective assumes that creative processes are the domain of genius and are present only as precursors to the creation of remarkably useful and universally novel products. The relativist perspective, on the other hand, allows for every individual to have moments of creativity that may, or may not, result in the creation of a product that may, or may not, be either useful or novel.

Between the work of a student who tries to solve a problem in geometry or algebra and a work of invention, one can say there is only a difference of degree. (Hadamard 1945 , p. 104).

Regardless of discourse, however, creativity is not “part of the theories of logical forms” (Dewey 1938 ). That is, creativity is not representative of the lock-step logic and deductive reasoning that mathematical problem solving is often presumed to embody (Bibby 2002 ; Burton 1999 ). Couple this with the aforementioned demanding constraints as to what constitutes a problem, where then does that leave problem solving heuristics? More specifically, are there creative problem solving heuristics that will allow us to resolve problems that require illumination to solve? The short answer to this question is yes—there does exist such problem solving heuristics. To understand these, however, we must first understand the routine problem solving heuristics they are built upon. In what follows, I walk through the work of key authors and researchers whose work offers us insights into progressively more creative problem solving heuristics for solving true problems.

1.2.4 Problem Solving by Design

In a general sense, design is defined as the algorithmic and deductive approach to solving a problem (Rusbult 2000 ). This process begins with a clearly defined goal or objective after which there is a great reliance on relevant past experience, referred to as repertoire (Bruner 1964 ; Schön 1987 ), to produce possible options that will lead towards a solution of the problem (Poincaré 1952 ). These options are then examined through a process of conscious evaluations (Dewey 1933 ) to determine their suitability for advancing the problem towards the final goal. In very simple terms, problem solving by design is the process of deducing the solution from that which is already known.

Mayer ( 1982 ), Schoenfeld ( 1982 ), and Silver ( 1982 ) state that prior knowledge is a key element in the problem solving process. Prior knowledge influences the problem solver’s understanding of the problem as well as the choice of strategies that will be called upon in trying to solve the problem. In fact, prior knowledge and prior experiences is all that a solver has to draw on when first attacking a problem. As a result, all problem solving heuristics incorporate this resource of past experiences and prior knowledge into their initial attack on a problem. Some heuristics refine these ideas, and some heuristics extend them (c.f. Kilpatrick 1985 ; Bruder 2000 ). Of the heuristics that refine, none is more influential than the one created by George Pólya (1887–1985).

1.2.5 George Pólya: How to Solve It

In his book How to Solve It (1949) Pólya lays out a problem solving heuristic that relies heavily on a repertoire of past experience. He summarizes the four-step process of his heuristic as follows:

Understanding the Problem

First. You have to understand the problem.

What is the unknown? What are the data? What is the condition?

Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?

Draw a figure. Introduce suitable notation.

Separate the various parts of the condition. Can you write them down?

Devising a Plan

Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

Have you seen it before? Or have you seen the same problem in a slightly different form?

Do you know a related problem? Do you know a theorem that could be useful?

Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?

Could you restate the problem? Could you restate it still differently? Go back to definitions.

If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?

Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

Carrying Out the Plan

Third. Carry out your plan.

Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

Looking Back

Fourth. Examine the solution obtained.

Can you check the result? Can you check the argument?

Can you derive the solution differently? Can you see it at a glance?

Can you use the result, or the method, for some other problem?

The emphasis on auxiliary problems, related problems, and analogous problems that are, in themselves, also familiar problems is an explicit manifestation of relying on a repertoire of past experience. This use of familiar problems also requires an ability to deduce from these related problems a recognizable and relevant attribute that will transfer to the problem at hand. The mechanism that allows for this transfer of knowledge between analogous problems is known as analogical reasoning (English 1997 , 1998 ; Novick 1988 , 1990 , 1995 ; Novick and Holyoak 1991 ) and has been shown to be an effective, but not always accessible, thinking strategy.

Step four in Pólya’s heuristic, looking back, is also a manifestation of utilizing prior knowledge to solve problems, albeit an implicit one. Looking back makes connections “in memory to previously acquired knowledge [..] and further establishes knowledge in long-term memory that may be elaborated in later problem-solving encounters” (Silver 1982 , p. 20). That is, looking back is a forward-looking investment into future problem solving encounters, it sets up connections that may later be needed.

Pólya’s heuristic is a refinement on the principles of problem solving by design. It not only makes explicit the focus on past experiences and prior knowledge, but also presents these ideas in a very succinct, digestible, and teachable manner. This heuristic has become a popular, if not the most popular, mechanism by which problem solving is taught and learned.

1.2.6 Alan Schoenfeld: Mathematical Problem Solving

The work of Alan Schoenfeld is also a refinement on the principles of problem solving by design. However, unlike Pólya ( 1949 ) who refined these principles at a theoretical level, Schoenfeld has refined them at a practical and empirical level. In addition to studying taught problem solving strategies he has also managed to identify and classify a variety of strategies, mostly ineffectual, that students invoke naturally (Schoenfeld 1985 , 1992 ). In so doing, he has created a better understanding of how students solve problems, as well as a better understanding of how problems should be solved and how problem solving should be taught.

For Schoenfeld, the problem solving process is ultimately a dialogue between the problem solver’s prior knowledge, his attempts, and his thoughts along the way (Schoenfeld 1982 ). As such, the solution path of a problem is an emerging and contextually dependent process. This is a departure from the predefined and contextually independent processes of Pólya’s ( 1949 ) heuristics. This can be seen in Schoenfeld’s ( 1982 ) description of a good problem solver.

To examine what accounts for expertise in problem solving, you would have to give the expert a problem for which he does not have access to a solution schema. His behavior in such circumstances is radically different from what you would see when he works on routine or familiar “non-routine” problems. On the surface his performance is no longer proficient; it may even seem clumsy. Without access to a solution schema, he has no clear indication of how to start. He may not fully understand the problem, and may simply “explore it for a while until he feels comfortable with it. He will probably try to “match” it to familiar problems, in the hope it can be transformed into a (nearly) schema-driven solution. He will bring up a variety of plausible things: related facts, related problems, tentative approaches, etc. All of these will have to be juggled and balanced. He may make an attempt solving it in a particular way, and then back off. He may try two or three things for a couple of minutes and then decide which to pursue. In the midst of pursuing one direction he may go back and say “that’s harder than it should be” and try something else. Or, after the comment, he may continue in the same direction. With luck, after some aborted attempts, he will solve the problem. (p. 32-33)

Aside from demonstrating the emergent nature of the problem solving process, this passage also brings forth two consequences of Schoenfeld’s work. The first of these is the existence of problems for which the solver does not have “access to a solution schema”. Unlike Pólya ( 1949 ), who’s heuristic is a ‘one size fits all (problems)’ heuristic, Schoenfeld acknowledges that problem solving heuristics are, in fact, personal entities that are dependent on the solver’s prior knowledge as well as their understanding of the problem at hand. Hence, the problems that a person can solve through his or her personal heuristic are finite and limited.

The second consequence that emerges from the above passage is that if a person lacks the solution schema to solve a given problem s/he may still solve the problem with the help of luck . This is an acknowledgement, if only indirectly so, of the difference between problem solving in an intentional and mechanical fashion verses problem solving in a more creative fashion, which is neither intentional nor mechanical (Pehkonen 1997 ).

1.2.7 David Perkins: Breakthrough Thinking

As mentioned, many consider a problem that can be solved by intentional and mechanical means to not be worthy of the title ‘problem’. As such, a repertoire of past experiences sufficient for dealing with such a ‘problem’ would disqualify it from the ranks of ‘problems’ and relegate it to that of ‘exercises’. For a problem to be classified as a ‘problem’, then, it must be ‘problematic’. Although such an argument is circular it is also effective in expressing the ontology of mathematical ‘problems’.

Perkins ( 2000 ) also requires problems to be problematic. His book Archimedes’ Bathtub: The Art and Logic of Breakthrough Thinking (2000) deals with situations in which the solver has gotten stuck and no amount of intentional or mechanical adherence to the principles of past experience and prior knowledge is going to get them unstuck. That is, he deals with problems that, by definition, cannot be solved through a process of design [or through the heuristics proposed by Pólya ( 1949 ) and Schoenfeld ( 1985 )]. Instead, the solver must rely on the extra-logical process of what Perkins ( 2000 ) calls breakthrough thinking .

Perkins ( 2000 ) begins by distinguishing between reasonable and unreasonable problems. Although both are solvable, only reasonable problems are solvable through reasoning. Unreasonable problems require a breakthrough in order to solve them. The problem, however, is itself inert. It is neither reasonable nor unreasonable. That quality is brought to the problem by the solver. That is, if a student cannot solve a problem by direct effort then that problem is deemed to be unreasonable for that student. Perkins ( 2000 ) also acknowledges that what is an unreasonable problem for one person is a perfectly reasonable problem for another person; reasonableness is dependent on the person.

This is not to say that, once found, the solution cannot be seen as accessible through reason. During the actual process of solving, however, direct and deductive reasoning does not work. Perkins ( 2000 ) uses several classic examples to demonstrate this, the most famous being the problem of connecting nine dots in a 3 × 3 array with four straight lines without removing pencil from paper, the solution to which is presented in Fig.  1 .

Nine dots—four lines problem and solution

To solve this problem, Perkins ( 2000 ) claims that the solver must recognize that the constraint of staying within the square created by the 3 × 3 array is a self-imposed constraint. He further claims that until this is recognized no amount of reasoning is going to solve the problem. That is, at this point in the problem solving process the problem is unreasonable. However, once this self-imposed constraint is recognized the problem, and the solution, are perfectly reasonable. Thus, the solution of an, initially, unreasonable problem is reasonable.

The problem solving heuristic that Perkins ( 2000 ) has constructed to deal with solvable, but unreasonable, problems revolves around the idea of breakthrough thinking and what he calls breakthrough problems . A breakthrough problem is a solvable problem in which the solver has gotten stuck and will require an AHA! to get unstuck and solve the problem. Perkins ( 2000 ) poses that there are only four types of solvable unreasonable problems, which he has named wilderness of possibilities , the clueless plateau , narrow canyon of exploration , and oasis of false promise . The names for the first three of these types of problems are related to the Klondike gold rush in Alaska, a time and place in which gold was found more by luck than by direct and systematic searching.

The wilderness of possibilities is a term given to a problem that has many tempting directions but few actual solutions. This is akin to a prospector searching for gold in the Klondike. There is a great wilderness in which to search, but very little gold to be found. The clueless plateau is given to problems that present the solver with few, if any, clues as to how to solve it. The narrow canyon of exploration is used to describe a problem that has become constrained in such a way that no solution now exists. The nine-dot problem presented above is such a problem. The imposed constraint that the lines must lie within the square created by the array makes a solution impossible. This is identical to the metaphor of a prospector searching for gold within a canyon where no gold exists. The final type of problem gets its name from the desert. An oasis of false promise is a problem that allows the solver to quickly get a solution that is close to the desired outcome; thereby tempting them to remain fixed on the strategy that they used to get this almost-answer. The problem is, that like the canyon, the solution does not exist at the oasis; the solution strategy that produced an almost-answer is incapable of producing a complete answer. Likewise, a desert oasis is a false promise in that it is only a reprieve from the desolation of the dessert and not a final destination.

Believing that there are only four ways to get stuck, Perkins ( 2000 ) has designed a problem solving heuristic that will “up the chances” of getting unstuck. This heuristic is based on what he refers to as “the logic of lucking out” (p. 44) and is built on the idea of introspection. By first recognizing that they are stuck, and then recognizing that the reason they are stuck can only be attributed to one of four reasons, the solver can access four strategies for getting unstuck, one each for the type of problem they are dealing with. If the reason they are stuck is because they are faced with a wilderness of possibilities they are to begin roaming far, wide, and systematically in the hope of reducing the possible solution space to one that is more manageable. If they find themselves on a clueless plateau they are to begin looking for clues, often in the wording of the problem. When stuck in a narrow canyon of possibilities they need to re-examine the problem and see if they have imposed any constraints. Finally, when in an oasis of false promise they need to re-attack the problem in such a way that they stay away from the oasis.

Of course, there are nuances and details associated with each of these types of problems and the strategies for dealing with them. However, nowhere within these details is there mention of the main difficulty inherent in introspection; that it is much easier for the solver to get stuck than it is for them to recognize that they are stuck. Once recognized, however, the details of Perkins’ ( 2000 ) heuristic offer the solver some ways for recognizing why they are stuck.

1.2.8 John Mason, Leone Burton, and Kaye Stacey: Thinking Mathematically

The work of Mason et al. in their book Thinking Mathematically ( 1982 ) also recognizes the fact that for each individual there exists problems that will not yield to their intentional and mechanical attack. The heuristic that they present for dealing with this has two main processes with a number of smaller phases, rubrics, and states. The main processes are what they refer to as specializing and generalizing. Specializing is the process of getting to know the problem and how it behaves through the examination of special instances of the problem. This process is synonymous with problem solving by design and involves the repeated oscillation between the entry and attack phases of Mason et al. ( 1982 ) heuristic. The entry phase is comprised of ‘getting started’ and ‘getting involved’ with the problem by using what is immediately known about it. Attacking the problem involves conjecturing and testing a number of hypotheses in an attempt to gain greater understanding of the problem and to move towards a solution.

At some point within this process of oscillating between entry and attack the solver will get stuck, which Mason et al. ( 1982 ) refer to as “an honourable and positive state, from which much can be learned” (p. 55). The authors dedicate an entire chapter to this state in which they acknowledge that getting stuck occurs long before an awareness of being stuck develops. They proposes that the first step to dealing with being stuck is the simple act of writing STUCK!

The act of expressing my feelings helps to distance me from my state of being stuck. It frees me from incapacitating emotions and reminds me of actions that I can take. (p. 56)

The next step is to reengage the problem by examining the details of what is known, what is wanted, what can be introduced into the problem, and what has been introduced into the problem (imposed assumptions). This process is engaged in until an AHA!, which advances the problem towards a solution, is encountered. If, at this point, the problem is not completely solved the oscillation is then resumed.

At some point in this process an attack on the problem will yield a solution and generalizing can begin. Generalizing is the process by which the specifics of a solution are examined and questions as to why it worked are investigated. This process is synonymous with the verification and elaboration stages of invention and creativity. Generalization may also include a phase of review that is similar to Pólya’s ( 1949 ) looking back.

1.2.9 Gestalt: The Psychology of Problem Solving

The Gestalt psychology of learning believes that all learning is based on insights (Koestler 1964 ). This psychology emerged as a response to behaviourism, which claimed that all learning was a response to external stimuli. Gestalt psychologists, on the other hand, believed that there was a cognitive process involved in learning as well. With regards to problem solving, the Gestalt school stands firm on the belief that problem solving, like learning, is a product of insight and as such, cannot be taught. In fact, the theory is that not only can problem solving not be taught, but also that attempting to adhere to any sort of heuristic will impede the working out of a correct solution (Krutestkii 1976 ). Thus, there exists no Gestalt problem solving heuristic. Instead, the practice is to focus on the problem and the solution rather than on the process of coming up with a solution. Problems are solved by turning them over and over in the mind until an insight, a viable avenue of attack, presents itself. At the same time, however, there is a great reliance on prior knowledge and past experiences. The Gestalt method of problem solving, then, is at the same time very different and very similar to the process of design.

Gestalt psychology has not fared well during the evolution of cognitive psychology. Although it honours the work of the unconscious mind it does so at the expense of practicality. If learning is, indeed, entirely based on insight then there is little point in continuing to study learning. “When one begins by assuming that the most important cognitive phenomena are inaccessible, there really is not much left to talk about” (Schoenfeld 1985 , p. 273). However, of interest here is the Gestalt psychologists’ claim that focus on problem solving methods creates functional fixedness (Ashcraft 1989 ). Mason et al. ( 1982 ), as well as Perkins ( 2000 ) deal with this in their work on getting unstuck.

1.2.10 Final Comments

Mathematics has often been characterized as the most precise of all sciences. Lost in such a misconception is the fact that mathematics often has its roots in the fires of creativity, being born of the extra-logical processes of illumination and intuition. Problem solving heuristics that are based solely on the processes of logical and deductive reasoning distort the true nature of problem solving. Certainly, there are problems in which logical deductive reasoning is sufficient for finding a solution. But these are not true problems. True problems need the extra-logical processes of creativity, insight, and illumination, in order to produce solutions.

Fortunately, as elusive as such processes are, there does exist problem solving heuristics that incorporate them into their strategies. Heuristics such as those by Perkins ( 2000 ) and Mason et al. ( 1982 ) have found a way of combining the intentional and mechanical processes of problem solving by design with the extra-logical processes of creativity, illumination, and the AHA!. Furthermore, they have managed to do so without having to fully comprehend the inner workings of this mysterious process.

1.3 Digital Technologies and Mathematical Problem Solving—Luz Manuel Santos-Trigo

Mathematical problem solving is a field of research that focuses on analysing the extent to which problem solving activities play a crucial role in learners’ understanding and use of mathematical knowledge. Mathematical problems are central in mathematical practice to develop the discipline and to foster students learning (Pólya 1945 ; Halmos 1994 ). Mason and Johnston-Wilder ( 2006 ) pointed out that “The purpose of a task is to initiate mathematically fruitful activity that leads to a transformation in what learners are sensitized to notice and competent to carry out” (p. 25). Tasks are essential for learners to elicit their ideas and to engage them in mathematical thinking. In a problem solving approach, what matters is the learners’ goals and ways to interact with the tasks. That is, even routine tasks can be a departure point for learners to extend initial conditions and transform them into some challenging activities.

Thus, analysing and characterizing ways in which mathematical problems are formulated (Singer et al. 2015 ) and the process involved in pursuing and solving those problems generate important information to frame and structure learning environments to guide and foster learners’ construction of mathematical concepts and problem solving competences (Santos-Trigo 2014 ). Furthermore, mathematicians or discipline practitioners have often been interested in unveiling and sharing their own experience while developing the discipline. As a results, they have provided valuable information to characterize mathematical practices and their relations to what learning processes of the discipline entails. It is recognized that the work of Pólya ( 1945 ) offered not only bases to launch several research programs in problem solving (Schoenfeld 1992 ; Mason et al. 1982 ); but also it became an essential resource for teachers to orient and structure their mathematical lessons (Krulik and Reys 1980 ).

1.3.1 Research Agenda

A salient feature of a problem solving approach to learn mathematics is that teachers and students develop and apply an enquiry or inquisitive method to delve into mathematical concepts and tasks. How are mathematical problems or concepts formulated? What types of problems are important for teachers/learners to discuss and engage in mathematical reasoning? What mathematical processes and ways of reasoning are involved in understanding mathematical concepts and solving problems? What are the features that distinguish an instructional environment that fosters problem-solving activities? How can learners’ problem solving competencies be assessed? How can learners’ problem solving competencies be characterized and explained? How can learners use digital technologies to understand mathematics and to develop problem-solving competencies? What ways of reasoning do learners construct when they use digital technologies in problem solving approaches? These types of questions have been important in the problem solving research agenda and delving into them has led researchers to generate information and results to support and frame curriculum proposals and learning scenarios. The purpose of this section is to present and discuss important themes that emerged in problem solving approaches that rely on the systematic use of several digital technologies.

In the last 40 years, the accumulated knowledge in the problem solving field has shed lights on both a characterization of what mathematical thinking involves and how learners can construct a robust knowledge in problem solving environments (Schoenfeld 1992 ). In this process, the field has contributed to identify what types of transformations traditional learning scenarios might consider when teachers and students incorporate the use of digital technologies in mathematical classrooms. In this context, it is important to briefly review what main themes and developments the field has addressed and achieved during the last 40 years.

1.3.2 Problem Solving Developments

There are traces of mathematical problems and solutions throughout the history of civilization that explain the humankind interest for identifying and exploring mathematical relations (Kline 1972 ). Pólya ( 1945 ) reflects on his own practice as a mathematician to characterize the process of solving mathematical problems through four main phases: Understanding the problem, devising a plan, carrying out the plan, and looking back. Likewise, Pólya ( 1945 ) presents and discusses the role played by heuristic methods throughout all problem solving phases. Schoenfeld ( 1985 ) presents a problem solving research program based on Pólya’s ( 1945 ) ideas to investigate the extent to which problem solving heuristics help university students to solve mathematical problems and to develop a way of thinking that shows consistently features of mathematical practices. As a result, he explains the learners’ success or failure in problem solving activities can be characterized in terms their mathematical resources and ways to access them, cognitive and metacognitive strategies used to represent and explore mathematical tasks, and systems of beliefs about mathematics and solving problems. In addition, Schoenfeld ( 1992 ) documented that heuristics methods as illustrated in Pólya’s ( 1945 ) book are ample and general and do not include clear information and directions about how learners could assimilate, learn, and use them in their problem solving experiences. He suggested that students need to discuss what it means, for example, to think of and examining special cases (one important heuristic) in finding a closed formula for series or sequences, analysing relationships of roots of polynomials, or focusing on regular polygons or equilateral/right triangles to find general relations about these figures. That is, learners need to work on examples that lead them to recognize that the use of a particular heuristic often involves thinking of different type of cases depending on the domain or content involved. Lester and Kehle ( 2003 ) summarize themes and methodological shifts in problem solving research up to 1995. Themes include what makes a problem difficult for students and what it means to be successful problem solvers; studying and contrasting experts and novices’ problem solving approaches; learners’ metacognitive, beliefs systems and the influence of affective behaviours; and the role of context; and social interactions in problem solving environments. Research methods in problem solving studies have gone from emphasizing quantitative or statistical design to the use of cases studies and ethnographic methods (Krutestkii ( 1976 ). Teaching strategies also evolved from being centred on teachers to the active students’ engagement and collaboration approaches (NCTM 2000 ). Lesh and Zawojewski ( 2007 ) propose to extend problem solving approaches beyond class setting and they introduce the construct “model eliciting activities” to delve into the learners’ ideas and thinking as a way to engage them in the development of problem solving experiences. To this end, learners develop and constantly refine problem-solving competencies as a part of a learning community that promotes and values modelling construction activities. Recently, English and Gainsburg ( 2016 ) have discussed the importance of modeling eliciting activities to prepare and develop students’ problem solving experiences for 21st Century challenges and demands.

Törner et al. ( 2007 ) invited mathematics educators worldwide to elaborate on the influence and developments of problem solving in their countries. Their contributions show a close relationship between countries mathematical education traditions and ways to frame and implement problem solving approaches. In Chinese classrooms, for example, three instructional strategies are used to structure problem solving lessons: one problem multiple solutions , multiple problems one solution , and one problem multiple changes . In the Netherlands, the realistic mathematical approach permeates the students’ development of problem solving competencies; while in France, problem solving activities are structured in terms of two influential frameworks: The theory of didactical situations and anthropological theory of didactics.

In general, problem solving frameworks and instructional approaches came from analysing students’ problem solving experiences that involve or rely mainly on the use of paper and pencil work. Thus, there is a need to re-examined principles and frameworks to explain what learners develop in learning environments that incorporate systematically the coordinated use of digital technologies (Hoyles and Lagrange 2010 ). In this perspective, it becomes important to briefly describe and identify what both multiple purpose and ad hoc technologies can offer to the students in terms of extending learning environments and representing and exploring mathematical tasks. Specifically, a task is used to identify features of mathematical reasoning that emerge through the use digital technologies that include both mathematical action and multiple purpose types of technologies.

1.3.3 Background

Digital technologies are omnipresent and their use permeates and shapes several social and academic events. Mobile devices such as tablets or smart phones are transforming the way people communicate, interact and carry out daily activities. Churchill et al. ( 2016 ) pointed out that mobile technologies provide a set of tools and affordances to structure and support learning environments in which learners continuously interact to construct knowledge and solve problems. The tools include resources or online materials, efficient connectivity to collaborate and discuss problems, ways to represent, explore and store information, and analytical and administration tools to management learning activities. Schmidt and Cohen ( 2013 ) stated that nowadays it is difficult to imagine a life without mobile devices, and communication technologies are playing a crucial role in generating both cultural and technical breakthroughs. In education, the use of mobile artefacts and computers offers learners the possibility of continuing and extending peers and groups’ mathematical discussions beyond formal settings. In this process, learners can also consult online materials and interact with experts, peers or more experienced students while working on mathematical tasks. In addition, dynamic geometry systems (GeoGebra) provide learners a set of affordances to represent and explore dynamically mathematical problems. Leung and Bolite-Frant ( 2015 ) pointed out that tools help activate an interactive environment in which teachers and students’ mathematical experiences get enriched. Thus, the digital age brings new challenges to the mathematics education community related to the changes that technologies produce to curriculum, learning scenarios, and ways to represent, explore mathematical situations. In particular, it is important to characterize the type of reasoning that learners can develop as a result of using digital technologies in their process of learning concepts and solving mathematical problems.

1.3.4 A Focus on Mathematical Tasks

Mathematical tasks are essential elements for engaging learners in mathematical reasoning which involves representing objects, identifying and exploring their properties in order to detect invariants or relationships and ways to support them. Watson and Ohtani ( 2015 ) stated that task design involves discussions about mathematical content and students’ learning (cognitive perspective), about the students’ experiences to understand the nature of mathematical activities; and about the role that tasks played in teaching practices. In this context, tasks are the vehicle to present and discuss theoretical frameworks for supporting the use of digital technology, to analyse the importance of using digital technologies in extending learners’ mathematical discussions beyond formal settings, and to design ways to foster and assess the use of technologies in learners’ problem solving environments. In addition, it is important to discuss contents, concepts, representations and strategies involved in the process of using digital technologies in approaching the tasks. Similarly, it becomes essential to discuss what types of activities students will do to learn and solve the problems in an environment where the use of technologies fosters and values the participation and collaboration of all students. What digital technologies are important to incorporate in problem solving approaches? Dynamic Geometry Systems can be considered as a milestone in the development of digital technologies. Objects or mathematical situations can be represented dynamically through the use of a Dynamic Geometry System and learners or problem solvers can identify and examine mathematical relations that emerge from moving objects within the dynamic model (Moreno-Armella and Santos-Trigo 2016 ).

Leung and Bolite-Frant ( 2015 ) stated that “dynamic geometry software can be used in task design to cover a large epistemic spectrum from drawing precise robust geometrical figures to exploration of new geometric theorems and development of argumentation discourse” (p. 195). As a result, learners not only need to develop skills and strategies to construct dynamic configuration of problems; but also ways of relying on the tool’s affordances (quantifying parameters or objects attributes, generating loci, graphing objects behaviours, using sliders, or dragging particular elements within the configuration) in order to identify and support mathematical relations. What does it mean to represent and explore an object or mathematical situation dynamically?

A simple task that involves a rhombus and its inscribed circle is used to illustrate how a dynamic representation of these objects and embedded elements can lead learners to identify and examine mathematical properties of those objects in the construction of the configuration. To this end, learners are encouraged to pose and pursue questions to explain the behaviours of parameters or attributes of the family of objects that is generated as a result of moving a particular element within the configuration.

1.3.5 A Task: A Dynamic Rhombus

Figure  2 represents a rhombus APDB and its inscribed circle (O is intersection of diagonals AD and BP and the radius of the inscribed circle is the perpendicular segment from any side of the rhombus to point O), vertex P lies on a circle c centred at point A. Circle c is only a heuristic to generate a family of rhombuses. Thus, point P can be moved along circle c to generate a family of rhombuses. Indeed, based on the symmetry of the circle it is sufficient to move P on the semicircle B’CA to draw such a family of rhombuses.

A dynamic construction of a rhombus

1.3.6 Posing Questions

A goal in constructing a dynamic model or configuration of problems is always to identify and explore mathematical properties and relations that might result from moving objects within the model. How do the areas of both the rhombus and the inscribed circle behave when point P is moved along the arc B’CB? At what position of point P does the area of the rhombus or inscribed circle reach the maximum value? The coordinates of points S and Q (Fig.  3 ) are the x -value of point P and as y -value the corresponding area values of rhombus ABDP and the inscribed circle respectively. Figure  2 shows the loci of points S and Q when point P is moved along arc B’CB. Here, finding the locus via the use of GeoGebra is another heuristic to graph the area behaviour without making explicit the algebraic model of the area.

Graphic representation of the area variation of the family of rhombuses and inscribed circles generated when P is moved through arc B’CB

The area graphs provide information to visualize that in that family of generated rhombuses the maximum area value of the inscribed circle and rhombus is reached when the rhombus becomes a square (Fig.  4 ). That is, the controlled movement of particular objects is an important strategy to analyse the area variation of the family of rhombuses and their inscribed circles.

Visualizing the rhombus and the inscribed circle with maximum area

It is important to observe the identification of points P and Q in terms of the position of point P and the corresponding areas and the movement of point P was sufficient to generate both area loci. That is, the graph representation of the areas is achieved without having an explicit algebraic expression of the area variation. Clearly, the graphic representations provide information regarding the increasing or decreasing interval of both areas; it is also important to explore what properties both graphic representations hold. The goal is to argue that the area variation of the rhombus represents an ellipse and the area of the inscribed circle represents a parabola. An initial argument might involve selecting five points on each locus and using the tool to draw the corresponding conic section (Fig.  5 ). In this case, the tool affordances play an important role in generating the graphic representation of the areas’ behaviours and in identifying properties of those representations. In this context, the use of the tool can offer learners the opportunity to problematize (Santos-Trigo 2007 ) a simple mathematical object (rhombus) as a means to search for mathematical relations and ways to support them.

Drawing the conic section that passes through five points

1.3.7 Looking for Different Solutions Methods

Another line of exploration might involve asking for ways to construct a rhombus and its inscribed circle: Suppose that the side of the rhombus and the circle are given, how can you construct the rhombus that has that circle inscribed? Figure  6 shows the given data, segment A 1 B 1 and circle centred at O and radius OD. The initial goal is to draw the circle tangent to the given segment. To this end, segment AB is congruent to segment A 1 B 1 and on this segment a point P is chosen and a perpendicular to segment AB that passes through point P is drawn. Point C is on this perpendicular and the centre of a circle with radius OD and h is the perpendicular to line PC that passes through point C. Angle ACB changes when point P is moved along segment AB and point E and F are the intersection of line h and the circle with centre M the midpoint of AB and radius MA (Fig.  6 ).

Drawing segment AB tangent to the given circle

Figure  7 a shows the right triangle AFB as the base to construct the rhombus and the inscribed circle and Fig.  7 b shows the second solution based on triangle AEB.

a Drawing the rhombus and the inscribed circle. b Drawing the second solution

Another approach might involve drawing the given circle centred at the origin and the segment as EF with point E on the y-axis. Line OC is perpendicular to segment EF and the locus of point C when point E moves along the y-axis intersects the given circle (Fig.  8 a, b). Both figures show two solutions to draw the rhombus that circumscribe the given circle.

a and b Another solution that involves finding a locus of point C

In this example, the GeoGebra affordances not only are important to construct a dynamic model of the task; but also offer learners and opportunity to explore relations that emerge from moving objects within the model. As a result, learners can rely on different concepts and strategies to solve the tasks. The idea in presenting this rhombus task is to illustrate that the use of a Dynamic Geometry System provides affordances for learners to construct dynamic representation of mathematical objects or problems, to move elements within the representation to pose questions or conjectures to explain invariants or patterns among involved parameters; to search for arguments to support emerging conjectures, and to develop a proper language to communicate results.

1.3.8 Looking Back

Conceptual frameworks used to explain learners’ construction of mathematical knowledge need to capture or take into account the different ways of reasoning that students might develop as a result of using a set of tools during the learning experiences. Figure  9 show some digital technologies that learners can use for specific purpose at the different stages of problem solving activities.

The coordinated use of digital tools to engage learners in problem solving experiences

The use of a dynamic system (GeoGebra) provides a set of affordances for learners to conceptualize and represent mathematical objects and tasks dynamically. In this process, affordances such as moving objects orderly (dragging), finding loci of objects, quantifying objects attributes (lengths, areas, angles, etc.), using sliders to vary parameters, and examining family of objects became important to look for invariance or objects relationships. Likewise, analysing the parameters or objects behaviours within the configuration might lead learners to identify properties to support emerging mathematical relations. Thus, with the use of the tool, learners might conceptualize mathematical tasks as an opportunity for them to engage in mathematical activities that include constructing dynamic models of tasks, formulating conjectures, and always looking for different arguments to support them. Similarly, learners can use an online platform to share their ideas, problem solutions or questions in a digital wall and others students can also share ideas or solution methods and engaged in mathematical discussions that extend mathematical classroom activities.

1.4 Problem Posing: An Overview for Further Progress—Uldarico Malaspina Jurado

Problem posing and problem solving are two essential aspects of the mathematical activity; however, researchers in mathematics education have not emphasized their attention on problem posing as much as problem solving. In that sense, due to its importance in the development of mathematical thinking in students since the first grades, we agree with Ellerton’s statement ( 2013 ): “for too long, successful problem solving has been lauded as the goal; the time has come for problem posing to be given a prominent but natural place in mathematics curricula and classrooms” (pp. 100–101); and due to its importance in teacher training, with Abu-Elwan’s statement ( 1999 ):

While teacher educators generally recognize that prospective teachers require guidance in mastering the ability to confront and solve problems, what is often overlooked is the critical fact that, as teachers, they must be able to go beyond the role as problem solvers. That is, in order to promote a classroom situation where creative problem solving is the central focus, the practitioner must become skillful in discovering and correctly posing problems that need solutions. (p. 1)

Scientists like Einstein and Infeld ( 1938 ), recognized not only for their notable contributions in the fields they worked, but also for their reflections on the scientific activity, pointed out the importance of problem posing; thus it is worthwhile to highlight their statement once again:

The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old questions from a new angle, requires creative imagination and marks real advance in science. (p. 92)

Certainly, it is also relevant to remember mathematician Halmos’s statement ( 1980 ): “I do believe that problems are the heart of mathematics, and I hope that as teachers (…) we will train our students to be better problem posers and problem solvers than we are” (p. 524).

An important number of researchers in mathematics education has focused on the importance of problem posing, and we currently have numerous, very important publications that deal with different aspects of problem posing related to the mathematics education of students in all educational levels and to teacher training.

1.4.1 A Retrospective Look

Kilpatrick ( 1987 ) marked a historical milestone in research related to problem posing and points out that “problem formulating should be viewed not only as a goal of instruction but also as a means of instruction” (Kilpatrick 1987 , p. 123); and he also emphasizes that, as part of students’ education, all of them should be given opportunities to live the experience of discovering and posing their own problems. Drawing attention to the few systematic studies on problem posing performed until then, Kilpatrick contributes defining some aspects that required studying and investigating as steps prior to a theoretical building, though he warns, “attempts to teach problem-formulating skills, of course, need not await a theory” (p. 124).

Kilpatrick refers to the “Source of problems” and points out how virtually all problems students solve have been posed by another person; however, in real life “many problems, if not most, must be created or discovered by the solver, who gives the problem an initial formulation” (p. 124). He also points out that problems are reformulated as they are being solved, and he relates this to investigation, reminding us what Davis ( 1985 ) states that, “what typically happens in a prolonged investigation is that problem formulation and problem solution go hand in hand, each eliciting the other as the investigation progresses” (p. 23). He also relates it to the experiences of software designers, who formulate an appropriate sequence of sub-problems to solve a problem. He poses that a subject to be examined by teachers and researchers “is whether, by drawing students’ attention to the reformulating process and given them practice in it, we can improve their problem solving performance” (p. 130). He also points out that problems may be a mathematical formulation as a result of exploring a situation and, in that sense, “school exercises in constructing mathematical models of a situation presented by the teacher are intended to provide students with experiences in formulating problems.” (p. 131).

Another important section of Kilpatrick’s work ( 1987 ) is Processes of Problem Formulating , in which he considers association, analogy, generalization and contradiction. He believes the use of concept maps to represent concept organization, as cognitive scientists Novak and Gowin suggest, might help to comprehend such concepts, stimulate creative thinking about them, and complement the ideas Brown and Walter ( 1983 ) give for problem posing by association. Further, in the section “Understanding and developing problem formulating abilities”, he poses several questions, which have not been completely answered yet, like “Perhaps the central issue from the point of view of cognitive science is what happens when someone formulates the problem? (…) What is the relation between problem formulating, problem solving and structured knowledge base? How rich a knowledge base is needed for problem formulating? (…) How does experience in problem formulating add to knowledge base? (…) What metacognitive processes are needed for problem formulating?”

It is interesting to realize that some of these questions are among the unanswered questions proposed and analyzed by Cai et al. ( 2015 ) in Chap. 1 of the book Mathematical Problem Posing (Singer et al. 2015 ). It is worth stressing the emphasis on the need to know the cognitive processes in problem posing, an aspect that Kilpatrick had already posed in 1987, as we just saw.

1.4.2 Researches and Didactic Experiences

Currently, there are a great number of publications related to problem posing, many of which are research and didactic experiences that gather the questions posed by Kilpatrick, which we just commented. Others came up naturally as reflections raised in the framework of problem solving, facing the natural requirement of having appropriate problems to use results and suggestions of researches on problem solving, or as a response to a thoughtful attitude not to resign to solving and asking students to solve problems that are always created by others. Why not learn and teach mathematics posing one’s own problems?

1.4.3 New Directions of Research

Singer et al. ( 2013 ) provides a broad view about problem posing that links problem posing experiences to general mathematics education; to the development of abilities, attitudes and creativity; and also to its interrelation with problem solving, and studies on when and how problem-solving sessions should take place. Likewise, it provides information about research done regarding ways to pose new problems and about the need for teachers to develop abilities to handle complex situations in problem posing contexts.

Singer et al. ( 2013 ) identify new directions in problem posing research that go from problem-posing task design to the development of problem-posing frameworks to structure and guide teachers and students’ problem posing experiences. In a chapter of this book, Leikin refers to three different types of problem posing activities, associated with school mathematics research: (a) problem posing through proving; (b) problem posing for investigation; and (c) problem posing through investigation. This classification becomes evident in the problems posed in a course for prospective secondary school mathematics teachers by using a dynamic geometry environment. Prospective teachers posed over 25 new problems, several of which are discussed in the article. The author considers that, by developing this type of problem posing activities, prospective mathematics teachers may pose different problems related to a geometric object, prepare more interesting lessons for their students, and thus gradually develop their mathematical competence and their creativity.

1.4.4 Final Comments

This overview, though incomplete, allows us to see a part of what problem posing experiences involve and the importance of this area in students mathematical learning. An important task is to continue reflecting on the questions posed by Kilpatrick ( 1987 ), as well as on the ones that come up in the different researches aforementioned. To continue progressing in research on problem posing and contribute to a greater consolidation of this research line, it will be really important that all mathematics educators pay more attention to problem posing, seek to integrate approaches and results, and promote joint and interdisciplinary works. As Singer et al. ( 2013 ) say, going back to Kilpatrick’s proposal ( 1987 ),

Problem posing is an old issue. What is new is the awareness that problem posing needs to pervade the education systems around the world, both as a means of instruction (…) and as an object of instruction (…) with important targets in real-life situations. (p. 5)

Although it can be argued that there is a difference between creativity, discovery, and invention (see Liljedahl and Allan 2014 ) for the purposes of this book these will be assumed to be interchangeable.

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Pólya, G. (1965). Mathematical discovery: On understanding, learning and teaching problem solving (Vol. 2). New York, NY: Wiley.

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Santos-Trigo, M. (2007). Mathematical problem solving: An evolving research and practice domain. ZDM—The International Journal on Mathematics Education , 39 (5, 6): 523–536.

Santos-Trigo, M. (2014). Problem solving in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 496–501). New York: Springer.

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Further Reading

Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex, and setting . Buckingham, PA: Open University Press.

Borwein, P., Liljedahl, P., & Zhai, H. (2014). Mathematicians on creativity. Mathematical Association of America.

Burton, L. (1984). Thinking things through . London, UK: Simon & Schuster Education.

Feynman, R. (1999). The pleasure of finding things out . Cambridge, MA: Perseus Publishing.

Gardner, M. (1978). Aha! insight . New York, NY: W. H. Freeman and Company.

Gardner, M. (1982). Aha! gotcha: Paradoxes to puzzle and delight . New York, NY: W. H. Freeman and Company.

Gardner, H. (1993). Creating minds: An anatomy of creativity seen through the lives of Freud, Einstein, Picasso, Stravinsky, Eliot, Graham, and Ghandi . New York, NY: Basic Books.

Glas, E. (2002). Klein’s model of mathematical creativity. Science & Education, 11 (1), 95–104.

Hersh, D. (1997). What is mathematics, really? . New York, NY: Oxford University Press.

Root-Bernstein, R., & Root-Bernstein, M. (1999). Sparks of genius: The thirteen thinking tools of the world’s most creative people . Boston, MA: Houghton Mifflin Company.

Zeitz, P. (2006). The art and craft of problem solving . New York, NY: Willey.

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Liljedahl, P., Santos-Trigo, M., Malaspina, U., Bruder, R. (2016). Problem Solving in Mathematics Education. In: Problem Solving in Mathematics Education. ICME-13 Topical Surveys. Springer, Cham. https://doi.org/10.1007/978-3-319-40730-2_1

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Introduction to mathematical thinking.

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The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years.

Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today's world. This course helps to develop that crucial way of thinking.

The course is offered in two versions. The eight-week-long Basic Course is designed for people who want to develop or improve mathematics-based, analytic thinking for professional or general life purposes. The ten-week-long Extended Course is aimed primarily at first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course.

Subtitles for all video lectures available in: Portuguese (provided by  The Lemann Foundation ), English

Course Syllabus

Instructor's welcome and introduction

•  Introductory material
•  Analysis of language – the logical combinators
•  Analysis of language – implication
•  Analysis of language – equivalence
•  Analysis of language – quantifiers
•  Working with quantifiers
•  Proofs
•  Proofs involving quantifiers
•  Elements of number theory
• Beginning real analysis

Recommended Background

High school mathematics. Specific requirements are familiarity with elementary symbolic algebra, the concept of a number system (in particular, the characteristics of, and distinctions between, the natural numbers, the integers, the rational numbers, and the real numbers), and some elementary set theory (including inequalities and intervals of the real line). Students whose familiarity with these topics is somewhat rusty typically find that with a little extra effort they can pick up what is required along the way. The only heavy use of these topics is in the (optional) final two weeks of the Extended Course.

A good way to assess if your  basic  school background is adequate (even if currently rusty) is to glance at the topics in the book  Adding It Up: Helping Children Learn Mathematics  (free download), published by the US National Academies Press in 2001. Though aimed at K-8 mathematics teachers and teacher educators, it provides an excellent coverage of what constitutes a good basic mathematics education for life in the Twenty-First Century (which was the National Academies' aim in producing it).

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Problem Solving in Mathematics

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.

Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition  problems:

Common clue words for  subtraction  problems:

• How much more

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:

• Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
• What did you need to do in that instance?
• What facts are you given about this problem?
• What facts do you still need to find out about this problem?

Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:

• Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
• If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

• Does your solution seem probable?
• Does it answer the initial question?
• Did you answer using the language in the question?
• Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

Tips and Hints

Some key questions to consider as you approach the problem may be:

• What are the keywords in the problem?
• Do I need a data visual, such as a diagram, list, table, chart, or graph?
• Is there a formula or equation that I'll need? If so, which one?
• Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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Khan Academy Blog

Unlocking the Power of Math Learning: Strategies and Tools for Success

posted on September 20, 2023

Mathematics, the foundation of all sciences and technology, plays a fundamental role in our everyday lives. Yet many students find the subject challenging, causing them to shy away from it altogether. This reluctance is often due to a lack of confidence, a misunderstanding of unclear concepts, a move ahead to more advanced skills before they are ready, and ineffective learning methods. However, with the right approach, math learning can be both rewarding and empowering. This post will explore different approaches to learning math, strategies for success, and cutting-edge tools to help you achieve your goals.

Math Learning

Math learning can take many forms, including traditional classroom instruction, online courses, and self-directed learning. A multifaceted approach to math learning can improve understanding, engage students, and promote subject mastery. A 2014 study by the National Council of Teachers of Mathematics found that the use of multiple representations, such as visual aids, graphs, and real-world examples, supports the development of mathematical connections, reasoning, and problem-solving skills.

Moreover, the importance of math learning goes beyond solving equations and formulas. Advanced math skills are essential for success in many fields, including science, engineering, finance, health care, and technology. In fact, a report by Burning Glass Technologies found that 71% of high-salary, entry-level positions require advanced math skills.

Benefits of Math Learning

In today’s 21st-century world, having a broad knowledge base and strong reading and math skills is essential. Mathematical literacy plays a crucial role in this success. It empowers individuals to comprehend the world around them and make well-informed decisions based on data-driven understanding. More than just earning good grades in math, mathematical literacy is a vital life skill that can open doors to economic opportunities, improve financial management, and foster critical thinking. We’re not the only ones who say so:

• Math learning enhances problem-solving skills, critical thinking, and logical reasoning abilities. (Source: National Council of Teachers of Mathematics )
• It improves analytical skills that can be applied in various real-life situations, such as budgeting or analyzing data. (Source: Southern New Hampshire University )
• Math learning promotes creativity and innovation by fostering a deep understanding of patterns and relationships. (Source: Purdue University )
• It provides a strong foundation for careers in fields such as engineering, finance, computer science, and more. These careers generally correlate to high wages. (Source: U.S. Bureau of Labor Statistics )
• Math skills are transferable and can be applied across different academic disciplines. (Source: Sydney School of Education and Social Work )

How to Know What Math You Need to Learn

Often students will find gaps in their math knowledge; this can occur at any age or skill level. As math learning is generally iterative, a solid foundation and understanding of the math skills that preceded current learning are key to success. The solution to these gaps is called mastery learning, the philosophy that underpins Khan Academy’s approach to education .

Mastery learning is an educational philosophy that emphasizes the importance of a student fully understanding a concept before moving on to the next one. Rather than rushing students through a curriculum, mastery learning asks educators to ensure that learners have “mastered” a topic or skill, showing a high level of proficiency and understanding, before progressing. This approach is rooted in the belief that all students can learn given the appropriate learning conditions and enough time, making it a markedly student-centered method. It promotes thoroughness over speed and encourages individualized learning paths, thus catering to the unique learning needs of each student.

Students will encounter mastery learning passively as they go through Khan Academy coursework, as our platform identifies gaps and systematically adjusts to support student learning outcomes. More details can be found in our Educators Hub . 

Try Our Free Confidence Boosters

How to learn math.

Learning at School

One of the most common methods of math instruction is classroom learning. In-class instruction provides students with real-time feedback, practical application, and a peer-learning environment. Teachers can personalize instruction by assessing students’ strengths and weaknesses, providing remediation when necessary, and offering advanced instruction to students who need it.

Learning at Home

Supplemental learning at home can complement traditional classroom instruction. For example, using online resources that provide additional practice opportunities, interactive games, and demonstrations, can help students consolidate learning outside of class. E-learning has become increasingly popular, with a wealth of online resources available to learners of all ages. The benefits of online learning include flexibility, customization, and the ability to work at one’s own pace. One excellent online learning platform is Khan Academy, which offers free video tutorials, interactive practice exercises, and a wealth of resources across a range of mathematical topics.

Moreover, parents can encourage and monitor progress, answer questions, and demonstrate practical applications of math in everyday life. For example, when at the grocery store, parents can ask their children to help calculate the price per ounce of two items to discover which one is the better deal. Cooking and baking with your children also provides a lot of opportunities to use math skills, like dividing a recipe in half or doubling the ingredients. 

Learning Math with the Help of Artificial Intelligence (AI) 

AI-powered tools are changing the way students learn math. Personalized feedback and adaptive practice help target individual needs. Virtual tutors offer real-time help with math concepts while AI algorithms identify areas for improvement. Custom math problems provide tailored practice, and natural language processing allows for instant question-and-answer sessions. 

Using Khan Academy’s AI Tutor, Khanmigo

Transform your child’s grasp of mathematics with Khanmigo , the 24/7 AI-powered tutor that specializes in tailored, one-on-one math instruction. Available at any time, Khanmigo provides personalized support that goes beyond mere answers to nurture genuine mathematical understanding and critical thinking. Khanmigo can track progress, identify strengths and weaknesses, and offer real-time feedback to help students stay on the right track. Within a secure and ethical AI framework, your child can tackle everything from basic arithmetic to complex calculus, all while you maintain oversight using robust parental controls.

Get Math Help with Khanmigo Right Now

You can learn anything .

Math learning is essential for success in the modern world, and with the right approach, it can also be enjoyable and rewarding. Learning math requires curiosity, diligence, and the ability to connect abstract concepts with real-world applications. Strategies for effective math learning include a multifaceted approach, including classroom instruction, online courses, homework, tutoring, and personalized AI support. 

So, don’t let math anxiety hold you back; take advantage of available resources and technology to enhance your knowledge base and enjoy the benefits of math learning.

National Council of Teachers of Mathematics, “Principles to Actions: Ensuring Mathematical Success for All” , April 2014

Project Lead The Way Research Report, “The Power of Transportable Skills: Assessing the Demand and Value of the Skills of the Future” , 2020

Page. M, “Why Develop Quantitative and Qualitative Data Analysis Skills?” , 2016

Mann. EL, Creativity: The Essence of Mathematics, Journal for the Education of the Gifted. Vol. 30, No. 2, 2006, pp. 236–260, http://www.prufrock.com ’

Nakakoji Y, Wilson R.” Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University ”. J Intell. 2020 Sep 1;8(3):32. doi: 10.3390/jintelligence8030032. PMID: 32882908; PMCID: PMC7555771.

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The Problem-solving Classroom

• Visualising
• Working backwards
• Reasoning logically
• Conjecturing
• Working systematically
• Looking for patterns
• Trial and improvement.

• stage of the lesson 
• level of thinking
• mathematical skill.
• The length of student response increases (300-700%)
• More responses are supported by logical argument.
• An increased number of speculative responses.
• The number of questions asked by students increases.
• Student - student exchanges increase (volleyball).
• Failures to respond decrease.
• 'Disciplinary moves' decrease.
• The variety of students participating increases.  As does the number of unsolicited, but appropriate contributions.
• Student confidence increases.
• conceptual understanding
• procedural fluency
• strategic competence
• adaptive reasoning
• productive disposition

• Our Mission

A Math Word Problem Framework That Fosters Conceptual Thinking

This strategy for selecting and teaching word problems guides students to develop their understanding of math concepts.

Word problems in mathematics are a powerful tool for helping students make sense of and reason with mathematical concepts. Many students, however, struggle with word problems because of the various cognitive demands. As districtwide STEAM professional development specialists, we’ve spent a lot of time focusing on supporting our colleagues and students to ensure their success with word problems. We found that selecting the right word problems, as well as focusing on conceptual understanding rather than procedural knowledge, provides our students with real growth.

As our thinking evolved, we began to instill a routine that supports teaching students to solve with grit by putting them in the driver’s seat of the thinking. Below you’ll find the routine that we’ve found successful in helping students overcome the challenges of solving word problems.

Not all word problems are created equal

Prior to any instruction, we always consider the quality of the task for teaching and learning. In our process, we use word problems as the path to mathematics instruction. When selecting the mathematical tasks for students, we always consider the following questions:

• Does the task align with the learning goals and standards?
• Will the task engage and challenge students at an appropriate level, providing both a sense of accomplishment and further opportunities for growth?
• Is the task open or closed? Open tasks provide multiple pathways to foster a deeper understanding of mathematical concepts and skills. Closed tasks can still provide a deep understanding of mathematical concepts and skills if the task requires a high level of cognitive demand. 
• Does the task encourage critical thinking and problem-solving skills?
• Will the task allow students to see the relevance of mathematics to real-world situations?
• Does the task promote creativity and encourage students to make connections between mathematical concepts and other areas of their lives?

If we can answer yes to as many of these questions as possible, we can be assured that our tasks are rich. There are further insights for rich math tasks on NRICH and sample tasks on Illustrative Mathematics and K-5 Math Teaching Resources .

Developing conceptual understanding

Once we’ve selected the rich math tasks, developing conceptual understanding becomes our instructional focus. We present students with Numberless Word Problems and simultaneously use a word problem framework to focus on analysis of the text and to build conceptual understanding, rather than just memorization of formulas and procedures. 

• First we remove all of the numbers and have students read the problem focusing on who or what the problem is about; they visualize and connect the scenario to their lives and experiences. 
• Next we have our students rewrite the question as a statement to ensure that they understand the questions.
• Then we have our students read the problem again and have them think analytically. They ask themselves these questions: Are there parts? Is there a whole? Are things joining or separating? Is there a comparison? 
• Once that’s completed, we reveal the numbers in the problem. We have the students read the problem again to determine if they have enough information to develop a model and translate it into an equation that can be solved.
• After they’ve solved their equation, we have students compare it against their model to check their answer.  

Collaboration and workspace are key to building the thinking

To build the thinking necessary in the math classroom , we have students work in visibly random collaborative groups (random groups of three for grades 3 through 12, random groups of two for grades 1 and 2). With random groupings, we’ve found that students don’t enter their groups with predetermined roles, and all students contribute to the thinking.

For reluctant learners, we make sure these students serve as the scribe within the group documenting each member’s contribution. We also make sure to use nonpermanent vertical workspaces (whiteboards, windows [using dry-erase markers], large adhesive-backed chart paper, etc.). The vertical workspace provides accessibility for our diverse learners and promotes problem-solving because our students break down complex problems into smaller, manageable steps. The vertical workspaces also provide a visually appealing and organized way for our students to show their work.  We’ve witnessed how these workspaces help hold their attention and improve their focus on the task at hand.

Facilitate and provide feedback to move the thinking along

As students grapple with the task, the teacher floats among the collaborative groups, facilitates conversations, and gives the students feedback. Students are encouraged to look at the work of other groups or to provide a second strategy or model to support their thinking. Students take ownership and make sense of the problem, attempt solutions, and try to support their thinking with models, equations, charts, graphs, words, etc. They work through the problem collaboratively, justifying their work in their small group. In essence, they’re constructing their knowledge and preparing to share their work with the rest of the class. 

Word problems are a powerful tool for teaching math concepts to students. They offer a practical and relatable approach to problem-solving, enabling students to understand the relevance of math in real-life situations. Through word problems, students learn to apply mathematical principles and logical reasoning to solve complex problems. 

Moreover, word problems also enhance critical thinking, analytical skills, and decision-making abilities. Incorporating word problems into math lessons is an effective way to make math engaging, meaningful, and applicable to everyday life.

Mathematical Problem-Solving: Techniques and Strategies

by Ali | Mar 8, 2023 | Blog Post , Blogs | 0 comments

Introduction to Mathematical Problem-Solving

Mathematical problem-solving is the process of using logical reasoning and critical thinking to find a solution to a mathematical problem. It is an essential skill that is required in a wide range of academic and professional fields, including science, technology, engineering, and mathematics (STEM).

Importance of Mathematical Problem-Solving Skills

Mathematical problem-solving skills are critical for success in many areas of life, including education, career, and daily life. It helps students to develop analytical and critical thinking skills, enhances their ability to reason logically, and encourages them to persevere when faced with challenges.

The Process of Mathematical Problem-Solving

The process of mathematical problem-solving involves several steps that include identifying the problem, understanding the problem, making a plan, carrying out the plan, and checking the answer.

Techniques and Strategies for Mathematical Problem-Solving

1.      identify the problem.

The first step in problem-solving is to identify the problem. It involves reading the problem carefully and determining what the problem is asking.

2.      Understand the problem

The next step is to understand the problem by breaking it down into smaller parts, identifying any relevant information, and determining what needs to be solved.

3.      Make a plan

After understanding the problem, the next step is to develop a plan to solve it. This may involve identifying a formula or method to use, drawing a diagram or chart, or making a list of steps to follow.

4.      Carry out the plan

Once a plan is developed, the next step is to carry out the plan by solving the problem using the chosen method. It is important to show all steps and work neatly to avoid making mistakes.

5.      Check the answer

Finally, it is essential to check the answer to ensure it is correct. This can be done by re-reading the problem, checking the solution for accuracy, and verifying that it makes sense.

Know About: HOW TO FIND PERFECT MATH TUTOR 

Importance of using online calculators while learning math.

Utilizing online calculators can prove to be a beneficial resource for learning mathematics. There are numerous reasons why incorporating them into your studies is a wise choice.

Firstly, online calculators offer the convenience of being easily accessible at any time and from anywhere. No longer do you need to carry a physical calculator with you; you can use them on any device that has internet connectivity.

In addition, online calculators excel in accuracy and can efficiently handle complex calculations that may be difficult to do manually. They can perform arithmetic at a faster speed, saving you time and increasing productivity.

Another advantage is that some online calculators include built-in visualizations such as graphs and charts, which can help students grasp mathematical concepts better.

Furthermore, feedback can be provided by certain online calculators, assisting students in identifying and rectifying errors in their calculations. This feature can be especially useful for students who are new to learning mathematics .

Online calculators have a versatile range of functions beyond basic arithmetic, including algebraic equations, trigonometry, and calculus . This makes them useful for students at all levels of math education.

Overall, online calculators are an invaluable tool for students learning math. They are convenient, accurate, efficient, and versatile, and aid in the understanding of mathematical concepts, making them an essential component of modern-day education.

Common Errors in Mathematical Problem-Solving

There are several common errors that can occur in mathematical problem-solving, including misunderstanding the problem, using incorrect formulas or methods, making computational errors, and not checking the answer. To avoid these errors, it is essential to read the problem carefully, use the correct formulas and methods, check all computations, and double-check the answer for accuracy.

Improving Mathematical Problem-Solving Skills

There are several ways to improve mathematical problem-solving skills, including practicing regularly, working with others, seeking help from a teacher or tutor, and reviewing past problems. It is also helpful to develop a positive attitude towards problem-solving, persevere through challenges, and learn from mistakes.

Must Know: WHICH IS THE BEST WAY OF LEARNING ONLINE TUTORING OR TRADITIONAL TUTORING

Mathematical problem-solving is a crucial skill that is required for success in many academic and professional fields. By following the process of problem-solving and using the techniques and strategies outlined in this article, individuals can improve their problem-solving skills and achieve success in their academic and professional endeavors.

Frequently Asked Questions

What is mathematical problem-solving.

Mathematical problem-solving is the process of using logical reasoning and critical thinking to find a solution to a mathematical problem.

Why are mathematical problem-solving skills important?

What are the steps involved in the process of mathematical problem-solving, how can online calculators aid in learning mathematics.

Online calculators can aid in learning mathematics by providing convenience, accuracy, and efficiency. They can also help students grasp mathematical concepts better through built-in visualizations and provide feedback to identify and rectify errors in their calculations.

What are common errors to avoid in mathematical problem-solving?

Common errors to avoid in mathematical problem-solving include misunderstanding the problem, using incorrect formulas or methods, making computational errors, and not checking the answer. To avoid these errors, it is essential to read the problem carefully, use the correct formulas and methods, check all computations, and double-check the answer for accuracy.

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• v.9(5); 2023 May
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Development and differences in mathematical problem-solving skills: A cross-sectional study of differences in demographic backgrounds

Ijtihadi kamilia amalina.

a Doctoral School of Education, University of Szeged, Hungary

Tibor Vidákovich

b Institute of Education, University of Szeged, Hungary

Associated Data

Data will be made available on request.

Problem-solving skills are the most applicable cognitive tool in mathematics, and improving the problem-solving skills of students is a primary aim of education. However, teachers need to know the best period of development and the differences among students to determine the best teaching and learning methods. This study aims to investigate the development and differences in mathematical problem-solving skills of students based on their grades, gender, and school locations. A scenario-based mathematical essay test was administered to 1067 students in grades 7–9 from schools in east Java, Indonesia, and their scores were converted into a logit scale for statistical analysis. The results of a one-way analysis of variance and an independent sample t -test showed that the students had an average level of mathematical problem-solving skills. The number of students who failed increased with the problem-solving phase. The students showed development of problem-solving skills from grade 7 to grade 8 but not in grade 9. A similar pattern of development was observed in the subsample of urban students, both male and female. The demographic background had a significant effect, as students from urban schools outperformed students from rural schools, and female students outperformed male students. The development of problem-solving skills in each phase as well as the effects of the demographic background of the participants were thoroughly examined. Further studies are needed with participants of more varied backgrounds.

1. Introduction

Problem-solving skills are a complex set of cognitive, behavioral, and attitudinal components that are situational and dependent on thorough knowledge and experience [ 1 , 2 ]. Problem-solving skills are acquired over time and are the most widely applicable cognitive tool [ 3 ]. Problem-solving skills are particularly important in mathematics education [ 3 , 4 ]. The development of mathematical problem-solving skills can differ based on age, gender stereotypes, and school locations [ [5] , [6] , [7] , [8] , [9] , [10] ]. Fostering the development of mathematical problem-solving skills is a major goal of educational systems because they provide a tool for success [ 3 , 11 ]. Mathematical problem-solving skills are developed through explicit training and enriching materials [ 12 ]. Teachers must understand how student profiles influence the development of mathematical problem-solving skills to optimize their teaching methods.

Various studies on the development of mathematical problem-solving skills have yielded mixed results. Grissom [ 13 ] concluded that problem-solving skills were fixed and immutable. Meanwhile, other researchers argued that problem-solving skills developed over time and were modifiable, providing an opportunity for their enhancement through targeted educational intervention when problem-solving skills developed quickly [ 3 , 4 , 12 ]. Tracing the development of mathematical problem-solving skills is crucial. Further, the results of previous studies are debatable, necessitating a comprehensive study in the development of students’ mathematical problem-solving skills.

Differences in mathematical problem-solving skills have been identified based on gender and school location [ [6] , [7] , [8] , [9] , [10] ]. School location affects school segregation and school quality [ 9 , 14 ]. The socioeconomic and sociocultural characteristics of a residential area where a school is located are the factors affecting academic achievement [ 14 ]. Studies in several countries have shown that students in urban schools demonstrated better performance and problem-solving skills in mathematics [ 9 , 10 , 15 ]. However, contradictory results have been obtained for other countries [ 6 , 10 ].

Studies on gender differences have shown that male students outperform female students in mathematics, which has piqued the interest of psychologists, sociologists, and educators [ 7 , 16 , 17 ]. The differences appear to be because of brain structure; however, sociologists argue that gender equality can be achieved by providing equal educational opportunities [ 8 , 16 , 18 , 19 ]. Because the results are debatable and no studies on gender differences across grades in schools have been conducted, it would be interesting to investigate gender differences in mathematical problem-solving skills.

Based on the previous explanations, teachers need to understand the best time for students to develop mathematical problem-solving skills because problem-solving is an obligatory mathematics skill to be mastered. However, no relevant studies focused on Indonesia have been conducted regarding the mathematical problem-solving skill development of students in middle school that can provide the necessary information for teachers. Further, middle school is the important first phase of developing critical thinking skills; thus relevant studies are required in this case [ 3 , 4 ]. In addition, a municipal policy-making system can raise differences in problem-solving skills based on different demographic backgrounds [ 10 ]. Moreover, the results of previous studies regarding the development and differences in mathematical problem-solving skills are debatable. Thus, the present study has been conducted to meet these gaps. This study investigated the development of mathematical problem-solving skills in students and the differences owing demographic backgrounds. Three aspects were considered: (1) student profiles of mathematical problem-solving skills, (2) development of their mathematical problem-solving skills across grades, and (3) significant differences in mathematical problem-solving skills based on gender and school location. The results of the present study will provide detailed information regarding the subsample that contributes to the development of mathematical problem-solving skills in students based on their demographic backgrounds. In addition, the description of the score is in the form of a logit scale from large-scale data providing consistent meaning and confident generalization. This study can be used to determine appropriate teaching and learning in the best period of students’ development in mathematical problem-solving skills as well as policies to achieve educational equality.

2. Theoretical background

2.1. mathematical problem-solving skills and their development.

Solving mathematical problems is a complex cognitive ability that requires students to understand the problem as well as apply mathematical concepts to them [ 20 ]. Researchers have described the phases of solving a mathematical problem as understanding the problem, devising a plan, conducting out the plan, and looking back [ [20] , [24] , [21] , [22] , [23] ]. Because mathematical problems are complex, students may struggle with several phases, including applying mathematical knowledge, determining the concepts to use, and stating mathematical sentences (e.g., arithmetic) [ 20 ]. Studies have concluded that more students fail at later stages of the solution process [ 25 , 26 ]. In other words, fewer students fail in the phase of understanding a problem than during the plan implementation phase. Different studies have stated that students face difficulties in understanding the problem, determining what to assume, and investigating relevant information [ 27 ]. This makes them unable to translate the problem into a mathematical form.

Age or grade is viewed as one factor that influences mathematical problem-solving skills because the skills of the students improve over time as a result of the teaching and learning processes [ 28 ]. Neuroscience research has shown that older students have fewer problems with arithmetic than younger students; however, the hemispheric asymmetry is reduced [ 29 ]. In other words, older students are more proficient, but their flexibility to switch among different strategies is less. Ameer & Sigh [ 28 ] obtained similar results and found a considerable difference in mathematical achievement; specifically, older students performed better than younger students in number sense and computation using one-way analysis of variance (ANOVA) ( F ) of F (2,411) = 4.82, p  < 0.01. Molnár et al. [ 3 ] found that the student grade affects domain-specific and complex problem-solving skills. They observed that the development of problem-solving skills was noticeable across grades in elementary school but stopped in secondary school. The fastest development of domain-specific problem-solving occurred in grades 7 and 8 [ 3 ], but the fastest development of complex problem-solving occurred in grades 5–7 [ 3 ]. No development was detected between grades 4 and 5 as well as grades 6 and 7 for domain-specific and complex problem-solving skills, respectively. Similarly, Greiff et al. [ 4 ] concluded that students developed problem-solving skills across grades 5–11 with older students being more skilled. However, the grade 9 students deviated from the development pattern, and their problem-solving skills dropped. The theories from Molnár et al. [ 3 ] and Greiff et al. [ 4 ] are the benchmark cases herein.

The above studies showed that problem-solving skills mostly developed during compulsory schooling and developed most quickly in specific grades. This indicates that specific development times can be targeted to enhance the problem-solving skills [ 3 ]. However, Jabor et al. [ 30 ] observed contradictory results showing statistically significant differences with small effects in mathematical performance between age groups: those under the age of 19 outperformed those over the age of 19 years old. Grissom [ 13 ] observed a negative correlation between age and school achievement that remained constant over time.

2.2. Effects of school location and gender on mathematical problem-solving skills

School location has been shown to affect mathematical achievement [ 9 , 14 ]. In 15 countries, students in rural schools performed considerably worse than students in urban schools in mathematics [ 9 , 10 ], science and reading [ 9 ]. In addition, Nepal [ 15 ] discovered that urban students significantly outperformed rural students in mathematical problem-solving skills ( t  = −5.11, p  < 0.001) and achievement ( t  = −4.45, p  < 0.001) using the results of an independent sample t -test (t). However, other countries have found that rural students outperformed urban students in mathematics [ 6 , 10 ]. These variations may be attributed to a lack of instructional resources (e.g., facilities, materials, and programs), professional training (e.g., poorly trained teachers), and progressive instruction [ 6 ]. The results of Williams's study [ 10 ] serve as the basis for the current study.

Gender differences in mathematics have received attention because studies show that male students outperform female students on higher-level cognitive tasks [ 31 ]. This is a shift from a meta-analysis study that found gender differences in mathematics to be insignificant and favored female students [ 32 ]. At the college level, female students slightly outperform male students in computation while male students outperform female students in problem solving. However, no gender differences have been observed among elementary and middle school students. This result was strengthened by other meta-analysis studies [ 7 , 8 ], which concluded that there was no gender difference in mathematical performance and problem-solving skills [ 15 , [33] , [35] , [34] ]. Gender similarity in mathematics is achieved when equal learning opportunities and educational choices are provided and the curriculum is expanded to include the needs and interests of the students [ 16 , 18 , 31 ].

From a sociological perspective, gender similarity in mathematics makes sense. If there is a gender difference in mathematics, this has been attributed to science, technology, engineering, and mathematics (STEM) being stereotyped as a male domain [ 8 ]. Stereotypes influence beliefs and self-efficacy of students and perceptions of their own abilities [ 8 , 19 ]. This is the reason for the low interest of female students in advanced mathematics courses [ 18 , 19 ]. However, Halpern et al. [ 16 ] found that more female students are entering many occupations that require a high level of mathematical knowledge. Moreover, Anjum [ 36 ] found that female students outperformed male students in mathematics. This may be because female students prepared better than the male students before the test and were more thorough [ 36 , 37 ]. The study of Anjum [ 36 ] is one of the basis cases of the current study.

Differences in brain structure support the argument that there are gender differences in mathematical performance [ 16 , 17 ]. Females have less brain lateralization (i.e., symmetric left and right hemispheres), which helps them perform better verbally. Meanwhile, males have more brain lateralization, which is important for spatial tasks [ 17 ]. In addition, the male hormone testosterone slows the development of the left hemisphere [ 16 ], which improves the performance of right brain-dominant mathematical reasoning and spatial tasks.

3.1. Instrumentation

In this study, a science-related mathematical problem-solving test was used. This is a mathematics essay test where the problems are in the form of scenarios related to environmental management. Problems are solved by using technology as a tool (e.g., calculator, grid paper). The test was developed in an interdisciplinary STEM framework, and it is targeted toward grades 7–9. There were six scenarios in total: some were given to multiple grades, and others were specific to a grade. They included ecofriendly packaging (grade 7), school park (grade 7), calorie vs. greenhouse gas emissions (grades 7–9), floodwater reservoir (grade 8), city park (grades 8–9), and infiltration well (grade 9). These scenarios cover topics such as number and measurement, ratio and proportion, geometry, and statistics. Every scenario had a challenge, and students were provided with eight metacognitive prompt items to help them explore their problem-solving skills.

The test was administered by using paper and pencils for a 3-h period with a break every hour. At the end of the test, students were asked to fill in their demographic information. Each prompt item had a maximum score of 5 points: a complete and correct answer (5 points), a complete answer with a minor error (4 points), an incomplete answer with a minor error (3 points), an incomplete answer with a major error (2 points), and a completely wrong and irrelevant answer (1 point). Each scenario had a maximum total score of 40 points.

The test was validated to determine whether it contained good and acceptable psychometric evidence. It had an acceptable content validity index (CVI >0.67), moderate intraclass correlation coefficient (ICC) (rxx = 0.63), and acceptable Cronbach's alpha (α = 0.84). The construct validity indicated all scenarios and prompt items were fit (0.77 ≤ weighted mean square ≤1.59) with an acceptable discrimination value (0.48 ≤ discrimination value ≤ 0.93), acceptable behavior of the rating score, and good reliability (scenario reliability = 0.86; prompt item reliability = 0.94).

3.2. Participants

The test was administered to grades 7–9 students in east Java, Indonesia (n = 1067). The students were selected from A-accreditation schools in urban and rural areas; random classes were selected for each grade. The majority of the students were Javanese (95.01%), with the remainder being Madurese (3.3%) and other ethnicities. Table 1 describes the demographics of the participants.

Demographic characteristics of participants.

3.3. Data analysis

Data were collected between July and September 2022. Prior to data collection, ethical approval was sought from the institutional review board (IRB) of the Doctoral School of Education, University of Szeged and was granted with the ethical approval number of 7/2022. In addition, permission letters were sent to several schools to request permission and confirm their participation. The test answers of the students were scored by two raters – the first author of this study and a rater with master's degree in mathematics education – to ensure that the rating scale was consistently implemented. The results showed good consistency with an ICC of 0.992 and Cronbach's alpha of 0.996.

The scores from one of the raters were converted to a logit scale by weighted likelihood estimation (WLE) using the ConQuest software. A logit scale provides a consistent value or meaning in the form of intervals. The logit scale represents the unit interval between locations on the person–item map. WLE was chosen rather than maximum likelihood estimation (MLE) because WLE is more central than MLE, which helps to correct for bias [ 38 ]. The WLE scale was represented by using descriptive statistics to profile the students' mathematical problem-solving skills in terms of the percentage, mean score ( M ) and standard deviation ( SD ) for each phase. The WLE scale was also used to describe common difficulties for each phase. The development of students’ mathematical problem-solving skills across grades was presented by a pirate plot, which is used in R to visualize the relationship between 1 and 3 categorical independent variables and 1 continuous dependent variable. It was chosen because it displays raw data, descriptive statistics, and inferential statistics at the same time. The data analysis was performed using R studio version 4.1.3 software with the YaRrr package. A one-way ANOVA was performed to find significant differences across grades. An independent sample t -test was used to analyze significant differences based on gender and school location. The descriptive statistics, one-way ANOVA test, and independent sample t -test were performed using the IBM SPSS Statistics 25 software.

4.1. Student profiles

The scores of students were converted to the WLE scale, where a score of zero represented a student with average ability, a positive score indicated above-average ability, and a negative score indicated below-average ability. A higher score indicated higher ability. The mean score represented a student with average mathematical problem-solving skills ( M  = 0.001, SD  = 0.39). Overall, 52.1% of students had a score below zero. The distribution of scores among students was predominantly in the interval between −1 and 0. When the problem-solving process was analyzed by phase, the results showed that exploring and understanding were the most mastered problem-solving skills ( M  = 0.24, SD  = 0.51). Only 27.9% of students had below-average scores for the exploring and understanding phases, which indicates that they mostly understood the given problem and recognized the important information. However, the problem-solving skills decreased with higher phases. The students had below-average abilities in the phases of representing and formulating ( M  = −0.01, SD  = 0.36), planning and executing ( M  = −0.15, SD  = 0.41), and monitoring and reflecting ( M  = −0.16, SD  = 0.36). About 57.9% of the students had below-average scores for the representing and formulating phase, which indicates that they had problems making hypotheses regarding science phenomena, representing problems in mathematical form, and designing a prototype. The obvious reason for their difficulty with making hypotheses was that they did not understand simple concepts of science (e.g., CO 2 vs. O 2 ). In the planning and executing phase, 66.8% of the students failed to achieve a score greater than zero. This happened because they failed to apply mathematical concepts and procedures. Because they were unable to plan and execute a strategy, this affected the next phase of the problem-solving process. In the monitoring and reflecting phase, 68.0% of the students had a below-average score.

4.2. Development of mathematical problem-solving skills across grades

The development of the mathematical problem-solving skills of the students across grades was observed based on the increase in the mean score. The problem-solving skills developed from grade 7 to grade 8. The students of grade 7 had a mean score of −0.04 while grade 8 students had the highest mean score of 0.03. The students in grades 7 and 8 also showed more varied problem-solving skills than the grade 9 students did. In contrast, the grade 9 students showed a different pattern of development, and their mean score dropped to 0.01. Although the difference was not large, further analysis was needed to determine its significance.

Fig. 1 displays the development of the mathematical problem-solving skills of the students. The dots represent raw data or WLE scores. The middle line shows the mean score. The beans represent a smoothed density curve showing the full data distribution. The scores of the students in grades 7 and 9 were concentrated in the interval between −0.5 and 0. However, the scores of the grade 8 students were concentrated in the interval between 0 and 0.5. The scores of the students in grades 7 and 8 showed a wider distribution than those of the grade 9 students. The bands which overlap with the line representing the mean score, define the inference around the mean (i.e., 95% of the data are in this interval). The inference of the WLE score was close to the mean.

Differences in students' mathematical problem-solving skills across grades.

Note : PS: Problem-Solving Skills of Students.

The one-way ANOVA results indicated a significant difference among the problem-solving skills of the students of grades 7–9 ( F (1,066) = 3.01, p  = 0.046). The students of grade 8 showed a significant difference in problem-solving skills and outperformed the other students. The students of grades 7 and 9 showed no significant difference in their mathematical problem-solving skills. Table 2 presents the one-way ANOVA results of the mathematical problem-solving skills across grades.

One-way ANOVA results of the mathematical problem-solving across grades.

Note. Post hoc test: Dunnett's T3. 7, 8, and 9: subsample grade. <: direction of significant difference ( p  < 0.05).

Fig. 2 shows the development of the mathematical problem-solving skills of the students across grades based on school location and gender. The problem-solving skills of the urban students increased from a mean score of 0.07 in grade 7 to 0.14 in grade 8. However, the mean score of urban students in grade 9 dropped. In contrast, the mean scores of the rural students increased continuously with grade. The improvements were significant for both the rural ( F (426) = 10.10, p  < 0.001) and urban ( F (639) = 6.10, p  < 0.01) students. For the rural students, grade 9 students showed a significant difference in problem-solving skills. In contrast, urban students in grades 8 and 9 showed significant differences in problem-solving skills but not in grade 7.

Differences in students' mathematical problem-solving skills across grades and different demographic backgrounds.

(a) Differences in students grade 7 of mathematical problem-solving skills across grades and different demographic backgrounds

(b) Differences in students grade 8 of mathematical problem-solving skills across grades and different demographic backgrounds

(c) Differences in students grade 9 of mathematical problem-solving skills across grades and different demographic backgrounds

Note: WLE_PS: The students' problem-solving skills in WLE scale; F: Female; M: Male; ScLoc: School location; R: Rural; U: Urban.

When divided by gender, both female and male students showed improvements in their problem-solving skills from grades 7 and 8. However, female students in grade 9 showed a stable score while the scores of male students in grade 9 declined. Only male students in grade 7 showed a significant difference in the mean score. In urban schools, the scores of male and female students increased and decreased, respectively, from grade 7 to grade 8. Male students in rural schools showed an increase in score from grade 7 to grade 9. However, the scores of female students in rural schools decreased from grade 7 to grade 8. Table 3 presents the one-way ANOVA results for the mathematical problem-solving skills of the students considering gender and school location.

One-way ANOVA results for mathematical problem-solving skills across grades and different demographic backgrounds.

Fig. 2 shows that the distributions of the male and female scores of students were similar for every grade except rural grade 9 students. The scores of the rural female students were concentrated in the interval between 0 and 0.5 while the scores of the rural male students were mostly below 0. The scores of rural students in grade 7 and urban students in grade 9 (both male and female) were concentrated in the interval between −0.5 and 0. The scores of urban students in grades 7 and 8 were concentrated in the interval between −0.5 and 0.5.

Fig. 3 shows a detailed analysis of the development of mathematical problem-solving skills across grades for each phase of the problem-solving process. Similar patterns were observed in the exploring and understanding and the representing and formulating phases: the mean score increased from grade 7 to grade 8 but decreased from grade 8 to grade 9. Grade 8 students had the highest mean score and differed significantly from the scores of students in other grades.

Differences in students' mathematical problem-solving skills in every phase across grades: (1) Exploring & understanding, (2) Representing & formulating, (3) Planning & executing, (4) Monitoring & reflecting.

(a) Differences in students' mathematical problem-solving skills in exploring and understanding phase

(b) Differences in students' mathematical problem-solving skills in representing and formulating phase

(c) Differences in students' mathematical problem-solving skills in planning and executing phase

(d) Differences in students' mathematical problem-solving skills in monitoring and reflecting phase

Note: WLE_Exp_Un: The WLE score in exploring and understanding; WLE_Rep_For: The WLE score in representing and formulating; WLE_Plan_Ex: The WLE score in planning and executing; WLE_Mon_Ref: The WLE score in monitoring and reflecting.

The scores of the students for the planning and executing phase increased with grade. However, the difference was only significant at grade 9. Grades 7 and 8 students showed an increase in score, but the improvement was not significant. There was no pattern detected in the monitoring and reflecting phase. The score was stable for grades 7 and 8 students but improved for grade 9 students. The mean score for each phase and the one-way ANOVA results are presented in Table 4 .

One-way ANOVA results for every phase of problem-solving across grades.

Fig. 3 shows that the distributions of the problem-solving skills of the students were similar across grades and phases. However, the distributions were different for grade 9 students in the representing and formulating, planning and executing, and monitoring and reflecting phases, where 95% of the data were in the interval between −0.5 and 0.5.

4.3. Effects of demographic background

4.3.1. school location.

The mathematical problem-solving skills of the students differed significantly based on school location. Urban students scored higher than rural students. The results of the t -test for mathematical problem-solving skills based on school location are presented in Table 5 .

T-test results for mathematical problem-solving skills based on school location.

The effects of the school's location on the performances of male and female students were analyzed. The results showed that the scores of the female students differed significantly based on school location ( t (613) = −6.09, p  < 0.001). Female students in urban schools ( M  = 0.18, SD  = 0.39) outperformed female students in rural schools ( M  = −0.08, SD  = 0.37). Similar results were observed for male students with urban students ( M  = −0.01, SD  = 0.35) outperforming rural students ( M  = −0.12, SD  = 0.39) by a significant margin ( t (382.764) = −3.25, p  < 0.01).

When analyzed by grade, grades 7 and 8 students contributed to the difference based on school location with t (377.952) = −6.34, p  < 0.001 and t (300.070) = −5.04, p  < 0.001, respectively. Urban students in grades 7 and 8 performed significantly better than their rural counterparts did. However, there was no significant difference between rural and urban students in grade 9 ( t (354) = 0.71, p  = 0.447).

4.3.2. Gender

Male and female students showed a significant difference in their mathematical problem-solving skills. Overall, female students outperformed male students. The detailed results of the independent sample t -test for mathematical problem-solving skills based on gender are presented in Table 6 .

T-test results for mathematical problem-solving skills based on gender.

The results were analyzed to determine whether the school location contributed to the gender difference. The gender difference was most significant among urban students ( t (596.796) = −4.36, p  < 0.001). Female students from urban schools ( M  = 0.12, SD  = 0.39) outperformed male students from urban schools ( M  = −0.01, SD  = 0.35). There was no significant difference between female and male students from rural schools ( t (425) = −1.31, p  = 0.191).

Grades 7 and 9 students contributed to the gender difference with t (372.996) = −3.90, p  < 0.001 and t (354) = −2.73, p  < 0.01, respectively. Female students in grades 7 and 9 outperformed their male counterparts. However, there was no significant gender difference among grade 8 students ( t (329) = −0.10, p  = 0.323).

5. Discussion

The mathematical problem-solving skills of the students were categorized as average. In addition, the difficulties of students increased in line with the problem-solving phase. Fewer students failed the exploring and understanding phase than the subsequent phases. This confirms the results of previous studies indicating that more students failed further along the problem-solving process [ 25 , 26 ]. Because the problem-solving process is sequential, students who have difficulty understanding a problem will fail the subsequent phases [ 27 ].

The development of mathematical problem-solving skills was evaluated according to the mean WLE score. The mathematical problem-solving skills of the students developed from grade 7 to grade 8 based on the increase in their mean scores. However, the development dropped in grade 9. This agrees with previous results that concluded that higher grades had the highest problem-solving skills, but the fastest skill development took place in grades 7–8 after which it dropped [ 3 , 4 ]. These results indicate that the mathematical problem-solving skills of the students should improve and be strengthened in grades 7–8, which will help them perform better in grade 9.

In this study, the effects of the demographic background of the students were analyzed in detail, which is an aspect missing from previous studies. The results showed that the mathematical problem-solving skills of urban students increased from grade 7 to grade 8 but decreased in grade 9. The same pattern was found among male and female students. However, a different pattern was observed for rural students, where the skills of grade 9 students continued to increase. The different patterns may be attributed to a structural reorganization of cognitive processes at a particular age [ 3 ]. However, more research is needed on the effects of the demographic backgrounds of students on mathematical problem-solving skills. These results were different from previous results because the previous studies only analyzed the development in general, without focusing on their demographic background. Hence, different patterns of development were observed when it was thoroughly examined.

Because solving problems is a cognitive process, the development of problem-solving skills for particular phases and processes needed to be analyzed. The students showed the same pattern for knowledge acquisition (i.e., exploring and understanding, and representing and formulating phases), with an increase in skill from grade 7 to grade 8 but a decrease in grade 9. However, the students showed increasing skill in knowledge application (i.e., planning and executing, as well as monitoring and reflecting phases) across grades. This means that the difference between the mean scores in grade 9 was not significant across phases. Grade 9 students had lower scores than grade 8 students for the knowledge acquisition phase but higher scores for the knowledge application phase. In contrast, the gap between the mean scores of grades 7 and 8 was large across phases.

These results proved that there is a significant difference in the mathematical problem-solving skills of students based on their demographic backgrounds. The urban students outperformed rural students, which confirms the results of previous studies [ 9 , 10 , 15 ]. The difference can be attributed to the availability of facilities, teacher quality, and interactive teaching and learning instruction [ 6 ]. In Indonesia, the policies for the public educational system for middle schools are set at the municipal level. This means that each city has its own policies for teacher training, teacher recruitment, teaching and learning processes, facilities, etc. Urban schools mostly have stricter policies as well as various programs to help students improve their knowledge and skills. In addition, they have supportive facilities for teaching and learning. This unequal environment is the strongest reason for the difference in mathematical problem-solving skills.

The results were analyzed in detail to observe which groups in the rural and urban schools contributed to the difference. Both male and female students in urban schools performed better than their counterparts in rural schools did. In addition, urban students in grades 7 and 8 outperformed their rural counterparts. There was no significant difference between urban and rural students in grade 9. This may be because grade 9 is the last grade in middle school, so students have to prepare for high school entrance requirements, including exam and/or grade point average scores. Hence, both rural and urban schools focus much effort on the teaching and learning process in this grade.

In this study, the female students surprisingly had better mathematical problem-solving skills than the male students did. This confirmed the results of the meta-analysis by Hyde et al. [ 32 ] and study by Anjum [ 36 ], which found that female students slightly outperformed male students in mathematics. This difference may be because of motivation and attitude [ 39 , 40 ]. Female Indonesian students are typically more diligent, thorough, responsible, persistent, and serious with their tasks.

A detailed analysis was performed to evaluate which group of students contributed to the gender differences. The results showed that female students outperformed male students in urban schools. This may be because male students at urban schools typically display an unserious attitude toward low-stake tests. In addition, female students outperformed their male counterparts in grades 7 and 9. The reason for this difference requires further analysis.

6. Conclusion

Studying the problem-solving skills of students is crucial to facilitating their development. In this study, the conclusions are presented as follows:

• • The mathematical problem-solving skills of the students were categorized as average. More students failed at higher phases of the problem-solving process.
• • Students showed development of their mathematical problem-solving skills from grade 7 to grade 8 but a decline in grade 9. The same pattern was detected across grades for urban students, both female and male. However, the problem-solving skills of rural students increased with the grade.
• • A similar development was observed for the individual problem-solving phases. In the knowledge acquisition phase, the problem-solving skills of the students developed from grade 7 to grade 8 but decreased in grade 9. However, problem-solving skills increased across grades in the knowledge application phase.
• • The school location was shown to have a significant effect on the mathematical problem-solving skills of the students. Urban students generally outperform students in rural schools. However, gender and grade contributed to differences in mathematical problem-solving skills based on school location. Female and male urban students in grades 7 and 8 outperformed their rural counterparts.
• • In general, female students outperformed male students in mathematical problem-solving skills, particularly those from urban schools and in grades 7 and 9.

The sampling method and the number of demographic backgrounds limited the scope of this study. Only students from A-accreditation schools were selected because higher-order problem-solving skills were considered assets. Moreover, the study only included three demographic factors: grade, gender, and school location. More demographic information, such as school type, can be added (public or private schools). Hence, future studies will need to broaden the sample size and consider more demographic factors. Despite these limitations, this study can help teachers determine the best period for enhancing the development of mathematical problem-solving skills. Moreover, the differences in mathematical problem-solving skills due to demographic background can be used as a basis for educational policymakers and teachers to provide equal opportunity and equitable education to students.

Author contribution statement

Ijtihadi Kamilia Amalina: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Tibor Vidákovich: Conceived and designed the experiments; Contributed reagents, materials, analysis tools or data.

Funding statement

This work was supported by University of Szeged Open Access Fund with the grant number of 6020.

Data availability statement

Additional information.

No additional information is available for this paper.

Declaration of competing interest

No potential conflict of interest was reported by the authors.

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37 Math Problems For 3rd Graders: Answers With Worked Examples

Michele Bell

3rd grade math problems formally introduce 3rd graders to math skills such as multiplication, division, and fractions. They build on learners’ conceptual understanding of partitioning shapes and using repeated addition with concrete models, drawings, and expressions, learned in previous grades, to make connections using symbols, letters for unknown numbers, and equations.

In this blog, we look closely at the essential math skills a third grader needs to know and provide math problems, worked examples, and teaching tips, to help teachers and educators support their third-grade students in mathematics.

What are math problems for 3rd graders?

3rd grade math problems are specific math problems suitable for 8-9 year olds. They include the following math concepts: 

• Subtraction
• Multiplication
• Place Value
• Measurement and Data
• Geometrical Shapes

16 Fun Math Games and Activities Pack for 3rd Grade

16 fun math games and activities for 3rd grade students to complete independently or with a partner. All games are printable and ready go. The perfect activity pack for 'fast finishers or morning work.

Benefits of math problems for 3rd graders

3rd graders can use these math problems to build on their prior knowledge of addition and subtraction developed in Kindergarten, 1st  and 2nd grade. 

The math concept of repeated addition, learned in 2nd grade, directly links to their understanding of multiplication through models learned in the 3rd grade such as equal groups and arrays. 

Many 3rd grade math concepts they focus on strengthen their foundation in fraction concepts and set them up for success with fraction operations explored in 4th grade and 5th grade .  

Word problems help students make connections between representations and develop students understanding of the relationship between addition and subtraction or multiplication and division.

Multiplication word problems are just one example of word problems used to deepen students’ understanding of each operation. Educators should incorporate word problems through all phases of teaching each concept, not just near the end of a unit when learners appear ready to “apply” their learning. 

3rd grade math curriculum 

The topics focused on in third-grade math include: 

• Addition within 1,000
• Subtraction within 1,000
• Whole number multiplication
• Whole number division
• Understanding fractions
• Rounding to the nearest 10 or 100
• Elapsed time
• Representing and Interpreting Data
• Describing Quadrilaterals

How the 3rd grade math curriculum fits into learners’ math development 

3rd grade students extend their 2nd grade knowledge of adding and subtracting within 100 to include adding and subtracting within 1,000 using place value strategies and properties of operations. 

Learners continue to build on equal sharing from 1st grade and learn about unit fractions and the meaning of the numerator and denominator. This sets them up for success with fraction operations in 4th and 5th grade.

Common challenges teaching third graders

As 3rd graders use all four operations, a common challenge for some is mixing up which operation to use when problem solving. They may focus on one word in a word problem instead of considering the entire problem. 

Concrete models and representations can help students work through problems by showing the actions happening in a problem situation. However, as they get older, some students may also want to reduce their use of math manipulatives to build concrete models or draw pictures to show their thinking leaving room for error. 

Teach students to visually show their thinking through models or pictures, or express it with words and numbers instead of keeping it in their head. 

Math problems for 3rd graders with answers

Here are 34 math problems for 3rd graders organized by topic, including:

Addition math problems for 3rd graders 

In 3rd grade, students build on addition skills from K-2nd grade and are expected to add fluently within 1,000. This can include one-digit numbers, two-digit numbers and three-digit numbers.

Question 1 

A local charity collected 356 canned food items in September. They collected 419 canned food items in October. How many canned food items did they collect during the two months?

Answer: 775

Students apply strategies based on place value when solving multi-digit addition problems. A place value chart is a great tool to help organize and record their thinking as they work with values that may require regrouping. 

In this problem, students solve an addition problem that involves regrouping in one place, from ten ones to one ten.

Jessica and Caleb were both sharing their baseball card collections with their class. Jessica has 435 baseball cards in her collection and Caleb has 482 baseball cards in his collection. How many total baseball cards did they both collect?  

Answer: 917

An open number line is another valuable tool that students can draw on anytime. They can use multiple addition strategies on a number line. For example, they might start with an amount and then add on the hundreds, tens and ones of the second number. 

Another strategy is adding the hundreds from both addends, then the tens, and the ones. No matter the strategy, number lines are great visuals for students to record their thinking and refer back to it in their explanations.

Sabrina had 267 beads to use for making bracelets. Her mom bought her a new pack of 135 beads. How many beads does Sabrina have now?

Answer: 402 beads

This problem requires regrouping of ones to tens, and tens to hundreds. Students should have experience regrouping in one or more places when adding within 1,000. 

The partial sum strategy helps set the foundation for using an algorithm in fourth grade to add greater multi-digit numbers.

Subtraction math problems for 3rd graders

As with addition, students build on their subtraction knowledge from K-2nd grade and use single-digit subtraction and subtracting 2-digit numbers within 100 to develop fluency in subtracting within 1,000. This can include up to 3-digit numbers. 

Although students have used subtraction since Kindergarten, it can still be difficult for many students. Providing students with multiple strategies and methods for solving with subtraction allows them to choose the most efficient method move towards fluency, which includes: 

• Efficiency 

283 guests were waiting to ride a ferry boat to get to the entrance of a theme park across the lake.  Only 155 guests can ride the ferry boat at a time. How many guests will have to wait for the next ferry boat to arrive?

Answer: 128 guests

When completing subtraction word problems, make connections between strategies students use to add within 1,000 and subtract within 1,000. This strengthens their understanding of the relationship between addition and subtraction, and place value. 

Similarly to addition, students can use a place value chart to help them with regrouping when subtracting 2 or 3-digit numbers. In this problem, students need to regroup 1 ten for 10 ones.

Andrew and his family drove 607 miles to get to their hotel. They made one stop after driving for 312 miles. How many more miles did they have to drive to get to the hotel?

Answer: 295

Using a number line to subtract, or count back, may confuse some students. Remind students to use strategies that make sense to them. They may start with one number and subtract the hundreds, tens, and then ones of the second number. 

Some students prefer to subtract using place value, while others subtract until they get to a multiple of 10 and continue subtracting from there.

Josie had $168 at the beginning of the week. She spent $27 at the mall on Tuesday. She worked on Thursday and earned some money. She ended the week with $195. How much money did she earn on Thursday?

Answer: $54

This problem includes both addition and subtraction and provides an opportunity for students to write an equation using a letter as the unknown value. Students can use the relationship between addition and subtraction to determine what the letter represents.

168 – 27 + s = 195

141 + s = 195

195 – 141 = s

Multiplication math problems for 3rd graders

Many third-grade standards group multiplication and division together. When students are officially introduced to both operations, typically multiplication is first. 

Learners develop an understanding of multiplication using concrete models, pictures, and equations when solving word problems. 

In 3rd grade, they solve word problems that lend themselves to making equal groups, arrays, or finding the area as a measurement. 

Other multiplication concepts that 3rd grade students will explore include:

• Properties of operations 
• Finding an unknown product 
• Solving one or two-step word problems that involve more than one operation

There are 8 dry-erase markers in one pack. How many dry-erase markers are in 3 packs?  

3rd graders explore equal groups. They can do this using an array. Students must understand that the one factor is the number of groups and the other is the number in each group.  

For example, 3 x 8 can be represented using the following array: 

Find the product of 5 x 3 x 4

Learners explore associative property in 3rd grade as well as commutative property and distributive property.

When evaluating multiplication expressions with three or more factors, students learn that the way they group the factors to multiply does not change the value of the product. 

Encourage students to start with the two factors that seem the friendliest to multiply, but to also keep in mind how confident they will feel when multiplying the product of those two factors with the next factor in the expression.

Other properties of multiplication that students will explore in 3rd grade include the:

• commutative property
• distributive property
• identity property
• zero property

Find the product of 7 x 40.

Answer: 280

7 x 40 = 7 x 4 tens 

7 x 4 tens = 28 tens 

28 tens = 280

7 x 40 = 280

Question 10

The school music teacher set up his classroom for an after-school chorus rehearsal. He formed 3 rows with 12 chairs in each row. How many chairs are in the classroom in total?

Answer: 36 chairs

Allow students to choose their method, whether it’s using manipulatives or drawing pictures to represent the information in word problems. 

Ask them where they see each value in their concrete model or picture.

Division math problems for 3rd graders

Many multiplication strategies are closely related to division problem strategies. Third graders must understand the relationship between multiplication and division to use it as a strategy for problem-solving. This helps with math problems that involve finding an unknown. 

Third graders learn to write a blank, question mark, or letter when there is an unknown product, quotient, or factor. This aids them when writing a related equation using a known math fact with an operation they are comfortable with. 

Question 11

Zoe has 42 stickers. She wants to give 7 stickers to each of her friends at her birthday party. How many friends are at her birthday party?

Answer: 6 friends

Once students begin working with division, they tend to focus on specific keywords such as “each” and think that the problem represents multiplication. Students must model the actions of what is happening in the problem. Giving the same amount out multiple times will highlight the division and sharing equal amounts. 

Encourage students to build and draw models to show their thinking. Ask them to explain their strategy to you, a partner, or the class for solving division word problems such as this one. 

Question 12

There are 3 boxes on each shelf in the backroom of a bookstore. 36 new boxes were delivered and must be placed on the 4 shelves. How many total boxes are on each shelf? 

Extend students’ understanding of multiplication and division by pairing multiplication and division together, or with addition or subtraction. This enables students to solve two-step problems represented as an expression or equation. 

Follow-up discussions to help students understand which operation they should solve first to set them up for success with the order of operations in fifth grade and beyond.

Question 13

George ran 63 miles last week. If he ran every day last week, how many miles did he run each day?

Answer: 9 miles

Writing an equation to represent math word problems supports learners’ understanding of the relationship between multiplication and division. 

When learners write a division equation using a letter or question mark for the unknown number, they can rewrite the expression as a multiplication expression if they aren’t confident with division facts. 

A familiar multiplication equation may lead third graders to use mental math once they’re fluent with multiplication facts.

Question 14

Determine what the ? represents in the equation. 48 6 = ?

48 ➗ 6 = ? 

6 x ? = 48 

6 x 8 = 48 

Rounding math problems for 3rd graders 

The third place value concept covered in third grade is rounding. Third-grade math requires students to use their place value understanding to round whole numbers to the nearest 10 or 100. 

Understanding the value of digits in a number is important when rounding because it helps to determine which 10 or 100 a whole number is closest to. 

Question 15

Round 27 to the nearest 10.

Using a number line with a midpoint and two endpoints helps students visualize which tens or hundreds a whole number falls between. 

Although number lines are not required to be used as a strategy in 3rd grade, they can be used to help students develop their rounding skills conceptually.

Question 16

Round 639 to the nearest 100

Answer: 600

Question 17

Round 450 to the nearest 100

Answer: 500

In some instances, a value is directly in the middle of the tens or hundreds. However, learners should know that if the digit being rounded is 5 or larger, then it must round up. 4 or less and the digit rounds down. 

Fraction math problems for 3rd graders

Students build on their fraction knowledge form 1st and 2nd grade, including: 

• Partitioning shapes into equal shares by exploring unit fractions 
• The meaning of the numerator and the denominator

3rd graders should have plenty of opportunities to represent fractions by building concrete models and drawing pictures before they are required to write fraction notations. 

Using fraction vocabulary such as halves, thirds, fourths, parts and parts of a whole helps students make sense of fractions as part of a number before writing the numerator above the denominator with a fraction bar in between. 

Facilitate discussions involving unit fractions and how the non-unit fractions are the sum of multiple unit fractions.

Question 18

Represent the fraction \frac{3}{4} by drawing a model.

Question 19

Count the parts and label each tick mark on the number line: 

\frac{1}{8} , \frac{2}{8} , \frac{3}{8} , \frac{4}{8} , \frac{5}{8} , \frac{6}{8} , \frac{7}{8} , \frac{8}{8}

Discuss with students that the intervals or spaces between the tick marks represent the distance from one tick mark to the next tick mark or a unit fraction such as \frac{1}{8} . 

The distance from zero to that point or tick mark is a different value labeled with a specific fractional value, such as \frac{5}{8} .

Fraction problems for 3rd graders

Question 20.

Select the equivalent models.

Answer: C and E

Equivalent fractions in third grade focus on visual models preparing them to explore finding equivalent fractions through operations in the fourth grade.

Question 21

Which is greater, \frac{4}{8} or \frac{4}{6} ? Write a comparison statement using the symbols >, =, or <.

Answer: \frac{4}{8} < \frac{4}{6}

Time math problems for 3rd graders

In third-grade math, students learn to tell time to the nearest minute on digital and analog clocks. 

They use their understanding of time to determine an elapsed time when given a start and stop time for an event or sequence of activities. 

Students explore using number lines to help them understand elapsed time word problems . These can include:

• Start and stop time given, elapsed time unknown
• Start and elapsed time given, stop time unknown
• Elapsed time and stop time given, start time unknown

Question 22

Represent the time shown on the digital clock by drawing on the analog clock. 

Answer: 

Question 23

It is 10:50 a.m. and lunch begins in 15 minutes. What time will it be? Use the number line to show the time.

Answer: 11:05 a.m.

Students should use the number line to determine time in a way that makes sense. 

Question 24

On Saturday, Angela left the park at 2:45 p.m. where she attended soccer practice for 75 minutes. What time did Soccer practice begin?

Answer: 1:30 p.m.

Working backwards to determine a time may seem challenging for some students. They may use a number line to subtract whole numbers. 

Pay attention to how students maneuver counting back in time from the 1 o’clock hour into the 12 o’clock or morning time.

Data math problems for 3rd graders

Third-grade math develops the concept of collecting and representing data from previous grades. Students must: 

• Ask and answer questions
• Ceate tally charts
• Draw picture graphs and bar graphs
• Learn about line plots

In third grade, students use larger values and scales to represent their data than in previous grades. They also answer one and two-step word problems about information presented in graphs.

Question 25 

 How many more cloudy days were there than rainy days in March?

Answer: 5 days

In grade 3, students create and read graphs with scales greater than 1. If they struggle to interpret the information on a bar or picture graph, they can add tick marks between the intervals to count accurately.

Question 26

The students in Mrs. Campbell’s class voted on which animal should be their class mascot. The votes are shown on the graph. 

How many fewer students voted for an animal that lives in the ocean than an animal that lives on land? 

Answer: 4 students

(8 + 6) – 10= 4

Provide students with opportunities to solve one and two-step problems involving a picture or bar graph. 

Problems should relate directly to “How many more?” and “How many less?” questions.

Question 27 

What is the length of the pencil measured to the nearest half or quarter of an inch?

Answer: 6 \frac{1}{2} inches

3rd grade students explore measuring objects with a ruler marked with halves and fourths of an inch. Examples like this bring math into real-life scenarios for working with fraction values.  Students can see and hear how the terms fourths and quarters are used interchangeably regarding measurement.

Question 28 

Create a line graph to display the collected data.

Students are introduced to line plots with whole number units in the 2nd grade. In the 3rd grade, students measure units to the nearest half or fourth and record the measurement of objects on line plots. 

They extend their understanding of representing information on a line plot in 4th and 5th grade where they solve problems involving fraction operations.

Area math problems for 3rd graders

Students apply their understanding of multiplication and division while exploring the area of two-dimensional shapes. Area is an attribute of 2-D shapes and multiple strategies can be used to find it. 

Question 29

Find the area of the rectangle.

Answer: 48 square units

Question 30

Jillian has a new painting to hang in her bedroom. The side lengths of the painting are 7 in. and 4 in. What is the area of the painting?

Answer: 28 square inches 

Students must understand that counting the number of tiles arranged as an array is the same as multiplying two of the given side lengths for a rectangle.

Question 31

Find the total area of the figure.

Answer: 10 square inches

(2 x 3) + (4 x 1)

Learners apply their understanding of area to determine the area of rectilinear figures: figures composed of more than one rectangle. 

A common misconception for some students is to multiply all of the side lengths. This is a great opportunity to relate the area to the distributive property by decomposing the figure into two separate rectangles. Then they can determine the area of each rectangle and add the two values together to find the total area. 

Perimeter math problems for 3rd graders

3rd graders explore the perimeter of shape, how to work it out and how it differs from the area. 

A common misconception for third graders, that sometimes carries on into fourth grade and fifth grade, is mixing up area and perimeter. Take the time to allow students to explore and discuss examples and non-examples of area and perimeter in the real world and around the classroom.

Question 32

Find the perimeter of the bulletin board.

Answer: 160 inches

Students need to add all of the side lengths together: 

45 + 45 + 35 +35 = 160

Question 33

The school garden has one side length of 13 feet. The perimeter of the garden is 40 feet. How long is the other side length?

Answer: 7 feet

13 + 13 + s + s = 40  

(2 x 13) + (2 x s ) = 40  

26 + (2 x s ) = 40 

40 – 26 = 2 x s   

Question 34

• Which rectangles have the same perimeters but different areas? 
• Which rectangles have the same area but different perimeters?

• Rectangles 3 and 4 have the same perimeters and different areas. 
• Rectangles 2 and 3 have the same area and different perimeters.

Reminder students the perimeter is the distance around the outside of a shape or object and the area is the number of square units needed to cover the space of the shape or object.

Rectangle 1 : side length = 2, side length = 12 

Perimeter: 12 + 12 + 2 + 2 = 28

Area : 2 x 12 = 24

Rectangle 2: side length = 6, side length = 6

Perimeter: 6 + 6 + 6 + 6 = 24

Area : 6 x 6 = 36

Rectangle 3: side length = 4, side length = 9

Perimeter: 9 + 9 + 4 + 4 = 26

Area : 4 x 9 = 36

Rectangle 4: side length = 6, side length = 7

Perimeter: 6 + 6 + 7 + 7 = 26

Area : 6 x 7 = 42

Geometry problems for 3rd graders

Describing, analyzing, and comparing properties of 2D shapes is one of the main concepts of geometry in third-grade math. 

3rd graders must determine whether shapes are considered quadrilaterals from the number of their sides. They describe and draw examples of specific quadrilaterals such as a rectangle and a rhombus and sort shapes according to their sides, angles, and other characteristics. 

This sets a foundation for further investigation of quadrilaterals and triangles in fourth grade and fifth grade where they learn about the hierarchy of quadrilaterals.

Question 35

How many of the shapes are quadrilaterals?

Answer: 5 are quadrilaterals

Provide printouts of the shapes and allow students to cut and sort the shapes into groups such as quadrilaterals or non-quadrilaterals. 

Students may also draw a circle around shapes with four sides.

Question 36

Count and record the number of quadrilaterals with:

• 0 pairs of parallel sides

1 pair of parallel sides

2 pairs of parallel sides

0 parallel sides 

Question 37

Draw the following 2D shapes: 

• A quadrilateral that is not a parallelogram or trapezoids.

Possible extension: ask students to explain why the shape they have drawn is an example of that specific shape.

3 top tips for teaching math problems to 3rd graders

• When exploring multiplication and division concepts, provide multiple opportunities for students to make connections between models, representations, and equations.  Present learners with real-life word problems. Have students use math manipulatives to build concrete models and draw pictures that represent their model and the word problem. Next, ask them to write an equation representing the word problem and their models. Last, prompt students to explain how all three pieces of their work represent the word problem and how they used each to solve the problem.
• Remember the CRA model: Concrete, Representation, Abstract. Understanding multiplication, division, and fractions starts at the concrete phase in 3rd grade, moves into the representation phase, and has many opportunities for the abstract phase. Other skills such as fluently adding and subtracting within 1,000 focus on the representation and abstract phases. There may be some students who need more time in the concrete phase, using manipulatives while others may be more comfortable with drawing pictures and number lines or using the relationship between addition and subtraction as their strategies. Students should always use strategies and models that make sense to them.
• Teach multiplication facts using positive experiences. Avoid focusing on memorization drill worksheets and multiplication tables. Instead provide opportunities to practice in meaningful ways using games, real-world problems, partner and group discussions, and reasoning with manipulatives.

How can Third Space Learning help with 3rd grade math?

STEM-specialist tutors help close learning gaps and address misconceptions for struggling 3rd grade math students. One-on-one online math tutoring sessions help students deepen their understanding of the math curriculum and keep up with difficult math concepts.

Each student works with a private tutor who adapts instruction and math lesson content in real-time according to the student’s needs to accelerate learning.

3rd grade math worksheets and resources 

Looking for more resources? Check out our math games and selection of second grade addition and subtraction worksheets, posters and activities covering the key 3rd grade math topics and more:

• 3rd Grade Fractions Error Analysis  
• 3rd Grade Addition And Subtraction Code Crackers
• 3rd Grade Place Value and Rounding Word Problems
• 3rd Grade CCSS Practice Test 

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The big myth that keeps people from loving math

• There is no secret code or single method to solve a mathematical problem.
• This “answer-getting” myth leads to disempowerment when learning math.
• To counteract the myth, don’t focus on the answer but the process.

Teaching math as if there’s only one correct way to solve a problem makes us think that we’re problem-solving, but instead, we’re “answer-getting.” I’ve seen it so many times, but none bothers me more than watching elementary and middle school students solve word problems in this way.

Consider this typical middle school prompt: A store is selling 6 bags of marbles for $18. What is the unit price for a bag of marbles? When I read this problem, I picture a child looking up at me and asking, “Does ‘of’ mean multiplication?” It has happened to me so many times when I visit math classrooms.

There’s no secret code. Of could mean “multiply,” but it might not. These are the highly counterproductive questions that children ask when they have been presented with a “single way” to solve word problems, such as looking for keywords. 

In this example, the students may immediately multiply 6 × 18. If you then ask them why the unit price of a single bag of marbles would cost $108 — and be so much more than the price of 6 bags of marbles — they will look at you with uncertainty. This is the end result of answer-getting.

Problem-solving is a distinct cognitive experience. Instead, we ask, what is happening in the problem? It is not mindlessly following a single prescriptive set of steps. The way to solve this problem, and every problem, is to understand what is happening. But that means there will be many paths to the answer. How I understand the problem might be quite unlike how you understand it.

The right way is the wrong way

When we are taught to rely on a singular, step-by-step process as the true way to solve a math problem, we turn off our problem-solving brain. These skills require continuous work to keep them sharp, and the constant reliance on someone else’s “exact right” method dulls them. Over the years, we may even lose at least some of our problem-solving acumen by not using it.

This reliance also discourages courage — we need to take chances to solve problems, and insistence on following a singular method prevents us from risking wrong answers via experimentation.

We can solve problems in many distinct ways. In fact, trying different approaches is fun as well as instructive, and it is necessary when problem-solving gets hard — which is often when the problems are most worth solving. Engineers who write software code or build bridges make a conscious attempt to solve problems in more than one way, even when a solution is readily available to them. 

Why not solve it and move on?

First of all, if you dig deeper to find more than one solution, you can decide which one among them is less expensive, more durable, or more elegant — whichever outcome matters most to you. Second, and perhaps more significant, when problem-solving gets really hard and the way ahead isn’t clear, you need to be ready to try anything. And the first step of the “try anything” approach is to back up and examine a problem from every angle, or at least from more angles than you initially see.

In the real world, of course, we often resort to looking at problems from fresh angles out of desperation. “Try anything” is the motto. As one member of a two-working-parents household with elementary school twins during the COVID-19 pandemic, we often were compelled to try anything to solve problems regarding work, social distancing protocols, on-and-off remote school, and limited childcare.

To expose the myth of a single correct method for the sham that it is, we need to understand the consequences of answer-getting versus problem-solving. Because we’ve been brainwashed into believing that answer-getting is good and because most of us spent years in answer-getting math curricula, we don’t realize the negative effects it has on us.

Here are typical ways we respond in an answer-getting environment:

• The mind goes blank. For a moment, nothing occurs to us because we’re not allowed to use our minds creatively.
• Racing heart. We react anxiously as we try to remember how the teacher did the math on the board. What was her first step again?
• Negative self-talk. For a moment, we have the germ of an idea, an instinct about how to start solving a challenging math problem, but because we’ve been conditioned to seek the answer only one way, we chastise ourselves for thinking we know better than what we’ve been taught, and we revert to standard operating procedure.
• Reluctance to talk through questions and concerns. We’re embarrassed to bring up these issues with others, assuming they are “right way” adherents. This reluctance to involve others is an obstacle to a creative, collaborative process.

The overarching effect of an answer-getting system is disempowerment. We feel defeated before even attempting to work on a problem.

Here are recent conversations I have had with children and adults on what this sort of math feels like:

“I want to use decimals. The teacher wants me to use fractions for no reason. I just have to do what he says. There is no freedom to do the math the way you want to do it, even if my way is easier for me. No one listens to me.”

“I actually remember getting dinged on a high school math test even when I had the right answer, but I had solved it my own way. As a teenager, that made me furious. Now looking back as an adult, I think about it like tennis. If you are drilling me so I learn or improve a new skill like backhand volleys, then I can understand the reasoning for forcing a specific approach. But if you have no reason whatsoever for forcing your way on me, it still steams me to think about it.”

A problem-solving approach conjures significantly contrasting responses — responses that reflect a sense of empowerment and courage. Ideally, schools would teach math with problem-solving as its driving principle rather than the myth of a single right way. To approach this ideal, however, we need to understand what problem-solving is all about.

Math should be taught as a collaborative process, much as other subjects are taught.

How to counteract the myth

We make math a performative rather than a learning experience. When the teacher asks the class, “What is the answer to 63 plus 37?” he turns math into an individual sport.

Add the myth of speed, and each student is scrambling to come up with the answer first and win the game. The answer becomes the only thing that matters, and both understanding and collaboration fall by the wayside.

No doubt, some of you might wonder if I’ve lost my math mind. After all, we need to get the answers right so that we can purchase the right amount of carpeting to cover a room’s floor or make sure that our rocket makes it to the moon. 

Again, this is an issue of integrative complexity. Of course, we need to know what 63 + 37 equals. But if that’s all we know, then we’re missing out on a lot of what math offers.

Fortunately, we can learn in a way that we obtain precision as well as other benefits. Consider again 63 + 37. What if the teacher framed the question this way: “Don’t tell me the answer. It’s 100. How would you start calculating 63 plus 37 in your head? What is your first step?” Now math is a process.

I have gotten the chance to hear second graders’ brains working at this moment many times. Each time is a joy. One might say, “I broke this up as 60 plus 30 plus 3 plus 7. And the next thing in my head I saw was that it was 93 plus 7. And then I knew that was 100.” Another second grader might offer another option: “I looked at this for a moment, and I saw that 3 and 7 make 10. So I knew I had a 60 plus 30 plus 10. And I know that’s 100.”

This is the math people need in their lives. This is what is needed to build bridges. It’s also how you build a deep math sense. 

Math should be taught as a collaborative process, much as other subjects are taught. We often view math as distinct from other subjects in K–8, as something that must be taught as an individual sport where everyone is on his or her own to come up with the right answer first. Other subjects are taught as team sports, ones where process matters, where students don’t rely on tricks, where students are encouraged to work together, and where a variety of ways to answer may be acceptable. But when it comes to math, collaboration and process work are subordinated or eliminated.

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Discover Frequently Asked Math Questions and Their Answers

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• How can we find the area of irregular shapes?
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• An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(7 ,1 )# to #(8 ,5 )# and the triangle's area is #27 #, what are the possible coordinates of the triangle's third corner?
• A triangle has corners at #(7 , 9 )#, #(3 ,7 )#, and #(4 ,8 )#. What is the radius of the triangle's inscribed circle?
• Circle A has a center at #(2 ,3 )# and a radius of #1 #. Circle B has a center at #(0 ,-2 )# and a radius of #4 #. Do the circles overlap? If not, what is the smallest distance between them?
• A parallelogram has sides A, B, C, and D. Sides A and B have a length of #2 # and sides C and D have a length of # 7 #. If the angle between sides A and C is #pi/4 #, what is the area of the parallelogram?
• What is a quadrilateral that is not a parallelogram and not a trapezoid?
• Your teacher made 8 triangles he need help to identify what type triangles they are. Help him?: 1) #12, 16, 20# 2) #15, 17, 22# 3) #6, 16, 26# 4) #12, 12, 15# 5) #5,12,13# 6) #7,24,25# 7) #8,15,17# 8) #9,40,41#
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How machines that can solve complex math problems might usher in more powerful AI

Google DeepMind’s AlphaProof and AlphaGeometry 2 are milestones for AI reasoning.

• Melissa Heikkilä archive page

This story originally appeared in The Algorithm, our weekly newsletter on AI. To get stories like this in your inbox first, sign up here .

It’s been another big week in AI. Meta updated its powerful new Llama model, which it’s handing out for free, and OpenAI said it is going to trial an AI-powered online search tool that you can chat with, called SearchGPT. 

But the news item that really stood out to me was one that didn’t get as much attention as it should have. It has the potential to usher in more powerful AI and scientific discovery than previously possible. 

Last Thursday, Google DeepMind announced it had built AI systems that can solve complex math problems. The systems—called AlphaProof and AlphaGeometry 2—worked together to successfully solve four out of six problems from this year’s International Mathematical Olympiad, a prestigious competition for high school students. Their performance was the equivalent of winning a silver medal. It’s the first time any AI system has ever achieved such a high success rate on these kinds of problems.  My colleague Rhiannon Williams has the news here . 

Math! I can already imagine your eyes glazing over. But bear with me. This announcement is not just about math.  In fact, it signals an exciting new development in the kind of AI we can now build.  AI search engines that you can chat with may add to the illusion of intelligence, but systems like Google DeepMind’s could improve the actual intelligence of AI. For that reason, building systems that are better at math has been a goal for many AI labs, such as  OpenAI .  

That’s because math is a benchmark for reasoning . To complete these exercises aimed at high school students, the AI system needed to do very complex things like planning to understand and solve abstract problems. The systems were also able to generalize, allowing them to solve a whole range of different problems in various  branches of mathematics. 

“What we’ve seen here is that you can combine [reinforcement learning] that was so successful in things like AlphaGo with large language models and produce something which is extremely capable in the space of text,” David Silver, principal research scientist at Google DeepMind and indisputably a pioneer of deep reinforcement learning, said in a press briefing. In this case, that capability was used to construct programs in the computer language Lean that represent mathematical proofs. He says the International Mathematical Olympiad represents a test for what’s possible and paves the way for further breakthroughs. 

This same recipe could be applied in any situation with really clear, verified reward signals for reinforcement-learning algorithms and an unambiguous way to measure correctness as you can in mathematics, said Silver. One potential application would be coding, for example. 

Now for a compulsory reality check:  AlphaProof and AlphaGeometry 2 can still only solve hard high-school-level problems. That’s a long way away from the extremely hard problems top human mathematicians can solve. Google DeepMind stressed that its tool did not, at this point, add anything to the body of mathematical knowledge humans have created. But that wasn’t the point. 

“We are aiming to provide a system that can prove anything,” Silver said. Think of an AI system as reliable as a calculator, for example, that can provide proofs for many challenging problems, or verify tests for computer software or scientific experiments. Or perhaps build better AI tutors that can give feedback on exam results, or fact-check news articles. 

But the thing that excites me most is what Katie Collins, a researcher at the University of Cambridge who specializes in math and AI (and was not involved in the project), told Rhiannon.  She says these tools create and evaluate new problems, motivate new people to enter the field, and spark more wonder. That’s something we definitely need more of in this world.

Now read the rest of The Algorithm

Deeper learning.

A new tool for copyright holders can show if their work is in AI training data

Since the beginning of the generative AI boom, content creators have argued that their work has been scraped into AI models without their consent. But until now, it has been difficult to know whether specific text has actually been used in a training data set. Now they have a new way to prove it: “copyright traps.” These are pieces of hidden text that let you mark written content in order to later detect whether it has been used in AI models or not. 

Why this matters:  Copyright traps tap into one of the biggest fights in AI. A number of publishers and writers are in the middle of litigation against tech companies, claiming their intellectual property has been scraped into AI training data sets without their permission. The idea is that these traps could help to nudge the balance a little more in the content creators’ favor.  Read more from me here . 

Bits and Bytes

AI trained on AI garbage spits out AI garbage New research published in Nature shows that the quality of AI models’ output gradually degrades when it’s trained on AI-generated data. As subsequent models produce output that is then used as training data for future models, the effect gets worse. ( MIT Technology Review ) 

OpenAI unveils SearchGPT  The company says it is testing new AI search features that give you fast and timely answers with clear and relevant sources cited. The idea is for the technology to eventually be incorporated into ChatGPT, and CEO  Sam Altman says  it’ll be possible to do voice searches. However, like many other AI-powered search services, including  Google’s , it’s already making errors, as  the Atlantic  reports.  ( OpenAI ) 

AI video generator Runway trained on thousands of YouTube videos without permission Leaked documents show that the company was secretly training its generative AI models by scraping thousands of videos from popular YouTube creators and brands, as well as pirated films. ( 404 media ) 

Artificial intelligence

How to opt out of meta’s ai training.

Your posts are a gold mine, especially as companies start to run out of AI training data.

Why does AI hallucinate?

The tendency to make things up is holding chatbots back. But that’s just what they do.

• Will Douglas Heaven archive page

How generative AI could reinvent what it means to play

AI-powered NPCs that don’t need a script could make games—and other worlds—deeply immersive.

• Niall Firth archive page

Synthesia’s hyperrealistic deepfakes will soon have full bodies

With bodies that move and hands that wave, deepfakes just got a whole lot more realistic.

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How to assess a general-purpose AI model’s reliability before it’s deployed

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Foundation models are massive deep-learning models that have been pretrained on an enormous amount of general-purpose, unlabeled data. They can be applied to a variety of tasks, like generating images or answering customer questions.

But these models, which serve as the backbone for powerful artificial intelligence tools like ChatGPT and DALL-E, can offer up incorrect or misleading information. In a safety-critical situation, such as a pedestrian approaching a self-driving car, these mistakes could have serious consequences.

To help prevent such mistakes, researchers from MIT and the MIT-IBM Watson AI Lab developed a technique to estimate the reliability of foundation models before they are deployed to a specific task.

They do this by considering a set of foundation models that are slightly different from one another. Then they use their algorithm to assess the consistency of the representations each model learns about the same test data point. If the representations are consistent, it means the model is reliable.

When they compared their technique to state-of-the-art baseline methods, it was better at capturing the reliability of foundation models on a variety of downstream classification tasks.

Someone could use this technique to decide if a model should be applied in a certain setting, without the need to test it on a real-world dataset. This could be especially useful when datasets may not be accessible due to privacy concerns, like in health care settings. In addition, the technique could be used to rank models based on reliability scores, enabling a user to select the best one for their task.

“All models can be wrong, but models that know when they are wrong are more useful. The problem of quantifying uncertainty or reliability is more challenging for these foundation models because their abstract representations are difficult to compare. Our method allows one to quantify how reliable a representation model is for any given input data,” says senior author Navid Azizan, the Esther and Harold E. Edgerton Assistant Professor in the MIT Department of Mechanical Engineering and the Institute for Data, Systems, and Society (IDSS), and a member of the Laboratory for Information and Decision Systems (LIDS).

He is joined on a paper about the work by lead author Young-Jin Park, a LIDS graduate student; Hao Wang, a research scientist at the MIT-IBM Watson AI Lab; and Shervin Ardeshir, a senior research scientist at Netflix. The paper will be presented at the Conference on Uncertainty in Artificial Intelligence.

Measuring consensus

Traditional machine-learning models are trained to perform a specific task. These models typically make a concrete prediction based on an input. For instance, the model might tell you whether a certain image contains a cat or a dog. In this case, assessing reliability could be a matter of looking at the final prediction to see if the model is right.

But foundation models are different. The model is pretrained using general data, in a setting where its creators don’t know all downstream tasks it will be applied to. Users adapt it to their specific tasks after it has already been trained.

Unlike traditional machine-learning models, foundation models don’t give concrete outputs like “cat” or “dog” labels. Instead, they generate an abstract representation based on an input data point.

To assess the reliability of a foundation model, the researchers used an ensemble approach by training several models which share many properties but are slightly different from one another.

“Our idea is like measuring the consensus. If all those foundation models are giving consistent representations for any data in our dataset, then we can say this model is reliable,” Park says.

But they ran into a problem: How could they compare abstract representations?

“These models just output a vector, comprised of some numbers, so we can’t compare them easily,” he adds.

They solved this problem using an idea called neighborhood consistency.

For their approach, the researchers prepare a set of reliable reference points to test on the ensemble of models. Then, for each model, they investigate the reference points located near that model’s representation of the test point.

By looking at the consistency of neighboring points, they can estimate the reliability of the models.

Aligning the representations

Foundation models map data points to what is known as a representation space. One way to think about this space is as a sphere. Each model maps similar data points to the same part of its sphere, so images of cats go in one place and images of dogs go in another.

But each model would map animals differently in its own sphere, so while cats may be grouped near the South Pole of one sphere, another model could map cats somewhere in the Northern Hemisphere.

The researchers use the neighboring points like anchors to align those spheres so they can make the representations comparable. If a data point’s neighbors are consistent across multiple representations, then one should be confident about the reliability of the model’s output for that point.

When they tested this approach on a wide range of classification tasks, they found that it was much more consistent than baselines. Plus, it wasn’t tripped up by challenging test points that caused other methods to fail.

Moreover, their approach can be used to assess reliability for any input data, so one could evaluate how well a model works for a particular type of individual, such as a patient with certain characteristics.

“Even if the models all have average performance overall, from an individual point of view, you’d prefer the one that works best for that individual,” Wang says.

However, one limitation comes from the fact that they must train an ensemble of foundation models, which is computationally expensive. In the future, they plan to find more efficient ways to build multiple models, perhaps by using small perturbations of a single model.

“With the current trend of using foundational models for their embeddings to support various downstream tasks — from fine-tuning to retrieval augmented generation — the topic of quantifying uncertainty at the representation level is increasingly important, but challenging, as embeddings on their own have no grounding. What matters instead is how embeddings of different inputs are related to one another, an idea that this work neatly captures through the proposed neighborhood consistency score,” says Marco Pavone, an associate professor in the Department of Aeronautics and Astronautics at Stanford University, who was not involved with this work. “This is a promising step towards high quality uncertainty quantifications for embedding models, and I’m excited to see future extensions which can operate without requiring model-ensembling to really enable this approach to scale to foundation-size models.”

This work is funded, in part, by the MIT-IBM Watson AI Lab, MathWorks, and Amazon.

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• Navid Azizan
• Young-Jin Park
• Laboratory for Information and Decision Systems
• Institute for Data, Systems, and Society
• Center for Computational Science and Engineering
• Department of Mechanical Engineering
• MIT-IBM Watson AI Lab

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• Artificial intelligence
• Computer science and technology
• Human-computer interaction
• Mechanical engineering
• Laboratory for Information and Decision Systems (LIDS)

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