problem solving method in teaching of general mathematics

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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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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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Article Contents

1 introduction, 2 background, 3 teaching mathematics with real data, 5 empirical model structures present challenges for teaching, 6 teaching functions with data: examples of modelling in schools, 7 discussion and conclusion.

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Teaching mathematical modelling: a framework to support teachers’ choice of resources

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Peter K Dunn, Margaret F Marshman, Teaching mathematical modelling: a framework to support teachers’ choice of resources, Teaching Mathematics and its Applications: An International Journal of the IMA , Volume 39, Issue 2, June 2020, Pages 127–144, https://doi.org/10.1093/teamat/hrz008

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Mathematics teachers are often keen to find ways of connecting mathematics with the real world. One way to do so is to teach mathematical modelling using real data. Mathematical models have two components: a model structure and parameters within that structure. Real data can be used in one of two ways for each component: (a) to validate what theory or context suggests or (b) to estimate from the data. It is crucial to understand the following: the implications of using data in these different ways, the differences between them, the implications for teaching and how this can influence students’ perceptions of the real-world relevance of mathematics. Inappropriately validating or estimating with data may unintentionally promote poor practice and (paradoxically) reinforce in students the incorrect idea that mathematics has no relevance to the real world. We recommend that teachers approach mathematical modelling through mathematizing the context. We suggest a framework to support teachers’ choice of modelling activities and demonstrate these using examples.

Connecting classroom learning to the real world is seen as important in teaching generally. This is especially true for mathematics ( Gainsburg, 2008 ; Smith & Morgan, 2016 ) where the connection increases students’ understanding of mathematics, increases motivation to study mathematics and helps students to apply mathematics ( Gainsburg, 2008 ). For example, one of the aims of the Australian Curriculum: Mathematics is to ensure that

students are confident, creative users and communicators of mathematics, able to investigate, represent and interpret situations in their personal and work lives and as active citizens. ( Australian Curriculum, Assessment and Reporting Authority, n.d. , p. 1)

That is, numeracy is inseparable from real-world contexts and so students must be exposed to mathematics in a wide range of real-world contexts ( Department of Education, Training and Youth Affairs, 2000 ).

In mathematics, many means are possible for making connections with the real-world ( Gainsburg, 2008 ), including using analogies, using word problems, facilitating discussions, using physical models, using simulations, analysing real data and developing mathematical models. One way to connect mathematics and the real world is to set the mathematics in real-world contexts:

Problem-solving in mathematics can be set in purely mathematical contexts or real-world contexts. When set in the real world, problem-solving in mathematics involves mathematical modelling. ( The State of Queensland (Queensland Curriculum and Assessment Authority), 2017 , p. 12)

The Common Core State Standards for Mathematics ( Common Core State Standards Initiative, 2010 ) from the USA identifies the steps involved in modelling as follows: specify: identifying variables, assumptions and approximations; formulate: formulating the mathematical model to identify relationships; analyse: analysing these relationships to draw conclusions; interpret: interpreting the solution to ensure it makes sense in the context of the original problem; and refine: refining the model if necessary. This is often interpreted as a cyclical process ( Fig. 1 ; Stillman et al. , 2007 ). One way to teach mathematical models with real-world applicability ( Kaiser, 2014 ) using this process is through using data.

Modelling as a cyclical process (adapted from Stillman et al. , 2007 ).

A framework to support teachers’ choice of resources for mathematical modelling. ST and SE means that the model structure is based on theoretical or empirical methods; PT and PE means that parameters are estimated using theoretical or empirical methods.

The marble sifter. (a) The `sifter’ made from a cardboard box lid; (b) students sifting marbles using the sifter.

This paper was motivated by seeing teachers struggle to teach mathematical modelling with the unfortunate outcome that their use of real data convinced students that mathematics had no practical purpose. In this paper, we first give some background and discuss the role and importance of real data and then describe mathematical modelling. We then present a framework with some examples for teachers to use when deciding how to incorporate real-world contexts and/or data in their classrooms.

Much has been written about teaching mathematical modelling. Galbraith (2011 ) describes six different meanings associated with mathematical modelling in schools: using real problem situations as a preliminary basis for abstraction, emergent modelling, modelling as curve fitting, word problems, modelling as a vehicle for teaching other mathematical material and modelling as real-world problem solving (pp. 280–282). In this paper, we are mainly focussing on the `modelling as a vehicle for teaching other mathematical material’ (p. 282). In this case, the curriculum determines the models chosen. This aligns with what Kaiser & Sriraman (2006 ) call an `educational modelling’ perspective (p. 301) where the purpose of the modelling is introducing and developing mathematical concepts and `the real world examples and their interrelations with mathematics become a central element for structuring of teaching and learning mathematics’ (p. 306).

Technology, including graphing calculators and software such as Excel, can be a useful tool for students to explore these mathematical models ( da Silva Soares, 2015 ) without the need to restrict examples to artificial data that is easy to manage computationally. Modelling as a vehicle for teaching other mathematical material can entail difficulties. One of these difficulties is that despite the intention to make real-world connections, students and pre-service teachers can `exclude real knowledge from their solutions’ ( Verschaffel et al. , 1997 , p. 339). For example,

[...] students demonstrated a very strong overall tendency to exclude real-world knowledge and realistic considerations [...] students’ lack of sense making when solving arithmetic word problems in a typical school setting is not caused by a strange cognitive deficit. Rather, the cause seems to be students’ beliefs about the role of mathematical word problems and how they should be treated and solved in the mathematics class. ( Dewolf et al. , 2014 , p. 105)

In other words, students often think that mathematics is not useful for realistically describing the real world ( Greer, 1997 ), and this may be reinforced by how mathematics is taught. Mathematical modelling includes `the connection between mathematical and non-mathematical contexts in this process’ ( Falsetti & Rodriguez, 2005 , p.15). Using mathematical modelling of real situations with real data can help students connect mathematics with the real world. In our experience though, modelling with real data can also have the unintended consequence of reinforcing that mathematics is disconnected from reality. The actual modelling process is vital for students’ learning and understanding:

The students’ responsibility is figuring out how to solve the problem as well as finding the solution. It is the strategies used for figuring out, rather than the answers, that are the site of the mathematical argument... ( Lampert, 1990 , p. 40)

[...] the demand that learners make connections between mathematics and the real world has been, and will continue to be, at the forefront of most major reform efforts ( Carrejo & Marshall, 2007 , p. 71)

Carreira & Baioa (2018 ) believe it is important to design tasks that `ascribe credibility to a modelling situation deliberately designed to stimulate what happens in a real situation’ (p. 202). Teachers must be honest with students when engaging them in experimental work to generate data or prototypes of actual objects. Students need to be aware of the authenticity of activities ( Jablonka, 2007 ).

This paper focuses on how to use real data and how to approach the modelling: both are critical to promoting that mathematics describes the real world. We develop a framework to support teachers make these choices.

The use of real data and context in teaching has many advantages. They can emphasize solving real (rather than artificial) problems rather than just `theoretical’ and computational problems ( Hand et al. , 1996 ); increase student interest and engagement ( Aliaga et al. , 2005 ); counter the idea that mathematics is dull and dry; enable students to learn to good quantitative thinking habits; help students to understand why we do mathematics, not just how ( Willett & Singer, 1992 ); ensure students remember the analysis as solving a real problem ( Bradstreet, 1996 ); and become a trigger for later recall of the techniques ( Singer & Willett, 1990 ). Real data can be found in books ( Hand et al. , 1993 ), online ( Smyth, 2011 ; JSE Data Archive: http://www.amstat.org/publications/jse/jse_data_archive.htm ) or in journal articles. Alternatively, students can collect their own data (for example, Carrejo & Marshall, 2007 ) or even obtain simulated data from more challenging and interesting situations based on realistic computer simulation ( Huynh et al. , 2016 ).

For these reasons, real data have a role to play in teaching mathematics, especially the mathematical modelling process. However, the role that data are allowed to play in a classroom impacts how students perceive the relevance of mathematics.

4.1. Mathematical models

Accuracy: A model agrees with reality in important (but not necessarily all) ways.

Simplicity: A model is generally simpler than reality (i.e. is an approximation).

Rodgers (2010 , p. 5) notes that a `mathematical model is one that captures these two features [accuracy; simplicity] within one or more mathematical equations’, while Blum & Ferri (2009 , p. 45) define mathematical modelling as `the process of translating between the real world and mathematics in both directions’. Kaiser (2014 ) notes that `to create a mathematical model, the real-world model has to be translated into mathematics’ (p. 399) or `mathematised’ ( Stillman et al. , 2007 ). Mathematical models are usually built on theoretical or scientific means of relating quantities using mathematical functions (for example, differential equations). A consequence of `mathematising’ is that mathematical functions are allowed to naturally emerge. For this reason, mathematical modelling is an excellent way of showing students the relevance of mathematics in, and emergence of mathematics from, the real world.

Creating a mathematical model may be difficult but is an important conduit to learning:

...our continuing efforts to bring the discovery method to the classroom naturally go hand-in-hand with attempts to bring genuine applications to the classroom. The two efforts reinforce each other, and both are essential for a complete and honest presentation of mathematics in our schools. ( Pollak, 1969 , p. 403)

To be useful and relevant for describing the real world, mathematical models need to have meaningful interpretations, allowing students to sensibly `interpret’ their mathematics in the real world ( Australian Curriculum, Assessment and Reporting Authority, n.d. , p. 1). Dewolf et al. (2014 ) discuss examples of students dismissing the real world when solving mathematical problems and not even expecting mathematical solutions to be sensible in the real world.

4.2. Mathematical models have a structure and parameters

Mathematical models are useful for describing the relationships between variables; in this paper, we focus on the case of two variables only (a response or dependent variable and an explanatory or independent variable), though the ideas extend to more than two. Mathematical models have two components: the structure of the model (for example, is the relationship between the variables best described by a linear or exponential function?) and the parameters in that model (for example, what is the slope of the linear function?). The structure of a mathematical model can often be developed purely from a theoretical understanding of a situation. Sometimes the structure of the model has no underlying theory and is established based on the data (for example, a graph of the data may appear linear). In either case, the values of the parameters in that model usually need to be assigned values, which can come from a theoretical description of the situation or from the data.

Different ways to use data when fitting models. Using data to validate or estimate can be done formally (using statistical methods) or informally

Establishing the structure of a model based on theory allows students to see the relevance of mathematics, where mathematical functions emerge naturally from the real-world development, understanding and description of the situation. Sometimes, the values of the parameters in the model also emerge through the same process. In contrast, developing a model (especially the model structure) purely based on the data presents numerous challenges, both for mathematicians in practice and especially for students in schools, as the suite of tools necessary to do this well are statistical, often complicated, not in the curriculum and perhaps produce a model that was not interpretable in the real-world context (Galbraith’s modelling as curve fitting).

4.3. Data can play two roles in model development

In model development, data can be used to validate or to estimate . Validation refers to the process of using theory and/or context to guide development of the model structure and/or the values of the parameters, then using the data confirms that the result is sensible and sufficiently accurate. That is

… the objective is not to confirm or deny the model (we already know it is not precisely correct because of the simplifying assumptions we have made), but rather to test its reasonableness [i.e. validity]. We may decide that the model is quite satisfactory and useful, and elect to accept it. Or we may decide that the model needs to be refined or simplified. In extreme cases, we may even need to redefine the problem, in a sense rejecting the model altogether. [...] this decision process really constitutes the heart of mathematical modelling. ( Giordano & Weir, 1997 , p. 38).

Estimation refers to using the data to determine the model structure or values of the parameters. If the structure is determined based on theory, then it can be validated by comparing to data; if no theory underpins the structure, then the data are used to estimate the structure. Similarly, if the values of the parameters are based on theory, then these can be validated using the data; if no guiding theory exists, then the data can be used to estimate the parameter values.

Validation and estimation can be performed using formal or informal tools. A crucial observation is that formal approaches using data require the use of statistical tools (such as residual analysis), which require experience to use well and appropriately and are usually not covered in the school curriculum. Even those few techniques that are covered in the school curriculum (such as R 2 ) are often used inappropriately even by experienced researchers ( Kvålseth, 1985 ). In many cases, informal processes might be suitable in the classroom (`there are no obvious departures from the proposed model’) and are often better didactically as they build conceptual understanding of the mathematics and ability to interpret the model to describe the real world. In Section 5, we discuss the challenges when teachers attempt to use statistical tools.

Hence, we see that real data have a role to play in the mathematical modelling process, but ensuring that the role is appropriate remains crucial. This has implications for classroom teaching.

4.4. Teaching implications

Theoretical structure and parameter values: Both the model structure and the parameters values are determined from underlying theory. Data are then used for validating the structure and the parameter values. This approach is an excellent choice for teachers (see Cramer, 2001 for examples) as the mathematics emerges naturally in the description of the real world, and real-world data are used for validating the model. Hence, the modelling is a `vehicle for teaching other mathematical material’ ( Galbraith, 2011 , p. 282).

Theoretical structure and empirical parameter values: The model structure is developed from an understanding or description of the context, and data are used to validate the structure. The unknown parameter values are then estimated using the data. This approach is also a good choice for teaching as students can see the mathematics emerging from the real-world situation and once again the modelling is a `vehicle for teaching other mathematical material’ ( Galbraith, 2011 , p. 282). Estimation of the parameters can be as simple or as complex as necessary and may not even require statistical tools for parameter estimation. Formally, a statistical approach would be used for model validation and parameter estimation, but in a school situation, this may be unnecessary and even counterproductive. In fact, a non-formal approach to parameter estimation is often preferred as this clearly shows how the mathematics describes the real world. Without a formal approach, students may develop many varied ways to estimate parameters based on the meaning of the parameters (Section 6.3).

[...] can be generated in complete ignorance of the principles underlying the real situation […] when used mindlessly it creates a dangerous aberration of the modelling concept. (p. 271).

Many reasons exist for why this approach should be avoided in schools, which we discuss in the next section.

The fourth combination—empirical model structure and theoretical parameter values—is unusual, if not impossible. If there is no guiding theory to suggest a model structure, then it is unlikely that theory will exist to determine the parameters of the structure with no theoretical basis. These approaches have been summarized in Table 1 .

Validation and estimation use data very differently, and the tools necessary are different for both. Formal (statistical) methods of validation and estimation require appropriate knowledge to do well, while informal methods are simple and retain connections with the real-world context.

As noted above, data can be used to validate or estimate the structure of a model and/or to validate or estimate the parameter values. We argue that adopting a theoretical approach to developing model structure is the best way for students to see how mathematical functions emerge naturally to describe the real world and hence are useful for showing the relevance of mathematics in the real world ( Galbraith, 2011 ). Using an empirical approach presents many challenges in the classroom. (Using a theoretical approach is also preferred in the real world, but sometimes an empirical approach is the only option.)

Firstly, an empirical approach to model structure is not usually required by the curriculum. The curricula require students to develop models by describing the real world but rarely cover formal empirical validation or estimation techniques for modelling with non-linear functions. The empirical approach imposes functions on a context, rather than having functions emerge from an understanding of the context. Commonly in Australian schools, R 2 is used to help with decision making as it is easily accessible (using Excel and graphing calculators), even when this is not appropriate. Without guidance from the real-world context, the resulting model may be a poor approximation to reality if the decision is made solely on which model has the `best’ fit. Furthermore, different students with different data from the same context may select different model structures. `Fitting the data well’ is not the only criterion for deciding on a model (see Dunn & Smyth, 2018 , Section 1.10).

Secondly, the resulting models may have no real-world interpretation if the model structure is not based on theory. When using mathematics to describe the real world, mathematical ideas and functions emerge naturally, and students need to think about the context to `mathematise’ it (`mathematising’ is just as important as `interpreting’: Wake, 2016 ). In contrast, when models are imposed upon data, no context is necessary, and the mathematics becomes subservient to the data (modelling as curve fitting Galbraith, 2011 ).

Thirdly, teachers are often not trained to teach empirical modelling. Determining model structures empirically requires statistical techniques, yet mathematics teachers are often not trained in the formal statistical techniques used to properly fit models. Furthermore, many teachers may not understand many basic statistical concepts ( Engel & Sedlmeier 2005 ) so that empirical modelling presents significant challenges. For example, one study ( Dunn et al. , 2015 ) asked teachers to define `regression’ and `correlation’. Although a small sample, the sample was a voluntary, self-selected group of keener mathematics teachers, so the results are optimistic. The results indicate that the teachers’ understanding of regression and correlation was poor. Only 32% gave a correct definition of regression and 16% for correlation (see Dunn et al ., 2015 for details of the study), yet an almost identical group of teachers expressed confidence in their knowledge of regression and correlation ( Marshman et al. , 2015 ). This aligns with other research showing that mathematics teachers may be proficient with the formulae and computations used with regression but do not necessarily understand the statistical concepts or the meaning of the formulae ( Engel & Sedlmeier, 2005 ) and hence the connections between the model and the real world.

Fourthly, teachers may not understand the difference between statistics and mathematics. Formal attempts to fit a model empirically (that is, using statistical approaches) require the use of statistical tools. However, mathematics and statistics have many differences. For example, teachers and researchers alike are often surprised to learn that the language and notation of statistics and mathematics may contradict each other ( Dunn et al. , 2016 ). Even the concept of `linear’ is different in mathematics and statistics ( Dunn et al. , 2016 ). Furthermore, a straight line has different presentations in mathematics and statistics. Straight lines in mathematics are presented deterministically as (say) |$y= mx+c$|⁠ , implying that every value of |$x$| is associated with a single possible value |$y$|⁠ . In statistics, the structure of the model is written as (say) |$\hat{y}=a+ bx$|⁠ , where the left-hand side shows that the model predicts a mean value |$\hat{y}$| for a given value of |$x$|⁠ , and the actual (observed) values of |$y$| for that value of |$x$| are generated randomly from the given (often normal) distribution of possible values around the given mean. Failure to understand these differences produces a clash of notation. Other examples of language inconsistencies between mathematics and statistics include the words `graph’, `estimate’, `significant’, `variable’ and the symbol |$\pm$| ( Dunn et al. , 2016 ).

Finally, teachers are not adequately supported to teach a statistical approach. Because many teachers lack an understanding of the empirical modelling approach (i.e. statistical techniques), teachers may reasonably be expected to turn to textbooks. However, many textbooks used in teaching mathematics contain errors and inconsistencies when discussing statistics ( Dunn et al. , 2015 ) or lack useful features to enhance learning ( Dunn et al. , 2017 ). Problems with textbooks include the following: errors in formulae ( Bland & Altman, 1988 ; Dunn et al. , 2015 ), cumbersome and misleading language, confusing or incorrect notation, unexplained language, misusing notation or not explaining what the model means. Furthermore, very few textbooks use real data, and many of those that do use real data (for example, Barnes et al. , 2016 ) use contrived examples such as the relationship between student height and hair length. Many examples and exercises are clearly made up or simply provide a list of |$x$| and |$y$| values and ask students to compute a regression line with no attempts at a context or to attach meaning (for example, Morris, 2016 , p. 598, 599). These approaches do not even try to help students see a connection between mathematics and the real world and may reinforce students’ belief that there is no connection (`these data are made up, because there are no real examples’). Other easily accessed resources, such as Wikipedia, also have significant problems explaining simple statistical concepts ( Dunn et al. , 2018 ).

6.1. Recommended approach for teaching modelling

We recommend approaching mathematical model by using data for validation of the model structure, for reasons described above, and suggest a framework to support teachers’ choice of resources for modelling activities ( Fig. 2 ). We then describe some examples of the modelling process (using Fig. 1 ) of how to do this in the classroom and some examples of how a different approach leads to poor experience of mathematics being useful in the real world.

6.2. Good practice: theoretical structure and parameter values (ST, PT)

A simple example that could be used with Year 11 and 12 (aged 16 and 17) students in their final 2 years of school is to simulate radioactive decay to introduce power functions. For example, consider repeatedly tossing coins on a table, where coins that turn up as `heads’ are considered to have radioactively decayed. A model is sought for the number of coins remaining after n tosses. After each toss, the expected number of coins removed at each toss is about one-half. Hence, the `half-life’ is one toss. Using dice and removing dice that land with a (say) six uppermost is similar but with a longer `half-life’ (about one-sixth of the dice are removed with each throw, and the `half-life’ is between three and four rolls). In both cases, a mathematical model can be developed with known parameters. Students can then validate the model by tossing coins (or rolling dice) to see how well the model fits the data. Students only need to be familiar with index notation (including fractional indices) and some guiding questions from the teacher to introduce power functions. This may also give a reason to introduce logarithms.

6.3. Good practice: theoretical structure and empirical parameter values (ST, PE)

An example that was used by one of the authors with Year 11 and 12 students both as an in-class activity and a take-home assessment task uses a `marble sifter’ in which holes just large enough for marbles to comfortably pass through are cut in the lid of a cardboard box (for example, printer paper box lid; Fig. 3a ). Students begin with 100 marbles and `sift’ them through the sifter by tilting the box ( Fig. 3b ) and counting how many remain unsifted after each tilt. The process is repeated for 11 to 20 tilts. We can define |$t$| as the number of tilts and |$N$| as the number of marbles (out of 100) that remain unsifted (specify: identifying variables). Some (real) example data are shown in Table 2 . Again, students could mindlessly fit many functional relationships to the data. However, this activity lends itself to a useful discussion of mathematical modelling to establish the model structure. The initial assumption is that we can model a discrete situation with a continuous model (specify: identifying initial assumptions).

To determine a suitable model structure for the marble sifter, students can graph the number of marbles remaining against the tilt number, so that they can visualize the data to help them understand the data (formulate). Then various model structures can be critiqued (analyse; interpret). For example, the model cannot be linear as the graph does not look linear. In addition, a linear model would imply that approximately the same number of marbles would fall through the sifter; however, many marbles began the sift at any iteration, which makes no sense. A linear model also implies that more and more sifting would eventually end up producing a negative number of remaining marbles.

The model cannot be quadratic, as the number of marbles remaining cannot start increasing as a quadratic would permit (marbles cannot jump back up through the holes). Furthermore, if students attempt to fit a quadratic, then they usually see a turning point occurring and the function increasing within the range of the data.

Attempting to fit a logarithmic model of the form |$N=a\ \log (t)+b$| will cause difficulties as the data begins at |$t=0$| and |${N}_0=100$| (since |$\log\ 0$| is undefined). A similar problem exists when trying to model with a power function (the initial conditions are undefined; |$N=a{t}^b$| can never satisfy the initial conditions).

An exponential relationship, however, makes sense. From Table 2 , the fraction remaining (say |$a$|⁠ ) at each tilt is approximately constant. For example, if we start with |${N}_0$| marbles, then after one tilt, we would have an average of |${N}_0a$| marbles remaining. After two tilts, an average of |${N}_0\ {a}^2$| would remain. Continuing, after |$t$| tilts, the predicted number of marbles remaining would be |$\hat{N}={N}_0\ {a}^t$|⁠ , for some value of |$a$|⁠ , and where |$\hat{N}$| is the average number of marbles left after |$t$| tilts. This is the process of mathematical modelling describing the real world.

Data from one example of conducting the marble-sifting study

Opportunities emerge to discuss the assumptions made in developing the model (interpret) (for example, is it reasonable to assume that decay rate is constant?) and strengths and limitations of the model (for example, the data are discrete, so does it matter that |$N$| is continuous as we develop the model?) ( The State of Queensland (Queensland Curriculum & Assessment Authority), 2017 ). A discussion of half-life also fits with the discussion.

To determine values for the parameters, first observe that the model has two unknown parameters to be estimated: |${N}_0$| and |$a$|⁠ , though |${N}_0$| could be constrained to 100 as it is the only known value. To estimate these values, students could use technology (such as graphics calculators or Excel) as a `black box’ or formal statistical tools. Technology allows students to determine equations and coefficients of determination, R 2 , for various regression lines from the scatterplot. However, if students are not encouraged to think about each model as it is generated, then the technology is a black box and opportunities for strengthening students’ understanding of modelling have been missed. Of concern is that the students (and most of their teachers) do not understand how the computer creates the estimates and the reason why it is important to refer each model generated to the real world for validation. Secondary school students were expected to evaluate every possible model before deciding on their final choice as part of the written assessment or so that they could participate in a whole class discussion.

An alternative approach to using a black-box approach, which has arisen when the activity was used with pre-service mathematics teachers, is for pre-service teachers to understand the meaning of the parameters, helping them connect the mathematics to the real world. For example, the value of |${N}_0$| is the initial number of marbles, so using |${N}_0=100$| seems sensible. An estimate of |$a$| can be found by combining results from all groups and estimating |$a$| as the fraction of marbles retained at each sift ( Table 2 ). Of course, the modelling can be rearranged to lead to the exponential function also.

For most students, this approach allows them to see explicitly how mathematics is used to describe the real world, how mathematics emerges naturally and that the parameters have physical meaning. However, students could also use some of the basic statistical modelling ideas that are in the curriculum to estimate the parameters, by first transforming to a linear model (taking logarithms of the modelling equation): |$\textrm{log}\ \hat{N}=\textrm{log}\{N}_0+t\ \textrm{log}\ a$|⁠ . This equation is now linear in |$\textrm{log}\ \hat{N}$| and |$t$| (a plot, not shown, of the data confirms this is approximately true), and the slope of the line is expected to be about |$\textrm{log}\ a$|⁠ . The slope can be estimated formally using technology or informally using a ruler ( ⁠|$a$| may be estimated using a graph and fixing the intercept at |$\textrm{log}\{N}_0$|⁠ , then a ruler used to estimate the slope |$\textrm{log}\ a$|⁠ ).

Estimating a by averaging the three values of a that emerge from each individual trial: |$\hat{N}=100\ {t}^{0.8890}$|

Fitting the model |$\hat{N}={N}_0{a}^t$| directly using a statistical approach: |$\hat{N}=98.892\ {t}^{0.8858}$|

Log-transform and the fitting a linear model statistically, fixing |${N}_0:\hat{N}=100\ {t}^{0.8854}$|

Fitting an exponential model: |$\hat{N}=98.0\ \exp (-0.119t)$|⁠ .

Fitting an exponential model, fixing |${N}_0$| as 100: |$\hat{N}=100\ \exp (-0.122t)$|

All produce similar, but not identical, values for |$a$|⁠ . This leads to useful discussions of why they are different, which are the `best’ estimates and what is meant by `best’. It is also useful and helpful for students to see that there is not just one possible answer ( Carrejo & Marshall, 2007 ). A discussion of these different methods would be dependent on whether students used these methods themselves and whether time was available.

6.4. Poor practice: empirical structure and empirical parameter values (SE, PE)

A child of one author was given a mathematics assignment in Year 11 where students were asked to generate data as follows. Stand 30 cm from a wastepaper basket, take 10 shots at landing a wad of paper in the bin and record the number of successful shots. Students were to repeat at other distances from the wastepaper basket (60 cm, 90 cm, 120 cm and up to 330 cm). Students were then asked to plot the data and fit linear, quadratic, cubic and reciprocal functions to the data using software. Students were told to vary the parameters of the model and then select a `best fitting’ model by eye (rather than by computer optimization, say). Students used their chosen model to predict what would happen at a different distance (say 105 cm). The data collected by the author’s child are plotted in Fig. 4 . Students can identify the variables as the distance from the bin |$x$| and the number of successful shots |$y$| out of 10 (step 1: specify; Fig. 1 ). They then defer to technology to fit a model (step 2: formulate), which they chose `by eye’ as the best (step 3: analyse).

Data collected from the paper toss study, with various fitted models shown over the data.

This task uses the empirical approach entirely: no attempt is made to mathematize the situation or even to understand what the fitted parameters mean. For example, none of the candidate functions restrict the values of |$y$| to between 0 and 10 even within the domain of the data. None of these proposed models sensibly describe reality, and none of the models or their parameters can be sensibly interpreted or have any physical meaning relating to the actual task. The mathematical functions have been imposed onto the data (the instructions tell students to fit specific functions), and none actually make sense. How can a student be expected to see mathematics being useful in this context, when the models are all clearly ridiculous?

This is an example of a task that `exclude[s] real knowledge from [the] solutions’ ( Verschaffel et al. , 1997 , p. 339). Despite intentions, this activity then could `actually teach students to suspend real-world sense making ( Greer, 1997 )’ ( Gainsburg, 2008 , p. 200; citation in the original) and reinforce the idea that mathematics has no connection to the real world (as it did for the author’s child):

Secondary mathematics teachers count a wide range of practices as real-world connections. Teachers make connections frequently, but most are brief and many appear to require no action or thinking on the students’ part. ( Gainsburg, 2008 , p. 215)

A sensible model would be a logistic regression model ( Fig. 1 , left panel), which is outside the scope of many curricula ( Morrell & Auer, 2007 ).

6.5. A more involved example: theoretical structure and empirical parameter values (ST, PE)

This example (from Dunn & Smyth, 2018 ) considers real data from small-leaved lime trees Tilia cordata grown in Russia ( Fig. 6 , left panel; Schepaschenko et al. , 2017 ). It has been used with university students but could also be used with Year 11 and 12 students (16 and 17 year olds). The interest is in estimating the oven-dried foliage biomass (hard to measure) from just the diameter of the tree (easy to measure). A data-based approach would require students to fit quadratic models, exponential models and other functional forms to model the structure of the data and then decide which fits the data best (Section 6.4). However, this approach is poor mathematical modelling and misses rich learning opportunities.

A small-leaved lime tree ( T. cordata). (Used under the Creative Commons Attribution-Share Alike 3.0 Austria licence. Photo by Haeferi, downloaded from https://commons.wikimedia.org/wiki/File:Feldkirchen_an_der_Donau_-_Naturdenkmal_nd325_-_Linde_in_Feldkirchen_-_Winter-Linde_(Tilia_cordata).jpg on 07 November 2017).

Of course, the value of the proportionality constant |$b$| is unknown, but can be estimated informally, as the modelling implies that |$y/{x}^2$| should be approximately constant (analyse). This estimation could be done formally using the data using statistical methods, but the complications this entails are easily avoided and teaching opportunities increased. For each tree, we can compute |$y/{x}^2$| and then find the mean of all these values; this produces an estimate of the proportionality constant of |$0.005309$|⁠ , so that |$y=0.005309\ {x}^2$|⁠ . This model can be plotted on the data ( Fig. 6 , right panel), and the fit appears to be reasonable, and so the mathematical modelling seems reasonable; this is informal validation (interpret). In addition, the students can see directly where the estimate of the proportionality constant comes from, and the parameter has meaning.

The oven-dried foliage biomass from 185 naturally growing lime trees in Russia. Left panel: the original data; right panel: some mathematical models plotted on the data.

Note that rather than suggesting using the mean of |$y/{x}^2$| as an estimate of the proportionality constant (Method 1), students may also suggest the mean of |$y$| divided by the mean of |${x}^2$| (Method 2; giving 0.004742) or the mean of |$y$| divided by the square of the mean of |$x$| (Method 3; giving 0.005766). These can all be drawn on the graph of the data and all seem reasonable. All these approaches—while not specifically statistical—demonstrate an understanding of the mathematics and the meaning of the parameters and connect the mathematics to the real world. Furthermore, all models are reasonable so there is not a single correct answer (which may surprise students in mathematics classes; Grootenboer & Marshman, 2016 ), and there is no obvious answer to which estimate is the `best’, or even an answer to what `best’ means in this context.

What else might impact foliage biomass, and how might these be incorporated into a model? (Soil type, hours of sunlight, amount of water received, genetics, planting density, competitor species present etc.)

In what ways is the spherical assumption for the canopy likely to break down? What might be the implication of this? (The canopy sphere may be not quite accurate; there is a `gap’ in the sphere where the trunk is; some foliage is inside the canopy. So maybe the `power’ may not actually be two, but the general structure of the relationship is probably sound.) This shows the balance of simplicity and accuracy, and the level of accuracy needed may depend on what the model will be used for.

What are the units of measurement for the proportionality constant, and what does it mean? (kg/cm 2 ; the average mass of foliage per unit surface area.)

Why are the estimates of the proportionality constant different? Which is the best estimate and why? What is meant by `best’?

Why is the power not two (using one of the approaches), when the modelling led us to expect a power of two? (The model is just an approximation.)

Connecting mathematics to the real world is important ( Smith & Morgan, 2016 ) for increasing students’ understanding of and motivation to study mathematics ( Gainsburg, 2008 ). Using real data and real contexts is a useful way of showing students the applicability of mathematics, and this can be done through mathematical modelling ( Jablonka, 2007 ). In this article, we considered mathematics modelling as a vehicle for teaching mathematical material ( Galbraith, 2011 ) to develop their students’ understanding of particular mathematical functions. We have provided a framework to support teachers as they choose resources and data sets for mathematical modelling.

Using examples where the model structure can be described theoretically (ST; Fig. 2 ) is a better approach for teaching (and in practice), as the mathematics is being used to describe the real world. Parameter estimates then are usually determined from the data (PE), and a theoretical basis for the model structure often leads to sensible ways to estimate the parameters that retain meaning in the parameters.

In contrast, asking students to fit many functional relationships to real (or artificial) data without any meaning behind the resulting models or their parameters (SE) can reinforce the misconception that mathematics is irrelevant in the real world (Section 6.4). Finding suitable real data sets—or having students generate suitable real datasets—is not always easy or straightforward. We have given some suggestions here for students to potentially generate different data sets (Sections 6.2 and 6.3).

We do not recommend fitting mathematical models by determining the model structure through empirical means. When teachers are trying to teach mathematical modelling this way, by using `modelling as curve fitting’ ( Galbraith, 2011 , p. 281), the applicability of mathematics is lost in the procedure and students forget about the real-world knowledge. Generally, students and teachers do not have the skills in statistical modelling to do this effectively as it is not generally included in the curriculum. As a consequence, we believe that if data are used in the teaching of mathematical modelling, then it should be used in a confirmatory (for validation) and not exploratory (for estimation) manner when applied to determining the model structure.

Teaching functions using real data is possible and sensible and can be abounding with meaning. Our framework is useful to remind teachers that the model should be derived from the real world rather than the data and the data then used for validation. If done well, using real data can encourage good mathematical modelling skills in students and help them see the utility of mathematics in describing the real world.

Dr Peter Dunn has broad expertise in the application of statistics in a variety of fields, with publications in diverse areas that includes the teaching of statistics, health, ecology, climatology, agriculture and mathematical statistics.

Dr Dunn has developed a unique R package (called tweedie) for understanding the Tweedie class of distributions. He has presented numerous conference papers (winning a prize at the 16th International Workshop on Statistical Modelling in Odense, Denmark; and the EJ Pitman Prize at the Australian Statistics Conference in 2002).

Dr Dunn is the co-author (with Prof. Gordon Smyth) of Generalized Linear Models With Examples in R, and developed the Dunn-Smyth (quantile) residuals used in statistical modelling. He has been invited to give presentations on the R statistical environment at conferences and workshops.

Dr Dunn was awarded a national Office of Learning and Teaching Citation in 2012, has worked as a climate scientist, and has over 20 years' experience as a lecturer in statistics and mathematics at university level.

Dr Margaret Marshman worked as a physicist in a hospital, laser physics and magnetic resonance imaging before becoming a secondary mathematics and science teacher and mathematics Head of Department. She is involved in teacher education in the undergraduate and Master of Teaching programs.

Dr Marshman supervises a range of doctoral and masters students in Education. Her research interests include Mathematics Education and Science Education and she is particularly interested in how students formulate and solve mathematical and scientific problems and peoples' beliefs about mathematics, its teaching and learning. Margaret also researches areas of collaborative learning, middle schooling, statistics education and first year in higher education.

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Center for Teaching

Teaching problem solving.

Print Version

Tips and Techniques

Expert vs. novice problem solvers, communicate.

• Have students  identify specific problems, difficulties, or confusions . Don’t waste time working through problems that students already understand.
• If students are unable to articulate their concerns, determine where they are having trouble by  asking them to identify the specific concepts or principles associated with the problem.
• In a one-on-one tutoring session, ask the student to  work his/her problem out loud . This slows down the thinking process, making it more accurate and allowing you to access understanding.
• When working with larger groups you can ask students to provide a written “two-column solution.” Have students write up their solution to a problem by putting all their calculations in one column and all of their reasoning (in complete sentences) in the other column. This helps them to think critically about their own problem solving and helps you to more easily identify where they may be having problems. Two-Column Solution (Math) Two-Column Solution (Physics)

Encourage Independence

• Model the problem solving process rather than just giving students the answer. As you work through the problem, consider how a novice might struggle with the concepts and make your thinking clear
• Have students work through problems on their own. Ask directing questions or give helpful suggestions, but  provide only minimal assistance and only when needed to overcome obstacles.
• Don’t fear  group work ! Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem. Additionally, group work helps students realize that problems often have multiple solution strategies, some that might be more effective than others

Be sensitive

• Frequently, when working problems, students are unsure of themselves. This lack of confidence may hamper their learning. It is important to recognize this when students come to us for help, and to give each student some feeling of mastery. Do this by providing  positive reinforcement to let students know when they have mastered a new concept or skill.

Encourage Thoroughness and Patience

• Try to communicate that  the process is more important than the answer so that the student learns that it is OK to not have an instant solution. This is learned through your acceptance of his/her pace of doing things, through your refusal to let anxiety pressure you into giving the right answer, and through your example of problem solving through a step-by step process.

Experts (teachers) in a particular field are often so fluent in solving problems from that field that they can find it difficult to articulate the problem solving principles and strategies they use to novices (students) in their field because these principles and strategies are second nature to the expert. To teach students problem solving skills,  a teacher should be aware of principles and strategies of good problem solving in his or her discipline .

The mathematician George Polya captured the problem solving principles and strategies he used in his discipline in the book  How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, 1957). The book includes  a summary of Polya’s problem solving heuristic as well as advice on the teaching of problem solving.

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Making Sense of Mathematics

Teaching Mathematics through Problem Solving- An Upside-Down Approach

By inviting children to solve problems in their own ways, we are initiating them into the community of mathematicians who engage in structuring and modeling their “lived worlds” mathematically.

 Fosnot and Jacob, 2007

Teaching mathematics through problem solving requires you to think about the types of tasks you pose to students, how you facilitate discourse in your classroom, and how you support students use of a variety of representations as tools for problem solving, reasoning, and communication.

This is a different approach from “do-as-I-show-you” approach where the teacher shows all the mathematics, demonstrates strategies to solve a problem, and then students just have to practice that exact same skill/strategy, perhaps using a similar problem.

Teaching mathematics through problem solving means that students solve problems to learn new mathematics through real contexts, problems, situations, and strategies and models that allow them to build concept and make connections on their own.

The main difference between the traditional approach “I-do-you-do” and teaching through problem solving, is that the problem is presented at the beginning of the lesson, and the skills, strategies and ideas emerge when students are working on the problem. The teacher listens to students’ responses and examine their work, determining the moment to extend students’ thinking and providing targeted feedback.

Here are the 4 essential moves in a math lesson using a student-centered approach or problem-solving approach:

• Number Talk (5-8 min) (Connection)

The mini-lesson starts with a Number Talk. The main purpose of a Number Talk is:

*to build number sense, and 

*to provide opportunities for students to explain their thinking and respond to the mathematical thinking of others.

Please refer to the document Int§roducing Number Talks . Or watch this video with Sherry Parrish to gain understanding about how Number Talks can build fluency with your students.

Here are some videos of Number Talks so you can observe some of the main teaching moves.

The role of the teacher during a number talk is crucial. He/she needs to listen carefully to the way student is explaining his/her reasoning, then use a visual representation of what the student said. Other students also share their strategies, and the teacher represents those strategies as well. Students then can visualize a variety of strategies to solve a problem. They learn how to use numbers flexibly, there is not just one way to solve a problem. When students have a variety if strategies in their math tool box, they can solve any problem, they can make connections with mathematical concepts.

There are a variety of resources that can be used for Math Talks. Note : the main difference between Number Talks and Math Talks, is that one allows students to use numbers flexibly leading them to fluency, develop number sense, and opportunities to communicate and reason with mathematics; the other allows for communicating and reasoning, building arguments to critique the reasoning of others, the use of logical thinking, and the ability to recognize different attributes to shapes and other figures and make sense of the mathematics involved.

• 2. Using problems to teach (5-8 min) Mini Lesson

Problems that can serve as effective tasks or activities for students to solve have common features. Use the following points as a guide to assess if the problem/task has the potential to be a genuine problem:

*Problem should be appropriate to their current understanding, and yet still find it challenging and interesting.

*The main focus of the problem should allow students to do the mathematics they need to learn, the emphasis should be on making sense of the problem, and developing the understanding of the mathematics. Any context should not overshadow the mathematics to be learned.

*Problems must require justification, students explain why their solution makes sense. It is not enough when the teacher tells them their answer is correct.

*Ideally, a problem/task should have multiple entries. For example “find 3 factors whose product is 108”, instead of just “multiplying 3 numbers. “

The most important part of the mini-lesson is to avoid teaching tricks or shortcuts, or plain algorithms. Our goal is always to help guide students to understand why the math works (conceptual understanding). And most importantly how different mathematical concepts/ideas are connected! “Math is a connected subject”  Jo Boaler’s video

“Students can learn mathematics through exploring and solving contextual and mathematical problems vs. students can learn to apply mathematics only after they have mastered the basic skills.” By Steve Leinwand author of Principles to Action .

• 3. Active Engagement (20-30 min)

This is the opportunity for students to work with partners or independently on the problem, making connections of what they know, and trying to use the strategy that makes sense to them. Always making sure to represent the problem with a visual representation. It can be any model that helps student understand what the problem is about.

The job of the teacher during this time, is to walk around asking questions to students to guide them in the right direction, but without telling too much. Allowing students to come up with their own solutions and justifications.

• Teacher can clarify any questions around the problem, not the solution.
• Teacher emphasizes reasoning to make sense of the problem/task.
• Teacher encourages student-student dialogue to help build a sense of self.

Some lessons will include a rich task, or a project based learning, or a number problem (find 3 numbers whose product is 108). There are a variety of learning target tasks to choose from, for each grade level on the Assessment Live Binders website created by Erma Anderson and Project AERO.

Again, keep in mind that some lessons will follow a different structure depending on the learning target for that day. Regardless of instructional design, the teacher should not be doing the thinking, reasoning, and connection building; it must be the students who are engaged in these activities

• 4. Share (8-12 min) (Link)

The most crucial part of the lesson is here. This is where the teaching/learning happens, not only learning from teacher, but learning from peers reaching their unique “zone of proximal development” (Vygotsky, 1978).

We bring back our students to share how they solved their problem. Sometimes they share with a partner first, to make sure they are using the right vocabulary, and to make sure they make sense of their answer. Then a few of them can share with the rest of the class. But sharing with a partner first is helpful so everyone has the opportunity to share.

“Talk to each other and the teacher about ideas – Why did I choose this method? Does it work in other cases? How is the method similar or different to methods other people used?” Jo Boaler’s article “How Students Should Be Taught Mathematics.”

Students make sense of their solution. The teacher listens and makes connections between different strategies that students are sharing. Teacher paraphrases the strategy student described, perhaps linking it with an efficient strategy.

“It is a misperception that student-centered classrooms don’t include any lecturing. At times it’s essential the teacher share his or her expertise with the larger group. Students could drive the discussion and the teacher guides and facilitates the learning.” Trevor MacKenzie

If the target for today’s lesson was to introduce the use a number line, for example, this is where the teacher will share that strategy as another possible way to solve today’s problem!

This could also be a good time for any formative assessment, using See Saw, using exit slips, or any kind of evidence of what they learned today.

References.

“Teaching Student-Centered Mathematics” Table 2.1 page 26 , Van de Walle, Karp, Lovin, Bay-Williams

“Number Talks” , Sherry Parrish

“How Students Should be Taught Mathematics: Reflections from Research and Practice” Jo Boaler

“Erma Anderson, Project AERO Assessments live binders

“Principles to Action” , Steve Leinwand

“ Turning Teaching Upside Down “, by Cathy Seeley

“Four Inquiry Qualities At The Heart of Student-Centered Teaching”

By Trevor MacKenzie

“The Zone of Proximal Development” Vygotsky, 1978

*** Here is a link to my favorite places to plan Math padlet, you will find a variety of resources, videos, articles, etc. By Caty Romero

***One more padlet for many resources to plan, teach, and assess mathematics that make sense: Making Sense of Mathematics Padlet.

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• learning math
• making sense of math
• number sense

Published by Caty Romero - Math Specialist

Passionate about learning and making sense of mathematics. Teacher, Math Learning Specialist, K-8 Math Consultant, and Instructional Coach. Student-Centered-Learning is my approach! Contact me at [email protected] or follow me on Twitter @catyrmath View all posts by Caty Romero - Math Specialist

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Teaching methods: Engaging students with problem solving tasks in maths

Thanks for downloading this podcast from Teacher magazine. I’m Dominique Russell.

Research shows that challenging problem solving tasks have a positive impact on student learning, but there is little evidence on student attitudes towards problem solving when it comes to doing this in the maths classroom.

My guest today is education consultant at Love Maths , Michael Minas. He works in primary schools to help improve learning in mathematics through professional development, classroom modelling and work with parents.

His areas of interest include problem solving and student engagement, and during his time working in primary school maths classrooms, he’s noticed anecdotally that students respond really positively to being presented with challenging problem solving tasks.

So, to formally investigate this, he conducted a study to assess student attitudes towards problem solving in maths alongside Dr James Russo from Monash University. This study focused on 52 students in two classrooms – a Year 3 and 4 class and a Year 5 and 6 class – in a primary school in Melbourne. Michael led a number of lessons in each classroom which presented challenging problem solving tasks to students. The classroom teachers observed these lessons, and then led these same tasks with the students. The lesson structure used was launch-explore-discuss/summarise, which Michael will go over in more detail in the episode.

After these lessons, the students completed a questionnaire to assess their opinions on the task. The results found that three-quarters of students reported unambiguously positive attitudes towards problem solving, the others were ambivalent, and no student expressed a negative attitude.

So, if you’re interested in implementing challenging problem solving tasks in your classroom, keep listening to hear Michael explain in detail the structure of these tasks, and what elements students enjoyed most. Let’s jump in.

Dominique Russell: Thanks for joining me Michael. I just thought it would be good to get a bit of background on the work you’re doing at the moment to start things off and why this research was important for you to conduct?

Michael Minas: Yeah I guess a lot of my work at the moment is in classrooms and one of the things that a lot of the schools are interested in is trying to get more problem solving happening in their mathematics classrooms. So a lot of the work that I do is in classrooms modelling problem solving lessons, working with teachers to sort of develop their sort of approach, their level of comfort with that style of teaching.

And so for me this was really interesting because, you know, I know anecdotally through my sort of experience with working with hundreds and hundreds of students, that I can see the positive responses. But, you know, it’s obviously an area that hasn’t had a lot of research done into it, so it’s good to be able to have, you know, the start of looking at it in a more formal way, of how do students actually feel in these types of lessons? What’s the experience like for them?

DR: And so the research obviously looks at two classrooms in particular in a primary school in Melbourne, looking at those middle years in primary school, which like you say, hasn’t really been looked at in much detail in the literature. So can you describe for me a bit about the school context of this particular school that you were doing the research in?

MM: Yeah, so I mean it’s a typical sort of primary school. It wasn’t, it didn’t have sort of anything outstanding in terms of the cohort of students, the size of the school – you know, 300 kids – it was a pretty sort of, demographically, a regular mix of students.

In terms of mathematics, it was philosophically quite a traditional environment for students to work in with a lot of sort of teacher directed work – you know, ‘I’ll show you how to do it, and then you go back to your tables and you reproduce what I’ve put on the board and maybe answer a series of questions using the approach that I’ve shown you’. So this style of lesson and learning was quite different for both the staff and the students at the school.

DR: And so obviously this style of learning that you exposed them to was received very positively from the students involved which we’ll talk about in a bit more detail soon, but I’m interested then in what the students opinions were of maths before this problem solving task was introduced to them. Do you have any concept of how they viewed maths in general? Did they enjoy it or were they enthusiastic about it?

MM: Yeah, so we had a couple things. So, you know, fairly informal, but when I arrived there one of the first things I did was I did some surveys with all of the students from Year 3-6. And the surveys were around their attitude to maths and also their self-perception (so, how they saw themselves as maths learners) and there were some really clear negative trends there.

So that was a starting point, you know, working with the leadership in the school to say, there are some issues here and you know, you can clearly sort of see that there are some issues here from the survey data.

But beyond that, I mean, anecdotally, my very first day at the school, I distinctly remember this. I was walking into one of the rooms at the school and a little Grade 3 girl said to me ‘oh, who are you?’ and I said, you know, ‘I’m Michael, I’m going to be here, I’m going to be working with you guys on maths’ and whatever. And her friend that was sitting with her, who wasn’t part of the conversation, inserted herself into the conversation to say, ‘oh, we hate maths. Both of us hate maths.’ Like really wanted to make a point of letting me know that she hated maths.

So that type of interaction was probably the most memorable, but I had lots of those types of interactions where people said ‘oh, you’re working with maths? Yeah I don’t like learning maths. I’m not interested in maths. I hate maths. Maths is my least favourite subject.’

So I had lots of those interactions with the kids and, you know, with that girl on the very first day and I said to her ‘well, you know, hopefully if we have this same conversation in November, that you will have shifted the way that you see maths. But that’s my job to do that, that’s not your job.’

DR: And so can you talk me through really the structure of these problem solving tasks that you led in the classrooms? Because I know you were leading them for a little while and the classroom teacher was observing the lessons that you were conducting. So what’s the structure of these tasks?

MM: Yeah, so one of the things is that the structure is a very, sort of central feature of this approach. And the idea is that, that structure is meant to be very predictable for both the staff and for the students.

You know, so we’d start with a warm up activity and the central idea of that is that you want that to be an engaging warm up to have the kids starting the lesson with, you know, a lot of energy and enthusiasm. And that would be followed by the launch of the problem. And, ideally, most problems, we want them to be launched with some of narrative link, some sort of connection to the real world. And we want that to be done in a concise way.

So the idea there is it’s not like a mini lesson of ‘let me get up the front and tell you everything I know about division for the next 15 minutes’. The idea is that we’re giving them a task that – maybe the task lends itself to multiplicative thinking and division, but we’re leaving space for the students to approach the task from their own perspective. So that, I will say to teachers ideally I want that launch to be sort of somewhere around the five minute mark. And for a lot of classroom teachers that’s a challenge, that sort of directly conflicts with the way they’re currently taking their maths lessons.

And then by extension, the shorter that launch time is, the more time the students have to be exploring, engaged with the task. So in order for, you know, students to stay working on a task for 35, 40 minutes, the task needs to be challenging, it needs to be cognitively engaging for them.

And so that explore time starts with five minutes of silent, independent work. And it’s really important that it is silent and is independent. And then from there, I’m a big advocate for actively encouraging collaboration in the classroom. So, not just saying ‘if you want to work with someone, you can’, but actively encouraging the kids; say: ‘hey, why don’t you go over and talk to Megan and see what she’s doing because she’s got some similar thoughts to you, but she’s approaching it a bit of a different way’.

And then the lessons will always finish with some sort of summary of what we’ve done and that again is student-centred. So the idea is that we’re (myself or whoever the teacher is) is looking for student examples to sort of showcase at the end of the lesson to say: ‘hey, you know, talk to me Dominique about what you’ve done’, and getting you to explain your thinking, but being really strategic about it who you select. So it’s not like ‘everyone come to the floor, okay, who’d like to share their work?’ and the same three kids put their hands up every day. It’s you as a teacher being really strategic about who you select and why.

DR: And so something that I found really interesting in the report, just as a bit of an example of how this structure plays out, the example of the chessboard tournament problem. So, the problem was launched with a short story about a family holiday and there was a big chessboard where they were staying. So you displayed a photo of one of the children playing on the chess board at the front of the classroom, and then you gave the task, which was if six children wanted to have a round robin tournament, how many games would need to be played? And then you had a prompt, which asked students to draw a diagram to show how they’d work this out, then you also provided some extending prompts. So can you give a quick run through of how that played out in the classroom? And did the students respond well to the extension prompts?

MM: Just to give you a bit of an idea about the narrative side of things – that task was based around a photo that I shared with the kids of my own family when we were on holidays playing chess on a chessboard. And I’ve had some really positive interactions with kids around that, where some kids will come and say to you ‘oh I play chess lots,’ and ‘I’m a big fan of chess’ so it’s building relationships there where they can say, you might share a common interest.

I mean I’ve had the other experience where I’ve been at a school and I’ve told the kids: ‘this giant chess board was at this particular holiday place in Queensland, and then I’ve had a student come back to me like two months later over the summer holidays and saying: ‘guess where I went over summer? We went and stayed at Paradise Resort and we played chess on that chessboard’ and the kid being really excited to share that with you.

So the narrative, that was the true part of the narrative. I mean the made-up part was – it talks about us having a round robin chess tournament. Now, we didn’t have a round robin chess tournament, we were actually there trying to enjoy, we didn’t spend all day at a giant chessboard playing chess.

So I think it’s teachers being able to feel comfortable taking parts of their life – you know, some real-world application – but also feeling free to be able to sort of elaborate, add to it, and make it work for the maths.

The task – yeah, it’s a really fantastic task because it’s got quite a low entry point, in that you could work on that task just sort of saying – you know it’s like the old problem where you say ‘there are eight people in a room and they all shake hands with each other. How many handshakes would there be?’ But it’s much more – I mean, the idea of playing chess against each other, students can visualise that a lot better and can sort of conceptualise it to say: ‘well if Nash plays against Isaiah, and then next Nash would play against Genevieve…’ so they can sort of work through all the combinations of who Nash would need to play.

Nearly every student you give that task to can enter the task and can have some level of success. But at the higher level it’s a very cognitively engaging task. I mean, the extension task is asking for them to basically find a formula of how to work out any triangular number. And so I used that task with Year 3/4 students and I’ve had students I’ve worked with in the Year 3/4 cohort who are able to sort of show you, ‘I can work out any triangular number and this is how you do it’. And they can show you visually how the formula works.

So I think that’s the beauty to this approach to teaching in that you’re really allowing for true differentiation. You’re presenting a task and there’s scope there for students to work there at a number of different levels.

DR: And as we mentioned very briefly before the classroom teachers were observing you first conducting these lessons before conducting the lessons themselves. Why was that important to do than just instructing teachers on how to run this and getting them to launch straight into it. Why was the observation element quite critical?

MM: I’m a big believer of if you want to get change happening within an organisation, it’s important to have buy-in from people. It’s important for people to actually believe that what you’re doing is going to be doing is beneficial. And for teachers, the vast majority of teachers, when they see that something is effective with their own students, you’ve won them over. So if they can see their own students being challenged in a way they previously haven’t been challenged.

I mean I had this experience yesterday when I was at a school in the western suburbs of Melbourne and I was working in a prep classroom and there was a prep student who, you know, traditionally didn’t really have a lot of success in the maths class and then this student produced some work and this classroom teacher was literally speechless. He was just blown away, he was like, ‘I cannot believe that he’s just done that, I’ve never seen him do that before’.

Now when I go back to work with that teacher in a fortnight’s time or whenever I’m back out there to work with them, they’re going to be much more receptive to this approach because they can see that it works.

And I think you’re also setting teachers up for success then. Because if they’ve seen that lesson structure a few times, the idea that it is very repetitive as a structure, it gives them something that they can sort of say, ‘right, now if I’m going to have a go at taking a lesson using this approach, these are the things that I want to do’. And it’s very easy to reproduce because they’ve seen it done a number of times.

So it’s both about supporting the teachers so they can have success, but also about generating that buy-in and I think that that comes – it’s one thing to deliver PD and to say ‘this is great’. It’s another thing for teachers to see it working with their own students.

DR: And so another big part of the study was how you actually measured the attitudes that students held towards these problem solving tasks and they were overwhelmingly positive. You’d mentioned before that this was kind of what you were expecting to happen because anecdotally you knew that students responded really well to these kinds of tasks. But something that I found really interesting was that they really enjoyed the challenging aspect of these problems and also the collaborative nature. So can you talk me through what the students said and wrote in their questionnaires about those two particular aspects?

MM: Yeah. I guess, I mean one thing that did surprise me was – I expected the results to be positive because that’s what I kind of see when I work not just at this school, but at lots of schools – I was surprised in the fact that of all the students that were involved in the study, that there was no one that expressed, like, negative attitude. Which, you know, was quite sort of gobsmacking for me.

But in terms of what they identified that made it enjoyable, engaging for them. Like you said, there was a couple things they touched on. So one was the idea of challenge. And I think this is something that sometimes teachers struggle with, this idea that: ‘if I make the work more challenging, the kids will disengage. They won’t persist, they won’t enjoy tackling the task’.

And I actually think that’s counter to everything we know about humans. If we think about ourselves as adults, if we’re given some sort of routine, mundane task to perform over and over again it’s every chance that we might do it if we have to do it, but we’re not going to enjoy it. But people love a challenge, people love being pushed cognitively and trying to see if they can be the first one to figure things out. I think humans love a challenge and if I enjoy a challenge as a 43-year-old, there’s no reason to think that like a six-year-old or a 12-year-old wouldn’t enjoy a challenge. So that’s come through to me anecdotally, you know, time and time again over the years, so it was good to see that come in through formally in the study that we did.

The other really big – and again, in some cases this really contrasts with the regular classroom practice – this idea of allowing the students to collaborate. And like I said before, not just allowing, but actively encouraging it. I think a lot of classroom teachers are concerned that if they let the kids move around the room and talk to each other, they’re going to lose control and it’s going to descend into chaos. But I think the two ideas that you’ve just asked about are connected. Like if they’re working on something that they think is worthwhile and challenging, they’re much more likely to stay on task.

And again, humans enjoy collaborating. Humans enjoy socialising, talking, sharing ideas. So if that’s the way, if I was to present PD [professional development] at a school and I was to do five hours of me talking and there’d be no opportunity for staff to actively engage and collaborate with each other, I mean, I would never be invited back to the school.

So then the question would be, well why do we get our students to do this? Why is a maths lesson me talking for 20 minutes telling you everything I know about place value, and then you working on a worksheet by yourself for half an hour and not being allowed to talk?

That’s not going to be enjoyable for us as adults. Why would it be enjoyable for an eight-year-old in a Year 2 class? So I think that in some cases the success that we have when we go and work in schools is partly because it’s such a sharp contrast to the regular practice in the school about the way maths is learned. And that if we can make mathematics more social, then we have much more chance of having students being engaged and wanting to learn.

DR: And is part of that as well – like you mentioned before – the fact that they have that five minutes at the beginning to concentrate on the problem as an individual and silently, but then they open up to the collaboration. Is that balance quite good and quite important?

MM: Yeah. It’s really crucial. And I always tell teachers that I’m working with that one’s not more important than the other, that they’re equally important. But if you let kids collaborate straight away then what you might find is that kids will just straight away – say you and I are working together, and you’re a stronger student in terms of your current performance in maths, well I might just be led by you, and you’ll just be telling me, ‘do this, do that’.

Much more likely if I’ve had some time to think and ponder on the task, that A, when I come to you, maybe I’ll have some questions about what I’m doing and you can guide me and direct me, rather than telling me what to do. But, B, there might be the chance that I may choose not to work with you, even thought we may be best friends, because I may see that someone else is approaching the problem with a similar mindset, a similar approach to me. Or I may choose to say in this instance: ‘I’m going to keep doing this by myself because I feel like I’m getting some momentum here. I can see that I’m making some progress’.

So I think that five minutes silent time is really crucial, and then it becomes really crucial (this becomes a classroom management thing) as a teacher, you have to be able to make sure that it is truly five minutes and it is truly silent and it is truly independent and also truly productive. Because it’s no good them just sitting silently looking at the clock, you know, looking at the stop watch counting down before they bang go into talking to each other.

So the way you know that’s productive is when you see the kids are on task. When you see the tops of their heads looking down at their page, and they’re thinking, and they’re gathering materials. And you can tell really clearly as a teacher when that’s not happening.

DR: And so just finally then, I’m thinking now for teachers who are listening to this episode who are thinking they want to implement a similar approach in their maths classroom for students of a similar age, is there anything that we haven’t covered already that would be good to keep in mind? Or perhaps some good first steps to take?

MM: Yeah, look I think that the model that I see that works really well is I think what we spoke about before. Is that you have to have people that are able to model what it should look like to be able to win teachers over, for them to be able to say ‘I can see the benefit of this, I can see how this works’. So whether that be – I mean, I’m definitely not trying to spruik for work – but whether that be internally – you know, like a lot of schools have really great classroom teachers. Some of those people have moved into learning specialist roles.

But whether that be internally with those people, like give them the time to go into other people’s classrooms and to be able to model this type of approach and to show the classroom teachers how it works and to be able to answer those questions. Or whether it be externally, by bringing in consultants who have the skill and expertise to do it, I think that’s really important. I think it’s important that people see it in practice first before they try to do it.

And it’s also really important, as well as seeing lessons, that people have time to then unpack the lesson and talk about it together. So if you’ve got a learning specialist at your school that’s modelling this type of lesson for, say a graduate teacher, there needs to be some time allocated for them to sit. Because the graduate teacher may walk away saying, ‘that was a great lesson’. But the next step is them being able to identify why was it a great lesson? What worked? And what can they do to plan a similar great lesson the following week?

Because, you know, if you just say ‘well, that was a great lesson, but I can’t do that lesson again because my kids have already done it, so where do I go with it?’ Whereas if you can identify and say: ‘oh I see what worked well. The thing that worked well is they were engaged with the problem.’ Why were they engaged in the problem? ‘It had a real world link’. Why was your questioning effective during the lesson? ‘Well, it was because you knew, you had a clear focus of what the content was’. What are we focusing here? What’s the mathematical concepts we’re focusing on?

So, as a classroom teacher you know the right question to ask and the right student at the right time and there’s a lot of work that goes into that, but like I said, it’s definitely something that’s attainable for all classroom teachers with the right support.

That’s all for this episode. Thanks for listening. Be sure to subscribe to our podcast channel on Spotify , Apple podcasts or SoundCloud , so you can be notified of any new episodes. While you're there, we'd love for you to rate and review the podcast in your podcast app.

Russo, J., & Minas, M. (2020). Student Attitudes Towards Learning Mathematics Through Challenging, Problem Solving Tasks: “It’s so Hard– in a Good Way”. International Electronic Journal of Elementary Education, 13 (2), 215-225. https://doi.org/10.26822/iejee.2021.185 .

Michael Minas says he believes sometimes teachers struggle with the idea that: ‘if I make the work more challenging, the kids will disengage. They won’t persist, they won’t enjoy tackling the task’.

Reflect on a recent lesson you taught. How challenged would you say students were? How do you know the level of challenge was appropriate? Do you think you could have challenged students further? Were there opportunities for students to participate in extension tasks?

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