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## Purpose of the Model

Philosophy of problem solving, fun: the bookworm, quick links.

This newsletter introduces the Problem Solving Model. This is a ten-step model to guide you (and your team) through a structured problem solving process. All too often, people jump from a problem to a solution. And it is often a solution that is short-lived or creates numerous other problems within the organization. The Problem Solving Model provides you a road map to continuous improvement.

As its name implies, this model is the road map to follow to solve problems. What makes something a problem?

a) When the process isn’t doing what it is supposed to and people don’t know why. b) When things keep going wrong no matter how hard everyone tries. c) When everyone believes that there is a problem to solve.

The first step in the model is to define the problem; it does not matter if it is late shipments, stock outs, computer downtime, typos, lost messages, or an agreed upon “red bead” that everyone keeps running into. Before you can solve the problem, you must truly understand what it is. This means brainstorming about the process, using a Pareto Diagram to prioritize potential obstacles and creating a process flow diagram of what is currently going on. After you have the problem defined, the model leads you through analyzing data you gather about the process, determining the root cause of the problem, and identifying possible solutions to the problem. Solutions to the problem will either be changes to the process which eliminate special causes of variation or changes which reduce common cause variation. After the best solution is implemented, the model leads the team to monitor the impact of its revisions to make sure that the problem is truly solved.

The problem-solving model, introduced below, incorporates an effective set of skills into a step-by-step process. The model combines the use of statistical tools, such as control charts and process flow diagrams, with group problem-solving skills, such as brainstorming and consensus decision-making. The statistical tools help us make data-based decisions at various points throughout the model. The group problem-solving skills help us draw on the benefits of working as a team.

Before we begin a discussion about the steps of the problem-solving model, we should talk a little about the philosophy that good problem solvers have about problems. Here are a number of ideas that are part of the philosophy.

Problem solving should occur at all levels of the organization. At every level, from top to bottom, problems occur. Everyone is an expert in the problems that occur in his or her own area and should address these problems. Problem solving is a part of everyone’s job.

All problems should not be addressed with the same approach. There are some problems that are easily and suitably tackled alone. Not all decisions need to be made by teams nor do all problems need to be solved by groups. However, groups of people help to break mental sets (i.e., figuring out new ways of doing things). In addition, people are more committed to figuring out and implementing a solution to a problem if they are involved in the problem solving.

Problems are normal. Problems occur in every organization. In excellent companies people constantly work on solving problems as they occur. Problems are opportunities to make things better and should be viewed as such.

Be hard on the problem and soft on the people involved. When working on a problem, we should focus on solving the problem, not on whose fault the problem is. We should avoid personalizing the problem and blaming others.

People should address the problems in their own areas. Everyone has problems associated with their work area, and they should take ownership for trying to solve these problems instead of waiting for their supervisors or another team to tell them what to do.

## Problem Solving Model

Step 1: define the problem..

Step 1 is a critical step; it determines the overall focus of the project. In this step, the team defines the problem as concretely and specifically as possible. Five SPC tools are helpful in defining the problem: brainstorming the problem’s characteristics, creating an affinity diagram, using a Pareto chart, creating an initial Process Flow Diagram of the present process, and Control Chart data. The process flow diagram (PFD) will help the team identify “start to finish” how the present process normally works. Often the PFD can dramatically help define the problem. After the problem is well defined, Step 2 helps the team measure the extent of the problem.

End Product = A clear definition of the problem to be studied, including measurable evidence that the problem exists.

## Step 2: Measure the Problem.

Baseline data are collected on the present process if they do not already exist. This permits measurement of the current level of performance so future gains can be subsequently measured. The team needs to make a decision on how to collect the present baseline data. In general, if data are collected daily, the time period should be a month. This way a standard control chart can be used. If data are collected weekly or once a month, baseline data will have only three or four points. Data collected less than once a month are of limited use; in such cases, historical data, if available, should be used. At this stage, the team must have measurable evidence that the problem exists. Opinions and anecdotes are a sound place to start, but eventually there needs to be concrete proof that there really is a problem.

End Product = A graph or chart with present baseline or historical data on how the process works; a collection of the present job instructions, job descriptions, and SOPs/JWIs (standard operating procedures and job work instructions).

## Step 3: Set the Goal.

Goals provide vision and direction and help the team make choices and know which path to take. Be sure to state your goal(s) in terms that are measurable. This way, the team can evaluate its progress toward the goal. As the team imagines the goal, it will identify benefits of achieving the solution to the problem. This inspires a higher commitment and support from all.

End Product = A goal statement that includes the what, when, where, why, who and how of the ideal solved problem situation.

## Step 4: Determine Root Causes.

In Step 4 the team studies why the process is working the way it is. If a control chart was developed in Step 2, determine whether the process is “in control” or “out of control.” If the process is “out of control,” the team should pinpoint the special causes and move to Step 5. If the process is “in control,” the team will need to use tools such as cause and effect analysis (fishbones), scatter plots and experimental design formats to identify root causes currently in the system producing common cause variation.

End Product = A list of most probable root causes of the problem (common and special cause variation); selection by team of the primary root cause of the problem to be eliminated.

## Step 5: Select Best Strategy.

The purpose of Step 5 is to select the strategy that best solves the problem. From the list of causes generated in Step 4, the team should brainstorm and strategically plan solution strategies. Fishbone diagrams and benchmarking can be helpful for this step. Then the team must reach consensus on the best possible strategy to solve the problem. This strategy should have the highest likelihood of success.

End Product = A well defined strategy to solve the problem is selected.

## Step 6: Implement Strategy.

An Action Plan is developed by team. This includes who will do what by when to implement the solution. The team sees to it that the Action Plan developed is carried out and documented.

End Product = The Action Plan is implemented.

## Step 7: Evaluate Results.

In Step 7 the team evaluates how effective the solution has been. Data must be collected to determine if the implemented strategy did, in fact, improve the process being studied. Performance must be clearly measured and evaluated. The team needs to monitor control chart data where appropriate and assess improvement; the process flow diagram should be checked for appropriate SOPs and JWIs. Additional feedback strategies such as histograms, process FMEAs, customer surveys and informal polls may also prove useful. What are the “customer” reactions (internal customer feedback)? What has produced measurable results? What hard data are available? Do people perceive an improvement? How have results matched customer needs? If the process did not improve, the team needs to discover if the wrong root cause(s) was identified or if the wrong solution was utilized. In either case, return to the steps above, beginning with Step 4. If the process improves, but the results are disappointing, there may be other root causes affecting the process. Again, return to Step 4 to further examine additional root causes. When the problem is solved (i.e. the “loop closed”), the team proceeds to Step 8.

End Product = The problem is solved; results of the improvement are measured.

## Step 8: Implement Appropriate Changes in the Process.

Step 8 develops an ongoing process to assure that the gains stay in place for the long term. Sometimes a problem is solved and then later resurfaces. This happens when a solution is determined, but a system or process to keep the problem solved has not been successfully adopted. Permanent changes need to be implemented. This means revising the existing procedures. The new improved process will need to be tracked over time; the process must be checked frequently to maintain improvement. This also helps everyone to stay aware of opportunities to continuously improve the process where the problem occurred.

End Product = A permanent change in the process, Quality Improvement, and people “closest to the job” monitoring the change.

## Step 9: Continuous Improvement.

This step is staying committed to continuous improvement in terms of this model – to remain actively alert to the ways the improved process can be made even better. This step is a conscious decision to allow others to innovate and to point out “red beads” in the process which the team has worked hard to improve. All involved, particularly those closest to the job, need to be encouraged to give constructive feedback and adjustments. Internal audits will monitor some processes to ensure effectiveness.

End Product = Commitment to continuous improvement.

## Step 10: Celebrate.

This last step includes a recognition celebration and the disbanding of the team. Always take time for this maintenance function; people have achieved an important goal. They have earned this moment of recognition and closure.

End Product = Closure for the team members; disbanding of the team.

Each of the four volumes in the picture has the same number of pages and the width from the first to the last page of each volume is two inches. Each volume has two covers and each cover is one-sixth of an inch thick.

Our microscopic bookworm was hatched on page one of volume one. During his life he ate a straight hole across the bottom of the volumes. He ate all the way to the last page of volume four. The bookworm ate in a straight line, without zigzagging. The volumes are in English and are right-side up on a bookcase shelf.

Challenge: how many inches did the bookworm travel during his lifetime? ____________

We will give the answer in next month’s newsletter.

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Thanks so much for reading our publication. We hope you find it informative and useful. Happy charting and may the data always support your position.

Dr. Bill McNeese BPI Consulting, LLC

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- Michael Gr. Voskoglou. Problem Solving and Mathematical Modelling. American Journal of Educational Research . Vol. 9, No. 2, 2021, pp 85-90. http://pubs.sciepub.com/education/9/2/6 ">Normal Style
- Voskoglou, Michael Gr.. 'Problem Solving and Mathematical Modelling.' American Journal of Educational Research 9.2 (2021): 85-90. ">MLA Style
- Voskoglou, M. G. (2021). Problem Solving and Mathematical Modelling. American Journal of Educational Research , 9 (2), 85-90. ">APA Style
- Voskoglou, Michael Gr.. 'Problem Solving and Mathematical Modelling.' American Journal of Educational Research 9, no. 2 (2021): 85-90. ">Chicago Style

## Problem Solving and Mathematical Modelling

Problem solving is one of the most important components of the human cognition that affects for ages the progress of the human society. Mathematical modelling is a special type of problem solving concerning problems related to science or everyday life situations. The present study is a review of the most important results reported in the literature from the 1950’s until nowadays on problem solving and mathematical modelling from the scope of Education. Its real goal is that it presents in a systematic way and in a few pages only the results of many years research on the subject. This helps the reader to get a comprehensive idea about a very important topic belonging to the core of Mathematics Education, which is very useful to those wanting to study deeper the subject and get directions for further research in the area.

## 1. Introduction

Problem solving (PS) is one of the most important activities of the human cognition. Volumes of research have been written about the steps, the mechanisms and the difficulties of the PS process, which affects the everyday life from the time that humans appeared on the Earth.

Most authors ( How I solve it: A new aspect of mathematical method. Princeton University Press: New Jersey, 1973." class="coltj"> 1 , For the Learning of Mathematics , 3, 40-47, 1983." class="coltj"> 2 , Cognitive Psychology , Braisby, N. and Gelatly, A. (Eds.); Oxford University Press: Oxford, 2005." class="coltj"> 3 , Phi Delta Kappan , 87(9), 696-714, 2007." class="coltj"> 4 , etc.) agree that a problem could be considered as an obstacle to be overpassed in order to achieve a desired goal. A problem, however, is mainly characterized by the fact that you don’t know exactly how to proceed about solving it. According to Schoenfeld 2 , if a problematic situation can be overpassed by routine or familiar procedures (no matter how difficult!), it is not a problem, but simply an exercise of the individual’s ability to tackle successfully this situation. The kind of a problem dictates the kind of the cognitive skills needed to solve it; e.g. linguistic skills are required to read and debate about a problem, memory skills to recall already existing knowledge being necessary to solve it, etc.

Mathematics by its nature is the subject whereby the PS process can be studied and analyzed in detail. Mathematical modelling (MM) , in particular, is a special kind of PS which formulates and solves mathematically real world problems connected to science and everyday life situations. The present review article studies the progress of research on PS from the time that Mathematics Education has been emerged as a self-sufficient science at the beginning of the 1950’s until recently, focusing in particular on the use of MM as a tool for teaching mathematics.

A lot of work has been done during those 70 years on PS and MM and the author had to consult hundreds of the existing in the literature references in order to be in position to present, in a few pages only, the most important findings on the subject in chronological order and connected to each other. And which is the meaning of this effort (?) one could ask. The answer is that the methodological presentation of the main results of research on PS and MM helps the reader to get a comprehensive idea about a very important topic belonging to the core of Mathematics Education. This could be very useful for those wanting to study deeper the subject and obtain directions for further research. Our approach could not contain of course all the relevant details, otherwise a whole book, and not a short article like the present one, should be needed! Our target was to include the central ideas only, managing in this way to show the evolution and the recent advances of research on PS and MM from the viewpoint of Mathematics Education.

The rest of the article is organized as follows. The next Section describes the development of PS in Mathematics Education and the main modes of thinking needed for PS. Section 3 analyzes the basic principles of MM and its advantages and disadvantages as a tool for teaching mathematics, while Section 4 studies some of the most important recent advances on PS and MM. The article closes with the general conclusions, which are presented in Section 5.

## 2. Problem Solving in Mathematics Education

Mathematics Education has been emerged as a self-sufficient mathematical topic during the 1970’s. The research methods applied on that time for the topic used to be almost exclusively statistical.

G. Polya (1887-1985), a Mathematics Professor of Hungarian origin at Stanford University, who introduced the use of heuristic strategies as the basic tool for tackling a problem’s solution How to solve it. Princeton Univ. Press: Princeton, 1945." class="coltj"> 5 , Mathematics and Plausible Reasoning . Princeton Univ. Press: Princeton, 1954." class="coltj"> 6 , is considered to be the pioneer of the systematization of the PS process. Polya proposed also Discovery as a method for teaching mathematics 7 , which is based on the idea that any new mathematical knowledge could be presented in the form of a suitably chosen problem related to already existing knowledge

Early work on PS focused mainly on the description and analysis of the PS process. The Schoenfeld’s expert performance model 8 is an improved version of the Polya’s framework for PS. Its real goal is that it provides a list of possible heuristics that could be used at each step of the PS process, which, according to Schoenfeld, are the analysis of the problem and the exploration, design, implementation and verification of its solution.

The assessment of the student PS skills and the effectiveness of the instructional treatments for improving those skills require measurement and several studies have been performed towards this direction ( J. of Educational Measurement , 14(2), 97-116, 19767." class="coltj"> 9 , J. Res. Math. Educ. , 13, 31-49, 1982." class="coltj"> 10 , etc.). Voskoglou and Perdikaris 11 introduced a Markov chain on the steps of the Schoenfeld’s expert performance model for PS and, by applying basic principles of the corresponding theory, obtained a measure of the student difficulties during the PS process. Voskoglou ( Turkish Journal of Fuzzy Systems , 3(1), 1-15, 2012." class="coltj"> 12 , Finite Markov Chains and Fuzzy Logic Assessment Models. Amazon-Create Space Independent Publishing Platform: Columbia, SC, 2017." class="coltj"> 13 : Chapter 7) used later principles of fuzzy logic too for modelling mathematically the PS process and for obtaining measures of student PS skills.

Much of the emphasis that has been placed during the 1980’s on the use of heuristics for PS was based on observations that students are often unable to use their existing knowledge to solve problems. It was concluded, therefore, that they lack suitable general PS strategies. Several other explanations were also presented later disputing the effectiveness of the extensive teaching of heuristics and giving more emphasis to other factors, like the acquisition of the proper schemas, the automation of rules, etc. 14 . What it has been agreed by many authors ( Cognitive Psychology , Braisby, N. and Gelatly, A. (Eds.); Oxford University Press: Oxford, 2005." class="coltj"> 3 , Thought and `knowledge: An introduction to critical thinking , Earlbaum: NJ, 4 TH Edition, 2003." class="coltj"> 15 , Cognition , Wiley & Sons: New York, 2005." class="coltj"> 16 , etc.), however, is that a problem consists of three main parts: The starting state, the goal state and the obstacles, i.e. the existing restrictions which make difficult the access from the starting to the goal state. And while, during the solution of the same problem, the first two parts are more or less the same for all solvers, the last part may hide many ways of tackling it, which could differ from solver to solver.

As a result, more recent studies have focused mainly on solvers’ behavior and required attributes during the PS process. The Multidimensional PS Framework of Carlson and Bloom 17 is based on the development of a broad taxonomy of PS attributes that have been identified as relevant to PS success. Schoenfeld 18 , after a many years research for building a theoretical framework, concluded that PS is an example of a goal-directed behavior , under which a solver’s “acting in the moment” can be explained and modelled by an architecture involving knowledge, goals, orientations and decision making which depends on subjective values that could differ from solver to solver.

In conclusion, PS is a complex cognitive action, directly related to the knowledge stored in the solver’s mind. Therefore, it requires a combination of several modes of thinking in order to be successful. Apart from the very simple and often automated thought (e.g. when performing calculations), other modes of thinking required for PS include critical, statistical, computational thinking and analogical reasoning . The nature of the problem dictates the mode(s) of thinking required in order to be solved.

Critical thinking (CrT) 15 is a higher mode of thinking involving analysis, synthesis and evaluation of the existing data, actions which give rise to other ones, like predicting, estimating, inferring and generalizing the corresponding situations. When a complex problem is encountered, it has to be critically analyzed: What is the problem, what is the given information and so on. CrT, therefore, is involved in application of knowledge to solve the problem. CrT plays also an important role in the transfer of knowledge, i.e. the use of already existing knowledge for producing new knowledge.

Statistical thinking (ST) 19 is the ability to use properly existing statistical data for solving problems related to randomness. Consider, for example, the case of a high school employing 40 in total teachers, 38 of which are good teachers, whereas the other two are not good. A parent, who happens to know only the two not good teachers, concludes that the school is not good and decides to choose another school for his child. This is obviously a statistically wrong decision that could jeopardize the future of the child.

ST, however, must be combined with CrT for obtaining the correct solution. In fact, going back to the previous example, assume that another parent, who knows the 38 good teachers, decides to choose that school for his child. His child, however, happens to be interested only for the lessons taught by the two not good teachers and not for those taught by the 38 good teachers. In this case, therefore, the parent’s decision is wrong again due to lack of CrT. In conclusion, CrT driven by logic and ST based on the rules of Probability and Statistics are necessary tools for PS.

If technology is added, however, those tools are not enough, since many technological problems are very complex. In such cases computational thinking (CT) becomes another prerequisite for PS. Although the term CT was introduced by S. Papert 20 , it has been brought to the forefront by J.M. Wing 21 , who describes it as “solving problems, designing systems and understanding human behavior by drawing on concepts fundamental to computer science”. This, however, does not mean that CT proposes that problems must be necessarily solved in the way that computers tackle them. What it really does is that encourages the use of CrT with the help of computer science methods and techniques.

Voskoglou and Buckley 22 developed a theoretical framework explaining the relationship between CT and CrT in PS. According to it, if there exists sufficient background knowledge, the new, necessary for the solution of the problem, knowledge is obtained with the help of CrT and then CT is applied to find a solution that might not be forthcoming under other circumstances.

Computer science does not concern only programing, it is an entire way of thinking, which has become now part of our lives. All of today’s students will go on to live a life heavily influenced by computing. Consequently, there is a need to be trained in thinking computationally as soon as possible, even before starting to learn programming 23 .

A particular attention has been also placed by the experts on the use of analogical reasoning for PS 24 . In fact, a given problem ( target problem ) can be frequently solved by looking back and by properly adapting the solution of a previously solved similar problem ( source problem ). The use of computers, in particular, enables the creation and maintenance of a continuously increasing “ library” of previously solved similar problems ( past cases ) and the retrieval of the proper one(s) for solving a new analogous problem. This approach, termed as Case-Based Reasoning (CBR) , is widely used nowadays in many sectors of the human activity including industry, commerce, healthcare, education, etc. 24 .

Computers facilitate also the creation of Communities of Practice (COPs) for teaching and learning including students and teachers from different places and cultures 25 . In the area of PS in particular, such COPs could help the exchange of innovative ideas and techniques on PS and problem-posing 26 .

More details about research and applications of PS can be found in earlier works of the present author ( Progress in Education, R.V. Nata (Ed.), Nova Science Publishers: NY, Vol. 22, Chapter 4, 65-82, 2011." class="coltj"> 27 , Int. J. Psychology Research , 10(4), 361-380, 2016." class="coltj"> 28 , etc.).

## 3. Mathematical Modelling in Classroom

A model is understood to be a simplified representation of a real system including only its characteristics which are related to a certain problem concerning the system ( assumed real system ); e.g. maximizing the system’s productivity, minimizing its functional costs, etc. The process of modelling is a fundamental principle of the systems’ theory, since the experimentation on the real system is usually difficult (and even impossible sometimes) requiring a lot of money and time. Modelling a system involves a deep abstracting process, which is graphically represented in Figure 1 29 .

- Figure 1 . Representation of the modelling process

There are several types of models to be used according to the form of the system and the corresponding problem to be solved. In simple cases iconic models may be used, like maps, bas-relief representations, etc. Analogical models, such as graphs, diagrams, etc., are frequently used when the corresponding problem concerns the study of the relationship between two (only) of the system’s variables; e.g. speed and time, temperature and pressure, etc. The mathematical or symbolic models use mathematical symbols and representations (functions, equations, inequalities, etc.) to describe the system’s behavior. This is the most important type of models, because they provide accurate and general (holding even if the system’s parameters are changed) solutions to the corresponding problems. In case of complex systems, however, like the biological ones, where the solution cannot be expressed in solvable mathematical terms or the mathematical solution requires laborious calculations, simulation models can be used. These models mimic the system’s behavior over a period of time with the help of a well organized set of logical orders, usually expressed in the form of a computer program. Also, heuristic models are used for improving already existing solutions, obtained either empirically or by using other types of models.

Mathematical modelling (MM) until the middle of the 1970’s used to be a tool mainly in hands of the scientists for solving problems related to their disciplines. The failure of the introduction of the “new mathematics” to school education, however, turned the attention of the specialists to PS activities as a more effective way for teaching and learning mathematics. MM in particular, has been widely used for connecting mathematics to everyday life situations, on the purpose of increasing the student interest on the subject.

One of the first who proposed the use of MM as a tool for teaching mathematics was H. O. Pollak 30 , who presented during the ICME-3 Conference in Karlsruhe (1976) the scheme of Figure 2 , known as the circle of modelling. In this scheme, given a problem of the everyday life or of a scientific topic different from mathematics (other world) for solution, the solver, following the direction of the arrows, is transferred to the “universe” of mathematics. There, the solver uses or creates suitable mathematics for the solution of the problem and then returns to the other world to check the validity of the mathematical solution obtained. If the verification of the solution is proved to be non-compatible to the existing real conditions, the same circle is repeated one or more times.

- Figure 2. The Pollak’s Circle of Modelling

Following the Pollak’s presentation, much effort has been placed by mathematics education researchers to study and analyze in detail the process of MM on the purpose of using it for teaching mathematics. Several models have been developed towards this direction, a brief but comprehensive account of which can be found in 31 , including a model of the present author ( Figure 3 ).

In fact, Voskoglou 32 described the MM process in terms of a Markov chain introduced on its main steps, which are: S 1 = Analysis of the problem, S 2 = Mathematization (formulation and construction of the model), S 3 = Solution of the model, S 4 = Validation of the solution and S 5 = Implementation of the solution to the real system. When the MM process is completed at step S 5 , it is assumed that a new problem is given to the class, which implies that the process restarts again from step S 1 .

- Figure 3. Flow-diagram of Voskoglou’s model for the MM process

Mathematization is the step of the MM process with the greatest gravity, since it involves a deep abstracting process, which is not always easy to be achieved by a non-expert. A solver who has obtained a mathematical solution of the model is normally expected to be able to “translate” it in terms of the corresponding real situation and to check its validity. There are, however, sometimes MM problems in which the validation of the model and/or the implementation of the final mathematical results to the real system hide surprises, which force solvers to “look back” to the construction of the model and make the necessary changes to it. The following example illustrates this situation:

Problem : We want to construct a channel to run water by folding the two edges of a rectangle metallic leaf having sides of length and , in such a way that they will be perpendicular to the other parts of the leaf. Assuming that the flow of water is constant, how we can run the maximum possible quantity of the water through the channel?

Solution : Folding the two edges of the metallic leaf by length x across its longer side the vertical cut of the constructed channel forms a rectangle with sides x and 32-2x ( Figure 4 ).

- Figure 4 . The vertical cut of the channel

Models like Voskoglou’s in Figure 3 are useful for describing the solvers’ ideal behavior when tackling MM problems. More recent researches Modelling and Mathematics Education: Applications in Science and Technology (ICTMA 9) , Matos, J.F. et al. (Eds.), Horwood Publishing: Chichester, 300-310, 2001." class="coltj"> 33 , Mathematical Modelling: Education, Engineering and Economics (ICTMA 12) , Chaines, C.; Galbraith, P.; Blum, W.; Khan, S. (Eds.), Horwood Publishing: Chichester, 260-270, 2007." class="coltj"> 34 , Modelling and Applications in Mathematics Education , Blum, W. et al. (Eds.), Springer: NY, 69-78, 2007." class="coltj"> 35 , however, report that the reality is not like that. In fact, modellers follow individual routes related to their learning styles and the level of their cognition. Consequently, from the teachers’ part there exists an uncertainty about the student way of thinking at each step of the MM process. Those findings inspired the present author to use principles of Fuzzy Logic for describing in a more realistic way the process of MM in the classroom on the purpose of understanding, and therefore treating better, the student reactions during the MM process 36 . The steps of the MM process in this model are represented as fuzzy sets on a set of linguistic labels characterizing the student performance in each step.

A complete methodology for teaching mathematics on the basis of MM has been eventually developed, which is usually referred as the application-oriented teaching of mathematics 37 . However, as the present author underlines in 38 , presenting also a representative example, teachers must be careful, because the extensive use of the application-oriented teaching as a general method for teaching mathematics could lead to far-fetched situations, in which more attention is given to the choice of the applications rather, than to the mathematical content!

More details about MM from the viewpoint of Education and representative examples can be found in earlier works of the author ( Quaderni di Ricerca in Didactica , 16, 53-60, 2006." class="coltj"> 39 , Journal of Research in Innovative Teaching , 8(1), 35-50, 2015." class="coltj"> 40 , etc.).

In the meantime, many textbooks devoted to applications of mathematical modelling in different areas of science and engineering have been published ( Guide to Mathematical Modelling . Macmillan: London, UK, Second Edition, 2001." class="coltj"> 41 , Principles of Mathematical Modelling: Ideas, Methods, Examples . CRC Press (Taylor & Francis Group): Boca Raton, FL 2001." class="coltj"> 42 , etc.). The general ideas about the MM process addressed in these books are very close to what it has been exposed above about MM in Education (circle of MM etc.), the main differences being in the methodology followed for each type of mathematical models discussed in them, from the elementary ones to those used for modelling complex objects.

## 4. Recent Advances on Problem Solving and Mathematical Modelling

According to new research from the London School of Economics and Political Science 43 , a mathematical theory of human behavior usually referred as Dual Inheritance Theory or Culture-Gene Coevolution , may re-organize the social sciences in the same way that Darwin’s Theory of Evolution re-organized the biological sciences. This theory, drawing together disconnected areas of research such as medicine, sociology, history and law, extends mathematical models of population biology and epidemiology to the social sciences and looks at how changes in our genes and changes in culture can interact.

In a recent paper presented in the ICME -13 Conference Leikin 44 describes neurocognitive studies that focus on mathematical processing and demonstrates that both mathematics education research and neuroscience research can derive from the integration of these two areas of research. In particular, mathematics education can contribute to the stages of research design, while neuroscience can validate theories in mathematics education and advance the interpretation of the research results. To make such an integration successful, Leikin notes that collaboration between mathematics educators and neuroscientists is crucial.

A recent study of the Russian researchers E. Kuznetsova and M. Matytcina 45 reveals specific ways of teaching mathematics to university students more effectively through the unity of social, psychological and pedagogical aspects. Many of the papers included in the list of the references of 45 could be also consulted by those willing to approach the subject of our discussion from the social and psychological point of view. Recent advances in mathematical PS as they were presented in research reports from the annual international conferences for the psychology of Mathematics Education are also studied in 46 .

Other recent works include studies on PS ability and creativity among the Higher Secondary students 47 , examples of PS strategies supporting the sustainability of 21 st Century skills 48 , etc.

## 5. Conclusions

The discussion performed in this study leads to the following conclusions:

• The failure of the introduction of the “new mathematics” in school education turned, from the late 1970’s, the attention of the specialists in Mathematics Education to the use of the problem as a tool for teaching and learning mathematics more effectively, with two components: Mathematical PS and MM.

• Polya, who proposed the use of the heuristic strategies, is the pioneer of the theoretical development of PS. The research on the subject was based for many years on his ideas, focusing mainly to the description of the PS process and the detailed analysis of its steps. More recent studies, however, have turned the attention mainly to the solvers’ behavior and required attributes during the PS process, which depend upon their personal style, their cognitive level and their subjective values and beliefs.

• PS is a complex cognitive action that needs the use of a variety of modes of thinking, according to the form of each problem, in order to be successful. Those modes include critical and statistical thinking, computational thinking and analogical reasoning.

• MM is a special type of PS concerning the solution of problems related to scientific applications or to everyday life situations. An integrated didactic approach has been eventually developed based on MM and termed as the application-oriented teaching of mathematics.

• Markov chain and Fuzzy Logic models have been developed in earlier works of the present author for a more effective description of the processes of PS and MM and for evaluating the corresponding student skills. The former type of models describes the ideal student behavior in the classroom, whereas the latter type attempts the description of their real behavior, which differs from student to student.

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## Problem Solving Methods for Mathematical Modelling

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Part of the International Perspectives on the Teaching and Learning of Mathematical Modelling book series (IPTL)

Providing a method for problem solving can support students working on modelling tasks. A few candidate methods are presented here. In a qualitative study, one of these problem solving methods was introduced to students in grades 4 and 6 (Germany), to be used in their work on modelling tasks. The students were observed as they worked and were subsequently interviewed. The results reveal differences between grades, and widely varying problem solving processes. The differences in written final solutions are considerably less pronounced.

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The DISUM Project (Didactical intervention modes for mathematics teaching oriented towards self-regulation and directed by tasks) led by W. Blum, R. Messner and R. Pekrun.

KOMMA (KOMpendium MAthematik) is a project aiming at the implementation and evaluation of a learning environment for mathematical modelling.

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Greefrath, G. (2015). Problem Solving Methods for Mathematical Modelling. In: Stillman, G., Blum, W., Salett Biembengut, M. (eds) Mathematical Modelling in Education Research and Practice. International Perspectives on the Teaching and Learning of Mathematical Modelling. Springer, Cham. https://doi.org/10.1007/978-3-319-18272-8_13

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