mathematical problem solving and modelling

Mathematical modeling and problem solving: from fundamentals to applications

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• Published: 15 March 2024

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• Masahito Ohue 1 ,
• Kotoyu Sasayama 2 &
• Masami Takata 3  

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The rapidly advancing fields of machine learning and mathematical modeling, greatly enhanced by the recent growth in artificial intelligence, are the focus of this special issue. This issue compiles extensively revised and improved versions of the top papers from the workshop on Mathematical Modeling and Problem Solving at PDPTA'23, the 29th International Conference on Parallel and Distributed Processing Techniques and Applications. Covering fundamental research in matrix operations and heuristic searches to real-world applications in computer vision and drug discovery, the issue underscores the crucial role of supercomputing and parallel and distributed computing infrastructure in research. Featuring nine key studies, this issue pushes forward computational technologies in mathematical modeling, refines techniques for analyzing images and time-series data, and introduces new methods in pharmaceutical and materials science, making significant contributions to these areas.

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The field of machine learning and mathematical modeling is rapidly evolving, significantly impacting diverse research areas. The recent surge in artificial intelligence technologies has further accelerated this trend, highlighting the growing importance of “mathematical modeling and problem solving” in scientific endeavors [ 1 ]. Modeling natural phenomena and engineering systems not only deepens our understanding of fundamental principles but also drives the development of innovative technologies for effective control. These advancements have considerable implications for both industry and academia.

This special issue showcases the latest advancements in mathematical modeling and problem solving across various disciplines. The scope of topics is wide, encompassing everything from foundational research in new matrix operation methods, heuristic search, and constrained optimization techniques to practical research in computer vision, drug discovery, materials science, financial engineering, and mechanical processes.

A key aspect of contemporary mathematical modeling research is its integration with supercomputing, which involves extensive parallel and distributed computing. The sheer volume and augmented data often require rapid computational strategies. The infrastructure, including hardware and software, supporting parallel and distributed computing is thus vital for applied research. This issue includes a selection of research presented at the “Mathematical Modeling and Problem Solving” workshop during the 29th International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA’23). After a thorough selection process, nine significant studies were chosen as articles on this issue.

Four papers focus on computational technologies foundational to mathematical modeling. Chiyonobu and colleagues enhance the two-sided Jacobi method for singular value decomposition for complex matrices, previously effective only for real matrices [ 2 ]. They incorporate QR decomposition for complex matrix scenarios, offering two distinct implementations for both complex and real matrices. Zhong et al. introduce a novel hyper-heuristic algorithm, the evolutionary multi-mode slime mold optimization (EMSMO), inspired by slime mold behaviors [ 3 ]. This algorithm demonstrates superior performance in benchmarks and engineering problems, outperforming traditional evolutionary and hyper-heuristic algorithms. Zhang et al. unveil the meta-generative data augmentation optimization (MGDAO), a method that advances data augmentation in foundational machine learning for image and natural language processing [ 4 ]. This technique surpasses standard auto-augmentation methods in few-shot image and text classification benchmarks. Matsuzaki and colleagues propose a mixed-integer programming (MIP)-based method for scheduling machining operations in automated manufacturing, considering worker conditions [ 5 ]. They validate this method through computer experiments modeled on real-world machining tasks.

Two papers address applications involving image and time-series data, traditional targets of mathematical modeling. Ishikawa et al. enhance concrete crack detection by using strongly blurred images in training data, improving recognizer accuracy [ 6 ]. Takata et al. develop a method for recommending stock combinations by analyzing price change waveforms, showing potential for diversifying portfolios and minimizing risks [ 7 ].

Last but not least, three papers focus on pharmaceutical and materials science applications. Ueki and Ohue assess AlphaFold2 and binder hallucination techniques for improving antibody binding affinity, indicating a more efficient method than traditional experimental approaches [ 8 ]. Morikawa et al. introduce a machine learning method using graph kernels for predicting metal–organic frameworks (MOFs) combinations, demonstrating accurate MOF structure prediction without physical synthesis [ 9 ]. Furui and Ohue present an enhanced version of the lead optimization mapper (Lomap) algorithm for drug discovery [ 10 ]. This improved algorithm offers a faster approach to create free energy perturbation (FEP) graphs for numerous compounds, while maintaining the quality of the output.

In summary, this special issue represents a significant contribution to the fields of mathematical modeling and application, providing innovative methods to the community. As editors, we extend our gratitude to all researchers who contributed to this collection, paving the way for the next era of mathematical modeling and problem solving.

Yüksel N, Börklü HR, Sezer HK, Canyurt OE (2023) Review of artificial intelligence applications in engineering design perspective. Eng Appl Artif Intell 118:105697

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Chiyonobu M, Miyamae T, Takata M, Harayama J, Kimura K, Nakamura Y (2024) Singular value decomposition for complex matrices using two-sided Jacobi method. J Supercomput. https://doi.org/10.1007/s11227-024-05903-6

Zhong R, Zhang E, Munetomo M (2024) Evolutionary multi-mode slime mold optimization: a hyper-heuristic algorithm inspired by slime mold foraging behaviors. J Supercomput. https://doi.org/10.1007/s11227-024-05909-0

Zhang E, Dong B, Wahib M, Zhong R, Munetomo M (2024) Meta generative image and text data augmentation optimization. J Supercomput. https://doi.org/10.1007/s11227-024-05912-5

Matsuzaki J, Sakakibara K, Nakamura M, Watanabe S (2024) Large neighborhood local search method with MIP techniques for large-scale machining scheduling with many constraints. J Supercomput. https://doi.org/10.1007/s11227-024-05912-5

Ishikawa S, Chiyonobu M, Iida S, Takata M (2024) Improvement of recognition rate using data augmentation with blurred images. J Supercomput. https://doi.org/10.1007/s11227-024-05901-8

Takata M, Kidoguchi N, Chiyonobu M (2024) Stock recommendation methods for stability. J Supercomput. https://doi.org/10.1007/s11227-024-05902-7

Ueki T, Ohue M (2024) Antibody complementarity-determining region design using AlphaFold2 and DDG Predictor. J Supercomput. https://doi.org/10.1007/s11227-023-05887-9

Morikawa Y, Shin K, Ohshima H, Kubouchi M (2024) Prediction of specific surface area of metal–organic frameworks by graph kernels. J Supercomput. https://doi.org/10.1007/s11227-024-05914-3

Furui K, Ohue M (2024) FastLomap: faster lead optimization mapper algorithm for large-scale relative free energy perturbation. J Supercomput

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This study was partially supported by JSPS KAKENHI (23H04887) (M.O.).

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Introduction to Mathematical Modelling

Modelling Process

Modelling process #.

The modelling process provides a systematic approach to solving complex problems:

Clearly state the problem

Identify variables and parameters

Make assumptions and identify constraints

Build solutions

Analyze and assess

Report results

Check out Math Modeling: Getting Started and Getting Solutions to read more about the mathematical modelling process.

Problem Statement #

Modelling problems are open-ended: there are many different solutions, different levels of complexity, and different tools that can be applied. It’s a challenge even to know where to start! To begin the modelling process, we need to clearly state the problem so that we know what we are trying to solve.

Variables and Parameters #

Independent variables are quantities that are input into the system and dependent variables are the quantities that are output from the system and that we are trying to predict. We should be able to indentify the varaibles from a clearly articulated problem statement. Parameters are quantities that appear in the relationships between variables. We must list all variables and parameters in the system, give each a name and symbol and identify their dimensions such as length, mass and time.

Assumptions and Constraints #

Assumptions reduce the complexity of the model and also help define relationships between variables and parameters in the system. For example, we often assume that the force of gravity is constant for an object moving near the surface of the Earth. However we would not assume that the gravitational forces of celestial bodies are always constant. Constraints describe the values that our variables and parameters are allowed to take. For example, the mass of an object is always positive.

Build Solutions #

Once we have a clear problem statement and lists of variables, parameters, assumptions and constraints, then we need to decide what mathematical tools to use to construct the model. It should be clear from the context if our model is deterministic, stochastic, data-driven or perhaps a combination. Once we have decided on the kind of model to use, we apply all the tools available.

Analyze and Assess #

Just because we can find a solution, does not mean that this is a meaningful result. We need to interpret the solution to see if it makes sense given the context of the problem. We will want to ask ourselves:

Does the solution make sense in the context of the problem?

Does the solution answer our problem statement?

Are the results obtained reasonable and practical?

If it does not make sense, then we need to critically analyze the process. Perhaps there is an algebraic error in the solution, perhaps an input is incorrect, perhaps we made an incorrect assumption. Analyzing and assessing the solution can be a difficult and tedius process.

Report Results #

The last step is to then share the results of our modelling efforts. We need to construct a clear and concise report of the model and how we implemented the model in our work. This report is how we share our findings with our research community and make contributions to the research area.

• Our Mission

Using Mathematical Modeling to Get Real With Students

Unlike canned word problems, mathematical modeling plunges students into the messy complexities of real-world problem solving.  

How do you bring math to life for kids? Illustrating the boundless possibilities of mathematics can be difficult if students are only asked to examine hypothetical situations like divvying up a dessert equally or determining how many apples are left after sharing with friends, writes third- and fourth- grade teacher Matthew Kandel for Mathematics Teacher: Learning and Teaching PK-12 .

In the early years of instruction, it’s not uncommon for students to think they’re learning math for the sole purpose of being able to solve word problems or help fictional characters troubleshoot issues in their imaginary lives, Kandel says. “A word problem is a one-dimensional world,” he writes. “Everything is distilled down to the quantities of interest. To solve a word problem, students can pick out the numbers and decide on an operation.” 

But through the use of mathematical modeling, students are plucked out of the hypothetical realm and plunged into the complexities of reality—presented with opportunities to help solve real-world problems with many variables by generating questions, making assumptions, learning and applying new skills, and ultimately arriving at an answer.

In Kandel’s classroom, this work begins with breaking students into small groups, providing them with an unsharpened pencil and a simple, guiding question: “How many times can a pencil be sharpened before it is too small to use?”

Setting the Stage for Inquiry 

The process of tackling the pencil question is not unlike the scientific method. After defining a question to investigate, students begin to wonder and hypothesize—what information do we need to know?—in order to identify a course of action. This step is unique to mathematical modeling: Whereas a word problem is formulaic, leading students down a pre-existing path toward a solution, a modeling task is “free-range,” empowering students to use their individual perspectives to guide them as they progress through their investigation, Kandel says. 

Modeling problems also have a number of variables, and students themselves have the agency to determine what to ignore and what to focus their attention on. 

After inter-group discussions, students in Kandel’s classroom came to the conclusion that they’d need answers to a host of other questions to proceed with answering their initial inquiry: 

• How much does the pencil sharpener remove? 
• What is the length of a brand new, unsharpened pencil? 
• Does the pencil sharpener remove the same amount of pencil each time it is used?

Introducing New Skills in Context

Once students have determined the first mathematical question they’d like to tackle (does the pencil sharpener remove the same amount of pencil each time it is used?), they are met with a roadblock. How were they to measure the pencil if the length did not fall conveniently on an inch or half inch? Kandel took the opportunity to introduce a new target skill which the class could begin using immediately: measuring to the nearest quarter inch. 

“One group of students was not satisfied with the precision of measuring to the nearest quarter inch and asked to learn how to measure to the nearest eighth of an inch,” Kandel explains. “The attention and motivation exhibited by students is unrivaled by the traditional class in which the skill comes first, the problem second.” 

Students reached a consensus and settled on taking six measurements total: the initial length of the new, unsharpened pencil as well as the lengths of the pencil after each of five sharpenings. To ensure all students can practice their newly acquired skill, Kandel tells the class that “all group members must share responsibility, taking turns measuring and checking the measurements of others.” 

Next, each group created a simple chart to record their measurements, then plotted their data as a line graph—though exploring other data visualization techniques or engaging students in alternative followup activities would work as well.

“We paused for a quick lesson on the number line and the introduction of a new term—mixed numbers,” Kandel explains. “Armed with this new information, students had no trouble marking their y-axis in half- or quarter-inch increments.” 

Sparking Mathematical Discussions

Mathematical modeling presents a multitude of opportunities for class-wide or small-group discussions, some which evolve into debates in which students state their hypotheses, then subsequently continue working to confirm or refute them. 

Kandel’s students, for example, had a wide range of opinions when it came to answering the question of how small of a pencil would be deemed unusable. Eventually, the class agreed that once a pencil reached 1 ¼ inch, it could no longer be sharpened—though some students said they would be able to still write with it. 

“This discussion helped us better understand what it means to make an assumption and how our assumptions affected our mathematical outcomes,” Kandel writes. Students then indicated the minimum size with a horizontal line across their respective graphs. 

Many students independently recognized the final step of extending their line while looking at their graphs. With each of the six points representing their measurements, the points descended downward toward the newly added horizontal “line of inoperability.” 

With mathematical modeling, Kandel says, there are no right answers, only models that are “more or less closely aligned with real-world observations.” Each group of students may come to a different conclusion, which can lead to a larger class discussion about accuracy. To prove their group had the most accurate conclusion, students needed to compare and contrast their methods as well as defend their final result. 

Developing Your Own Mathematical Models

The pencil problem is a great starting point for introducing mathematical modeling and free-range problem solving to your students, but you can customize based on what you have available and the particular needs of each group of students.

Depending on the type of pencil sharpener you have, for example, students can determine what constitutes a “fair test” and set the terms of their own inquiry. 

Additionally, Kandel suggests putting scaffolds in place to allow students who are struggling with certain elements to participate: Simplified rulers can be provided for students who need accommodations; charts can be provided for students who struggle with data collection; graphs with prelabeled x- and y-axes can be prepared in advance.

.css-1sk4066:hover{background:#d1ecfa;} 7 Real-World Math Strategies

Students can also explore completely different free-range problem solving and real world applications for math . At North Agincourt Jr. Public School in Scarborough, Canada, kids in grades 1-6 learn to conduct water audits. By adding, subtracting, finding averages, and measuring liquids—like the flow rate of all the water foundations, toilets, and urinals—students measure the amount of water used in their school or home in a single day. 

Or you can ask older students to bring in common household items—anything from a measuring cup to a recipe card—and identify three ways the item relates to math. At Woodrow Petty Elementary School in Taft, Texas, fifth-grade students display their chosen objects on the class’s “real-world math wall.” Even acting out restaurant scenarios can provide students with an opportunity to reinforce critical mathematical skills like addition and subtraction, while bolstering an understanding of decimals and percentages. At Suzhou Singapore International School in China, third- to fifth- graders role play with menus, ordering fictional meals and learning how to split the check when the bill arrives. 

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Unit 12: Modeling

About this unit.

Let's dive even deeper into the world of modeling. We'll take our knowledge about all the different function types we were exposed to so far, and use it to model and analyze various phenomena, from heart rates to business profits.

Modeling with function combination

• Modeling with function combination (Opens a modal)
• Model with function combination Get 3 of 4 questions to level up!

Interpreting features of functions

• Periodicity of algebraic models (Opens a modal)
• End behavior of algebraic models (Opens a modal)
• Symmetry of algebraic models (Opens a modal)
• Periodicity of algebraic models Get 3 of 4 questions to level up!
• End behavior of algebraic models Get 3 of 4 questions to level up!

Manipulating formulas

• Manipulating formulas: perimeter (Opens a modal)
• Manipulating formulas: area (Opens a modal)
• Manipulating formulas: temperature (Opens a modal)
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Modeling with two variables

• Graph labels and scales (Opens a modal)
• Rational equation word problem (Opens a modal)
• Quadratic inequality word problem (Opens a modal)
• Exponential equation word problem (Opens a modal)
• Graph labels and scales Get 3 of 4 questions to level up!
• Equations & inequalities word problems Get 3 of 4 questions to level up!

Modeling with multiple variables

• Modeling with multiple variables: Pancakes (Opens a modal)
• Modeling with multiple variables: Roller coaster (Opens a modal)
• Modeling with multiple variables: Taco stand (Opens a modal)
• Modeling with multiple variables: Ice cream (Opens a modal)
• Interpreting expressions with multiple variables: Resistors (Opens a modal)
• Interpreting expressions with multiple variables: Cylinder (Opens a modal)
• Modeling: FAQ (Opens a modal)
• Modeling with multiple variables Get 3 of 4 questions to level up!
• Interpreting expressions with multiple variables Get 3 of 4 questions to level up!

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Original research article, mathematical problem-solving through cooperative learning—the importance of peer acceptance and friendships.

• 1 Department of Education, Uppsala University, Uppsala, Sweden
• 2 Department of Education, Culture and Communication, Malardalen University, Vasteras, Sweden
• 3 School of Natural Sciences, Technology and Environmental Studies, Sodertorn University, Huddinge, Sweden
• 4 Faculty of Education, Gothenburg University, Gothenburg, Sweden

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students’ mathematical problem-solving in heterogeneous classrooms in grade five, in which students with special needs are educated alongside with their peers. The intervention combined a cooperative learning approach with instruction in problem-solving strategies including mathematical models of multiplication/division, proportionality, and geometry. The teachers in the experimental group received training in cooperative learning and mathematical problem-solving, and implemented the intervention for 15 weeks. The teachers in the control group received training in mathematical problem-solving and provided instruction as they would usually. Students (269 in the intervention and 312 in the control group) participated in tests of mathematical problem-solving in the areas of multiplication/division, proportionality, and geometry before and after the intervention. The results revealed significant effects of the intervention on student performance in overall problem-solving and problem-solving in geometry. The students who received higher scores on social acceptance and friendships for the pre-test also received higher scores on the selected tests of mathematical problem-solving. Thus, the cooperative learning approach may lead to gains in mathematical problem-solving in heterogeneous classrooms, but social acceptance and friendships may also greatly impact students’ results.

Introduction

The research on instruction in mathematical problem-solving has progressed considerably during recent decades. Yet, there is still a need to advance our knowledge on how teachers can support their students in carrying out this complex activity ( Lester and Cai, 2016 ). Results from the Program for International Student Assessment (PISA) show that only 53% of students from the participating countries could solve problems requiring more than direct inference and using representations from different information sources ( OECD, 2019 ). In addition, OECD (2019) reported a large variation in achievement with regard to students’ diverse backgrounds. Thus, there is a need for instructional approaches to promote students’ problem-solving in mathematics, especially in heterogeneous classrooms in which students with diverse backgrounds and needs are educated together. Small group instructional approaches have been suggested as important to promote learning of low-achieving students and students with special needs ( Kunsch et al., 2007 ). One such approach is cooperative learning (CL), which involves structured collaboration in heterogeneous groups, guided by five principles to enhance group cohesion ( Johnson et al., 1993 ; Johnson et al., 2009 ; Gillies, 2016 ). While CL has been well-researched in whole classroom approaches ( Capar and Tarim, 2015 ), few studies of the approach exist with regard to students with special educational needs (SEN; McMaster and Fuchs, 2002 ). This study contributes to previous research by studying the effects of the CL approach on students’ mathematical problem-solving in heterogeneous classrooms, in which students with special needs are educated alongside with their peers.

Group collaboration through the CL approach is structured in accordance with five principles of collaboration: positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing ( Johnson et al., 1993 ). First, the group tasks need to be structured so that all group members feel dependent on each other in the completion of the task, thus promoting positive interdependence. Second, for individual accountability, the teacher needs to assure that each group member feels responsible for his or her share of work, by providing opportunities for individual reports or evaluations. Third, the students need explicit instruction in social skills that are necessary for collaboration. Fourth, the tasks and seat arrangements should be designed to promote interaction among group members. Fifth, time needs to be allocated to group processing, through which group members can evaluate their collaborative work to plan future actions. Using these principles for cooperation leads to gains in mathematics, according to Capar and Tarim (2015) , who conducted a meta-analysis on studies of cooperative learning and mathematics, and found an increase of .59 on students’ mathematics achievement scores in general. However, the number of reviewed studies was limited, and researchers suggested a need for more research. In the current study, we focused on the effect of CL approach in a specific area of mathematics: problem-solving.

Mathematical problem-solving is a central area of mathematics instruction, constituting an important part of preparing students to function in modern society ( Gravemeijer et al., 2017 ). In fact, problem-solving instruction creates opportunities for students to apply their knowledge of mathematical concepts, integrate and connect isolated pieces of mathematical knowledge, and attain a deeper conceptual understanding of mathematics as a subject ( Lester and Cai, 2016 ). Some researchers suggest that mathematics itself is a science of problem-solving and of developing theories and methods for problem-solving ( Hamilton, 2007 ; Davydov, 2008 ).

Problem-solving processes have been studied from different perspectives ( Lesh and Zawojewski, 2007 ). Problem-solving heuristics Pólya, (1948) has largely influenced our perceptions of problem-solving, including four principles: understanding the problem, devising a plan, carrying out the plan, and looking back and reflecting upon the suggested solution. Schoenfield, (2016) suggested the use of specific problem-solving strategies for different types of problems, which take into consideration metacognitive processes and students’ beliefs about problem-solving. Further, models and modelling perspectives on mathematics ( Lesh and Doerr, 2003 ; Lesh and Zawojewski, 2007 ) emphasize the importance of engaging students in model-eliciting activities in which problem situations are interpreted mathematically, as students make connections between problem information and knowledge of mathematical operations, patterns, and rules ( Mousoulides et al., 2010 ; Stohlmann and Albarracín, 2016 ).

Not all students, however, find it easy to solve complex mathematical problems. Students may experience difficulties in identifying solution-relevant elements in a problem or visualizing appropriate solution to a problem situation. Furthermore, students may need help recognizing the underlying model in problems. For example, in two studies by Degrande et al. (2016) , students in grades four to six were presented with mathematical problems in the context of proportional reasoning. The authors found that the students, when presented with a word problem, could not identify an underlying model, but rather focused on superficial characteristics of the problem. Although the students in the study showed more success when presented with a problem formulated in symbols, the authors pointed out a need for activities that help students distinguish between different proportional problem types. Furthermore, students exhibiting specific learning difficulties may need additional support in both general problem-solving strategies ( Lein et al., 2020 ; Montague et al., 2014 ) and specific strategies pertaining to underlying models in problems. The CL intervention in the present study focused on supporting students in problem-solving, through instruction in problem-solving principles ( Pólya, 1948 ), specifically applied to three models of mathematical problem-solving—multiplication/division, geometry, and proportionality.

Students’ problem-solving may be enhanced through participation in small group discussions. In a small group setting, all the students have the opportunity to explain their solutions, clarify their thinking, and enhance understanding of a problem at hand ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ). In fact, small group instruction promotes students’ learning in mathematics by providing students with opportunities to use language for reasoning and conceptual understanding ( Mercer and Sams, 2006 ), to exchange different representations of the problem at hand ( Fujita et al., 2019 ), and to become aware of and understand groupmates’ perspectives in thinking ( Kazak et al., 2015 ). These opportunities for learning are created through dialogic spaces characterized by openness to each other’s perspectives and solutions to mathematical problems ( Wegerif, 2011 ).

However, group collaboration is not only associated with positive experiences. In fact, studies show that some students may not be given equal opportunities to voice their opinions, due to academic status differences ( Langer-Osuna, 2016 ). Indeed, problem-solvers struggling with complex tasks may experience negative emotions, leading to uncertainty of not knowing the definite answer, which places demands on peer support ( Jordan and McDaniel, 2014 ; Hannula, 2015 ). Thus, especially in heterogeneous groups, students may need additional support to promote group interaction. Therefore, in this study, we used a cooperative learning approach, which, in contrast to collaborative learning approaches, puts greater focus on supporting group cohesion through instruction in social skills and time for reflection on group work ( Davidson and Major, 2014 ).

Although cooperative learning approach is intended to promote cohesion and peer acceptance in heterogeneous groups ( Rzoska and Ward, 1991 ), previous studies indicate that challenges in group dynamics may lead to unequal participation ( Mulryan, 1992 ; Cohen, 1994 ). Peer-learning behaviours may impact students’ problem-solving ( Hwang and Hu, 2013 ) and working in groups with peers who are seen as friends may enhance students’ motivation to learn mathematics ( Deacon and Edwards, 2012 ). With the importance of peer support in mind, this study set out to investigate whether the results of the intervention using the CL approach are associated with students’ peer acceptance and friendships.

The Present Study

In previous research, the CL approach has shown to be a promising approach in teaching and learning mathematics ( Capar and Tarim, 2015 ), but fewer studies have been conducted in whole-class approaches in general and students with SEN in particular ( McMaster and Fuchs, 2002 ). This study aims to contribute to previous research by investigating the effect of CL intervention on students’ mathematical problem-solving in grade 5. With regard to the complexity of mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach in this study was combined with problem-solving principles pertaining to three underlying models of problem-solving—multiplication/division, geometry, and proportionality. Furthermore, considering the importance of peer support in problem-solving in small groups ( Mulryan, 1992 ; Cohen, 1994 ; Hwang and Hu, 2013 ), the study investigated how peer acceptance and friendships were associated with the effect of the CL approach on students’ problem-solving abilities. The study aimed to find answers to the following research questions:

a) What is the effect of CL approach on students’ problem-solving in mathematics?

b) Are social acceptance and friendship associated with the effect of CL on students’ problem-solving in mathematics?

Participants

The participants were 958 students in grade 5 and their teachers. According to power analyses prior to the start of the study, 1,020 students and 51 classes were required, with an expected effect size of 0.30 and power of 80%, provided that there are 20 students per class and intraclass correlation is 0.10. An invitation to participate in the project was sent to teachers in five municipalities via e-mail. Furthermore, the information was posted on the website of Uppsala university and distributed via Facebook interest groups. As shown in Figure 1 , teachers of 1,165 students agreed to participate in the study, but informed consent was obtained only for 958 students (463 in the intervention and 495 in the control group). Further attrition occurred at pre- and post-measurement, resulting in 581 students’ tests as a basis for analyses (269 in the intervention and 312 in the control group). Fewer students (n = 493) were finally included in the analyses of the association of students’ social acceptance and friendships and the effect of CL on students’ mathematical problem-solving (219 in the intervention and 274 in the control group). The reasons for attrition included teacher drop out due to sick leave or personal circumstances (two teachers in the control group and five teachers in the intervention group). Furthermore, some students were sick on the day of data collection and some teachers did not send the test results to the researchers.

FIGURE 1 . Flow chart for participants included in data collection and data analysis.

As seen in Table 1 , classes in both intervention and control groups included 27 students on average. For 75% of the classes, there were 33–36% of students with SEN. In Sweden, no formal medical diagnosis is required for the identification of students with SEN. It is teachers and school welfare teams who decide students’ need for extra adaptations or special support ( Swedish National Educational Agency, 2014 ). The information on individual students’ type of SEN could not be obtained due to regulations on the protection of information about individuals ( SFS 2009 ). Therefore, the information on the number of students with SEN on class level was obtained through teacher reports.

TABLE 1 . Background characteristics of classes and teachers in intervention and control groups.

Intervention

The intervention using the CL approach lasted for 15 weeks and the teachers worked with the CL approach three to four lessons per week. First, the teachers participated in two-days training on the CL approach, using an especially elaborated CL manual ( Klang et al., 2018 ). The training focused on the five principles of the CL approach (positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing). Following the training, the teachers introduced the CL approach in their classes and focused on group-building activities for 7 weeks. Then, 2 days of training were provided to teachers, in which the CL approach was embedded in activities in mathematical problem-solving and reading comprehension. Educational materials containing mathematical problems in the areas of multiplication and division, geometry, and proportionality were distributed to the teachers ( Karlsson and Kilborn, 2018a ). In addition to the specific problems, adapted for the CL approach, the educational materials contained guidance for the teachers, in which problem-solving principles ( Pólya, 1948 ) were presented as steps in problem-solving. Following the training, the teachers applied the CL approach in mathematical problem-solving lessons for 8 weeks.

Solving a problem is a matter of goal-oriented reasoning, starting from the understanding of the problem to devising its solution by using known mathematical models. This presupposes that the current problem is chosen from a known context ( Stillman et al., 2008 ; Zawojewski, 2010 ). This differs from the problem-solving of the textbooks, which is based on an aim to train already known formulas and procedures ( Hamilton, 2007 ). Moreover, it is important that students learn modelling according to their current abilities and conditions ( Russel, 1991 ).

In order to create similar conditions in the experiment group and the control group, the teachers were supposed to use the same educational material ( Karlsson and Kilborn, 2018a ; Karlsson and Kilborn, 2018b ), written in light of the specified view of problem-solving. The educational material is divided into three areas—multiplication/division, geometry, and proportionality—and begins with a short teachers’ guide, where a view of problem solving is presented, which is based on the work of Polya (1948) and Lester and Cai (2016) . The tasks are constructed in such a way that conceptual knowledge was in focus, not formulas and procedural knowledge.

Implementation of the Intervention

To ensure the implementation of the intervention, the researchers visited each teachers’ classroom twice during the two phases of the intervention period, as described above. During each visit, the researchers observed the lesson, using a checklist comprising the five principles of the CL approach. After the lesson, the researchers gave written and oral feedback to each teacher. As seen in Table 1 , in 18 of the 23 classes, the teachers implemented the intervention in accordance with the principles of CL. In addition, the teachers were asked to report on the use of the CL approach in their teaching and the use of problem-solving activities embedding CL during the intervention period. As shown in Table 1 , teachers in only 11 of 23 classes reported using the CL approach and problem-solving activities embedded in the CL approach at least once a week.

Control Group

The teachers in the control group received 2 days of instruction in enhancing students’ problem-solving and reading comprehension. The teachers were also supported with educational materials including mathematical problems Karlsson and Kilborn (2018b) and problem-solving principles ( Pólya, 1948 ). However, none of the activities during training or in educational materials included the CL approach. As seen in Table 1 , only 10 of 25 teachers reported devoting at least one lesson per week to mathematical problem-solving.

Tests of Mathematical Problem-Solving

Tests of mathematical problem-solving were administered before and after the intervention, which lasted for 15 weeks. The tests were focused on the models of multiplication/division, geometry, and proportionality. The three models were chosen based on the syllabus of the subject of mathematics in grades 4 to 6 in the Swedish National Curriculum ( Swedish National Educational Agency, 2018 ). In addition, the intention was to create a variation of types of problems to solve. For each of these three models, there were two tests, a pre-test and a post-test. Each test contained three tasks with increasing difficulty ( Supplementary Appendix SA ).

The tests of multiplication and division (Ma1) were chosen from different contexts and began with a one-step problem, while the following two tasks were multi-step problems. Concerning multiplication, many students in grade 5 still understand multiplication as repeated addition, causing significant problems, as this conception is not applicable to multiplication beyond natural numbers ( Verschaffel et al., 2007 ). This might be a hindrance in developing multiplicative reasoning ( Barmby et al., 2009 ). The multi-step problems in this study were constructed to support the students in multiplicative reasoning.

Concerning the geometry tests (Ma2), it was important to consider a paradigm shift concerning geometry in education that occurred in the mid-20th century, when strict Euclidean geometry gave way to other aspects of geometry like symmetry, transformation, and patterns. van Hiele (1986) prepared a new taxonomy for geometry in five steps, from a visual to a logical level. Therefore, in the tests there was a focus on properties of quadrangles and triangles, and how to determine areas by reorganising figures into new patterns. This means that structure was more important than formulas.

The construction of tests of proportionality (M3) was more complicated. Firstly, tasks on proportionality can be found in many different contexts, such as prescriptions, scales, speeds, discounts, interest, etc. Secondly, the mathematical model is complex and requires good knowledge of rational numbers and ratios ( Lesh et al., 1988 ). It also requires a developed view of multiplication, useful in operations with real numbers, not only as repeated addition, an operation limited to natural numbers ( Lybeck, 1981 ; Degrande et al., 2016 ). A linear structure of multiplication as repeated addition leads to limitations in terms of generalization and development of the concept of multiplication. This became evident in a study carried out in a Swedish context ( Karlsson and Kilborn, 2018c ). Proportionality can be expressed as a/b = c/d or as a/b = k. The latter can also be expressed as a = b∙k, where k is a constant that determines the relationship between a and b. Common examples of k are speed (km/h), scale, and interest (%). An important pre-knowledge in order to deal with proportions is to master fractions as equivalence classes like 1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 6/18 = 7/21 = 8/24 … ( Karlsson and Kilborn, 2020 ). It was important to take all these aspects into account when constructing and assessing the solutions of the tasks.

The tests were graded by an experienced teacher of mathematics (4 th author) and two students in their final year of teacher training. Prior to grading, acceptable levels of inter-rater reliability were achieved by independent rating of students’ solutions and discussions in which differences between the graders were resolved. Each student response was to be assigned one point when it contained a correct answer and two points when the student provided argumentation for the correct answer and elaborated on explanation of his or her solution. The assessment was thus based on quality aspects with a focus on conceptual knowledge. As each subtest contained three questions, it generated three student solutions. So, scores for each subtest ranged from 0 to 6 points and for the total scores from 0 to 18 points. To ascertain that pre- and post-tests were equivalent in degree of difficulty, the tests were administered to an additional sample of 169 students in grade 5. Test for each model was conducted separately, as students participated in pre- and post-test for each model during the same lesson. The order of tests was switched for half of the students in order to avoid the effect of the order in which the pre- and post-tests were presented. Correlation between students’ performance on pre- and post-test was .39 ( p < 0.000) for tests of multiplication/division; .48 ( p < 0.000) for tests of geometry; and .56 ( p < 0.000) for tests of proportionality. Thus, the degree of difficulty may have differed between pre- and post-test.

Measures of Peer Acceptance and Friendships

To investigate students’ peer acceptance and friendships, peer nominations rated pre- and post-intervention were used. Students were asked to nominate peers who they preferred to work in groups with and who they preferred to be friends with. Negative peer nominations were avoided due to ethical considerations raised by teachers and parents ( Child and Nind, 2013 ). Unlimited nominations were used, as these are considered to have high ecological validity ( Cillessen and Marks, 2017 ). Peer nominations were used as a measure of social acceptance, and reciprocated nominations were used as a measure of friendship. The number of nominations for each student were aggregated and divided by the number of nominators to create a proportion of nominations for each student ( Velásquez et al., 2013 ).

Statistical Analyses

Multilevel regression analyses were conducted in R, lme4 package Bates et al. (2015) to account for nestedness in the data. Students’ classroom belonging was considered as a level 2 variable. First, we used a model in which students’ results on tests of problem-solving were studied as a function of time (pre- and post) and group belonging (intervention and control group). Second, the same model was applied to subgroups of students who performed above and below median at pre-test, to explore whether the CL intervention had a differential effect on student performance. In this second model, the results for subgroups of students could not be obtained for geometry tests for subgroup below median and for tests of proportionality for subgroup above median. A possible reason for this must have been the skewed distribution of the students in these subgroups. Therefore, another model was applied that investigated students’ performances in math at both pre- and post-test as a function of group belonging. Third, the students’ scores on social acceptance and friendships were added as an interaction term to the first model. In our previous study, students’ social acceptance changed as a result of the same CL intervention ( Klang et al., 2020 ).

The assumptions for the multilevel regression were assured during the analyses ( Snijders and Bosker, 2012 ). The assumption of normality of residuals were met, as controlled by visual inspection of quantile-quantile plots. For subgroups, however, the plotted residuals deviated somewhat from the straight line. The number of outliers, which had a studentized residual value greater than ±3, varied from 0 to 5, but none of the outliers had a Cook’s distance value larger than 1. The assumption of multicollinearity was met, as the variance inflation factors (VIF) did not exceed a value of 10. Before the analyses, the cases with missing data were deleted listwise.

What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

As seen in the regression coefficients in Table 2 , the CL intervention had a significant effect on students’ mathematical problem-solving total scores and students’ scores in problem solving in geometry (Ma2). Judging by mean values, students in the intervention group appeared to have low scores on problem-solving in geometry but reached the levels of problem-solving of the control group by the end of the intervention. The intervention did not have a significant effect on students’ performance in problem-solving related to models of multiplication/division and proportionality.

TABLE 2 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving.

The question is, however, whether CL intervention affected students with different pre-test scores differently. Table 2 includes the regression coefficients for subgroups of students who performed below and above median at pre-test. As seen in the table, the CL approach did not have a significant effect on students’ problem-solving, when the sample was divided into these subgroups. A small negative effect was found for intervention group in comparison to control group, but confidence intervals (CI) for the effect indicate that it was not significant.

Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

As seen in Table 3 , students’ peer acceptance and friendship at pre-test were significantly associated with the effect of the CL approach on students’ mathematical problem-solving scores. Changes in students’ peer acceptance and friendships were not significantly associated with the effect of the CL approach on students’ mathematical problem-solving. Consequently, it can be concluded that being nominated by one’s peers and having friends at the start of the intervention may be an important factor when participation in group work, structured in accordance with the CL approach, leads to gains in mathematical problem-solving.

TABLE 3 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving, including scores of social acceptance and friendship in the model.

In light of the limited number of studies on the effects of CL on students’ problem-solving in whole classrooms ( Capar and Tarim, 2015 ), and for students with SEN in particular ( McMaster and Fuchs, 2002 ), this study sought to investigate whether the CL approach embedded in problem-solving activities has an effect on students’ problem-solving in heterogeneous classrooms. The need for the study was justified by the challenge of providing equitable mathematics instruction to heterogeneous student populations ( OECD, 2019 ). Small group instructional approaches as CL are considered as promising approaches in this regard ( Kunsch et al., 2007 ). The results showed a significant effect of the CL approach on students’ problem-solving in geometry and total problem-solving scores. In addition, with regard to the importance of peer support in problem-solving ( Deacon and Edwards, 2012 ; Hwang and Hu, 2013 ), the study explored whether the effect of CL on students’ problem-solving was associated with students’ social acceptance and friendships. The results showed that students’ peer acceptance and friendships at pre-test were significantly associated with the effect of the CL approach, while change in students’ peer acceptance and friendships from pre- to post-test was not.

The results of the study confirm previous research on the effect of the CL approach on students’ mathematical achievement ( Capar and Tarim, 2015 ). The specific contribution of the study is that it was conducted in classrooms, 75% of which were composed of 33–36% of students with SEN. Thus, while a previous review revealed inconclusive findings on the effects of CL on student achievement ( McMaster and Fuchs, 2002 ), the current study adds to the evidence of the effect of the CL approach in heterogeneous classrooms, in which students with special needs are educated alongside with their peers. In a small group setting, the students have opportunities to discuss their ideas of solutions to the problem at hand, providing explanations and clarifications, thus enhancing their understanding of problem-solving ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ).

In this study, in accordance with previous research on mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach was combined with training in problem-solving principles Pólya (1948) and educational materials, providing support in instruction in underlying mathematical models. The intention of the study was to provide evidence for the effectiveness of the CL approach above instruction in problem-solving, as problem-solving materials were accessible to teachers of both the intervention and control groups. However, due to implementation challenges, not all teachers in the intervention and control groups reported using educational materials and training as expected. Thus, it is not possible to draw conclusions of the effectiveness of the CL approach alone. However, in everyday classroom instruction it may be difficult to separate the content of instruction from the activities that are used to mediate this content ( Doerr and Tripp, 1999 ; Gravemeijer, 1999 ).

Furthermore, for successful instruction in mathematical problem-solving, scaffolding for content needs to be combined with scaffolding for dialogue ( Kazak et al., 2015 ). From a dialogical perspective ( Wegerif, 2011 ), students may need scaffolding in new ways of thinking, involving questioning their understandings and providing arguments for their solutions, in order to create dialogic spaces in which different solutions are voiced and negotiated. In this study, small group instruction through CL approach aimed to support discussions in small groups, but the study relies solely on quantitative measures of students’ mathematical performance. Video-recordings of students’ discussions may have yielded important insights into the dialogic relationships that arose in group discussions.

Despite the positive findings of the CL approach on students’ problem-solving, it is important to note that the intervention did not have an effect on students’ problem-solving pertaining to models of multiplication/division and proportionality. Although CL is assumed to be a promising instructional approach, the number of studies on its effect on students’ mathematical achievement is still limited ( Capar and Tarim, 2015 ). Thus, further research is needed on how CL intervention can be designed to promote students’ problem-solving in other areas of mathematics.

The results of this study show that the effect of the CL intervention on students’ problem-solving was associated with students’ initial scores of social acceptance and friendships. Thus, it is possible to assume that students who were popular among their classmates and had friends at the start of the intervention also made greater gains in mathematical problem-solving as a result of the CL intervention. This finding is in line with Deacon and Edwards’ study of the importance of friendships for students’ motivation to learn mathematics in small groups ( Deacon and Edwards, 2012 ). However, the effect of the CL intervention was not associated with change in students’ social acceptance and friendship scores. These results indicate that students who were nominated by a greater number of students and who received a greater number of friends did not benefit to a great extent from the CL intervention. With regard to previously reported inequalities in cooperation in heterogeneous groups ( Cohen, 1994 ; Mulryan, 1992 ; Langer Osuna, 2016 ) and the importance of peer behaviours for problem-solving ( Hwang and Hu, 2013 ), teachers should consider creating inclusive norms and supportive peer relationships when using the CL approach. The demands of solving complex problems may create negative emotions and uncertainty ( Hannula, 2015 ; Jordan and McDaniel, 2014 ), and peer support may be essential in such situations.

Limitations

The conclusions from the study must be interpreted with caution, due to a number of limitations. First, due to the regulation of protection of individuals ( SFS 2009 ), the researchers could not get information on type of SEN for individual students, which limited the possibilities of the study for investigating the effects of the CL approach for these students. Second, not all teachers in the intervention group implemented the CL approach embedded in problem-solving activities and not all teachers in the control group reported using educational materials on problem-solving. The insufficient levels of implementation pose a significant challenge to the internal validity of the study. Third, the additional investigation to explore the equivalence in difficulty between pre- and post-test, including 169 students, revealed weak to moderate correlation in students’ performance scores, which may indicate challenges to the internal validity of the study.

Implications

The results of the study have some implications for practice. Based on the results of the significant effect of the CL intervention on students’ problem-solving, the CL approach appears to be a promising instructional approach in promoting students’ problem-solving. However, as the results of the CL approach were not significant for all subtests of problem-solving, and due to insufficient levels of implementation, it is not possible to conclude on the importance of the CL intervention for students’ problem-solving. Furthermore, it appears to be important to create opportunities for peer contacts and friendships when the CL approach is used in mathematical problem-solving activities.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics Statement

The studies involving human participants were reviewed and approved by the Uppsala Ethical Regional Committee, Dnr. 2017/372. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

Author Contributions

NiK was responsible for the project, and participated in data collection and data analyses. NaK and WK were responsible for intervention with special focus on the educational materials and tests in mathematical problem-solving. PE participated in the planning of the study and the data analyses, including coordinating analyses of students’ tests. MK participated in the designing and planning the study as well as data collection and data analyses.

The project was funded by the Swedish Research Council under Grant 2016-04,679.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

We would like to express our gratitude to teachers who participated in the project.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2021.710296/full#supplementary-material

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Verschaffel, L., Greer, B., and De Corte, E. (2007). “Whole number concepts and operations,” in Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics . Editor F. K. Lester (Charlotte, NC: Information Age Pub ), 557–628.

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Wegerif, R. (2011). “Theories of Learning and Studies of Instructional Practice,” in Theories of learning and studies of instructional Practice. Explorations in the learning sciences, instructional systems and Performance technologies . Editor T. Koschmann (Berlin, Germany: Springer ). doi:10.1007/978-1-4419-7582-9

Yackel, E., Cobb, P., and Wood, T. (1991). Small-group interactions as a source of learning opportunities in second-grade mathematics. J. Res. Math. Edu. 22 (5), 390–408. doi:10.2307/749187

Zawojewski, J. (2010). Problem Solving versus Modeling. In R. Lesch, P. Galbraith, C. R. Haines, and A. Hurford (red.), Modelling student’s mathematical modelling competencies: ICTMA , p. 237–243. New York, NY: Springer .doi:10.1007/978-1-4419-0561-1_20

Keywords: cooperative learning, mathematical problem-solving, intervention, heterogeneous classrooms, hierarchical linear regression analysis

Citation: Klang N, Karlsson N, Kilborn W, Eriksson P and Karlberg M (2021) Mathematical Problem-Solving Through Cooperative Learning—The Importance of Peer Acceptance and Friendships. Front. Educ. 6:710296. doi: 10.3389/feduc.2021.710296

Received: 15 May 2021; Accepted: 09 August 2021; Published: 24 August 2021.

Reviewed by:

Copyright © 2021 Klang, Karlsson, Kilborn, Eriksson and Karlberg. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Nina Klang, [email protected]

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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. https://pubs.sciepub.com/education/9/2/6 ">Normal Style
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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.

Published with license by Science and Education Publishing, Copyright © 2021 Michael Gr. Voskoglou

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Softening the sharp edges in mathematics.

For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start — to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics — a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

“Mathematics should be flexible, conceptual, a place where we play with ideas and make connections," says Professor Jo Boaler. (Photo: Robert Houser Photography)

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced “How to Learn Math,” the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users. 

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer. 

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 — but the most common answer 13-year-olds gave was 19. The second most common was 21. 

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important? 

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mind-set. I think we all need a little more ish in our lives. 

You also argue that mathematics should be taught in more visual ways. What do you mean by that? 

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things — all of that contributes to our understanding of how it works. 

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking. 

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, ‘I’m in geometry class now, and I still remember that  sugar cube, what it looked like and felt like.’ His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing — elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

Click to enlarge: A depiction of various ways to calculate 38 x 5, numerically and visually. (Image: Courtesy of Jo Boaler)

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them. 

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

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Why math is important if you want to become a programmer

In the realm of programming, mathematics serves as both a cornerstone and a catalyst for innovation. At its core, programming is a discipline rooted in logical problem-solving, requiring practitioners to navigate complex algorithms and data structures with precision and efficiency. Thus, a solid understanding of mathematical principles is not only beneficial but often essential for success in the field.

At the heart of this digital frontier lies the art of problem solving in math , a skill as ancient as civilization itself yet as vital as ever in the age of technology. In this introductory exploration, we delve into the integral role that mathematics plays in programming, examining its influence on areas such as algorithmic design, computational thinking, and optimization strategies. By uncovering the interconnectedness of mathematics and programming, we gain insight into how mathematical proficiency empowers programmers to tackle challenges with clarity and creativity, ultimately driving forward the evolution of technology.

Table of Contents

Foundation of Logic and Problem-Solving

Mathematics serves as a crucible for the development of logical thinking skills, offering a structured framework through which individuals learn to analyze problems, identify patterns, and derive conclusions based on evidence and reasoning. From the rudimentary stages of arithmetic to the complexities of calculus, mathematical concepts encourage the cultivation of deductive and inductive reasoning abilities, fostering a mindset characterized by precision, clarity, and systematic problem-solving.

Problem-Solving in Mathematics and Programming

The problem-solving techniques inherent in mathematics bear a striking resemblance to those employed in programming. Both disciplines require individuals to break down complex problems into manageable components, formulate strategies to address them systematically, and iterate through potential solutions until an optimal outcome is achieved. Whether deciphering mathematical proofs or debugging lines of code, the analytical skills honed through mathematical practice provide programmers with a solid foundation upon which to build their coding expertise. Thus, the symbiotic relationship between mathematics and programming underscores the intrinsic value of mathematical proficiency in the pursuit of computational mastery.

Data Structures and Mathematics

Data structures serve as the backbone of programming, facilitating the organization, storage, and manipulation of information in a systematic manner. Common data structures include arrays, linked lists, stacks, queues, trees, and graphs, each tailored to specific computational tasks and requirements. Arrays provide a contiguous block of memory for storing elements of the same type, while linked lists offer flexibility through dynamic memory allocation. Stacks and queues adhere to the principles of last-in, first-out (LIFO) and first-in, first-out (FIFO) access, respectively, making them ideal for managing sequential data operations. Trees and graphs, on the other hand, enable hierarchical and non-linear representations of data, fostering diverse applications ranging from database management to network modeling.

Mathematical concepts such as sets, arrays, and graphs form the conceptual underpinnings of many data structures utilized in programming. Sets, characterized by their collection of distinct elements, find resonance in data structures such as hash tables and dictionaries, which leverage set theory principles to achieve efficient data retrieval and manipulation. Arrays, with their ordered collection of elements, serve as the foundation for array-based data structures like stacks and queues, where elements are accessed sequentially. Graphs, a fundamental mathematical abstraction representing relationships between objects, manifest in data structures like adjacency matrices and adjacency lists, facilitating efficient traversal and manipulation of interconnected data elements. Thus, the synergy between mathematics and data structures underscores the intrinsic relationship between abstract mathematical concepts and their tangible manifestations in the realm of programming.

Complex Systems and Mathematical Modeling

In the intricate tapestry of programming, mathematical modeling emerges as a powerful tool for understanding and simulating complex systems. From the dynamics of weather patterns to the behavior of financial markets, complex systems exhibit nonlinear interactions and emergent properties that defy simple explanations. Through mathematical modeling, programmers can distill these intricate phenomena into formal representations, using mathematical equations and algorithms to capture the underlying dynamics and predict future states. Whether modeling the spread of infectious diseases or simulating the behavior of artificial intelligence algorithms, mathematics provides a rigorous framework for grappling with the complexities of the world and transforming them into actionable insights.

Mathematical modeling finds widespread application across diverse domains, underpinning critical decision-making processes and driving innovation in programming. In the realm of environmental science, mathematical models are employed to forecast climate change, predict natural disasters, and optimize resource management strategies. In finance, complex mathematical models underpin algorithmic trading systems, risk assessment tools , and portfolio optimization algorithms, enabling investors to make informed decisions in volatile markets. Moreover, in fields such as engineering, healthcare, and telecommunications, mathematical modeling plays a pivotal role in optimizing system performance, predicting outcomes, and guiding the design of innovative solutions to complex problems. Thus, the real-world applications of mathematical modeling underscore its indispensable role in programming, serving as a bridge between abstract mathematical concepts and tangible technological advancements.

Specialized Fields of Programming and Mathematics

Within the expansive landscape of programming, specialized fields emerge as focal points of innovation and exploration, each characterized by its unique challenges and opportunities. Machine learning, a subfield of artificial intelligence, revolutionizes the way computers learn from data, enabling systems to recognize patterns, make predictions, and adapt to changing environments autonomously. Cryptography, the art of secure communication, relies on mathematical principles such as number theory and modular arithmetic to safeguard sensitive information and protect digital transactions from prying eyes. Meanwhile, game development combines elements of art, design, and mathematics to create immersive virtual worlds where players can escape reality and embark on thrilling adventures. Through an exploration of these specialized domains, programmers gain insights into the diverse applications of mathematical concepts and their transformative potential in shaping the future of technology.

In specialized fields such as machine learning, cryptography, and game development, advanced mathematical concepts serve as the cornerstone of innovation and progress. In machine learning, algorithms rooted in linear algebra, calculus, and probability theory empower computers to analyze vast datasets, extract meaningful insights, and make intelligent decisions in real-time. Cryptography leverages advanced mathematical concepts such as elliptic curve cryptography and discrete logarithm problems to create unbreakable encryption schemes, ensuring the confidentiality and integrity of sensitive information in digital communications. Similarly, game development harnesses mathematical concepts such as vector calculus, physics simulations, and geometry to create realistic graphics, simulate dynamic environments, and implement complex gameplay mechanics. Thus, the application of advanced mathematical concepts in specialized fields of programming underscores the symbiotic relationship between theory and practice, driving innovation and pushing the boundaries of technological possibility.

The symbiotic relationship between mathematics and programming underscores the intrinsic connection between abstract theoretical concepts and tangible technological advancements. From laying the foundation for logical thinking and problem-solving to enabling the creation of sophisticated algorithms and models, mathematics serves as a guiding light in the ever-evolving landscape of programming. Through our exploration of the vital role that mathematics plays in areas such as data structures, mathematical modeling, and specialized fields of programming, we gain insight into the transformative power of mathematical proficiency in shaping the future of technology. As we navigate the complexities of the digital age, let us embrace the synergy between mathematics and programming, leveraging their combined strengths to drive innovation, solve complex problems, and create a brighter future for generations to come.

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Researchers from Alibaba Group have introduced a novel approach named AlphaMath that leverages the Monte Carlo Tree Search (MCTS) to automate the generation and refinement of training data for LLMs in mathematical reasoning. This method uniquely eliminates the need for manual data annotation, a common bottleneck in traditional model training, by using a combination of pre-trained LLMs and algorithmic enhancements to autonomously produce and improve training inputs.

The methodology of AlphaMath hinges on integrating MCTS with a policy model and a value model. Initially, these models use a dataset comprising only questions and their final answers, avoiding detailed solution paths. The MCTS algorithm iteratively develops and evaluates potential solution paths, refining them based on the estimated values from the value model. This continuous process not only generates high-quality training data but also optimizes the model’s problem-solving strategies. The training and evaluation are conducted using the MATH dataset, renowned for its complexity, thereby testing the models’ proficiency under challenging conditions.

The application of the MCTS methodology in AlphaMath has yielded significant improvements in the model’s performance on the MATH dataset. Specifically, the enhanced models demonstrated a solution accuracy rate that exceeded 90% on complex problem sets, an increase from the baseline accuracy rates previously recorded. These results indicate a substantial advancement in the model’s ability to solve intricate mathematical problems with minimal error autonomously, validating the effectiveness of the MCTS integration in reducing the need for manual data annotation while maintaining high levels of accuracy and reliability in mathematical reasoning tasks.

To summarize, the research by Alibaba Group introduces a novel approach, Alphamath, using MCTS to enhance large language models’ capabilities in mathematical reasoning. By automating the generation of training data and refining solution paths without manual annotation, this methodology significantly improves model accuracy on complex mathematical problems, as evidenced by its performance on the MATH dataset. This advancement not only reduces the reliance on costly human intervention but also sets a new standard for efficiency and scalability in the development of intelligent computational models.

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Nikhil is an intern consultant at Marktechpost. He is pursuing an integrated dual degree in Materials at the Indian Institute of Technology, Kharagpur. Nikhil is an AI/ML enthusiast who is always researching applications in fields like biomaterials and biomedical science. With a strong background in Material Science, he is exploring new advancements and creating opportunities to contribute.

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ChatGPT’s Next Magic Trick Is Singing and Solving Math Problems With Your Phone Camera

GPT-4o responds as fast as humans.

ChatGPT 4 may still be relatively new, but OpenAI is already iterating with an upgrade that can respond as quickly as humans do in normal conversation. The company showed off GPT-4o in a live demo , showing off its ability to use your phone’s camera to solve math equations and deliver a much more conversational voice assistant experience.

While we only have the event demo to go off of, GPT-4o looks impressive. It doesn’t even have to wait for you to finish your request and can roll with interruptions mid-prompt, bringing one step closer to living out Her in real life.

Even Faster Response Times

According to OpenAI, the GPT-4o model can respond as fast as 232 milliseconds to audio inputs. More realistically, it averages around 320 milliseconds to respond, which OpenAI said is similar to how fast humans respond in conversation.

On top of the speed, GPT-4o can handle interruptions and any adjustment requests. As seen in the bedtime story demo, GPT-4o immediately stopped talking when interrupted and quickly handled requests like adding more dramatic inflections, narrating in a robot voice, and even singing the entire prompt out loud. If that demo doesn’t convince you, two GPT-4o models improvising a song together should.

GPT-4o isn’t just more responsive to voice, it can also see better. The new vision features allow it to see through your device’s camera and understand things like handwritten math equations or messages . It’s eerie how genuinely touched GPT-4o sounds when it sees and understands a message that says “I Heart ChatGPT.” Even more impressive, GPT-4o can handle coding tasks and live translations between two people. This should feel way more natural than Google Translate when you’re trying to have a conversation in a foreign country.

Available for Free

OpenAI said the text and image capabilities for GPT-4o roll out today, but the voice feature will be coming to alpha within ChatGPT Plus in the coming weeks. Once it’s fully ready, the upgraded ChatGPT model will be available to all users, subscribers or otherwise. However, if you pay $20 per month for ChatGPT Plus , you’ll get five times the message limits of GPT-4o compared to the free version.

Anytime a large language model gets such an impressive update, we have to consider the potential for misuse . Considering how smoothly the live demo for solving the equation went, it looks like an even better way to help students get out of their math homework. However, OpenAI said that GPT-4o was built with new safety systems to offer guardrails on voice outputs. We’ll have to wait and see if these guardrails are enough.