Problem solving is central to mathematics and can challenge students’ curiosities and interest in mathematics. Polya’s (1945) seminal book, How to Solve It stated,

A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development and misuses opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of independent thinking. (Polya, 1945, p. v)

Polya suggested four steps to problem solving, which included (1) understanding the problem; (2) devising a plan; (3) carrying out the plan; and (4) looking back. Problem solving is also one of the important NCTM mathematical process standards along with communication, representations, connections, and reasoning. The NCTM (2000) defined problem solving as:

Engaging in a task for which the solution method is not known in advance. In order to find a solution students must draw on their knowledge, and through this process, they will often develop new mathematical understandings. Solving problems is not only a goal of mathematics but also a major means of doing so. (p. 52)

The Common Core State Standards for Mathematics (CCSSM) encourages a problemsolving approach to teaching the standards in order to develop mathematically proficient students and emphasizes the importance of developing students’ metacognition, self-monitoring, and self-reflection.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.. .[They] check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. (NGACBP & CCSSO, 2010, p. 9)

Think about the many ways that you can encourage the use of models and math modeling in your teaching. Some examples may include finding the height of a monument or finding the mass of the earth or estimating the population of a town in 50 years. In each case, “modeling the physical situation” would involve identifying the appropriate variables to work with that represent known and unknown quantities in the problem; establishing relationships between the variables identified and finally be able to solve for the unknown under some given constraints.

For example, one may try to express the height of the monument in the first example in terms of some distance and angles, which can be measured on ground. For finding the mass of the earth, one may employ masses of known objects that are similar in shape to the earth and invoke proportional reasoning arguments. To estimate population, one can extrapolate from a previous census or possibly develop a model expressing the population of the town as a function of time.

Polya’s Four Steps to Problem Solving have similar processes that are employed as big ideas in mathematical modeling. The first idea involves observation, which helps us to recognize the problem. In this step, the variables describe the attributes in the problem as well as any implicit or explicit relationships between the variables. The second idea includes the formulation of the mathematical model. In this step, one makes the necessary approximations and assumptions under which one can try to obtain a reasonable model for the problem.

These approximations and assumptions may be based on experiments or observations and are often necessary to develop, simplify, and understand the mathematical model. The third idea addresses computation that involves solving the mathematical model for the physical problem formulated. It is then important as a final step to then perform validation of the solution obtained that helps to interpret the solution in the context of the physical problem. If the solution is not reasonable, then one can attempt to modify any part of the three steps including observation, formulation, and computation, and continue to validate the updated solution until a reasonable solution to the given physical problem is obtained. Mathematical modeling is a very distinct class of problems that are very open (example of a mathematical modeling research lesson in chapter 3).

In any given mathematics classroom, problems might be presented along a continuum ranging from the mathematics being presented in a structured word problem to the mathematics being presented in an unstructured real-world problem. The difference along this continuum can be seen as the degree of open-ness (unstructured) or closed-ness (structured) of the application. Each of these different types of problems along the continuum has its place in the mathematics classroom.

Depending on the specific learning goals, teachers may want to use a variety of problem types for different purposes. When learning about the different multiplication and division structures and interpretations, one might use structured word problems to be able to distinguish from equal group problems and multiplicative comparison problems. For example, Zachary has 3 bags with 4 apples in each bag. How many apples does Zachary have? versus Jeremy has 3 apples and Zachary has 4 times more than Jeremy. How many apples does Zachary have? Learning multiple interpretations for multiplication can help students be flexible in the way that they use these operations with meaning.

On the other hand, one may pose a rich task like the handshake problem to teach problem-solving strategies and the notion of algebraic reasoning in finding a pattern of the number of handshakes given any number of people. Furthermore, one might use an open-ended real-world problem for students to see how they go through the process of defining the variables, making assumptions to build a mathematical model or solution to a nonroutine problem. There are several unique affordances of mathematical modeling problems that are important to consider.

Mathematical modeling requires reasoning about several ideas or quantities simultaneously. It requires thinking about situations in relative rather than absolute terms. For example, if the number of students in a middle school grows from 500 to 800 and another middle school grows from 300 to 600, a student thinking in absolute terms (or additively) might answer that both schools had the same amount of increase. On the other hand, a student that is taught to think in relative terms might argue that the second middle school saw more increase since it doubled the number of students unlike the first school which would have needed to end up with 1000 students to grow by the same relative amount. While both answers seem reasonable, it is the relative multiplicative thinking that is essential for proportional reasoning. This ability to think and reason proportionally is very important in the development of a student’s ability to understand and apply mathematical modeling to real-world problem solving.

The goal for the lesson can determine what type of problems that students are working through. It is important for educators to support students in the discovery of connections that help them comprehend the applications of mathematical concepts to real-world problems in a way that is beyond what is taught in a traditional classroom.

Making such connections in the middle grades would require the teachers to begin a process of investigation, which may take multiple forms. Such forms may include anything from random guessing to a more systematic and organized approach, but inherent to this problem-solving investigation process would be to make algebraic connections between these various forms. Good problems are those that are mystifying and interesting. They are mystifying when they are not easy to solve using traditional textbook formulas and approaches. The handshake problem is a prime example of a task that can engage students in a rich problem-solving investigation and algebraic thinking process as they make important connections through various models.

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