Math Instruction: What Do You Need To Know To Solve Problems?

What does a student need to know to solve math problems? is one of the most frequently asked questions in the realm of math instruction. And it is that this subject usually presents a multitude of problems for students. Therefore, to what extent is it adequately taught?

For this, it is important to take into account which are the fundamental components that students have to develop to learn and understand mathematics and also how this process develops. Only in this way can adequate and adapted mathematics instruction be exercised.

Thus, to understand mathematical functioning, the student has to master four fundamental components:

• The linguistic and factual knowledge appropriate to construct the mental representation of the problems.

• Know how to build schematic knowledge to integrate all accessible information.
• Possess strategic and meta-strategic skills to guide the solution of the problem.
• Have the procedural knowledge to solve the problem.

In addition, it is important to bear in mind that these four components are developed throughout four different phases in the tasks of solving mathematical problems. Below we will explain the processes involved in each of them:

• Translation of the problem.
• Integration of the problem.
• Planning the solution.
• Execution of the solution.

1- Translation of the problem

The first thing the student has to do when faced with a mathematical problem is to translate it into an internal representation.  In this way, you will have a picture of the available data and its objectives. However, for the statements to be translated correctly, the student must know both the specific language and the appropriate factual knowledge. For example, that the square has four equal sides.

Through the investigation, we can observe that the students are guided many times by the superficial and insignificant aspects of the statements. This technique can be useful when superficial text matches the problem. However, when it is not, there are a number of problems with this approach. In general, the most serious is that  students do not understand what is being asked of them. The battle is lost before it begins. If a person does not know what he has to achieve, it is impossible for him to carry it out.

Therefore, instruction in mathematics must begin by educating in the translation of problems. A multitude of research has shown that specific training in creating good mental representations of problems improves mathematical ability.

2- Integration of the problem

Once the problem statement has been translated into a mental representation, the next step is integration into a whole. To carry out this task it is very important to know the real objective of the problem. In addition, you have to know what resources we have when dealing with it. In short, this task requires that you get an overview of the mathematical problem.

Any mistake when integrating the various data will lead to a feeling of lack of understanding and of being lost. In the worst case, it will result in solving it in a totally wrong way. Therefore, it is essential to emphasize this aspect in math instruction because it is the key to understanding a problem.

As in the previous phase, students tend to focus more on superficial aspects than on deep ones. When determining the type of problem, instead of looking at its objective, they look at the less relevant characteristics. Fortunately, this can be solved through specific instruction and by accustoming students to the fact that the same problem can present itself in different ways.

3- Planning and monitoring the solution

If the students have been able to understand the problem in depth, the next step is to  generate an action plan to find the solution. Now is the time to subdivide the problem into small actions that allow you to progressively approach the solution.

This is, perhaps, the most complex part when solving a math exercise. It requires great cognitive flexibility together with an executive effort, especially if we are faced with a new problem.

Math instruction around this may seem impossible. But research has shown us that through various methods we can achieve increased planning performance. They are based on three essential principles:

• Generative learning.  Students learn best when they are the ones who actively build their knowledge. A key aspect in constructivist theories.
• Contextualized instruction.  Solving problems in a meaningful and useful context greatly aids students’ understanding.
• Cooperative learning.  Cooperation can help students share their ideas and be reinforced by those of others. This, in turn, encourages generative learning.

4- Execution of the solution

The last step in solving a problem is finding the solution to it. To do this, we have to use our prior knowledge of how certain operations or parts of a problem are solved. The key to good execution is to have basic internalized skills that allow us to solve the problem without interfering with other cognitive processes.

Practice and repetition are a good way to proceduralize these skills, but there are a few more. If we introduce other methods into mathematics instruction (such as teaching about the notion of number, counting, and number lines), learning will be highly reinforced.

As we can see, solving mathematical problems is a complex mental exercise made up of a multitude of related processes. Trying to instruct in this subject in a systematic and rigid way is one of the worst mistakes that can be made. If we want students with great mathematical ability, it is necessary to be flexible and center the instruction around the processes involved.