Teaching Math: What Students Must Know to Solve a Problem

To successfully teach children how to solve problems in math, you need to make sure there is problem translation, problem integration, solution planning, and problem execution.

What does someone studying math need to know in order to solve a math problem? This is surely one of the most common questions in the math field. Math is usually a difficult subject for students. So, how do you make sure you’re teaching math the right way?

It’s important that you keep in mind the fundamental qualities students must develop in order to learn and understand math, and also how the learning process unravels. Only this way, you’ll be able to teach math properly.

In order to understand how math works, students must first dominate four different aspects:

• Linguistic and factual knowledge to build a mental representation of the problem in question.
• Building their own schematic knowledge to cover all the available information.
• Strategies to identify what the problem is asking.
• Having procedural knowledge that allows them to solve the problem.

Furthermore, it’s important to keep in mind that these four aspects are developed in four steps as well, which are:

• Problem translation
• Problem integration
• Solution planning
• Problem execution

1. Problem translation

The first thing students have to do to solve a math problem is to translate it into an inner representation. This way, they’ll have an overall picture of available data and the objectives.

But in order for the text to be translated correctly, students must know the specific language and the proper factual knowledge. For example, they should know that a square has four equal straight sides.

Research suggests that students often focus on the superficial aspects of the problem’s text. This technique can be useful when the superficial words go in the direction of the solution, but when it’s not like that, this approach leads to more obstacles.

And it’s even worse if students don’t even understand what the problem is asking them to do. It’s no use that they try to solve something they don’t comprehend.

That’s why teaching math must begin by educating about problem translation and explaining the language of the word problems. Many studies have demonstrated that specific training to create good mental representations of problems can improve math skills.

2. Problem integration

Once the student translates the problem into a mental representation, the next step is to “tie” all the data together. In order to do this, it’s important to recognize the problem’s objective. Besides, students must know what resources they have to tackle it. In other words, this stage requires them to have a general perspective of the entire mathematical problem.

Any mistake they make when integrating the data will lead them to feel that they’re lost and that there’s something they’re not entirely understanding. In the worst case scenario, the approach to the problem will go in the completely wrong direction. So it’s essential to emphasize this aspect when teaching math because it’s the key to fully understanding the problem.

Just like in the last step, students tend to focus on the superficial rather than the important aspects. When it’s time to determine the problem’s nature, instead of seeing the objective itself, they reach out for the least relevant data.

However, this can be solved through specific instruction and teaching students that the same problem can be presented in other ways.

3. Solution planning and supervision

If students manage to fully comprehend the problem, the next step is to create an execution plan to find the solution. This is the time to split the problem into small tasks that make it easier to get to the solution progressively. 

This is probably the most difficult part of solving a math problem. It requires cognitive flexibility along with effort, especially when facing a new problem.

It may seem impossible to teach math around this aspect, but research suggests that through different methods, it’s possible to improve planning skills. There are three essential principles to do this:

Generative learning

Students learn better when they’re actively building their own knowledge. It’s a key aspect in constructivist theories.

Contextualized instruction

Solving problems in a meaningful and useful context helps students understand better.

Cooperative learning

Cooperation might help students share their common ideas and reinforce their knowledge with other ideas. This also encourages generative learning.

4. Problem execution

The last step to properly solve a math problem is, of course, finding the solution. To do this, you must resort to your previous knowledge about how certain operations or parts of a problem can be solved. The key to a good execution is to internalize basic skills that allow you to solve the problem without interfering with other cognitive processes.

Practice and repetition are good methods to internalize those skills, but there are many more. If we apply these other methods to math (like the notion of numbers, counting, etc.), you’ll reinforce your learning process.

As you can see, solving math problems is a complex mental exercise that involves many cognitive processes. Teaching math in a systematic and rigid way is one of the worst mistakes you can make.

Most importantly, if you want highly skilled students, you must teach them to be flexible and approach the problem using these four aspects.