If math is the study of patterns for the purposes of problem-solving then problem-solving is a major activity that we learn in math. And we observe and learn how to work with patterns so that we can solve problems!

It has been found that rote learning of math does not result in all students understanding how to solve problems which is the purpose of math in the first place. In the traditional method of teaching the teacher shows exactly how to do it the student learns how to do that exact problem. The issue with that is when faced with other types of problems the student may not know how to solve them.

The student needs to learn how to think with the concept of mathematical patterns. And more importantly learn many ways using these patterns to solve problems.

An effective four step problem-solving process by George Pyrola a famous mathematician is:

1. understanding the problem

2. devising a plan

3. carrying out the plan

4. looking back to see if you have now solved the problem.

Teachers can and should pose tasks or problems that engage students in thinking about and developing important mathematics they need to learn. Problems need to take into consideration what the students level of understanding currently is, what will engage them in the subject matter they need to learn, what type of problem, and ways that they can justify or explain their answers and methods.

One issue that comes up with my students repeatedly is: the word problem when not understood makes solving the problem impossible. The solution for this is to clarify the verbal problem and also devise a similar problem — Let the student come up with the problem from real life and then work together to help the student define a problem and then let the student have at it!

One of the keynotes of this new method is that students are not constantly interrupted with judgment as to whether they are solving the problems correctly or not. Let them have at it and see what they can come up with because there are many ways to solve the problem. We want them to make mistakes so that they can learn from their mistakes and to be guided how to solve problems, not put Ina rote learning one way only sort of box. They can learn to think outside the box.

This method of allowing them time to think through problems develops their fundamental understanding of seeing patterns because you can refer back to the patterns that apply to that problem. And more importantly it helps students to learn to solve problems by using patterns and mathematical reasoning