The heart of mathematics

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Axioms, theorems, proofs, definitions, methods, are just some of the sacred words in mathematics. These words command respect and create awe  especially to mathematicians but deliver shock to many students. P.R. Halmos argued that not even one of these sacred words is the heart of mathematics. Then, what is? problem solving. Solving problems is at the heart of mathematics. For many mathematicians, it is the main reason for their existence.


Indeed, can you imagine mathematics without problem solving? It might as well be dead! But why is it that problem solving tasks are relegated as end of lesson activity? When it’s almost end of the term and the teacher’s in a hurry to finish their budget of work, the first to go are the problem solving activities. And when time allows the teacher to engage students in problems solving, the typical teaching sequence goes like this based on my observation in many math classes and from the teaching plans made by teachers.
1.Teacher reviews the computational procedures needed to solve the problem.
2.Teacher solves a sample problem first usually very neatly and algebraically (especially in high school)

3.Teacher asks the class to solve a similar problem using the teacher’s solution

4.Students practice solving problems using the teacher’s method.

Even textbooks are organized this way! We call this strategy Teaching for Problem Solving. In this strategy, students are given problem solving tasks only after having learned all the concepts and skills needed to solve the problem. Most often than not, they are also shown a sample method for solving the problem before they are given a set of similar problems to work on. I will not even call this a problem solving activity/lesson. How can a problem be a problem if you already know how to solve it? Of course, this particular strategy also gives the students the opportunity to deepen, consolidate and synthesize the new math concepts they just learned. But it also deprives them the opportunity to engage in real problem solving where they themselves figure out methods for solving the problem and using knowledge they already possess.

Another approach to increase students engagement with problem solving is to teach mathematics through problem solving. This is discussed in the next post.

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