Tag: teaching properties of equality

Posted in Teaching mathematics

Use of exercises and problem solving in math teaching

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Mathematical tasks can be classified broadly in two general types: exercises and problem solving tasks. Exercises are tasks used for practice and mastery of skills. Here, students already know how to complete the tasks. Problem solving on the other hand are tasks in which the solution or answer are not readily apparent. Students need to strategize – to understand the situation, to plan and think of mathematical model, and to carry-out and evaluate their method and answer.

Exercises and problem solving in teaching

Problem solving is at the heart of mathematics yet in many mathematics classes ( and textbooks) problem solving activities are relegated at the end of the unit and therefore are usually not taught and given emphasis because the teacher needs to finish the syllabus. The graph below represents the distribution of the two types of tasks in many of our mathematics classes in my part of the globe. It is not based on any formal empirical surveys but almost all the teachers attending our teacher-training seminars describe their use of problem solving and exercises like the one shown in the graph. We have also observed this  distribution in many of the math classes we visit.

The graph shows that most of the time students are doing practice exercises. So, one should not be surprised that students think of mathematics as a a bunch of rules and procedures. Very little time is devoted to problem solving activities in school mathematics and they are usually at the end of the lesson. The little time devoted to problem solving communicates to students that problem solving is not an important part of mathematical activity.

Exercises are important. One need to acquire a certain degree of fluency in basic mathematical procedures. But far more important to learn in mathematics is for students to learn to think mathematically and to have conceptual understanding of mathematical concepts. Conceptual understanding involves knowing what, knowing how, knowing why, and knowing when (to apply). What could be a better context for learning this than in the context of solving problems. In the words of S. L. Rubinshtein (1989, 369) “thinking usually starts from a problem or question, from surprise or bewilderment, from a contradiction”.

My ideal distribution of exercises and problem solving activities in mathematics classes is shown in the the following graph.

What is teaching for and teaching through problem solving?

Problems in mathematics need not always have to be an application problem. These types of problems are the ones we usually give at the end of the unit. When we do this we are teaching for problem solving. But there are problem solving tasks that are best given at the start of the unit. These are the ones that can be solved by previously learned concepts and would involve solutions that teachers can use to introduce a new mathematical concept. This strategy of structuring a lesson is called Teaching through Problem Solving. In this kind of lesson, the structure of the task is king. I described the characteristics of this task in Features of Good Problem Solving Tasks. Most, if not all of the lessons contained in this blog are of this type. Some examples:

  1. Teaching triangle congruence through problem solving
  2. Teaching the properties of equality through problem solving
Click the links for more readings about Problem Solving:
Posted in Algebra

Teaching the properties of equality through problem solving

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problem solvingI like to teach mathematical concepts via problem solving. It right away engages students mind. It creates a need for learning a more systematic way of doing things and hence a reason for learning the concept. It provides a context for making connections. Most important of all, it gives students opportunity to learn before they are taught.

One of the ways of creating a problem solving task for this kind of lesson is to start by giving the solution/answer.  For example, a standard textbook task is to solve for x in an equation.  Why not do it the other way and ask students to find the equation given the solution?

Here is my favorite problem for introducing the properties of equality:

Write three equations in x with solution x = 5.

Fresh from their elementary school math experience of solving equation of the type  15 – ___ = 20 or x + 12.3 = 20, students will generate equations by trial and adjust. That is, they think of an equation then check if it gives x = 5 by substituting the value of x to the equation.  It will not take long for them to realize that this is something very tedious.

Other students will verify their equations by expressing the terms into equivalent structure. I explained this method in my previous post. This method is not also very efficient for some equations. But students have to experience these so that they will be able to appreciate the efficiency of solving equations using the properties of equality.

Usually when they already have at least two equations I will extend the task to:

Two of the equations should have x on both sides of the equal sign .

Challenge them further to think of a more systematic way for making the equations. You will not hear the bell ring before students will realize that all they need to do to have an x on both sides is to add or subtract x on both sides of the equal sign! (Multiplying or dividing by x is a different matter as there is a possibility that x can be zero). Now they know that those ubiquitous equations in their algebra textbooks did not just drop from they sky. They can even create one themselves. The students are applying the properties of equality before they even know what they are! Speaking of constructivist teaching here.

Teachers I shared this with were also very happy because they no longer have to do trial and adjust when they make exercises for solving equations.