Tag: importance of problem solving

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, Mathematics education

When is a math problem a problem?

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One of the main objectives of mathematics education is for students to acquire mathematical habits of mind. One of the ways of achieving this objective is to engage students in problem solving tasks. What is a problem solving task? And when is a math problem a problem and not an exercise?

What  is a problem solving task?
A problem solving task refers to a task requiring a solution or answer, the strategy for finding such is still unknown to the solver. The solver still has to think of a strategy. For example, if the task,

If x^2 - 7 = 18, what is x^2 - 9 equal to?

is given before the lesson on solving equation, then clearly it is a problem to the students. However, if this is given after the lesson on solving equation and students have been exposed to a problem similar in structure, then it cease to be a problem for the students because they have been taught a procedure for solving it. All the students need to do is to practice the algorithm to get the answer.

What is a good math problem?

The ideal math problem for teaching mathematics through problem solving is one that can be solved using the students’ previously learned concepts/skills but can still be solved more efficiently using a new algorithm or new concept that they will be learning later in the lesson. If the example above is given before the lesson about the properties of equality, the students can still solve this by their knowledge of the concept of subtraction and the meaning of the equal sign even if they have not been taught the properties of equality or solving quadratic equation (Most teachers I give this question to will plunge right away to solving for x. They always have a good laugh when they realize as they solve the problem that they don’t even have to do it. They say, “ah, … habit”.)

Given enough time, a Year 7 student can solve this problem with this reasoning: If I take away 7 from x^2 and gives me 18 then if I take away a bigger number from x^2 it should give me something less than 18. Because 9 is 2 more than 7 then x^2 - 9 should be 2 less than 18. This is 16.

Why use problem solving as context to teach mathematics?

You may ask why let the students go through all these when we there is a shorter way. Why not teach them first the properties of equality so it would be easier for them to solve this problem? All they need to do is to subtract 2 from both sides of the equal sign and this will yield x^2 - 9 = 16. True. But teaching mathematics is not only about teaching students how to get an answer or find the shortest way of getting an answer. Teaching mathematics is about building a powerful form of mathematical knowledge. Mathematical knowledge is powerful when it is deeply understood, when concepts are connected with other concepts. In the example above, the problem has given the students the opportunity to use their understanding of the concept of subtraction and equality in a problem that one will later solve without even being conscious of the operation that is involved. Yet, it is precisely equations like these that they need to learn to construct in order to represent problems usually presented in words. These expressions should therefore be meaningful. Translating phrases to sentences will not be enough develop this skill. Every opportunity need to be taken to make algebraic expressions meaningful to students especially in beginning algebra course. More importantly, teaching mathematics is not also only about acquiring mathematical knowledge but more about acquiring the thinking skills and disposition for solving problems and problem posing. This can only happen when they are engage in these kind of activities. For sample lesson, read how to teach the properties of equality through problem solving.

Finally, and I know teachers already know this but I’m going to say it just the same. Not all ‘word problems’ are problems. If a teacher solves a problem in the class and then gives a similar ‘problem’ changing only the situation or the given ‘numbers’ but not the structure of the problem or some of the condition then the latter is no longer a problem but an exercise for practicing a particular solution to a ‘problem’. It may still be a problem of course to those students who did not understand the teacher’s solution. I’m not saying that this is not a good practice, I am just saying that this is not problem solving but an exercise.

You may also want to read How to Solve It: Modern Heuristicsto further develop your problem solving skills.