Tag: why teach 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, Assessment

Assessing understanding of graphs of functions

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Problems about graphs of functions can be grouped into interpretation or construction tasks. The tasks may involve interpreting individual points, an interval, or the entire graph. The same may be said about construction tasks. It may involve point-plotting,  a part of the graph, or constructing the whole graph.

Tasks involving constructing graphs are considered more difficult than interpreting graphs tasks but with the available graphing technology, constructing graphs is now easy.  But not when you have to construct a relationship, not just graphs! In fact, I would consider it as an indicator of students deep understanding of graphs and functions when he or she can interpret and reason in terms of relationship shown in the graphs and from these be able to construct a new relationship, a new function. Here is a task you can use to assess this level of understanding. Note that in this task the graphs are not on grids to encourage holistic analysis of the graph rather than point-by-point. Interpreting graphs not on grids encourages algebraic thinking.

graphs
Relating graphs

Below is a a sample a Year 8 student solution to the task above. This answer indicates that the student understands graphs and the function it is representing but  he/she could still not reason in terms of relationship so resorted to interpreting individual points in x vs y and y vs z in order to relate x and z.

solutions by point-by-point analysis

The figure below shows a solution of a Year 10 student who could reason in terms of the relationships of the variables represented by the graphs.

reasoning in terms of relationship

A similar solution to this would be “x is directly proportional y but y is inversely proportional to z hence x would also be inversely proportional to z”.

Both solutions are correct and both solved the problem completely. Note that initially students will use the first solution just like the Year 8 student. The Year 10 however should be expected and encouraged to reason in terms of relationship.

A good assessment task not only assesses students’ mathematical knowledge and skills but also assesses the level of thinking and reasoning students are operating on. See posts on features of good problem solving tasks.

Posted in What is mathematics

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.


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!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.