I have put together in this post some of the ideas behind the kind of mathematics teaching I promote. As I stated in the subheadings of this blog, the articles and lessons I write here are not about making mathematics easy because it isn’t but about making mathematics makes sense because it does. Before reading any of the articles below, I suggest read the About page first and what I think mathematics is. I hope I make sense in the following articles. Click here for the list of math lessons.
This is a collection of math lessons posted in this blog. Most if not all of the lessons use the strategy teaching through problem solving or through mathematical investigation. I believe that school mathematics is about teaching students how to think mathematically first and learning the mathematics second so math lessons should be designed so that students are engaged in thinking mathematically. This is something that should not be left to chance.
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:
In this post I propose a way of teaching the concept of triangle congruence. Like most of the lessons I share in this blog, the teaching strategy for this lesson is Teaching through Problem Solving. In a TtPS lesson, the lesson starts with a situation that students will problematize. The problems either have many correct answers or have multiple solutions and can always be solved by previously learned concepts and skills. Problems like these help students to make connections among the concepts they already know and the new concept that they will be learning in the present lesson. The ensuing discourse among students and between teacher and students during the discussions of the different solutions and answers trains students to reason and communicate mathematically and thereby help them to appreciate the power of mathematics as a language and a way of thinking. In mathematics, language is precise and concise.
Here’s the sequence of my proposed lesson:
1. Setting the Problem:
Myra draw a triangle in a 1-cm grid paper. Without showing the triangle, she challenged her friends to draw exactly the same triangle with these properties: QR is 4 cm long. The perpendicular line from P to QR is 3 cm.
Pose this question: Can you draw Myra’s triangle?
Give students enough time to think. When each of them already have at least one triangle, encourage the class to discuss their solutions with their seat mates. Challenge the class to draw as many triangles satisfying the properties Myra gave.
2. Processing of solutions: Ask volunteers to show their solutions on the board. Questions for discussion: (1) Which of these satisfy the information that Myra gave? (2) What is the same among all the correct answers? [They all have the same area]. Possible solutions are shown below.
3. Introducing the idea of congruence: Question: If we are going to cut-out all the triangles, which of them can be made to coincide or would fit exactly? [When done, introduce the word congruence then give the definition.]
Tell the class that Myra only drew one triangle. Show the class Myra’s drawing. Question: In order to draw a triangle congruent to Myra’s triangle, what conditions or properties of the triangle Myra should have told us?
Myra’s triangle
Possible answers:
QR is 4 cm long. The perpendicular line PQ is 3 cm.
QR is 4 cm long. PQ is 3 cm and forms a right angle with QR.
PQR is a right triangle with right angle at Q. QR is 4 cm and PQ is 3 cm.
4. Extending the problem solving activity: Which of the following sets of conditions will always give triangles congruent to each other?
In triangle ABC, AB and BC are each 5 cm long.
ABC is a right triangle. Two of its shorter sides have lengths of 4 cm and 5 cm.
I would appreciate feedback so I can improve the lesson. You feedback will inform the sequel to this lesson.. Thank you.
