Giving examples, sometimes tons of them, is not an uncommon practice in teaching mathematics. How do we use examples? When do we use them? In his paper, The purpose, design, and use of examples in the teaching of elementary mathematics, Tim Rowland considers the different purposes for which teachers use example in mathematics teaching and examine how well these examples were achieving the objective of the lesson. He classified the use of examples in two types – deductive and inductive.
Types of examples
Examples are used deductively when they are given as ‘exercises’. These examples are usually given after teaching a particular procedure. The initial purpose is to assist retention by repetition of procedure and then eventually for students to develop fluency with it. It is hoped that through working with these examples, new awareness and new understanding of the preocedure and the concepts involved will be created (I’m not sure if many teachers do something to make this explicit). In using examples for this purpose, the teachers should not just haphazardly give examples. For instance, practice examples on subtraction by decomposition ought to include some possibilities for zeros in the minuend. For practice in subtracting integers, the range of examples should include all the possible cases such as minuend and subtrahend both positive; minuend and subtrahend both negative with minuend greater than subtrahend and vice versa, etc.
The second type of examples is done more inductively. Here, examples are used to teach a particular concept. Their role in concept development is to provoke or facilitate abstraction. The teacher’s choice of examples for the purpose of abstraction reflects his/her awareness of the nature of the concept and the category of things included in it, which of these categories may be considered exceptional and the dimensions of possible variation within a particular category. In other words, teachers must not only give examples but give nonexamples of the concept as well.
Sequencing examples
It is not only the example but also the sequence that they are given that affect the kind of mathematics that is learned. Rowland reports in his paper a Grade 1 lesson about numbers that add up to 10. The teacher asked “If we have nine, how may more to make 10?”. The subsequent examples after 9 are as follows: 8, 5, 7, 4, 10, 8, 2, 1, 7, 3. This looks like random examples but in the analysis of Rowland it was not. The teacher had a purpose in each example. It was not random.
- 8: the teacher knows that the pupils usually uses the strategy of counting up so they will have success here
- 5: this will bring up the strategy of a well-known double – doubling being a key strategy for mental calculation
- 7: same as in 8 but this time, pupils have to count up a little bit further
- 4: for the more able students
- 10: to point to the fact that zero is also a number which can be added to another number
- 8: strange to repeat an example but the teacher used this to ask the pupil who answered 2 “If I’ve got 2, how many more do I need to make 10?” which was the next example.
- 2: here the teacher said based on previous interaction “2 add to 8, 8 add to 2, it’s the same thing (commutative property and counting up from larger number)
- 1: the teacher did not ask how many more to make 10 as this will trigger counting up but instead related it to 2 and 8 to make obvious the efficiency of the strategy of counting up from a bigger number and perhaps to make the children be aware of commutativity.
- 7 and 3: to reinforce the strategies made explicit in using 8 and 2 as examples.
Every classroom should have an integer number line. Number line should run from -40 or so up to +100. These are commercially available. Perhaps students can construct the pieces, tape them together, have the line at eraser tray level, and then hoist it up and secure it with tape or velcro.
The number line comes in handy for addition, subtraction, and one digit multiplication. Students can see that 23-8=15, 33-8=25…
Adding sevens is 7,14,21, etc. If you start at 5 and add sevens it is 12, 19, 26, etc.
This visualization really helps!
A ruler is sort of like a number line…and we can introduce fractions using rulers. Meter sticks with cm up to 100 work well for this, though inches and feet are important. Vertical meter sticks and yard sticks starting at the floor, kids measure their heights in inches and cm’s, write them next to each other in vertical columns, then conjecture on how to convert inches into cm’s or cm’s into inches.
Kids can construct vertical or horizontal c and f thermometers. Keep track of the days temperature (low and high and current) and see how to convert c into f. This comes a little later… The original number line can be used to track temperature. A digital classroom thermometer that switches from c to f is useful.
Math was invented to be useful. It was concrete (until the 1600’s) and imho is best learned in concrete, useful ways.