Why reasoning?
Stacey, Kaye
Reasoning is one of the proficiency strands of the new Australian Curriculum. Of course, reasoning has always been important in
mathematics and its importance has always been recognised in mathematics
curricula across Australia. However, the new proficiency strand provides
an opportunity for all teachers to reconsider how they teach this
essential aspect of mathematics. There are many aspects to reasoning in
mathematics, but this focuses on the reasoning that establishes why
mathematical results are true.
Mathematics is distinguished amongst the areas of human knowledge
by the special way in which claims of what is true are justified. The
assumptions (technically called axioms) and definitions are stated, and
gradually, piece by piece, all other mathematical knowledge is built up
using the rules for logical deduction. It is an enormously complex web,
but the consequence is that mathematical results can be definitely
proved. This is not true to nearly the same extent for any other
subject.
How can this fundamental characteristic of mathematics be conveyed
at school?
Consider the observation that an even number plus an even number is
odd, that odd plus odd is even and that even plus odd is odd. It is
useful to know these even-odd adding rules. After all these years of
calculating, I still use them as a quick and virtually automatic check
when adding and subtracting: for example if I need to add three numbers
and I notice they are all odd, then I know that my answer must be an odd
number.
There are several ways in which the even-odd adding rules can be
justified in the classroom. One way is to rely on the superior knowledge
of the teacher, without other reasons. "My teacher said so, so they
are correct." This approach does not help students to develop
reasoning or independent mathematical thinking.
Another way is to have children look at many examples: 3 + 5 = 8, 4
+ 6 = 10, 1 + 7 = 8 and so on, and then to find the rules for
themselves. This has definite advantages--children are actively engaged
in making and testing generalisations and can experience the joy of
discovering a pattern and sharing it. But in a classroom that values
mathematical reasoning, even young children can do more than guess
patterns based on evidence from examples. Mathematical truth is not
established just by looking at examples.
To uncover the reasons for the even-odd adding rules, we need to
understand better what an even number is, not just know that 2, 4, 6, 8
and the rest are called even numbers. In adult language, even numbers
are multiples of 2; in the language of small children, even numbers of
objects can be put into pairs (groups of two). With odd numbers, the
pairing always leaves one left over. The top arrows of Figure 1
illustrate this.
Equipped with this mathematical definition, we can see why the
even-odd adding rules must be true. To add an odd and an even number of
blocks, we have one set of blocks that consist of pairs, and another
that consists of pairs with one left over. When we add by combining the
two sets, we have pairs with one block left over. So even plus odd is
odd, as in Figure 1.
To show that an odd number plus an odd number is even involves just
one extra step, shown in Figure 2. The two sets of blocks are both made
up of pairs with one block left over. After combining the blocks
(adding), the two left over blocks can be put together to make another
pair. So an odd number plus an odd number is even.
A serious mathematical proof would be written using symbols to
stand for generalised numbers rather than illustrated with blocks, but
the essential idea of the proof (combining those left over blocks to
make another pair) is just the same. Quite young students can see that
this idea generalises--that it works for any number of blocks. There is
nothing special about the numbers of blocks chosen in the figures.
The example of the even-odd adding rules illustrates several
important points:
1. Even very young children can engage in mathematical reasoning,
so logical reasoning can and should feature in mathematics classes at
all levels.
2. Reasoning should be age appropriate, so in the primary grades it
will frequently involve reasoning with models like blocks.
3. As students progress, their reasoning will become more
sophisticated across many dimensions. This needs systematic attention
throughout the years of school. One dimension for growth is appreciating
the nature of the evidence given for why mathematical statements are
true. Examples confirm in mathematics, but only a logical argument can
provide convincing evidence that a mathematical claim is true for an
infinite number of cases.
4. Teachers should not be afraid to explain age-appropriate reasons
why mathematical results are true. It is hard to do this well. Students
need to be prepared, for example, by gathering evidence (e.g., by
measuring) and looking for patterns so that they can guess results and
get a sense of discovery.
5. Look at the resources used at your school--the textbooks,
worksheets, computer programs or lesson plans. Make sure that there is
attention to explaining reasoning.
6. Set tasks that require students to explain their thinking and
why their discoveries might be true. Discussion is a good way to clarify
reasons before writing them down.
7. Right from the start, students can get to know mathematics as
the subject where they do not need to remember rules without reasons.
They should not focus on memorising what the teacher says, but know that
they can often think things through for themselves. Mathematics makes
sense and is a coherent whole.
[FIGURE 1 OMITTED]
Wherever possible, give reasons--maybe using a model, as in the
case of the even-odd addition rules. Mathematical reasoning at school
will not be the same as a mathematician would use--we need explanations
appropriate to students' development. However, all mathematics
lessons, at whatever level, need to convey the impression that knowing
the reasons why mathematical rules are true is important and the key to
learning it well.
[FIGURE 2 OMITTED]
Kaye Stacey
University of Melbourne <
[email protected]>
