In my previous columns I said that learning mathematics is one great big investigation and investigations characterise the classroom in which mathematics is regarded as a meaningful, sense-making, problem-solving activity. In that column we looked at the process of investigating in general by considering a specific problem.
In this column I want to look at how we can use investigations and looking for patterns to establish important mathematical concepts.
Teachers often bemoan the fact that their learners can recite the multiplication tables with confidence and yet do not have any sense of what 5×3 is when asked to calculate it as in: 5×3 = ? or to solve a problem as in: if five children each get three apples, how many apples are needed altogether?
My first response is actually delight. I am delighted that these teachers are recognising that simply reciting a multiplication table does not mean much. Reciting a rhyme by heart does not mean that you know the meaning of the rhyme. Many of us can sing the national anthem without really knowing the meaning of all of the words!
Exactly the same is true of the children who recite their multiplication tables: they know the words, but the words have no meaning and the children do not see the relationship between the rhyme they are saying and the mathematics they are doing.
Are multiplication tables important? Of course they are — we want children to know their multiplication facts. We also want children to know their number facts (number bonds) and we want them to have efficient mental maths strategies.
Where teachers differ is on how they believe children come to know these facts. There are those who believe in memorisation. I do not agree with them. I believe that children will come to know the facts by being exposed to them in a large variety of different ways and by recognising the patterns inherent in the tables.
Let us begin with the easiest of all of the facts — multiplication by 10. There are a number of different ways in which we can help children to observe “the pattern”. In our work we like to use flow diagrams.
After children have completed a number of these types of flow diagrams they come to recognise that there is a pattern associated with multiplication by 10 — a pattern that involves “adding a zero”.
The teacher has two important roles to play here. First, (s)he must create a large number of situations through which the child is exposed to the pattern.
Second, (s)he must help children to reflect on the situations and contexts helping them to become aware of the patterns and to articulate them in their own words.
Having established the “10 pattern” we are ready to look at multiplication by five — a closely related pattern. We can use a series of flow diagrams to expose children to the relationship between the two.
Notice how many, but not all, of the input numbers of the two flow diagrams are the same. What we hope is that the children will recognise that multiplying a number by five is much the same as multiplying the number by 10 and then halving the answer. Mutiplying by 10 and halving the answer is often a lot easier than multiplying by five which, for many people, relies entirely on memory.
I need to make a very important cautionary remark here: please do not expect that children will “know” and/or “remember” the pattern after a single activity such as this. Children need frequent and repeated exposure to many situations such as these before they begin to recognise and then remember and finally use with confidence the relationships between the numbers and the operations.
The point is that we cannot do a series of activities such as those above on Monday and then believe that we have dealt with it by ticking it off in our work schedule and expecting children to use it with confidence from Tuesday morning onwards. Learning is a slow process that relies on frequent exposure and lots of practice. Mathematics is not learned in isolated lessons — it is learned over time, through repetition and practice. Teachers need to expose children to many opportunities that will allow them to see patterns and relationships before they can expect that the children will begin to assimilate them.
So far, we have dealt with multiplying by 10 and by five. Because we can see a pattern, we don’t have to remember “all of the sums”. Instead, we can multiply any number by 10 and five: 22×10 = 220, therefore 22×5 = half of 220 = 110.
So what is next? Multiplying by two, four and eight. Multiplying by two is simply doubling the number and multiplying by four is to double again and multiplying by eight is to double yet again. Again, we can use activities that expose the pattern — rather than telling it. This is why doubling is an important skill.
Exposure to many exercise opportunities such as these will help children to recognise patterns — patterns that will help them to develop a far more general and deeper sense of number than the mere recitation of the multiplication tables ever will.
Investigations are central to classrooms where we encourage children to: understand what they are doing; apply what they have learned to solve unfamiliar problems; reason about what they have done; compute with confidence; and engage enthusiastically.
The activities discussed in this column are also investigations that reveal the relationships that teachers want children to know. In many ways, effective mathematics teaching is about setting a trap — creating a situation from which the children can extract the mathematical patterns, relationships and concepts that we want them to develop.
The teacher’s role is both to create the situations and to facilitate the reflection which leads to the development of insight and understanding.
Aarnout Brombacher is a private maths consultant. For more information go to www.brombacher.co.za