A Mathematical Mind
Raising a Numerate Child Without Agony, Hopefully

By Denise Lai, BA, BSocSc (Hons), MEd

I remember how one of my classmates at college tried to persuade me to keep on studying Economics, a supposed compulsory subject for government scholars then. “You have such a huge emotional block about it,” he said, in reply to my strong insistence that I was going to spend my life helping humanity instead. Not surprisingly, I never spoke much with the chap again!

But I continued to be haunted by Mathematics in all of its various forms long after I left college. Soon after admission into University, when I thought I had left algebra and geometry behind, I was made to study Basic (and later, Advanced) Statistics as part of my Psychology course! It was a true case of "Bye, pï. Hi, chï." It seemed as if there was no running away. Whether I liked it or not, whether I would accept it as a friend, or fight it like a foe, Mathematics was going to be a part of my life… forever…!

And of course, to cut the rhetoric short, Mathematics really is a part of our lives. For all of us, regardless of what we do, within the confines of our homes, or within the four walls of our office, the need to count, think numerically, and problem-solve, is an absolute given. We need an understanding of Mathematics to tell the time, use the telephone, look for a unit in an apartment block (especially in Singapore!), sew a set of curtains, cook laksa, buy petrol, or calculate the cost of ten apples at Cold Storage (and then determine whether we have enough money to make the purchase, and how much change would come back, if any).

The Dismal State of Mathematics Education
Sadly though, the research literature on Mathematics achievement and the teaching-learning processes associated with it, are downright dismal. Apparently, my experience disliking the subject throughout school was in no way singular or exceptional. A Dutch scholar by the name of Wolters writes,

"There are far too many children who dislike arithmetic or worse, children who think it is a stupid school subject. With relatively few exceptions, this situation is quite general and has to be taken for granted."

(cited in Wood, 1994)

Wood himself declares,

"Mathematics is difficult to learn and hard to teach… the early foundations of mathematical knowledge are often very shaky"

Such words bode poorly for a teacher like me now, where on the other side of the fence, I both deeply empathize with the lack of motivation my young charges display, yet recognize the need to equip them with the numeric skills that they must have in order to thrive in the future, both at school and in life.

The Objectives of Early Mathematics Instruction
After all, if you really think carefully about what an understanding of Early Numeracy involves, you will have to admit that it is not that easy to acquire or to convey. By the age of 6, the young pre-school child is expected to be able to:

And if you multiply these abilities in sets of ten to a hundred (which is what many school-teachers expect of our little ones to be fairly aware of within their first year in primary school), as well as consider the other associated concepts that they must learn (such as weight, volume, length, money and time), then you know that you, and your child, may well be in for a pretty rough road ahead.

Positive News about Effective Teaching Approaches¹
Interestingly enough though, many educational researchers and theorists are of the view that Mathematics need not be a torture to learn or to teach. In a seminal study conducted in 1965, the psychologist Jerome Bruner and his colleague, Kenney, showed, in a series of experiments, how even children as young as 8 years could be taught to understand and to use the quadratic equation, as long as they had had enough time exploring and manipulating a concrete illustration of the concept; in this case, wooden blocks and a balance beam (materials that in turn, had been invented by a mathematician and teacher, Dienes).

(1) The results from these experiments were used to de-bunk a previously and widely accepted Piagetian belief that children at certain stages of development could not acquire concepts from a higher or later stage of development ("readiness" theory). They also showed how "procedures must ultimately be grounded in practical activities if they are to be understood." In other words, the use of paper and pencil activities such as sheets and sheets of assessment papers, rote-counting or traditional blackboard instruction, do nothing for abstract learning if they are not accompanied by practical experiences that illustrate the mathematical concept being taught. Have you ever seen the Number Rods or Golden Beads that are used in Montessori classrooms? Well, they are just some of the ways in which teachers have sought to implement this practice-oriented idea.

Bruner however, mentions yet another important principle that is required for the real and effective learning of Mathematical concepts, a principle that is not shared, unfortunately, by teachers trained in the Montessori method. He realized that when probed on their thought processes whilst solving mathematical problems, many children in his studies (a) did not have a mathematical vocabulary and (b) could not regulate their own thinking to pursue a solution logically. To put this in simpler language, not only could the children not solve mathematical problems presented to them in word sums, they were also impulsive in coming up with the answers to these problems.

A 'simple' request such as "take 3 apples away from 5 apples" or "share 8 apples between 4 children," for example, demand an understanding that in the first instance, the child is required to subtract or to minus 3 from 5. In the second case, the child is required to divide 8 by 4. Time and time again, researchers observed that, "many children in the early stages… do not even recognise which operations they should perform on a verbally stated problem…" They may subtract 5 from 3, or subtract 4 from 8 in the examples given above, without "asking oneself questions, reminding oneself, looking for new evidence (or) trying to view the problem from a different angle" (Wood, p.197).

To overcome these learning hurdles, Bruner made another significant recommendation in the teaching and learning of Mathematics. (2) Children, as novices in our world, must interact with expert others – their parents, teachers and other adults – in an individualized way as far as possible, to (a) negotiate (discuss) the meanings and interpretations of the vocabulary that is distinct in Mathematics concepts and (b) internalize the higher-level thinking (meta-cognitive) skills required to problem-solve. Only then would they be able to answer future and challenging questions such as, "If it is 5 kilometres to Jalan Ahmad in the west, and 2 kilometres to Jalan Hussein in the east, what is the distance between Jalan Ahmad and Jalan Hussein?"

To cut it short then, one simply cannot discount the effects of concrete (play) materials and adult involvement in helping the young child become numerate and able. Concrete materials aid abstract thinking by making ideas/problems more plausible and easier to visualize. Adult involvement in instructing, guiding, suggesting and reviewing, equip the young child to use the right sorts of thinking strategies to understand and to solve problems.

Food for thought, surely, and implications for how we can better enhance our lessons, both at home, in school, and when out shopping!

1 I am indebted to a paper written by David Wood, Professor of Psychology at Nottingham University, for many of the views and ideas put forth in this article. References available upon request.

If you have found the information in this article useful, please pass it on to your friends.

For more information on the Bright Starts Programme for ages 2-4yo, please visit our website at www.weecare.com.sg.