# Teaching Preschoolers The Algebra of Mathematics

Here’s a question I spotted from a primary one CA test paper :

Susan and Mary have 8 chocolate cookies altogether. Mary has 4 more cookies than Susan. How many chocolate cookies does Susan have?

When I showed this question to fellow parents, the most common reaction I get from these parents is perplexity at solving this without having to resort to algebra.

“Isn’t this something we learn in secondary school?” “How can a seven year-old work out a problem sum like this?” were among some common complaints.

Well, in fact, I have proven that even 5 years old can solve problem sums like this if they have been given the right tools and foundation in learning. In my years of studying and learning about early childhood teaching (I’m still learning every day), I have been heavily influenced by Lev Vygotsky, the revolutionary Russian thinker whose study on developmental psychology inspired generations of educationists. It is through the study of Vygotsky’s theories and the under the mentorship of Galina Dolya, a Russian educationist currently based in UK, that helped me understand that the first step towards teaching mathematics is to help children first develop their understanding of mathematical relationships. In other words, before we even teach children to count, we teach them algebra – mathematical relationships.

What do I mean by this? First and foremost, we as adults have to realize that a number is not just a digit, a label, it is the expression of a relationship between the unit of quantification and the objects or features quantified. For example, if I were to place a picture of a group of butterflies in front of you, you could say 3 butterflies but another person could say 6 wings. The digit 3 or 6 have no meaning on its own unless we understand the underlying relationships and it is exactly this grasp of the underlying relationships that we need to help our children to master so as to unlock the mystery of number for them.

So, how do we do it, some might ask. For children, the most important tool we can give to our children to help them grasp the underlying mathematical relationships is visual models. And there are many, many different kinds of visual models. Visual models start off with replacing concrete objects with symbols or tokens – for example, yellow circles to represent chocolate cookies. Progressively, we then introduce other visual models such as correspondence grids, number lines, venn diagram and decision-making trees. As children internalize the various visual models, they practice solving various problems by “looking” with their “mind’s eye”.

So, to solve a problem mentioned at the beginning of the article, the children first need to understand the most basic but also the most fundamental mathematical relationships : more than, less than, equal to. Some children get confused when they are introduced the idea that a number can be “less than” one number and “greater than” another simultaneously. In such cases, the combined use of number line and venn diagram is useful to visually show the children the relationship between numbers so that they can “see” the relationship for themselves.

This is but one example of how visual models can be made use of to develop children’s understanding of the algebra of mathematics. What is important to remember, as parents, is to encourage children to develop quality understanding of the relationships between numbers, rather than focusing on rote counting of numbers. Even if they can count all the way up to 100 at the age of 3, it doesn’t mean they can solve problems later on.