- Background to the NumberSense Mathematics Programme
To understand the NumberSense Mathematics Programme, it is important to understand and have a sense of what it means to do mathematics at school. There are, broadly speaking, two views. On one hand, mathematics is regarded as a collection of rules, formulae and procedures to be learned, remembered and used to produce answers. This view is particularly popular in school mathematics. On the other hand, there is the view that mathematics is a sense-making, problem-solving activity in which children are expected to develop a body of knowledge that they can apply in unfamiliar situations with understanding and reasoning.
In contexts where mathematics is regarded as a collection of rules, formulae and procedures, teaching is focused on developing children’s confidence in answering questions (examination type), using standardised approaches (methods) and setting out their work in a particular way. This approach does not work. In South Africa 1,2 mil. children start school every year. 12 years later there are in the order of half a million learners that attempt the school leaving certificate examination – with only 200 000 offering mathematics. Of the learners that offer mathematics, 4 415 learners (less than one half of a percent of the children that started school 12 years earlier) achieved an 80% or higher score. The children that start school are better than this and capable of more. The approach to teaching mathematics lets them down. For most children, their mathematics careers end in or about Grade 4.
To understand the reason why many children’s mathematics careers end in Grade 4, it is helpful to explore how children learn to read. To read fluently and with comprehension we need to reach a level of automaticity. That is, many of the words we read need to be sight words. However, we cannot teach words as sight words because the reader simply cannot cope with the cognitive load, and, the reader will not be equipped to deal with an unfamiliar word when confronted with one. For this reason, we must teach young readers to read by breaking words up into their constituent sounds and putting the sounds back together to form words. As children use this technique and read a great deal, some of the words will become more familiar, more automatic and finally, sight words. Children move from sounding words to reading them as sight words because they read a lot, not because they memorised the words as sight words.
So too with mathematical concepts. Consider number bonds. These are the foundational “facts” that form the basis of arithmetic and even algebra. We need children to reach a point where they simply know without having to think about it, that 6 + 2 is 8. More so, when they know that 6 + 2 is 8, they should be able to apply that fact to calculate 26 + 2, 60 + 20, 600 + 200, and even 7 + 1. Coming to know that 6 + 2 is 8 (as a fact) involves using the information in lots of different ways (as in reading words in lots of different contexts/passages contributes to the word developing into a sight word). This is precisely where early mathematics teaching fails: it focuses on working with numbers in a low number range (where you can use your fingers) and without using concepts in a range of different situations that would contribute to the number fact being made concrete and not needing to be reconstructed every time. Just as in reading, words become sight words when the same words are read in many different passages. There are no short cuts. And, no amount of lessons (and extra lessons) that focus on drill and the reproduction of methods and procedures will develop this deeper understanding.
To make the point in a different way, being able to recite one’s multiplication tables does not develop the flexibility to apply the multiplication facts (which you “know”) with reasoning and understanding in unfamiliar situations. By contrast, knowing what it means to multiply by 10 equips children to multiply any number by 15 (for example: 23 × 15 = 23 × 10 + 23 × 10 ÷ 2 = 230 + 115 = 345); being able to double numbers equips children to multiply any number by 2, 4, 8 and even 16 (for example: 23 × 16 = 23 × 2 × 2 × 2 × 2 = 46 × 2 × 2 × 2 = 92 × 2 × 2 = 184 × 2 = 368) and so on. To be clear, this is not to say that children should not be expected to know their “number bonds” or “multiplication facts” – they should! The issue is with how they come to know these facts and whether or not they can apply the facts with fluency and flexibility in unfamiliar situations all the while reasoning about what it is that they are doing.
Before getting to the specifics of the NumberSense Mathematics Programme, there is one more idea that is important. While in the senior years of high school (and beyond) it makes sense to dedicate different blocks of time and even courses to different topics (e.g. trigonometry; calculus; and probability), the same does not apply to early grade mathematics. If we teach addition, subtraction, multiplication and division as separate topics each with their own rules and procedures, children will not develop an understanding of and appreciation for the interrelated nature of the four “basic” operations. Almost all early grade subtraction problems can be solved by addition (counting up), almost all multiplication problems can be solved by counting in groups or repeated addition, and, the same applies for division which can be thought of in at least two different ways … sharing and grouping, with the latter involving repeated addition.