- 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 the 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 students 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 to be learned, remembered and used to produce answers, teaching is focussed on developing students’ confidence in answering (examination type) questions, using standardised approaches (methods), and setting out their work in a particular way. This approach does not work. In South Africa 1,2mil children start school every year, 12 years later there are in the order of half a million students that attempt the school leaving certificate examination with only 200 000 offering mathematics. Of the students that offer mathematics 4 415 students (less than one half of a percent of the students that started school 12 years earlier) achieved an 80% or better score in 2019. The students that start school are better than this and capable of more. The approach to teaching mathematics lets them down. For most students their mathematics careers end in or about Grade 4.
To understand the reason why for many students their 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 she is 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, and not because they learned the words as sight words.
By using the analogy with reading it is exactly the same with mathematical concepts. Consider number bonds. 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). And, this is precisely where early mathematics teaching fails, it focusses first on working with numbers in a low number range (where you can use your fingers) and without using the concepts/knowledge in a range of different situations that contribute to the reification/objectification of the number fact so that it does not have to be reconstructed every time (just as in reading, reading the same words in many different passages contributes to the words becoming sight word). 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/recognising what it means to multiply by 10, equips the student to multiply any number by 15 (for example: 23 × 15 = 23 × 10 + 23 × 10 ÷ 2 = 230 + 115 = 345), being able to double numbers equips student 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 students 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 (explaining) 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, students 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.