Mathematics Part 1: The mathematics pendulum

I have long wanted to have Charlotte Wilkinson, an independent mathematics consultant, set out her ideas on mathematics but, in the previous education environment, any association with me would have been dangerous for her work. With that changed, I am delighted to present two writings from her which are an overview of nearly everything in mathematics. 

This is the second major writing on mathematics in The File

In Attacks! 15, 28, 29, there is a detailed description of Dan Murphy’s non-streamed, problem-based way to undertake mathematics. There are other ways of taking mathematics, but they all have problem-solving at their heart as the way to discern and understand patterns.

Attack! 15 Holistic mathematics Part 1

Attack! 28 Holistic mathematics Part 2

Attack! 29 Holistic mathematics Part 3

In Dan Murphy’s mathematics, the problems are, in a sense, bigger; Charlotte Wilkinson’s (if you go through her programme) are smaller and run faster together. I was involved with a school in implementing Dan Murphy’s mathematics and found it was first-rate for getting children into mathematics from which principles could be extracted. The same can certainly be said for Charlotte Wilkinson’s programme. What Charlotte has provided here are mathematics’ principles in concise and powerful form – a careful allocation and application of those principles will take teachers and children to the heart of mathematics. 

The pendulum on the best way to teach mathematics began 200 years ago in the publication of two opposing philosophies of teaching methods published just over 30 years apart. In the 30 years between 1788 and the text written by Pike, and 1821 and the text written by Colburn, the world had greatly changed. It was on the brink of the industrial revolution. To create that revolution, people were required who were able to think critically, creatively, strategically, and logically.

No longer was the philosophy of the teaching process based on state a rule; give an example; then complete practice exercises.

Colburn’s text set out the need for sequenced questions using concrete materials, also to postpone practice until children understood the mathematics.

If the goal is to get answers correct then, as in Pike’s publication, all children require is instructional understanding that met the requirements of mathematics 230 years ago. The philosophy being that mathematics is about getting correct answers and children needing to be able to recall facts and apply set procedures to familiar problems from everyday life.

If, however, the philosophy is based on the belief that students need to build an understanding of mathematical concepts, to explore why processes or formulas work, and to connect pieces of knowledge together – then the pendulum swings toward Colburn’s publication and the recognised requirements for a rapidly changing world.

For many children who struggle with mathematics, teaching instruction is limited to instructional understanding. Because they are getting answers right in tests, learning facts by rote, and recalling them at speed, this leads them to be appearing to make progress. But with no real understanding, the inevitable outcome is mathematics’ anxiety, confusion, and failure.

The one aspect both sides of the pendulum agree on is the need for practice but practice for improvement requires intrinsic motivation and the desire to improve.

Children whose school mathematics learning experiences have been limited to instructional understanding will often be defeated in later mathematics and end up seeing themselves as ‘no good at maths’, contributing to the fallacy of brains that are no good at mathematics.

Teaching that focuses on children’s understanding both in what to do and why, leads to transferable mathematics knowledge. As new knowledge is connected to knowledge children already know, the new learning becomes easier to remember. This is a highly satisfying intellectually experience. But there are important provisos, such an experience only develops over time and becomes more complex as children increase their connections between ideas.

Our mathematics curriculum document (p. 26) states that mathematics is the exploration and use of patterns and relationships in quantities, space, and time, also that statistics is the study of patterns in data. It states the reason for studying mathematics is for children to develop the ability to think creatively, critically, strategically, and logically. They should also learn to structure and to organise, to carry out procedures flexibly and accurately, to process and communicate information, and to enjoy intellectual challenge.

The curriculum document points us firmly in the direction of teaching for deeper understanding but, as emphasised, that will take time, it is not for rushing.

Two reasons why this deeper understanding has not happened for all of our students:

1. National standards and the assessment tools available for mathematics. The GloSS interview requires children to be able to perform specific procedures at particular stages of learning. If children can perform the procedures and, in many cases name and describe the procedure they are using, they are assessed as at that level of understanding. While national standards did require overall teacher judgement, many teachers put the emphasis on GloSS results.

‘When an area of children’s learning is measured in a high stakes environment, what children learn is not learning occurring then being measured, it is learning being changed for the measurement and by the measurement’ (from a previous posting by Kelvin).

Under the pressure of national standards and immediate measurement, the hoped-for real understanding was changed for many children to instructional understanding.

2. Teachers’ conceptual understanding of mathematics. Teachers are the product of an education system that left many of them with only an instructional understanding of mathematics. Most teachers state they can do mathematics, many have secondary qualifications in mathematics, but that does not necessarily mean they have the conceptual understanding of the mathematics, or sufficiently made those connections themselves to enable them to make those vital mathematical connections in children. This is especially true for the early levels of mathematics, as many of early concepts are in the unconscious and often not recognised as early mathematical conceptual building blocks. Connecting those ideas is how concepts deepen and expand and become more complex. If early ideas are missing then further development is very difficult. The idea that teacher training (as a previous minister of education informed me) ensures teachers leave teachers college with everything they will ever need to know to teach mathematics was never realistic.

The roll out of the Numeracy Project started with the good intentions of improving teachers’ conceptual understanding. But because developing conceptual understanding takes time, very early on it became clear (2002) that this was going to be an enormous task. A series of five three-hour workshops and three in-class observations was never going to be sufficient. To keep the minister interested, the project became dependent on immediate and measureable results, leading to the Numeracy Project becoming the victim of a quick fix programme. It was, as a result, delivered to teachers at an instructional understanding level, with GloSS further perpetuating the status quo.

Schools in countries that achieve highly, value their teachers as learners, with professional learning built into their working week. Without regular professional learning to meet teachers’ individual learning needs, the development of mathematical conceptual understanding will continue to be limited, meaning the number of numerate adults required for our rapidly changing future will be compromised.

Charlotte Wilkinson is an independent education consultant (MOE Accredited #654) and resource developer (The Wilkie Way & Pearson Mathematics, Primary Mathematics Assessment Tool [Available from]) specialising in Primary Mathematics

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