Abstract concepts are better for teaching maths than real-world ones, according to a study reported on in the New York Times.
Teasing out the principles behind equivalent numbers may not be as straightforward when you’re thinking about bread and fruit, but at least you have a jumping-off point, a handle for the young mind to grasp. We explore the world at first using all our senses – touch, vision, hearing, smell and even taste – and many students (and adults) continue to understand the world in terms of physical analogies. As an educational researcher commented in the NY Times piece: “Some children need manipulatives to learn math basics… but only as a starting point.”
Indeed, it would be absurd for an accountant to have to count out pound coins and pennies whenever she audited a company – she has to be able to abstract and apply general mathematical rules to specific problems. Taking aside facetious thoughts that some banking experts might have failed basic counting and adding up, it seems only right that educational methods help train young brains to think abstractly.
Teaching maths through history
Ancient Greek mathematicians famously had difficulty thinking of numbers larger than ten thousand (known, then, as a ‘myriad’) and solved the problem by just multiplying myriads. It is thought the base 10 number system has arisen because, in short, we have ten fingers and thumbs. Both these examples hint that we have historically started with concrete examples and moved to the abstract as science and mathematics have matured. Science often works by collecting testable, real-world examples and building those into a framework that extends from the specific to the more general. If maths is said to be the science of patterns, this method can apply to numbers as well as it can to biology or physics.
This process of moving from concrete to abstract is reflected in the Maths-Whizz curriculum. We begin, at Foundation (roughly age 5 equivalent), by giving examples of ducks getting on and off buses, with eggs in baskets and buckets of water. Moving into Key Stage 2 our students have greater understanding of the number system and confidence with numerals and symbols. We can start to present worded problems, pencil-and-paper style problems, and basic algebra.
Finally, at Key Stage 3 we encourage students to create their own equations from given scenarios and extend given equations into new scenarios. The topics added at year 7 – Probability; Equations Formulae and Identities; Integers, Powers and Roots; Sequences, Functions and Graphs – reflect this new emphasis.
Even if we represent those scenarios visually – goats and fences one such example – Maths-Whizz students must always be able to give more formal, abstract answers. Our exam-style tests that follow each of our animated exercises are valuable tools for consolidating and reinforcing maths, even maths learned with jumping dogs and flying cakes…