Abstraction and Subtraction
Abstract concepts are better for teaching maths than real-world ones, according to a study reported on in a recent New York Times article. What does this mean for teaching maths and what does it mean for Maths-Whizz?
Rather than offer endless examples of trains setting off from stations, apples in baskets, bars of chocolate and slices of pizza, teachers might well be encouraged to have their students think of problems less in practical terms and more in terms of non-specific abstractions.
The study in question, conducted by researchers at Ohio State University, presented university students with a set of mathematical rules, expressed either with abstract symbols or with concrete examples (tennis balls, and so forth):
“Then the students were tested on a different situation — what they were told was a children’s game — that used the same math.
“The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.”
The implication here is that abstract examples are transferable; students who study concrete examples are less able to ascertain the underlying principles, or apply those principles to other problems.
The article goes on to say that this issue is as relevant for younger brains as those of university age, and that this has implications for how elementary, or primary, maths is taught:
“…researchers suggested that their findings might also be true for math education in elementary through high school, the subject of decades of debates about the best teaching methods.
“The researchers said they had experimental evidence showing a similar effect with 11-year-old children. The findings run counter to what Dr. Kaminski said was a “pervasive assumption” among math educators that concrete examples help more children better understand math.”
Whilst not all researchers in this field might agree with the implications for primary education, the principle seems sound. After all, it is quicker to write an equation than it is to count out apples or cut up loaves of bread.
But, and here’s the caveat from this corner, it is natural to analogise in education. In other words, we readily use specific examples from life to extrapolate to the general. In some disciplines (like economics or history) particular examples can be less instructive when looking at the general issue. But it is just as true that 2 and a half apples are equivalent to 5 half-apples as it is true that 12 quarter-loaves are equivalent to 3 whole loaves.
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.
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…