Welcome to ‘mathsland’
Should we be teaching Year 8 mathematics students in a way that suggests to them that maths comprises a collection of rules, tricks, rote procedures, and ‘my favourite way of doing it’, where the experience and experimentation that were the basis for early years learning, are no longer important? Welcome to ‘Mathsland’, where this is typical teaching practice.
I have just experienced the fourth in a weekly series of online revision classes for Year 8 maths students. The first session was interesting – I was keen to see how the teacher approached the task, used technology, interacted with the students, presented resources, and how the students responded to this learning environment. Subsequent sessions have been less than riveting, indeed rather disturbing.
Today’s lesson was on elementary algebraic manipulation – defining ‘like terms’, and then working with expressions involving addition, multiplication, and division of mixed algebraic terms. A puzzle was used as a motivator, and was revisited a couple of times later in the lesson – you know the kind of thing: a graphic image shown in four rows, with Row 1 showing 3 chickens said to be ‘equal’ to 60; Row 2 showing 1 chicken plus 2 nests (containing eggs) said to be equal to 26; Row 3 with 1 chicken plus 1 nest with eggs plus 1 bunch of bananas, together said to equal another number, and finally, Row 4 showing an expression (using images) to evaluate:
chicken + nest (with eggs) × partial bunch of bananas.
I can appreciate these types of puzzles, but the whole scenario is unrealistic. The relationship between the objects used in the puzzle and the mathematics to be used is non-existent. Solving the problem depends, first, on a complete suspension of all sense of reality (that is, we have entered Mathsland, where strange things happen for no apparent reason, and I am expected to decipher obscure clues to come up with an answer that will be deemed ‘correct’ or ‘incorrect’). Second, solving the problems depends on application of a mathematical rule, or convention, which also bears absolutely no connection with the objects of the puzzle. By the way, this Mathsland is the same place where people buy 69 watermelons, and nobody wonders why. A colleague recently gave me another example of the extremes possible in Mathsland with questions like this: If a person weighing 100kg needs a chair with four legs, how many legs should a chair for a 50kg person have?
The lesson I watched was filled with ‘algebraic expressions’ that related to nothing in particular, yet students were expected to decode the information, apply some rules or conventions, or to proceed using ‘the way I [the teacher] usually like to do these problems’. The language was loose. For example, in establishing that the expression pq is equivalent to the expression qp, the commutativity of multiplication was invoked and explained by saying that the letters in these expressions represent numbers. If you worked out a similar problem with numbers, such as 4 × 3, you realise it has the same value as 3 × 4, so because the numbers are commutative it is safe to assume the letters are also commutative. However, I note that it is the operation that is commutative, not the letters or numbers. This imprecision with language can so easily cause confusion and almost always needs to be repaired later.
If I was a student who loved engaging with abstract systems that worked by applying a set of rules and procedures with their own internal logic, perhaps I would get through Year 8 maths easily and look eagerly forward to Year 9, when the teacher has promised that I will be allowed to look at an expanded set of expressions that also include ‘powers’. This is the ‘lock-step’ approach to curriculum typically seen in Mathsland.
Unfortunately, many students in Year 8 are not immediately excited by abstraction. The fact that they are all expected to go through the same set of learning steps that assume they all relate fulsomely to these types of abstractions, is likely why so many of them come to hate maths, think it makes little sense and does not connect to anything they understand about their lives. Dropping out of maths at the earliest opportunity is the consequence of this disaffection. Even worse, the students may be receiving the message that they are not capable of learning the stuff that everyone says is so important.
When I was a student (I went to school in Mathsland), I was taught the mnemonic BODMAS (standing for Brackets, Of, Division, Multiplication, Addition, Subtraction) that was intended to help me remember the mathematical convention for the correct order in which multiple arithmetic operations should be performed. For example, BODMAS would tell you to evaluate the expression 5 + 2 × 3 by doing the multiplication first (2 × 3), and then the addition (5 + 6), giving a result of 11. If you did the addition first, you would get 21, and would probably be marked incorrect.
Nowadays, I think of calculations like this a bit differently. The experiences people have can sometimes lead to the need to carry out a sequence of calculations that would produce a different result depending on the order in which those calculations were performed. For example, a taxi fare typically involves a flat charge (the ‘flag fall’, let’s say it is $5) and a charge for each kilometre travelled (let’s say I want to go three km, and the rate is $2 per km). If one wanted to formulate an expression to calculate the cost, the formulation would be firmly based on the problem situation being considered. The cost (C) of the fare would be $5 plus two lots of $3. Written as a formula,
C = 5 + 2 × 3.
Of course, this formula could easily be generalised to help calculate the cost of a taxi fare for a different company, or for a different time of day, perhaps as follows:
C = f + k × d
(where the cost C is calculated as the fixed flagfall amount f, plus the product of the kilometre rate k and the distance travelled, in kilometres).
Each element of the mathematical formulation would relate to some feature of the problem being explored. The way in which the mathematical formulation would be processed to solve the problem would be clear from the demands and nature of the problem. No BODMAS needed.
The issues I highlight here are not just restricted to Year 8 nor to Year 8 maths teachers. They are relevant right through schooling. Abstraction is a very important part of mathematics – you could say an essential and defining feature. However, the ability to abstract and to use abstraction is built on the same fundamental experiential learning that helps young children learn to count. First, they notice the separation of objects in their world, which is an essential precondition for countability (‘mum’ and ‘dog’ are different objects). They learn the names and symbols for numeration, and that there is meaning in the ordering of those names and symbols (the numbers displayed in an elevator are ordered, and all elevators use the same ordering). Young children develop an idea of correspondence between objects and their labels (this is 1 apple; here are 2 apples). They learn that the last number of a count represents the number of objects counted (1, 2, 3: there are 3 apples). They learn that the number of objects in a set is an abstract feature not fundamentally related to the characteristics of the objects counted (the colour of the apples doesn’t affect the count). And they learn that when counting a set of objects, they can start with any of them – the order in which the objects are counted doesn’t affect the count.
These are fundamental aspects of counting, an abstract activity that is firmly rooted in experience and experimentation. None of us learned that in Mathsland.
Perhaps maths instruction right through school can continue to emphasise experiential learning, and development of mathematical understanding based on underlying meaning, thereby building abstract principles from a firmer conceptual base?
ROSS TURNER
ACER, April 2020