Do We Want Our Students to Understand What They Are Doing or Follow a “Math(s) Recipe”?
In a previous post, I explained some aspects of the exam obsession in English schools. I want to show here with a few examples how math(s) educators sometimes choose to stick to a “recipe” in whatever situation, thinking it’s the best method to prepare their students to the exam, but at the expense of stimulating thinking and understanding.
The standard algorithm to solve linear equations is called the balance method. It is based on the fact that the equation remains true (balanced) as long as we perform exactly the same operations on both sides of the equation. The idea is therefore to carry out a series of operations on both sides of the equations to end up with “unknown = a number”. So, for the equation x -5 = 0, we add 5 to both sides, which gives x -5 + 5 = 0 + 5, i.e., x = 5. However, there is no need to apply the standard algorithm here since the question “What is the number that gives 0 when we take away 5 from it?” is easily answered by young children long before they can use algebra. But many teachers will stick to the ‘solving procedure’ whatever comes, to be sure students know how to solve an equation, even if, by doing so, they prevent them from thinking about what the equation actually says and linking it with their basic number sense.
When solving equations involving absolute values, the general method is to derive two equations from the original equation: one in which the term inside the absolute value symbol is supposed to be positive, so its absolute value is equal to itself, and another in which this term is supposed to be negative, so its absolute value is equal to its opposite (i.e., itself multiplied by -1). So, for an equation of the form |x + 6| = x + 6, we would do: 1) If x + 6 ≥ 0, that is, if x ≥-6, then x + 6 = x + 6, so any value of x such that x ≥ -6 is a solution of the equation |x + 6| = x + 6. 2) If x + 6 < 0, that is, if x < -6, then –(x + 6) = x + 6, i.e., -x -6 = x + 6, 2 x = -12, x = -6. As x = -6 is not compatible with x < -6, there is no solution to the equation |x + 6| = x + 6 if x + 6 < 0. The final solution is therefore x ≥ -6.
The method works well. However, thinking about what the equation is saying makes solving it much more interesting — and actually faster. |x + 6| = x + 6 means that the number x + 6 is equal to its absolute value. In other words, it’s positive or zero, hence x + 6 ≥ 0, x ≥ -6.
Often, when children learn about multiplication of fractions, they will first study the case of a fraction multiplied by an integer, e.g. 3 × 2/5 = 6/5. Then, they will learn that to multiply two fractions together we multiply the numerators together and the denominators together. (And there are many ways to show why it is so.) The same applies for dividing fractions; students learn first what happens when we divide a fraction by a whole number. For instance, ¼ ÷ 3 means dividing a quarter into three equal parts, each of them being one twelfth (one can visualise this as splitting each quarter of the whole into 3, which gives 12 twelfths for the whole) — or, it is simply 1/3 of ¼. Then, they might see the case of dividing a whole number by a fraction. For instance, dividing 2 by ¼ is equivalent to answering the question “How many quarters are there in 2?”. There are 4 quarters in 1 (here, we consider 1 as being the whole), so there are 8 quarters in 2. They may then learn more complex divisions such as 4 ÷ 2/5, seeing that because there are 20 fifths in 4, this division is equivalent to 20 ÷ 2. However, as soon as children are taught the method for dividing fractions by fractions as “a/b ÷ c/d = a/b × d/c” (“flip and multiply”), they are strangely regarded as not being able to deal any more with a ÷ b/c without rewriting it as a/1 ÷ b/c = a/1 × c/b. This is as if multiplication and division of an integer by a fraction couldn’t be worked out anymore without using the corresponding general ‘method’.
The gradient (or slope in American English) of a line quantifies the steepness of a line, namely, it gives by how much the line goes up or down (vertical change) going from left to right between two points 1 unit apart horizontally. The gradient is the number in the equation of a line written in the form y = mx + c (or y = mx + b in the USA). There are different ways to show that m is the coefficient of proportionality between any change in x and the corresponding change in y. One is geometric; it shows that whatever two points we choose on the line, the right triangles formed with the line segment between the two points, a horizontal and a vertical segments (see the diagram below) are all similar (they all have the same angle measures). It follows that their sides are in the same proportions. We can start with two points that are 1 unit apart horizontally and call m the change in y going from the point on the left to that on the right. Then, choosing a third point that is at a distance of k units from the first point on the left, then this third point is at a vertical distance k × m from the first point. To illustrate the meaning of the gradient (slope) m, we can therefore draw arrows that show that a difference Δx between the x-coordinates of two points on the line corresponds to a difference in their y-coordinates Δy = m Δx.
However, in many textbooks and websites, the arrows are drawn on the ‘other’ right triangle, so that we ‘read’ the change in y first, and then the change in x, as shown below.
It doesn’t make much sense from a conceptual point of view, so why is it used? Because it’s much closer to the formula to calculate m: m = Δy/Δx; as Δy comes first in the equation, it’s better to indicate it first… and never mind if we remove the opportunity from students to consolidate their understanding of the meaning of m, which explains why we calculate it this way.
Powers of powers. When a monomial (a product of a constant and variables raised to different powers) is raised to a power, then each factor of the monomial is raised to that power. It becomes very clear when e.g. (a⁴b²)³ is rewritten as (a⁴b²) (a⁴b²) (a⁴b²), which gives, using the commutativity and associativity of multiplication: (a⁴)³×(b²)³=a¹²b⁶. The general rule is then written as
However, when solving e.g. (3x³y⁵)², some students may be taught to simply apply the general rule and be told by their teacher to first rewrite this as (3¹x³y⁵)² — because the risk is high they forget to square the 3. It is as if students can only apply the general rule without knowing what they’re doing and don’t understand that we are here squaring each factor inside the brackets, and so 3 squared is… 3².
The general method to rationalise a fraction, that is, to get rid of any surds in its denominator, is to multiply the numerator and denominator of the fraction by a number such that the denominator will become rational. The simplest case is when the denominator is just √a. Multiplying the fraction by √a/√a will then give a fraction with a as denominator. With the fraction a/√a, however, we have a particular case. While the general method can be of course applied, I find it is a shame not to prompt students to think about the actual numbers and recognise that, by definition, a is the square of √a, , and so a/√a= √a.
Finally, acronyms are obviously believed by some to be the best insurance that students will remember what to do. We didn’t use them in my secondary education in France. This may explain why I have been surprised to encounter so many of them in maths education. But the thing is, I find it is often more difficult to remember the acronym than simply understand what we’re talking about. Take BIDMAS to start with. The only thing that one needs to understand and remember about the order of operations is that multiplication and division have priority over addition and subtraction (while the order between multiplication and division, or between addition and subtraction, doesn’t matter, e.g. 3 × 8 ÷ 4 = 3 ÷ 4 × 8) , unless there are brackets, in which case the operation(s) inside the brackets has (have) to be performed first. Powers (the “I” in BIDMAS stands for indices, which is commonly used to designate powers in the UK) are repeated multiplications, so they will have priority over addition and subtraction. However, the “I” is just after the “B”, which could be misleading for students who know the acronym but don’t understand the general idea. For instance, for calculating (3 + 5²) × 6, the brackets indicate that the addition is performed before the multiplication by 6. Inside the brackets, however, one has to work out 5² before performing the addition because it is a multiplication (5 × 5) and so has priority over the addition. So, we’ll actually carry out the ’indices’ before the ‘brackets’ here. Now, with (3 + 2)⁴, the brackets mean that the addition 3 + 2 is carried out before raising to the fourth power, which is simply (3 + 2) × (3 + 2) × (3 + 2) × (3 + 2), i.e., a repeated multiplication. We can just apply the rule that the addition inside the brackets has priority over the multiplication outside the brackets, so no need for BIDMAS really.
Take FOIL as a second example, which stands for First, Outside, Inside, Last, meaning “multiply first terms in both pairs of brackets together, then both extreme terms (‘outside’ terms are the first in the first brackets and the second in the second brackets), then both terms ‘inside’, that is, the second in the first brackets and the first in the second brackets, and finally both ‘last terms’, i.e. both second terms.”. Is it really useful to remember this acronym? I would much prefer using the area model of multiplication for (a + b) × (c + d) with a diagram like this:
With this model, students understand the general rule: each term inside the first pair of brackets is multiplied with each term inside the second pair of brackets. The risk when expanding brackets is to forget one product. For this, students can use arrows to develop the habit of proceeding systematically:
Most importantly, if students have understood why we multiply binomials in this way, they can transfer this skill to the multiplication of e.g. a trinomial by a binomial.
A last example of acronym is SATC, which stands for Sine, All, Tangent, Cosine (remembered as “Sugar Add To Coffee”), also called CAST, and is supposed to help students remember for which angles the values of the sine, cosine, and tangent functions are positive. The acronym is associated with the four quadrants in this way
If you have never seen this diagram — lucky you! If you have, then maybe you can explain to me in which way it is supposed to help students. If quadrants are used, then I suppose that students have learned how to use the unit circle to associate an angle θ to a point P on its circumference, namely the angle between the positive direction of the x-axis and vector OP. When this has been taught properly, there is no need for SATC. The only thing students need to remember is that the cosine value of angle θ is given by the x-coordinate of P, the sine value by its y-coordinate, and the tangent value by the ratio sinθ/cosθ, which is the x-coordinate of the intersection of the line OP with the axis parallel to the y-axis that passes through point (1,0) (therefore tangent to the unit circle). Once students see the trigonometric functions in this way, they won’t need SATC, because they will easily mentally visualise these diagrams:
AND they will much better understand trigonometry and its use with vector components!
