If you have an algebraic expression of the form:
You can use the distributive property of multiplication to multiply the two brackets (i.e. multiple each term in the first bracket by each term in the second). When you do, you will usually produce a ‘trinomial’ i.e. an expression with and unit terms. For example:
However, going in the reverse direction from a trinomial to two sets of multiplied brackets or ‘factors’ is often the more useful thing to be able to do and it is harder. For a start, if you simply dream up any old trinomial, it may not factorise at all or, if it does, the numbers that go into the brackets might be fractions or even irrational numbers such as the square root of 2.
Typically, high school students are taught how to factorise trinomials that do readily factorise and there are two levels of difficulty – those where you have only a single term and those with and so on. I prefer a strategy called ‘grouping’ for the harder kind of factorising, but this is always a cause of some debate within my department. Regardless of the approach, the ability to factorise depends a great deal on knowledge of multiplication facts (times tables).
The interesting stuff happens around the special cases such as the ‘difference of two squares’ or when the trinomial will not readily factorise. There is plenty of scope for exploring important ideas such as irrational numbers, linking to graphs and functions, linking to other approaches such as ‘completing the square’ and generally building a deeper, more interconnected understanding of mathematics.
It is therefore quite baffling that at some kind of forum hosted by Stanford University, Jo Boaler, a professor of mathematics education, and Professor Keith Devlin, a mathematician, came out against teaching students how to factorise trinomials:
In the ensuing Twitter discussion, Boaler and Devlin suggest there are few ‘real-life’ applications for factorising trinomials and even if they did discover one, they would let software factorise the trinomial for them, presumably no wiser about what the software was actually doing.
The justification appears to rest on drawing a distinction between knowing things like how to factorise trinomials and being a ‘creative, flexible thinker’, which seems to be some kind of generic capability that does not rest on such knowledge.
This distinction is obviously false. How can you be creative and flexible if you don’t have this kind of knowledge to draw upon? What exactly are you going to do?
The result is a paradox. I am sure, if asked, that Boaler and Devlin would make a case for students understanding the mathematics they use. Indeed, Boaler and colleagues have argued for the use of visual methods in maths specifically to aid understanding. Yet, in this case, Boaler and Devlin seem content for the understanding to lie encoded in the zeros and ones of the Wolfram Alpha software rather than in the long-term memories of mathematics students.
The argument belies an oddly functional view of mathematics education. Nobody would argue that young people do not need to learn to cook because McDonald’s or the local Greek restaurant can do that for them. Nobody says, “I know, let’s not teach children to draw any more because they can all take photos on their phones”. And yet, when it comes to maths, some people lack the imagination to see past the immediate practical uses of the subject to such an extent that once a bit of tech comes along that can factorise a trinomial or solve an integral, they suggest that we might as well pack it in and focus on vague, new age ideas about creativity instead.
Mathematics is one of the great ways of seeing the world that has been developed by human culture and is a gift from one generation to the next, whether it is initially received with enthusiasm or not. It will survive into the future, despite the best efforts of false prophets, because ordinary mathematics teachers will continue to teach their students how to factorise trinomials.