Here, Debra Kidd writes about her concerns with respect to the idea of a mastery curriculum.
I should begin by noting that I agree entirely with the sentiment of wanting children to love and enjoy mathematics. Also, I sympathise with the mistake being made in this post, because I made it myself. In fact, almost every newly minted, naive maths teacher I know has made the same mistake. Those with more experience tend to fit into two categories: they feel guilty because they don’t understand why following the vision isn’t working, or it was explained to them why following the vision won’t work, and the inordinate complexity of mathematics teaching is still unravelling before them.
If you want to reach for the moon, you don’t point your rocket at it and fire; if you do, you miss. Instead you have to point it somewhere else, compensating for a staggering range of complex variables, but in the end you reach the vision of landing on the moon.
Why is it wrong?
First, the post represents a misunderstanding of what mastery teaching is. This isn’t wholly unreasonable; as noted both by Mark McCourt and me at the last La Salle mathematics conference, ‘mastery curriculum’ is a term that is being interpreted in a few different ways. Still though, the definition as suggested in the blog post, that mastery is all about ceaseless repetition of the same content and prep for exams, is so grossly far from any fundamental intention of the ‘mastery curriculum’ concept as to be, simply, wrong.
It goes on to suggest repeated practice of basic arithmetic is a bad thing. The post notes that when asked what they did enjoy about maths in the past:
…they start to talk about investigations and inquiries they’ve done in the past. Lessons where “you had to work out codes and clues and find out things for yourself – it makes you think”. And they mention lessons where “we had to pull together as a team to figure it out – working in a group to get the answers”
I recall with any clarity only one maths lesson from primary school. We were asked to investigate something about the relationship between the perimeter of shapes and the number of squares they were made up of. I started finding so many patterns I wanted to keep exploring that it was not only my most enjoyable lesson, I was truly dismayed when the teacher said I had to stop!
So why is it wrong to pursue this? Here’s where it gets very tricky, and deeply nuanced…
It’s not wrong, per se. It’s wrong if you think ‘ergo this is better than practice.’ It’s wrong if you think ‘this is how kids should learn things.’
Running an investigation because you want children in school to experience a mathematical investigation is almost certainly a good thing, and probably something we should be taking pains to do more often. Running an investigation with the intention that children learn a new piece of mathematical knowledge is almost universally doomed to fail. There are several reasons why, but let me just articulate one of them in the hope that it’s enough for now: if you are investigating something in a lesson, then (at best!) you arrive at the conclusion – that thing that the teacher wanted you to learn – only at the very end of the lesson. This means that you spent most of the lesson *not* thinking about the thing you were supposed to be learning. If you instead were taught that thing explicitly right at the start, then you can go on to spend the entire lesson thinking about it. More time thinking about the idea, greater initial storage strength (the whole ‘memory is the residue of thought’ thing.)
There are so many shades of complexity to the above paragraph that I am omitting; but suffice it to say that ‘investigation good, explicit instruction and practice bad’ is in error to the point of absurdity.
Next up, the novice-expert continuum. It was mentioned that the group in question were a top set. Higher achievers notoriously tire of simple drill practices much more quickly than others, and they do generally need less of it (though, typically still more than they *think* they need.) They are also more able to take on and learn from more complex forms of unguided instruction.
Furthermore, there are boring ways to practice, and there are fun ways to practice. Just look to all the incredible work Bruno Reddy did on Times Table Rockstars for examples of fun ways, and I’ll say no more.
Why is it dangerous?
The ideas expressed in the post are not new, they are in fact very common. A similar view was expressed by Conrad Wolfram only a few months ago in this article. This notion has actually been kicking around for decades now, as noted by Professor Askey – apparent veteran of the US ‘math wars’ – when he said the following back in 1997:
“Mathematics is like a stool; it sits on three legs. In the New Math period the only leg used was the structure of mathematics. The feeling was that if you understood the structure of mathematics, then you could compute and solve problems. That turned out to be false for all except a small group. Then we got ‘Back to Basics’, which was founded on computation. However, the level was too low, and good problems and structure were both ignored. This failed badly. Then NCTM tried Agenda for Action and later the Standards. Both of these were built on the idea that if you could solve problems, then you could do mathematics. You can, but at too low a level. All three are needed—problems, technique, and structure. I fail to see why this obvious fact is not appreciated, but it does not seem to be. The New Math failed for good reasons, and the New New Math will fail for good but different reasons. Maybe then we can start to try to do this right.”
The assertion from the post that ‘We need more Mystery than Mastery’ is an intellectually deficient idea. Although it was no doubt intended as a pun on some earlier commentary in the post as to the distinction between ‘master’ and ‘mistress,’ and a not uncommon response by the author might be to suggest that it was only meant as some light-hearted fun, it’s only once you’ve seen what ideas of that ilk do both to maths teachers and to their poor suffering pupils that you can grasp the danger in treating it as such. Mathematics education is so curiously complex that while we’ve been uncovering more and more of its nature for more than half a century now, we still haven’t yet pieced it all together into a structure that works. I agree with the sentiment that asks:
What is the point in taking children on a learning journey in which they feel like passengers trapped in a repetitive hell?
Obviously this is not a vision of mathematics education to which we wish to aspire, but the ideas expressed in the post are dangerous precisely because they risk once again moving us away from achieving that vision. They are dangerous because they take the name of ‘mastery curriculum’ and then slander it, twisting its intent (probably through ignorance, not deliberate slander.) I’m all in favour of Mill’s notion of free speech as a means to ensure we don’t stifle truth. Where previous articles such as Wolfram’s have equally missed the mark in their ideas, however, it is this defamation of a named concept that meant I felt the need to respond now in particular, equally in the spirit of Mill’s freedom of speech.
Debra’s post isn’t the real danger, though. It’s the ecosystem of thought that surrounds it, the idea professed by others with the heft of academic credential behind them, or capital investment, who would also repeat the now decades-faded echo that any and all ‘drill’ must equate to ‘kill’. How could they be wrong?