A quick heads up: Throughout this blog I’m going to refer to “trendy maths” and “traditional maths”, for lack of better terms. I’m going to polarise and make some assumptions about both groups’ teaching style along the way. It isn’t meant to be offensive. It’s just me trying to sort out the mess of thoughts in my head.
I’ve been told “cheer up! It might never happen!” more on my commute this past week than I have been in the past 23 years. Clearly I’ve been frowning a lot. This is because I’ve been spending my journeys to and from school thinking long and hard about Dan Meyer and co. I’ve spent an inordinate amount of time reading many of the blogs on Meyer’s blogroll and trying to see what I can learn.
So. Where to begin? I believe there is a problem with many secondary school pupils’ grasp of mathematics. Well, there are multiple problems. But one of the problems does relate to this notion of conceptual understanding, though I don’t like that term because I think it comes with a lot of baggage.
I think what I mean is best illustrated with an example.
Most students know how to find the area of a rectangle in two ways:
1. Count the squares if the rectangle is drawn on a grid for you.
2. If there’s no grid, multiply the two lengths they’ve written on the sides.
This serves them very well for nearly all the questions they will ever encounter on the topic. Bosh. Job done. They know how to find the area of a rectangle.
Do they, though?
Give them an unlabelled rectangle, and what do they do? They sit, staring blankly, asking where the numbers are.
Give them a question on grid paper that isn’t a centimetre by a centimetre sized, and what do they do? They still count the squares and write the answer with “cm²” after for good measure.
This is not good.
This is the entirely predictable response of teaching to two specific question types that come up on the exam.
I believe students in, say, Fawn’s class are far more likely to get their rulers out when confronted with an unlabelled square.
This is the entirely predictable response of giving lots of problems that involve students having to ask for or figure out extra information, or rejecting extraneous information.
Another thing I think Fawn’s students would be better at is recognising when an answer to an area calculation is absurd. That’s because students do a lot more work on estimating and thinking about the real life implications of their answers in her class than students in my class do.
I think both those things are great. I definitely want my children to have the ability to select the information required before implementing an algorithm and the ability to check their answers.
Where trendy maths gets it wrong
So far, so good. But what always seems to accompany this exposure to a greater range of problem types, sense-checking and other good stuff, as sure as PJ accompanies Duncan, is a bucket load of rhetoric around discovery learning. Statements like “the less I speak, the better the lesson is” or “I’ve had to make peace with the fact some students will leave my lesson still not really understanding what’s going on”.
I see no reason why explicit instruction cannot be used to develop the skills Meyer espouses.
I think explicit instruction would do it better.
Where trad maths gets it wrong
I have a few tentative ideas as to why the traditional maths classroom might not be getting it right at the moment:
1. We are not good enough at naming the steps
I think experienced teachers probably don’t make this mistake, but I’m only just starting to realise how many steps I take when I solve a problem. I start with a ball park answer in my head as I’m answering; before I take a certain step I remind myself not to make that common error; I stop partway through problems at strategic points and apply certain tests to see if my answer is plausible; I visualise my final answer and compare to my estimate at the end. I do all this fluently because I’m an expert, which makes it easy to forget to include when I’m naming the steps. But when I leave out those steps, I am failing to teach the procedure fully. It’s not enough to say “check your answer” at the end. We need to be explicit as to how and when. We need to bang on about it.
2. We spend too little time on the basic knowledge
I use the term knowledge carefully. I mean facts and definitions. Too few children have a precise definition of area in their arsenal. Too few know what we mean by a dimension. They deserve a crystal clear idea of this. Otherwise they don’t have a fighting chance of building up the algorithms they know into a body of relational knowledge. To make this a reality, we need to spend time on it! It’s not level 7. It’s not high level Bloom’s. That does not, however, make it lesser – as Daisy Christodoulou explains in Seven Myths.
3. Our curriculum is far too jumpy
In our current year 7 scheme of work, I have one lesson on measures; one lesson on area; one on perimeter; one on surface area; and one of volume. Then, kids, it’s onto fractions! It’s not enough time to practise much of anything to mastery. And in the age of colour coded trackers and performance-related pay it would take a brave teacher to veer away from the few predictable problem types that will be on the half-termly assessment. We can’t spend time thinking carefully about sequencing when it’s just rush, rush, rush.
So after my week of trendy maths soul searching, what am I planning on changing in my practice?
I’ll be explicitly teaching estimation and sense checking. Lots and lots.
I’ll be teaching problem types as normal. Once those are mastered, I’ll introduce problems with too much or too little information for students to tackle so they develop the ability to select information and articulate what information they need for a problem. Hopefully only doing this after they’ve mastered the algorithm itself should avoid cognitive overload.
I’ll be teaching definitions. Talking about them. Making them chant ’em. Quizzing them on them.
What I’ll not be doing is breaking out the sugar paper.