Closing the gap: differentiation by time

There’s a lot of talk about differentiation by outcome, differentiation of explanations, differentiation of support, differentiation of task, etc. etc. etc.. Heck, there’s eighty purported differentiation strategies here.

I worry sometimes about doing differentiation for its own sake. We don’t talk so much about the goals or the assumptions behind differentiation, when those, to me, seem to be the much bigger questions. In this post I’ll explore some of those questions, and present my view of the conditions when, and how, differentiation makes sense. (The title gives you a hint.)

I remember watching this episode from the Simpsons on BBC2 when I was at school myself. This particular section has lingered in my mind since then, and since becoming a teacher it naturally came back to mind. As I watched it again, it’s even more devastating than I remembered.

While that clip has so much worth commenting on, Bart’s quote says it all:

“Let me get this straight. We’re behind the rest of our class and we’re going to catch up to them by going slower than they are?”

Differentiation via a slower remedial class: clearly a ridiculous idea – or is it?

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The descriptive principle that learners minimize their cognitive load

I’ve written before on the normative principles behind minimizing cognitive load – i.e. to what extent should teachers design teaching so that learners’ cognitive load is minimized? This post, however, will focus on something foundational I’ve observed, that’s been in the background of these previous posts.

In my previous post I described the phenomenon where:

students are able to accurately reproduce a process, while being simultaneously completely unaware of the WHAT they are doing.

Similarly, the conclusion of another post was:

to enable students to learn x, I believe we should include desirable difficulties in tasks to ensure that students are required to use everything they need to know about x in order to solve the task; students are thus forced to think about everything they’re meant to be think about to understand x correctly, and they thus embed those thoughts.

I believe the underlying principle that explains these two recommendations is the following descriptive observation: whilst listening to instruction and doing tasks, learners aim to achieve ‘success’ while experiencing as little cognitive load as possible. This often has negative learning consequences and must be counteracted.


Someone else who’s noticed this phenomenon.

Here are some examples and suggestions related to this observation:

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Germane load: linking processes with their names


Bokhove had some great comments on my post on minimizing cognitive load vs. desirable difficulties which clarified the debate for me. As he says,

Squaring [the circle of minimizing load vs. desirable difficulties] is not necessary.
Cognitive Load Theory does not state that less load=best… it should be about successful integration into schemas.
This fits in with all kinds of worthwhile processes Didau set out in his blog on fading, scaffolding etc. One mechanism can also be ‘desirable difficulties’ or intentional crises (perturbations, cognitive conflicts, productive failure). Yes they might (intentionally) spark load but they are conducive to schema building. (Emphases mine)

The phrase of ‘germane load’ stands out to me. It was emphasized again in this great blog post, which goes into much more detail:

encouraging learners to engage in conscious cognitive processing that is directly relevant to the construction of schemas benefits learning. For example, varying the conditions of practice appears to have beneficial effects upon learning, despite the fact that the presence of that variety would raise the loading on working memory. They called this germane cognitive load.

Since then post I’ve been thinking a lot about such day-to-day teaching strategies that might be conducive to sparking germane load: i.e. strategies which build students’ mathematical schemas. Hopefully there’ll be more such ideas in this series.

I’m sure I’m not alone in seeing this phenomenon: encountering a student(/whole classes) struggle with a starter question that was literally done yesterday, even when the whole class were getting it 100% right yesterday.

As many have said, evidence of a student performing a process accurately does not imply they’ve learnt it securely. Performance and learning just aren’t the same. Memory is crucial too.

Due to the fact we did it just yesterday, I think it’s more complex than a memory problem. Sometimes the problem starts even whilst students were getting all the questions right. From some of my observations, sometimes the problem is that students are able to accurately reproduce a process, while being simultaneously completely unaware of the WHAT they are doing.

Let me be clearer: here I’m not even talking about ‘understanding’ what they’re doing. What I mean is, students can be thinking deeply ‘so I multiply this number here with this one, then I add these ones all up, then I divide that by this total… let’s check if I have the same as my partner – yes! Great, I’m doing it right!’ And yet, all the while, they have no trace of the thought ‘I am currently finding the mean from a frequency table’.

This happened for some of my students in a recent lesson. Their pages were filled with ticks, yet when I was asking them ‘so what is this number? What have you just found?’, such students had absolutely no idea. It’s no surprise, then, when it comes to next lesson’s question asking them to ‘find the mean’, they’ve forgotten what to do. They never thought of it as finding the mean in the first place. In the terms used in the literature, they were failing to even begin building a schema; they were simply replicating a process, detached from any sense of what it was for.

I decided to blog on this as it seemed such an obvious and yet subtle issue with my pedagogy. And a problem that’s so easily correctable as well.

Next time round, I taught finding both the mean and the mode from frequency tables in one go; then the questions required them to find both measures from each. The students were then forced explicitly to label their answers: ‘mean = ….’ and ‘mode = ….’; they thus had to link the process of finding the mean with its name.

This solution can be used where there’s another similar easy-to-do process that can be lumped alongside the other process you’re trying to teach. I mentioned a very similar one in my previous post on finding areas and perimeters together in the same task. I think one could do similar things with fraction operations, angle facts, index laws, other statistics, and undoubtedly many other topics.

I do have some slight reservations with this strategy, however, due to it violating the principle of ‘separating minimally different concepts’. But from what I’ve seen so far, it’s worked better for medium/long-term retention than whatever I did before.

Undoubtedly there are other ways of linking processes with their names. I think this principle might lend itself usefully to plenaries too: e.g. a 1-minute ‘discuss with your partner: what have we learnt to do today? So, how do I find the MEAN and the MODE from a frequency table?’

I’m also a big fan of rhymes and chants as memory aids, and I think these have an even more powerful effect than simply lumping several processes together. I also think testing students on knowledge organisers might be the best way to do this in general (give them and expect them to learn the exact schema you want them to have in their mind!). However, effective memory aids and knowledge organisers take some careful thought and energy. When it comes to daily lesson planning in a sequence of work, lumping together several similar processes together is relatively easier to implement.

To bring it back to my original post, this strategy undoubtedly increases cognitive load, yet in a way that encourages schema-building. It seems to be working well so far.


Efficacy, efficiency, and mastery

Efficacy: ‘the ability to produce a desired or intended result.’

Efficiency: ‘achieving maximum productivity with minimum wasted effort or expense.’

Mastery: …

In this post I’ll try and explore an aspect of mastery in mathematics that I’ve been thinking about: the fundamental (and often implicit) understandings that underpin effective and efficient strategies.

There are many ways to solve mathematics problems. To take a previously used example, consider the area of a triangle. The GCSE formula booklet gives it as: 1/2 x b x h. Yet in the past I have usually begin by teaching it as ‘b x h divided by two’. Why?

  1. It doesn’t require pre-knowledge of fraction multiplication
  2. It explicitly encourages students to multiply the lengths first, before halving; this avoids error-prone situations where students start by halving an odd-numbered length and then have to attempt to multiply this by another integer. (Of course, the commutativity of multiplication means the GCSE formula doesn’t necessarily lead you to these errors; nonetheless, such an understanding is implicit rather than explicit within the formula, compared to the formula I teach.)

For these two reasons, I think my version of the formula minimizes cognitive load (though that might not always be a good thing) and is a good place to start. In other words, I think this formula has greater efficacy than the alternative: it is more likely to get my students to the right answer, and less likely to lead them into making mistakes.

Yet this strategy isn’t always very efficient; and I think this reveals something important in teaching for mastery.

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Reflections on learning Chinese 2: how I memorised characters

[This is the second post in a series that began here. This post assumes you’ve read the first one, which contains a summary of how written Chinese words are formed.]

In my last post I talked about the methods that Chinese people use to learn Chinese characters: namely, rote learning and regular testing. I mentioned that while this approach is clearly successful for many East Asian children, I did not use that same method myself.

The main reason is personal experience. I grew up attending a weekly hour-long Chinese class, which was done in a very traditional Chinese style with weekly dictation & spelling tests. However, because I only attended once a week, the frequency of rote learning and testing was far less than a local would do. The night before class, I would copy out the brushstrokes of a character and until I could repeat it consistently without looking. Yet over the longer term, I regularly forgot what I had crammed and became discouraged by my complete lack of anything resembling literacy, despite years of these weekly classes.

Firstly, note that this is not a criticism of the methods per se. The main reason for failure was my infrequency in testing myself. Nonetheless, as I restarted learning Chinese as an adult, I knew personally how much work the local method entailed, and how mindless that work was. I was reluctant to use it again without first seeing if there was a better way.

Then I came across this book in my local library, which changed everything.



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Reflections on learning Chinese 1: rote learning, regular testing and literacy

A few years ago I spent a year in Beijing learning Mandarin Chinese, starting pretty much from scratch (I spoke some basic Cantonese, but had next to no reading or writing ability). Chinese is a famously difficult language (this is a fun read), but I really enjoyed learning it and was pretty successful in doing so. This was in large part thanks to a growing online community of language ‘hackers’ who actively apply modern theories of learning and memory to foreign languges. As a keen learner, I quickly dove into applying such theories to myself. Of course, looking back, it was also a fantastic introduction to teaching, and trying to get others to learn effectively. The knowldege I gained of the education system as a whole, my experiences as a learner, and my reflections on them as a teacher form the basis of this series.

First up: an introduction to the difficulty written Chinese, and some reflections on how locals learn the language.

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Mathagogy: adding and subtracting negative numbers

Adding and subtracting negative numbers – a vital topic. Yet even my sixth formers make mistakes using them, for example in differentiating reciprocals, or solving simultaneous equations. Much of this comes down to the classic ‘two minuses make a plus’ memory aid – a classic case of over-generalization – yet what can replace it? As Rob Brown tweeted, it’s not just hard to learn, but ‘hard to teach’. There seem to be so many different rules and exceptions.

Here’s my take: a few teaching approaches I’ve tried, my reflections, and my current approach (and the lesson Powerpoint I currently use – though it makes much more sense if you read on!).

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