*“Which would you hate more: to overhear someone say you were stupid or say you were boring?”*

*“Oh definitely boring! That’s got to be the worst thing to hear.”*

*“You don’t mind someone thinking you’re stupid?” *

*“If someone thought I was stupid I’d assume they lack the intelligence to realise how clever I am.”*

I won’t embarrass the speaker by naming them, but suffice to say that everyone present agreed that being called boring is the most crushing of criticisms.

—

Our most recent maths survey (we do one each term) yielded mostly positive results: over 90% saying they’d take maths if it were optional; the modal response to ‘Do you like maths as a subject?’ was ‘I love it’; only 1 of 259 respondents disagreed that ‘there is usually a positive atmosphere in lessons’; only 1 disagreed with the statement ‘my teacher tries her best to be fair’; the modal description of work in lessons was ‘challenging, but I can do it.’

The results don’t vary with significance when filtering for different teachers or sets, which reflects the hard work of my colleagues in the department as we try to offer the students a consistent experience of maths (we all believe in allowing group 4 to access the same curriculum and expectations as group 1, even if it can feel like a profound challenge at times).

Obviously, I should be delighted with this. I am. I feel so blessed to have colleagues that let their students feel so positive about maths.

But…

In the free-text section for ‘What 3 words do you associate with maths and maths lessons?’, 12 students said ‘boring’. My immediate reaction was to feel wracked with guilt and feel we’d let them down. Maths is amazing and beautiful, we work hard to convey our enthusiasm for it and help all students access it. We must have failed these 12 students! Even though I know it’s a huge overreaction, it niggles at part of my brain.

In a recent review of the school (as part of the Bradford Partnership), the observers commented that students were very serious-minded in maths lessons and expressed some dismay that we never rarely have occasion for the students to work in groups. Retorts spring to mind: “Of course they were serious when there was a stranger in the room!” “You only saw 10 minutes!” “There’s a lot of evidence in favour of a serious atmosphere in maths learning!” and so on, ad infinitum. I’m trying very hard not to reject this feedback just because it doesn’t fit with my current conclusions about how best to teach maths.

Are we missing an obvious solution?

I was intrigued to read a blogpost by Matthew Smith, arguing – amongst broader points – against what he perceives to be needlessly boring lessons:

“When planning a lesson on adding fractions, a bland, one dimensional lesson might involve a few examples by the teacher and a worksheet with a range of questions. A *highly effective *lesson that encourages **all** learners to make progress and see the beauty of maths might start with the following question; “Can all unit fractions be expressed as the sum of two other unit fractions?” This leads to an investigative lesson, no one knows the answer straight away and it is accessible to all students. In short, mixed ability teaching can and does work.” [his emphases]

I have selected Matthew’s post not because of anything specific about him or his school – I haven’t neither met him nor spoken with him; his posts suggest he is a committed teacher who works hard for his students – but because I think that the passage is very representative of a dominant school of thought around maths teaching.

I think it’s completely incorrect. The question – can all unit fractions be expressed as the sum of other unit fractions? – is definitely very interesting and even as I type I feel a little bubble of excitement thinking about working on it later. It’s easy to convey that excitement to students and to work together to answer that question. There is definitely a place for these questions in maths lessons – and not just for higher groups or the keenest students – for all students’ lessons.

But…

How on earth can a student ‘investigate’ that question without knowing how to add and subtract fractions? It is not an easily-learned process. It’s fiddly to actually do – selecting an appropriate denominator, forming equivalent fractions, resisting an understandable instinct to add the denominators together – and it takes a significant understanding of proportion to make the leap from grasping the initial ‘why’ (e.g. when exploring diagrams) to seeing its link to the how (having a common denominator). Even a student who can confidently explain why we need common denominators to add will occasionally relapse into ‘add everything.’ It’s not that they don’t get it, it’s that they don’t remember it at all times. It’s not obvious. It takes a huge chunk of working memory, even for A level students. Insofar as I can claim to understand psychology, it strikes me as exemplary of how System 1 can trump System 2 unless we are in a state of serious-minded vigilance as we work.

All my experience so far suggests that, particularly for students lacking mathematical confidence or mathematical knowledge (the latter label encompasses almost all students), it is profoundly unhelpful to offer too rich a context as a vehicle for learning, especially for elements of maths that require consistent, fluent and (necessarily) fiddly processes. This doesn’t mean you don’t share the opportunities to then use and enjoy that knowledge, it just means that they don’t have to happen in tandem.

My (limited) experience so far suggests this:

- Convey that the topic or skill is interesting (not necessarily useful. It usually isn’t and I’m cool with that). If it can be introduced with a ‘huh???’ moment, then that is great. It’s not always possible and it’s better to just dive in than to create a pseudo-hook.
- Show them how to do it. Scaffold it carefully, being sensitive to your group. For the love of God please don’t ask them to discover it. It’s great to ask them to notice and think about patterns, and to explain what is happening, but it is a terrible disservice not to also explain it very clearly and very carefully. And to check that they can also explain it very clearly and very carefully.
- Never stop telling them about how awesome you think this bit of maths is, how proud they’ll feel when they can do it fluently and connect it with other things.
- Let them practice. LOTS.
- Use formative assessment throughout to unpick and discuss misconceptions.
- Let them keep practising. LOTS. On their own. In purposeful silence in a supportive, happy atmosphere.
- Keep using real-time assessment to match the pitch and pace of the work to avoid students being stranded or coasting (constrained by what is possible without being torn in 30 directions).
- As more of the process is chunked into long-term memory, and confidence in the idea ‘I can actually do this’ grows, build the links between the how and the why. For example, if learning to divide by a decimal, this is the time to cement the link between the idea of ‘3÷0.5 is 6 because I am seeing how many half-cakes are in 3 cakes’ and what is happening when 3.4÷0.2 is rewritten as a fraction, multiplied by 10/10, and simplified.
- Keep returning to it throughout the year – and their time in school – so that they don’t lose what they learned. Interleave it with increasingly complex and rich contexts (not necessarily ‘real-life’. Learning to appreciate the beauty of the abstract is a great gift to see them through the tedium of adult life). Let them experience the delight that their hard work in learning how to divide by a decimal, or add a fraction, or form and solve an equation, is allowing them to work on an intriguing puzzle or fascinating context.

Maybe it’s fine that it’s boring sometimes.

When I practise A-Level topics, or practise French, I do sometimes find it boring. Not dislikeable, just sometimes tedious. But it’s an acceptable – even welcome – boredom; it’s like the way that exercise is sometimes boring. There are more fun forms of exercise, and there are more fun forms of maths, but I can’t really do either without making sure the basics are solid, and examined in isolation without the trappings and context of more exciting or intriguing problems. It’s a satisfying boredom that is part of the journey to a worthwhile goal.

—

In the same free-text question over half of students described maths as ‘fun’, and 49% made ‘interesting’ their word of choice. Their later comments suggested that the teachers and the atmosphere are the drivers behind this, as few made any mention of the activities in lessons. These are reasonably representative comments, to give an idea (currently all the teachers are female – hence the default to ‘she’) –

*“she brings a huge buzz to the class and tries to make it as enjoyable as possible” (a Y9 who is pretty clear that he doesn’t actually like maths itself)*

*“They all talk about how much they like maths and when they explain something to you, you understand perfectly” (I should employ this kid to write my blogs, as he’s clearly able to say the same thing but in fewer words).*

*“My teacher(s) do like maths because they have a such pleasure and passion in teaching it and also they go over something more than once for those who do not understand which proves FAIRNESS.”*

*“they all are very passionate about it, for example refer it as cool constantly”*

*“For me maths is difficult however on the other hand I feel happier every time I do it because I then learn and understand something new which is very important to me.”*

* “I love maths because I know that DTA does it for our benefits and the teachers make us feel comfortable with maths.”*

In summary:

- Love maths. Show it.
- Teach the processes explicitly.
- Let them practise it lots, and in a context- and confusion- free way.
- Interleave and revisit. Build up the confusion and context. NOW is the time to decide if there is time and merit in more ambitious activities (I am partial to a Barbie bungee, but not as a vehicle for teaching).

If you think this sounds awful, I am interested to hear from you. If you think this sounds awesome, please apply to work at our school. Bradford is great and so is working at our school.

I really like this post and totally agree with what you say.

This is the most brilliant maths pedagogy I hv ever read. Thank you. Packing my bags, will knock on your door. Want to work with such passion. Can I bring all my maths toys? Please?

Reblogged this on The Echo Chamber.

I’ve got some praise and criticism all mixed together. I hope you’ll recognize and accept the praise without being bothered too much by the criticism.

First, as a general observation people do not exercise. Relative to what we understand is optimal for health, they do too little, with too little intensity, too infrequently. Boredom is one reason why. this analogy does not support the case that boring is ok for math classes, rather it supports the opposite.

Language lessons are similar. In the US, foreign language classes in school produce an insignificant number of foreign language speakers. Perhaps results are somewhat better in the UK where the motivation (by geographic proximity) is higher.

Next, can we really believe your maths lessons are boring? I find the evidence does not support this claim.

Only 12 out of 259 students said “boring?!” That’s less than 5% and a rate i bet almost all teachers/schools would love to have. Heck, pick any amazing thing in the world and normally 10% of kids will say it is boring, 25% if they encounter it in a school context. This is strong evidence that your lessons aren’t boring.

Next, look at all the other responses that indicate enthusiasm, passion, excitement, fun are present in their math lessons. more strong evidence.

Third, the fact that you are writing this blog, reflecting on your practice as a teacher, considering what works and doesn’t, soliciting feedback. Again, signs that this teacher is unlikely to be leading many boring classes.

Ok, I’ve made my case, now here’s the punchline: I doubt you can provide convincing evidence that boring works. The risk is that someone else gets this idea and sees it as an excuse for boring lessons. Even worse, as an excuse for why they aren’t passionate and excited about the subject, reflecting on their practice, considering what works and doesn’t, etc etc.

Dan Meyer has a series trying to explore what questions are engaging. You are talking about what methods are effective. I think the emphasis is misplaced in both cases and that, with the right teacher, almost any question can be engaging and almost any method can be effective.

Take your example of fractions with unlike denominators. Two possible lessons:

(1) explain what we’re going to do

(2) show process

(3) give some more examples

(4) students work examples

(5) student questions and teacher observations turn up misconceptions/mistakes students are making

(6) address misconceptions/mistakes

(7) talk about why: why do we have to do it like this, why doesn’t the obvious/intuitive approach work, what variations on the method work, are there associated pictures we can draw/explanations we can give, etc?

Or, do it in order: (1), (7*), (2) – (6), with

(7*) talk about why: what do these denominators mean, why doesn’t the obvious/intuitive approach work,

Or even, give Matthew Smith’s challenge and, after explaining what it all means, say, “whoah, how do we even add these things?” and have discussion to tease out issues and methods. Then, clearly describe the methods and have opportunity to practice.

Which would work, the first approach, second, third, all, none?

That’s an interesting observation; I hadn’t thought about it in that way.

You’re probably right that, for most disciplines (music, exercise, languages, maths) we do far from the optimal amount of practice needed to reach proficiency.

I don’t think the lessons are ‘boring’, per se; we do really want to convey a love of maths to our students and do use our personalities a lot in lessons to show that we’re happy to see them, have playful jokes and comments, etc.

BUT, judged by the received wisdom of what’s considered fun or engaging in British education (group work, hands-on tasks, investigations, discovery learning), our pedagogy is a failure and very boring. There’s a lot of structure in lessons, it’s fairly teacher-led with a focus on direct instruction, there is plenty of silent and focused practice (coupled with questioning and support). I don’t think it’s boring, but some people would think so. I guess my conclusion is that I don’t mind if people think it’s boring, but do think it’s important to show enthusiasm for maths and for your students, and to bring personality and joy to your lessons.

I understand that you were trying to show that Matt’s approach is, fundamentally, not that different, but I disagree. You mentioned ‘clearly describe the methods and have opportunity to practise’ in a somewhat throwaway manner (I get you were trying to make a point!), but that’s doing a disservice to how massive that part is! ‘Clearly describe the methods and have opportunity to practise’ IS the bulk of the lesson and the core experience (i.e. quite close to the description of ‘teacher shows a few examples and students do worksheets with a range of questions’). The intrigue and questions can be a way in, but they’re not the vehicle and structure for how we teach but are used as a vehicle and structure for teachers approaching in the way he describes.

So I think I may still disagree with you, but your comment has definitely made me think. Thanks 🙂

Agree with this, but unless I’m missing something, the question about unit fractions doesn’t have much to it. Every unit fraction is the sum of half of itself and half of itself, where half of itself will be a unit fraction. Is that what they are meant to discover? That seems a pretty tedious problem, easy for those who can add fractions, difficult for those who can’t.

Ha-I just assumed it was “two different unit fractions” (that is genuinely interesting and a nice proof is possible also, I have since discovered).

Isn’t this what I said??? It’s what I meant anyway…

The task is a good one. I too assume different unit fraction ( Egyptian fractions). However I also agree with you about the difficulty of the task if they haven’t learnt + in the first place.

Basically there is too much ‘ running before walking ‘ going on

You could extend the problem to can *any* fraction be written as the sum of unit fractions, which can include some interesting discussion about Egyptian mathematics too.

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Really disagree with this-your post about being boring certainly applies if we are training our students to be successful in exams but for schools who are not running an exam factory it’s important that we show them how creative maths is and allow students to develop as mathematicians. Maths is not a boring subject and this is what you seem to be suggesting.