What can I learn from trendy maths?

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.

20 thoughts on “What can I learn from trendy maths?

  1. Michael Tidd

    Will you be teaching longer blocks rather than chopping about, too? That’s what I’m doing this year. Screw the coverage issues in September; let’s worry about actually getting them to securely do what it is we’ve said we’ve taught them.

    Reply
    1. RedGreen Post author

      I am trying to as far as possible without getting in my head of department’s bad books. I hate the staffroom politics of teaching!

  2. MaryUYSEG

    I have just been reading Randall Knight’s Five Easy Lessons, a great book about teaching physics. He talks there about articulating the thought processes when problem solving on the board, which chimes exactly with your first point.

    Reply
  3. Alaric Thompson

    As a physics teacher I know that students understand very little of the maths they do, even if they can ‘do’ it. I despair of their lack of understanding but haven’t yet striven to find out why they appear to be just taught recipes for getting the answer. I resolve to speak to the maths department.

    Reply
  4. missquinnmaths

    Heartily agree with this all. It’s a relief-and no surprise-to me that DM, Sam Shah and FN all reference “lecture blocks” as part of how they teach/taught….the fastest way to pupils grappling and applying and thinking hard is to make sure they first are taught directly and accurately and to check they can explain it back and stretch it and do it quickly/correctly.

    For me, DM’s approach helps me think about how to build intrinsic motivation to understand things clearly and get it right each time. Still doesn’t justify group work though, only pairs at a push 🙂

    AGREE TOTALLY re maths schemes going too fast. It’s so easy to create the facade of “working at L7” provided no one asks any probing questions….

    Reply
  5. Syd (@FairgroundTown)

    Not especially relevant, but I am reminded of my son’s maths homework last week – lots of angles in triangles. He measured them all VERY carefully, using his protractor, down to half a degree of accuracy… and got them all wrong because the diagrams weren’t to scale and he was supposed to calculate them!! Whose fault? The text book gave no indication that the diagrams weren’t to scale, and they HAD been using a protractor to measure angles that week. Yes, it was “obvious” to me that the diagrams weren’t to scale, because “given” angles were supplied on the diagrams, but it is another one of those things that is only obvious when you know it. So, as you say, we need to be so careful to include every step in the method we teach – and thinking “ah – there are ‘given’ angles, so the diagrams probably aren’t to scale” is a step… just one we never notice because we have the KNOWLEDGE to take it as read.

    Reply
    1. TeacherP

      Syd, when I was reading RoGP’s post I was thinking of this – sure, it’s the ideal that kids will understand area (say) so well that they’ll make use of all of the information at their disposal. Counting squares if that’s the right thing to do, using written dimensions or perhaps measuring.

      Thing is though, measuring is NEVER going to be the correct thing to do in an exam. So have “Fawn’s class” really been served by being taught to get the rulers out?

      It’s so easy for us to forget that kids are not just studying maths; at KS4, some of them will be doing 13 different subjects a fortnight. Remembering they can use scale drawing in geography but not in maths (except in scale drawing questions!) is a tiny part of one a heck of a workload, so anything that aims to “deepen understanding” but perhaps results in exam confusion has to be considered very, very carefully.

  6. Dan Meyer

    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.

    Neat trick there. Almost makes it look like those were my quotes.

    I don’t mean to be too tetchy about it, especially when you disclaim up top that you’re about to conflate a bunch of sources. That conflation is going to obscure more than it reveals, though. I suspect you’ll find my blogroll buddies differ quite a bit on one measure – how comfortable they are taxing a students working memory. I’m rather uncomfortable. My work considers less the avoidance of lecture (see Fawn’s quotes) and more the best pre-conditions for lecture.

    Estimation generates interest in knowing. Asking students what information seems important helps them establish a schema for knowing what to do with that information. Those are great preconditions for an explanation. Both place a rather low tax on a student’s working memory where “Figure out what to do now.” is a much higher tax.

    It’s really, really hard for me to find good foils on this stuff in the blogosphere. I’ll be tuning back in. To end with a question: how do you intend to teach estimation through direct instruction?

    Reply
    1. RedGreen Post author

      I’m sorry Dan – I genuinely didn’t mean to misrepresent your position and am happy to edit the blog post.

      I know you’ve said in your blogs about pennies that you would teach reasonably didactically at one point in a three acts lesson. However, I’ve struggled to really picture explicit instruction and practise to mastery in a three acts lesson. Let’s go back to the film metaphor: the film has a great hook, the tension has been built and you’re desperate to find the answer. Then halfway through the film goes off on a crucial but comparatively mundane subplot. All the viewer can think about is “when are we getting back to the drama?” Actually, that mundane subplot is the meat and bones of the lesson. I want them thinking about that subplot carefully for an extended period of time.

      Whilst this may not be your intention, the relative time and weight you give to the hook and the subplot on your blog suggested to me that you weren’t particularly concerned about making that subplot bit fantastic. I reckon making the subplot clear and pacy enough is a more important endeavour than making the hook incredible.

      On the estimation point, I need to sit down and properly think through what I do (and talk to other experts about what they do) when they estimate, and teach that. I definitely think we need to give students a sort of mathematical cultural capital so they have reference points: this is how much a litre is, here are three things that weigh about a tonne etc. The bog standard method of rounding to 1 sf is valuable much of the time. I also think it will involve sharpening up their ability to multiply/divide by powers of 10, comparing fractions, converting between FDP and the like.

    2. Dan Meyer

      Let’s go back to the film metaphor: the film has a great hook, the tension has been built and you’re desperate to find the answer. Then halfway through the film goes off on a crucial but comparatively mundane subplot.

      I think you’re onto one of the real challenges of my approach, but it’s hard for me to think of a subplot in any movie I’d describe as “crucial but comparatively mundane.” I mean, “crucial” and “mundane” are almost antonyms.

      I reckon making the subplot clear and pacy enough is a more important endeavour than making the hook incredible.

      It just isn’t something I struggle with, I guess, which is why I don’t chat much about it on my blog. Get organized beforehand. Stay concise. Involve the students to the extent that it doesn’t bog down the class. Have them apply their new knowledge quickly after. I’m sure I’m not telling you anything you don’t already know.

      If I had to speculate on why there’s any interest in my work, it’s that teachers understand how to lecture but we all struggle to give students reasons to care about them.

    3. RedGreen Post author

      Crucial but mundane: necessary to tie up lose ends and fill holes in the story line, but not really interesting in and of itself.

      I think people know how to lecture, but not how to lecture well. Making complex and abstract material understandable and memorable to students is something I think many teachers want to know more about.

      Is there not a danger that the hook becomes a distraction, obscuring the main point?

    4. danmeyer55351818

      Making complex and abstract material understandable and memorable to students is something I think many teachers want to know more about.

      Certainly, you’re right. And it isn’t a trivial skill. But it’s the other skill that interests me and it’s a huge blogosphere out there with room for both sorts. My only point here is that direct instruction is an essential element of my lesson sequences. But I’m trying to locate the best pre-conditions for that instruction.

      Is there not a danger that the hook becomes a distraction, obscuring the main point?

      Definitely. When I use a question like “How many pennies are in the penny pyramid?” as the motivator for a lesson on structuring and computing sequences, if I’m not careful, students can walk away with the impression that this one skill was isolated to a particular context. It was “the penny skill,” basically. So I assign structured practice, decontextualized exercises, assessments, and metacognitive questions to account for that possibility. It’s a danger, sure, but I don’t want to overstate it – particularly next to the danger that kids might come to see math class as a place where their teachers just talk to them about stuff that they write down for a grade.

    5. Kris Boulton

      Interesting exchange!

      Re: how to explicitly teach estimation:

      Something I’ve thought about in the background of my mind a little. It seems straight forward enough to me though. I learnt how to solve management consultancy revenue estimation, market sizing and other miscellaneous Fermi estimation problems by studying worked examples, and then practising with friends. If/when I try to do this, that’s where I’d start. There are other varieties of estimation then possible, but again, teach by example – I can’t see any immediate reason why that wouldn’t work.

      David Thomas (@dmthomas90) should be experimenting with this in more detail later in the year.

  7. educationrealist

    I always find myself nowhere in these discussions. I absolutely agree with Harry; there’s no evidence that Dan’s method works, and he’s not explicit about what it is, exactly, he did. I teach concepts in a way very similar to Fawn Nguyen, but it’s clear that she spends 2-3 days, sometimes, on work that only half the class gets, because she wants the kids who can relate to open-ended questions to take the time they need. This, I find inexplicable and inexcusable.

    At the same time, I don’t lecture. My kids rarely take notes. I use graphic organizers for most important topics, and only occasionally, as a learning exercise, give formal notes on a topic–and then only at the end of a long discussion. I use manipulatives, goofy non-tech activities that illustrate a point. I teach linear, quadratic, and exponential equations by modeling, only eventually moving to the equations. I very rarely tell my kids to memorize information (absolute value is an exception), and approach concepts from many different directions. I don’t use books teaching algebra, geometry, or algebra 2.

    Precalc, sometimes, I do something close to a lecture, but primarily because I’m new at it. My second time through, I’m now much more likely to throw problems on the board and get the kids up and working on them, which they like. But I do use the book; the problem sets are fantastic.

    Five years in, I could give a crap about schedules. My kids are going to score low on state tests, for the most part. What I want them to do is get better at math, and have a fighting shot at their placement tests in college. In algebra 2, I teach mostly second semester algebra I and the rudiments of second year. In precalc, I teach a lot of algebra II. I fully expect the calc teacher to mostly cover pre-calc.

    I don’t want a seventh grade teacher to try and cover everything, and for god’s sake, I don’t want them spending time on measurement. (It may have some sciency purpose of which I’m unaware). I want them getting the kids to have internalized some sort of proportional thinking, a reasonable awareness of equations, some sense of the pythagorean theorem. More than that, most of them will forget.

    I could care less about estimation per se. If you want them to learn how to estimate, give them multiple choice tests and show them how to use the answers to test against their own sense of things.

    What I want, most of all, is for my top kids to be able to look at a problem that requires multiple skills–a pythagorean triangle problem that uses both systems of equations and a quadratic equation. Isometries using analytic geometry, and so on. For my middle to low ability kids, I want them to be able to use their internal sense of math to work on problems when all else is forgotten, I want them to have more confidence because they’ve spent a year feeling productive and competent, and I want them to think word problems are slightly easier than simple equations. Ideally, I’d like them to remember how to graph a line.

    Sorry for this meandering, but it really seems to me that kids can learn math in all sorts of ways, and there’s no one way we need to use. But we should not be wasting our students’ time. That, I see as a big problem with both reform and traditionalist approaches.

    Reply
  8. bt0558

    For me an excellent post. I had an AHA moment while reading this one. I think I understand for the first time ( I am getting old) some of what the knowledge advocators are getting at.

    I disagree with you a bit about Bloom’s taxonomy. Bloom’s book did explain how knowledge of facts, concepts and procedures was the foundation of learning and that understanding and application were not possible without the basic knowledge first. So much so that Anderson and Krathwohl took the original taxonomy and created a new one with two dimensions…knowledge and cognitive process.In their book on the revised taxonomy they explain that knowledge is fundamental and that cognitive processes including understanding, application analysis and evaluation use knowledge. A&K also agreed with Bloom that the various cognitive processes from remember to create did not necessarily follow in order. A small point but one I like to make whenever I can.

    I think I can see what you are saying about Knowledge and the NC levels if I assume correctly that the level 7 applies to NC and not Bloom.

    My big beef is that I will have to go back and re-read the Daisy book where it refers to this issue. I am not sure I will be a complete convert but I think I can see the point. I teach a non NC subject(s) and tend to dictate my own knowledge base but I have taught some NC stuff at KS3 and KS4 but have not really concerned myself with the detail, just followed the SOW.

    I will also have to go back and review the Dan Mayer stuff as a result of the comments above and your replies. I think maybe I can start to see the extent to which knowledge may have been reduced in importance (which I would agree is misguided), especially in core subjects and within the NC.

    I arrived at the last 8 lines and thought “this is how I approach a good deal of my teaching, how can ‘Red or Green Pen’ and I be so far apart when it comes to knowledge/skills”. Reread the post again twice and I think I see the point ( a bit ).

    Thanks

    Reply

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