Teaching math in the 21st century: A review

I have not known what to think about the U.S. Common Core initiative for some time now. When first mooted, it seemed like a good idea. I subscribe to E D Hirsch Jr’s view of the importance of knowledge. Part of that argument is that we should identify a common curriculum that students follow in order to ensure systematic exposure to ideas. This is something that many high performing states already have in place and, to my view, a positive feature of the U.K. and Australian systems. Don’t mistake me, there are many and various problems with the Australian Curriculum as currently composed and yet it is far better to have something to argue about than nothing at all. Well, that’s what I thought.

In the U.S., for some reason that I still do not understand, the idea of a common ‘curriculum’ is politically impossible. ‘Standards’, however, appear more palatable, even though standards and a common curriculum pretty much amount to the same thing (the U.K. and Australian curriculums are actually expressed in terms of standards with plenty of flexibility in how these are interpreted on the ground). Despite the window-dressing, Common Core has been hugely divisive. Given the polarisation of U.S. politics, I couldn’t quite decide whether the concerns were legitimate or whether they were those peculiarly American ‘concerns’ about the ‘government’ and how it wants to take away people’s God-given right to carry automatic weapons and shoot lots of people when they feel sad. But I digress…

Thanks to Barry Garelick and his book, “Teaching math in the 21st century,” I now have a window into the world of common core mathematics. And it’ not looking too healthy. Michael Gove, erstwhile U.K. education minister utilised the title of the movie ‘The Blob’ to describe the workings of the education establishment and how, whatever you did to fight it, it would ooze back in around you and finish you off. The emotive nature of this description is problematic but it is an eerily accurate description of the capture of Common Core described in Garelick’s book.

Common Core mathematics was clearly intended to raise the standard of maths teaching in the U.S. to bring it more in line with higher performing countries. That was and is the plan. As a set of standards, it wasn’t supposed to imply a particular teaching method. Yet we hear the tale of Calvin who is struggling a bit in maths and has been diagnosed with ADHD. His counsellor, Teresa, hails the coming dawn of Common Core, “So no more ‘here’s the assignment from the book, and there’ll be a test on the material next week. It’s more about understanding.” This would be a good thing, she claims, because Calvin won’t have to memorize procedures. Calvin objects that he likes memorizing procedures and has memorized the quadratic formula. Garelick – Calvin’s maths teacher – offers praise for this feat but Teresa chides that this won’t be required under Common Core. With understatement, Garelick echoes my thoughts: “How a student could be deemed to understand the quadratic formula without knowing it was puzzling.”

Similar doublethink is on display when Sally from the district turns-up for one of the Monday morning meetings. A grizzled old teacher mentions that some of the ‘below grade’ students have gaps in their knowledge. “Some of them don’t know the basic math facts, or how to do basic operations,” he complains.

“That’s because they haven’t been taught how to think,” Is Sally’s utterly ridiculous answer. I am reminded of Carl Bereiter’s quip that trying to teach children how to think is like trying to teach them how to digest.

The grizzled old teacher bites back but it goes nowhere. The power lies with Sally.

You might ask what Garelick is doing all this time. Surely, he’s issuing hard truths and fighting the good fight? Well, no. Garelick is mainly trying to stay employed. He is a late entrant into the teaching profession after a full career in the real world and nobody seems to want to give him a permanent contract. This is why he ends-up with various long-term cover positions. He is a model of diplomacy, trying his best to follow the plans of the regular class teachers. You can tell that he’d rather keep his counsel and stay in work doing a job he enjoys.

And it’s clear that he enjoys it even if, like most new teachers, some of his classes make him nervous. Much of the book consists of anecdotal tales of his interactions with students; of his frustrations and his palpable excitement when he hits on a way of making progress with them. He confides his scepticism about Common Core in us, the reader, but you can see that he is genuinely trying to do his best. Like any new teacher, he is trying to figure out an effective way to run a class – one day, teachers will be actually trained in such things. In his struggles, he has to confront a discipline system that is almost impossible to use. “There are a variety of methods one can use to discipline students: detentions, referrals, sending the student outside of class, contacting the parents. I was confused about using them.” Of course you were, Barry. Can an experienced old fox let you in on a little secret? You are meant to be confused about them because you are not supposed to use them. Put up a motivational poster instead.

Garelick gives Common Core more credence than I probably would at this stage in the process. He knows it’s daft but you can detect a sense of guilt; he’s not quite sure if he’s right. And so he has to keep proving to himself that kids really do understand things better if they practice and master basic procedures. “See!” he pleads with us, “It really works. Will anybody listen?”

And that is the charm of the book. Self-effacing and humble, Garelick just wants to take us to the places he has been and check his own thinking.

Yet mixed with the naivety of someone new to the profession, there are some flashes of wisdom that can only come from someone who has seen a lot of life outside the classroom. At one point, he makes me think in a new way about a common trope. Enacted Common Core, as one of its things-we-fluffy-non-mathematicians-imagine-real-mathematicians-do, prioritises the ability to solve ‘non-routine’ problems, otherwise known as ‘problems that students haven’t been taught how to solve’.

“What I find inauthentic is the prevailing group-think which holds that judging math ability should be based on how well students in K-12 are able to apply prior knowledge to problems that are substantially different than what they have seen before. In the working world (which the education establishment tries to emulate by insisting that students be given “real-world” problems) most people employed in technical fields are expected to apply their skills to variants of well-studied problems. For those who need to solve problems of a substantially new nature, it takes weeks, months and years.”

Quite right, Barry. This fluffy kind of Common Core is never, I’m afraid, going to bring the U.S. up to the same standard as the Far East with its rigour and procedural fluency. The strange obsession with eccentric definitions of ‘understanding’ and the commitment to problem-solving as some kind of generic, trainable skill will misdirect teachers from what is important. What is more, in those states that had good standards prior to its implementation, Common Core may represent a dumbing-down.

I fear for the future of  maths in America. You should read this book and see whether you agree.

Note: The Author was kind enough to send me a review copy of his book.

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10 Comments on “Teaching math in the 21st century: A review”

  1. teachwell says:

    I don’t get it Greg – I thought Common Core was to raise standards? I thought it was more knowledge oriented? Certainly that’s the way that Hirsch would have intended it. Has it been subverted along the way a la national curriculum in 1988?

  2. CC has been interpreted along reform math ideologies, per the reform “dog whistles” embedded in the standards, as Tom Loveless describes it. One can interpret CC along more knowledge-based traditional means, but that’s not the prevailing interpretation in the US. Rather, it has thrown gasoline on the ideological fire of math reform that has been raging for the last 20+ years.

    • teachwell says:

      I am afraid the same thing is happening here not just for the curriculum but the free school movement actually involves many who want to embed progressivism at all costs. However, at least if there is something to compare it will not be in vain. Other reforms have not allowed even this. I will read your book and get more of an understanding.

      The sad thing is I was teaching intervention groups for maths and it was all about teaching them the knowledge they lacked or techniques that are dismissed (drill and kill indeed – its just practice). The strongest impression was the fact that the children loved the lessons because missing pieces of the jigsaw were being added and one of them was fascinated by whether it would now mean they knew their times tables forever. It seems to me that children’s input is only valued if they are reinforcing the progressive ideals. Otherwise. they would appear to be misguided like the traditional teachers…

      • barryg99 says:

        Yes, despite what the ed schools and the newly minted disciples tell us, students actually like doing problems if they are given proper instruction on how to do them–in fact they even like it when the problems start getting harder as well-scaffolded problems should do. The reformers would tsk-tsk this, saying, “Well, yes, kids like rote memorization because it’s the same problem over and over.” Not if properly scaffolded. And the problems that get the Dan Meyer seal of approval in my opinion are fairly tedious and don’t really do a hell of a lot to broaden students’ math knowledge. A one-off experience is just that.

      • teachwell says:

        Indeed – yes children want to repeat things until they have got them – but don’t we all? I mean when I started to learn to code, I started with the manual and was excited by it. If I had tried to figure it out for myself, I would still be here scratching my head!! Yes there is the odd person that can do that but even then practice makes perfect.

        Here’s the thing that makes me laugh when people say children don’t like doing the same thing or memorising – I used to get pages (and I mean pages) of numeracy work that the children did off their own bat. What was it? Always number bonds or times tables!! They went home and wanted to remember it and to grasp it. I have never had a child come in with a word problem that they created themselves. In fact most of the time the children in class used to hate doing this as they couldn’t work out what to do so did a version of what was there. There isn’t anything wrong with that per se but the truth is that children want to know things – we are forgetting what its like to know significantly less than those older than us and wanting to be able to do the same as them.

        I wanted to read, do hard calculations, etc because my brothers were. This whole child led thing has me perplexed because it doesn’t seem to be based on any actual children. Just some concept of children and how they should act and be by people who don’t wish to grow up. This is hardly a reason to teach children accordingly.

  3. Hi Greg — it is not that a common curriculum is unpalatable in the U.S., only that having such dictated to the states by the Federal Government is unconstitutional (unlike in GB and OZ, but Canada also separates powers and the central government may not dictate education policy as in the U.S.). It would be illegal for such to be mandated. There is much discussion surrounding whether CC as implemented is overly intrusive on State education jurisdiction and therefore over the line into unconstitutionality.

    In contrast, in Canada we have the WNCP (which is problematic but for entirely different reasons), an agreement among several provinces, and adopted by a total of 8 provinces so far, to a common “Curriculum Framework” — upon which each participating provinces develops its own provincial curriculum, although in places you need a microscope to distinguish the framework from the curriculum.

    I agree in principle that something like Hirsch has laid out ought to be the universal expectation in education, with some minor local variants and occasionally updated, but in small increments. There simply is no reason to be reinventing the wheel and for curriculum developers around the country, or world, to put on blinders and pretend nobody elsewhere is asking (and answering) exactly the same questions about what children should be learning — or that answers arrived at elsewhere are unsuitable locally merely because of geography or culture.

  4. vlorbik says:

    with or without a reason, “reinventing the wheel”
    is pretty much the crux of the biscuit in “education”
    (insofar as it exists as a subobject of “politics”).

    a lot of people seem to believe that learning to endure
    nonsense-presented-with-authority is what schools are *for*.
    indeed, they’re running the show and have been for decades.

    it oughta be different for math (since math people are so
    good at admitting when they’re wrong). but no “should”
    implies an “is” (and literate people have understood this
    for centuries). so, to some extent, at least until you get
    to the “advanced” stuff, it *ain’t* (much) different.

    one has known how to do math-ed right
    since the sputnik crisis if only the political
    framework allowed it. but “mass literacy”
    isn’t on anybody’s horizon around here
    and mostly these days this kind of thing
    just seems comical (when it has any interest
    for me at all). thank you and good day.

  5. paulmartin42 says:

    ” I am reminded of Carl Bereiter’s quip that trying to teach children how to think is like trying to teach them how to digest.”

    The trouble with Obesity is that few teach kids how to think about what they digest. There is too much … to learn. So I can see why Common Core has come about but one problem with initiatives, such as CfE in Scotland, is everyone is discussing a different agenda (such is life). The benefit of Mao’s little red book, the 10 commandments (I can only remember three BTW) is that at least you have a fair idea what they are. When I taught in FE for 20 years or so I was astonished to find that no (struggling) student led me to believe they understood the concept of percentage – no wonder we had sub-prime.


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