Matthew Oldridge, a maths teacher on secondment to the Ontario Ministry of Education in Canada, has just published an Edutopia article on The Playful Approach to Math.
Interestingly, Ontario finds itself at a crossroads. For many years, it has been a standard bearer for fuzzy maths, eschewing the supposedly rote memorisation of procedures in favour of an approach where a basic arithmetic problem is over-intellectualised and performed in five different ways, all in the context of problem-based or inquiry learning. Predictably, this approach has coincided with a decline in maths performance.
In this context, it is therefore interesting that Oldridge rows back a little from the extremes of fuzzy maths. It is barely detectable among all the talk about how, “Play moves math instruction beyond rote memorization to a more expansive understanding of mathematics,” and yet, at one point, Oldridge does suggest the need to memorise multiplication facts. Well, at least as far as six times six. This limitation is because multiplication fact practice has to be performed by rolling two dice and dice only go up as far as six (Oldridge is clearly unfamiliar with Dungeons and Dragons).
The reason for the dice-rolling is Oldridge’s claim that, “Play is irrepressibly human, and we can play to learn”. This is true, up to a point. Play is perfect for learning what evolutionary psychologist David C. Geary refers to as ‘biologically primary knowledge’ – language, basic social skills, folk psychology and so on. This is knowledge we have needed to acquire since the dawn of humanity and so we have evolved mechanisms such as play as a means for acquiring it.
Maths is different. It is a far more recent cultural invention that Geary would term ‘biologically secondary knowledge’. This distinction is key to understanding cognitive load theory. When dealing with biologically secondary knowledge, we have to process every new item in our limited working memory. Presenting new biologically secondary knowledge alongside all the variables associated with a play situation is an obvious way to overload working memory.
To be fair to Oldridge, this does not preclude play from being used as a form of practice once students have grasped the relevant concepts. However, although Oldridge describes using play for practice, he does not make this distinction.
Oldridge is also ambiguous on the use of guidance. Briefly, when Kirschner, Sweller and Clark famously criticised approaches such as problem-based learning as deploying ‘minimal guidance’, adherents of these methods retorted that they use loads of guidance. For novices, a more guided approach will always be better than a less guided one, which is why fully guided, explicit teaching is the most effective approach of all (note that explicit teaching is a whole system and not just a bit of direct instruction here and there).
So when Oldridge promotes ‘guided play’ it can be seen as partly an attempt to preempt this criticism. However, we can get a sense of what he really means when he suggests, “There is a sweet spot, often about 30 minutes into working on an interesting problem, where ideas start to become solutions.”
This is an oddly faith-based approach. I would never let my students struggle without a solution to a problem for this amount of time. That’s my definition of ‘minimal guidance’. There is no reason to believe they will eventually figure it out or that there is something magical about 30 minutes. Instead, there are plenty of reasons to predict confusion, frustration, demotivation and distraction. And even if they do eventually figure it out, that’s 30 minutes of time wasted.
Being in favour of play is a lot like being in favour of peace and motherhood, and it will certainly get you published in Edutopia. However, when you examine it more closely, it’s merely a repackaging of the same bad ideas that have harmed students’ understanding of maths in Ontario and across the English-speaking world.
Let’s not play dice with our children’s futures.
Filling the pail 
One of my strong reservations about using play to teach Maths, is the lack of oversight by the teacher. If two or three students are playing a card game on negatives and positives, how can any of their errors be corrected? All they will do is bed in any misconceptions deeper and deeper. Because if one student says -6+-5 = +11, his playmates are likely to agree with him. And so you get 13 year-old students who, after years of being “taught” this way, still cannot add two negatives.
Teaching Maths is about overcoming misconceptions more than anything else. Any strategy that does not actively do that is likely to lead to poor results.
Rather than rely on play, which is indeed a driver for humans, traditional teaching often uses competition, which is also a very natural drive. Competitions allow for any errors to be actively spotted, and hence fixed. But the progressives tend to shy from any taint of real competition — it’s a purely political choice to avoid a very useful technique.
Ah but…won’t he make a terrific ed consultant?
If you title your blog with the guy’s name, followed by an insult, it makes the entire thing ad hominem, regardless of what’s within. That’s pretty basic. Surprised in the era of clickbait that you think otherwise. Change the title, and I mostly agree it stays in bounds. Except you are kind of leading your readers to think there is something up in Ontario-whereas I just support schools and have nothing to do with the curriculum. (last Fall’s Fundamentals of Math document writtten by the current Conservative government is relevant though-in going back to basics, I believe they even mention using games to practice basic facts.) The link you are seeking here is between me and Ontario- and you reader is led to make that link, I do believe. That is not a fair characterization- there is no real link between this blog and any provincial directions- especially with the current government.
Edutopia is fair game (I could talk about the editorial process and chopping down to 800 words, and that I actually had referenced Sweller, KIrschner, Clark, 2006, but they cut it), public articles are obviously fair game-I don’t get why the odd personalization? I wasn’t making it up that we find it creepy. It is also a direct attack on my character, despite being somewhat witty, and very hyperbolic. Also my true mea culpa here is that “rote and expansive understanding” slipped in at the end, and I missed it-not words I would really ever use.
PS: I most definitely do play dice with my childrens’ future- my 9 year old knows up to 12 x 12 already…
I likely won’t hear from you again, and that is okay-I just wanted you to think about it. To each his own conscience.
Thank you, and best regards,
Matt.
I am sorry you have been upset by this blog post. I do not agree that it is ad hominem. The post directly tackles your arguments and does not comment on you as a person.
The description that you are on, ‘secondment to the Ontario Ministry of Education in Canada,’ is one I have taken from your public Twitter account. Given that you blocked me a while ago on Twitter, I logged out in order to ensure that the description was accurate. I have blogged a lot about maths in Ontario and so I used that as some context for my regular readers.
Normally, I would alert someone to a blog post that mentions them by tagging them on Twitter, but blocking makes this pointless. You have every right to block whoever you want – I am just explaining why you did not get tagged.
If you write articles about how to teach maths and you get these published in forums such as Edutopia then you really do need to accept that your arguments will be criticised. I understand your frustration with editors – I have been annoyed by an editor putting scare quotes around a word in the past and I am now highly vigilant for subtle changes.
I personally think ideas such as allowing students to struggle for 30 minutes are extremely damaging and, when widely enacted, harm educational outcomes. I do not think you are a bad person. I am sure you are a great guy, but I think these ideas are bad. The subtitle reflects this while playing on this idea of dice-rolling that you use in the Edutopia post.
I often use people’s names in blog posts. Here are some examples:
https://gregashman.wordpress.com/2019/07/05/keith-devlin-is-wrong-about-teaching-mathematics/
https://gregashman.wordpress.com/2015/11/17/can-jo-boaler-grow-your-brain/
Although my name is not in the title, for several months, the blog post below was in the top three search results if you Googled my name. It is far more of a personal attack than anything I have written:
View at Medium.com
So I suppose I am saying that criticism comes with the territory and you are not always going to like it. It is part of the transition from a private person to more of a public figure (even if it is within our rather rarefied world of education commentary).
I am minded to remove your name from the title of this post given that you find it upsetting. However, in the context of your recent Twitter comments, I am concerned that this may be seen as an acknowledgement that I have done something wrong rather than a goodwill gesture.
When Oldridge finally defines what he means by play he ends up describing playfulness. Exploring ideas, talking about solutions and generally “playing around” in an explority sense. He dosn’t realise that explicit teaching permits this as a form of modelling. The difference being that the tutor demonstrates and encourages this aspect of mathematics while still guiding and explains the specific problem.
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I just stumbled on this blog in search of more information about Matthew Oldridge. I agree that research is often misconstrued when it comes into contact with large educational organizations and that educational trends have a tendency to wax and wane leaving the public confused and frustrated. But does this not say more about the learning structures currently in place than the actual ideas themselves?
In terms of learning, before we make any claims about any particular type of instruction or teaching, one ought first define the purpose of learning. This is sorely missing from this article and a few others I have read on your blog but I confess I have likely missed your viewpoint elsewhere. Along those lines, Kirschner, Sweller, and Clark have received significant criticism for their work you cite here, with the understanding these indictments are not necessarily evidence of guilt – though your readers might also benefit from understanding the valid criticisms directed at their research.
Nonetheless, thanks for your perspective on mathematics.
Kirschner, Sweller and Clark define learning as a change in long-term memory. I would suggest it is about developing schemas in long-term memory. I have discussed these ideas at length.
I have also discussed critiques of Kirschner, Sweller and Clark’s paper e.g. here
https://gregashman.wordpress.com/2016/09/14/that-minimal-guidance-paper/
Thanks for your reply. First of all, I think you ought to know many of your links no longer link to the studies and critiques cited. Nonetheless, I am wondering if there is not more relevant criticisms of cognitive load theory needed as Kuhn’s paper is nearly fifteen years old (which also makes me wonder how old the research might be for the unavailable links).
For example, Stewart Martin wrote a thoughtful synopsis and criticism of cognitive load theory research in Cognitive Load Measurement and Application: A Theoretical Framework for Meaningful Research and Practice (ed. Robert Zeng, 2018) In the same book, Zheng and Kevin Greenberg also provide useful insights into the limitations of measurement within the research.
Subsequently, it is hard to for me to accept a theory of learning that is based on the measure of some thing yet it cannot state with any sort of reasonable, scientific certainty what is actually being measured. As Stewart points there is currently no way to measure schema development and attempts to measure loads are generally unreliable. Even Zheng and Greenberg, who appear to be supporters of the theory, acknowledge that load measurement seems to depend more on convenience than methodology. Additionally, it appears our understanding of working memory is insufficient to support some of the claims Sweller et al. are making. I find all of this highly problematic, particularly if we are to use it as a basis for teaching and learning (with the understanding that just because something cannot be measured now, does not mean it cannot be measured in the future). Not that other theories of learning are free from limitations or uncertainty but parallels to Gardner’s multiple intelligences springs to mind here (I concede I may be overstating the situation).
Alas, I have only recently stumbled onto this theory and I am wondering if you could point me in the direction of further reading that might frame cognitive load theory in a more promising light. I feel like there is something missing from my reading to date.
Hmmm… this doesn’t come across as an entirely impartial request for more information, but here’s a lot of free CLT readings:
https://gregashman.wordpress.com/2018/11/16/free-cognitive-load-theory-resources/
Yeah, links break in old blogs. You get that.
On your comments on directly measuring schema development and/or load: Presumably, you would have been unconvinced by Darwin’s theory of evolution by natural selection until the discovery and isolation of DNA.
Not certain why requests for information for a position you support need to be impartial or how speculating on intentions is even relevant; nonetheless, thanks for the link.
Fair point about Darwin as his ideas were not initially well received. Of course, the body of evidence developed prior to the discovery of DNA was/is tremendous – perhaps that is where your analogy breaks down. That said, I think there are points of interest in the theory that need to be considered, such as the interactions between working memory and long-term memory and learning.
Just want to add here (since there does not seem to be a way to edit a previous post) that I fear my word choice has gotten me off on the wrong foot. I think my comments about the links and the possible age of the cited works likely comes off as an insinuation and it is easy to see now why that might have raised your eyebrow. They were mental notes that became unproductive comments and so I apologize for the misstep. In any future post, I strive to be more clear and concise with my statements.