I originally posted the following on my old blog back in 2013. I thought it was worth reposting given recent discussions.
For those of you who don’t know, Dan Meyer is a bit of a phenomenon. On the basis of having taught maths for six years in California, he now travels the world, holding professional development sessions on his approach to teaching. He has even given a TED Talk.
I was recently made aware of Onalytica’s list of the most influential education blogs. I’m not clear how Onalytica rates influence and it seems to throw-up some rather odd results. However, Dan Meyer’s blog is rated number 1 by this approach and it has been for some time.
I think it interesting that this should be the case because;
1. I disagree with Dan Meyer’s analysis of the teaching of mathematics
2. Dan Meyer presents little evidence to support his case
Dan Meyer has an aphorism that sits beneath the title of his website; “less helpful.” This really summarises his approach. For instance, in his TED talk, he discusses a textbook question about a water tank. The original question has a lot of structure. It is split into multiple parts that lead students through the question in a particular order. This is not accidental. The textbook will have been designed by writers who have an implicit or explicit understanding of cognitive load. Novice learners need this structure because the capacity of the working memory is limited and so it enables novices to focus on a few salient points at a time.
However, Meyer says, “The question is: How long will it take you to fill it up? First things first, we eliminate all the sub-steps. Students have to develop those, they have to formulate those. And then notice that all the information written on there is stuff you’ll need. None of it’s a distractor, so we lose that. Students need to decide, “All right, well, does the height matter? Does the side of it matter? Does the color of the valve matter? What matters here?” Such an under-represented question in math curriculum. So now we have a water tank. How long will it take you to fill it up? And that’s it.”
This is quite poor advice. It is conceivable that some students who posses more mathematical knowledge will be able to cope – this is known as the expert reversal effect. But, for novice learners, this is going to lead to confusion and frustration. I am sure than Meyer is a talented and inspirational teacher who has strategies for mitigating these issues but a less experienced teacher following this advice will find it a struggle. Little learning will take place. Given that conventional means are more consistent with both cognitive science and the findings of educational research, it is odd that a strategy would be suggested that throws up such problems.
My second point about the TED talk in particular – and Meyer’s approach more generally – is the odd nature of the discourse. Meyer does not seem to feel the need to justify any of his claims with reference to any relevant evidence. At one point in the TED talk, we hear a reference to something said by a TV producer but that’s about it. Are we to assume that this whole edifice is constructed solely upon what Dan Meyer reckons?
An interesting contrast would be with Dylan Wiliam. He is one of the most respected and influential educationalists in the world right now, having been deputy director of the Institute of Education in London. Yet, he never presents anything without discussing the research evidence behind it. He doesn’t expect us to accept an argument from authority, even from him.
I do not claim that all assertions are unacceptable. Some points in an argument are obvious and it would be tedious to reference every statement that is made. Some points are clearly a matter of opinion (e.g. my own assertion that thinking hats are silly) and should be taken as such. However, to build an entire methodology around assertions seems a little bit much. Most posts on Meyer’s blog are descriptions of the application of his methods with the central tenets seemingly assumed. Some plucky individuals have made reference to cognitive science in the forums but these are treated largely as an interesting curiosity.
I don’t actually hold Meyer responsible for any of this. Teachers have views which they are entitled to share. However, it is an indictment of our profession that assertions can elevate you to such dizzy heights, even when they appear at odds with the available evidence.
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Additionally, that water tank, without the structure, becomes a very different problem. The initial problem is to find the volume of the octagonal prism and then rate of inflow is used to find the time to fill. In Dan’s re-creation, the time to fill a portion is easy to measure (the video has a timer) and the students easily jump to “We’ve taken 2 minutes to watch it fill 25%, so 8 minutes,” which is a valid answer to the question but it ignores all of the intended learning.
I spoke about it here: http://mathcurmudgeon.blogspot.com/2010/07/curriculum-makeover-and-problem-11.html
Like most things I find myself in the middle of the possible ends here. There is evidence from the work of Boaler and Skemp and Pratt that there is merit in getting children to work on problems that have some problems solving element rather than what Pratt calls “dressed up problems” e.g. as Curmudgeon above points out the ‘water tank’ problem is really about the volume of an octagonal prism.
I agree that you would not want to leave mathematically weaker pupils without any structure or support but I do not think that that is what Meyer is saying (having heard him talk and read some of his work) – I think what he want pupils to do is to consider the nature of the problem rather than focus on the calculation (and the reduction of mathematics to calculation) – which in many ways is the least interesting part of the problem. Skemp differentiates this as instrumental learning rather than relational (conceptual) learning.
I think that Meyer would argue that the choice of the task should be appropriate to the mathematic skills of the learning – and that some of these will need to be taught more instrumentally – but that there should be an iterative approach to the teaching.
You will be aware that when children are asked about the subjects that they like Maths often scores as a Marmite subject (i.e. the Gallop polls among American high schoolers) so there are some questions to be asked why.
In terms of your last point, I suspect that all has to do with facility in maths.
I don’t think there is much evidence for problem solving in the way that you suggest but at least you have cited some names. Do you have any links to papers? I am aware of the ‘product failure’ research but most of this is not properly controlled.
I just learned a new phrase–“Marmite subject.” The first hit when I searched was this (https://goo.gl/V12ozZ), which indicates that it means “a subject that students either love or hate.”
Anyway, I had two thoughts:
First, there is genuine and legitimate confusion among both detractors and supporters of methods like the one in the video as to whether those methods are tactical or strategic suggestions. That is, it is left unsaid as to whether teachers are being advised to create “this kind” of classroom environment (strategic) or to use these methods as appropriate (tactics). And because this is left unsaid, the message gets to shape-shift depending on the circumstances and the audience.
Secondly, and relatedly, this is worrisome to me because I have heard now from more than a few teacher friends and colleagues who work in professional development (PD) that information about the effectiveness of explicit instruction is deliberately downplayed or left out of PD presentations out of fear that teachers would simply go back to teaching how they were taught. I really hope that this kind of patronization, condescension, and distrust of teachers via PD is not widespread, but given that, in my experience, messages like the one in the video routinely make no time at all for even a mention of the robust body of research leaning against them, it is hard to not come to that conclusion.
People who give professional development to teachers who do not inform their audiences about the wealth of research regarding explicit instruction–in the presentation, not in isolated side conversations afterwards (i.e., in the blog, not in the comments; in the Tweet, not in the DM)–or at least allow that research to temper their rhetoric, are either embarrassingly uninformed or are being deliberately dishonest.
Meyer argues that students do not like math because they cannot connect what they are learning to how it will be used. This is not my experience, nor does it turn out to be Meyers’. I recall a post on his blog and dialogue. It is here: http://blog.mrmeyer.com/2011/ps-the-progress-weve-made-in-34-years-2/
I’ll reproduce part of it here:
“Christopher Danielson finds a text in his college library called How to Solve Word Problems in Algebra: A Solved Problem Approach (Johnson, 1976). A sample problem: Mrs. Mahoney went shopping for some canned goods which were on sale. She bought three times as many cans of tomatoes as cans of peaches. The number of cans of tuna was twice the number of cans of peaches. If Mrs. Mahoney purchased a total of 24 cans, how many of each did she buy? (p. 14)”
A dialogue with comments then follows but the one that grabbed my attention was this:
“PedagoNet: I liked that problem.
” Dan: Me too. But I had really strong mathematical schemas (for a high school student) to fall back on. I wasn’t still trying to figure out the answer to the question, “Where does this math stuff figure into my life?” If I were trying to answer that question, this weird contrivance where Mrs. Mahoney knows all these relationships and facts about the cans of tuna, tomatoes, and peaches she bought, but not the number of cans themselves would pretty well answer the question for me.
“PedagoNet: Have you asked students for their opinion?
“Dan: Not explicitly, if only because the answer has been really implicitly clear.
“jg: Well… if we stick with peoples’ actual experience (especially kids’), then there will be almost no problem solving at all. They’re not used to actually thinking.
“Dan: This mistakes cause and effect. If students are told on a daily basis that they have to shut off their prior experience and common sense to succeed with pseudocontextual assignments, then they’ll oblige. It’s the wrong response to then say, “Look at them! They don’t think! It’s pointless to assign them thoughtful problems!”
Dan’s responses are interesting. PedagoNet liked the problem and Dan admits to liking it too but then said he had really strong mathematical schemas (for a high school student). Why then is Dan not interested in building up similar schemas in his students instead of giving them tedious, number crunching exercises that pose as real-life problems? Why does he consider the schemas that he had strong for a high school student? They used to be quite common. Traditional algebra was focused on developing such skills in students, and Dan clearly benefitted from them. Why does he deny these in the name of ridding math of “pseudocontextual assignments”?
The remainder of the dialogue I reproduced shows that Dan and others are aware that students actually like solving such problems if they are able to do so. He of course qualifies that by saying they have to “shut off their prior experience and common sense”. Kids play video games that are unrealistic as well; is he against those? If students can do something with success and feel good about it, most really don’t care whether it’s relevant or not–though in more recent posts he has started to hit back against that notion.
The algebraic skills involved with the tomato can problem are valuable and generalizable; more so, in my opinion and in the opinion of others I happen to know, than having students gather their own data so they can tediously crunch it and feel like they are solving “real” and “relevant” problems.
I think the underlying issue here is that the mathematical schemas that are strong in Meyer were there in part to a pre-disposition towards them. The cans problem appeals to anyone who already has the inclination to see the math problem through the words regardless of what those words may be.
The problems we face are that more often we are pushing students to achieve higher levels of math competency as a whole regardless of predisposition towards it. This has created larger populations of students who see these “equations that use words for the sake of delivering implied equations” as unnecessary and contrived, rather than jumping on the chance to solve the problem.
What I’m seeing Meyer do is make the question more appealing to curiosity and let their inherent logic take root, supplementing it with more complex algebraic thinking as necessary, something that’s hard to convey in a TED talk. The implication here being that you engender a sense of “wanting to know the answer” to situations that may arise in their non-school lives instead of the head scratch followed by a shrug because it seems too complicated. If they think that math class problems are contrived and only for math class, the chance of them having this reaction is reduced.
I understand that the intent of the water tank problem is to calculate the volume of said tank as a prism and use the rate of water inflow as a result. How often, though, will someone actually do this? If you are filling a tank and need to walk away, will you take the time to measure the tank and run calculations? Or will you eyeball it and reasonably determine for how long you can walk away?
If we want to push certain skills, then we should come up with problems that require them, not contrived sets of words that only serve to force an equation. Find a reason that the student has value in that skill, otherwise retention is low.
There are two problems with this argument. Firstly, it’s all supposition. Secondly, it represents a purely instrumental view of education. We might as well as what’s the use in everyday life of learning about the civil war / oxbow lakes / Shakespeare / photosynthesis.
https://xkcd.com/1050/
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Reblogged this on The Echo Chamber.
Forgive me for saying so, but I wasn’t aware that supposition was completely invalid for this conversation. I would say that your entire first point of contention with the approach is supposition (I have numbered the statements that are particularly demonstrative of my observation)
“This is quite poor advice. It is conceivable that some students who posses more mathematical knowledge will be able to cope – this is known as the expert reversal effect. (1)But, for novice learners, this is going to lead to confusion and frustration. I am sure than Meyer is a talented and inspirational teacher who has strategies for mitigating these issues but (2)a less experienced teacher following this advice will find it a struggle. (3)Little learning will take place. Given that conventional means are more consistent with both cognitive science and the findings of educational research, it is odd that a strategy would be suggested that throws up such problems.”
The initial 60% of that paragraph is all supposition. You eventually reference that conventional means are more consistent with cognitive science and educational research, when there is evidence to contrary. To name a few:
http://www.researchgate.net/publication/41846896_Effect_of_Using_Problem_Solving_Method_in_Teaching_Mathematics_on_the_Achievement_of_Mathematics_Students
http://digitalcommons.cedarville.edu/cgi/viewcontent.cgi?article=1025&context=education_theses
http://journals.lww.com/academicmedicine/Abstract/1993/07000/Does_problem_based_learning_work__A_meta_analysis.15.aspx
Secondly, education is meant to be instrumental in some way. The civil war informs the state of the nation which informs future decisions that have led to where we are today. Oxbow lakes and photosynthesis are instrumental to understanding that the world isn’t magic. Shakespeare is instrumental in understanding the influence that written work in the more era has experienced through its dramatic and comedic devices.
Even those who pursue higher education out of pure enjoyment do so to further their comprehensive view of the subject at hand. It is instrumental in terms of personal growth.
Back to the actual point of the matter, Meyer is basing his message on experience (both his own and others’), which is informing his approach, as he stated in the talk. The message resonates with a community of math teachers who have experienced the same. This has also created enough curiosity as to the validity of such approaches, which has spawned numerous studies that support such a claim.
My comments are not based upon supposition at all. They are based on research on cognitive load. If you would like to read more about the expertise reversal effect – which you seem to think is supposition – then here is a useful paper:
http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1141&context=edupapers
Let us deal with the papers that you have provided.
The first paper is a reasonably large trial. It seems that they have used a matched allocation based upon pre-tests rather than a totally random allocation. It appears that the two groups also had different teachers although the paper stresses that they had the same level of experience. Teacher effects in education are quite large so this means it’s not a fair test – a fair test requires only one factor to be varied at a time. Also, there will be an expectation effect – the authors clearly think problem based learning is superior and this is likely to affect levels of expectation, enthusiasm etc that could filter down to the students. This is one of the reasons why John Hattie cites an effect size of 0.40 as a cut-off for such studies rather than 0.00. In this case, I cannot see a reported effect size.
The second paper is a small-scale study that finds a ‘significant’ i.e. statistically significant effect. Again, I cannot find an effect size. However, I note that this experiment was conducted in two classes that were not randomised. It is also likely that there would be an expectation effect as in the previous study.
The third paper appears to be a meta-analysis of PBL in undergraduate medical education that finds low effect sizes in favour of it. Undergraduates have more expertise than high school students so I’m not sure that this is particularly relevant. Hattie has performed a review of all of the available PBL evidence, alongside evidence for more didactic forms of teaching, finding that the effect size is generally larger for the latter. I am not entirely convinced by meta-analysis in education but, if we are going to use it, Hattie would seem to be the most authoritative.
If you would like to review the (superior) evidence for my position then you can find it in this post:
https://gregashman.wordpress.com/2015/07/31/nothing-to-prove-but-i-will-anyway/
You argument that oxbow lakes are somehow ‘instrumental’ in that they enable students to understand that the world is ‘not magic’ is extremely weak. If this is the case then we can make such vague claims about anything, including highly abstract mathematics.
I reject an instrumental view of education as training for future work or life. I prefer Michael Young’s conception of the purpose of school being to impart powerful knowledge:
“Knowledge is ‘powerful’ if it predicts, if it explains, if it enables people to envisage alternatives, if it helps people to think in new ways. If these are some of the ‘powers’ of powerful knowledge, how might it then be distinguished from the everyday or commonsense knowledge that we all have as members of society?”
You are right that Meyer’s argument is from experience. I prefer arguments based upon evidence. It is a shocking indictment of the state of educational discussions that an argument based on experience can be so influential.
I don’t support everything Mr. Meyer says, but I have some questions: How can you get statistical evidence when no one wants to try something new? Don’t you need to spread your hypothesis to find enough experienced teachers to research with? After all, is Michael Young the one that wants people to “think in new ways”. I don’t think that you can measure the effectiveness of any method in less than, say, 12 years, or some generations… (I know that’s a supposition too)
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I agree with some of his comments, especially about the way the textbooks lay everything out to get answers, not to teach processes. Math, like science, is about a process, and the more practice a student can get following a process, the easier they can solve any problem. Repetition will get students a passing grade, but not a lifetime learned lesson.
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