Elizabeth Green, of Chalkbeat has a new piece out on teaching fractions. She is the one who wrote the piece “Why Americans Stink at Math” which appeared in the NY Times Magazine and was riddled with errors and assumptions that the cognescenti and punditry were totally unaware of and just assumed she was right. And when people pointed out her errors, (see, for example here and here) the errors continued to go unnoticed.
Now she writes about a “Math Lab” experience in NY:
“The Math Lab’s emphasis on learning math by talking and thinking about it is clear almost as soon as the students enter the room. A list of the class’ “math community agreements,” posted on a board, reminds students to “add onto each other’s thinking” and “analyze and observe each other’s work.”
“To help students internalize that philosophy, Van Duzer led an activity called “convincing a skeptic,” where students were asked to fold pieces of green paper into squares one quarter the size of the original and then convince their partner that the new shape was, in fact, one-fourth of the original.”
Sorry; that’s about as far as I got. (I recognize the “convince me” gambit from Steve Leinwand’s promotion of the technique). Being able to explain and convince rather than be instructed in the basics and given scaffolded problems to help reinforce procedures and understanding is off the table. Such techniques in the worldview of reformers is apparently too procedural and rote-like.
Then there’s this:
“As teachers reflected on Tuesday’s lesson, a debate of their own emerged. They began wondering about how Cipparone handled what the group would begin calling “Kris’ problem” — the moment that morning when Kris misplaced five-thirds on the number line and Cipparone had to make a split-second decision about whether to correct him before the students left for the day. Deirdre Flood, a teacher at Brooklyn’s P.S. 11, said it could make sense to end the lesson ambiguously if “every single [student] made a decision before they left, so they were thinking about it on their way out.” ”
Because ambiguity and not teaching students what they need to know (particularly when they need to know it) has just worked out great for the last 25+ years, now hasn’t it?
Just call me unconvinced.
traditional math 
“Being able to explain and convince rather than be instructed in the basics and given scaffolded problems to help reinforce procedures and undersanding [sic] is off the table.”
Can you elaborate on this particular gripe? You seem to think that exercises in explanation preclude other kinds of instruction. Why the reflex? In another post, you were concerned that explanations weren’t rigorous enough – inductive rather than deductive. How do you propose students get great at rigorous explanation without practicing any kind of explanation?
“Because ambiguity and not teaching students what they need to know (particularly when they need to know it) has just worked out great for the last 25+ years, now hasn’t it?”
Your suggestion is that ambiguity & discovery has been the dominant pedagogy for 25 years? How do you figure?
It’s interesting to me that people on both sides of the pedagogy discussion will point to the last 25 years as evidence that “things haven’t been working.”
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I’m happy to see that you read this blog.
With respect to my objection to “explanation” it depends on what we are asking students to explain. Students in lower grades lack skill in articulating ideas,and such explanations often have little mathematical value unless guided carefully. It is different in high school as students gain in maturity, mathematical and otherwise. But what I see is the “convince me” type argument. Leaving things ambiguous so the student doesn’t know if his answer is right or not does not strike me as a good educational practice, particularly in those lower grades. Yes, practice increases the skill in explanation but from what I see, there is too much emphasis on explanation in lower grades that is not adding to proficiency in explanation, and which takes away time from basic instruction that in my opinion would be more constructive.
The notion that failure to provide an explanation is an indicator of lack of understanding is overplayed in lower grades. A question here and there to check on reasoning is a bit different than “convince me” or not telling a student whether they’re right or wrong. At the middle and high school levels, students have more tools and more developed articulartion skills. Like many things, such practices can be done well or poorly, and my “reflex” that you cite is the danger that the practice becomes faddish, not well executed and results in students engaging in the exercise of guessing (or learning) what the teacher wants to hear–more “rote understanding” via “rote explaining”.
The “induction” questions you mention when they are “find the next number in the pattern” type question are misleading and give the impression that all that is needed is a conjecture. Yes, much of math revolves around conjectures that are inductive in nature, but at some point, the conjecture must be proven, and such proof relies on deductive reasoning. When the teaching infers that all you need to do is make an inductive conjecture, then I feel that that is a poor message to send to the student. (And I’m not saying that you do this or advocate it.)
As far as discovery and other practices that have been in vogue, they have increasingly manifested themselves in the lower grades. Thirty or so years ago, it was not unusual to have students in first year algebra classes who had mastered foundational procedures including decimals, percents, fractions, not to mention proficiency in basic multiplication/division, etc. Over the past two decades, there has been a noticeable decline. Although some point to changing demographics as an explanation, the decline is seen in areas where the demographics have remained relatively stable such as in Iowa.
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I think there are a number of issues here.
Firstly, where is the evidence that ‘convincing a skeptic’ improves a student’s knowledge or understanding of maths? In terms of the maths itself, folding a sheet of paper into four is very basic. I can’t see that it adds much to understanding. One of the key misconceptions that students have about fractions is that a fourth can only be a fourth if the object is divided into four identical pieces – Dylan Wiliam has a great multiple choice question on this issue in ‘Embedded Formative Assessment’ – this exercise does nothing to address that. Instead, it seems to rely on valorising discussion, persuasion etc.
This is more based upon the ‘I reckon’ school of maths pedagogy than anything else and yet it is presented as if it is best practice.
The problem in schools is not that teachers are following this approach to the exclusion of a traditional one. The problem is that most teachers are guiltily using a mix of suboptimal explicit instruction alongside these rather pointless activities. They are being pulled in a direction by the experts that experience shows them doesn’t really work and so they are caught in the middle.
What we really need is to inform teachers about what the research says on how to make default explicit instruction better. They need to know about highly interactive whole-class teaching and what research into the most effective classroom teachers – the process-product research of the 1960s-1970s – showed that they tend to do i.e. be very explicit and ask lots of questions of the students.
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“Like many things, such practices can be done well or poorly, and my “reflex” that you cite is the danger that the practice becomes faddish, not well executed and results in students engaging in the exercise of guessing (or learning) what the teacher wants to hear–more “rote understanding” via “rote explaining”.”
Rather than complaining whenever you see “explain” in a news article or blog post because the technique might become faddish or incompetently executed (what couldn’t!) why not complain when you see the the technique incompetently executed? Or compliment when you see it competently executed? All of us progressive squishes need the best critics we can get.
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Good idea; if I see it executed competently and beneficially I will indeed comment. From what I see, however, more often than not, many educational ideas become prone to what Eric Kalenze calls “over-correction”. But I appreciate your suggestion; thanks.
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Barry – you need to be aware of the second rule of education research: If it doesn’t work then that’s because it wasn’t implemented properly.
(the first rule is to design your study to ensure that it works)
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Thanks, Greg. I keep forgetting about that rule.
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Hard to pick out a point worth commenting on here, but I agree with Greg that this sort of approach certainly does appear to invest more in “valorizing discussion, persuasion, etc.” than in any adequate sort of formative assessment or (more important to me) progress toward genuine understanding of correct mathematical argumentation.
It is not that having students work in pairs or engage in conversation or debate about a topic of study has no value, but that relying on this as (what I understand in this piece) an epitome of the *primary, dominant* mode of instruction is problematic. For the teacher is in that room if for no other reason than to impose and pass along a standard of the discipline that is extremely unlikely to be found by social exploration by children in a classroom setting without a major amount of guidance.
I could see such an exercise as a *prelude* to a proper lesson, or interspersed with explicit instruction showing students what a correct mathematical demonstration would look like — even pointing them the way to getting one going. We do this all the time, and it is a classical technique in the math class. In such a formulation there is nothing owned by “progressive squishes” in the use of dialogue to provide a tangible experience that reinforces directly instructed material — the key element here is the apparent assumption that the instruction will spontaneously take place in these little private conversations with no knowledgeable guide. That’s simply a fantasy.
Here’s what WILL happen. Lots — probably ALL, if the kids are at all streetwise — of these pairs will report “success”. But what does that mean? What went on that was of value?
In one group (these are the kids I would give an A+ to because they honestly understand that there’s no intrinsic value in the lesson and choose to act accordingly … good for them) the first child will say “It’s 4x the area because Mr Duver just said it is”. The second child will say, “I’m convinced. My turn. It’s 4x the area because if you don’t think so I’ll punch you at recess”. The first child will be convinced. Lesson done. They’ll also get laughs from the whole group if Mr Duver decides to waste the rest of the hour by turning it into a “pair and share” exercise and you hear from each group in turn. Social status bonus!
In the keeners’ group you’ll get two kids who are really intend. One of these kids is from an immigrant home where academics are taken very seriously. They debate Plato and Satre around the table. The parents have already drilled them in this year’s math. And next years’. And they’ve read Euclid’s elements as bedtime stories. He’ll lay out the definition of area, explain a ruler-and-compass construction and develop the logic in verbal two-column format. The other kid will grasp the ideas immediately. But will decide to go at the problem completely differently, arguing that mass of a piece of paper is additive by the principle of conservation of mass, and so on … (If they’re made to share they’ll not win any social bonus points but they won’t care).
Others will just stare at each other, mumble something and agree that the other has convinced them … so they get the marks … or don’t stand out as the dumb kids. And so on.
The point is, the teacher is a homogenizing influence — the teacher can ensure that every child has an educational experience, and is exposed to appropriate standards, content and rigor so that it can be said that those leaving the class potentially have what they ought to have received from the class — though of course it is then upon their shoulders to learn.
Without the teacher’s guidance at that critical stage of building insight and understanding you have divergent outcomes. If anyone wants to know where inequity in learning happens … well, this is one of those places. The teacher has encoded inequality into the lesson. Your learning experience hangs significantly upon your partner and your own predisposition, and except if it’s a class of clones … that will tend toward inequality.
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A tag to that last thought: I don’t style or label myself a “progressive”. Honestly I consider myself a small-c conservative. I think the label “progressive” is ironic — in more ways than are worth listing here. But one of those ways is that what is often labelled “progressive education” is — upon examining its effects — quite inimical toward the concerns political progressives profess to value.
Explicitly, I make the case (in this one instance) that this “progressive” technique tends to contribute to educational inequity. As I read modern progressives, inequity is very the root of all evil.
I am by no means the only person to observe this wee contradiction between “progressive education” and “progressivism”. Indeed, four of those I consider to be leading authors in opposition to progressive education also openly profess to be progressives in the sense of political philosophy and orientation:
Daisy Christodoulou
Robert Peal
Tom Bennett
E. D. Hirsch
I suspect we could add our friend Greg to this list — correct me if I’m wrong, Greg! I won’t speak for Barry; the number of examples, in any case, is irrelevant, only that a lot of intelligent progressives are of the view that “progressive education” is not an instance of Progressivism (even though they do have a common historical origin, arising out of the Democratic Socialism movement).
Which brought me to mind of Hirsch’s beautiful essay on this point
http://www.huffingtonpost.com/e-d-hirsch-jr/why-was-it-called-progressive-education_b_2866999.html
The other three authors all touch on the point in their respective books.
So I’m a little unsure of how to take Meyer’s self-identification as a “progressive squish” here. Educational, or Political? Or do you reject the dichotomy?
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I would consider myself politically progressive. I am on the centre-left of politics. I have written about the fact that political progressivism should not imply educational progressivism here:
http://www.labourteachers.org.uk/do-lefties-have-to-love-inquiry-learning-greg_ashman/
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Thanks Greg. As I believed. Perhaps I have a dim recollection of reading that, as I was pretty sure of it, but couldn’t put my finger on the reason.
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What bothers me the most about these types of discussions, is that the intrinsic beauty, and system of mathematics, which took a few millennia to create, is so easily dismissed as being old fashioned. There is no consideration being given to how efficient, professional, and also how successful the methodology of mathematics really is! Instead, we now have students who did poorly in math class, go on to become math teachers, or better yet, education consultants, who want to reinvent mathematics and make it better based on their own experience…all the while ignoring how fantastic these former mathematicians really were, in developing a system easily recognizable in all 4 corners of the globe, which built our civilizations and sent a man to the moon. Better yet, why not just acknowledge that math is hard. It’s really, really hard, but is that anymore reason to change it all around?
We now have many who point fingers and say they got it all wrong…because kids didn’t “explain” their work.
Instead of reinventing the educational wheel…one more time…why not just acknowledge what works. American students suck at math when comparing them to many others around the globe. The way kids learn arithmetic, then mathematics, in school, is putrid. And yet, all we have are more excuses and reasons why we need to change mathematics. No attention is being paid to Cognitive Science, suggesting that kids first need to master their arithmetic facts in order to understand higher order mathematical concepts. (This, by the way, is also called common sense. Just sayin’…) No mention is ever given to conventional methods or the success they elicit in the classroom. And never, ever, are any of the empirical studies acknowledged which time and again, have proven that explicit, teacher led instruction, mastery of core facts, and successful teaching methods, are successful. They are never mentioned at a teacher workshop, at a School District meeting, or at a parent information night. No wonder we now have upwards of 50% of kids attending tutoring centres to learn their fundamentals – it’s outrageous!
So kids, keep esplaining yourself in class as it will lead to a nice job on the factory floor, or something where you get to say, “would you like fries with that?” once you’ve graduated. That’s about the only benefit I can see of “explaining how you got that” over actually knowing, and doing the work correctly.
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Wow, very well put, Tara! Indeed, with so many great examples of classical math instruction it is so maddening to see these shallow gimmicks promoted as deep, revolutionary or insightful. No, they’re bad pop pscychology ideas dressed up as innovative pedagogy. And innovative they’re not.
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“Firstly, where is the evidence that ‘convincing a skeptic’ improves a student’s knowledge or understanding of maths?”
Please see David & Roger Johnson’s ample research into constructive controversy.
But there is the other supposition – common to many comments here – that a) “mathematics” has a single definition, and b) that definition doesn’t prominently feature discussion, persuasion, argumentation, etc.
But the definition of mathematics is contested even by mathematicians. Moreover, even if they had a common definition, the question of how we allocate classroom time in the service of that definition is a political question, not a question with an empirical answer. Time spent on discussion is time not spent on notes, worked examples, explicit instruction, etc.
Politically, in the US, if you’re anti-argumentation you’re an outsider party. The Common Core standards for ELA, Math, and the Next Generation Science Standards have exactly one common standard and that’s being able to argue from evidence. If you’re anti-arguing, you have a lot of options, just like any outsider party. (Ironically, they’ll probably involve a lot of persuasion.) Convince the insider party they’re wrong about argumentation. Help teachers get argumentation right. Advocate a balance with other approaches. But arguing from a definition that doesn’t actually exist isn’t going to help anybody.
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“Please see David & Roger Johnson’s ample research into constructive controversy.”
You have raised this research with me before. Without a reference to a specific paper, I went looking and all I could find was some research based in social studies classrooms. I am not sure what this has to say about maths teaching. It’s an obvious point to make but maths is very different to social studies and I see no reason to treat them the same. The link between this research and the ‘convincing a skeptic’ activity outlined above is tenuous as best.
I discussed what I found out about the Johnson’s research here:
https://gregashman.wordpress.com/2016/01/23/the-evidence-for-dan-meyers-ted-talk/
I am not much interested in whether I am an ‘outsider party’ or an ‘insider party’. I am interested in the best ways to teach students mathematics. We can argue about definitions but I’d rather not. Most people know what mathematics is and it’s not really about mounting social studies type arguments. Even professional mathematicians who attend conferences and debate their ideas still have to be able to do a lot of detailed, technical maths. In fact, that is what they spend the bulk of their time doing.
It’s odd that international tests designed on such different premises as TIMSS maths and PISA maths – the former decontextualised and traditional and the latter all about applications and calculator use – show a similar pattern with countries of the Far East performing best. This may be due to cultural factors but it should give us pause before we change the definition of mathematics and assume that this implies the need for a different teaching style.
One thing I’ve noticed is that the education establishment tend to get together to write a set of standards and then argue that something must be done because it is in the standards. I recently wrote a piece that was critical of the idea of differentiation as it is popularly conceived and one response I had from an academic was that teachers had to make use of differentiated instruction because it was listed in the Australian Professional Standards for Teaching. As if that was an argument.
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Hi Dan. Few mathematicians bother to formally define the subject area of mathematics; they regard it as self-evident. However, I am one of those few. Mathematics is the science of structure, form and relationship. I have discussed this definition with many of my colleagues and it is never controversial, although it is quite often that others propose different takes on the content or focus. A common one is Keith Devlin’s “Mathematics is the science (or, sometimes “study”) of patterns”. I find this inadequate, but understand what folks are getting at — in fact, mathematics is also the study of that which is not a pattern, and of many things which do not follow patterns. When you try to define “pattern” generally enough for mathematics you find that the meaning must be stripped of the term. Which is why I prefer my definition as names those particular abstract qualities which are the focus of mathematical understanding.
The activity of formulating and presenting proofs is indeed fundamental in the field of mathematics. However the exercises described do not lend themselves to this, and in the hands of a teacher untrained in mathematics is likely to do nothing more than waste a lot of class time on a ludicrous non-mathematical, and mathematically uninformative, social engagement. Proper rigorous development of proofs and the standard approaches to proofs is a worthwhile investment in the classroom.
From what I have seen of Johnson & Johnson, they are largely concerned with the social dynamic of argumentation in, say political discourse. Can you show a paper in which they study a large sample in mathematics classrooms? I’m interested in what devices they use to check their own experimenter biases — something one should always watch for when the researcher is long known as a huge cheerleader for his or her own conclusions.
Perhaps there is civic value in teaching civil discussion practices. But this is not tantamount to learning what constitutes a proper mathematical demonstration. And without modelling and instruction from a teacher well-versed in the subject I am skeptical that this will happen in today’s public school classes.
If you want to see students learning proper proof techniques, then you could hardly do better than to reintroduce proper (Axiomatic) Euclidean geometry — a time-proven approach. Other subject matters would be as appropriate (such as basic axiomatic field theory) but hard to motivate with Jr High students, which is the appropriate point to be teaching this stuff.
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My understanding is that you believe math is motivating. Why makes axiomatic field theory exceptional? Do you think it should be a requirement for high school graduation? University matriculation? Do you have any ideas for making it motivating?
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Vern Williams, who has taught the accelerated math course for gifted middle school students for many years, gave me a text book that had been used in Montgomery County, Maryland in the 70’s in the last dying days of the 60’s New Math era. The book was part of a series called “Unified Math” and was for gifted students. It was written by Howard F. Fehr, James T. Fey, and Thomas J. Hill. Fehr and Fey were active in the 60’s New Math era, and Fey was actually Vern’s advisor when he was in ed school. The book he gave me covers sets, subsets, relations, mapping, binary operations, elementary group theory and field theory, and axiomatic affine geometry.
Vern taught some of these concepts in his middle school courses for gifted students, as well as Cantor’s theories on infinity, and what limits are. Vern’s course was obviously not for everyone. Having sat in on some of his classes, I can tell you his students were very motivated. So it can be done given the right students.
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Due respect to Williams, but he and people who work with honors students, university students, private school students, and students in other self-selecting environments, don’t offer useful case studies for motivation.
My personal interest isn’t in creating more math olympians but rather more students with above-average interest in math and numeracy that extends into adulthood. That might explain my interest in motivation, curiosity, and other affective issues that this group considers non sequiturs.
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I start to lose interest when the discussion grinds down to arguing about argumentation vs. anti-argumentation. I thought this was all about interpreting the evidence behind effective practices…of which there is plenty of that, yet acknowledging how to do it right seems to be evasive.
Education is the only profession which many choose to argue or anti argue on the evidence supporting best practices. This is a shame. Because when evidence suggests what is most effective, and we still have “professionals” choosing to fiddle and faddle rather than follow through accordingly, the students lose out every single time.
But progressives seem to like it that way. Tweet and twat while Rome burns, and never mind the students falling through the cracks. No wonder so many parents continue to send their kids to Kumon…
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““With rules-based teaching, if you don’t understand why you’re doing what you’re doing, then that rule only applies to that very specific type of problem,” explains Kim Van Duzer, a teacher at Brooklyn’s P.S. 29 who co-founded the lab program along with Cipparone and Kate Abell, who have both taught in New York City schools.”
This is the big assumption lie or stupidity. Which is it?
They never define what understand means and they never show that their top-down process works better than “rules-based teaching” done right for a curriculum. It’s been 20+ years of TERC, MathLand and Everyday Math, but they still pull out that old “traditional” rules-based teaching excuse. One can always slow down coverage of the material, but that can improve any pedagogy. What we have is a fundamental mess in K-6 in terms of full inclusion and “trust the spiral” that cannot ensure any particular level of mastery. (CCSS offers NO STEM-level preparation in K-6. It’s official now.) Tracking is hidden at home – I had to ensure “basic math facts” (we parents got notes from the school to do that!) at home for my math brain. I got to calculus in high school in the 60’s with absolutely no help from my parents. That is almost impossible now.
What’s being done now is to redefine learning to make full inclusion and spiraling differentiated instruction (our school actually called it differentiated learning) seem to work. It doesn’t. Tracking is hidden at home and schools don’t dare ask parents what they had to do. Proper learning and understanding is built from the bottom up by pushing everyone and ensuring grade-by-grade mastery of basic skills. Figuring out how to fix math does not come by looking at the details of a class of very mixed ability students after many years of a screwed up curriculum.
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