Best Practices In Education
Most of us remember that (a+b)^2 = a^2 + 2ab + b^2
But why?
Via François Taddei comes this wonderful video:
I’m willing to bet none of us were taught it like that, and yet once you see it it’s so obvious. It’s one of those “A-HA!” type moments.
It reminds me of reading about KIPP schools in Work Hard, Be Nice and reading that the schools had turned the multiplication tables into sing-song ditties that the entire class sings, which of course makes it much easier for children to remember. (I still remember the torture it was to learn multiplication tables as a child, and I still don’t know them. If you ask me what 4×8 is, I do the math in my head.)
There’s of course an important policy point to be made about all this: I’m willing to bet all the money in my pocket that if a fry cook at McDonald’s comes up with a faster way to make a Big Mac, his manager will notice, who will get his team to use it, and the information will trickle up to his manager and so on, and then trickle down and a year later all McDonald’s fry cooks around the world will be using the new, faster Big Mac cooking technique.
Such are the virtues of the competitive sector. Not all private sector companies are like that. Much has been written in the business literature about kanban kaizen the process of continuous incremental improvement which was so instrumental in the trouncing of US automakers by Japanese automakers. Elsewhere, I’ve described bureaucracies as “Institutions which do not see themselves as being under competitive pressure.” In the case of Toyota vs GM, GM became a bureaucracy because it forgot it was under competitive pressure, while Toyota was very aware that it was competing with GM and so the impetus to innovate and then broadly apply best practices was felt throughout the organization.
Forgive me for thinking this is precisely a feature public educational systems lack.
Actually I’m willing to bet most of us were taught that expansion that way; you seem to be an oddity….
— Kieselguhr Kid · Feb 9, 03:37 PM · #
@Kieselguhr Kid
Actually, you’d be surprised how many of us were, in fact, not taught the principle behind that “math fact” — it was just a given that we were taught to memorize where I was educated.
From high school and all through college, I never had a teacher or prof expand it out to its principle….
— JW · Feb 9, 05:07 PM · #
In fact I should say — I’m watching this in a place where I can’t stream video (usually I can’t), so I can’t see what PEG is referencing, I made this assumption: somebody’s drawing an (a+b) by (a+b) square and showing the subdomain areas. I guess if that’s NOT what PEG is referencing then JW’s comment makes sense (although, damn, I can’t imagine a simpler illustration). Note it’s not really a “proof” — the underlying principal is just algebra.
— Kieselguhr Kid · Feb 9, 07:48 PM · #
Kieselguhr Kid: That’s right. Well, good for you, you had a good teacher! But for me and JW and the people I saw responding to the video on Twitter, not everyone was so lucky.
— PEG · Feb 10, 01:23 PM · #
It’s not obvious to me that competitive pressure is what the public schools need.
I also thought the guru’s graphic was cool— and I did not learn the equation that way— but I’m skeptical that a profit-based education model necessarily promotes more effective techniques.
Most of us who attended public schools remember more than a few uninspired Krabappels, but we also recall some extraordinary, interesting teachers. The motivation of the latter, or the indifference of the former, was not evidently a function of the system. Rather, it has mostly to do with the uneven performances you can expect from employees whose positions are not regarded as specially prestigious and who are paid modestly, at best.
Think of universities— public or private— or the justice system. Professors and judges tend to rise to secure, long-term positions where the pressure to compete with peers is minimized. Yet, the system of tenure or lifetime appointment, while often scrutinized, is not maligned with nearly the intensity and consistency critics reserve for the public school system. Part of this has to do with prestige and pay— tenured professors and judges will tend to be effective because they are paid a lot better and held in higher esteem. Those positions, therefore, attract a higher percentage of motivated, talented individuals. And in the case of professors, the results obviously look better because their students go through a process of selection that does not occur in public schools. Regardless of teacher innovation, private/charter schools are going to see uneven results, similar to those of public schools, as soon as they accept a broad population of students, for many of whom home life is dicey/difficult/unsupportive.
I don’t mean to claim that an infusion of new ideas, possibly spurred by a marketplace environment, can’t direct a school in a positive direction. I just think this assumption— typically, but now always, held by conservatives— is vasty overrated.
— Pamela · Feb 10, 06:29 PM · #
Really? Having actually worked in chain food service, I’m willing to be you all the money in my pocket that if a fry cook came up with a faster way to make a Big Mac, his manager would tell him to stop fucking around and get back to making Big Mac’s the way he’s supposed to – somebody from Corporate might be watching!
Kid, of course, is chiming in with his own brand of aggressive idiocy – almost nobody gets taught polynomial expansion this way. We get taught FOIL – first, outside, inside, last. I’d be surprised if the geometric way is even in a textbook. And while it makes the expansion intuitive in a way that the mechanical FOIL algebra (which is what PEG and JW and I all learned) doesn’t, it’s not clear that the geometric example helps you apply multiplication of polynomials in the general case. And how do you make a square for (a-b)^2, or (a-b)(a+b)? Now you’re dealing with lines of negative length and rectangles with negative area. Doable, but easier than FOIL? I don’t see it.
Getting back to PEG’s point, schools already are under a significant degree of competition, and competition in schools resolves itself much the same as it does when health insurers compete – health insurers compete to only insure the healthiest, and schools compete to only educate the best students. It’s more cost-effective to make a needy customer your competitor’s problem.
— Chet · Feb 11, 12:30 AM · #
I don’t remember learning this at all, but this approach has a very Montessori-esque quality to it. Well done.
— Derek Scruggs · Feb 11, 02:29 AM · #
Actually it’s pretty easy to see how you’d use an analogous framework for (a-b)(a-b) or (a+b)(a-b), but I tend to agree that it’s a terrible way, in general, to teach this process, unless, ironically, you want to do what PEG seems to be looking down on and simply memorize one formula, which isn’t very useful. The geometry doesn’t really “prove” the algebra, and you want to be able to generalize to more and more binomials.
I’m thoroughly unimpressed by this video. I’m betting those impressed Twitterers are — a bunch of pundits and political commentators, not guys who use a lot of math?
The larger point is that while it’s certainly true a lot of teaching methods have ossified in standard school curricula, and surely those methods don’t evolve as quickly as one might like, they’re usually the way they are for at least some reason; there is some consensus behind those methods.
PEG, man, I never had a good math teacher. Well, not for this stuff.
— Kieselguhr Kid · Feb 11, 06:20 PM · #
Actually, following up that thought, I think it’s easy to explain why what PEG’s saying here is badly misguided, and if it helps, I can open with an attack on the political left.
When Al Gore was running for President in 2000, he made some comments on school choice I found thoroughly irritating. He was reiterating his opposition to school choice, and saying, look, what we need to do is invest in those dysfunctional schools, and really work hard over years to build them into world-class institutions. And I thought, man, that’s crap: maybe you can do it, but you have to tell parents: hey, you can’t move your kid somewhere else, you keep him in a failing school, but we’re going to fix that school eventually, after your kid’s out of it and his education is over. Whee.
When PEG talks about innovation, it doesn’t acknowledge the same problem Gore didn’t acknowledge. Look, innovation is hard because most innovation is bad. I had a friend in Chicago who was pissed because his daughter’s (public-school) math teacher was full of wonderful wacky innovative ideas and had decided to teach intro algebra by starting out with the field axioms — which I agree is probably not a good idea. But here’s the thing: she’s not going to take intro algebra again.
That’s why the analogy to a Big Mac fry cook is terrible. Most of the ideas your fry cooks have about how to change the Big Mac are going to be bad. But that’s OK: I try the Big Mac, I’m out a couple bucks, and I know immediately whether the idea’s any good and if I ever want to try it again. This is the argument for buying cheap wines!
But if I make some profound change to how I do education, I don’t know how to evaluate it until I have a lot of statistics — kids vary a lot! — and I look some years down the road when I try to see how the cohort does working with follow-on concepts. If they do badly, there’s no do-over or redo: you can’t toss the kid like a nasty (-ier) Big Mac. Try selling that to a parent. You’re in the Gore position.
I am a parent. My kids seems skilled and hardworking academically, so if the school pretty much does the same-old same-old, they’ll be OK. I’d certainly like the school to try some experimentation and innovation, but just around the edges, and in consultation with me — and when you consider the number of parents, that’s going to be a significant brake on the changes you can make. PEG is treating that risk-aversion like a bug; it’s not, it’s a feature! Significant experimentation is only going to happen at the edges and in small groups and as parents in general that’s how we want it. If you are advocating for some system that feels free to experiment with process more widely and commonly, it’s not one we want.
— Kieselguhr Kid · Feb 11, 06:55 PM · #
It’s kaizen, not kanban. Kanban is a production system based on pull rather than push, and can enable continuous improvement (kaizen).
— Bruce Hughes · Feb 11, 07:35 PM · #
I know how this will go over because of the virulently reactionary crowd in comments, and because you are a person who never even remotely challenges his own preconceptions, but look. I’m a researcher in educational and pedagogical policy. The constant attitude from people on the outside is that these problems are very simple. So you look at this video, and you, being an educated person from a stable background and reasonable natural intelligence, say, aha, this is how to do it.
Here’s the problem: the vast majority of empirical evidence demonstrates that pedagogical practices (such as what’s happening in this video) are far less determinative than demographic factors. People like you— well meaning, I’m sure— vastly overestimate the impact of different pedagogical techniques on disadvantaged children. The fact is, most students have remarkably static educational performance throughout their lives, and this performance is remarkably unaffected by major changes in their education: moving from state to state, switching from public to private, going from traditional pedagogies to nontraditional like Montessori, whatever. People point to efforts like KIPP, but KIPP’s differences have never been demonstrated to scale adequately (in fact the evidence suggests just the opposite), and in fact the biggest differences in KIPP schools are the infrastructural differences. I’m not kidding. Take your average student’s performance at 7 and you can make startlingly accurate projections, for 10, 13, 15, 18.
You have a very common, cheery, and totally false vision of education: that students are endlessly mutable and improvable. As time goes on, the empirical case gets stronger and stronger: most of educational output is purely student-input dependent, based on the material conditions of a given students life, and on their inherent intelligence. Not popular, with either liberals (who want to believe that everyone is of equal ability, even though that’s nonsense) or conservatives like you, who just want an ax to grind against public schools. But true,
— Freddie · Feb 11, 10:02 PM · #
Hold up, Freddy, that’s not entirely true: I seem to remember that last month everybody was buzzing about a study from Chetty, Friedman and Rockoff that showed that teacher performance actually did subtantially affect student outcome (I remember the names, sad to say, not from innate brilliance but because Rockoff was on “Diane Rehm” Thursday, and in fact the education researchers on that program seemed unanimous that there is considerable empirical support for the proposition that changes in teaching do in fact affect educational outcome.)
That’s not to discount the fact that the most significant factors in educational outcome, in bulk, are surely social and economic background. But I think it’s pretty easy to find serious education researchers who disagree with what you just asserted, and Chetty et al. (and other research groups) seem confident that the gains are greatest at the lower end (unsurprisingly).
Although, again, it’s sort of moot. Kids of parents with good educational backgrounds and modest-to-substantial means — kids like mine — are going to come out ahead even when run through a pretty standard, by-the-book curriculum, so if Freddy opens up Freddy’s Awesome School or PEG opens up PEGs Awesome School which uses some amazing new teaching technique (although, I can see the video now, and that guy’s a bit creepy), I would be an ass to send my kids there: the potential upside is small since I already can reasonably expect good outcomes, and the potential downside large and irreversible. Again, that’s why the widget manufacturing analogy is silly.
— Kieselguhr Kid · Feb 12, 12:56 AM · #
and in fact the biggest differences in KIPP schools are the infrastructural differences
Freddie, I’m not familiar with this. Could you explain or point to more information on just which infrastructural differences these are?
— The Reticulator · Feb 12, 02:55 AM · #
BTW, back when I was a teacher, a few decades ago, I was pretty good at the kind of explanation shown in the video, except I tried to draw the students more into it, inductive-learning style. While I got a few plaudits for that from peers, students, and others, it does not by any means mean I was a good teacher, or that the students learned as much as they should have, or that even in better hands this sort of thing is the key to fixing what ails education.
— The Reticulator · Feb 12, 03:03 AM · #
“I’m willing to bet all the money in my pocket that if a fry cook at McDonald’s comes up with a faster way to make a Big Mac, his manager will notice, who will get his team to use it, and the information will trickle up to his manager and so on, and then trickle down and a year later all McDonald’s fry cooks around the world will be using the new, faster Big Mac cooking technique.”
This is hilarious nonsense. There isn’t a large company in America that works this way. I’ve done a lot of work as a business consultant for a lot of different companies (banks, retailers, you name it) and not one single company I’ve ever worked for had a ‘trickle-up’ mechanism. As one of the previous commenters noted, the relationship of large companies to their grass roots is entirely about compliance to top-down standards.
The way you get a better method of making anything is for a small-scale innovator to create and implement it, show success, and then have the idea bought, stolen, or copied by everyone else who can use it.
— tim b · Feb 13, 02:32 AM · #
Pamela: I explicitly mentioned the need for competition, as opposed to the profit motive, because the two are different. The profit motive is indeed a powerful spur to competition in many cases, but not all. Meanwhile, the “weighted student formula” that some US school districts have, where public schools compete for education dollars based on how many children they attract, can help spur competition without a profit motive.
The point is—let a thousand flowers bloom. Let people experiment. Let’s have public schools, and let’s have private schools, for profit and non for profit, and let anyone attend them. And let’s see what happens. I don’t see what we have to lose.
You say one reason why we have poor teachers is because it’s a job that’s not very well paid and not very prestigious—I agree! But to me this is an argument for more competition and deregulation.
Chet: So, a few points there:
- There’s extensive evidence in the business research literature that in the fast food sector the chains that thrive, which in the past few years has included McDonald’s, combine a high degree of standardization with high flexibility and openness to “bottom-up innovation” or what I’ve referred to as kaizen, ie seeing what works at the local level and then spreading best practices through the organization. I’m sorry that your own experience doesn’t reflect that and I’m sure that many fast food chains/franchises don’t reflect that. But it doesn’t mean it’s not happening.
- See, I wasn’t taught FOIL either. But this isn’t the case of either/or. The FOIL method might be more effective over time, but the square method is intuitive. You can use the square method to get that “A-ha!” moment and then segue into the FOIL method. The point is that we need a system where incentives are aligned so that a) lots of experiments are made and b) the results of successful experiments are spread rapidly through organizations.
- I find your analysis of how the private sector functions to be frankly lacking. Schools compete to educate the best students because there’s little incentive to work on these other students. There are many things one may imagine to fix that. And I don’t think the US health insurance market, riddled as it is with regulations and inefficiencies, is an example of anything except that if you work very hard at it, you can get a system with the worst features of the public sector and the worst features of the private sector. More broadly, you’ll find that in the competitive for-profit sector, the competition is hardest to serve the underserved. In the retail sector, Wal-Mart and Amazon don’t compete for the biggest spenders, they compete to offer the lowest prices. In the automotive industry, the sector where the competition is most cutthroat is at the lower-end, not the high end. Same in consumer electronics.
Derek Scruggs: Thanks!
Bruce Hughes: Whoops! You’re right! Embarrassing. Fixing, thanks!
Kieselguhr Kid: I agree that innovation is hard, and is made harder in the educational system because measuring things is even harder. But I think it only strengthens the argument that we need to make the system more flexible so that failing innovations can die off faster and successful innovations take off faster.
Freddie: Thanks for your comment.
I don’t necessarily entirely disagree with what you’re saying. The point I was making isn’t “Hey, if you teach maths this way, presto, everything is going to be ok and children from awful backgrounds will do as well as the children of professors!” The point I was making is: we can surely improve things at the margin.
While I’m not sure that the evidence is as one-sided as you make it out to be (I do believe there’s evidence that excellent education can change outcomes) even if it were I would question the direction of the correlation. If you take widely different inputs and then put everything through the same cookie-cutter machines then you’re going to get widely different outputs. Maybe it means you need to change the inputs (and I certainly think we could have better social policy!) but maybe it means you also need to change the machine.
— PEG · Feb 16, 11:41 AM · #
PEG, I think you missed the point. It’s not simply that innovation is hard. It’s that it is — at least for middle-class-and-up parents, undesireable. You’ve dedicated a post to trying to suggest that the public school system lacks an ability to swiftly innovate, but that lack is a feature not a bug. Why wpould I send my kids to an “innovative” school?
Noah had the same problem once bemoaning a lack of innovation in how colleges operate — it has the same adverse selection problem.
There is, in fact, a reason why things like the Harlem Children’s Zone or KIPP operate first in heavily disadvantaged populations.
— Kieselguhr Kid · Feb 16, 08:58 PM · #
It’s not simply that innovation is hard. It’s that it is — at least for middle-class-and-up parents, undesireable.
What the Kid said.
At least since Dewey the cry has been for innovation. The more innovation we’ve got, the worse the results. This naturally leads to a call for more innovation. And so on.
Unfortunately, I’m the kind of person who can’t stand not to innovate. And of course there is such a thing as evaluating innovations to see if they are any good. But such evaluations as are usually done tend to assume an infinite amount of time and money, and also assume that each individual child has infinite amounts of time to discover from scratch the knowledge that billions of people have accumulated over thousands of years.
— The Reticulator · Feb 17, 05:41 AM · #
Kieselghur Kid & The Reticulator: I definitely agree that what you’re describing is part of the problem. The main political force against school vouchers isn’t teachers’ unions, it’s suburban parents who don’t want poor/brown kids to be in the schools that they bought a voucher for through the purchase of land. (I don’t necessarily equate school vouchers with innovation, but it’s another avatar of the risk/change-aversion of bourgeois parents.)
— PEG · Feb 17, 10:39 AM · #
PEG, not only have you fallen back on a weird left-wing resentment trope — look, man, I have no problem with minorites doing well for themselves, in fact I deliberately chose to live in an area with more minorities than the surrounding areas, where al the white folks seemed likely to make life hard for my kids! — but I feel like something Retic and I were saying sort of clearly has been overlooked. Let me try to more clearly voice two reasons why I think what you’re saying is nonsense. I don’t think I’m saying anything here I haven’t already, but.
1) You are ignoring substantial costs to individuals. If ten fry cooks radically innovate and, as you might expect, nine produce horrible stuff and one produces something great, then you know when the customer tastes the burger, you throw the bad ones away and everybody shifts to the good technique. If ten schools radically innovate, you know years down the line — if then! — when the students are competing in the workplace. Then maybe you can make corrections. But what do you do with the students to whom you’ve given a less-than-standard education? How do you fairly decide who pays that cost? Want your kid to be in an experimental school given reasonable odds? For those reasons, applying private sector-type innovation to education on any large scale is nuts — the product is really valuable, can’t be remade, and needs to be integrated into society. You need most of the educational system output to be predictable.
2) The system may be failing the country but it isn’t failing the bulk of parents. I’m not wowed by my kids’ school. The teachers seem kind of dim, the curriculum repetitive — it’s holding them back, far as I can tell, and boring them — and there are some stupid innovations (they remember complicated, one-off unintuitive stories for each item in the multiplication table). But here’s the thing: if the school does more or less what schools do my kids can reasonably expect to go to an Ivy League college and do well for themselves, and if they don’t it’s probably because of external things beyond the school’s power. This goes to Freddie’s point: the parents’ socioeconomic status is the major determinant, so basically every parent can expect the kids will do as well as he/she has done. Maybe a bit better. Since having kids is fundamentally optimistic, let’s say, we’re expecting a little better. Why would I want to mess with that? That’s not about not wanting poor folks to do well. It’s that, rationally, all you can offer me is ways to worsen my kids’ outcome. And the likelihood for any given approach is, you won’t even improve, and may worsen, their education. It’s not wrong of a parent to say, that’s nuts. It is nuts. What we want is for those disadvataged kids to get exactly the same blah education our kids get, and the big big problem is there’s considerable evidence that they’re getting something worse — something less resourced with still less competent teachers in a less safe environment. I can address all three of those problems without “innovation” in education at all!
The system is resistant to change because it should be resistant to change: what we want is an education system that changes slowly and at the margins and then those ideas can diffuse through, slowly. Proposals to “fix” that are just silly.
— Kieselguhr Kid · Feb 17, 02:50 PM · #
Oh, I’m sure McDonalds has a test kitchen or two where they prototype new technologies, new methods, etc. Obviously fast food is as innovative as any other American manufacturing. But it’s not the “bottom-up” kaizen process where a line cook figures something out and it’s passed up the chain, like at Toyota. That’s just not something that happens – the purpose of the managers at a McDonalds is to ensure uniformity of product and prevent employees from varying their techniques.
But, fine, here’s how it’s going to work – I’ll provide examples of my own experience and the experience of others where legitimate innovation was immediately punished by managers, and you’ll just say that was at one of the chains that doesn’t do kaizen, or that it was a bad manager, or something. All of which you’ll do by referring to the “evidence” that this is happening, but without providing even a single example of it.
Sure, if you already get algebra. If not, the part right at the beginning where he draws a line and calls it “a + b” is going to throw a lot of kids for a loop. Adding two letters and getting a line? I don’t see how that’s a big step up in the “intuitive” department, and I doubt you do, either, unless you have zero ability to remember what it was like before you were taught to work with equations.
Thank you for your opinion, but I find “my analysis” to be truer than anything you’ve ever said on the subject.
Funny that you don’t mention even a one.
But that’s tautological – “cutthroat competition” means “competing in a low-margins market segment.” There’s no support there for your notion that businesses are falling all over themselves to make the small bucks serving the hardest cases. How does that make any goddamn sense?
And the fact that retail is a segment where you’re paid by your customers, but health insurance and education aren’t, seems to completely disappear from your analysis. Health insurers are paid by the healthy to treat the sick. Educators are paid by adults to teach children.
That matters, don’t you think?
— Chet · Feb 17, 06:03 PM · #