I think the wisdom of student engagement is best determined on a case-by-case basis.

I think that Boaler's research could easily set off someone who was hostile to the complex of weenie-ass ideas associated with phrases like "different ways of knowing." The kind of person drawn to mathematics tends to be about hard facts and have fairly rigid ideas about epistemology. I have no trouble seeing how an academic feud could develop. My bullshit detector goes off when I encounter that particular phrase.

Nothing about Boaler seems particularly bad to me based on a quick reading of her faculty page and publication list, but the magical bullshit detector activation phrase does show up in the titles of her work. I'm inclined to give her the benefit of the doubt, assuming that she means what I would call 'ways of learning' or 'ways of understanding,' but still, most often when I've encountered that phrase what follows is wooly headed bollocks.

Good point.

I'm put off by the racism claim, because the specific she quotes looks like Bishop objecting to other people (not clear specifically who) being racist. That doesn't make Bishop right, but someone who lumps together (possibly objectionable) complaints of racism with affirmative racism as the same sort of thing is someone who I regard with suspicion.

Generally, I obviously don't know a thing about this, but my instant reaction is that Boaler is at the least not unambiguously in the right. You know what her tone reminds me of? David Graeber losing his shit at Crooked Timber.

Plausibly topical!

I felt like if I came down hard on her in the OP I'd be accused of blaming the victim, but the sniff test is off and I didn't have time to really poke around. But yes.

Is this her faculty bio page? Is this a normal use of a faculty bio page?

Sure. Except for people trying to get a consulting gig, they are all used to vent.

5: Interesting article. I'd like to know more about the claim that there's "no relationship between the learners' preferences and the instruction style" because the experiment they describe leaves me with a lot of questions: Is a learner's expressed preference the best indicator or should we consider some external evaluation of what style best suits that learner? Would the findings hold with other subjects, other ages, other side-by-side methods? Are there longitudinal studies? Etc.

(These are in addition to the obvious questions about sample size, control groups, etc.)

The sniff test is indeed off (speaking as STEM faculty here). Perhaps, like that breastfeeding prof recently, she's just not skilled at public/inter-colleague debate, and underestimated her colleagues' desire to understand her work from top to bottom. That said: using "racist language" (as Bishop does when erecting a racist straw man to direct fire upon) is not the same as making racist statements. It's shameful for her to use this tactic.

One possibility is that Milgram & Bishop might be the sort of people who (in physics) insist that their lecture style is not going to ever change from its zero-interaction zero-feedback level, despite all evidence (thanks, physics education research!) that suggests that standard lectures are essentially useless, and occasionally worse.

One interesting point a colleague made recently is that it is a privilege for a student to know at some level that lectures are useless, and that the real learning comes from elsewhere. It is also a privilege to have the time and environment to do that learning. So restructuring STEM class time so that it is actually *useful* should not surprisingly show stronger results in less privileged academic settings.

Longest post ever. Back to lurking.

Boaler's statements unfortunately seem justified to me. I'm a little surprised that Heebie seems hostile to her. It looks like Boaler has been the target of a sustained campaign of harassment that goes beyond academic disagreement about her results and attempts to sabotage her career by accusing her of serious misconduct.

Also, the purely academic issues at stake here seem pretty significant and go well beyond cheap 'different ways of learning' BS -- from a super quick look it seems like Boaler is saying that the emphasis on standardized tests, hierarchical tracking, and inflexible curriculums in math create crippling math anxiety in a lot of students and removing some of these pressures can make kids perform better in math, her opponents are saying that she is essentially dumbing down the math so her metrics of 'perform better' are wrong. That is a real substantive disagreement which you could explore without calling people liars and trying to get them fired for professional misconduct.

9: what would an external evaluation of learning styles consist of? If there is no obvious difference between different approaches, or if there aren't strong individual differences in the efficacy of different approaches, how would you develop an external measure?

To the other questions, I don't honestly know, but I think it's worthwhile to keep in mind that we're looking at an 800 word popular-press summary of a large research literature. I think if you are trying to judge veracity based on this article it's pretty much going to depend on whether you trust the authors or not (they can make gorillas disappear).

If you want to dig down into the review article on learning styles that they mention I think it's probably this one, with a PDF available (if that one is paywalled; can't tell from here) here.

Those links in 12.last are ugly but I guess they work okay.

I pretty much wholly agree with 11. I speculate that the thing which set Milgram and Bishop off is a perception (accurate or not) that her work is wooly headed bollocks, and once that perception is engrained (the way it tends to become once a shot or two has been fired) they decided to go after her whole hog. The accusations of dishonesty are a bit much, especially after she's been cleared. The whole thing seems to me a bit typical of the sort of petty feuds I don't miss in academia, though taken to a bit of an extreme with the accusations of fraud

I once had a department chair (a full blown psychopath) dangle a carrot in front of the faculty if someone could put together a proposal for a grant that involved extensive use of the catchphrase "different ways of knowing." I decided to reach for that carrot. I thought long and hard and decided that there really were different ways of knowing, but to see what was really going on here, you had to abandon the idea that "different ways of knowing" was just a code phrase for "oh, the humanities can give you knowledge, too." I remember spending a lot of time thinking about the differences between verbal and pictorial representations of the world and the differences between knowing how and knowing that.

Of course, it was all a waste of time. I had asked specifically whether the proposal was just about saying "oh the humanities can give you knowledge, too," and was told "no, no, this isn't just another defense of the humanities." But, of course, it was just a generic defense of the humanities, but no one was allowed to say that.

To this day, though, I think that the phrase "different ways of knowing" is not bullshit. Nor is it a funny way of saying "different ways of learning," as if the content of knowledge were always the same, and there were only different ways to it.

Having not read the link this sounds like small potatoes compared to the average linguistics fight.

Based on a quick skim I'm going with both (a) has a real and legitimate grievance and was unfairly attacked and (b) is a bit of a nutter. Not that I know anything of course, but I'm calibrating my sense of smell.

I'd be more inclined to take it seriously if she linked to the unwarranted attack.

I've only encountered the phrase "different ways of knowing" in the context of ye olde Science Wars back in the 90s.

Here's the Milgram paper. A quick scan doesn't make it look terribly objectionable, but I'd have to read it carefully to actually have an opinion.

I'm completely puzzled by the ethics of "here's my data, but it's from schools I'm not going to identify so you can't check it," and then having someone else figure out which schools they were. The initial unidentified schools thing seems sketchy to me, but I just don't know the norms.

Here is one of the papers that Boal is complaining about. This is the paper where her opponents apparently got the data by calling all the school districts in CA and threatening them with legal action unless they revealed whether they were one of the anonymized schools in her study. There are clearly real substantive issues here. But if you read the first few pages it also seems pretty clear that her opponents have lost perspective and are starting to foam at the mouth. They did try to get her fired and disciplined. THey stop just short of saying she's a liar but I wonder if earlier drafts did.

To this day, though, I think that the phrase "different ways of knowing" is not bullshit.

I agree with this completely, I was just trying to refer to the many forms of bullshit that have unfortunately been enabled by the rhetoric of 'different ways of knowing'. If you go too far down the line of denying the range of cognitive styles you end up in the stupidity of 'G' and IQ being all-important.

I think it's worthwhile to keep in mind that we're looking at an 800 word popular-press summary of a large research literature.

Oh, definitely. I'm not passing judgment on the authors' conclusions, just wondering about the underlying research. Questions about learning styles are interesting and important to me.

"Foaming at the mouth"? It doesn't look all that heated to me.

22: I would absolutely check out that review article, in that case. My instinct is to take it as a basically accurate summary current dominant thinking in the field because I basically trust those authors, but that is obviously not enormously useful to you.

I would read the review article myself but I should probably finish these other eight billion pages I need to read first.

23: yes, I guess you're right. By academic standards it is heated (they consistently insinuate that she deliberately cut corners to get her results) but foaming at the mouth was a big overstatement. My initial reading was colored by her description of how they got the data -- I think it's illegitimate to try to chase down people or institutions who participated in an academic study on a guarantee of anonymity, which is what she claims they did.

That they haven't been able to publish their article attacking her in a peer-reviewed journal is a pretty good sign that the attack is bullshitonious or somehow improper. Again, just using the smell test.

But the way in which she put together that website is classic bad PR for someone who (maybe) is unfairly attacked.

what would an external evaluation of learning styles consist of?

I want to know whether visual learning in fact works better than lecture for Student K, not whether Student K likes visual better.

This may be clearly addressed in the research.

Thanks for the links in 12.

I think it's illegitimate to try to chase down people or institutions who participated in an academic study on a guarantee of anonymity

What puzzles me about this is that I don't understand why a school would want to be anonymous under these circumstances. Boaler's data shows that they have a mathematics curriculum that works great -- the interest in anonymity (given that all the information seems to be public anyway) is mysterious to me. Which makes me sympathize with the de-anonymizing.

I'm surprised by heebie's surprise. I haven't read any of Boaler's work, but I can easily imagine that some of the methods she suggests for encouraging student engagement would be perceived by cranky mathematicians as "coddling the students."

30: I don't understand it either, but Boaler presumably promised them anonymity up front before the results were clear. Boaler does not have the option of breaking that promise unilaterally. Doing that would make it more difficult for her to recruit subjects in the future (as does the fact that these guys forced disclosure).

Sir Kraab, I don't know a ton about this but think that in some cases the "learning style" is about making up for deficits in other areas. Nia is just now beginning to read a little and so she missed everything in the kindergarten class she was in last year that assumed basic reading skills but can learn by listening if she's willing to shut her mouth for 5 seconds, which is rare. Does that make her an auditory learner? Dunno; she just doesn't have other options yet. I personally think this sort of thing tends to spiral, but that's my prejudice rather than anything legitimately scientific.

I'd be interested to read what people who think it's not bollocks mean by 'different ways of knowing.' I can speculate endlessly, but I'd love an example of a particular thing and the different ways people can know it in a nontrivial way. I can come up with trivial examples that might with a pass muster with a quick squinty glance, but nothing that's really good and solid.

On the anonymizing: almost certainly a function of the promises she made to the IRB to get permission to do the original research. The Stanford IRB in particular is not well set up for social scientists (or, at least, was not roughly 8 years ago). Although a lot of ed research should be IRB exempt, you still have to get the IRB to declare it exempt, and they are often very strict on reporting anything non-anonymous about children. The school might not care, but if you are using case studies, you might describe a classroom situation in which individual children could be identified if the school were known. While this is not a big deal to anyone in the real world, it freaks the IRB out.

32: But what exactly does anonymity mean? Boaler didn't reveal any non-public information about the school's results, as far as I can tell. The only anonymity is which school's publicly available data she was referring to. I really am confused about this, there could be something I'm missing, but I can't see any negative impact on the school at all.

Oh, 35 might answer 36.

But still, once the cat's out of the bag, she's got Milgram saying that the available data doesn't match what she said about it. Now, either it does or it doesn't, and there's something strange about not addressing the substantive criticism, and instead complaining about Milgram's bullying and unethical behavior: it's perfectly possible for Milgram to be an unethical bully and her study to be a dishonest presentation of the results she was touting, enabled by her not having identified the school.

43: Theoretical and practical knowledge is a good place to start. The two have different success criteria: there are kinds of things that make a theory a good theory that actually make it harder to use in practice. The two have different vehicles as well. Ultimately, practical knowledge is embodied by the actual people (and perhaps objects) who can accomplish a task. On the other hand it is reasonable to say that theoretical knowledge is identifyable with the abstract model. It is more public that way. Nancy Cartwright is good on the differences here. Also, Aristotle.

33: I don't tend to think that people fall rigidly into one category or another, just that different styles will work better for different people at different times, which is so vague that I assume it's uncontroversial.

Rather than "Nia is an auditory learner," I think in terms of if Nia is struggling with learning or retention, I should consider whether a different teaching style is a way to address it rather than just trying to make her more successful with whatever teaching style I've been using. (As you point out, Nia's not the best example since auditory's been the only thing open to her.)

but I'd love an example of a particular thing and the different ways people can know it in a nontrivial way.

You can know a car or train is moving by looking forward at the trajectory or through the side windows at the landscapes. Although these don't really even tell you that:the car or train could be stationary and the "world" (or a passing car or train) could be moving.

And yes, part of the problem is that different ways of knowing are also looking at slightly different things and are incommensurable.

38 -- that article basically accuses her of making shit up (that is, doctoring her research, and focusing on different subsets of student population at the different schools). She says the allegations that she made stuff up, focused on different populations, etc., are totally false, and that there was a 2006 investigation that looked into these charges and concluded they were false. In addition, the attack article hasn't been peer reviewed or published anywhere, despite being around since 2006.

I mean, again, I have no idea what's going on, but unless she's flat-out lying it seems pretty likely that she has a legitimate basis for attacking these guys. And the institutional support seems to be lining up with her, not her critics.

This was also interesting, if not conclusive of anything.

I'd like to ditch this business of associating learnign styles with sensory modes. It's a pop psych add-on that has little evidence and oversimplifies dramatically.

There is a difference between verbal and visual, but this is really the difference between the linguistic and the spatial. It's a left brain/right brain thing, and has nothing to do with how well you hear. Indeed, if you get your linguistic input visually, you could be completely deaf and still a "verbal" learner.

The biggest thing for learning styles is about interpersonal style, not sensation: do you need someone to hold your hand through a process, or does having someone hanging over your head bug the fuck out of you.

Also, there is something to be said for making students get better at acquiring knowledge in ways they don't like or aren't used to. A language-oriented person could benefit from visual training.

42: Yeah, you could be right. I'm still leaning against her -- she seems to me to be leaning too heavily on "There was an investigation and I was cleared of misconduct," which could mean anything or nothing, rather than "Their presentation of problems with my research was incorrect for these reasons." But I don't know, and I could be jumping in the wrong direction.

I'll take back "incommensurable" from 41, tentatively.

Exploded View Drawings do not use one- or two-point perspective

And then there are the American maps we have seen recently:the red-blue, the purple county-level, the resized according to population.

"Different ways of knowing" "different ways of seeing something known in the only way something can be known", "ways of seeing different things" ...whatever

Cinematic, animetic, database, exploded view

40.2 I agree and thing rob is right at 43.1, which is why I used her as the example. I think there are lots of ways she could be thrown in categories that don't actually describe her well and that makes me wonder about the other students who get put in those categories.

Since I'm declaring allegiances, my inclination is to follow Halford's instincts pretty closely here, but unbelievably petty vendettas hit a little too close to home right now and I don't have much distance.

But still, once the cat's out of the bag, she's got Milgram saying that the available data doesn't match what she said about it.

That's not true. That Milgram paper is hard to understand, but basically he seems to be comparing various California standardized test scores for the schools in question to her results, which are scores for her own tests of the specific set of students within the schools that she was tracking. That's apples and oranges, neither the students nor the tests are the same. His findings can raise some doubts but he does not seem to have data that can directly refute her findings.

Then he goes off into why her tests are bad tests. That would be one explanation for differences in findings.

I think that paper would need substantial editing before it was anything a third party could easily make sense of.

I've only read her side of this, but she doesn't sound particularly crazy. A couple of old established guys decide she's a problem because they don't like her methodology and that it would be good to attempt to destroy her career, it got as far as a panel that had to decide whether her research is fraudulent, a panel agreed with her, and we're quibbling because she's not carefully demonstrating the ways in which their non-peer-reviewed "article" is incorrect, which she can't do without blowing anonymity?

It's a little odd that it's on her webpage, but it may well be the kind of thing where she has to fight back to protect her reputation via Google.

it may well be the kind of thing where she has to fight back to protect her reputation via Google.

Sounds like it. But like most amateurs faced with similar situations, she did a pretty lousy job of doing so.

48: I'm arguing to argue, rather than because I have any real knowledge here. But isn't there a lot of room between research that a college is willing to discipline as fraudulent and research that doesn't fairly or competently represent the data it purports to? Relying on "The investigation said I didn't commit fraud" seems weak to me.

I also sent an email to my math ed friend, who's at Disneyworld for the week, to get her inside scoop. She wrote back that it's complicated and she'll tell me more later.

It's a pop psych add-on that has little evidence and oversimplifies dramatically.
[...] It's a left brain/right brain thing

C'mon, dude.

50: The methodology in their study seems to be "Well, we don't know what schools she studied, but we're pretty sure that it's these ones, and the standardized test scores from these schools show that she committed fraud because her data is different from our data." A university committee investigating that isn't going to say "On the contrary, we proved that her research is correct." That's not their job. All they can see is whether she can back up how her data was collected. She apparently did that. She might be wrong for other reasons -- I don't know her or her work -- but fraud's a pretty strong allegation and beating that back does show something.

This sounds a lot to me like a vendetta, and perhaps it's just my priors here, but it's not hard for me to believe that an old guy who doesn't like the new hire might throw his weight around inappropriately.

52: You got me.

39: But that's different ways of knowing different things, which is easy. The thing that gets me worked up is when people talk about different ways of knowing the same thing, in this case some subset of mathematics. I can see different ways of knowing how to do a thing like multiplication, but knowing that 0*1=0 you either know or don't.

There's certainly many different ways to prove that 0*1=0, and any one of those ways would be considered "knowing" it by the most hardcore of mathematicians.

55: I dunno, you could have a set theoretic understand, or a number theoretic understanding, or a rote understanding or whatever, right? Even within math there are plenty of relatively disparate ways of talking about the same thing.

Category theory is totes an existence proof of different ways of knowing.

My reaction is that fraud's a very strong allegation, so beating it back doesn't show much at all about weaker allegations (and I don't know any details of what allegations were actually made in the Stanford investigation, or what the results were other than 'not fraud'). Milgram's version of the situation seems to be something like: Boaler said that students in school [X] were taught using her favored curriculum, and did better than comparable students taught with other curricula, as measured by her own test of the material; Milgram identified school [X], and standardized test results show them doing worse than comparable students taught with other curricula; and an analysis of her test shows how it might give misleading results.

That seems compatible with Boaler not having committed fraud, but with her research being not good evidence for her point of view, and in a way that she should have known about. I obviously don't know that Milgram's version is true (or, honestly, that I've correctly understood what it is), but I think there's room for her research to be very bad but not fraudulent at all.

There can also be a number of different theories that will rationalize or forecast the same set of empirical observations. Those are 'different ways of knowing' and it is an interesting question whether one is knowing 'the same thing' in such a case. I suspect that is an even more ubiquitous form of different knowing than different routes that take you to the same place.

55: As I recall, Cartwright's examples include things like theoretic vs. engineering understandings of fluid dynamics. Does that count as the same thing?

55 is me.

Category theory is totes an existence proof of different ways of knowing.
On the contrary, category theory is a proof that all ways of knowing are equivalent. Or maybe that there is only one thing to know. Or something. To be honest, my eyes kind of glaze over when I hit that phrase. I do not have a category theoretic learning style.

Yes, but his reasons for thinking that it's not good research don't look like good reasons.

This makes me very uncomfortable: big tenured guy cries that there is fraud for what turn out to be bad reasons, there is no fraud at all and his reason for thinking the research is bad are still bad reasons, so the reasonable conclusion held by sensible people becomes that there musta been something going on, so she's probably bad at her research.

59: if Boaler's finding disagree with standardized tests for a possibly similar but not identical population, that doesn't seem like evidence that her work is 'very bad'. It might be sort of bad, or else overreliance on standardized tests to the exclusion of other forms of testing might be sort of bad too. There's probably value to diversity and to multiple lines of research in such cases.

Also, real science, particularly the softer sciences, has plenty of mediocre work in it. In my experience in social sciences it is quite common for people to subtly tilt or misrepresent the data in various ways, even without knowing they are doing it. I suspect this happens in harder sciences too. It's human nature to start to advocate for positions you have invested in or that you have an intuitive tendency to believe. The question would be not whether her research is perfect but whether it features deliberate fraud or lying, and also how it compares to the imperfections of the research her opponents are championing. Science is a communal process and not a matter of perfect individual scientists, and there should be a reasonably high bar for trying to throw people out of the community entirely.

64.1: I'd have to understand the merits of it better to have a real opinion, but the reasons set forth in the Milgram paper for calling Boaler's research bad seem compelling if they're true. If the students she's touting as a success for her methods on the basis of her own testing instrument in fact do poorly compared to other students based on standardized tests, that makes her research sound weak, and the anonymity of the school sound like a shield for the weakness of the research -- if there's some reason the standardized test measures are irrelevant, that seems like the sort of thing that she should either have addressed in her original work, or at least now that it's been brought up.

59.last: but I think there's room for her research to be very bad but not fraudulent at all.

At some level if you go right into a standard scientific discussion with Milgram then his terrorism has won. He has pretty much disqualified himself from further participation by his initial approach.

And, yes, her research might suck. Heaven forfend!

65, 67: Is there something on how evil Milgram's been beyond Boaler's linked page?

This is just another volley in the Math Wars.

LB has nailed the synopsis of Milgram's side, and I would love to see a point-by-point response from Boaler that isn't "they investigated me 6 years ago and didn't find fraud". The vast majority of Milgram's paper is not devoted to "we called up the schools and discovered your secret", but rather to "we think your homemade tests are way too easy and occasionally poorly written".

My knee-jerk reaction is to assume all social science is bullshit, and all old math professors with unusual opinions are totally batshit crazy. I don't know which reaction to go with, so I might need to actually read about this specific situation before forming an opinion, which isn't much fun.

65.1

But if you're going to use a test to evaluate learning, surely we can agree it should be standardized? And if not, how can you tell?

The paper she wrote based on her study, and that her enemies were attacking, is here. It talks about the testing methods used at the three schools.

61: I wouldn't consider that knowing the same thing, but I could see how a not-crazy person might. The objections to my claim that 0*1=0 is knowable in only one way seem stronger to me, but not in the context of teaching math to kids.

I can come up with examples of how to know how to do something that are very different - I know how to tie my shoes because I do it every day, but I would be sorely pressed to come up with a verbal description of how to tie shoes, though such a thing no doubt exists and a person who read and understood it could follow it and end up with their shoes properly tied. That's two different ways of knowing the same thing that are more or less orthogonal to each other, but even that involves merely executing an algorithm that's stored in different parts of the brain but is in some deeper way the same algorithm.

And now I'm bothered by the fact that the more I think about it the more confused I am about how my shoes end up tied every morning.

I'd be very surprised if the body of her work doesn't address standardized testing. And maybe she's done bad research; I mentioned that above. But there are ways for bad research to be criticized that don't involve getting the university ethics committee involved or running people out of the school, and that's what's ringing weirdly about this whole case.

No one's been able to prove in 25 years or so of trying that the "learning styles" theory amounts to much of anything. Here's a useful article on the subject: http://elearnmag.acm.org/featured.cfm?aid=2070611

68: Is there something on how evil Milgram's been beyond Boaler's linked page?

Admittedly my priors are set by a dislike for his bullying, activist role in the Math Wars in general.

Just to be clear, it doesn't look like (based on the article linked in 72) that Boaler is interested in promoting a "learning styles" theory.

The more people talk about category theory here, the more I think I should actually learn about it. Recommendations on where to start?

I feel like the Math Wars kind of miss the point: elementary and middle school teachers don't understand or like math, and it no curriculum tricks are going to change that.

I would agree with that. I thought the elementary school my kids went to was largely great, but there were a number of incidents where it was clear that some of the teachers were uncomfortable with grade-school math, and for a kid who was having any trouble at all, they might easily have been put off by it.

73: Instead of category theory maybe translation theory or pragmatics?

I disagree that the different proofs of 0*1 = 0 prove the same thing, just as I would claim that you cannot really translate from French to English with perfect precision and communication.

And the claims of epistemology and the goals and methods of education differ, and should, wildly depending on the purported "object of knowledge:" a Euclidean Theorem, this chair, free will, the class struggle.

Category theory is a bit difficult because the really important examples are things that you don't learn until graduate school. Unless you've already thought about covering spaces and tensor products, and already know half-a-dozen different "free" constructions it's going to be a bit tough to get the point.

The standard text is probably "Categories for the working mathematician," but you could also start with a modern graduate algebra textbook (say Jacobson's "Modern Algebra 2").

Is there any "I'm not a scientist, man" thread here?

Category theory is a bit difficult because the really important examples are things that you don't learn until graduate school.

But don't people also talk about it as if it could work like set theory as a fundation for mathematics? Doesn't that tie it back to more basic stuff somehow?

Got the tidbit from Well-Informed Friend At Disney World that her sympathies are with Boaler, and that while her research may have some flaws, their tactics are totally evil.

I feel like the Math Wars kind of miss the point: elementary and middle school teachers don't understand or like math, and it no curriculum tricks are going to change that.

Also even with teachers who understand and like math, there is no way of having continuity between one year and the next. Maybe the person who taught me set theory was intending it to be a basis for future study, but it was actually just a couple months of interesting trivia with new "vocab words" such as ∪. No one ever mentioned it again.

"Categories for the working mathematician,"

Lessee, "sellout", uh, "pure", "applied", come one someone make this funny.

87: "Just broke the water pitcher".

I'd also say actual set theory is really hard...

Category theory certainly ties back into basic stuff (e.g. the fact that the empty set is an initial object and that the one point set is a terminal object is elementary), but to understand the actual content you really need some advanced examples.

The basic idea though is easy to explain. Traditionally you think about mathematical objects by looking *inside* them at their elements, but category theory says that it can be fruitful to ignore their insides and instead look *outside* your objects at their maps to other objects.

no way of having continuity between one year and the next
Is looping catching on at all in grade school?

81.2: Is that distinct from a claim that different proofs may teach different things?

An axiomatic system is significant precisely because the meaning of a proposition is independent of the context of the proposition, and instead the system as a whole provides the context in which we understand its parts.

For example, "Socrates is mortal" as a natural language proposition is highly context-dependent. Which Socrates? In what respect? What does "mortal" mean? Is it the "is" of identity, membership, or something else entirely? If I show that Socrates is mortal by killing him, then I mean something different than if I show that Socrates has a strong resemblance to other persons who have died, and no differences that would plausibly lead to a material difference in mortality.

On the other hand, if you're in the same axiomatic arithmetic (and that's a big if), "1*0=0" is the same proposition no matter what propositions precede or follow it in treatment. Anything that follows from "1*0=0" based on one proof, follows from it based on any proof that rests on the same ultimate axioms.

OTOH in practice it is easy to lose track of exactly which axioms you're relying on, and for some pairs of proofs, it must be the case that either one of them starts with a different set of axioms then the other (even when they define isomorphic systems), or one of them assumes the thing being proved. So in that sense, you could say that there's a difference between proving that the statement "1*0=0" follows from set of axioms A, and that it follows from set of axioms B, and that proving the axiomatic systems are isomorphic is yet a third thing to be proved. Because the purely mathematical fact that "1*0=0" is a property of the axiomatic system, not of the world.

Catching up on the thread, my priors say Cala has the right attitude and LB's is surprisingly, horribly wrong. Cranky old dudes being cranky is not usually a reason to suspect that they're right, or even half right.

86, 90:

no way of having continuity between one year and the next

Because continuity of self is an illusion? Or because of the Planck length?

Or because of the Planck length?

Sigh.

90: Is looping catching on at all in grade school?

GO TO new teacher considered harmful.

More to 81: And that's just in the space of axiomatic systems. If you admit mathematical intuition or a practical knack for numbers as "knowing", then those are even more different.

94: OK, in 30 words or less, what did I get wrong?

97: It's true that it's hard to measure distances shorter than the Planck scale, and spacetime is in some sense fundamentally not classical at short distances. But it isn't a regular lattice or anything that is, in any straightforward way, not continuous.

I think I went over 30 words but they're mostly small words.

There was a bizarre paper on the arxiv today by Bekenstein about the possibility of experimental quantum gravity that I think has to be falling prey to some version of the idea that spacetime is discrete at the Planck scale, which is a widespread mental virus, but it was a weird enough example that it was annoyingly difficult to figure out precisely where it was going off the rails.

95 is funny.

Hey Essear, speaking of different forms of knowing, have you seen that book "The Trouble WIth Physics" by Lee Smolin? There was an interesting part at the end about different types of scientific styles and how they are rewarded or penalized by existing academic structures.

addendum to 100.1: and if so, what did you think of it?

But it isn't a regular lattice or anything that is, in any straightforward way, not continuous.

Well, until plasma annealing.

Hey Essear, speaking of different forms of knowing, have you seen that book "The Trouble WIth Physics" by Lee Smolin?

Arrrrrrrrrrgh.

So, largely about pirates?

Man. I was trying to make essear's head explode but everybody else is way better at it.

Yay! It's annoy essear day.

Quarks aren't real. Einstein showed that everything is relative. Quantum mechanics shows that post-modernism is right! String theorists are the modern equivalent of alchemists.

Hey essear have you seen that movie What The Bleep Do We Know?

What's fascinating is that the whole universe should have this E8 shape.

Aristotle conclusively demonstrated that science can't explain everything.

Linked without comment.

I can immediately tell Essear is not a visionary. No Perimeter Institute fellowship for you, Essear!

No Perimeter Institute fellowship for you, Essear!

This is funny.

I want to learn something deep about physics, should I start with Fritjof Capra or Gary Zukav?

110: haven't read the article, but there is a a long tradition of economists taking models from physics so it doesn't seem like an outrageous move to take whatever mathematical tricks you use in your physics research and dump them on economics. Arrow-Debreu should hardly be sacred anyway.

The thing about interdisciplinarity is if you can convince the people in Field A that you're really good in Field B and the people in Field B that the opposite is true, you can get away with not understanding anything at all!

I do think that being half-good at physics is the best setup for living your life out as an academic scoundrel in other fields (starting in other disciplines leave you more vulnerable to being found out and exposed).

PI has actually improved dramatically in recent years by hiring a lot of people who are less "visionary" and more "reasonable" or "hard-working" or "not crazy". It was a good move for them. I have kind of a love/hate relationship with the place-- mostly love, despite its less than optimal location. There's a coffee shop a few blocks away with a surprisingly diverse selection of whiskey.

What's fascinating is that the whole universe should have this E8 shape.

Stupid. That's infinity shape.

I have no doubt Smolin would be a lousy economist in the sense of truly fruitful theorizing about the economy, but to the extent there is an area of economics that is mostly about playing math games then it seems fine for him to introduce his physics math games to the sandbox. Especially because he is probably a better mathematician then a lot of mathematical economists.

Physicists playing that game sometimes end up at the Santa Fe Institute, which does have some very good people but is not really mainstream in the discipline. They could probably judge whether he has added any value.

or: I shouldn't say 'I have no doubt Smolin would be lousy" since I haven't read any of his work. I was being defensive!

His math games don't have any relevance for physics, so I hope they have relevance somewhere.

Hey essear have you seen that movie What The Bleep Do We Know Mindwalk?

(Or perhaps that's from the "things that annoy NickS file")

if you can convince the people in Field A that you're really good in Field B and the people in Field B that the opposite is true, you can get away with not understanding anything at all!

Wait. So if you can convince people in Field B that you are really bad in Field B, you can get away with not understanding anything at all?

So, what? Fail the entrance exam?

119.2: I couldn't possibly comment.

Did every field have a "[field] wars" over curricula in the late 80s/early 90s? That's about when the history standards stuff exploded too.

Here is Bishop and Milgram's paper.

ftp://math.stanford.edu/pub/papers/milgram/combined-evaluations-version3.pdf

this doesn't sound so good:

In terms of outcomes, Dr. Boaler points out that 41% of the students in the cohort she studied at Railside were taking "pre-calculus or calculus" in their final years at the school. This number was confirmed by Railside's principal in a phone conversation with one of us. The principal also stated that no Railside student had taken an AP Calculus exam during the last five years, and this was confirmed by the actual data.

Don't know if this was linked above, but here's a summary of the case from Inside Higher Ed.

I'm assuming no one has read Boaler's book for parents and teachers, but I think I'll see if the library has it. I've been looking for good writings on teaching the very very very basics.

116 makes me wonder what would have happened had I followed my housemate into neuroscience. He's done well, but I think I would have sucked at all the icky gooey bits. Chopping up critters isn't really my style.

From the comments in the link from Halford's 42, there's a note from Bishop in which he claims that Dr. Boaler had him investigated by the Stanford police as a potential terrorist for his "nuke 'em all" comment on a private listserve (in respect to how to treat schools of education). If true, while that was probably in response to her feeling harassed and attacked by Bishop and Milgram, it probably didn't help keep the situation from escalating further.

On 126:last, those two claims are not necessarily inconsistent, particularly if most of the students were taking precalc rather than calculus, and the calculus course was not aimed specifically at prepping students for the AP exam. In a school where most of the students may be steered towards community college (if anything) rather than Harvard, there might not be a lot of students who are motivated to spend the time and money to take the AP exam, particularly if the school hasn't been offering it in the past. If there are a few who do want to take the exam, they might do so at another school that the principal doesn't know about. But it does mean that we don't have an easy national measure of how much calculus those students learned.

In a school where most of the students may be steered towards community college (if anything) rather than Harvard

Well, you transfer to Harvard from the community college.

Anyhow Wave is entirely correct about the AP exam, and not just for dumpy schools where kids go to community college; I took calculus in high school and no fucking way did I intend to take that exam. Nobody in my math class did. There was a more advanced AP math class where I think most or all of the people did.

130, cont.: Which doesn't mean that the school shouldn't be encouraging calculus students to take the AP exam, of course. But there can be a lot of institutional inertia there, especially if they are mostly focused on the students at the bottom end of the curve.

The thing about interdisciplinarity is if you can convince the people in Field A that you're really good in Field B and the people in Field B that the opposite is true, you can get away with not understanding anything at all!

My dream!

I can't remember if my school had both Calc class types, but I know people who did the AB(?) and didn't take the BC(?) exam. My school did have a class so AP-focused the class basically ended in May around the time of the test.

That is, there was a Calc AB and Calc BC.

My school had both.

I don't know what those letters mean, but anyhow we had two levels of math that got to at least some calculus.

Not that it's relevant.

130/131 hadn't occurred to me; without thinking about it, I assumed that a high school calculus class would naturally be feeding toward one or the other AP exam and that something is odd if it doesn't.

137: I think AB calculus is supposed to substitute for a semester of college calc, and BC for a year of college calc? Something like that? They're two different AP calculus exams, and AB covers less material than BC.

139 is more or less correct, though I think/home/dream that a year of college calc would go into e.g. taylor series or whatever in greater depth than my hs bc calc course did.

Right, I meant that I knew some people who did an AB class but didn't do the AB test. The only people I've met who took the AP - a small number, to be sure - took the BC version.

Also, a math teacher I know said the math department where he majored required everyone to take calculus even if they'd done the AP and had a high score. He might have taken the AP anyway but he said there wasn't much reason to. He also thought he understood calculus a lot better after taking it again in college, fwiw.

Right, I meant that I knew some people who did an AB class but didn't do the AB test. The only people I've met who took the AP - a small number, to be sure - took the BC version.

Also, a math teacher I know said the math department where he majored required everyone to take calculus even if they'd done the AP and had a high score. He might have taken the AP anyway but he said there wasn't much reason to. He also thought he understood calculus a lot better after taking it again in college, fwiw.

The second post was for college credit.

I like to point out at this point in the conversation that I did not take calculus in high school, a decision reached after being advised by my junior year teacher. Despite being super engaged in her class and doing very well there. No, no chip at all on my shoulder for this one of many light discouragements not to pursue math.

I actually am not convinced my high school class covered a semester of college calculus. I'm not sure we even did integration. Maybe my weirdo public school is not a good sample.

144: my friend who ended up majoring in math at MIT got there without ever having seen calculus. I think he must have sorta stood out that first semester.

I barely get to integration in one semester of calculus. Maybe 1-2 weeks at the end.

I don't think I ever stood out in math, actually. Which isn't self-deprecating - I don't have a super high opinion of the type of flashy talent that stands out.

146: The beauty of 300-student lectures is that no one else has any reason to know if you're struggling.

Actually I'm a little jealous of it. But it doesn't keep me up at night.

144: I had to whine and threaten to be allowed to take AB rather than BC Calc -- I'd been tracked into the ordinary rather than the advanced math track after a disagreement with my seventh grade teacher about whether I should do any homework, ever, and she was the head of the math department and remembered me five years later. But in my case it was deserved suspicion rather than sexism, almost certainly.

I mean BC rather than AB.

144

I like to point out at this point in the conversation that I did not take calculus in high school, a decision reached after being advised by my junior year teacher. Despite being super engaged in her class and doing very well there. No, no chip at all on my shoulder for this one of many light discouragements not to pursue math.

So you admired your teacher and thought her advice was worth taking?

What the fuck. Are you taking a swipe at heebie's career?

Odds are he's doing what he usually does, saying something that could be interpreted as wildly obnoxious or innocuous depending on how you take it, and waiting to enjoy being unjustly misinterpreted.

Hey, Shearer: do you have a point that can be made non-Socraticly?

I never heard of any of these people before but this article by Boaler sets off my BS detector.

I am a math ed professor at a different institution who has been following this story closely. I don't know Boaler personally, but have interacted with her briefly in professional settings and have followed her work for over a decade. A few weeks ago Jo gave a plenary at one of our national conferences and discussed some aspects of this series of events.

To clear up some confusion about about the anonymity of these schools, when you do this type of research, you must get it approved by your institution's IRB. In conducting any school-related research, you must ensure both privacy and confidentiality of the participants. Even if your research reveals only positive things about the schools or the students, that's irrelevant. The schools, teachers, and students must remain anonymous and you are not legally allowed to disclose identifying characteristics. Boaler is not even allowed to address whether Milgram and Bishop were correct in potentially identifying the schools in question. Milgram and Bishop engaged in severe academic misconduct by violating human subjects regulations in their attempts to identify any of the participants.

A number of Boaler's proponents have addressed and disputed the errors in Milgram & Bishop's essay over the years, but I'm not surprised Boaler isn't bothering to even address it. It'd be like a mathematician bothering to address the crank who claims he can trisect the angle. Their essay isn't ethical, it's not research, it's not even coherent, and it rests on potential identification of the schools which Boaler cannot legally discuss. The fact that they were never able to get it published speaks for itself (and is wholly unsurprising). You can't just turn a paper into a PDF and pretend it's an article.

I have enjoyed a number of positive and collegial collaborations with mathematicians over the years, and continue to do so, but my experiences with the math wars (and in particular with Milgram's tactics) leave me completely unsurprised at this story.

155

Hey, Shearer: do you have a point that can be made non-Socraticly?

I am curious as to why Heebie-Geebie paid any attention to her teacher. I wouldn't have (assuming the advice was to the effect that she wasn't smart enough to take calculus).

157

To clear up some confusion about about the anonymity of these schools, when you do this type of research, you must get it approved by your institution's IRB. In conducting any school-related research, you must ensure both privacy and confidentiality of the participants. Even if your research reveals only positive things about the schools or the students, that's irrelevant. The schools, teachers, and students must remain anonymous and you are not legally allowed to disclose identifying characteristics. Boaler is not even allowed to address whether Milgram and Bishop were correct in potentially identifying the schools in question. Milgram and Bishop engaged in severe academic misconduct by violating human subjects regulations in their attempts to identify any of the participants.

Perhaps there are good reasons for this but it is awfully convenient for concealing fraud (or lesser problems).

"I am curious as to why Heebie-Geebie paid any attention to her teacher. I wouldn't have (assuming the advice was to the effect that she wasn't smart enough to take calculus). "

Some do extremely well. Their indifference to what others think makes them indifferent to the intense peer pressure of adolescence. They can flourish within their specialty, and become accomplished musicians, historians, mathematicians

I knew what this discussion needed was more Shearer cowbell.

144: Basically, yes. I was scared of calculus and would have needed encouragement to go for it.

there's a note from Bishop in which he claims that Dr. Boaler had him investigated by the Stanford police as a potential terrorist for his "nuke 'em all" comment on a private listserve (in respect to how to treat schools of education). If true

She mentions it on her site as well.

In 2003 Bishop discussed Schools of Education in the US and suggested to readers that they "nuke 'em all dammit". This, alongside his personal attacks on my work, prompted Stanford's police department to travel to LA to speak to Wayne Bishop.

This makes me leery of how she characterizes the rest of this conflict. Seriously, you thought that comment was an actual threat to do you physical harm that warranted a call to the police? And even worse, if they really did travel down to So Cal over this then they're even bigger idiots. And what 159 said. Really, this is a standard practice? How are results reported under these conditions to be reviewed by other parties?

I am disappointed by the Shearer comments on this thread. I'd have thought the combo of defending math rigor, hating on secondary school teachers, and opportunity to bash the social sciences would have been like catnip, but instead he just went personal and went home. Needs more black people I guess.

Did every field have a "[field] wars" over curricula in the late 80s/early 90s? That's about when the history standards stuff exploded too.

The famous Canon Wars were going on around then as well. Must have been something in the water.

In terms of the math pedagogy gang fight, I'm inclined to put the primary blame on cranky old dudes being jerks, but it sounds like everyone involved started getting weird after things heated up.

It's very common for mediocre schools to have "AP classes" where few or no students take the exam. At my local high school in the history classes I took there, only a handful of the top students actually took the test. (Remember that AP exams cost money, in my district the school paid that money, but whether the student or school is paying that's an incentive not to have as many people take the test.)

98: Thanks, that's helpful.

When you say "hard to measure", does that mean it's definitely possible but a pain to do, or does that mean we aren't 100% sure how to do it yet but based on the most likely theory we think we can?

144: I thought students had to be turned away three times before they could become real mathematicians.

Maybe that's just the orthodox mathematicians...

I feel as if I should say that I'm being gradually persuaded that I jumped the wrong way on this one. There still seems to be psycho behavior on both sides, but 157 and earlier posts making similar points are persuasive.

I kind of thought you were going to say "I feel as if I should say that I'm being gradually persuaded BUT HELL NO NEVER SURRENDER DIE COMMIE SCUM."

Yeah LB, Heebie gets it right. Never surrender. As Iron Maiden once sang, if you're gonna die, die with your boots on.

I kind of thought you were going to say "I feel as if I should say that I'm being gradually persuaded BUT HELL NO NEVER SURRENDER DIE COMMIE SCUM."

I think you're forgetting that LizardBreath isn't an academic.

That's always my first instinct, of course. Getting myself to actually say that someone I'm arguing with has a point requires making it through the initial stage where I think "Dammit, it's going to look bad to third parties that they've said something persuasive that I haven't refuted." It takes a surprisingly long time after that to get to "Huh. If I haven't refuted it, possibly that means that it's convincing."

Unless you are trying to vanquish your enemies with FOOT SMELL ASSAULT.

What they meant was that as you accumulate more risk factors for heart disease and diabetes, boots become your ideal fashion choice. Especially keep them on during surgery to prevent edema.

175 to whatever.

At this very moment, I'm wearing jackboots. Someone give me a hill to die on, and I'm there.

This is horribly off-topic, but: I'm somehow not in a mood to get real work done so I decided to download and install one of those paper-and-citation-management-software things I keep hearing so much about. So first I downloaded ReadCube and tried to import some papers and it couldn't figure out what they were, and then I tried to use it to search Google Scholar and it told me I don't have an internet connection. But I clearly do! So then I gave up and decided to try Mendeley, but when I click the download button on its website nothing happens. What's the deal? Do all these things just suck?

From the link in 127:

In an interview Sunday afternoon, Milgram said that by "HR" in the above quote, he meant "human research," referring to the office at Stanford that works to protect human subjects in research. He also said that since it was only those issues that prevented publication, his critique was in fact peer-reviewed, just not published.

How convenient. If you're inclined to be suspicious, it looks like everyone involved is leaning on the anonymity requirements to cover something or other.

I have Mendeley installed on my computers. I rarely use it because I'm not actually a researcher, but what experience I have has been positive. I certainly had no trouble downloading it and importing all my .pdfs.

Oh good, an off-topic complaining thread. Why does my pdf viewer not scroll smoothly between pages, but instead abruptly jump so that the new page starts at the top, and making it really finicky about actually staying put at the bottom of a page? I just want to keep scrolling so that the bottom of the page isn't at the bottom of my screen. Grrr.

I like Mendeley ok, but I don't use it much. Science-y people seem to like it more than the alternatives. It didn't convince me to stop using Zotero.

179: I use Papers and dig it but there are ways in which it probably objectively sucks.

@182

If you're using Adobe, under "view"-->"Page Display" there's an option to select "single page view" or "enable scrolling".

182: There's probably a viewing setting for "single-page continuous" (or something like that) somewhere that you can turn on.

It's a viewer built right into Firefox? There seem to be very limited options.

Oh hey, there's a button to make all the options available to you. I thought it was just an inactive adobe icon. Well, my problems are all fixed.

180: Yeah, the IRB requirements, while I'm sure there's a good rationale for them, seem to make this kind of situation difficult to resolve.

Oh good, I can still complain. Every time I open a new document - and each part of each job application is a new document so there are like 200 documents - I have to go reactivate the options and pick continuous scrolling. Why does God forsake me?

This weird reference told us that 'we don't have to worry that the candidate would "cause problems".' I hadn't been?

@190

Are you using windows? Go to "Advanced System Settings-->Environmental Variables" and just start randomly changing stuff. That should help.

I change the settings to force browsers to open pdfs in an external viewer if I can, so I don't know if this will work, but you might be able to solve your problem by opening your reader on its own and changing the default view in the preferences and then restarting the browser and trying again.

Some documents/websites seem to force a particular way of viewing pdfs and it usually drives me crazy.

179: I use Papers and dig it but there are ways in which it probably objectively sucks.

I'm playing with it now. At least it is capable of searching. ReadCube claims to be able to do things like turn references in PDFs into clickable links, which Papers doesn't seem to do (?), but that's not very useful if ReadCube can't connect to the internet. The lack of reference and citation tracking in general seems like a problem with Papers, unless I'm just overlooking it?

There's a just-for-particle-physics app I use that connects to a database hosted at SLAC for things like references and citations, so I was hoping for at least equivalent functionality. I'm not sure any of these programs have it? The other issue with Papers is that it's not free, but I can live with that....

191: There's an implied second half to the sentence. You don't have to worry, but you probably should.

I use Sente and like it a lot, but it's Mac-only.

Go to "Advanced System Settings-->Environmental Variables" and just start randomly changing stuff. That should help.

Yeah, or cause another frankenstorm.

Storms use PDF settings to gain superpowers?

As a mathematician lurker (check out my proof of the Pythagorean theorem), I have little doubt that older mathematicians could be inappropriately dismissive of what she's done, and attack her personally because of it.

I have no clue how I would deal with such a nightmare situation, but I don't think it would linking a "graphic novel" summary of the events from my blog

I use and like Bibdesk (also Mac-only, I think). One can search some scholarly databases from within it, and I have Google Scholar set to offer a .bib citation and can get that into Bibdesk with keyboard-only, so.

---

Here's a clay-pigeon: all fields had curriculum wars in the 1980s because of feminism. We had a hundred years of public education being subsidized by women's artificially restricted labor. When the labor markets opened, the quality of teaching labor almost has to have gone down, because we never did pay or respect it very well, so teaching probably got worse. The flipside was education becoming more academic* in an attempt to either respect women's work or justify open wages; but since we don't actually agree on who we need to learn what, it's vexed curriculum wars all the way down.

* Don't know about this, really, would like to know more about the Normal schools.

I use Papers and am a big fan. I export a bibtex file from it. I have a reasonably nice workflow for going from website/citation/etc to papers import. Mendeley has potential but so far has not convinced me.

To Essear and others: I should say that Lee Smolin and I have already coauthored a mathematics paper so I am somewhat prejudiced in this case.

196: The prettiness of the Sente website intrigues me.

And "Mac-only" sounds like a good thing to me...

202 is great.

Smolin's assistance was invaluable in showing how several of my conjectures could be seen as examples of symmetry breaking and reframed in the language of gauge theory. I don't really expect a non-visionary such as yourself to understand the significance of this work.

Along with Mendeley, Zotero, Papers, ReadCube, and Sente, there's yet another one called Colwiz. Too many choices! Surely one of them will do what I want.

There's also Citavi, but I don't think they've done a Mac version.

Here's a clay-pigeon: all fields had curriculum wars in the 1980s because of feminism.

Interesting theory. I don't know enough, personally, to judge how accurate that gloss is, but I'm hoping somebody responds because it seems like a fair starting point.

Sente seems to be winning on the "easily add PDFs from random places on the web to a library" front.

Here's a clay-pigeon: all fields had curriculum wars in the 1980s because of feminism.

Here's my own pet theory. By the mid-late 80s the first big wave of non-white men who entered academia in the 70s were getting senior enough to actually have an influence on things like curriculum structure. Much pearl clutching ensued.

I'm not saying that some wrong headed things weren't proposed by the new wave*, but I suspect if those same things had been proposed by tweedy caucasian men, they wouldn't have been treated like !!!The End of Civilization!!! in quite the manner that they were.

*There definitely were.

||

Wes Unseld (left) was not an attractive man.

|>

For the humanities and social sciences, at least, you probably also have people saying, "hey, scholarship has really changed since the 1950s, but textbooks and surveys have not changed so much. We need to do something about that."

210: for that with Papers I either drag 'em on to it or search within it (you can add your institution's proxy to its setup).

Huh, missing apostrophe. That's a new one.

What? What is wrong with you, Sifu?

Shut up shut up shut UP

Hey, ease up on the little feller.

162

144: Basically, yes. I was scared of calculus and would have needed encouragement to go for it.

I expect your teacher was scared of calculus also (probably with more reason). Didn't you realize you were smarter than her?

And what changed your mind in college?

180

How convenient. If you're inclined to be suspicious, it looks like everyone involved is leaning on the anonymity requirements to cover something or other.

What's Milgram hiding? As I understand it his paper is readily available.

He appears to be claiming that the paper passed peer review for publication but Stanford HR refused to sign off. You may doubt this but it doesn't seem inherently deceptive.

Surely 213 has always been true. 212 confused me until I recast it as non-(white men), and is a nice hammer to the anvil of K-12 teaching supply. (Are we generally thinking of K-12, which is what I think of for Math Wars, or of college curricula?)

I think captive overqualified teachers should also have taken a lot of the strain off curricula, by heroically adapting goony, incomplete, or overambitious curriculums.

Curriculi curricula. Stalks at midnight.

Didn't you realize you were smarter than her?...
And what changed your mind in college?

To the former, not exactly. I remember her being a pretty good teacher.

To the latter, I just enjoyed taking math classes, so I kept taking them, and eventually just decided I enjoyed them enough to ignore whether or not I was good at math.

I actually believed I was good at math after I passed my prelims in grad school.

220: I don't know why I'm replying, but I was careful to not say that I personally thought anyone was being deceptive. What I did say was that the anonymity cuts all ways and provides grounds for anyone to doubt things if they're inclined to doubt. We have no idea what corrections he didn't make because, he says, he got busy doing other things at that time and I guess never got around to it since then because he's always been too busy and will be forever or something, but we do know that, according to him, the school-lurkers have since supported him in email.

222 con't: at which point I was actually not especially good compared to my peers, but solidly middle of the pack. All I mean is that after taking prelims I felt it was legitimate for me to be in grad school.

166

It's very common for mediocre schools to have "AP classes" where few or no students take the exam. ...

And I suspect it is common for such classes to be a farce.

222

To the former, not exactly. I remember her being a pretty good teacher.

She couldn't have been that good if she didn't see you had talent.

She might have been a perfectly good teacher in other regards, and yet poor at identifying mathematically talented students if they didn't fit her preconceptions of what talented students were like. Figuring out in what regard Heebie might have failed to match up with her search image is left as an exercise for the reader.

227

She might have been a perfectly good teacher in other regards, and yet poor at identifying mathematically talented students if they didn't fit her preconceptions of what talented students were like. Figuring out in what regard Heebie might have failed to match up with her search image is left as an exercise for the reader.

This is silly, we aren't talking about the first day of class. If after teaching Heebie for a year the teacher still thought calculus would be too much for Heebie because Heebie was a girl, the teacher really wasn't very good. Being able to evaluate what your students are good and bad at is a key part of being a teacher.

The other question is why Heebie went into her class afraid of calculus.

This is like the argument that there can't actually be employment discrimination because some sensibly non discriminatory employer would take advantage of the lower pay members of disfavored groups can be offered.

Obviously, I wasn't there in Heebie's math class, so I don't know what happened. But I wouldn't be surprised if she fit neatly into a particular sexist stereotype: she's pretty, well-mannered, plays sports -- girls like that get good grades because they're compliant and hardworking, not out of real merit. A boy pulling a B in a math class and not doing the homework is obviously much more talented than a good-girl type who's just getting A's by doing things like studying, and who are engaged in class because they're kissing up, not out of actual interest. If you're thinking that way, you can discount a lot of demonstrated merit very easily.

Another possibility is that the teacher identified with Heebie, and passed on sexist discouragement she'd gotten at a similar stage and internalized: "I remember thinking I was interested in advanced math at that age, and then when I tried it in college, professors were clear that even though my grades were good, I didn't have what it takes to do math for real. Heebie likes math, but like I did, not like someone who's really good at it -- I should let her down easy now, rather than letting her head down a road she's not suited for."

229

This is like the argument that there can't actually be employment discrimination because some sensibly non discriminatory employer would take advantage of the lower pay members of disfavored groups can be offered.

Huh? I am perfectly prepared to believe Heebie's teacher was a sexist idiot, just not that the teacher was simultaneously a sexist idiot and a good teacher.

Obviously, I wasn't there in Heebie's math class, so I don't know what happened. But I wouldn't be surprised if she fit neatly into a particular sexist stereotype: she's pretty, well-mannered, plays sports -- girls like that get good grades because they're compliant and hardworking, not out of real merit. A boy pulling a B in a math class and not doing the homework is obviously much more talented than a good-girl type who's just getting A's by doing things like studying, and who are engaged in class because they're kissing up, not out of actual interest. If you're thinking that way, you can discount a lot of demonstrated merit very easily.

This is more plausible for say a college admissions officer not someone who taught Heebie for a year.

Another possibility is that the teacher identified with Heebie, and passed on sexist discouragement she'd gotten at a similar stage and internalized: "I remember thinking I was interested in advanced math at that age, and then when I tried it in college, professors were clear that even though my grades were good, I didn't have what it takes to do math for real. Heebie likes math, but like I did, not like someone who's really good at it -- I should let her down easy now, rather than letting her head down a road she's not suited for."

Or maybe the teacher flunked calculus (or barely passed). And we are just talking about a high school class not a career choice.

As a junior in high school, I was evaluating teachers by how much I enjoyed the class. I remember enjoying her class, vaguely.

The only vivid memory I have is getting really worked up in a debate about whether or not there were as many numbers between 0 and 1 as there are in the whole real line. I am pleased that my memory places me on the right side of the debate.

(A data point against the mathematical insight of the teacher: she didn't have any context to place that debate, and seemingly hadn't ever heard of it before, and I think was on the wrong side. I think some student brought it in from their parent. Of course, I thought it was a completely isolated, unresolvable problem and didn't know enough to count it against her.)

I don't see why someone can't be a good teacher and slightly biased against women in math on the margin. Research suggests that just about everyone is at least a little biased against women on the margin, and yet half the teachers out there are above average.

On the other hand, I agree with Shearer that the most likely scenario here is that the teacher was afraid of Calculus. I just don't think being afraid of calculus and not being particularly good at analyzing talent differences among good students necessarily makes someone a bad teacher.

Also, I'm not quick at arithmetic, and it's common to use arithmetic-quickness as a yardstick for good at math.

(For the record, I wasn't particularly well-mannered and compliant. I never shut up. I was usually participating, eagerly, but it was actively difficult to get me to shut up and let someone else contribute.)(My theory was that it was so boring to listen to anyone besides myself or the teacher and so if I was going to enjoy myself I needed to dominate the show. I might have been a major pain in the ass.)

Evidence in favor of the instructor being scared of Calculus: she herself never taught it; a different teacher always did.

I am perfectly prepared to believe Heebie's teacher was a sexist idiot, just not that the teacher was simultaneously a sexist idiot and a good teacher.

My point wasn't that you were denying that Heebie's teacher was sexist, it was that you can be sexist and not otherwise transparently an idiot, just like you can practice employment discrimination and still be a profitable company.

As Upetgi said in 232.

232

I don't see why someone can't be a good teacher and slightly biased against women in math on the margin. ...

It doesn't sound like a slight bias to me, the teacher discouraged Heebie from taking a class she could have easily passed.

... Research suggests that just about everyone is at least a little biased against women on the margin, and yet half the teachers out there are above average.

My standard for good is more than above average.

233

... I just don't think being afraid of calculus and not being particularly good at analyzing talent differences among good students necessarily makes someone a bad teacher.

It seems to have led to a rather serious failure in this case.

Keep in mind, though, that she wasn't the only teacher to do so. I was discouraged from taking Real Analysis in college. (By someone who never had me in class, and I had all As on my transcript.) Those are the most overt examples, but I was definitely never selected as someone to groom or encouraged to compete in some extra competition, etc. Also keep in mind that I reinforced this, explicitly, as it matched my own opinion.

236

My point wasn't that you were denying that Heebie's teacher was sexist, it was that you can be sexist and not otherwise transparently an idiot, just like you can practice employment discrimination and still be a profitable company.

It's true the standard for high school teachers isn't very high. I wonder though how many would have told Heebie she was too dumb to take calculus.

I was usually participating, eagerly, but it was actively difficult to get me to shut up and let someone else contribute.

At that point the stereotype turns into Reese Witherspoon in Election: kiss-ass showoff who should be taken down a notch for her own good, no? But still not authentically talented.

I wonder though how many would have told Heebie she was too dumb to take calculus.

I doubt the teacher's thinking, if asked explicitly, would have been that Heebie was literally too dumb to pass. More that there's no use to taking calculus unless you're going to progress to specializing in some math-heavy field, and without a genuine math aptitude there's no point to taking the class at all. Same with the professor who discouraged her from taking Real Analysis: probably if you'd asked them whether someone with a straight A record in their college math classes could pass R.A., they'd say yes. But if they could look at her and tell that she didn't have that mathy spark that somehow women tend not to have, there'd be no use in her taking it.

At that point the stereotype turns into Reese Witherspoon in Election: kiss-ass showoff who should be taken down a notch for her own good, no?

This resonates more. I definitely had teachers on occasion who wanted to take me down a peg, sometimes quite explicitly, like telling me I wouldn't be able to get away with this kind of thing at [next academic level] and I'd be exposed for whatever I really am.

231

(A data point against the mathematical insight of the teacher: she didn't have any context to place that debate, and seemingly hadn't ever heard of it before, and I think was on the wrong side. ...

It's tough to be a good teacher when your students (or at least the best ones) know more than you do.

239

... Also keep in mind that I reinforced this, explicitly, as it matched my own opinion.

And yet you somehow ended up a tenured professor of mathematics.

Well, sure. In graduate school my opinion changed.

242: Man, people suck.

Didn't Silvana (possibly while being M. Leblanc) have similar experiences? Strong interest in math, consistent good grades, but was told that she wouldn't be able to handle the next level up and drifted off to major in something else?

The time that hurt my feelings quite badly was in 5th grade (which I posted about somewhere in the archives here) with the school play that I was gradually blacklisted from. In middle school, it didn't happen so much because I was meeker and cowed and quieter. When it started happening in high school again I didn't care that much about the opinions of the teachers once they leveled the accusation. Or rather, the accusation just didn't land on a vulnerable spot - other things hurt my feelings, but that "taking down a peg" was surprisingly easy to shrug off.

(One of the most vicious was actually the calculus teacher, who I had for not-quite-calculus as a senior. I just thought he was batty.)

when i was graduating my secondary school, our teachers all wanted me to pursue their own respective fields when choosing my future profession, chem or math or philosophy which was called at that time " scientific communism"
if i chose that at that time could have become a politolog and be now something like a public figure criticizing our MPs
like the girl who chose that assignment
that heebie became what she became could be thanks to her teacher discouraging her since one tends to do one's best going against hurdles and obstacles and proving the bias wrong

247: Yes, I think you're right.

She went to college at State U in Heebie Town, actually. At one point I was in conversation with someone who might have likely known her, and I tried to ask them if they did. I couldn't generate the name "Silvana" and it ended up sounding like I was asking about the guy from Friends and I aborted the mission.

Most people's experience with math is that at some point they hit a math class that they struggled with and then they gave up on math. This is true also of most math teachers. If your picture of math is "there's some level of hardness which outmatches your math ability and once you hit that you're screwed" then you're going to be wary about recommending people take harder math classes even if they've done well on math classes so far.

Of course at a higher level everyone has hit that point and fought through it and learned how to solve problems that you didn't already know how to do. Even though that's key to actually being good at math, it doesn't match with the popular conception of what being "good at math" means.

238: Heebie never said that the teacher was good by Shearer standards.

Here's her story. Actually, not so much that she was discouraged, but that she wasn't encouraged at all.

251 (and similar things that have been said in the past) interests me, because that is pretty much how I ended up leaving physics; I hit a class where the math didn't click for me, and didn't recover from it. I still get kind of wistful about that in retrospect.

yet half the teachers out there are above average.

Above the median, sure.

253. Physics got defunded after the cold war ended, leading to an exodus of people from physics into fields where there were jobs.

254: At or above the median, surely.

True, true.

Is it true that math departments do a lot more service teaching than research? If yes, and if identifying and training the next generation of influential researchers who will stay in math is a career goal, then there are incentives that favor this behavior. It's still a shitty outlook, since getting the chance to pass knowledge on generously is good, and being stingy with knowledge is bad. But, for instance, in much of academia under communism, academicians were very stingy with knowledge and mentoring as a viable career strategy. Training a student who leaves the field, leaves the country, makes trouble is self-damaging behavior under this outlook. Basically I'm saying that there's a systematic problem caused by academic careerism in addition to the frequent personal inadequacies among mathy types. Maybe a weak effect, hard to be sure.

Certainly this does not apply at the HS teaching level (where discouragement sucks, sympathies to anyone on the receiving end).

Is it true that math departments do a lot more service teaching than research?

All math departments have a lot of service courses, compared to almost anything except English/Composition. But to what degree this is at the expense of research varies widely by institution.

Along with math and writing, foreign languages are the other high-service fields.

If yes, and if identifying and training the next generation of influential researchers who will stay in math is a career goal, then there are incentives that favor this behavior.

Define "this behavior"? If there's a gender discrepancy in the discouragement, then incentives that don't relate to gender don't explain it. I mean, it's perfectly plausible that all the male mathematicians out there were also discouraged/not encouraged, but I don't think we have anecdotes to that effect.

Personally, I feel like math was just incredibly badly taught in high school across the board. IIRC the teaching method was basically to put a bunch of algorithms or principles on the board, without any explanation as to why they might be interesting or important. And then to respond immediately to people who seemed to get the idea super-quickly without responding to anyone else at all. Making the whole thing seem like a pointless exercise designed only to allow the jerks who were quick at it (with "it" I think including a lot of arithmetic) to show off. It's been a long time since I've been in a class so its hard to remember, but I remember math generally being by far the worst taught subject, infinitely worse than the sciences.

I agree that that's true -- there's a combination of some teachers who are really spooked by math (this is more at the younger levels, but there's some of it all the way up) and a widely held belief that math is something you get or you don't, so if it doesn't come naturally there's no sense bothering.

Identifying the kids who seem quickest because they correspond to some stereotype and also test well would have discrimination as a byproduct.

The stereotype that math is something you get or you don't get is particularly problematic in relationship to the stereotype that girls aren't naturally smart they just work hard.

242: Man, people suck.

Actually, not so much that she was discouraged, but that she wasn't encouraged at all.

Don't forget that every student has a lot of teachers, and there will inevitably be a range of reactions ranging from discouraging to encouraging. That isn't to minimize the impact of sexism, particularly in mathematics, But even somebody who has generally received encouragement from their teachers is likely to have some significant memories of being discouraged.

[Personal experience: I remember being dissapointed that I never had any professors in college encourage me to go to grad school. I talked to friends who would mention conversations in which one of their teachers would say something, and I never got that. But in my case I suspect the dynamic at work was (a) they all assumed that I'd go to grad school anyway and (b) I was both shy and mostly self-confident so I didn't seek out encouragement. As it turns out I didn't go to grad school and I think that was the correct decision, so it worked out well enough in the end but it felt like a bit of a sleight, even though I knew my professors thought highly of my work -- students are neurotic.]

The woman who taught AP Calculus at my high school also taught "remedial" math - I'm not sure what it covered, but pre-algebra - and I don't think she taught anything in between. She was great.

Don't forget that every student has a lot of teachers, and there will inevitably be a range of reactions ranging from discouraging to encouraging. That isn't to minimize the impact of sexism, particularly in mathematics, But even somebody who has generally received encouragement from their teachers is likely to have some significant memories of being discouraged.

Seriously? I'd expect most people getting A's in a subject to have no memories of being actively discouraged by a teacher from pursuing that subject further (from incapacity, rather than general 'don't go to grad school/law school because no one gets jobs'). Not that it's impossible, it happens sometimes as we've been discussing, but it would seem to me to always require some kind of explanation.

More of 268: Not encouraged is different -- like you, I don't recall ever being encouraged by a teacher or a professor to do anything at all at the high school/college level. I stayed pretty much under the radar. (I did get some active attention in law school.) I don't really know how much mentoring/encouragement most students who head off to grad school get.

Further searching indicates that my former teacher left teaching about ten years later and now runs a boutique clothing store.

Also, I thought the school was having trouble while I was there, but it seems to have gone even more to shit in the intervening years.

Another problem (this wasn't Heebie's problem) is that people who aren't going to be superstars also need "encouragement" of a different kind even if they aren't that talented and arent going to end up as Math PhDs. For example, it's true I am just not that (relatively) talented at Math and there's no way I would have ended up as a tenured math professor (or even in a math-heavy field) so in that sense my teachers were probably *right* to not encourage me in Math. On the other hand, I'm pretty comfident that with a bit more encouragement and work I could have gotten to the point where I didn't affirmatively hate math and felt comfortable with it, and with a slight change in instruction and effort could have gotten enough math to at least think plausibly about some kind of more STEM field. I imagine a lot of people are in roughly the same boat, and it's worse if you're a woman and also have to battle the sexism factor.

On the other hand, I'm pretty comfident that with a bit more encouragement and work I could have gotten to the point where I didn't affirmatively hate math and felt comfortable with it, and with a slight change in instruction and effort could have gotten enough math to at least think plausibly about some kind of more STEM field.

Or to even be more comfortable with the sort of math that's useful outside STEM fields; I don't know about you, but I know lots of lawyers who really shut down at the kind of low-end algebra you need to work out even moderately complicated financial stuff. There's a real benefit to almost anyone from being comfortable with high school math, and you're right that better teaching could get most ordinarily intelligent people there.

I teach logic to people with math panic so crippling they can't bear to take college algebra. People definitely believe that you can do math or you can't do math; no one would happily admit being able to read at a sixth grade level, but will say "oh, I just can't do math" like I might say "oh, I can't run a seven-minute mile. Yet most of them seem to be capable of doing the same kinds of manipulations, if you tell them it's logic and there are no numbers.

I'm pretty psyched that I misspelled "confident" there. I am so comfident that I'm smart!

I've forced myself to learn basic financial algebra through necessity, and can usually figure what's right or wrong with a damages calculation pretty quickly, but you're right that even in law there's a lot of basic math that comes up and having a background level of "comfortable" would be nice.

I don't think I've ever asked this (and if it's a rude question, please place it in the context of it being asked by a historian, who constantly hears some version of "while history is a lovely avocation -- 'all those battles! fascinating!' -- it's not really a vocation in the sense of, well, a science, the sort of thing that generates immense gobs of grant money and happens in labs with teams of grad students and whatnot, is it?"), but I'm wondering how hiring at heebie u works. Did they, as most schools like heebie u seem to, look for the person likely to be the best math scholar? Or did they, as would make much more sense for a school like heebie u, look for someone who has a passion for teaching (as heebie seems to -- note also that I admire the fact that heebie apparently maintains a research program while teaching at a school that demands that she devote an immense amount of her time to pedagogy, not to mention service*).

* Tons of that here, actually, but undergraduate teaching can take relatively little of one's time, if one makes that choice. Graduate teaching, on the other hand, oy vey, don't get me started.

Shorter VW: I'm fascinated by hiring processes at schools where research isn't by any rational measure a priority, but yet hiring pivots on finding the person most likely to be a star scholar.

re: 276

I've been through [in the UK] the recruitment process for a part-time fixed-term contract teaching-only post, and yet they still seemed to care more about research potential than ability to teach. It's stupid.

More of 268: Not encouraged is different

I agree, and I should have made clearer that I wasn't specifically responding to Heebie's story, I was just vaguely pontificating.

I think anything more explicitly discouraging than, "you'll have to work harder when you get to X" is notable, and should require explanation.

I also remember that, as a student, it was easy to feel encouraged or discouraged by very minor interactions, but perhaps that's just me.

I don't really know how much mentoring/encouragement most students who head off to grad school get.

Nor do I . . .

277: right. It's insane and yet happens ALL THE TIME in my field and apparently others.

People definitely believe that you can do math or you can't do math

I'm told this is particularly American and is also associated with low median math achievement (duh). Presumably profitable for those of us who don't suffer it, in a crabs-in-a-bucket way, but I do not want to live in a crab bucket.

276, etc: I taught at a school where you had a 4/4 load and had to reapply for your job every year and justify your appointment based on your research.

271 is very important. See what happens when you don't encourage folks in math? You get copyright lawyers, that's what.

but I'm wondering how hiring at heebie u works. Did they, as most schools like heebie u seem to, look for the person likely to be the best math scholar? Or did they, as would make much more sense for a school like heebie u, look for someone who has a passion for teaching (as heebie seems to -- note also that I admire the fact that heebie apparently maintains a research program while teaching at a school that demands that she devote an immense amount of her time to pedagogy, not to mention service*).

We heavily prioritize teaching in the hiring process and later in the tenure process. You can't be absentee as a researcher, but your hiring committee is going to watch you teach a class and take your teaching statement very seriously. Similarly, for tenure, if you've got enough compensating great qualities, your research could be limited to research with students and attending a conference now and then. But mediocre teaching is a deal-breaker.

283: that's so...rational. Surely there's a movement afoot to reform this practice?

We feel it's best sabotaged by harnessing the power of microscopic levels of assessment.

261

... I mean, it's perfectly plausible that all the male mathematicians out there were also discouraged/not encouraged, but I don't think we have anecdotes to that effect.

1. My high school didn't offer calculus.
2. When I asked my father (physics PhD) to teach me calculus he declined.
3. In ninth grade I was strongly discouraged by a teacher from taking the MAA high school math contest test (which IIRC at my school was mostly taken by seniors) but was eventually allowed to take it.

Of course I also had positive experiences. And as NickS noted above it is unrealistic to expect all your teachers to be great. Especially since a teacher might be good with some types of students while being bad with other types.

277

I've been through [in the UK] the recruitment process for a part-time fixed-term contract teaching-only post, and yet they still seemed to care more about research potential than ability to teach. It's stupid.

The thing is, it tends to be easier to evaluate research potential than teaching ability. There are a few obvious desirable qualities for teachers such as subject matter knowledge, fluency in the language of instruction, willingness to make an effort and a little experience. Past that I doubt the ability of the recruiters to tell good teachers from bad (even assuming that there is a lot of difference).