That’s interesting. Thanks. [Aside: I wonder if that’s what happened to Australia’s generation long superiority at tennis ? And if so, why it hasn’t happened like that to swimming at which Australia is still world competitive ? Maybe because many more Aussies swim than play tennis nowadays, so there’s a bigger ‘pool’ (so to speak) to choose from ?]

But personally, that’s exactly what I would expect with maths and music (two of the core things the entire human race has in common). But then, many people can enjoy music for itself, even if they aren’t at the professional forefront.

How many people nowadays do maths at home as a preferred leisure activity simply because they enjoy doing it ? How many ever did, other than perhaps Fermat ?

Thanks. Maybe, yes, that’s the real secret: requiring that every student achieve ‘mastery’ instead of the tried (and tried and tried) and failed (and failed and failed) approach of “Let’s run it up the flapole to see who can salute.” And those who, for whatever reason, can’t salute are declared “underachievers” and henceforth ignored.

Salient @85

Yes, some ‘salient’ research (ha ha) from you could be enlightening. Perhaps in terms of Matt @77’s point about the earlier you have subject-specialist teachers, the better is your final result, at least for some (though I doubt if the majority of Russkis are any better at understanding the difference between the arithmetic mean and the median than the majority of Americans, Britains or Aussies are. I wonder if the majority of Finns are any better).

Perhaps in terms of your own experience, generalised. Were you part of a ‘whole of class mastery’ approach, or just the usual ‘flagpole’ approach ? Are you going to go on to do original mathematics research, or just ‘teach’ ? Or will you become the ‘new Tom Lehrer’ ? :-)

If we (ie homo sapiens) need original mathematics researchers, is it ok to concentrate on that for the few in each year that will go on to accomplish this, even if it means ‘failing’ the majority of students ? Or would a more determined approach of ‘whole of class mastery’, no matter how much it holds back the potential ‘overachievers’, be justified ?

Or can we somehow achieve both ? Would a determined ‘whole of class mastery’ approach overcome the ‘lack of talent’ problem with those who turn up at the Ed. schools’ doors wanting to become teachers ?

Even so, I guarantee you that every single sport instructor on the face of the earth past or present has one word in their vocabulary that they deploy ten times more than any other, and that word is “hustle!” These instructors still need to encourage and motivate their charges, but because most of their charges in some sense already want to be there badly enough to be there, the instructors have cruder tools at their disposal. That’s a blessing.

Otherwise, it appears to me, if there are problems with how well teachers teach kids, then maybe there’s even worse problems with how well educators teach teachers.

In a way, yep, though another problem is recruiting. As an administrator very patiently explained to me on this end not long ago—not long after my CT comment on the subject :)—each state needs x many new educators, and most teachers go to university in-state, so the state schools need to graduate x many new educators. There must be sufficient supply to meet demand each year, give or take a few percent, and it’s basically up to the universities to ensure supply, and to ensure that that supply is as well-prepared as we can get them. So we can’t require all entrants to the major to be superstars. But when only x + 31 many people are even interested in the major across the state, recruitment becomes a huge concern. So let’s trace this back to roots; Why don’t more college students want to become teachers?

(I don’t know of any other major which tracks its number of graduates across-state and the number of open positions across-state so meticulously. I don’t even know of any other major which could do this. I really do think it’s neat that we are able to observe what open positions tend to go chronically unfilled, and then guide students toward those career opportunities. Unlike most majors, I think, the education major really can be geared toward career preparation for a very narrow range of public service jobs. This is potentially a big thing in our favor.)

Maybe, if all students are taught as though they were going on to ‘higher maths’, it would greatly support those who do without permanently injuring those who don’t.

Like you, I doubt this, but perhaps unlike you, I think it comes down to a question of what the definition of “prepare for higher maths” is. Put a bit paradoxically, because I’m up against the limits of grammar here, I don’t think that preparation for higher maths helps prepare students for higher maths.

A lot of ‘higher maths’ can be cruised through—e.g. blue Rudin—by students with (a) powerful reasoning skills, like the ability to make comparisons and notice particular parts of a diagram, and (b) a big tinker-toolbox, problem solving tricks that commonly work in a given subfield.

And given solidity in (a), the size of (b) that a student is less important. Or at least, it’s easy to plunk more stuff down in (b) in college. Heck, I’m still adding to my tinker-toolbox as a master’s student in maths. But of course the huge difficulty is both in defining what the heck (a) means exactly (thankfully this job is handled admirably well by the NCTM), and then, in determining how best to get students to develop (a), which is hard to even measure/assess.

Plenty of time for open-ended exploration, with conjecture and follow-up to see if the conjectures are true, with the teacher walking around and guiding students to interesting results, helps students develop (a) no matter what their level of facility. This benefit’s fairly well-established, with certain constraints, i.e. assuming a given baseline of teacher preparation and comfort, and overall classroom management facility, etc. Worth acknowledging that lots of teachers are amazing at providing this environment. And lots of teachers who are too scared of maths to feel secure providing this environment.

Oops, this is becoming a long low-substance comment. I think I’m going to start collecting and collating good research/information links to share for future CT posts, and maybe be useful for once, promise!

But yes, music teachers have figured out how to get high levels of performance from students who seem to have little discernible “talent” to begin with. One reason the Suzuki method works well is that it entails three or more lessons a week, for example. And I understand that Asian (non-Suzuki) music classes have daily practice in school on desktop keyboards, or so I have heard. Where there’s a will, there’s a way. It is understood that music instruction from the earliest years develops the brain and helps in other subjects. In the Waldorf Schools early math is combined with movement—throwing a bean bag while practicing the times tables—dance-like moving in groups, and the like. These methods use mastery learning, which means that everyone (not just the “talented” is supposed to master the subject before moving on—rather than “failing” and being eliminated.

Though I have to say that the discussion is something of a reminder of a paper that was influential in my former profession titled “Archipelagos of Information”. It strikes me that there is much valuable information held in some quite distinct islands, or, if you prefer a traditional description “the right hand doesn’t know what the left hand is doing” (and vice versa, of course).

What we seem to have is the age old problem (to quote myself): “if you don’t know what to do, do what you know”, which, as usual, doesn’t work very well in complex situations. So some, like Lemov and Ball, due to both personal qualities (mainly intelligence, motivation and persistence) and external cicumstances (ie funded opportunities), have worked long and hard to come up with approaches, techniques etc that, at least in part (though I share Madeleine’s concerns about Lemov), could make for better teaching which in turn could make for more accomplished students.

I tend to share Matt’s view of emphasising subject-matter expertise and early concentration thereupon, but then again, as I said in a previous thread: ” for most students, maths education is too much too soon, but for those who will go on to higher maths, it’s much too little too late”. Maybe, if all students are taught as though they were going on to ‘higher maths’, it would greatly support those who do without permanently injuring those who don’t. But how many subject areas could be given this kind of treatment without seriously overloading the kids (and the teachers, I suspect) ?

Perhaps the secret to Finland is that it is an extraordinarily unified country because of its history of external threat. Maybe the Finns just think that “no Finn should be left behind” or something.

Otherwise, it appears to me, if there are problems with how well teachers teach kids, then maybe there’s even worse problems with how well educators teach teachers. Precisely what Lemov and Ball (et al) are trying to address, each in their ‘separate island’ way: if you can’t rely on the inventiveness and ‘native talent’ of proto-teachers, then train ‘em like you would a Navy Seal … or something like that, anyway.

Just one thing though: does anybody here teach music or sport ? Doesn’t seem to be any shortage of candidates for the Julliard School, or for football, baseball or basketball teams ? Do the music and sport teachers know something that the ‘academic’ teachers don’t ?

Following Finland’s continuing success in the PISA tests, the World Bank published a report (Aho Pitkanen and Sahlberg 2006) on Finland’s educational system, and it gives clear guidance on what works – high social status and professional respect for teachers, a holistic approach to schools as part of their community – healthcare and law enforcement representatives visiting frequently, interaction between the community and the school, and a system which allows students to switch streams between academic and vocational and across disciplines until they are 18/19. There is far less focus on standardised testing than in the US and the UK.

There is plenty of research out there to tell us what is necessary to improve education – investing in teachers is critical, encouraging teachers to believe in themselves and to develop both their skills and their subject knowledge is critical – but ultimately, in market-driven economies like the UK and the US, taxpayers are simply not willing to foot the bill, even when it comes to helping their children to be the best they can be.

Not really, but here are some disconnected-yet-related observations.

It occurs to me that my class is (probably) the last math class any elementary education or special education major at this university ever takes, in their life. Now bear in mind, I’m not some Ph.D. in instructive awesomeness. I don’t have decades of experience to bring to bear. I’m a graduate student. And most of the mathematics-education reading/research I do is entirely of my own volition.

We’re accepting students into the program who struggle with core skills—not just in mathematics, but also spelling, writing, etc—and there’s only so much one year of mathematics instruction can do for them (elementary ed majors are only required to take two math courses, which begin with core-skill adding and multiplying and estimating techniques, and end with three-dimensional geometry—nothing that hasn’t already been taught in the high schools, and most of which was taught at the grade levels they’ll be teaching).

Many students go into the program, by their own admission, because they “like working with kids” and making kids happy (the most common reason we receive, regarding why students entered the program, is students reporting positive babysitting and/or day-care and/or after-school-YMCA-program experiences, and second in frequency, students reporting that one of their relatives, who they loved, was a teacher).

Of course, these kids got straight As in high school and are somehow managing nearly straight As here at university. I was surprised to learn just how high the proportion of GPA-based scholarships going to education majors is. It seems the average GPA of an education major is substantially higher than the average GPA overall.

Again, something that seems to be important in Russian (and before that Soviet) schools doing quite well in teaching math is that they have specialized teachers for different classes much sooner than in the US. I’m pretty sure that having only one teacher for all (or nearly all) subjects stops at 2nd grade, if that late. Math is taught be people who studied math at the pedagogical university, even to pretty young kids. Some teachers are better than others, of course, but teaching kids in general isn’t a refuge for those who can’t do even low-level math. This would be a pretty big change from the US system, where I think having one teacher is still common through at least 5th, often 6th grade, but would be a clear improvement, I think. (Not just for math, either, but also other subjects as well.)

Yes, we all get mugged by reality from time to time, and it’s been a long time (if ever), since the ‘best and brightest’ beat a path to the Ed. school.

The main question as I see it is whether any of the approaches, techniques, methods etc of the very talented – such as Deborah Ball or your grad school economist friend – can somehow be imparted to the very large numbers of not so talented who do find their way into education and without whom there simply would be no public education.

If it requires the trainee educators to be “excellent mathematician[s]”, then the attempt is doomed from the start. But Deborah Ball did not start out as an “excellent mathematician”, she merely had the motivation and talent required to become good enough to be able to focus on her students, rather than on herself.

Can any of that actually be taught to the majority of Ed. students ?

Any natural number, that is, except 1. (Or 0, if you count that as a natural number.) I don’t disagree with any point you make, Walt, except your implied claim to have banished arbitrariness.

The ed profs I mentioned weren’t cynics but they were realists. They spent a lot of the time in their methods classes on math content because so many of their students simply couldn’t do basic math. They couldn’t just flunk out unprepared students without destroying their programs—back in the 70s, or so they told me, the quality of ed majors declined. The usual explanation was that the bright women who formerly went into elementary education had better options while students in general were less idealistic than earlier generations.

Things may or may not be better now than a few years ago. My evidence is certainly dated, and I’d be interested if anybody could make a more recent report on the view from education departments, especially departments in non-elite institutions.