A while back we had a vertical meeting with math teachers from grades 4-8. The agenda was to familiarize ourselves with the standards for fractions outside of our own grade levels. The facilitators cut all the relevant standards into little strips, removing any identity as to the grade levels they came from. Our job was to put them back together again in the proper 3-8 sequence.

It couldn't be done except at a gross level. Each year, and within each year as well, there are nuances on the nuances on the nuances. It was a good exercise in cross polination. In hind sight it was also a dramatic demonstration of the ridiculous nature of those standards. If the teachers charged with delivering them can't see clear annual goals then how can the kids?

I have our cities math curriculum from 1958 (I was in middle school then) and it reads precisely like the A+ international standard in the link. Interestingly, it is riddled with demands for mastery at key milestones, something that is totally absent in our present day constructivist, spiral curriculum.

Funny how 50 years of 'research' has taught us how to do what we already forgot.

And here's Barry:

Also funny how the math of 50 years ago is said (by those pushing reform math) to have failed large numbers of students.

That is funny!

data driven loops and noise

data driven instruction redux

## 7 comments:

What I don't understand is this: did the curriculum of 50 years ago really work? Or more broadly, did math education then work? Were high school graduates of 50 years ago truly numerate?

When I was in graduate school teaching calculus, badly, a version of calculus was being introduced that focused on conceptual understanding and real-worldish applications. The idea was that rote learning of derivative and integration rules had left generations of students unable to understand calculus, and at best able to apply a bunch of rules they would never really need. (That will all sound familiar to KTM readers if we substitute all the terminology of calculus with fractions and arithmetic.)

I taught the old curriculum one semester and the new the next semester. As far as I could tell, my poor students learned didley squat in either case (and no, I did not continue with calculus teaching as a trade). But I don't think the whole problem was with me. The old plug-n-chug curriculum did not help students develop intuition, and it seemed like students were not willing to do enough practice to really learn the techniques. The so called "new wave" calculus seemed not to teach people any more of the underlying concepts either, and still they couldn't work out integrals. (Although I heard many times, "I get the concepts, I just can't do the problems." I always interpreted this to mean "I don't get the concepts.")

It seemed to me that teaching calculus to young adults was inherently difficult. Some approaches handle these difficulties (*) and some do not. Schools are doing a bad job of teaching math to kids now, using one set of fuzzy methods. Back when they were teaching with well-defined standards and coherent curriculum, were they actually teaching people math? Who knows the answer and how do you know it?

(*) Difficulties are to be handled; problems are to be solved. E.g. diabetes is a difficulty; pneumonia is a problem.

I have written about this issue extensively and will continue to do so in an article that will be in American Educator next year. It is a complex issue and I don't mean to distill it down to test scores, but an examination of SAT math scores from the 60's shows, in general, that they declined over the next 20 years and are slowly making their way back up. There were aspects to both teaching and texts of the past that could definitely be improved. But it is important to look at what worked in the past, (as well as what is working in countries such as Singapore and Japan) rather than dismissing these as "traditional approaches" that rely solely on "rote memorization" etc etc, and which don't work. There are many aspects of math teaching and textbooks in the past that have worked; that they can work better is not a reason for absolute rejection.

I am familiar with the reform calculus movement that you describe. I am also a product of calculus taught in the 60's. We were taught the reasons behind the methods for taking derivatives as well as integration techniques. I felt at the time that my high school math background (which did not offer calculus at the time) prepared me well to understand that course. Could it have been improved? Yes. Was it a total disaster? No.

The mischaracterization of traditional math fails to capture the nuance of instruction and scaffolding that many teachers (including mine) engaged in.

Upon honest reflection, I can say I never really became expert at any subject until I finished the next one. That is to say I only excelled at arithmetic after algebra, algebra after trig, trig after calculus, calculus after physics, physics after fields and wave theory, fields and waves after a stint at physical oceanography. And all of that was certainly enhanced by 30 years in engineering practice. Each bit of hierarchical learning gets enlightened by the intense glare focused on it by the next in a line of succession.

The 'traditional', at least from my perspective, was more layered so that you left a subject enough of a journeyman to successfully apply it in the next. It is that next application that creates true mastery. What has transpired over the last 50 years is a sort of compression (diguised as a spiral) that attempts to deliver mastery concurrent with the apprenticeship.

I fear I've just jumped the shark on metaphor overload. Yikes!

Anyway, the compression leaves many with the false impression that we are connecting math to the real world thus creating more creative connected thinkers at an earlier age. But by comingling the layers delivered at any one time you must reduce the level of mastery demanded. This leads to a dilution of your ability to apply the 'last' subject. Plus, the proffered 'connections' are tenuous and at times very artificial (anything truly deep can't be navigated successfully yet).

It was dismaying at first when I entered teaching to be told I didn't have enough math courses to cut it. I thought every single course I ever took was a math course. They were of course. These were the applied math courses that made me into a mathematician, not the math courses.

I just read the links that Catherine placed at the end of the post. I'm reading along thinking wow, whoever wrote this stuff is thinking about data just like I do. It was posted by Anonymous.

So I'm reading along getting ever more taken with the synchronicity thing until I finally realized it was my comments, made when I was a chicken, afraid to come out into the open. What a dope!

When you read something you wrote and the scales don't come off until you've gone through a dozen or so paragraphs, it's embarassing; like getting caught talking to yourself.

At least we didn't have an argument.

Now THAT would be funny, flaming yourself in a blog. Priceless!

---As far as I could tell, my poor students learned didley squat in either case ...the curriculum did not help students develop intuition, and ...students were not willing to do enough practice to learn the techniques...

The problem with this statement is that once we are adept at a subject, our goalposts for "didley squat" move. A LOT.

Barry's articles can probably point to some actual normed data to show whether or not students are learning more math now than before, or even if we have a baseline for knowing what they knew in the past. But our IMPRESSIONS are that "students never learned a darn thing in our course" and This is ALMOST ALWAYS TRUE. And it doesn't mean we're doing it wrong.

Because the truth is, "students never learned a darn thing" is true of any course. Of any population of students. This is the truth of human beings and the way their minds work. It's also almost always true, that most students aren't doing enough problems to really practice a technique.

Paul gives the example that he never "really" learned one subject until seeing the next. I had similar phenomena--never understanding a subject til the 2nd time or 3rd time through it, however obliquely. Obviously, lots of people have had this experience, because it is one of the main thrusts in reform math--"it's when we saw the same material a 2nd time, a 3rd time, a 4th time, that we got so much more out of it, understood it! So if we teach them in a Spiral, where they see it several times over the years, THEY'LL get it!"

But I found out something even more amusing. After taking much time off from academic math and physics and CS, I found that going back a few years later, I had learned these subjects more than I EVER HAD at the time even though I DIDN'T take the next one! Time alone had matured my understanding of these concepts!

Our intuition isn't just built by doing problems. It's built by doing problems, and doing them some more, and doing them some more, and then cooking-until-done. Our brains keep cooking, connecting up concepts to other ones, chunking more and more stuff into higher level abstractions. It all takes time, and experience, and good teaching supports that process but still doesn't reduce the time to do it down to zero.

And now that I DO understand these concepts, it's hard to imagine how trivial my old "understanding" was; it's hard to remember what was so hard to understand, or how I possibly made sense of those concepts before my current way. So it's really easy to look at those just entering these fields and think "they don't know didley squat." True--but it doesn't meant that they didn't learn a great deal, and it doesn't mean they weren't well prepared. It probably means that you just can't *do* enough work in 16 weeks to grok all of the intricacies of a subject, even if you do a LOT of problems.

back to Bky: "were graduates truly numerate 50 years ago?" Again, this is too easily subjective. Barry mentions that he felt prepared for college calculus--but maybe his professors felt all of their students learned didley squat, too. We don't have an objective defn of "truly numerate", and part of the never ending debate in math education is that no one seems to agree on whether objective defns of arithmetic knowledge are really getting at what people want from a notion of "numerate" in the first place.

Hey Paul,

I've had that experience here, too. One time I dashed something off and Catherine pulled it up front a week or so later. I remember thinking, "Who wrote this incoherent mess?" I should have paid her to fix it.

My defense is usually a lack of caffeine.

SusanS

Barry mentions that he felt prepared for college calculus--but maybe his professors felt all of their students learned didley squat, too.And it's a time-honored tradition that professors feel that incoming freshman are not as well prepared as they were when they were that age, or as the previous generation, etc etc. Putting aside definitions of "numeracy" and lack thereof, the primary complaint I hear from math professors who teach freshman calc, is the alarming number of students who don't have mastery of 1) basic algebra skills and 2) basic arithmetic. The increase in the number of students in remediation math classes, as well as the increase in such classes themselves points to something going awry. The set of "something going awry", is likely to contain more elements than the convenient excuse of an ever-increasing population of students.

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