If you are fighting a war then you need to understand which side to be on. This is hard, the maths war is complex, with wars within wars, false friends (often well intentioned), propaganda of varying quality etc.

A brief history of american K-12 mathematics education in the 20th Century by David Klein

This argues that progressivism or discovery learning or constructivism (fuzzy descriptors that are used interchangeably) has done enormous harm to maths education

How?

- reducing academic content
- capitulating to utilitarianism
- restricting the teaching of algebra
- teaching the child and not the subject
- integrating subjects
- focusing on everyday living
- allowing children to learn what they want
- promoting freedom as profound and revolutionary ("Summerhill")
- reducing pencil and paper computation
- excessive use of calculators
- too much real world problem solving focus
- teamwork focus
- de-emphasising calculus and its prerequisites (algebra, geometry, trigonometry)
- offering a curriculum smorgasboard
- vague curriculum goals, eg. "to be a maths problem solver"
- too much learning by doing, hands on, inquiry based
- encouraging students to invent their own algorithms
- definitions and proofs gone missing
- books downgraded or missing
- promoting a false dichotomy between basic skills (bad) and conceptual understanding (good)

"Sifting through the claims and counterclaims, journalists of the 1990s tended to portray the math wars as an extended disagreement between those who wanted basic skills versus those who favored conceptual understanding of mathematics. The parents and mathematicians who criticized the NCTM aligned curricula were portrayed as proponents of basic skills, while educational administrators, professors of education, and other defenders of these programs, were portrayed as proponents of conceptual understanding, and sometimes even "higher order thinking." This dichotomy is implausible. The parents leading the opposition to the NCTM Standards, as discussed below, had considerable expertise in mathematics, generally exceeding that of the education professionals. This was even more the case of the large number of mathematicians who criticized these programs. Among them were some of the world's most distinguished mathematicians, in some cases with mathematical capabilities near the very limits of human ability. By contrast, many of the education professionals who spoke of "conceptual understanding" lacked even a rudimentary knowledge of mathematics.What I have learnt from this:

More fundamentally, the separation of conceptual understanding from basic skills in mathematics is misguided. It is not possible to teach conceptual understanding in mathematics without the supporting basic skills, and basic skills are weakened by a lack of understanding ..."

In war language is distorted as the advocates take a good idea and turn it into propaganda. Understanding will not come through repeating slogans like constructivism, either for or against. It will only come by delving more deeply into the actual curriculum and evaluating it from these points of view:

- does the author understand mathematics deeply?
- does the author understand child development, what is age appropriate?

I became confused about this myself. This article:

Basic Skills versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education by professor H. Wu.

The title is great, the author is a maths professor, his writing is open and transparent, he makes some valid criticisms of some aspects of discovery learning being taken too far, he makes a brave effort to make algorithms more understandable but falls down on the age appropriate child development criteria. I changed my initial favourable evaluation after being helped out by this comment on the squeakland list.

The evaluation process is not easy. To be good at both maths and child development takes some effort. The real danger is that you'll end on the wrong side of a fuzzy war.

## 5 comments:

I think in some places the fuzzy maths wars have spilled out into a fuzzy science war.

This post from Stressed Teacher was one I came across when I was working a (still un-written) piece on the value of rote learning.

The Stressed Teacher post describes how unrealistic demands wrt experimental design are placed on students and teachers are not allowed to provide direct guidance. It does seem to me to be a clear case of misguided "discovery learning". What I call "good constructionism" (desperate terminology) does require scaffolding to be in place. Since "good constructionism" requires deep thought and is hard to design correctly then its going to happen that a lot of what goes under the name of constructionism or discovery learning is bound to be deficient in some way. This ends up fueling the false dichotomies and a "back to basics" movement which calls for more rote learning.

from stressed teacher:

"Under our new School-based Practical Assessment (SPA), students are now asked to design their own experiments to verify certain Physics formulas or concepts. Instructions are vague, and no table is provided for students to write down their data.

Based on the vague information provided, students now are expected, within one hour, to come up with their own experiment, carry out the experiment, analyze their data and draw certain obvious conclusions.

Under SPA, the Science teachers serve mainly as facilitators. We are not allowed to guide the students, since each SPA session is supposed to be carried out under exam conditions. Yes, it also means that every student does his or her own work. No group work allowed."

Note, however, that IMHO Summerhill was more about having kids stop being afraid of adults in the context of the UK cultural background.

Thanks,

Andres.

This list is interesting. I'm surprised by the de-emphasis on algebra. As a programmer I've used it frequently. I've used it even in daily life.

I remember reading some teacher comments some years ago that basically argued for this kind of teaching, that it was more incumbent on teachers to teach things that kids will actually need to learn for life, rather than "pie in the sky" stuff like Calculus. A common refrain from them was "When I was in school I had to learn X. It was totally useless to me. It was boring. I never used it again."

What it might be is a "pedestrianizing" of high schools. In the U.S., high schools used to be known as "preparatory schools", as in "preparing for college". I don't know what the ratio is now. About half of my high school graduating class went to college almost 20 years ago, but only a third of them were expected to graduate from college with a degree. So most students taken as a whole didn't get through college. I suspect what's happening is high schools are recognizing this fact and are just consigned to it: Most of our students are not going to, or graduating from college. Let's educate them with that in mind.

Bill Gates made the rounds a couple years ago calling the current high school system in the U.S. "obsolete". He made clear he wasn't just making a hyperbolic statement. He said, "Even if every high school in the country worked as it was designed, it still would not be good enough," for educating today's children for tomorrow's jobs. When he said "as it was designed", he meant that the high school systems were designed with the idea that only a certain percentage of students were expected to go on to college. He said the ratio needed to be a lot higher. So he argued for drastic reform, not mere tweaking, though I never got the details of what he had in mind.

Re: Stressed Teacher

This reminds me of other discovery-based methods. It's unfortunate that some people have taken this up as a religion: all discovery all the time; as if adult teachers don't have a role in a child's education at all. I've always liked classes that have discovery as an element in them, which allow me to learn my own way, but I agree with Stressed Teacher that if the entire class was that way I'd probably be frustrated as well. I've mentioned this to you, Bill, before. I believe Mark Guzdial uses the constructivist method in his books on Squeak. I think that's how he describes it. His approach is to start out with teaching tools, and some examples, and then "lets you go" to solve some problems he gives you. The problems are like goals to reach. He gives you some mental tools and templates to work with before you do this. His approach promotes more of an exploratory model, where you go down some dead ends, backtrack, and try something else. You learn something every step of the way, stuff you didn't expect to learn. The difference with Squeak is you have something to explore, not a blank piece of paper. It provides content and some constraints.

An approach I saw used (rarely) in school in the 1980s in junior high, high school, and college is a more gradual approach. The beginning of the class starts with strong guidance from the teacher. As a knowledge base is built, the teacher starts backing away, allowing the students to explore the subject more on their own. I remember I liked this approach.

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