Unified Modern Mathematics

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Does anybody have any experience with the Unified Modern Mathematics course from the 1960s and 70s that is based on set theory?

http://en.wikipedia.org/wiki/Secondary_ ... ment_Study

Download the books in pdf form from

http://eric.ed.gov/?q=unified+modern+mathematics

That approach to teaching mathematics turned out to be a failure. Ramping the kiddies up on set theory before they developed mathematical maturity and intuition turned out to be a bust.

ruveyn

Set theory in the 4th grade ... It made perfect sense to me. I used to blurt out the correct answers without raising my hand. The other kids hated me for it, and the teacher told me to shut up.

UMM was a failure because it seemed to rely on an "intuitive" approach to maths - it seemed to be expected that everyone already knew the subject, and that all that was necessary was to present the material and the students would automatically "get it".

AFAIK, only one in seventy did, and the teachers already thought of that kid as a problem child.

Someone told me once that this program was conceived as part of the US response to the mathematics teaching system in schools in the East block, which was known for its set-theoretic approach. Is that indeed the case?

In other words, it is better to learn the basic foundations before progressing on to abstractions? I would completely agree with that.

Without a solid grasp of the foundations, it seems like it would be really tough to grasp the abstract because one would not really understand the reasons for the abstract.

UMM is undeniably a course written by mathematicians for mathematicians rather than for people who use mathematics in scientific or real world applications. Set theory is the language of UMM which runs through most of the subsequent topics. This means that if students fail to master set theory – complete with all the symbolic notation – then they have no chance of being able to understand anything else. The course includes some A Level topics such as combinatorics and proof by induction; deeply abstract concepts like groups and fields; and highly technical terminology like axiomatic affine geometry.

What age group was UMM targeted at? Was there an examination at the end of the course?

Set theory is the foundation of mathematics.

Yes and no. It's the foundation for real mathematics, but it is not the foundation that young students need to worry much about to learn their simple arithmetic. The foundations for learning mathematics are the very concrete type of mathematics found in arithmetic and then expand into more and more abstractions.

Yes, in that sense I think I agree with you.

The reason that I say "I think", is because for me personally mathematics would have been a lot more appealing in school if it had been presented using abstractions. Here in my country it is presented in a very concrete way, using many examples from everyday life. To me this obscured the structure that lies underneath and it made me dislike mathematics quite a bit. Only later, in university, did I fall in love.

With me, it was a big help to learn arithmetic first. If we had started with abstractions, I suspect that I would not have had much interest because it wouldn't be clear that it was tied back to the real world. We did have some simple set theory as we went, but so simple that I doubt that the teachers even knew it was set theory.

Imagine learning math by starting off with abstract algebra. Instead of talking of adding 1 and 2 to get three, you would learn all about groups and the properties of groups. You'd learn how there was exactly one identify element in a group operation without having any background to tie the notion to in real life. You'd learn about Albelian and non-Albelian groups. These may be relatively simple, but I think they would be more difficult to pick up without a solid basis in everyday math.

That makes perfect sense, I agree. Of course one should first be proficient in basic arithmetic before moving to abstract structures such as groups. What I'm opposing, is the fear for abstraction that seems to exist from primary school all the way to the final grades of highschool, at least in the Netherlands. It leads to a misrepresentation of mathematics as a grab bag of tricks useful for making calculations without giving even the slightest hint of the abstract structure underneath.

This is bad. In my opinion, at least at some point during their school career, students should be exposed to examples of axiomatic thinking. I'm convinced that this is of use to everybody, not only to those that already plan to study math in the first place.

That's a good point. By the time that students are in high school, if they have a decent introduction to math in lower grades, they should be ready for some abstractions.

By the way, if you want to see something truly beautiful, check out the US Mathematical Talent Search for high school students at http://www.usamts.org/. They have all the tests going back for year and those are pretty challenging tests. If you join the mailing list, you get notifications of the new test when it is released.

That looks very interesting, thank you. I like the set-up as described on the main page, found it noteworthy that this competition does not primarily reward speed. Excellent initialive, I will certainly have a closer look.

I like to print out the problems from the Mathematical Talent Search and work on them while eating at restaurants.

Wouldn't a Scottish café be more appropriate?

The biggest problem with New mathematics is the way that it dived into abstractions before the kids had mastered basic arithmetic and geometry. It was also written in a way that ignored the chronological development of math over the centuries. Mathematicians who devised modern math topics during the 19th and 20th centuries already had a reasonable knowledge of most traditional math topics and were fully proficient in arithmetic. The end result was that New math wasn't just a case of adding new topics into an existing math course, it was a complete rewrite of the math course starting from the most elementary level that required a completely different way of thinking to that of a traditional math course.

Did UMM result in students having difficulties with physics which requires a high ability in basic algebra and some knowledge of trigonometry and calculus?

Something that hasn't received the attention that it deserves is that a 9 year old from the 1960s and 1970s would most likely have found set theory very abstract because there were very few things in the real world that he could relate it to, but a 9 year old today could relate set theory to relational databases which are taught in British primary schools as part of the ICT course.