# Teaching math: Balance and discrete topics

I am a big fan of balance in teaching mathematics. I don’t much appreciate the extremes that either the constructivists or the back to basics folks are currently offering us. And, honestly, most math teachers, most good ones, find ways to balance between the two extremes.

Let’s remove some discrete topics from K-12 mathematics

So in response to my old new math post, when e wrote: “I just don’t understand why the concept of balance seems to be foreign…” I was mostly inclined to agree with her. But there is a bit more here.

In this case there is a question of whether the topics should be taught at all. Topics from discrete mathematics were not traditionally part of the K-12 mathematics curriculum in the United States.

(more below the fold –>)

Fundamentally I am uneasy with teaching probability to little kids, at all. I would advocate teaching set theory, and young, but not to great depth, and reinforcing with little bits of set, subset, empty set, intersection, and union each year, but without advancing the topics. I am uneasy with teaching stats, at all, to little kids. I object to teaching major chunks of logic, but advocate a reasonable unit to go along with geometry, perhaps a year early, or at the start of the year.

I think that by throwing bits of every discrete topic into the curriculum, and dividing them into tiny pieces each year, we are contributing to the “mile wide, inch deep” feel of US mathematics instruction. Taste all, master none. This is the one place I may oppose balance.

When I spoke of balance I had in mind more the way we teach and what we deem important in a sense of skills vs conceptual understanding, formalism vs handwaving, proof vs examples, drill vs problems that require more than just a recall, etc. I guess I was thinking more of how than what. I do agree that “mile wide, inch deep” is not what we should aim for. However, I am not sure that throwing all the topics one can think of into each grade curriculum should be called balanced curriculum :) Maybe a hodge podge would be a better word.

So which ones should be removed? What a hard question! No one wants to let go of anything.

So I started. Take out all probability. Take out all statistics.

That would be a controversial start, no?

Not for me :) Well, maybe I should clarify what you’re saying before I commit. Do you truly mean all of probability and statistics out of all the grades?

yup. take them out. Maybe introduce them in a course beyond geometry…

It seems to me there are some interesting developmental questions here…

Probability and statistics are once of the most important concepts for educated adults to understand, in part because the results aren’t always intuitively obvious. But kids have to develop a certain trust in the logic of math before counter-intuitive results will make sense (and some never do develop that trust).

My sense is that before age 10 or so, very few kids have much ability to deal with abstractions. And I have to say that the difficulty small children have with basic logic has been one of the shocks of parenting :-)

But, watching my daughter’s education (she’s in 5th grade now), I like the approach of introducing concepts a bit at a time. I learned fractions in a couple of very concentrated doses in 5th and 6th grade, with greatest common factors, least common denominators, and improper fractions and reciprocals all appearing the the space of a couple of months, and I have a feeling some of my classmates never managed to make sense of math after that.

My daughter has gotten fractions a bit at at time — the concept in 2nd grade, simple addition of fractions in 3rd, etc, and she’s comfortable with them as just ordinary numbers.

The probability she’s done has all made sense to her — and had some connection to the reality of board games and dice. But it probably shouldn’t be one of the core standards for 5th grade math.

— Rachel

I just read the Data Analysis and Probability standards (well, the abbreviated version). I almost agree with you. I’d toss out almost all of it. I think I would leave “Understand and apply basic concepts of probability” in 6-12, but would put it all somewhere in 9-12. I further think that anybody who reads papers today, and I am hoping we agree that we want our students to be informed individuals, should be able to navigate various statistics and data that are presented (especially since it’s not always clear that the data supports the writer’s claim). I would imagine that could be achieved without heavy machinery. To be honest, I don’t know that my college stats class covered what’s in k-12 stats standards.

Well, you’ve got my attention.

What do you propose for teaching set theory? Would this be a tool for understanding algebra later on? If taught as a separate topic, at what grade would you advance it and how far?

I had very good look this year teaching set theory to my son from a 1966 SMSG text. We covered empty set, null set, closure, set equality, as well as union, intersection. It was very different from anything that he had done and he liked it a lot. It will be continued next year.

This topic, what ought to be removed from the math curriculum, was the topic of a 1980 article in Mathematics Teacher by Zal Usiskin (the UCSMP guy). It was reprinted in this year’s 100th anniversary issue. I think it was chosen because people recognize that we’ve gotten too broad and sacrificed depth, but no one really knows what to do about it.

I grabbed the PDF of the article in case you’re interested:

mrc,

I am just about positive I read the Usiskin piece when I was in grad school. For the record, I disagree with almost all of his recommendations (though I remember liking them back then)

He is laying the ground for a semi-academic mathematics… math for those intent on not being engineers, or studying significantly more math…

For sets, Myrtle, probably exactly what you have done, but I wouldn’t push it much further, just refresh it periodically.

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