## Tuesday, June 05, 2007

### Improving math education: a conjecture

When I lived and worked in Germany, I noticed to my surprise that my engineering colleagues couldn't do fractions. They had no concept of whether 9/16 was bigger or smaller than 35/64, and they couldn't easily solve "simple" problems such as

`  3          5  -  +  ------------  4     4 +    2             -----             4 + 3`

Of course, they could figure out that 9/16 is 0.5625, that 35/64 is 0.546875, and that the former is thus bigger than the latter, but they wouldn't immediately convert 9/16 to 36/64 and see it was bigger.

When I expressed my surprise, we discovered that they never studied fractions in school. We decided the only likely reason US students studied fractions in elementary school was to be able to deal with inches, gallons, and the like. From vague recollection, I must have spent the better part of a couple of years in elementary school arithmetic studying fractions plus memorizing how many feet in a mile, pints in a gallon, and teaspoons in a cup (not to mention conversion factors from those units to metric). By comparison, my European colleagues had to learn a set of metric prefixes, the names of basic units of measurement, and the universal conversion factor of 10 (just to show there's a Wikipedia page on almost anything!).

As it's often written that students in other countries are often, on average, well ahead of US students in math skills and that such a gap makes us less competitive in world markets, what if we switched to the metric system (or, more precisely, the International System of Units)? Would that make a year or two long hole in math education that could be filled in with more advanced topics? Would that help us in the USA catch up?

I don't know. I recognize that it's a challenging problem, and there is no silver bullet, not even in my idea. I would be curious to know if anyone has evaluated this approach.

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John F said...

Fascinating. So maybe fractions are just unnecessary when we can always convert to decimals. Yet fractions seem so fundamental too.

How about skipping fractions as you suggest, then adding them in during about 11th grade when those two years worth of fractions could probably be learned in about a month?

06 June, 2007 22:32
Bill Harris said...

Hi, John. I see your point, but think a minute. When do you really use fractions where decimals wouldn't suffice? When I went through the exercise, I came up with things about inches, gallons, and the like. The only other example I could think of had to do with hours, but the most we need there are quarter and half hours, and I suspect that's relatively trivial—no significant math involved in that. I finally came down to fractions just being a comfort concept, one I could get rid of if I switched to metric.

We do use fractions again for continued fraction expansions and for things such as Laplace transforms, but, as you say, the concepts become pretty easy at that age.

Thanks for the tip about the comments. I don't know how that happened; I must have clicked on the wrong thing. I've fixed it now.

26 June, 2007 22:22
Mom4MathMastery said...

For a great discussion of the teaching of fraction, H. Wu has a great discussion about this topic

http://math.berkeley.edu/~wu/EMI2a.pdf

In fact, his writing on the teaching of mathematics is articulate and of significant interest.

http://math.berkeley.edu/~wu/

06 July, 2007 12:02
Bill Harris said...

Mom4MathMastery, thanks for stopping by and leaving that reference. I skimmed the discussion on fractions, and the approach does seem intriguing. The discussion brought back memories of not knowing what a ratio was years ago when a teacher was trying to explain fractions.

I still wonder, though, if we'd be ahead by switching to the metric system and then skipping most work on fractions. While they intuitively seem useful, I can't figure out why except that I know them and I use them when figuring out cups, quarts, inches, feet, and the like. It seems like a great productivity gain in arithmetic: we can skip learning something, and we can skip all the conversions the rest of our lives.

06 July, 2007 14:31
Raybro said...

Bill,

I had the dubious pleasure of working in a cabinet shop for 2 years, programming a CNC machine to cut cabinet parts and millwork. Fractions are absolutely necessary in this case: there is just no sense in designing parts to .0004" (the limit of the machine) when a day's humidity change can change the overall part's dimensions by 1/32". Fractional tolerances which are standard in the wood industry are meaningless decimals, even to the metrically-trained. And when we dealt with 'modern' cabinetry, faceless frame cabinets, for instance, the tolerances were illogical collections of milimeters, as were wood thicknesses.

Another realm where fractions just make sense is acoustics. Harmonics are based on ratios, and fractions represent ratios in a way that makes comparisons sensible. If I told you that the relation between two harmonics was 1.25, it is unlikely that you would make the connection that the two harmonics make a ratio of 5/4, and that this meant you were dealing with the fifth and fourth harmonics, which are 5 times and 4 times (respectively) the frequency of the fundamental.

I wonder, if the tests which show so conclusively that the US lags so much in math were to contain questions involving fractions, if those other countries would shine so brightly.

15 December, 2008 06:56
Bill Harris said...

Thanks for your comments; you have come up with good uses for fractions, especially with acoustics.

I still want the SI system, though! :-)

15 December, 2008 19:24
Raybro said...

You can have SI, no problem. For acoustics, you still need fractions. It makes woodwork more complicated, but that can be lived with. To be honest, for woodwork, getting 3mm and 5mm drill standardizations is almost enough to make up for the vagueries of wood.

So SI is fine, but fractions gotta stay. Both your European coworkers and your American ones need to be a bit more flexible (and many of us already have!)

16 December, 2008 05:17