Ranting about division in education and stumbling into an old friend while doing it

I hang out a lot on study aid forums these days which I enjoy, both as a means of continuously refreshing what I already know as well as keeping track of how the curriculum has been changing since I moved on to university studies. One of the hot topics these days in Swedish media is the discussion of the state of our schools and how we are falling behind in international rankings and so on and every year since 2008 there has been some half-assed attempt to ‘fix’ it by changing up the curriculum or moving obligations to and from teachers. The perhaps most major reform took place in 2011 when the gymnasium (highschool equivalent) curriculum was revamped. Something which mostly entailed ditching much of the classical geometry for some reason and making up the difference with more (elementary) calculus and making some areas of discreet math and number theory mandatory to ‘seek to smoothen the transition to higher education.

I don’t particularly agree with all of the priorities but at least it has meant that I get to work with helping gymnasium students with some new types of problems as the confused students bring their problems to the forums. One the things which have entered the standard curriculum are things like modular arithmetic and prime factorization and the students are  just awful at it. Now I too had problems with understanding residues and work with modular arithmetic at that stage but having moved on it really makes no sense at all. I mean you work with residues in elementary school for Christ sake. The first thing you do when learning to divide whole numbers is to separate a number into a quotient and a residue like 13 = 4 \cdot 3 + 1 although you tend to write it like 13/3 = 4 + 1/3 often just discarding the ‘indivisible part’ as an uncomfortable absurdity to be dealt with later. I think this desire to want to be able to divide everything is the principal reason why kids are often very willing to accept rational and by extension real numbers because they solve this problem for them. But in the process everyone seems to forget that integers aren’t just real numbers but form their own domain of logic and have their own special rules. This rush in education to move on to real numbers before really developing integers properly is understandable and it works out more or less fine but it is kind of pathetic that we end up with 16 to 18 year olds who can’t conceptualize euclidean division anymore even though it should have been the first thing they learned once upon a time. The main perpetrator is of course memory decay but I also suspect that calculators too play a part in this corruption of the concept of division as at a certain point and onward students aren’t asked to be able to do division by hand anymore and so division itself as an operation fades into obscurity.

Now I don’t actually think that a person needs to be able to perform difficult number arithmetic by hand and minds (I for one am terrible at it) but these things are at heart algorithms and the ability to understand and develop algorithms is priceless. Long division for example is a beautiful algorithm and understanding it requires one to apply it throughout ones education that one is given the opportunity to discuss the logic behind it.

That rounds up my little rant but consequently I felt the need to just playing around with division and trying to find more varied and hopefully good examples of problems which force a student to work out residues of division and actively work with decomposition of the form p = nq + r. Now I have never actually done any work with continued fractions since as far as I know they are completely ignored by physics and I’m far from good mathematician who has studies every math thing there is in detail but I had been looking at them lately since I have been interested in the representations of irrational numbers. Anyhow I found them to be an excellent example of a situation where one naturally finds the need to perform euclidean division.

So just to be consistent I should say that a continued fraction is a rational number written in a particular form

a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{\ddots + \cfrac{1}{a_n}}}}= [a_0;a_1,a_2,...,a_n]

For example

\cfrac{13}{17} = 0 + \cfrac{1}{\cfrac{17}{13}} = 0 + \cfrac{1}{1 + \cfrac{4}{13}} = 0 + \cfrac{1}{1 + \cfrac{1}{3 + \cfrac{1}{4}}} = [0,1,3,4]

A rational number can be rewritten in this way algorithmically and the relevant part is simply that in each step you’re doing euclidean division in the denominator with respect to the numerator as illustrated by:

\cfrac{a}{b} = \cfrac{a}{na + r} = \cfrac{1}{n + \cfrac{r}{a}}

Now that closes the book on this theme, the rest of this post should just outline my embarrassment and joy at accidentally chosing a particular fraction. So I mentioned I don’t work with continued fractions much? In fact never. Well instead of reading anything on them I decided to just play with it and see if I could figure out anything interesting on my own. (It is a weekend after all) I just wanted to see what some infinite continued fraction would turn out to be so I picked the most obvious infinite continuous fraction I could imagine and set out to work it out. I chose [1,1,1,1,...] because that just seemed reasonable somehow. I got to 8 ones ([1;1,1,1,1,1,1,1]) before I realized what it was turning out to be.

It’s kind of awkward to stumble into the one number I find to be the most missrepresented and overrated number in mathematics education on the first go at evaluating an infinite continued fraction. Makes me look like an idiot for calling it so.

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2 thoughts on “Ranting about division in education and stumbling into an old friend while doing it

  1. Continued fractions are one of those things that it seems just don’t quite fit anywhere in a standard curriculum. They don’t seem to lead to too much that’s useful in primary school, and they look too basic to be useful when you’re in college, so it sort of falls to people who run across them in their spare time and find them fascinating to do anything with.

    I just have the lurking suspicion that there’s something like a neatly stable method of solving partial differential equations or the like waiting to be discovered in continued-fraction methods. They just seem too accessible not to have some staggering usefulness too.

    • True, and I can’t see a reason to introduce a kid or adult to work with them unless he or she would find them interesting for their own sake. The original reason I have been looking at them lately however is in pondering different alternatives for characterising irrational numbers, the teaching of which I find rather unpleasant. Their existence is obvious enough as geometry generates lengths and areas which aren’t rational and the negative definition of ‘not rational’ is perfectly adequate for logical reasoning but it does not on its own give you much insight into what they are when viewed separate from the rationals. My personal preffered conceptualization are as the aperiodic decimal expansions because the they end up nicely filling the holes in the space of decimals left by the periodic rational expansions. I’m just wondering whether or not a kid would be more inclined to accept this description or that of infinite continued fractions. (There is also the constructive variant but it’s not that appealing for forming intuitive explanations to most people)

      I have seen analogs of continued fractions in quantum perturbation theory but it’s only a similarity in that’s repeated division rather than a number heretic thing.

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