### Prakash Chandrasekaran

Shared publicly -I've been thinking about this since yesterday. The author finds it amusing that his students prefer to write 1/3 as 0.3̅ (might not render properly in this font: that's 3 with a bar on top, i.e. standing for 0.333333333…).

It wasn't clear to me yesterday, but after some discussion I realize there are two questions here:

1. Is there a reason in general that students prefer to use the decimal representation?

2. Is there any good reason, for the specific fraction 1/3?

It is a well-attested experience of many people that they were doing well in mathematics until they reached some point at which they seemed to hit a ceiling, and suddenly everything after that was hard. (Asimov has said it (his ceiling was integral calculus: http://mathforum.org/kb/message.jspa?messageID=7004907), IIRC Martin Gardner has said it too.)

It is also known that many schoolchildren struggle with algebra, and many with fractions. While thinking about this post (specifically about the question 1) I thought I had an epiphany about one aspect of it, which may explain both students' discomfort with fractions (usually learned first?) and with algebra (usually comes later?): it is that students have not yet become comfortable with the abstraction that a "number" exists, and is not merely an adjective denoting a magnitude. (This is one of the early abstractions in mathematics: to go from "three apples", "three houses", "three words" etc. to an abstract "three". Of course it's far from the earliest abstraction: I remember reading somewhere, probably in Lawvere and Schanuel's book, that when a child goes to kindergarten and learns that other children have grandmothers too, and generalizes from grandma being a specific someone, to being a relative anyone can have, the child is already doing mathematics.) If a student never gets comfortable working with a number without being conscious at every moment exactly how big it is, they'll already have trouble with numbers like, say, 78 (definitely harder to visualize than, say, 5), then go on to have trouble with fractions like 3/5 or 2/11, and finally struggle a fair amount with algebra, where one starts treating letters as if they were numbers, and constructing nonsensical (to them) expressions like x^2 + 2x + 1. Of course everyone gets used to them eventually, but if they didn't already seem natural when they were first introduced—if the idea appeared to come out of nowhere—then it's hard to ever get really comfortable with it.

This perhaps explains why students want to immediately convert 3/8 to 0.375 or √563 to 23.7276…. There is another explanation, which is that often in maths class, being already a traumatic environment for many (it can seem one has to carry out some arcane ritual of steps and come up "the right answer" for the teacher's approval (though if you understand it well you can love maths class precisely because you

*don't*need the teacher in order to know that you're right), doing well in maths class is treated by some as a stand-in for intelligence, etc.), maths problems often turn into stimulus-response learned behaviours: you see two numbers and you write them one below the other, draw a line, manipulate these two digits to get that, etc (Tim Gowers has an example: "if one asks children a question such as the following: a number 35 bus pulls up at a bus stop and 8 passengers get on; what is the age of the bus driver? A large percentage of children, their minds numbed by years of symbol manipulation, will give the answer 43. This is a tragedy: rather than being trained to think, these children have been trained to do the opposite" via https://plus.google.com/+TimothyGowers0/posts/4bkfusUoXot), and particular, expressions like 3/8 or √563 or even 1/3 seem to them the "stimulus", an invitation to solve a (nonexistent) problem and do

*something*with it, until it's flattened down neatly on a line.

(There are of course cases, such as when it's the final answer to a real-life problem, that one

*should*seek the magnitude, as a sanity check and for whatever purpose the problem was being solved. But even for doing real-world mathematics or physics one often has to go through abstraction; you can't do much mathematics without buying into it.)

(I think the hypothesis I'm proposing is even falsiable by an experiment: figure out how often students are uncomfortable with fractions and try to write a fraction as its decimal representation immediately, and see if there's a correlation with being uncomfortable with algebra. But I guess the correlation may stem from general mathematical difficulty and it's hard to design controls for it.)

(There are also other reasons for using decimals of course, proposed in the comments on the post: that they're easier to compare, that this is the representation calculators give, that there's a veneer of sophistication about them, etc...)

Having said all this (which was new to me), I only now understood Ben Orlin's other point, which is that, even though there are a lot of good reasons in general for using decimal notation, very few of them apply to 1/3 specifically: its magnitude is not exactly hard to visualize: see the "pie chart" in the image attached to this post. Its magnitude is in fact

**more**immediately clear to the eye from "1/3" (if you don't like circles, take the line segment from 0 to 1 and divide it into three parts) than it is from 0.33333….

So the "floundering students seek a sense of magnitude" theory, even if it explains a general preference for decimal notation, doesn't really explain a preference for 0.(3) over 1/3 in particular. At this point I have to go back to the stimulus-response model, and guess that maybe students don't look at 1/3 and visualize a quantity, but instead see it as a division problem: they have in fact, over their years of converting fractions into decimals (e.g. with their calculators) actually

*lost*the habit of visualizing fractions as magnitudes, and are instead preferring 0.(3) not because its magnitude is clearer but because it's in a form easy for them to manipulate. Also perhaps they have had more facility working with decimal numbers than with adding and subtracting fractions? (Similarly students may not be seeing 75% as three-fourths: there's someone I know, a typical graduate of a typical engineering college, who after being told the autorickshaw fares had increased by 20% didn't even want to try calculating what the new fare for Rs 30 should be.)

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Over the years I've slowly come to realize that one's understanding of mathematics proceeds not like a bucket filling up with water (either smoothly or in spurts), but perhaps more like someone walking, two legs alternating. These legs are conceptual understanding and computational fluency (e.g. what fractions really mean, and how to do perform arithmetic operations on them): sometimes understanding precedes fluency and sometimes fluency precedes understanding (https://shreevatsa.wordpress.com/2009/12/01/is-it-important-to-understand-first-2/), but you can best proceed to the next topic only when both of them catch up somewhat, else you'll be dragging one leg behind.

Returning to the "mathematical barrier" I mentioned earlier, in my case I've never understood Galois theory despite several attempts. (We studied it for an entire semester at college; later I got books about it and read them; the problem being solved I'm motivated about so that's not the problem.) Now I think I can guess why: I don't have a good conceptual grounding in group theory in the first place. I can tell you the definition of a normal group, and given typical undergraduate exercises accompanying the introduction of normal groups I can pull out my well-honed middle-school algebra skills, play with the definitions, and probably solve them (I did get through those early semesters somehow after all). But I think it's never really sunk in. (I still remember the feeling when they were introduced that it seemed an arbitrary definition and not very natural from my experience—now I know that it's not at all an arbitrary definition; it's actually very, haha, normal—I was just not sufficiently prepared for them.) Perhaps after enough computational fluency the understanding will come… meanwhile it's not enough to know the definitions and be able to solve problems; one has to actually think like a group-theorist; there is such a thing as thinking like a probabilist and probability theory is not simply measure theory of spaces with total measure 1 any more than number theory is the study of strings that terminate (or something like that: Terence Tao); and all I can say to conclude is that teaching mathematics is hard and it takes a great mastery of abstraction to be able to generalize from one's mathematical difficulties and apply it to help the students find a way out of theirs.

Reading the original article, all the comments and Shreevatsa's view I gathered one thing - the number line is more primal in our understanding of maths than sectors in a circle (for odd-numbered denominator) or segments in a rectangle (even-numbered). In other words, the order in which I was taught these at school is correct!! Hurrah to our education system 😉☺