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Prakash Chandrasekaran
Works at Nanyang University
Attended Chennai Mathematical Institute
Lives in Singapore
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Prakash Chandrasekaran

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(sleepy, probably incoherent post)

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.)

---

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.
My students have lost their grip on mathematical reality.
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OMG, if I could give 10 +1s to this post, I would!
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 😉☺
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Watch this balloon-like drone 'absorb' and deliver objects

http://flip.it/wRFKh
Festo's adorably-named FreeMotionHandling drone is a helium-filled balloon that can move through the air and pick up as well as drop off objects.
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Balloon mail - the latest parcel delivery service 👌
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"There are only four circles here and they don't touch. C'mon brain you can do this." — From https://twitter.com/Sci_Phile/status/546089757755703296

See also the discussion at https://plus.google.com/+TimothyGowers0/posts/PYETJvHzAxH where it appears younger children have less trouble seeing it clearly for what it is!
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Thomas Yang, creative director at advertising agency DDB Singapore, has found a new use for bicycle tyres, but not in a way you would expect.
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+Google​ the new launcher icons are horrible! 
We're still digging into the new Android N preview to see what little things have changed, and one of the most immediately apparent tweaks is the new home... by Ryan Whitwam in Android N, News
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IDK... The existing design didn't preview more than 4 apps anyway, now it's 3/4th lesser. Don't think it's much of a problem as you've the folder name and can see the first (main) app. What's more important is the animation involved in leafing through the folder and choosing the app you want. I imagine it like using an expanding file where swiping up or down moves through the apps, previewing the one stopped at. Tapping on the preview should then launch the app, tapping elsewhere on the screen cancels action.
Can't comment on looks as that's subjective. Functionality is more important, at least where I'm concerned 😊
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Internet billionaires flew 14 scientists to Silicon Valley last week to turn them into “superheroes,” into “rock stars.” Tech investor Yuri Milner, who created the first of the Breakthrough Prizes two years ago and co-sponsors all of them, called them the “Oscars of science,” and the black-tie affair honoring the...
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Fresh Guacamole: The Shortest Film that has ever been Nominated for an Oscar
Fresh Guacamole by PES is the shortest film that has every been nominated for an Oscar. It is animated using pixilation a stop motion technique. I guess some of you know it already as it has so many views but it is still beautiful to watch even if you have already seen it.

Fresh Guacamole by PES

#Oscar   #PES   #shortfilm   #stopmotion   #beautiful  
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Have him in circles
157 people
Vipul Naik's profile photo
Sanjay Ghosh's profile photo
Kate Winston's profile photo
remish minz's profile photo
Kriti Chidambaram-Kharade's profile photo
Vineet Kumar's profile photo
Ramen Ghosh's profile photo
Pathan Shajahan's profile photo
Anshul Khandelwal's profile photo
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Currently
Singapore
Previously
Bangalore - Paris, France - Chennai - Chennai - Trento - Paris - Bordeaux
Work
Occupation
Researcher / Consultant / Technologist
Employment
  • Nanyang University
    Research Fellow, 2014 - present
    Rolls-Royce @ NTU Corporate Labs
  • Multicore RTOS Project
    Consultant, 2013 - 2014
  • Caliverse
    Founding Member, 2013 - 2014
  • ABB Ltd
    Associate Scientist, 2012 - 2013
  • CEA
    Post-Doc, 2010 - 2011
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  • Chennai Mathematical Institute
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Prakash Chandrasekaran's +1's are the things they like, agree with, or want to recommend.
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TuneIn es la verdadera radio. Descubre, sigue y escucha lo más importante para ti, desde la colección más grande del mundo de estaciones de

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This app is a front-end for the Android file hosting site, Goo.im. It allows for downloads of ROMs, Google Apps packages, kernels, apps, and

ixigo indian rail trains irctc
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The best Indian Railways / IRCTC train info. The only ads-free rail app ! ** Winner of the 2013 Mobby's Award for the best app. **Remember t

Hungry Shark Evolution
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***Jogo do ano de 2013!*** (Prêmio TIGA) Experimente a vida como um Tubarão no maior teste de sobrevivência de todos os tempos. Gráficos 3D

Métropolisson | Janol Apin Photographe
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La photographie est un grand labyrinthe où les regards se croisent. Janol Apin nous invite à un voyage souterrain parfois candide, souvent d

Are investigators to be blamed for policy paralysis?
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Corruption of the kinds that have been uncovered in India have primarily exposed the deep rot of crony capitalism in the country

Calculator (CyanogenMod)
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A calculator that tries to include graphing and matrix math without losing its simplicity. Slide left and right to access more panels.Requir

Flipboard: Your News Magazine
market.android.com

Flipboard brings together world news and social news in a beautiful magazine designed for your Android phone and tablet. Once you pick a few

Google Search
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Google Search app for Android: The fastest, easiest way to find what you need on the web and on your device.* Quickly search the web and you

Chrome Browser - Google
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Browse fast with the Chrome web browser on your Android phone and tablet. Sign in to sync your Chrome browser experience from your computer

Gmail
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Gmail is built on the idea that email can be more intuitive, efficient, and useful. And maybe even fun. Get your email instantly via push no

Google Nose BETA
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The new scentsation in search. Coming to your senses: go beyond type, talk, and touch for a new notation of sensation. Your internet sommeli

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Know, share, and map your world.

I analyzed the chords of 1300 popular songs for patterns. This is what I...
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For many people, listening to music elicits such an emotional response that the idea of dredging it for statistics and structure can seem od

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I work in an office with no windows and I have no idea what the weather is like now. Tell me if it is raining outside, because I'm too lazy

It's very interesting concept and execution. But, quite short to stay back in the island to watch.
Public - 5 months ago
reviewed 5 months ago
Dining in the dark is hard to describe, and one has to experience to understand. It's pitch black, except for the tiny red dots of the IR cameras, and a very faint sign of a glow from the kitchen. The food is very good, but the portions are rather small. They tend give you more time between courses, which would be good if you are not very comfortable with the dark.
Public - 5 months ago
reviewed 5 months ago
It's a small but decent mall with ample food options. Also houses a NLB branch, which is a great plus. For larger/weekend shopping one would still need to visit a larger mall.
Public - 5 months ago
reviewed 5 months ago
5 reviews
Map
Map
Map
One of the very kid friendly malls.
Public - 5 months ago
reviewed 5 months ago
The food is just awesome and service is superb. And, their philosophy of pay what to feel like, is to be experienced to understand.
Public - 5 months ago
reviewed 5 months ago