The Power of Simple Representations

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Keith Devlin is one of today's most important popularizers of mathematics, having written a column for many years called Devlin's Angle for the Mathematical Association of America. Many years ago, I found his "Letter to a Calculus Student", and made it required reading in my Calc I classes (although it works in any Calc class). He expounds on the beauty to be found in calculus, and on that most beautiful equation of calculus, the limit definition of the derivative:


In his 2017 article The Power of Simple Representations, he points me to another mathematician he feels that we should get to know: James Tanton, who made a video called Exploding Dots!.

Please read the two articles, and especially watch the video (and learn the math salute!). Then respond on our Canvas discussion.


  1. Devlin, in his Letter to a Calculus Student, waxes poetic about the limit definition of the derivative (quoting the poet William Blake):

    To see a World in a Grain of Sand
    And a Heaven in a Wild Flower
    Hold Infinity in the palm of your hand
    And Eternity in an hour

    Your challenge: find some poetry to describe another aspect of calculus (or any math/stats, really). Or something poetic -- sometimes it's a good joke!


  2. Devlin says this in his article The Power of Simple Representations: "Of course, there is a sense in which representations do not matter to mathematics." Clearly he goes on to change his mind (or perhaps James Tanton does it!). Can you think of a simpler representation of something you feel is presented in an overly complicated way in mathematics/statistics/cs/etc.? (You'll hear James Tanton talk more about this problem in his video. You might think about what he has to say about this before responding -- I found it thoroughly thought provoking!)
  3. James Tanton does various remarkable things in this talk -- the breadth is amazing. What strikes you as the most interesting or exciting thing he does with "Exploding Dots"?
  4. "The math salute" is an interesting problem of topology (much like my infinite bacon). What do you think of it? Did you figure it out? Can you think of similar sorts of interesting tricks you've learned? (The joking definition of a topologist is someone who can't tell their doughnut from their coffee cup...:)


Question 1 Response

Most people came up with some pretty good jokes or poems to capture the artistic math side. Most people referenced poems that put into perspective the poems central idea and make it easier for them to understand. Hearing complex mathematical ideas in a lighter, more easier to understand form really allows one to understand and comprehend it faster and more accurately so that they can succeed at the topic.

Question 2 Response

Simple Representations:

A wide variety of ideas were brought up. Madison Goodwin talked about the object oriented programming model, and how the model she showed help understand object oriented programming and thinking, as apposed to functional programming, Chelsea Debord talked about how some people were confused about square roots, before realizing they are simply the reverse of squaring a number, and Blake Weimer even showed a GIF to help visualize the Pythagorean formula.

As the conversations continued, the idea that schools and the education system make concepts too complicated, and causes more children to struggle. Using these simpler representations not only will allow more students to learn concepts, but also can help gain a deeper understanding of them.

Question 3 Response

James Tanton's wonderful video on exploding dots was very interesting, and many people in the class enjoyed different parts about it.

Chelsea Debord mentioned how she thought the subtraction portion was interesting, and how he introduced tods as the opposite of dots.

A couple people, like Shawn Heuesman and Madison Goodwin enjoyed James Tanton's perspective on math, and how he thinks even answers that might not be framed in the "usual" way are still correct, and can be valued. The especially liked his quote that says "All correct math is correct", which drives home the point that solutions can be found in many different ways, and as long as it is correct, the way you got to the solution in completely valid.

Many other people, such as Blake Weimer and Madison Goodwin enjoyed how he showed you can divide with this dot method, and they were surprised by its effectiveness.

Austin Paolucci enjoyed the section about "making it happen, but accepting the consequences", which is when you can add however many dots to a box to help you get a solution, as long as you balance it out with an equal number of tods.

Question 4 Response

Interesting Tricks

One common trick multiple people shared was how to calculate multiples of nine. One way they calculated them was using the finger method where if you had to calculate 9 x 5 you would put down the fifth finger from the left and the amount of fingers on the left side of the finger you put down is the number in the tenths place and the amount of fingers to the right is the number in the ones place. The other way is a simple logic to the multiples where the tens would be n-1 where n is the multiple of 9 and the ones place is n-10 where n is the multiple of 9.

Another common trick is variations in the box method. This box method can be used to calculate high value multiplication as well as polynomial multiplication.

One more common trick that was shared by multiple people is where a chocolate is cut up and put back together only to have extra chocolate somehow leftover. Even Professor Long mentioned how mesmerizing it is!

The rest talked about the argument of how many holes a human has (according to him and the video he referenced, we actually have 4! or maybe 9? We'll never know). As well as a multiple guessing method which guarantees a better chance than the usual 20% (or 25% depending on the amount of choices obviously).