Impossible Learning: watch someone trying to learn calculus from scratch

David Wees retweeted this from Jared Cosulich:

It turns out Impossible Learning is a just-started-last-month blog where Jared is trying to learn Calculus and post about his struggles. It’s terrific and you should read it. See him ask the perennial question When Will I Use This?

And I immediately found myself saying “come on, when am I ever going to have to find the limit of this random equation”.

I felt like I was back in High School again.

But seriously, why is this one of the first things I’m directed to learn when I want to know more about Calculus? Why is it so hard for me to find some practical applications of this material? I know there is value in understanding the abstract math, but I’d like to balance that with at least some understanding of how this works practically…

Or his attempt at explaining the power rule:

I don’t think this actually counts as a proof, but it definitely made the “Power Rule” click for me a bit more. Basically it’s saying that the derivative of a square (x²) is two lines (2x) and the derivative of a cube (x³) is three squares (3x²).

So for a square to get a tiny bit bigger you need to add on two lines (one to the top and one to a side). Similarly for a cube to get a tiny bit bigger you need to add a square to three sides (e.g. top, right, and front).

Counting puzzle from the first US puzzle championship

My students had fun with this one today. Part d is what showed up in the actual championship.

One night, in an attempt to combat insomnia, you begin counting (1, 2, 3, …), but you decide to do it digit by digit. As you go along, for example, the 15th digit you count is the 2 of the number 12.
1 2 3 4 5 6 7 8 9 10 11 12

a.) What’s the 50th digit you count?
b.) What’s the 100th digit you count?
c.) What’s the 1000th digit you count?
d.) What’s the 1,000,000th digit you count?

Commentary on “Wrong Answer: The Case Against Algebra II”

There’s a new article by the author Nicholson Baker that is not raising as much a fuss as “Is Algebra Necessary?” from The New York Times last year, probably because it’s at Harper’s behind a paywall. Also, as I write this children are fleeing from algebra all over magazine stands:


The title strikes me as odd, given that the general argument is aimed at all algebra, and the text constantly references algebra as a whole rather than the quite US-specific version of Algebra II. (As far as I’m aware no other country has such a thing; most of them integrate all forms of mathematics together.) What is Algebra II anyway, and where is the cutoff point where Nicholson Baker considers algebra to be too hard to handle? He hints at what he’d like as a replacement, which strikes me as a math appreciation analogous to music appreciation courses:

We should, I think, create a new, one-year teaser course for ninth graders, which would briefly cover a few techniques of algebraic manipulation, some mindstretching geometric proofs, some nifty things about parabolas and conic sections, and even perhaps a soft-core hint of the innitesimal, change-explaining powers of calculus. Throw in some scatter plots and data analysis, a touch of mathematical logic, and several representative topics in math history and math appreciation.

This seems like a truly random collection of topics, but it sounds like Mr. Baker is basing his list off the notion of using The Joy of x by Steven Strogatz as a possible text.

I have had some personal experience with using popular texts; when I team-taught statistics at the University of Arizona we used The Drunkard’s Walk, Fooled by Randomness, and Struck by Lightning: The Curious World of Probabilities. While it led to interesting discussions, there just wasn’t enough meat to have students do problem solving based solely on the text. The math from the books was purely a passive experience. (We still did just fine by augmenting with our own material.)

I still hold forward the absurd idea that students still solve math problems in a math class. If you’re designing a freshman mathematics-teaser course, I might humbly suggest Problem Solving Strategies: Crossing the River with Dogs, which has the virtue of steering away from algebra as the sole touchstone for problem solving.

Back to Mr. Baker’s attempt to define Algebra I:

Six weeks of factoring and solving simple equations is enough to give any student a rough idea of what the algebraic ars magna is really like, and whether he or she has any head for it.

Mr. Baker himself seems to have a confused idea of what algebra is like, but he’s not alone. (See: my prior post on what is algebra and why you might have the wrong idea.) Also of note: some countries don’t bother to factor quadratics. (I haven’t made a map comparing countries, but it seems to be continental Europe that ignores it and just says “use the quadratic formula”.)

I could see solving equations managed in six weeks, but in a turgid, just-the-rules style that would be the opposite of what we’d want in this kind of appreciation class in the first place. Just the concept of a variable can take some students a month to wrap their head around, making me disturbed by the notion that six weeks would be enough for a student to find “whether he or she has any head for it”. (This isn’t even touching the issue of just how much is internal to the student. I’ve heard dyscalculia estimates of up to 7% of all students, but excluding that group a great deal of the attitude seems culturally specific. Allegedly in Hungary [I don't have a hard research paper or anything, this is just from personal anecdotes] it is quite common for folks to say mathematics was their favorite subject in school and the level of disdain isn’t remotely comparable to the US.)

They are forced, repeatedly, to stare at hairy, squarerooted, polynomialed horseradish clumps of mute symbology that irritate them

The article’s invective along these lines makes me wonder again how much the visual aspect to mathematics is the source of hatred, as opposed to the mathematics itself.


This picture is from the Adventure Time episode “Slumber Party Panic” and is supposed to represent the ultimate in math difficulty. However, the math symbolism is strewn truly at random and there is no real problem here. I am guessing a good chunk of the audience couldn’t even tell the difference and for them, any difficult math problem looks like random symbolic gibberish.

This is related to another issue, that of bad writing. Here’s Mr. Baker quoting a textbook:

A rational function is a function that you can write in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions. The domain of f(x) is all real numbers except those for which Q(x) = 0.

I claim the above definition is simply bad writing, and a cursory check of the Internet reveals several definitions I’d peg as clearer in a students-who-don’t-like-math-are-trying-to-read sense:

Google’s dictionary:

any function which can be written as the ratio of two polynomial functions.


a function that is the quotient of two polynomials

Some random webpage I found at Oregon State:

“Rational function” is the name given to a function which can be represented as the quotient of polynomials, just as a rational number is a number which can be expressed as a quotient of whole numbers.

Hardy’s A Course of Pure Mathematics, page 38:

the quotient of one polynomial by another

Godfrey Harold Hardy admittedly goes on from there, but his text is written for mathematicians. High school texts have no such excuse.

In a similar vein, the article later quotes a 7th grade Common Core standard:

solve word problems leading to equations of the form px + q = r and p(x+q)=r, where p, q, and r are specific rational numbers

which I suppose is meant to be horrifying, but in this case the standards are written for the teachers and aim to remove any ambiguity. Here’s problems that matches the standard:

1. You bought 3 sodas for 99 cents each and paid 10 cents in tax. How much money did you pay?
2. You bought 4 candies for 1.50 each and paid $6.20. How much was tax?

The standards can’t simply produce many examples and gesture vaguely. They must be exact. Just because 4th grade specifies that students “Explain why a fraction a/b is equivalent to a fraction (n×a)/(n×b)” does not mean students are using variables to do so. (In case you’re curious what it does look like to the 4th graders, Illustrative Mathematics has tasks here and here matching the standard.)

There’s lots more to comment on, but let me leave off for the moment on this quote, because I’m curious…

Math-intensive education hasn’t done much for Russia, as it turns out.

…is this statement (in the last paragraph of the article) accurate?

The origami proof that the square root of 2 is irrational

Start with a single paper square:


Fold from upper right to lower left (colors added to both sides of the paper for clarity):


Follow up with one more fold:


Voila, a proof that the square root of 2 is irrational.

(Mind you, there is some reasoning involved, but what’s the fun in giving that away? Start with “if the square root of 2 is rational, then there is some isosceles right triangle where the sides are the smallest possible integers.”)

This proof first appears in slightly different form in 1892. The paper folding version is from Conway & Guy in 1996.

Guess the year of the quote

Let’s say plus or minus 5 years:

Teaching is more than telling and explaining, and learning is more than imitating and memorizing. During the last 60 years teachers of mathematics have gradually sensed that, above all else, their pupils should learn the meaning of mathematical terms, principles, operations, and patterns of thought.

EDIT: Answer in the comments.

The difference between game and drill

So in my last post I opined that the optimal mathematics game in the Tiny Games spirit should “incidentally have some mathematics in them and are the sorts of games one might play even outside of a mathematics classroom.” That led to some confusion.


Let me try a do-over:

During a game, when the primary action of the players is indistinguishable from doing traditional homework or test problems, it is a gamified drill.

Gamified drills are not always bad. However, they’re not the sort of thing I’d say counters the notion “that students must learn and practice the basic skills of mathematics before they can do anything interesting with them.” They are what Keith Devlin calls a “1st generation educational game”.

There’s lots of gamified drills. It’s easy to do: just take what you normally would do in a math problem review and tack on a game element somewhere (for me it’s usually Math Basketball). To be integrated the primary actions of the players will require using mathematics in a way that is linked with the context of the game. That 1-2 Nim requires understanding multiples of 3 is inextricable from the game itself and not interchangeable the same way Math Basketball can be easily switched to Math Darts.

Tiny Games, mathematics edition?

So there’s a Kickstarter project closing today which has me wondering about mathematics potential.

Tiny Games: Hundreds of real-world games, inside your phone.


The concept here is to have games suited for different settings that can be described in only a few sentences.


Could one make an all-mathematics variant — mathematical scrimmages, so to speak? The only games I could think of offhand in the same spirit as Tiny Games were some Nim variants and Fizz-buzz.

1-2 Nim (for two players): Start with a row of coins. Alternate turns with your opponent. On your turn you can take either 1 or 2 of the coins. The person who takes the last coin wins.

Fizz-buzz (for a group): Players pick an order. The first player says the number “1″, and then the players count in turn. Numbers divisible by 3 should be replaced by “fizz”. Numbers divisible by 5 should be replaced by “buzz”. Numbers divisible by both should be replaced by “fizz-buzz”. Players who make a mistake are out. Last one in wins the game.

Anyone have some more?

EXTRA NOTE: One condition I’d add is the games need to work as games and not as glorified practice. “Challenge a friend to factor a quadratic you made” meets the “Tiny” but not the “Game” requirement.

EXTRA EXTRA NOTE: Dan Meyer asks “Aside from the counterexample that follows, what are the qualities that make Fizz-Buzz and Nim gamelike and not, say, exerciselike?” In both cases the games incidentally have some mathematics in them and are the sorts of games one might play even outside of a mathematics classroom. Even though Wikipedia claims Fizz-buzz was invented for children to “teach them about division” (?), my first encounter was from The World’s Best Party Games. (This still doesn’t totally answer the question, I know. A related question is: what is the difference between a puzzle and a math problem?)


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