By William McCallum, IM President

Big ideas are popular in mathematics education, and you can find many lists of big ideas on the web. Some are more thoughtful than others, and I can see how some might be useful for organizing a curriculum. But few of the ideas I see in these lists really get me excited, or really capture what I love about the subject. I am a big fan of small ideas; like intricate joints in a fine piece of carpentry, small ideas often evade the eye, but are crucial to the beauty and structural integrity of the finished product. I’d like to mention a few of my favorite small ideas.

Using a letter to stand for a number. When I first was shown this idea as a child, I thought it was amazing. I used to love those “think of a number” puzzles where you always knew the answer was going to be 17: think of a number, add 3, double that, subtract 2, halve the result, subtract the number you thought of, add 15. Your answer is 17. I discovered that you could make up your own puzzles of this sort by letting $x$ stand for the number the person thought, building an expression that represents the steps in the puzzle, and then making sure that the expression is equivalent to single number, with all the $x$ terms cancelling out. The puzzle above would be represented by $\frac12 (2(x+3)-2) -x + 15 = 17.$

Now, this small idea sometimes occurs in lists of big ideas under the heading of variable. The idea of a variable is sometimes treated as a somewhat mysterious concept, linked to the idea of varying quantities, such as time and distance, in a dynamic relationship. This is far too much weight for such a small and simple idea to carry. I think of the word variable as linked to the idea that the letter you are using can take on various different values, rather than to the idea of a varying quantity. The latter idea gets us into the territory of relationships and functions, where the simple idea of letting a letter stand for a number continues to be useful, because we can express the relationship between the input and the output of a function by choosing two letters, one for the input and one for the output, and writing an equation relating them. Functions are indeed a genuine big idea, but there’s no need to let that get in the way of appreciating the simplicity of the small idea.

Exponential notation. Whoever thought of writing $2^5$ instead of $2 \times 2 \times 2 \times 2 \times 2$ was a genius. Choosing to write 100 as $10^2$ and 1,000 as $10^3$ brings a cascade of notational innovations. First we discover the rule that $10^n \times 10^m$ is $10^{n+m}$. Oh, that means that 10 must be $10^1$, because $10 \times 10^2$ is $10^3$, so to make the rule work I need to say $10$ is $10^1$. Similar reasoning tells me that $10^0$ must be 1, because according to the rule multiplying by $10^0$ just adds 0 to the exponent, and so doesn’t change anything. Extending this rule also gives a meaning to negative exponents. And extending the other rule, $(10^n)^m = 10^{nm}$, gives a meaning to fractional exponents. And once we have a number in the exponent, we can use the previous small idea and let a letter stand for that number, leading to the idea of an exponential function.

Completing the square. By now you must think I’m crazy. Completing the square? How could anybody love that? Generations of schoolchildren have been tortured with endless worksheets on completing the square. If you think of completing the square as a method, then I am inclined to agree. Thinking of it as an idea changes everything, however. What is the idea? That you can transform a quadratic equation $ax^2 + bx + c = 0$ to an equation of the form $(x+p)^2= q$. The latter equation is easy to solve: $x + p$ must be one of the two square roots of $q$. It’s a miracle that you can make that pesky $b$ disappear. I was once teaching this idea to a college precalculus class and a student put up their hand and said, “Oh, I learned a different method.” “What method was that?” “I was taught to do it using the $-b/2a$ method.” It’s sad to see this beautiful idea become a method.

There are many other beautiful small ideas. Thinking of 0 as a number, for example, is foundational to our base 10 system, because it enables us to write a number like 103 and think of it as $1\times 100 + 0 \times 10 + 3 \times 1$, living in the same family as all other 3-digit numbers and not some degenerate case.

Small does not mean inconsequential. Some of the small ideas I have mentioned here have huge consequences, so I suppose you could see them as big in the sense that they have a big effect. But they are consequential in the same way that a jewel might be consequential: simple but profound, and not overburdened by the panoply of other ideas that we might, in our desire for bigness, be tempted to attach to them.

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What ideas, big or small, excited you when you were a student of mathematics? ##### William McCallum
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Bill McCallum, founder of Illustrative Mathematics, is a University Distinguished Professor of Mathematics at the University of Arizona. He has worked in both mathematics research, in the area of number theory and arithmetical algebraic geometry, and mathematics education, writing textbooks and advising researchers and policy makers. He is a founding member of the Harvard Calculus Consortium and lead author of its college algebra and multivariable calculus texts. In 2009–2010 he was one of the lead writers for the Common Core State Standards in Mathematics. He holds a Ph. D. in Mathematics from Harvard University and a B.Sc. from the University of New South Wales.

## 4 thoughts on “The Power of Small Ideas”

1. Phil Daro says:

Lovely piece on small ideas. Shows how the coherence of mathematics often pivots on small ideas. The coherence is so valuable for students because there are fewer pieces to learn when the pieces attach to each other… like Bill’s simple “using a letter to stand for a number” pivots so sweetly to the use of letters to express functions. Nice work Bill, as usual.

2. Joe Swinehart says:

This was a great read, and I like the idea of focusing in on the importance of small ideas as well as zooming out to capture the big ideas as we so often do..
One small idea that I’ve appreciated is that we can see the infinite in the concept that any number or amount can always have one more added to it. Conversely, to make something infinitely small, any quantity can always (theoretically) be cut in half and shared between two people.
The idea is explained simply and eloquently in one of my favorite math texts, Hans Magnus Enzensberger‘s delightful book “The Number Devil.”
It’s a small idea to explain the biggest of concepts, and I always enjoy sharing that with students.
Thanks for the thoughtful post!

3. xiousgeonz says:

For me, the relationship between words and symbols: that “of” connected to “multiply” (14 percent of something… ) even if the answer was getting smaller, and “is” was about equal value…

4. Cathy Kessel says:

You can see use of a letter for a number as an instance of the heuristic of working backward—which is how the people who developed the idea of using a letter for a number saw it (Grabiner, 1995, pp. 85–86). Grabiner remarks, “To say ‘let x = ‘ the unknown, and then calculate with x—square it, add it to itself, etc., as if it were known—is a powerful technique when applied to word problems both in and outside of geometry.” (See https://scholarship.claremont.edu/cgi/viewcontent.cgi?referer=https://scholar.google.com/&httpsredir=1&article=1120&context=pitzer_fac_pub.)

How exponential notation is connected with working backward and use of letters for numbers may not be as obvious. In discussing mathematical notation of the 1600s, Florian Cajori says, “As long as literal coefficients were not used and numbers were no generally represented by letters. . . . [t]here was no pressing need of indicating the powers of a given number, say the cube of twelve, they could be computed at once. . . . Moreover, as only the unknown quantity was raised to powers which could not be computed on the spot, why should one go to the trouble of writing down the base? . . . But when . . . several unknowns or variables came to be used . . . then the omission of the base came to be seen as a serious defect in the symbolism” (1993, pp. 344–345) and notations that indicated both exponent and base received more attention. (See https://archive.org/details/historyofmathema031756mbp/page/n245.)

Completing the square can be seen as working backward, e.g., if x^2 + 3x came from squaring x + a, what would a be? Squaring x + a (calculating with it as if it were known, which involves raising two different unknowns to a power) gives x^2 + 2ax + a^2, so a must be 3/2.

So . . . you might see the three small ideas as part of one idea: using symbols to work backward.