## The Many Uses of =

The equals sign is a sophisticated, subtle tool.  You may not think of it as such, but it is deep.  It is hard to learn and use and understand.

I first understood the challenge while teaching a calculus class.  I had two expressions, something like (x – 2)(x + 3)(x – 7) and (x – 2)^3 – x + 50, and our goal was to show that they were equal.  I took one, and I simplified it to x^3 – 6x^2 – 13x + 42, a chain of equals signs spread across the blackboard.  I took the other, and simplified it to the same thing.  So I proudly declared that they were equal, and thanks to the magic of my boardwork, the two identical expressions had ended up next to each other.   I drew an equals sign between them, creating a giant chain of equals signs connecting the two original expressions.  I thought this was a very clever presentation.

They didn’t get it.

I’d heard before that this might be a challenge.  They’d all been drilled in elementary school on problems like 5 + 2 = ?.  They thought of equals as meaning the outcome of an operation, which is why they felt comfortable using it sloppily.  (You know what this looks like: asked to compute “five squared plus two”, they write 5^2 = 25 + 2 = 27.)

I thought, then, that it was time to correct this misunderstanding.  So I speechified at length about how they had been tragically mis-taught about the equals sign.  It actually means that two things are the same thing!  I pointed to (x – 2)(x + 3)(x – 7) = (x – 2)(x^2 – 4x – 21) and said “these two things are the same thing!”  Then I pointed to (x – 2)(x^2 – 4x – 21) = x(x^2 – 4x – 21) – 2(x^2 – 4x – 21) and proclaimed, “these two things are the same thing too!”  Since the first expression was the same as the second, and the second was the same as the third, the first and third must be the same, and a pointed and said it.  I went on, through all the equals signs, until I had discovered that the two things on opposite sides of equals signs were the same thing.  I discussed why I didn’t like the way they did a problem like “five squares plus two”.  I was proud of my improvisation, but somehow, it didn’t seem like the students had the great moment of realization (and repentance for their mathematical sins!) that I wanted.

I say it again: the equals sign is a sophisticated, subtle tool.  Despite my oratorical skill, I had failed to bring together a deeper understanding of all the possible uses of equals.  There are at least four:

• As the outcome of an operation (used in school, but mathematically the wrong way to look at it).
• To declare that two things are equal all the time, as in 4(x + 2) = 4x + 8.  This is an equvalence.  There is a hidden universal quantifier.
• To declare that two things will be equal for the right value of x, as in 4x + 2 = 3x + 9.  (This is disambiguated from the previous case by explicitly saying, “Find x so that…”  To a mathematically sophisticated reader, it is clear that this is the same = sign, just with an existential quantifier first.  This is not clear to someone just learning mathematics.)
• As a definition, such as f(x) = 3x^2 – 4x.

Without any explicit guidance, we expect students to recognize these different situations.  To recognize quantifiers and definitions without any discussion of what “equals” really means.  All this while learning how to “solve” these problems.  Also without discussions of the other subtleties of equals: symmetry, transitivity, reflexivity.

Embedded here is also the understanding that two things can look completely different and can be the same thing, can be equal.  30/3 and 10 are different representations of the same number.  7 + 3 and 5 + 5 are all different representations of that same number.  All of these things are equal!  They are also equal to 2x when x = 5.  The mathematical symbols alone are not enough; the words nearby must be used to interpret them.

Worse, uses of the equals sign get conflated in different settings.  For example, suppose that you are trying to find an x so that 4x + 2 = (x + 2)^2.  You might write that (x + 2)^2 = x^2 + 4x + 4.  One of your equals signs is only true for a few values of x; the other equals sign is true for all values of x, all in the same math problem.  Imagine how confusing it would be if you wrote 4x + 2 = (x + 2)^2 = x^2 + 4x + 4!

The same thing can happen when you are defining a function.  You might say “Let f(x) = (x + 2)^2 = x^2 + 4x + 4”.  In one line, two different uses of the equals sign.

To me personally, these uses of = are all the same thing, stemming from the same definition.  To a student just learning algebra, however, there are layers upon layers of subtlety, and they must be addressed explicitly or students will not truly understand the mathematics they are doing.

## What should we teach young students?

Recently, Paul Tough released a new book about non-cognitive skills (link to a highly-recommended This American Life episode where Tough discusses the book).  Tough emphasizes the importance of things like grit, the ability to deal with setbacks, the ability to postpone gratification (whose importance can be seen in the remarkable Stanford Marshmallow Experiment), curiosity, self-confidence, and so forth.  The book is meant to present a new view of education: that, like KIPP schools, we must focus on teaching kids these essential skills for success.  It also emphasizes that kids generally have to learn these skills quite early.

There was also recently a New York Times story about admissions for the specialized high schools, emphasizing the word gap: that kids with higher socioeconomic status just plain hear a lot more words growing up.  Like any social science experiment, it’s hard to formally establish a causal link, but it’s intuitively clear that a word gap would impact their vocabulary, comprehension, and communication skills from kindergarten on up.

In E.D. Hirsch’s review of Tough’s Book in Education Next, Hirsch emphasizes that Tough has only part of the story.  He says that vocabulary—and also basic information (such as if a student can locate Africa on the map)—is the major determinant of a student’s outcomes.

Everyone agrees that good early education is essential.  Many kids do start kindergarten well behind their peers.  They’re behind in reading and math.  Even in early childhood, there’s an illusory IQ gap (I say “illusory” because it becomes smaller when adopted students from low-income backgrounds are raised in more affluent homes).  Underserved students do worse in school and often act out more.  In the end, they have just an 8% graduation rate from college.  (Compared to the population at large, an also-depressing 33%.)

So what should we teach kids?  Should we teach them non-cognitive skills, or should we focus on building their vocabulary?  Is the bigger challenge that underserved students have few non-cognitive skills, or less basic knowledge?  What are the raw materials that allow students to learn better later?

Tough’s answer is to focus on non-cognitive skills.  The New York Times editorial seems to suggest just the opposite: focus on building vocabulary.  Hirsch would add that we must focus on cultural literacy, which enables communication and further learning.

A better answer, though, is obvious: the “all of the above” strategy.  There’s evidence that all of these kinds of early learning are important, and so we should design early experiences that develop all of them, either through direct work with students or through parent education, like what Geoffrey Canada does with the Harlem Children’s Zone’s Baby College (another great This American Life episode).

I claim, further, that this whole argument is a red herring.  The best way to teach both non-cognitive skills and basic knowledge is to do both at once.  In fact, I’ve never seen a content-less effort to build skills succeed.  You learn to study hard by studying something hard.  You learn to solve hard problems by tackling a real problem—but real problems require serious background knowledge.  In other words, these different skills are not in conflict at all.  They are best taught together, with conscious awareness of all the elements that are being incorporated into a curriculum or program.  You still need a rich environment to impart simple facts that do not come from traditional studying: the location of Africa or a large vocabulary.  But beyond that, deep material learning should be paired with skills building, and any other way is a major failure of education.

A real education requires learning facts and learning skills to learn more facts.  While it is possible to design a class that teaches facts without teaching skills to learn more facts, it’s dumb and it doesn’t work in the long term except for a select few students who pick up the necessary skills.  Cognitive and non-cognitive skills must come together, and by putting them together in a conscious way we will develop the strongest educational experiences for students.