In their rush to implement Universal Design for Learning (UDL), a federally approved framework intended to maximize learning outcomes for *all* students, schools have hired gymnasts, carpenters, and political philosophers to provide multiple representations of the equals sign.

According to UDL, “An equals sign (=) might help some learners understand that the two sides of the equation need to be balanced, but might cause confusion to a student who does not understand what it means. … An important instructional strategy is to ensure that alternative representations are provided not only for accessibility, but for clarity and comprehensibility across all learners.”

“I thought the equals sign was pretty clear,” said John Knap, a high school mathematics teacher. “Not sure why we have to represent it in other ways. Yesterday two gymnasts came to my class to perform double flips, and the kids were supposed to grasp that the two routines were ‘equal.’ They weren’t equal. One was a little faster than the other. And of course the kids wanted to see more routines. We couldn’t get to the lesson.”

Kelly McEwen, a carpenter hired to provide alternative representations of the equals sign at elementary schools in San Diego, expressed misgivings over the project. “The pay’s great,” she said, “but I’m not sure I’m doing the right thing. I’m supposed to show them the spirit level and pretend it’s just like the equals sign. It isn’t just like the equals sign. I end up doing a lot of qualifying and explaining, and the teachers and kids get anxious. Plus, they’re waiting for me to take out the saw, which I never bring, for safety reasons.”

Political philosophers seem especially disgruntled with the project. “I was invited to come to Inspiration Academy to talk about political equality,” reported Andrew Ravny, author of numerous books on William Hazlitt and Thomas Jefferson. “I accepted gladly. When I arrived, I was told to draw stick figures and put a smiley face between them to show that they were equal. I did this and went on to say that two equals three in such a scheme, because both numbers have the same inherent dignity and rights. I don’t think I’ll be invited back.” He chuckled grimly. “Which is just as well, since I need to focus on my next book.”

We had the pleasure of interviewing the inventor of the equals sign, Robert Recorde, whom we heard stirring in his grave. We asked him why he had chosen to represent mathematical equality with two parallel lines. He replied that he did it “to auoide the tediouse repetition of these woordes : is equalle to.”

But why the two parallel lines? we asked.

“Bicause noe 2 thynges can be moare equalle,” was his reply.

We thought that Recorde would be pleased to learn that his equals sign was now inspiring multiple representations. As we told him about the reforms, we watched and listened closely for his reaction. But he replied in cryptic verse and then faded from our midst:

*One thyng is nothyng, the prouerbe is,
Whiche in some cases doeth not misse.
Yet here by woorking with one thyng,
Soche knowledge doeth from one roote spryng,
That one thyng maie with right good skille,
Compare with all thyng: And you will
The practice learne, you shall sone see,
What thynges by one thyng knowen maie bee.**

“It’s a nice poem, but I’m not sure how it applies to classroom practice,” said Mercy Trout, director of instructional services in Boise, Idaho, who had accompanied us for the interview. “Is he saying kids should study math as math? Or is he saying all things are connected? What are the policy implications for school improvement?”

“Where are you, Recorde, and where’s the whetstone of witte when we need it?” cried another.

“I think he’s saying that if people *do* study math in a focused way, *then* they will see….” a third member of our party ventured. But it had grown dark and windy, and conversation turned to our flight back home and whether it would depart on time.

*The verse appears in Recorde’s preface to his *Whetstone of Witte* (1557).

## ashanam

/ September 3, 2012Many children really do not understand that the equals sign means that two things have the same value. They understand it as a “to do” sign. They also don’t understand operations as instructions to do something because those signs just sit there quietly until the equals sign tells you to do it. I am not kidding about this. It has to do with conventions for writing math in early grades, and that there are usually few discussions about equal quantities much before 5th grade. Equality really is not a difficult concept, but we teach math as almost pure procedure in elementary school and then are surprised that students have difficulty thinking of math in terms of a logical system later on. It’s our own fault.

## Diana Senechal

/ September 3, 2012Of course teachers should explain clearly, early on, what the equals sign means (and review it at various levels through elementary school). But one it’s explained, the students should work with the symbol itself. If teachers must

continuallyaccommodate students who don’t know what an equals sign means, then neither they nor the students will learn to work within mathematical language; they will depend on something outside of it. It’s like translating “je” for students in their third and fourth year of French.I agree that these things should be explained well at the outset (and occasionally after that)–but students should also take the explanations into their minds and make them second nature. When teachers are expected to represent even basic symbols in multiple ways, students gather that they need not become fluent; the picture, explanation, or demonstration will always be there for them.

## ashanam

/ September 3, 2012Mathematics is, in some ways, a language, and is learned in a way that is similar to how children learn any language: by how it is used. Children’s math problems typically look like this: 6 + 4 = , where the expected answer is 10. In reality, other perfectly valid completions to the equation are 7 + 3, 10-0, and 5×2. Ten is expected because the equals sign is being used here to mean simplify. The fact that it is consistently (mis)used in this way through several grades of math instruction means that children internalize the meaning of equals as being to simplify an expression, regardless of what they are told about the equals sign. Contextual meaning has a much greater significance for people than an explanation.

When students begin to work with algebra in upper elementary grades, and the equals sign really does just mean two things have the same value that may or may not be in fully simplified form, it’s terribly confusing So, although gymnasts and carpenters are a little bit silly as a response (and expensive, I would imagine, as well), the meaning of the equals sign is a real problem for teachers and students.

## Diana Senechal

/ September 3, 2012I see your point. Wouldn’t it make sense, then, to introduce students to a variety of problems using the equals sign, but to use the equals sign throughout?

The piece is satirical; schools aren’t really hiring gymnasts, carpenters, or political philosophers to represent the equals sign (as far as I know). But the UDL quotation is real, and I find it misleading. The equals sign needs no alternative representations. Students do need to learn what it means.

## Marge

/ August 25, 2014Do you know Rick Lavoie? Maybe check into what he’s written about educating kids.