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).