Are Mathematicians Uncomfortable With Ambiguity?

If you follow me on Twitter, then you know I’m an idiot.

Let me rephrase.

If you follow me on Twitter, then you know I loathe Buzzfeed.

And if you follow me on Twitter, then you also know I sometimes share Buzzfeed articles anyway.

Which means at some point I saw a link to a site that I know makes me angry and I thought “Eh, I’ll check it out anyway.”

I refer you to the first sentence.

Anyway. Right now, Buzzfeed quizzes are fairly popular on Facebook. (Another reason to be annoyed with both of those sites. Besides, any good “What Parks and Rec Character Are You” Quiz would have known that I am RON FREAKING SWANSON not Larry-Jerry-Gerry-Gary Gurgich. Sheesh.)

I recently saw one pertaining to math floating around. Would You Pass School Maths Now? This particular quiz was either meant to shame you into forgetting minutiae about middle school mathematics or give you a falsely inflated sense of superiority as you share your score with your facebook friends, further reinforcing the idea that “math people” are smart and everyone else can suck it. Or hope for extreme brain trauma.

Here’s what I mean. Take a look at the feedback for your chosen answer to the 7th question about correlations. Right or wrong, the explanation is, “I mean, it’s basically a straight line.” Yeah, man! DUH. A straight line. There is no explanation about why it matters that it is a straight line, only that apparently its important. Now I understand this is a short quiz and lengthy mathematical explanations that give the quiz taker a deeper understanding and better appreciation of math isn’t the point and it isn’t gonna get the click-traffic that Buzzfeed is looking for with this stuff. But that is exactly the problem. And I’m not trying to shame the author this quiz, who is on staff for Buzzfeed. This junk bugs the crap out of me as math educator. These kind of quizzes aren’t too different from those images of basic arithmetic problems that people will share that are essentially the social media math equivalent of “gotcha!” journalism that are based on that fact that people have been trained to evaluate algebraic expressions by memorizing a specific set of “rules” of “order of operations” without really understanding what they are doing when they perform operations on the symbols. But I want to talk about that more in a moment.

I took the quiz. (And received a perfect score. Neener neener, kiss my math degree, etc, etc.) What bothered me the most was the explanation for the correct answer to the 6th question, which was along the lines of y2y3=? When you chose your answer, the explanation simply said “Correct. Add the indices. 2+3=5.”

There are two things I dislike about that explanation.

The first was the use of the word indices. The little snarky math pedant in me went, “Ha-HA, you ignorant click-baity website, the word for those is ‘exponents’, not ‘indices’! Indices are used to keep track of ordered numbers.” But then I was unsure of myself so I googled the term and then turned to Twitter.

I learned from @Mythagon and @redorgreenpen that while the usage I espoused was common in the US, other countries use both terms. The UK in  particular–if you notice the title of the quiz used maths instead of math. I think that being specific with our terms, that a certain notation communicates a certain meaning or idea, in math is important. Especially in math education. To me, the words index and indices denote certain ideas about lists or ordered numbers and give me nightmarish-y flashbacks to linear algebra (we had a poorly written textbook). This seems to be what @evelynjlamb was thinking as well. But this preference most likely comes from my US education and therefore I cannot say absolutely that this terminology is better. After all, mathematicians are comfortable with notational ambiguity in other contexts such as using the letter z to denote the set of all integers, the normal distribution, or a complex number. So while I would prefer to say exponents has a meaning separate from indices, I cannot fault the author for using a term that is commonly accepted where she grew.

But even so, I think ambiguity in math education is dangerous.

Let me rephrase.

I think ambiguity in teaching notation in mathematics is dangerous.

I’m all for giving students problems that are, as Dan Meyer says, perplexing. Problems where the strategy to find the answer isn’t obvious or easily found and there may be multiple paths to a solution that may or may not be correct. That kind of ambiguity in math education is great. (But this is related to the second reason I dislike the explanation.)

I’ve seen articles before that tell teachers not to say “cancel” when they are teaching students how to manipulate algebraic expressions. I understand the reasons and I think those reasons are excellent–we want students to understand what is going on when we say “cancel”–most often we have multiplied by a number’s multiplicative inverse (the reciprocal) to get the multiplicative identity (which has always been and will most likely continue to only be 1 for anyone who hasn’t taken an abstract algebra course) or we added a number’s additive inverse to get the additive identity (0). It’s good for students to understand what they are doing when they divide both sides of 3x=11 by 3, but cancel is convenient to say and it is hard for me to get away from it. This may be because it was what I heard growing up and learning math or it might be that I explain a lot of basic algebra problems and cancel isn’t really a bad way to describe what is happening. In any case, I have made a habit of repeating what is actually happening whenever I am tutoring a student and I say “cancel” or some other synonym.

To get the point of this subpoint (I’m sorry. I ramble. If you follow me on Twitter made it this far in the post then you know this), when we teach students to work with algebraic expressions, we need to help them make sense of what they are writing down or reading. Many of the students I help don’t understand that the numbers and symbols convey a specific meaning. A sentence I hear often is “I can never remember which one is which” in reference to the ordering symbols, < and >. Students struggle with interpreting the meaning of 10 < x. They often get it backwards or will have it right one line and wrong the next. I tell them to literally read the sentence left to right as they would as speakers of English and many other languages. “Okay, so this symbol (< ) always means ‘is less than’. So what does (15 < x -1 < 24) say?” A co-worker of mine likes to say “Math has a certain specificity of language”. We need to train students to be fluent in communicating their own mathematical ideas through symbols and decoding the ideas of others.

This takes me back to Buzzfeed’s explanation of the answer to the question y2y3=? and the second reason I dislike it, which I will talk about in Part 2 of this post.


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