When teaching an entry level math class [Algebra, Precalculus, Calculus] I sometimes like to ask:

Does \(8 + 3 = 11\)?

These students have already had enough exposure to arithmetic to not be surprised by this question. Initially, there is some hesitation in that they think that the question is somehow a trick question, but pretty quickly, they can’t find anything wrong and agree that \(8 + 3 = 11\).

Next question:

Does \(11 = 8 + 3\)?

And now, I receive interesting responses. First, and for a while, there is a general hesitation to answer this and we all wait through the awkward silence. I don’t attempt to coax an answer out other than periodically repeating the question: Does \(11 = 8 + 3\)?

Eventually, someone feels brave enough to respond. The responses I have received are often of the following form:

- Yes, \(11 = 8 + 3\) but it could also be \(7 + 4\) or \(6 + 5\) or …
- No. Eleven can’t equal eight plus three. It’s the other way around.
- Yes, but not always.

And so begins a mini-discussion about implicit biases in our [English] language.

The problem, if there were one, is that there is an implicit *causality* bias when we speak out the mathematics. When we write \(8 + 3 = 11\), we say “eight plus three equals eleven”; that is to say that when eight and three are added together the *result* is eleven [eight and three caused eleven]. Thus, when we write \(11 = 8 + 3\), we say “eleven equals eight plus three” which can sound strange since eleven isn’t causing eight plus three. This is a subtle, but pervasive bias in how we speak and explain [elementary] math concepts. If there are doubts, run this experiment for yourself. Ask your class if eleven equals eight plus three and see what the responses are when compared to eight plus three equals eleven.

In any case, I point this out to students and explain that both statements: \(8 + 3 = 11\) and \(11 = 8 + 3\) are the same and they are both true, not just sometimes. Further, I offer an alternative way to understand statements of equality such as \(8 + 3 = 11\). Rather than speak it as “eight plus three equals eleven”, we can say “eight plus three and eleven are equal to each other”. [And perhaps you may have picked up on yet another bias / overuse in the word “and”, where some students can reasonably interpret the “and” to be synonymous to addition. Thus, proceed with care!] The point though is the emphasis on “equal to each other”.

Extensions of this bias show up in Algebra classes and beyond, the general convention is to write something like this: \(x = 3\) instead of \(3 = x\) — the left-hand side is what we want and the right-hand side is the result. Though I was told of one story of a classroom of students who felt more comfortable with \(3 = x\) than the traditional \(x = 3\). In other words, the right-hand side is the conclusion or the answer to the question “what is \(x\)?”. One leads to another. [As a large parenthetical, this bias can be reinforced still further in programming languages where the equal sign does not actually function as an equal sign, but rather as an assignment operator. Interestingly, students who have never programmed before often remark in a math class that \(3 = x\) isn’t correct because “three can’t equal \(x\)”, but rather \(x = 3\) is correct since “\(x\) can indeed equal three”. Again, an implied causality.]

There are a few more points regarding using the phrase “equal to each other”. One is that it has a different ring to it and gets around the perceived causality of one side being made to equal the other. But another is that it calls to attention another standard way of communicating — left to right [or more generally directional]. In English, and many other languages, we read from left to right. It’s then natural to read \(8 + 3 = 11\) from left to right and be seduced into believing that there is a difference with \(11 = 8 + 3\). But the transition that we generally fail to impress on our students is that reading mathematics *can be* bidirectional [or not in any particular direction!]. Equality is a good starter example of this. But consider also something like $$5 > 3$$ which we are inclined to read as “five is greater than three”. While this reading is correct, “three is less than five” is also correct. Yet, many students find this discomforting. Interestingly, the expression $$3 < 5$$ is readily accepted as "three is less than five". But again, reading it in the other direction as "five is greater than three" becomes problematic for many students. Some of this probably can't be helped in any other way but through regular emphasis on the nature of reading mathematical symbolism.
Here is an expression that no matter how well we were to try, we couldn't sensibly read it in the same way we would read English text $$\Big(\sum_{n = 1}^{k}(n + 3)^{4}\Big)^{\frac{5}{2}}$$
One way to read this expression is "the five halves power of the sum from \(n\) equals 1 to \(k\) of the fourth power of the sum of \(n\) and three", which is more of the pattern of "outer to inner and bottom to top". Instead, we often end up saying the above expression piecemeal, perhaps like this: "take the sum from \(n\) equals one to \(k\) of \(n\) plus three to the fourth power and then raise that whole thing to the five halves power". But that statement translated back into math symbolism has multiple interpretations. As a for instance, was it \(n + 3^{4}\) or \((n+3)^{4}\)? [Also, notice that \(n = 1\) for the lower boundary of the summation is actually an assignment since we're beginning our counter which we name to be \(n\) to begin at one, rather than "\(n\) and one are equal to each other".]

### Language

Mathematics probably doesn’t fit a linguistic definition of “language”, but we can still view the translation and synthesis of mathematical symbolism through a language acquisition lens. While we can get away with inaccuracies in our non-mathematical speech or writing — that is there is not a loss of meaning — mathematics is less forgiving. A statement like \(x – 3 = 7 = 4\) is nonsensical mathematically, but completely sensible for the beginner student. They are writing their thoughts — namely, that the solution to \(x – 3 = 7\) is 4, where “is” is the second equal sign referring to the solution for \(x\). I bought orange juice and dishwasher detergent and drank it for breakfast. In both the Algebra problem and the previous sentence we lost reference to the object to which we were referring.

Notice that while I have argued that we read English from left to right, I also want to emphasize that we don’t process it left to right. We do have to employ some amount of memory to discern which objects are being referenced [direct vs indirect objects, for example]. This actually adds to the difficulty of processing mathematical statements. There are a lot of “possessives” as in the case of “the five halves power OF the sum from \(n\) equals 1 to \(k\) OF the fourth power OF the sum OF \(n\) and three”, which we could rewrite more confusingly as “\(n\) plus three’s sum’s fourth power’s sum from \(n\) equals 1 to \(k\)’s five halves power”. And I’m not sure if that actually makes any sense. The symbolism on the other hand, while being dense is devoid of inaccuracy $$\Big(\sum_{n = 1}^{k}(n + 3)^{4}\Big)^{\frac{5}{2}}$$ with the hard part being that we have to be able to read, translate, and understand the jumble.

Being conscious of the translational biases that can influence how we think about mathematics when we are parsing the symbolism would greatly benefit teacher and student alike. I encourage teachers, parents, and students to think about this when there is confusion about the symbolism or when “it doesn’t make sense”. Often times, “it” is generic and hard to pin down to one concrete aspect, which then leads to blanket statements that “math is hard” or “math sucks”. Instead, I would lean more towards attending to a broader confusion related to translating between the constructs of the dialect of mathematics and the student’s colloquially spoken language. I do not believe that there is enough of a linguistic focus on understanding mathematics.

Grace This Blog, Leave A Comment!