Mathematics isn’t about finding answers. It’s about asking questions.
As a mathematician, here’s a question I usually find boring: What’s the answer?
Consider this manifestation of a sort of meme that wanders the internet:
The most likely intended answer is 73, just to get that out of the way. I’ll call this “Witch”.
This is an evolution of a type of worksheet properly popular in elementary school, such as this one. I’ll call it “Candy”.
In “Candy”, the idea is that each image stands for some numeric value, and the goal is to find the numeric values. So “green watermelon candy” is worth 1, “Swedish fish” is worth 5, and so on.
This is a fun way to introduce the concept of simultaneous solutions in a way that avoids letters for variables. The assumption is that each “green watermelon candy” has the same value throughout, a key assumption we want to carry over to traditional algebraic notation. I would much rather solve “Candy” than this equivalent:
This sort of system (five variables, five equations) is generally introduced in middle or high school, but “Candy” is marked (properly, I think) as a Grade 1–3 level problem. So yay!
But let’s go back to “Witch”. There are a few tricks that make this more difficult:
- The witch in the last line isn’t holding anything.
- The central image on the third line and the last image on the fourth line are doubled.
So how does the presence of these changes affect the mathematics?
That, to me, is a more interesting conversation than “what’s the answer?”
It’s fair to ask if either of these changes are “fair”. Do they make for a more challenging problem that shows the intricacies of mathematics, or are they frustrating “gotchas” that teach children that math is for trick questions?
The answer is: I don’t know. My gut tells me that the first change (where the witch isn’t holding anything) is a fun reminder to pay attention to details, while the second change (barely visible doubling of objects) is a “gotcha”, but that’s just my opinion.
Either way, though, what does it mean? How do we interpret the difference between the witch with objects and without? The most common assumption is that size or degree of visibility is irrelevant and that what we’re doing is assigning a numeric value to every distinct object. Also, when multiple objects appear in the same image, we’re to add their values.
Under these assumptions, a star wand has a numeric value of 7 (because there are three in the second row) and a broom has a numeric value of 3 (because there are four in the third row). A witch holding a star wand and a broom has a numeric value of 15, so a witch alone has a numeric value of 15–7–3=5. Hence, the last row is 3 + 5 * (2 * 7) = 73.
But what does that say about what numbers are, and what numbers are for, and what symbolic representations of numeric values are all about? Is there another way to interpret the difference between a witch with and without objects?
Let’s say that the star wand in the witch image is half the length of the star wand alone. Does that mean that the small wands have half the numeric value of the large ones? Or, since it’s a two-dimensional image, does it have a quarter the value (having a quarter the area)? Perhaps it has an eighth the value, since it’s meant to be a three-dimensional object (and hence our relevant consideration is volume or weight).
Perhaps combining images (such as the witch, broom, and wand) is a multiplicative process instead of an additive one. That would make the third line equivalent to x + x² + x = 12, that is, “broom” = √13–1, or about 2.6.
We could even decide “broom behind broom” means exponentiation, meaning that “broom” would be worth approximately 2.34.
This would naturally make for a much more complicated problem, but without some sort of agreement, those are possible assumptions.
So this is where the conversation gets interesting, and useful in a mathematics class: How do we come to these agreements? At what point can we safely stop making explicit statements and simply agree on default interpretations?
That’s a key stage in understanding mathematical notation, after all. So much of our standard notation involves assumptions that we rarely make explicit. Why does 2x represent multiplication while 2½ represents addition?
And then there’s this meme:
The relevant text of the image reads: “The Battle of English and Mathematics. Question: One rabbit saw six elephants while going to the river. Every elephant saw two monkeys going towards the river. Every monkey holds one parrot in their hands. How many animals are going towards the river?”
This one overtly says in the title that it’s intended to be as much about English as about mathematics, and I’m still seeing people on social media insisting they know the One True Answer.
Spoiler alert: There isn’t one. But instead of arguing about whether five or nineteen or some other number is “correct”, we could be talking about how this problem illustrates the challenge of writing mathematics story problems in a way that is simultaneously rigorous enough to avoid ambiguity and natural enough to seem meaningful.
Here are the ambiguities I’ve found:
- Since “Animals” is capitalized in the image, it could conceivably refer to the rock band, in which case the answer is zero. Psych!
- Did each elephant see the same two monkeys, or did they see different ones? I’ve seen people point to the word “every” as meaning something different than “each”, which it does… except, in this context, that’s a distinction without a difference. So there are somewhere between two and twelve monkeys (or possibly more that are unseen by any elephants, so we’re assuming that the question is asking specifically about animals mentioned in the story).
- Is the rabbit currently going toward the river? Does “going to” require that the vector of travel is pointing at the river, or does it simply require that the intended endpoint is the river? What about “going toward”?
- And speaking of intention, does “going toward” imply intention? If so, the parrots aren’t necessarily going toward the river. But under a more typical interpretation, they are.
- How many parrots are there, anyway? It’s certainly possible for two monkeys to carry the same parrot, inasmuch as not a lot of parrots will cooperate with being carried by monkeys at all. So there might be one parrot (jointly carried by the two monkeys), or there might be as many as twelve (one per monkey, with twelve monkeys). It’s also possible that at this moment of time, the uncooperative parrot has made a break for it, and is trying to fly away from the river while the monkeys hold on to it.
- We know nothing about where the elephants are going.
So assuming “animals” wasn’t meant to be capitalized, that’s a minimum of two (two monkeys deliberately going toward the river, while the rabbit wanders in the general direction of the river and the parrot is just along for the ride) and a maximum of 1 + 6 + 12 + 12 = 31. We’re not even clear how many animals are mentioned in the story, which is somewhere between 1 + 6 + 2 + 1 = 10 and 31.
What does it mean? How can we interpret it? What assumptions have we made? What assumptions do we need to make?
These are more important questions to ask than “What’s the answer?”
When we’re speaking of proper mathematical notation, there’s often little room for interpretation. -1² = -1, even though there’s a strong urge to think -1² = 1; this is a convention about which there is little dispute.
These “problems”, though are just as much about interpretation of symbols and story problems as they are about pure mathematics, and as such they’re fruitful opportunities to discuss mathematical representation and even to prepare students for real world modeling.
So stop stressing about “the answer” and reflect more on the question.