# The absence of number

One of my favorite conversations to have with my students (Grades 9-11) has quickly become centered around the concept of zero.

It first arose naturally last semester, when a Grade 9 student blurted out: “But 0’s not a number, it’s the absence of number!” (I’ve forgotten the context of this remark.) I remember a few other students laughing at this, and I know that this was unfortunately a negative formative experience for her, because she wrote about it in her end-of-year reflection. And yet, when I prodded further at a later point in time, I found that many other students, if not most, held the same belief. We had a whole-group conversation about it, talked about how 0 behaves in a similar way as other numbers in many ways (you can add it, subtract it, multiply it…), but still, I think a lot of students held on to this conception. This was especially clear when the students were solving equations and came to a problem such as this:

2x + 7 = 7

Had the problem been 2x + 9  = 7, they would have easily breezed on through. And yet, when it came to subtracting 7 from both sides in the above equation, many students hit what they felt was a wall: when you subtract 7 from the right side, nothing’s left! On assessments, some wrote “no solution” or simply did not go further. I wasn’t sure how to help these students go beyond this wall, besides repeating that “zero is just another number!” (in hindsight, I should have gone back to the “cover-up” method and asked them “what plus 7 gives you 7?”).

Today, the same question arose in my Grade 10 class in the context of a discussion about prime numbers. (I also posed the question in a Grade 11 class later in the day and received very similar responses, save for that of a very advanced student who brought up the notions of cardinality and measure, which both imply that 0 is a well-defined number.) After debating whether 0 was prime or not, I posed the question: “Is zero even a number?” Here are some of the responses I received from students of all skill levels:

• “No, it’s the absence of number!” (heard from a few students)
• “No – with other numbers like 2 or 3, you can multiply them by other numbers and their value changes, but not with zero. It always stays the same.”
• “No, it’s a placeholder – for example, in 10, or 0.1”
• “Google says it’s a number”
• “No, can’t divide by zero”
• “Yes, but it’s a weird number; it’s an exception”
• “No, it’s nothing!”
• “Yes. Like if you have 17 minus 17, that has to be a number, because 17 is a number. And 17-17 is 0, so 0 is a number.”
• “My brain hurts”

I found it really interesting that students echoed many of the same conceptions about zero that humans have held throughout history, such as 0 being simply a placeholder. (On the recommendation of several MTBoSers, I’ve ordered Zero: The Biography of a Dangerous Idea to learn more about the historical development of the notion.) The bolded response, though, blew my mind: Offered by a student who normally participates little in class discussions, it used the property that the integers are closed under subtraction to give the most concise, beautiful, compelling argument I have heard from a student so far for why 0 is a number. I’ve been thinking about it all day.

I’m not sure that other students found the argument as compelling as I did, though, and the question was left open, to be discussed again tomorrow. (We also dived into the problem of dividing by zero, when one student showed us how 10*0 = 5*0 implied that 10 = 5, if you divide both sides by 0! Another student suggested that something divided by 0 is infinity, which did sound reasonable when we tried dividing 1 by increasingly smaller and smaller numbers… Tonight’s homework question: “Try to figure out what ***** broke by dividing by 0. Either 5 = 10 and all the math we know is wrong, or something isn’t right…”)

Anyway, my question for the #MBToS is: What’s my move here? What might convince students that zero is indeed a real number (in the precise mathematical sense, and otherwise)… and how hard should I try to get them there? I’ve seen that confusion about zero has caused trouble when solving equations, so my gut says that this is an important concept to develop. At the same time, I love this ambiguous space we’re occupying, where an idea students have played with since their primary years still seems strange and up for debate. The following remark by Tracy Zager still comes to my mind in these kinds of situations since I read it last year: “Whatever I do, I’m in no rush to define this loveliness away.” Christopher Danielson agrees:

Thoughts? I’d love to hear about similar (or different) experiences and where you took these conversations with your own students.