Wednesday, December 12, 2012

Zero and Negative Zero

Today a random 11th-grader asked me what the difference is between zero and negative zero. He sounded so sure that there was a difference, that for a fraction of a second I had to double-check my entire logical understanding of numbers before answering him. I asked him what the difference is between -2 and 2, and he said that they're "...of course different. They are opposites!" And I asked him what the '2' means. He said, "Well, they are both 2 units from the 'center'." So, "OK," I said. "And -0 and 0 are both how many units away from the 'center'?" He said, "Zero units, but in opposite directions!" We went back and forth like this for a bit until I convinced him that -0 and 0 are the same using the number line as a reference frame.

Then, my colleague comes in and this student asks the same question to him. I didn't say anything because I was a bit curious what my colleague would say. My colleague invoked reflection on the number line to explain geometrically why 0 and -0 are the same, which is exactly the same explanation that I had given! Afterwards, we were both amused. My colleague says, "You see, there are some things that all math teachers can agree upon."

But, we are both people who are comfortable reasoning through number lines and relationships of numbers. In the end, we are able to provide an answer to the kid that is logically sound and coherent with other concepts that the kid knows and understands. The same question could easily have come up in a classroom (for example, of younger children perhaps) where the teacher is multiple subject-certified and perhaps not quite as comfortable with mathematics as they are with other subjects. In that case, what systems can we put in place in order to support those teachers to answering conceptual questions such as this? (I realize that elementary-school age children do not typically learn about negative numbers, but the same types of innocent questions can still very easily arise, with other math topics that they do learn about.) We want to encourage questioning and robust reasoning in mathematics, and that mode of thinking should be instilled starting at a young age. What can I do as a math department head of a K-12 school, in order to ensure of this and to help all teachers feel equipped to answer conceptual questions from curious learners?*

*For example, the art department head of our school regularly models lessons in the elementary school, in order to show the teachers how to deliver art lessons using the same general approach as in the middle- and high- school. But, I don't feel confident that I can manage young children or that I would be equipped to explain concepts at their young comprehension level. So, if that is not an option, then what is??

What does your school have in place in order to support vertical alignment and conceptual development at different ages, not just on paper but in tangible terms?


  1. Hi Mimi - I like the way you framed this question so concretely. I hope you get some good responses. I'm experiencing some of this frustration here, where there appears to be little alignment in any subject especially between buildings, and teachers in the different buildings don't talk in any formal way about content or pedagogy.

  2. I think it's a difficult problem that comes (at least partly) from the fact that at international schools, people are hired only after they have had some years of teaching experience, which means that everyone is comfortable in their own style of teaching and thinks communication is nice to have but not always so necessary. I think my current school is much better than my previous school about this, but we still have a lot of room to grow.

  3. Hi Mimi,

    the part about -0 and 0 made me think of something else concerning vocabulary and translations (as in language translation and not as in, well, plane geometry).

    In French, zero is considered both positive and negative whereas in English it is considered to be neither. The "positive whole numbers" are {1, 2, 3, 4, ...} while the "nombres entiers positifs" are {0, 1, 2, 3, 4, ...} and, as you are aware, this set of "nombres entiers positifs" is the set of whole numbers being called "non-negative".

    We translate accordingly "positif" by "non-negative" and "négatif" by "non-positive". Positive is then translated in French as "strictement positif" and negative as "strictement négatif". This is quite confusing when switching to proofs written in English then in French, then in English again, etc.

    I was wondering what about German ? Being closer to English than French at its roots, what are the words for positive, negative and non-positive and non-negative ? Is 0 a "positive Zahl" or is it considered something else ?

    Cheers !