I have taught the second half of a Math 9 Enriched course for the last three years. The students generally finish two-thirds of the curricular outcomes during the first semester (with an different teacher). This alleviates the perpetual nemesis of time, and leaves me with no excuse to stretch the boundaries of what is possible in a classroom.

I spend most of the time developing a classroom ecology focused around conjecture, community, and curiosity. The result is a constant focus on problem shaping, solving, and re-posing.

At the end of the semester, I ask students to respond to a simple prompt. They have ten minutes to answer:

**What is Mathematics?**The responses are incredibly thoughtful (mimicking their work on the mathematics tasks throughout the semester). A selection of them are reproduced below. It goes without saying that I am incredibly proud of what we were able to accomplish as a group. They create an image of possibility. Often times, as teachers, we concern ourselves with the restrictions of schooling. Here, we contemplate the opportunities.

“Mathematics is a way of thinking and problem solving with an argument to back you up of why you know it’s true. You can create, analyze, and solve problems. With mathematics, it’s not always the work you put down on the page, it’s also how you present the solution to the problem with words in an argument. You think in mathematics; you do mathematics.”

“Mathematics is figuring out the world around you. It’s stretching your brain to not only remember formulas but to know where they come from and what they represent. It’s a language you first hear people using, then become immersed in it and learn it yourself. It’s a new point of view where anything is possible–even infinity.”

“Mathematics is, to me, reasoning, wondering, asking questions and being mathematical. To me, being mathematical is having your own solid reasoning. Some things like the formula for the area of a circle, it doesn’t really make sense. That’s why you wonder and ask why it works. In this class, we focused more on thinking and reasoning with problems than using formulas to solve the problem. To me, being mathematical is more important because the answer you got by doing mathematics is no use until you become mathematical and reason.”

“Mathematics is the asking of questions to help you better understand the main question. It involves numbers, discussions, and opinions. It’s not about the answer, it’s how you get there. We go through school having to take math classes, we are told how to answer the questions and we are expected to know why and how it works without really asking questions and questioning what we are doing and why. But is that really mathematical? Mathematics is as much asking questions as it is equations and using formulas.”

“Mathematics is the study of numbers and everything involved in it. It is solving problems, but more importantly knowing why and how you did them.

“Mathematics is the study of numbers but I think there is a lot more to it. It’s also a way of thinking. Questioning and learning why things do what they do is mathematics. Finding out why things work like they do. It’s not just about doing work from a text book like I thought it was.”

NatBanting

“Mathematics is figuring out the world around you. It’s stretching your brain to not only remember formulas but to know where they come from and what they represent. It’s a language you first hear people using, then become immersed in it and learn it yourself. It’s a new point of view where anything is possible–even infinity.”

“Mathematics is, to me, reasoning, wondering, asking questions and being mathematical. To me, being mathematical is having your own solid reasoning. Some things like the formula for the area of a circle, it doesn’t really make sense. That’s why you wonder and ask why it works. In this class, we focused more on thinking and reasoning with problems than using formulas to solve the problem. To me, being mathematical is more important because the answer you got by doing mathematics is no use until you become mathematical and reason.”

“Mathematics is the asking of questions to help you better understand the main question. It involves numbers, discussions, and opinions. It’s not about the answer, it’s how you get there. We go through school having to take math classes, we are told how to answer the questions and we are expected to know why and how it works without really asking questions and questioning what we are doing and why. But is that really mathematical? Mathematics is as much asking questions as it is equations and using formulas.”

“Mathematics is the study of numbers and everything involved in it. It is solving problems, but more importantly knowing why and how you did them.

__Mathematics is a way of looking at the world__[*emphasis in the original*]. Mathematics is exploring what you did and how you got there.”“Mathematics is the study of numbers but I think there is a lot more to it. It’s also a way of thinking. Questioning and learning why things do what they do is mathematics. Finding out why things work like they do. It’s not just about doing work from a text book like I thought it was.”

NatBanting

Outstanding, Nat … you are right to be proud of these kids, and congratulations on the good work you (and others) have done with them!

Too often, people would have these same kids say "I practiced getting Answers until I could do that really well", in the hope that repetition would lead to enlightenment. While repetition leads to many worthy things (fluency, speed, accuracy, and others), it seldom is sufficient for learning how to develop a new "way of looking at the world".

Thanks for sharing this!

#verycool

Great answers, Nat. Love the focus on problem solving and reasoning.

Yes, you must be doing something right if this is your students' views of mathematics!

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