This is fun…but is it math?

A few months ago, I got an email from one of the art teachers in the school.  She was offering her classroom, a makerspace of sorts, and her creative expertise to any teacher that wanted to bring their class down.

I jumped at the opportunity.  After all, what have I been enjoying most about math lately? Making and appreciating #mathart.

So. We brainstormed and ultimately settled on doing a project that would lead into our upcoming similarity unit.  It would have the kids constructing equilateral triangles descending in size, alternating colors, each triangle with a side length the same as the height of the previous triangle.

The students had never done constructions before, so it was a good introduction to using a compass, and it was also a chance for the students to play with the artistic concepts of structure, contrasting color choice, and composition.  The kids had fun, and the results were stunning!

This whole project took about two 50 minute class periods.  Toward the end of the second day, one of the students said: “Mrs. von Oy. This is fun and all, but it doesn’t seem like we’re really doing much math.”

Well. Turns out we are. Lots of it.

And also? Maybe math just is fun, and we teachers need to bring the fun out a little more often.

So here’s the plan for the “triangle art” debrief discussion:

  1. Why did that particular compass construction create an equilateral triangle? (and did it matter if your compass was too loose to keep its radius the same?)
  2. Did it matter what size triangle you started with?
  3. How many triangles could you make if you were to keep going? (Most students made about 10 triangles. This is a great time to begin getting kids to think about asymptotic behavior, even to think about limits)
  4. What is the difference between how many triangles you could make out of paper, and how many you could imagine mathematically? (a question to provoke conversation about the nature of mathematical abstraction. Perhaps I will have students read this blog post by Ben Orlin in class or for homework: There’s No Such Thing as Triangles)
  5. If you made the spiral pattern (like the one below), could you ever get all the way around, and how do you know? (A student conjectured today that it would be impossible because the triangles were getting smaller. That led to a great discussion of side lengths getting shorter vs. angles getting smaller…or do they stay the same measure? Would love to have the whole class in on this conversation)wp-1488329994162.jpg
  6. If the side length of the first triangle was 10 cm, what would the side length of the second triangle be?  (Let the students struggle with this one. They know the Pythagorean Theorem, but haven’t spent much time using it for interesting problems yet. I want to see them figure this out on their own.)

What are some questions you would ask your students? Send some comments my way…I’d love to facilitate an even more robust discussion.

What’s in a name?

This year, I didn’t teach students to factor the difference of squares.  Don’t get me wrong, I did teach how to factor the difference of squares, but I held off on giving this particular case of a quadratic a label.

For the past several years I have been reflecting a great deal on the pedagogy of teaching mathematics and how precise language factors into best teaching practices.  So I want to talk about this decision around difference of squares.

Let me start by saying I am a huge believer in using precise and robust mathematical vocabulary.  Being able to name something allows us to organize our thoughts better and to communicate effectively with others.

Case in point: this year, I used the language of input-output extensively in reference to functions of all sorts and in all representations.  When we got to the midterm exam and students were identifying a function by looking at graphs, I expected an explanation of “it passed the vertical line test.” But to my surprise and delight, out of 30+ students, fewer than 5 left their reason as simply “it passed the vertical line test.”  The rest went beyond that to give some variant of “it passed the vertical line test, which showed that for each input there was exactly one output.”  Bravo, students! Bravo!  I never emphasized memorizing the exact words of the definition of a function, but through using language carefully and consistently, students were able to recall this definition with ease.

Let me mention a second example. Continue reading

Tiny Change, Big Difference.

One of the jobs I have as a math teacher is to coach students to become more and more confident in their own ability to think and reason.

Recently, I was working with a student.  She is an incredibly hard worker, but lacks confidence when it comes to math.  So naturally, when she comes for extra help, she looks to me constantly to reassure her that she is doing the “right thing.”  She will say things like:

“Would I add x to both sides now?”

“Should I set them equal to 180?”

“Do I use CPCTC now?”

And yet, I don’t want her to be looking to me for whether or not she’s on the right track.  I want her to first look to her own reasoning abilities.  So we have worked to change her language from “Should I,” “Would I,” and “Do I” to:

“I think  _______________ because _______________.”

Starting that very day, she really took my suggestion to heart.  For example: she was working with a complicated diagram trying to find angle measures. She started to say: “Would I add these up to equal 180?” But she caught herself and said instead: “I think I should add these angles up to equal 180 because they make a straight line.” To which she responded (to herself) “oh wait…no they don’t.  They make a line with a third angle, though.  I could add all 3 up to equal 180.”

It seems like a small thing, but a teacher is there to coach students through the process of learning math, not to be the arbiter of all truth.  Students are empowered when they learn to think of themselves as critical thinkers, capable of reasoning through complicated problems.