I think every IB teacher approaches calculus the same way: look, average rate of change. Look, instantaneous rate of change. Secant, tangent, gradient of tangent. Previous years I've tried to make this introduction come alive by giving students a function describing the distance fallen by a parachute-jumper, and having students discover themselves what the average speed would be, and the instantaneous speed at certain moments.

This has worked... well, let's just say students got the main ideas but not at an intuitive level, neither did they retain their discoveries more than a week or so. The whole thing seemed very artificial, pseudo-contextual.

So this year I decided to try graphing stories, choosing one about distance and one about speed. I downloaded graphing paper for graphing stories from Dan Meyer. I played the first video, asked students to graph it, and showed the answer. It was like lighting a fuse and seeing the classroom first fill with expectancy and then erupt in a constructive chaos of discussions about distance, displacement, speed, velocity, mathematical modelling, acceleration... "Look, the steepness is the same on the way from the camera as on the way back, so the guy who made the video assumes the speed of the dog was the same in both directions!" is one memorable quote.

It was a weird and funny experience to have the class completely ignore me and all my efforts to bring order into their discussions simply because they were so incredibly involved in figuring out how the video related to the graph and what we could infer from the graph about the beliefs of the grapher.

Then I showed the second video, about speed. Students immediately connected steepness or graph to acceleration and after a brief discussion when I asked "so what is the distance that the runner covered?" some students immediately replied that we must look at the area under the curve. I asked how we could calculate that area and students proposed dividing it into small sections, rectangles, trapezoids, triangles.

I believe that students developed a really strong intuitive connection to the main concepts of calculus, through their understanding of distance, speed, and acceleration. We will follow this up with more structured work using the graphing stories graphs to figure out approximations to average and instantaneous rates of change. That will take a full lesson. And then we're ready for functions and the limit definition of derivative.

It might seem like a long time (more than one week of class) to build up to the definition of derivative. In my opinion, conceptual understanding of derivative is necessary for everything that follows, especially when using derivatives (and second derivatives) to graph functions and for optimization problems. Without the conceptual understanding nothing else is going to make sense, will just be an imposing set of strange rules and lengthy procedures. With a solid understanding of derivative (and later integral) calculus comes alive, becomes beautiful, and leads students to ask, as one senior student just did: "What profession should I choose that lets me use calculus as much as possible?" :) :) :)

## Sunday, March 25, 2012

## Saturday, March 24, 2012

### Most challenging: homework

Three and a half years into my teaching, I've now tried four ways of motivating students to do homework.

- Hands off - my first year I was too overwhelmed with lesson planning and simply told the class "I expect you to master lesson content before the next lesson". Results in this class were the highest I've ever seen, but that may have other reasons than my homework (non-)strategy.
- Binder checks - this simply required way too much organization on behalf of myself and the students.
- Weekly quizzes - this works wonders in psychology, less so in mathematics. In maths, I'm concerned that weekly quizzes give some students weekly opportunities to fail, demotivating them further.
- Flipped classroom - didn't work, see previous post.

I'm thinking in part that this is an uphill battle. In Sweden, attitudes towards children and childhood is very "let children be children" = much play, little work, minimum pressure. In a different culture maybe all the above strategies would work, maybe they wouldn't, I can't know. I do have the luxury of having students from all over the world in my lessons, and I've asked them about their previous school (and homework) experiences. Without exception, students from Russia and East Asia report doing much more homework (double or triple) in their previous countries compared to now in Sweden.

They also report that teachers would:

- assign much more homework, often give out exercise papers each lesson
- collect homework every lesson from every student
- mark every exercise from every student every day and give it back almost immediately (latest next day)
- have almost no exams, letting homework be the assessment of choice

Considering that classes in East Asia are rarely smaller than 35 students, I have absolutely no idea how teachers find time for so much marking. Also not sure how students used to a more laid back system would react to a sudden change in this direction. Worth thinking about though.

### Flipped classroom - no more

It seemed like a good idea.

It worked for a while. How do I know? Because in class, more kids would take the initiative to come to me with questions regarding understanding rather than procedure. Because kids were getting group practice on sophisticated (more or less) questions done in class. Because I could see on thatquiz that students were doing their assignments and I loved seeing what was going well and needed re-teaching.

Then it didn't work anymore. Maybe it was in part because I was gone for a week and lost the connection with the class somehow. Students also say watching youtube videos was good (because you can replay) but also bad because you couldn't interact with the teacher to ask questions, etc. Of course, it's a much less social type of learning, and losing the social side can be hugely demotivational. The thatquiz homeworks, while done by many students, were sometimes done in something like 2 seconds per question - indicating that just maybe the students were copying other students' answers or in other ways avoiding actually thinking about the exercises.

The test results were horrible. We had a wonderful constructive chat afterwards, with the class unanimously agreeing to go back to the "old way", with going over new content in class and independent practice/review at home.

Even though it's disappointing that the flipped classroom strategy didn't work, I'm happy we tried it. It's this kind of experimentation that makes teaching fun, and the only way to eventually find more effective teaching strategies. If we hadn't tried it, I would always have wondered if we were missing out on something spectacular. Also, the kids learned a lot from this. Many of them still want me to post videos and they use thatquiz for extra practice with direct feedback, strategies they learned from our flipped classroom experiment.

Afterwards, I talked at length about why we tried flipped classroom (to get more time for problem solving / exercises in class because the students were not doing them at home). We talked about homework and how much they were doing. Very little, it turns out, and is rather unsurprising. We talked about if they

It worked for a while. How do I know? Because in class, more kids would take the initiative to come to me with questions regarding understanding rather than procedure. Because kids were getting group practice on sophisticated (more or less) questions done in class. Because I could see on thatquiz that students were doing their assignments and I loved seeing what was going well and needed re-teaching.

Then it didn't work anymore. Maybe it was in part because I was gone for a week and lost the connection with the class somehow. Students also say watching youtube videos was good (because you can replay) but also bad because you couldn't interact with the teacher to ask questions, etc. Of course, it's a much less social type of learning, and losing the social side can be hugely demotivational. The thatquiz homeworks, while done by many students, were sometimes done in something like 2 seconds per question - indicating that just maybe the students were copying other students' answers or in other ways avoiding actually thinking about the exercises.

The test results were horrible. We had a wonderful constructive chat afterwards, with the class unanimously agreeing to go back to the "old way", with going over new content in class and independent practice/review at home.

Even though it's disappointing that the flipped classroom strategy didn't work, I'm happy we tried it. It's this kind of experimentation that makes teaching fun, and the only way to eventually find more effective teaching strategies. If we hadn't tried it, I would always have wondered if we were missing out on something spectacular. Also, the kids learned a lot from this. Many of them still want me to post videos and they use thatquiz for extra practice with direct feedback, strategies they learned from our flipped classroom experiment.

Afterwards, I talked at length about why we tried flipped classroom (to get more time for problem solving / exercises in class because the students were not doing them at home). We talked about homework and how much they were doing. Very little, it turns out, and is rather unsurprising. We talked about if they

*want*to do homework (unanimous yes) and how I can support them in doing it. I proposed, and the kiddos agreed, to try Mimi's strategy. In short, we agreed that the students will do at least 12 exercises from the book every week, I will collect them and choose one exercise per student to mark and give feedback on the overall level of difficulty of the exercises the students have chosen to do. Students seem enthusiastic about this idea, and I'm excited to try out yet another homework strategy. :)
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