High School Math Teachers Learn About Dolphins: The Recap

Googling "math whale"
is just as dangerous.

I tried to do a recap of last week's workshop for high school math teachers in my last post, but was easily distracted by a misconception-full image I found while googling "math dolphin" images. I'm going to try debriefing on the workshop again today.  I was invited to the workshop on the basis of my post about using whales and dolphin science to teach high school math.

At the start of the workshop, Michelle (a professor from the math department and fellow GWIS member) did an overview on trigonometry.  I followed with a quick talk about whales and dolphins, adapted from the talk I give to the university of Hawaii Marine Biology Class and to some of the high school classes I've visited.  I start by going through all the different cetacean families, to give people some idea of the diversity of these animals, and for background in case I happen to mention a more obscure species later on.

Ganges river dolphins and Amazon river dolphins are
in totally different families than bottlenose dolphins,
 From Nikaido et al (2000). 

Then I talked a little bit about why whales and dolphins make sound underwater (communication and foraging).  Some species spend SO much time underwater that they are actually easier to find by listening for them than by looking for them at the surface.  Beaked whales are a great example of an animal that can be easier to find using sound.  Here is some tagging data from a Cuvier's beaked whale.  You can see that the whale spends most of its time underwater, only very briefly appearing at the surface. Some of the gaps between surface intervals are as long as 80 minutes.

 In the open ocean, it's easy to lose something you haven't seen for an hour.  The ocean is big, and if you are not looking where the whale is when it pops back up again, you've missed it.  This is why stellar beaked whale researchers like Robin Baird will have 4+ people scanning the water 360 degrees around the boat.  Another way keep track of whales is by following the sounds they make underwater (places where the whale was making sounds are shown in yellow above).

At this point, we stopped talking about whales and Michelle  handed out review worksheets with practice problems on Trigonometry.  While the teachers worked on them, Michelle told me about how much she loves working with math teachers: "They're the best group - they love figuring out problems, and if one of them understands and someone else doesn't, they teach each other!"

After the teachers were done with their worksheets, we took a quick break.  There were snacks,

After the break, we gave the teachers three pieces of information:

1) The speed of sound underwater (1500 m/s)
2) The arrangement of the hydrophone array:

3) The difference in time between when the whale sounds arrived at hydrophone 1 and 2 (0.06), and between hydrophone 1 and 3 (0.07).

That's all we gave them.  With this information, some trigonometry, and a little critical thinking, it is possible to find the location of the whale.

How did it go?

At first, there were a lot of questions about how the time difference between H1 and H2 could have been 5x larger than the time difference between H1 and H3.  Although interesting, this question is only marginally relevant to solving the problem.  The answer is that it depends on the location of the whale and thus shape of the triangles we are using to find its location:

At some point, one of the math teachers came up with the idea of starting with first principles and the Pythagorean theorum to find the location of the whale.

x distance to the whale^2 +y distance to the whale^2 = distance to the whale^2

We don't know x or y, but we do know that

Distance to the Whale = Time for the Sound to Travel from Whale to Hydrophone * 1500 m/s

so

x^2 +y^2 = time*speed^2

After these ideas, the math teachers whizzed along until we ran out of time, and Michelle went through the answer to the problem.  Then we asked for comments.  Here is some of what I remember. It's been a week, so I've forgotten a lot, and I also may be remembering things I didn't actually say at the time, but thought of later.  But you get the gist.

"That was really hard!"

Michelle: "Last year a lot of the comments that you guys gave me about the workshop were that you wanted more real-world examples of math. This is a great example of why we often DON'T use real-world examples of math - because applied math can be really complicated! When we're teaching math, we often try to use simplified examples to easily express concepts." (Me: Hmm, so maybe my idea about teaching about dolphins in high school math isn't so good).

"I'm not sure how this helps me.  I teach remedial middle-school math - this math is way beyond what my class knows."

"That may be true, but I think it is still important to have examples of how math is used in the real world.  One of the complaints that many students have is that they will never really use math.  This is an example of how math is used to study something lots of people are interested in - whales!  You can also edit the math we used today to fit your student's level of expertise - we started at a high level of math because we knew you are good at it."

"Want to come talk to my high school students?"

YES!!!

In summary, I had a really great time talking to the math teachers.  They were a lot of fun, and they asked great questions. Outreach = da bomb.

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