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Posts Tagged ‘Pedagogy’

This is an activity a group of teachers I was working with fleshed out based on ideas on Mark Wahl’s website, and here.  This activity is appropriate for Math 10-3 (Measurement) and Math 10C (Measurement or Real Numbers).

Statues of human bodies that the ancient Greeks considered most “perfect” embodied many Golden Ratios. It turns out that the “perfect” human face also contains many Golden Ratios.  This task allows students to take measurements of an ancient statue, and check whether these measurements approach the Golden Ratio.  Then two celebrity photos are provided and the students check how beautiful the celebrities are based on these calculations.  Finally, students can use pictures of their own faces and GeoGebra to check how they match the Golden Ratio.

Part 1 Students perform the measurements on a photo of a statue, and calculate the ratios requested.  These measurements could be made by printing the picture and physically measuring, or by inserting the picture into GeoGebra and using the software to measure.

Part 2 Students complete the same measurements, either on a provided celebrity photo, or one of their own choice.

Part 3 Students then take their own picture and insert it into GeoGebra.  They can then measure, and check how closely the ratios in their own faces match the Golden Ratio.  Students who are not comfortable using their own pictures can choose other celebrities from the internet. (I didn’t teach Cameron Diaz, but her face is much more likely to be beautiful using this test than my own…)

The complete student package in Word format is available for download here.

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I had a long drive home from an assessment session yesterday, and I was reflecting on some of the conversations we had. We talked about how some teachers justify strange assessment practices because they sincerely believe that they are helping prepare their students for the real world.

It occurred to me that this view of our role in preparing students for life after school treats the real world like a minimum wage job at a fast food restaurant.  This view of the real world values things like showing up on time, having a great attitude, and working hard. Those are admirable qualities in the guy who serves me my curly fries. They should have nothing to do with my assessment of students’ mathematical abilities.

A better view of our role in preparing kids for the real world would be to treat the real world like a contract job.  This view of the real world values things like taking responsibility to finish what you start, taking as long as you need to do the job right, seeking help from others when needed, and producing good work every time. These are great qualities in the guy who designs the bridge I drive across every day. These are the things I should be encouraging in  my students, and should be reflected in my assessment practices.

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At a session I facilitated today, I was going over our old (1996) curriculum as compared to our revised (2010) curriculum.  Many of the outcomes are similar, but the big difference is in the wording of the outcomes.  I asked the participants to compare the two below.

1996

13. Use the Pythagorean relationship to calculate the measure of the third side, of a right triangle, given the other two sides in 2-D applications.

2010

2.  Demonstrate an understanding of the Pythagorean theorem.

The outcomes get at essentially the same thing.  In my opinion, though, the big difference is in the word “understanding” in our 2010 curriculum.  Back in 1996, we don’t appear to have been too concerned about whether or not the students understood it, as long as they could do it.  I threw this thought out in my session, and one person challenged it and said that they were the same thing.  He insisted that a student couldn’t do something unless that student actually understood it.

I disagreed, and he claimed I was arguing semantics (which I had to look up later).  I had to move on, but I certainly thought about what he had said on my flight home.  In the end, though, I stand by my original assertion.  I think that too often we ask kids to “do” things in math class without really trying to get at understanding.  Some kids imitate our algorithms, score well on tests, without having a true understanding of the math involved.

I tried to think of a real-world example of this distinction, and so far the best one I have come up with is as follows.  I know that if I push the gas pedal on my car, it will move forward (if I put that stick thing in the spot marked “D”, which I assume stands for “Go”).  This is something I am confident I can do.  I do not, however, have any understanding as to why the car goes forward or what is going on to make that happen.  I do it without any understanding at all.

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Eric Mazur is a Harvard Physics professor.  I came across this video when somebody posted it on Dan Meyer’s blog in response to another person who was demanding data to prove that Dan’s methods would work.  Professor Mazur has data to prove the merit of his shift from being a lecture style instructor to an instructor who facilitates peer teaching.  Peer teaching is definitely a strategy we could employ in our revised curriculum in Alberta.  If a University Physics professor can facilitate learning without having to tell his students everything, then certainly a high school math teacher who wanted to teach for understanding could emulate this process.  This video is about 80 minutes long, but is worth the time if you can spare it.  For those who can’t, I’ve summarized the key points below.

The reason that Professor Mazur switched his style was because his students were not showing gains when he compared pre-test results to post-test results.  Despite going through an entire term of his lecture style physics class, students could not show a significant improvement in their understanding of basic physics.

His Key Messages:

  • Students who have recently learned something are better at explaining it to other students than a teacher who learned and mastered it years ago. It is difficult for a teacher who has mastery of a concept to be aware of the conceptual difficulties of the beginning learner.
  • Give students more responsibility for gathering information and make it our job to help them with assimilation.
  • You can’t learn Physics (or Math or anything) by watching someone else solve problems.  You wouldn’t learn to pay the  piano by watching someone else play.  You wouldn’t train for a marathon by watching other people run.  If you want to learn problem solving, you have to do the problems.
  • Better understanding leads to better problem solving.  The converse of this statement is not necessarily true.  Better problem solving does not necessarily indicate better understanding.
  • Education is no longer about information transfer.
  • He says that in his original methods he covered a lot, but the students didn’t retain much so the coverage was basically meaningless.  In his new method, he has relaxed the coverage a little bit, but increased the comprehension enormously.

The Peer Instruction Process

  • Students pre-read the lecture notes or text.
  • Class is then used for depth, rather than coverage.
  • Depth is attained through what he calls a concept test.

The Concept Test:

  1. A question is posed.
  2. The students think silently about the question for a minute or so and it must be completely silent in the class.
  3. Students answer individually and vote by show of hands or by SMART Response systems.
  4. Peer discussion.  Defend your answer.
  5. Revised group answer.
  6. Explanation

Then he will “lecture” for a couple of minutes, and repeat the process with his class.

Benefits of his process:

  • This process promotes active engagement.  “It is impossible to sleep through class, because every few minutes, your neighbor will start talking to you.”
  • He can continually assess where his students are.

Challenges:

  • Teachers have to find the right questions to ask in their classes.
  • Students will write on their evaluations that professor Mazur isn’t teaching them anything and that they have to learn everything themselves.

My take on all this:

I see this process as being one we could employ in our high school classes.  I don’t think we’d expect kids to pre-read sections of the textbook, but we could certainly present problems to them in this manner and allow them to explore and construct solutions without having to tell them so much.

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