Posts Tagged ‘Teaching’

I started teaching in the dark ages.  When I began, there was no internet, email, social media, or cell phones.  We had just gotten computers in classrooms for teacher use for things like attendance and marks entry.  Our PD focused on how to use technology like PowerPoint and Word.  I started teaching way back in 1992. My first computer, purchased in the mid-80’s, looked like this:

Recently, I had the opportunity to meet Dan Meyer.  He was younger than I expected, and I got thinking about how different the start to his teaching career was than the start to mine.  He never taught in a world that didn’t include the internet.  He probably can’t even remember a world that didn’t have internet.  He has been connected in ways I only dreamed of in 1992.

When I started teaching, I was in a small rural school.  I was the only high school math teacher.  My two resources were a curriculum guide and a textbook.  When I got stuck, there was no other high school math teacher within a half hour drive that I could talk to. I couldn’t email anyone, because email didn’t exist.  We had to use long distance calls sparingly, because the cost was high.  Essentially, I spent the first 8 years of my career alone in my planning, preparation and reflection.  It seriously hindered my growth.

I hope you young teachers know how good you have it these days.  Young teachers like Sam Shah, Kate Nowak, Jason Buell, Sarcasymptote (Greg), Dan Meyer, Shawn Cornally, and Dan Anderson are connected in ways I couldn’t have imagined in 1992. Many of them have probably never met each other in person, and yet they remain in close contact through their blogs and Twitter.  When Dan and I talked about some of them, it seemed strange to me that he had never met them.  In my mind, they exist in the same world and should know each other.

These virtual networks of  math teachers are powerful, and lead to rich discussion and professional learning.  We are in an exciting time in education.  We are all getting better because of the willingness of these young teachers to explore and share.  What strikes us older folks (and probably puts fear into the hearts of textbook publishers) is that these people are not sharing their stuff for profit.  They are sharing for the good of their profession.  I commend them for that.  I would be a better teacher today if such opportunities had been available to me in 1992.

Thank you for sharing.

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October’s issue of the NCTM Mathematics Teacher magazine included an excerpt from an interesting article in their Media Clips section.  They used a Forbes article that dealt with tween earnings, and the portion they included is:

Hollywood’s 10 top-earning tweens collectively pulled down $107 million between June 1, 2007 and June 1, 2008.   According to Alloy Media and Marketing, it’s a rabid and often indulgent fan base: America’s 20 million kids aged 8 to 12 spend $51 billion of their own money annually and influence $150 billion more in spending by their doting parents. Tied for the top spot: Hannah Montana star Miley Cyrus and Harry Potter lead Daniel Radcliffe. The young talents each banked a cool $25 million in the last year.

I love this article for the questions it could elicit.  The problem with the Mathematics Teacher Magazine treatment of this is that it follows the article with 8 questions we are supposed to ask kids.  Teachers might see this, and defer to the questions provided, thereby missing a great opportunity.  I’d much rather show students the article, and have the kids decide what would be interesting to explore further.  There are lots of potential rate and ratio questions that could come from student exploration here.

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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.


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.


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.


  • 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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For your viewing pleasure, here are what I believe to be all the entries in the binomial expansion contest.  They are presented without bias, and in no particular order other than putting the best one at the end.


James Tanton’s Part 1

James Tanton’s Part 2

Jason Dyer’s


Edit (May 30):  Jason Dyer’s was judged the winner in the competition.  See here for the rationale for the decision.  Congratulations, Jason.

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