Archive for the ‘Pedagogy’ Category

Last week, I had the pleasure of attending a workshop with Peter Liljedahl from Simon Fraser University. The session was on assessment, but he opened with his take on problem solving. I’m going to discuss the assessment stuff in a later post, but I really wanted to talk about his problem solving process while it was fresh in my mind.

Peter grouped us randomly by having us select cards, and then gave us the following problem.

1001 pennies are lined up in a row. Every second penny is replaced with a nickel. Then every third coin (might be a penny or a nickel) is replaced with a dime. Then every fourth coin is replaced with a quarter. What is the total value of the coins in the row?

He gave each group a whiteboard marker, and a window or small whiteboard to work on. We were to solve the problem on the space we were given. Peter circulated and encouraged us to look at what was written on the other windows if we were stuck. In the end, most of us were satisfied that we were right. Peter never told us the answer or revealed it in any way.

Here’s what interested me about Peter’s take on problem solving, which is very similar to what I have advocated as Learning Through Problem Solving.

  1. The problem wasn’t written down any where. He just told us the story, clarified, and put us to work. This isn’t a textbook problem.
  2. He grouped us randomly, rather than letting us sit with the people we came with. Throughout the day, he kept on re-grouping us. His feeling is that this frequent changing of the groups leads to more productive classes, and helps include all students. Kids are more willing to work with someone they don’t like when they know that it won’t be for long. This process helps include even the shy and “outcast” students in the classes. These students are included because the people working with them know that the groups will change again very soon, and so they can be patient for that long. Peter also has a way to assess group processes, which I will discuss in the subsequent post on his assessment material.
  3. Writing on non-permanent surfaces is critical to Peter’s process. He has observed that when kids work on paper, they take a long time to start. When they work on something like a window or a whiteboard, they start right away, because they know they can erase any mistakes easily. Groups that are stuck can step back and see what the other groups are doing as a way to help them get started. When finished, each group’s work  is displayed prominently around the room so that students can share solutions with each other in a non-threatening way. Students could do a gallery walk and discuss the different solutions, or sit down and write down one that works for them individually.
  4. This is the third time I have had the pleasure of working with Peter. He has never once told us the right answer. This drives teachers nuts. I’ve tried it in classrooms, and it drives kids nuts. The message, though, is that they have to figure out the answer. The teacher isn’t going to let them off the hook. An even more important message is that the teacher is less concerned about the answer than she is about the process. These are two pretty nice messages.

Peter’s problem solving process fits really nicely with the Learning Through Problem Solving discussed frequently on this blog. There’s always a “yeah, but” when I share some of this with teachers. They have one whiteboard, no windows, and nothing else to write on in their classrooms. Let me solve that problem for you.

Superstore has $20 whiteboards that are just the right size. Staples has tons of different sizes available for less than $30 each. I’m confident you could get enough whiteboard space in your classroom for 15 groups to work for less than $300. Your principal will find that kind of money in a hurry, especially when you explain that you are going to use it to enhance mathematics instruction by engaging students in group processes that will lead to better problem solving skills. I’ll write you a letter if you need one.

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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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I can’t believe I’m about to wade into the homework debate, but here goes.

When I became a consultant, they spent a great deal of time teaching me how to coach and facilitate.  One thing they taught me was to be up front about my assumptions and biases.  So here’s three pretty important ones.

  1. As a kid, I hated doing homework.
  2. As a teacher, I hated chasing kids to get their homework done.
  3. As a parent, I am learning to hate fighting with my kid to do her homework.

A few years ago, while tutoring a student, I came to the realization that most of the math homework we assign is a waste of time.  Her teacher had assigned her questions 1-15, parts a, c, and e.  She did 1a and knew how to do it.  I suggested that we move on to #2, but she was the kind of kid who did every question assigned, so she did 1c and 1e even though they were exactly like part a, but with different numbers.  It occurred to me at that point that I needed to look at how much and what kind of homework I was assigning.

I started paying more attention to which students were completing what homework assignments.  Some kids did it all, and some kids did none.  The problem is that the students who needed the practice, didn’t do the work, and the students who didn’t need the practice, diligently did every single question I assigned.

At about that time, some teachers in my department were complaining about how kids didn’t do homework.  I threw out the idea of just not assigning it at all to alleviate everyone’s (teacher, student and parent) stress.  That suggestion didn’t go over well.  In my classes, I consciously started giving less homework.  My thought was that if we assigned 4 useful questions and the kids did 3, we were way better off than if we assigned 20 rote practice questions and the kids did 6.

I also started differentiating my assignments.  Kids who needed practice got a manageable amount of practice.  Kids who needed deeper thinking got richer questions.  As a kid, I hated it when I got done an assignment, only to have the teacher give me more of the same.  I learned not to get done.  We do that to our strongest kids.  Instead of giving them better assignments, we just make them do more work, and because they’re good kids, they tend to do it.

Another change I made was to move away from giving my class practice time at the end of the lesson.  I tend to talk less than most high school math teachers.  In an 80 minute block, I try hard to keep lessons to no more than 40 minutes, so that kids have at least 40 minutes to work.  I used to give that work time at the end.  Kids were tired after a 40 minute lesson, and it was difficult to focus them on the practice questions.  I started breaking my lessons into smaller sections, so that I’d teach for 10 minutes, and then assign 2 question, then teach for 10 minutes and assign 2 questions, and so on.  By doing this, the kids really felt they had to do the questions, because it seemed like part of the lesson.  As such, I assigned far fewer questions at the end of the lesson.  Most kids had nothing to take home at all, which I felt good about, and I’m sure they felt good about.  More importantly to the people who will suggest kids need lots of practice to be successful, my student’s grades didn’t suffer by doing less homework.

If I get brave later on, I’ll add a post about how I feel about assessing homework.

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Here’s a WCYDWT for Alberta’s Math 10-3 on Ratios and Rates.

This fish, a 60 pound sturgeon, was recently caught and released in a small self-contained pond on a golf course near the Red Deer River.  When local biologists heard about the catch, they decided to investigate.  They surmise he (or she) came into the pond when the river flooded five years ago, and then spent the next five years feeding happily on the stocked trout.  Since sturgeon are endangered, it was important to return him (or her) to the river, which they were able to do.

This story reminded me that about ten years ago, I was fortunate enough to be present when a team of biologists surveyed a cutthroat trout population on a remote mountain stream.  Biologists typically use a method called electroshock fishing to stun fish so they can be tagged and released. They had a boat with a generator on it and a long metal prod which they poked around in the water.  When the current got near enough to a fish, it would be momentarily stunned, and float to the surface.  The biologists then weighed, measured and tagged the fish before returning it to the stream.  Any fish that had been previously tagged had its number recorded and it was again weighed and measured.  It was a humbling experience to see that there were actually abundant numbers of fish in that stream, because I had been unable to catch any earlier that day with my fly rod, but I digress.

I have used the story of watching the biologists electroshock fish several times in my classes as a ratio problem and asked students determine the population of fish in that stream.  Sometimes I gave my students the number of fish tagged on the first day, and then the number of tagged fish caught on a second run of the same stretch of river later on.  Other times (and probably a better problem), I had my students design an experiment to predict the population of fish before providing them the actual numbers.

Red Deer Advocate story about the fisherman who first caught the fish

Red Deer Advocate story about the biologists capturing and releasing the fish back to the river

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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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A few weeks ago, Kate Nowak posted on her blog that she was frustrated at the lack of online resources to support the teaching of the binomial expansion.  She threw out a challenge asking people to create a better way to teach the binomial expansion.

Here’s her challenge:

“Objective : Present the binomial expansion in a way that makes sense. Bonus points if students are able to as a result completely expand a power of a binomial and find a specific term in an expansion.”

Our revised curriculum in Alberta includes a pedagogical shift that asks teachers to create opportunities for students to establish their own meaning through exploration, problem solving, investigation and developing personal strategies.  One of the common concerns I hear from teachers is that we can’t teach higher level mathematics this way, because the material is so hard that students won’t be able to construct their own meaning.

Kate’s challenge gave me an opportunity to try to teach in this manner in one of our academic courses.  In Alberta, binomial expansion shows up in a pre-calculus course called Math 30 Pure.  Students go from that course into Math 31, which is a calculus course covering limits, derivatives, application of derivatives, and basic integration.  Binomial expansion, and the binomial theorem are two pretty dry topics.  I took the challenge and attempted to create a lesson that was both engaging, and allowed students to construct their own meaning through looking at patterns.  A colleague was nice enough to let me try the lesson in his class.

My lesson went over fairly well with the students, and I think I met all of Kate’s objectives.  The introduction/hook was more compelling than anything I had done in the past on that topic.  Students were able develop the patterns in the binomial expansion on their own, and were able to apply those patterns to expand more complicated binomials.  What surprised me was that they were able to come up with the formula to find any term in an expansion all on their own.

I’d love to hear some feedback on the lesson.

An 11 minute compressed version of the lesson can be watched here:  Binomial Expansion Lesson

If you’d prefer to just read the lesson plan, here it is.

Lesson Plan

Introduction:  I showed them a mathemagician video from TED.

Hook:  I claimed that I was a mathemagician, too. I wrote (a + b)^2 on the board, and asked them to expand it. Then I wrote (a + b)^3 on the board and had them expand it. Then I wrote (a + b)^4 and (a + b)^5 and challenged them to a race. They could use friends, pencil, paper, calculators or whatever they wanted, and I would just use my brain. I pretended to struggle, and then wrote down the answers as quickly as I could. I had to hurry because one kid was darn quick, and was getting at (a + b)^4 by multiplying the answer to (a + b)^3 by (a + b) rather than expanding the whole thing as I had expected them to do.  If you watch the video, we appear to tie on that one, but he already started while I was blabbering.

Students look for patterns:  I explained that I am not really that smart, and that I was cheating using a pattern. I wrote “In the expansion of (a + b)^n, ” on the board, and asked them to spend a few minutes together coming up with ways to complete that statement. They got that each term had the same degree as the exponent on the binomial, and that there were n+1 terms in the expansion. It took a little longer and some direction from me for them to notice that the a’s started at a^n and decreased to a^0, while the b’s did the same thing in the opposite direction. One girl put Pascal’s triangle on the board when she noticed the pattern of coefficients.

Expand a binomial:   I asked them to use all that they had learned to expand (x + 2y)^5. They worked together and there was much discussion about how to handle the 2y part. They managed to figure it out.

Develop a formula for a general term:  I gave them one like (x + 2y)^12 and asked them if they could figure out the 8th term without writing out the first 7. I gave them three blanks to fill in: coefficient, “a”, and “b”. They got the a and b part just fine, but struggled with whether the coefficient should be 12 choose 7 or 12 choose 8. They figured it out by counting.   I told them that they had just figured out the formula. I wrote the formula tk+1 = nCk x^(n-k) y^k on the board.

(At this point I fell back into my old bad lecture style, and did two examples with them using the formula, rather than making them do it themselves.)

Example: Find the term containing y^7 in the expansion of (x – 3y)^7.

Example: Find the constant term in the expansion of (3x – 2/x^5)^12.

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