Sunday, September 18, 2011


I am not very good at implementing new procedures in my classes, and I haven't been reinforcing the roles students are supposed to take on in groups (according to the Complex Instruction philosophy). I've also been using groups sometimes for work on exercises (rather than deep, rich problems). Even so, I'm still finding the positive effects pretty substantial.

I've often had students work in ad hoc groups in the past, but there'd always be a few huge groups (where some of the students were just leaning on the industrious ones) and a few students who preferred to work alone. With the set groups of 4, there has been a different dynamic. They develop a concern for each other, and are working together so much more effectively. This new dynamic has improved student learning in all 3 of my classes, I think. Along with allowing retesting, I think groups have changed my classes so that students take lots more responsibility for their learning. My job continues to be to make it interesting...

On Wednesday evening I was worrying about my next day's lesson in trig intresting. I had spent much of our Wednesday hour on the proof of the Law of Cosines. It was the least interactive class we'd had, I think. I'd like to come up with a way to get them walking through a proof like this, with a little more action on their parts. Any ideas? (I'll be teaching this again next semester... And of course there are lots more proofs to come in this course.)

I didn't come up with anything brilliant, but had them work in groups on a lake problem I had given them earlier in the week - before they really had the tools to solve it. Their second task was to carefully draw a triangle, measure its sides, and solve for the angles. (Then pass it to another person in the group, and have them solve it too.)

I was enjoying going around and helping people find their mistakes. One person had just eyeballed his sides, and written 'measurements' from his rough estimation. His triangle didn't work out right somehow. I think my slight scorn may have gotten him to try again.

One student wanted me to check his work. I scanned it and said it seemed reasonable. He had gotten 37 degrees for one of the angles. I used my circle with angle lines to measure and it was right on. His response:  "Wow, math is real!" That just about made my day. (Kind of shocking that he'd be surprised though...)

Factoring Polynomials
In Calc II, we're working on partial fractions, and I wanted to talk about the consequences of the Fundamental Theorem of Algebra - that any real polynomial can be factored into linear complex factors or into linear and quadratic real factors. I've never worked my way through the proof of that, and still feel a bit mystified that at the same time there is no formula possible to solve 5th degree polynomials. I'd never thought much about it, and the mystery of it (for me) makes me want to learn more about all this. (I may not ever get to it, though...) I went into class feeling high on math.

Systems of Equations
My evening class went pretty well too. I was showing my Intermediate Algebra students the process for solving 3 equations in 3 variables. I made up another silly coin problem, just so they could see the possibility of 3 equations in 3 variables being meaningful. (I have pennies, nickels, and dimes in my pocket. My 32 coins are worth $1.87, and I know I have twice as many nickels as pennies.) I also showed them a 3D coordinate system in the air, with one side of my desk being the x-axis, the front of it being the y-axis, and a line straight up from the corner being the z-axis. I place a few points in the air from that. I usually stop there. This time, I pointed to the corner of the room (floor), and talked about where negative values on each axis would put us (outside, in a closet, in the foundation), and then walked to (10,15,4), by pacing 10 feet along the wall, 15 feet into the room, and putting my finger 4 feet high. I then had two students do (10,20,3) and (12,20,3) at the same time.

When we did the system of equations, I didn't do anything new, but I felt like they were approaching it more sensibly than past classes. Most students want to write down a bunch of rules. I talk about figuring out which variable is easiest to get rid of by addition method, and then we get two new equations and solve the simpler problem. They seemed to be getting into it.

I felt so lucky that day to have work I love.


  1. For trig, for example with the law of cosines proof, I often give a problem with numbers in which they have to calculate the length of the third side, then gradually replace some of the numbers with variables, and eventually have them go through the steps with variables everywhere.

    Or, sometimes I give the big outline and then make them fill in the details; for example "We're going to calculate the length of this side. Because we love the Pythagorean theorem so much, that means we're going to draw this perpendicular here and then find those two lengths. We can find this length pretty easily if we just ignore the rest of the picture -- how long is it? We can find that length by doing a little subtraction -- what two pieces can you subtract, and what final expression do you get? Now can you use the Pythagorean theorem to finish the problem?"

  2. Hmm, sounds like your proof flows a bit differently than the one I used from the textbook. I think I'll play around with it a bit.

  3. I always work to arrange groups with a heterogeneous ability level and an homogeneous leadership level. I do this for many reasons including hoping to level the playing field when it comes to airtime. I want all members to be able to have the ability to contribute. There is much to talk with about this idea. I know I have a few posts hanging around about this topic somewhere...

  4. Sorry, my URL was missing an "L" try me at for more about groups...

  5. Jim, I'm interested, but I don't see a groups post at your blog, nor a search feature.

  6. Hi Sue, I previously solved a nth degree polynomials which I posted on my site: (please see worked problems numbers 1, 5 and 6).

    These are related to the Cambridge A level maths syllabus. Hope it helps. Peace.

  7. In high school, I always found working in groups to be very helpful. Not only do you learn better by helping others (peer teaching) but you also get to see the same problem from another point of view. Sometimes this helps when confronting a difficult problem. I know that in some of my college courses, it's only through group work or discussing the problems with others that I am able to finally see what to do and actually figure out the problem (or in most cases, the proof).


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