The Reflective Educator

Education ∪ Math ∪ Technology

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Day: January 22, 2013

Learning outside of school

I realized that I have done a lot of learning outside of school as a kid. Here are some examples.

  • I took apart almost every piece of electronics we owned (except the television as I was told this was dangerous) and then put it back together, most of the time.
  • I explored every inch of the island I grew up on within 5 kilometres of our house, usually on bike, often with my dog.
  • We had a library of many hundreds of books in our house, and I read most of them.
  • I taught myself how to program, and I created a computer game in grade 10.
  • I (re)discovered the sum of triangular numbers formula, and the quadratic formula on my own.
  • I planned a trip to Mars in as much detail as I could, right down to where the bathrooms would be located on my rotating space-ship.
  • I collected a sample of every different kind of grass I could find on my island. I ended up with over 50 different types.
  • I played a lot of games, including Dungeons and Dragons and many exciting computer games.
  • I watched way too much television (with a strong preference for science fiction). I don’t know how valuable a learning experience this was, but in the interest of being honest, I thought I should include it.
  • I went for regular jogs around town. One time I finished my run, wasn’t tired enough, and so I went for another loop.
  • I wrote poetry, sometimes in the middle of the night.

I’m not offering this list to brag about my childhood, but simply to suggest that there are plenty of ways kids can learn outside of school, but they must have the time, space, and support to do so.

A factoring success story

I covered a couple of my colleague’s classes yesterday so he could attend a math conference. The afternoon class was a somewhat boisterous grade 10 group. I was asked to teach students how to find the greatest common factor, and if I had time, introduce them to more general factoring techniques.

I decided that the greatest common factor is a topic students find relatively easy, and so I just showed some examples of how to do it (actually, I drew the "how to" out of the class by asking them questions, but this is my standard technique) after verifying that they understood the distributive principle. I then assigned some practice problems, which then each student wrote their solution up on the board, and we discussed. I then showed students a couple of different techniques for multiplying binomials (like (x+2)(x+3) for example).

Next, I put up the following 4 questions.

1. x2 + 7x + 12

2. 2x2 + 7x + 3

3. x2 – 25

4. x3 + 8

I asked students to try and figure out how to write these expressions as one set of brackets times another, just like with the example from before, but I suggested to them that what we are trying to do is undo the distributive rule.

I went around the room and encouraged students, gave them hints when they needed them, asked them questions to prod their thinking, and observed their problem solving strategies. Students were engaged in the problem solving activity for a good 30 minutes. Once some of the students’ attentions started to wane a bit, I gave them a sheet with a description of how to do factoring by grouping and some problems to work on the back.

A group of students though really dove into question 4, which, as you may notice is actually quite a bit more difficult than the other three problems. I ended up having to give students two hints: I told them that the expression broke into two factors, one of which was (x+2) and the other of which was three terms long. The group of students worked feverishly on solving the 4th problem for a good twenty minutes, and then all of a sudden, one of the girls in the group leapt out of her seat and screamed, "I GOT IT!! YES!!" I circled around to see if she had the right answer, asked her how she was so sure it was right (she had multiplied everything back through using the distributive rule), and then gave her group x3+27 to solve (which she did quickly) and then x3 + a3 to solve.

At 5:30pm that night, I received an email from the girl, excitedly telling me how she had an inspiration while she was on the bus home on how to solve the general question, and had then figured out the general formula for how to factor a sum of cubes.

I emailed her back and congratulated her on becoming a mathematician.

Investigation into scoring systems

I played ultimate tonight, and we usually keep score with shoes. Our normal scoring system is to count in base 5. Tonight, I tried to use binary, but at half-time I switched back to base 5 when most of our team struggled to read our score quickly.

I took some pictures of the arrangement of shoes during the game (when I wasn’t playing).

01000

11000

01100

00010

10010

11010

 

I can imagine some investigations could be made out of these photos.

  • Given the numbers associated with each photo, try and determine how to count in this number system,
  • More challenging: From these photos, try and determine the missing numbers.

If you want a project that might take a while:

  • Design a scoring system using shoes. It should be easy to maintain and not require too many shoes.