10/17/06 Students worked in pairs on individual “white boards”: transparent page protectors containing white card stock. Students could write on these and erase them. I supplied exactly one tissue and one pen for each team. We wrote down equations as they were modeled on the overhead (in +/- diagrams), and it was semi-competitive as students spontaneously tried to get the answers before the other teams. This method could be used for all kinds of learning – which I guess is what they were thinking of 150 years ago when individuals all had slates to work on. I’ll be doing this again in the future. There was no pen theft!
10/19/06 Students made playing cards and played in pairs to win tardy passes. Each team folded one blank piece of paper twice and tore on the seams to make four two-sided cards. The cards were labeled with eight potential “moves” in solving linear equations: + (meaning you will add a number to both sides of an equation), –, ×, ÷, CLT (combine like terms), DP (distributive property), MBR (multiply both sides by the reciprocal of a fraction that would otherwise be distributed), and CHECK (indicating solution is complete and all that’s left is to check our work. I displayed equations on the board in various stages of completion, and the team which displayed the correct next step first won two points. (I announced taking away a single point for displaying the wrong next step, but that turned out to be redundant.) During one period, I had the students just hold up their hands when they had the answer – first one up would get to announce their guess – and this turned out to be unrealistic, since most students quickly figured out that they could hold up their hands immediately and then use the next couple seconds to figure out the answer before they had to say it out loud. I like the game as originally planned except you need enough eyes to see everywhere in the room. Having a scorekeeper helped, but then that student is excluded from the game.
10/25/06 Reflections on problem solving. I present puzzles occasionally for transitions or sometimes for warm-ups. The students enjoy them. I also try to lead students to discover the skills and concepts we’re studying – with limited success. They get impatient with the indirect approach. Their attitude is, Can you just tell us how to do it? They’ve been conditioned over the years to believe that math is procedural: to get the answer, use these steps. They’re pretty good at learning this way: we just finished the unit on solving linear equations, and I’m happy with our progress. However, there’s no understanding behind most of their knowledge, and so the students on the fringes (who’ve been absent physically or mentally) have picked up a bit of information but have huge gaps in their ability to adapt or, some cases, to even do the basics. They will attempt to collect variable terms on one side of an equation, but will write down “–2” under each side of the equation, rather than “–2x”; or they’ll subtract a number from each side of an equation, producing a zero on one side but not writing down the zero because they don’t perceive it as even a number.
10/27/06 I presented a puzzle today – the fishing puzzle from our class Wednesday. A few students worked on the problem quietly, a few rowdy students blew off the assignment completely, and a few of the “troublemakers” got into it. Of course, it was hard for them to explain their thinking process, and nobody actually got the answer, but I think if you do this regularly, they might begin to recognize this kind of work as an actual math activity. In general, I’m not happy with any of the groups’ willingness to take turns talking and listen to each other. (One problem: as the level of discussion rises, the students who aren’t interested find the background noise serves as an excellent mask for their off-topic conversations.)
11/1/06 I keep looking for excuses to do origami in my algebra classroom. Rewards? Problem-solving? Instruction-following practice? (They’re so bad at following instructions. It’s all part of a general problem with listening to others – to me and each other.) I briefly taught a daily origami class to middle-school kids. It was delightful: almost 100% engagement each time. We worked simple models for a week and were just about ready to move into modular origami when the class ended. I realized as I was doing the class that we should have supplemented the geometry lessons the previous year with origami. My students never understood classical constructions: they would draw arcs free-hand, use the ruled markings on the straightedges, and pick endpoints arbitrarily. It was very frustrating. Would they have followed the origami instructions better? I know from experience that a lot of novices will go ahead and eyeball the origami folds rather than line up edges and creases per the directions. But I think the average user is more comfortable folding paper than using a compass, so we’d probably get more commitment.
11/3/06 What about using origami for teaching “non-graphic” math? Of course, you can use origami for various geometric ideas, but what about things like algebra and number theory? I researched this a bit and found a lot of discussion by mathematicians, math teachers and expert origamists regarding geometry (constructions and proofs), 2nd-year algebra (solving 4th-degree polynomials, folding parabolas), and various advanced topics, but really nothing about beginning algebra. Is all lost? Maybe not, if I make origami an ancillary activity and consider it valuable for teaching problem solving and prepping the algebra students for geometry.
11/7/06 Really bad lesson today on solving for a given variable in a formula. For starters, the students didn’t know how to use a formula, and then the first example in the book was the Fahrenheit/Celsius conversion – which I thought was a good example and which I illustrated with a nice table of temperatures but which completely overwhelmed the students. They didn’t know any of the basics about temperature, including where water boils and freezes. One girl even said that Celsius was “colder” than Fahrenheit. So, we were distracted by what I thought was fundamental. And then, although the kids were comfortable with the two basic equation-solving moves – adding/subtracting on both sides, and multiplying/dividing on both sides – they had completely forgotten (or never mastered in the first place) the alternate move of multiplying both sides by the reciprocal of a fraction, when the fraction “wants” to be distributed. (Many of them saw the “– 32” inside some parentheses and wanted to add 32 to both sides.) So, they got confused by the by-now unfamiliar multiply-by-a-reciprocal move. And then at some point they saw a variable in a denominator and really freaked out. I’m going to back up and re-teach tomorrow, focusing on the simpler examples, without any applications. Just: solve “y – 3 = x” for y.