Proportional reasoning is my personal focus for the year. It’s a major unifying trend throughout all of 8th grade math, but it’s extremely hard to teach. Last year I just forced the kids through cross-multiplying and solving for x. This year I’m trying to at least do a little better than that.

We’re currently working on probability. Today, we were comparing experimental probability to theoretical probability. “You roll a six-sided number cube 30 times and it lands on the number four 7 times. How does this compare to the theoretical probability of landing on a four?” The kids have to know that theoretical probability is 1/6, and that means that in 6 rolls, you should see four 1 time. They then have to extend that to realize that over 30 rolls, they should see it 5 times. Since they actually saw it 7 times, their experimental probability is larger than their theoretical probability. If you’re good at math, you did all this without even realizing you were calculating. If you aren’t good at math, I just made about a billion leaps and you’re lost.

I’m reading a really interesting piece on proportional reasoning, and it explains that there are multiple levels of proportional reasoning and students can fall anywhere on a broad spectrum of how well they can do it. You’re supposed to let the kids improve along the spectrum from wherever they are, instead of just pushing them all from Point A to Point W. This has made me much more patient with the kids, and much more intrigued by what level each of them is on.

So today I took time to sit with random students and give them questions to push their thinking. I let them answer however they wanted and then got to just sit back and see how their brains work. It’s a pretty cool thing to be able to see. Stop reading now if you’re not a math teacher, but I can’t resist sharing this.

One kid is at such a low reasoning level that he needed to start in concrete terms. We were discussing a spinner split into four colors, and he could tell me the theoretical probability of landing on blue was 1/4. He couldn’t tell me how many timesĀ it should land on blue in 4 spins, so we had to make a table and make up data for 4 trials. (I didn’t want to confuse him with an experiment in case the results didn’t work out, and he luckily knew to fairly distribute his results among the colors.) He figured out that it was 1. I asked him how many times we’d see it in 8 trials, and he guessed 4… so we extended our table to 8, and he realized it was only 2 blues. Then we extended our table again to 12 trials and saw blue 3 times. We were eventually able to just make a trials/blue table, counting by 4s on our trials column and by 1s on our blues column, but it took him awhile to get there and we couldn’t get farther.

When I asked other kids how many times the spinner should land on blue for 100 trials, I usually got the right answer. Some kids were guessing and checking numbers that added to 25, and showed me work that said “25+25+25+25 = 100″. Others were dividing 100 by 4. Others were patiently extending a table until it reached 100 trials. One kid was using equivalent fractions, explaining that a 1/4 chance is equal to 25/100. It was especially interesting that most kids could tell me that a coin would land on heads 50 times out of 100 because there are only 2 possible outcomes that need to be split evenly, but most did not articulate that same logic for the spinner and even fewer used that strategy when I asked about rolling a die.

I’m pretty sure being good at proportional reasoning is strongly aided by having a good foundation in multiplication. All those patterns and tables and the understanding of how given numbers relate to one another is based in multiplying. The kid who could barely make a table is also the kid who can barely multiply 2 x 3. Yet another way that missing basic skills makes the upper grades SO MUCH HARDER!