When I mentioned yesterday to the SuperGirl 5 Math Group that the next unit coming up in their school curriculum will be adding fractions, there was a huge chorus of groans, and they took a minute to vent. They all complained that “Adding fractions is soooo boring.” and they seemed to use ‘boring’ in three different senses:
• This is not challenging us. We’ve already done this before.
• This is pointless. There is no good reason to be adding fractions anyway.
• This is treating us like little kids, insulting our intelligence. We’ve been doing fractions since second grade.
Of course, whether these perceptions are true or not matters less than that they are indeed the girls’ perceptions. There’s also a paradox here, that they may be bored with adding fractions but I know from SGM Summer Camp that they are shaky on even understanding what fractions mean, and in a well-sequenced curriculum, understanding what fractions mean comes before adding them up.
But maybe it would be better to not to have a well-sequenced curriculum. As luck would have it, I had planned two activities to teach the meaning of fractions, but mixed up were a bunch of other topics they won’t see till sixth grade or so. The first was a poker-like card game. One round, the winner was the person who had the best cards to fulfill
1 / (x + 2y) = 0
(Winner is she who comes closest to 0)
This led to the tricky strategy that if you want the fraction to be as small as possible, then the denominator, (x + 2y), should be as big as possible — which I could only explain by reaching for those boring pizza slices they shared back in second grade to demonstrate the meaning of fractions.
In another round, three girls dropped out of the bidding because they claimed they had poor hands for
(x + y) / 2 (x + y) = 1/2
(Winner is she who comes closest to 1/2)
What they didn’t realize is that any two cards substituting for x and y will always give 1/2 for the whole expression. Everybody is a winner! I connected this to the boring fourth grade idea of equivalent fractions.
In the second activity, each girl had a number taped to her back and had to deduce what that number was. The only clue was that if each added their hidden number to the number on the back of some other girl, the two would add up to 1, so they had to figure out who was paired with whom. In one round, Laila deduced, for example, that 200% must equal 2, since that person had to be paired with someone who had −1 on her back, and 2 + −1 = 1. Although none of them could have solved any whole round individually, they could solve each round together in about ten minutes.
At this stage in their curriculum, they haven’t been officially taught percentages, but they’ve certainly seen then around, and maybe the girls are right, maybe it would insult their intelligence to pretend that they can’t do any problem-solving with percentages. And the same with the several other topics that I had jumbled into the mix, including all of the algebra topics that we discovered in our poker game.
The fun of problem-solving isn’t just for advanced students, and this group is more or less average, “at grade level”. We still ran into such misconceptions as 1/8 = .8, or 1/0 = 1, each of which had to be carefully debunked, again. But there appeared to be a lot of excitement and a lot of messy learning yesterday, and perhaps that was more important than the sometimes bumpy ride.
Students have taught me something similar in chess class. There is a months-long stretch after learning how the pieces move but before learning to play a whole game, and years ago I designed a curriculum to cover that stretch. But students rebelled (boys especially), saying in effect, “I don’t want you to teach me all these clever ideas. I would rather have the excitement of playing many games, losing many games, and slowly bumping into those same edeas. Then will be the right time to teach me.” The style is more ad hoc and opportunistic, like yesterday when we figured out that Naomi would have edged Meliya by .1 in a poker round if she had only played her cards right. After that motivating loss, I know she’ll nail it the next time around. But in chess, it took me about two years to ease up on my carefully planned curriculum, because I can be quite stubborn when I know I am right.