SOLVE - CRUMPLE - TOSS in Algebra 1: hommage à Kate Nowak

Kate Nowak creates some of the most innovative and engaging practice activities anywhere -- especially for those skill/concept areas that are more like scales and arpeggios than like discovery/inquiry lessons. Some skills, like basic math facts, simply need to be practiced. This is true not because students need to be worn down but rather because it takes the mind and body time and first-hand experience to process these as matters of technique. It takes time to get used to the new realities they represent.

Nowhere is this more true than in tinkering with the multiple different forms and components of linear equations in Algebra 1. No sooner have students gotten the hang of finding the intercepts of a line than they're asked to find the slope. They figure out how to find the slope and the y-intercept, and they're given the slope and a non-intercept point. They figure out how to crawl toward slope-intercept form, but fall on their faces when asked to convert to standard form. Standard form, point-slope form, slope-intercept form, two points and no slope, it's a lot of abstraction to juggle. Mastery is part vocabulary work, part detective work, part scales and arpeggios, and part alchemy of different forms. It's a lot to take in.

Enter Kate's Solve - Crumple - Toss activity. I have loved this practice structure since the day I first read about it, but I have struggled with the fact that the most engaging part of the activity destroys the paper trail/evidence. This was less important with high school students, but it is really important with middle schoolers, I find, because they are so much more literal.

For today's linear equation-palooza in class, I created a basic "score sheet" for each student and I numbered each of the quarter sheets on which I glued blocks of problems (4-6 problems per mini-sheet). I also differentiated them from "Basic" level (Basic-1 through -4, Level 2-1 through -4, etc), so that students could choose their own levels. Students were also invited to work in pairs or groups of three because I find it encourages mathematical language use and increases risk-taking. It also seems to be more fun.

After the "Solve" part of the activity, students brought their solved mini-sheets to me to be checked. If they completed the problems correctly, they got a stamp on their score sheet and proceeded to the back of the classroom where I'd set up the Tiny Tykes basketball hoop over the recycling bin. There they completed the "Crumple" and "Toss" stages, awarding themselves a bonus point on the honor system if they made the shot. Then they returned to the buffet table of problems and chose a new mini-sheet.

Because my middle school students like to bank extra credit points toward a test wherever they can, I like to attach these to practice activities such as this one or Dan Meyer's math basketball. Being more literal and concrete than high school students, middle schoolers seem to find great comfort in the idea that they can earn extra credit points ahead of time in case they implode on a quiz or test. What they don't seem to realize -- or maybe they do realize and they just aren't bothered by it -- is that if they participate in the process, they win no matter what. Either they strengthen their skill/concept muscles and perform better and more confidently on the test; or they feel more confident and less pressured because they have banked a few extra-credit points for a rainy day; or both.

It was fun to hear my previously less-engaged students infused with a rush of sudden, unanticipated motivation to tease apart a tangled ball of yarn they have previously been unmotivated -- or uncurious -- to unravel. And something about the arbitrary time pressure of trying to complete as many problem sheets as possible in a short period was also fun for them. I'm feeling a little ambivalent about not having found the secret ingredient of intrinsic motivation in this required blob of material. But I am grateful that, once again, an unexpected game structure generated what the late Gillian Hatch called "an unreasonable amount of practice."

The last word goes to the one student who put it best: "The crumpling is definitely the most satisfying part."