Tl;dr: authentic opportunities for kids to pay attention to detail in ways that actually matter via Estimation 180.
Let’s rewind almost two years. I’m student teaching, getting close to certification, already landed my first full-time job. I’m also delving into the MathTwitterBlogosphere for the first time. I stumble across these things called the “Standards for Mathematical Practice“. Well, I had barely read the other standards (you know, the ones about the slope of the function and why that stuff is congruent to those things and etc). I read them, and my first thought was along the lines of “these sound cool”.
Fast forward six months. A few weeks into my first year teaching, I feel like I have a better grasp of what they mean. I’m looking for chances to engage in modeling, picking tasks to provide opportunities to see structure. But I’m stuck in the Stone Age of Attend to Precision. I was convinced it meant little more than “and don’t forget to write your units!”
Fast forward another year. Now I see Attend to Precision everywhere. Everywhere. It’s what mathematicians do. It’s what non-mathematicians do wrong and get in trouble with their boss for. And I’m thinking every day about how I can create meaningful opportunities for my students to Attend to Precision (yes, I’m going to capitalize it, probably forever).
Anyway, I traded number talks for Estimation 180 a few weeks ago. I was losing engagement, winter was coming, and we needed a change to the routine. Right now we’re diving into this sequence of estimation tasks:
So let’s flashback again. December-ish. Systems of equations. Something about Betsy buying tulips and roses and the tulips cost $3 and the roses cost $2 and she bought 30 flowers and spent $81 and for some reason she doesn’t know how many roses she bought. Help Betsy out.
And little Johnny says “oh! 9!”.
And of course I say “9 what? How is Betsy supposed to know what she has 9 of?” in my best funny voice.
I really said that. Seems stupid now. The question was asking about roses. Anyone could have said 9.And who cares about Betsy’s roses anyway?
Ok, back to this week. I throw up the first estimation task, the one with the pennies. Kids think, kids discuss, I pick a few popsicle sticks. Kid says “102”. And the whole class kindof chuckles quietly. “102 what? And this time, I really meant it. It mattered. Kids had discussed the task, they wanted to know the answer, and there was tension over the difference between pennies and dollars.
I actually debated this leading into the task. Do I make sure they understand what is meant by value? Do I give them an example or non-example? Or do I just casually throw it at them? This is the central question, for me, both of this task and teaching tasks in general – how much is too much? In this case, nothing was just fine. A bunch of kids were thinking in terms of pennies instead of dollars, but that was fine — they worked it out in the discussion. And the next day they got another chance to try with nickels. Tomorrow, dimes. Thursday, quarters.
Next, the numbers all work out really well (spoiler alert). The pennies are worth $0.50, the nickels are worth $2.00, the dimes are worth $5.00, and the quarters are worth $10.00. No weird numbers here. And that reflects the world — we tend not to package things in groups of 73 when we can avoid it. More importantly, it provides opportunities for kids to Attend to Precision again. They notice that the pennies work out to a nice even number, then they use that to refine their estimate for the nickels. Or they don’t notice — and that’s fine, they realize when all of their peers’ guesses for nickels are $1.50, $2.00 and $3.00 that some estimates are more logical than others. And they want to get the answer right. It means something to them. I’m not really sure why it does, but it does. And Andrew Stadel put together a whole slate of great money estimation challenges, with a variety of structure. Check them out!
Anyway, this, for me, is Attend to Precision. Give students a reason, a reason they care about, to pay attention to detail and communicate precisely. And this week, I feel good about making some small strides in that direction.