As usual, there was a lot of warm and fluffy language about finding things that different students had shown us in their pieces of worksheet work that they are "smart at," but nobody mentioned the elephant in the room — namely, that students were not demonstrating any form of active engagement with the actual problem situations at hand.
Where, I kept trying to ask, is the sense-making???
Now, don't get me wrong — I know that kids are smart. Very smart. And these kids clearly understood that they were being measured on whether or not they slapped numbers into the quadratic formula or came up with enough lines' worth of symbols to create the appearance of the desired forms of problem-solving.
Unfortunately, that also meant that they clearly understood that, if they had taken the time instead to write out words, phrases, or whole sentences; to challenge the diagram by drawing a different diagram; or try to figure out what variables in the text go with what parts of the diagram, they were going to get marked down. A lot.
These thoughts left me wondering, where are these kids being challenged to engage in sense-making?
This has been on my mind a lot lately because over the past week, we've been working on systems of linear equations in Algebra 1. Textbook explanations simply suck. They are vast, multi-colored makeup experiments on dogs wearing bandannas, with slashes of red and blue arrows pointing everywhere and nowhere.
Textbook explanations do active harm to student reasoning and sense-making.
We needed to escape from the textbook.
I did a lot of thinking and reflecting and being willing to ask stupid questions over the course of a week and I came to a few important realizations. First of all, there are only three main categories of linear systems problems that show up to torment Algebra 1 students: (1) upstream/downstream problems, (2) mixture problems, and (3) number and digit problems.
So I decided we would take some time to work our way through the sense-making aspects of each of these types of problems. If nothing else, at least my students would have some skills for actively making sense of the crap we throw at them. But at best, they would have tools to connect their mathematics to their beautiful, intuitive common sense about the world around them.
I was rewarded after today's test with smiles and much more confident statements of belief by students about their own increasing success.
All of this has confirmed my newest hypothesis, which is as follows:
if I'm not teaching sense-making, then I'm not actually teaching modeling.
The process of decoding and translation and re-encoding into symbolic form was so powerful for my students that I felt a need to document it for myself, so I do not forget or lose sight of this fact. I discovered that, if I spend time working through guided interpretation and translation of situations as a means of scaffolding the up-front part of the modeling process, it pays us all dividends in student engagement and clarity and success.
Here is how we started.
You are allowed to use your own vocabulary!
It is often super-valuable to use a table to organize your info and build out your equations!
r is the trickiest part here. It consists of two elements: (1) the protagonist's basic speed, which I declared to be "b"— "b" for "basic speed." There is also the idea of (2) a current (air, water, wind, whatevs), which I labeled as "c" for "current."
The secret of these problems, my students explained to me today on our test, is figuring out what the relationship between b and c is. Is c helping the protagonist's progress... or is it hurting?
"Helping or hurting?" over and over again as we modeled different situations.
This required a lot of small-group and whole-class discussion and work on vocabulary. We made a word board (we don't have a whole wall):
helping = downstream, tailwind, downhill, with the current
hurting = upstream, headwind, uphill, against the current
If c is helping the protagonist's progress, then r = (b + c)
If c is hurting the protagonist's progress, then r = (b — c)
Once you figure that stuff out, you can make a table.
And at this point, students can usually build their equations and solve with minimal help. But they really need some scaffolding and guidance about how to take their common sense and mathematical understanding and bring this modeling problem to the next level, where they know what to do.
I was wracking my brain until my colleague Robert said, "Oh — I just draw it for them this way:"
Sometimes a simple solution is the best.
NUMBER AND DIGIT PROBLEMS
The hardest part of these is organizing the given information and identifying what you want or need to turn into a variable.
STEP 1: Name the numbers (I use capital letters A and B to start with)
STEP 2: Rewrite A & B using your knowledge of place value
A = _x__ _y__ = 10 (_x_) + 1 (_y_) ; this means that A = 10x + y
B = __y_ __x_ = 10 (_y_) + 1 (_x_) ; this means that B = 10 y + x
x + y = 7
B = A + 27
STEP 4: Substitute what you figured out in Step 2 into Step 3
x + y = 7
10y + x = 10x + y + 27
STEP 5: Simplify
x + y = 7
9y — 9x = 27
Can you divide through by 9?
Final equations: x + y = 7 AND y — x = 3
STEP 6: Solve
STEP 7: What was the problem asking for — digits or numbers?
A = 25
B = 52