In this issue we look at the second Common Core Mathematical Practice, “Reason abstractly and quantitatively.” What does it mean to reason abstractly and quantitatively as students work on solving a Problem of the Week?
I’ll start with a story that helped me think about what “concrete” and “abstract” mean when solving math problems with students:
The Story
I was teaching 5th grade kids about area and perimeter using this scenario: you have 36 meters of fencing and want to build a rectangular frog pen using all of it. What are some different pens you could make? If each frog needs 1 square meter of space to flourish, how many frogs can your pen designs hold? Which design holds the most?
One traditional model of teaching suggests that what’s hard for students when solving word problems is getting rid of the fluff and decoding the underlying abstract mathematics hidden in the context, and that if the teacher can restate the problem in mathematical language, it will support the students to solve successfully. Here’s what I observed when we used that model:
Students’ Concrete Action |
Teachers’ Abstract Response |
Student’s Concrete Response |
Mention 36 meters of fence | Re-state the idea as “the perimeter is 36 meters” | Ignore the word perimeter, not use any of the teachers’ taught strategies for finding side lengths of a given perimeter. |
Use guess and check and drawing pictures to try to find different shaped rectangles that would use 36 feet of fencing; it’s taking a while. | Remind the student of the “hint” that the first step is to “divide it [perimeter] in half. What is half of 36? Can you find two numbers that add to 18?” | The students can, but as soon as the teacher leaves, they start looking for 4 numbers that add to 18 because they look at the picture and remember that rectangles have 4 sides. |
Mention that each frog needs one square meter | Ask, “great, what do square meters measure? Area? Yes! Now you need to find the area of each pen you came up with in part 1.” | Ignore the suggestion to find area; give up on the problem; raise their hand to ask for more help. One student tells me, “I know how to find area, but I don’t get what that has to do with how many frogs can fit.” |
The next period we tried an alternate model, in which the context was used to elicit the students’ concrete ideas, and the concrete ideas were valued. We helped the students organize their ideas and look for patterns. In short, we avoided abstraction that the students didn’t suggest, and instead supported organization, pattern recognition, and referring back to the concrete.
Once we established that when frog farmers say “pen” they mean fenced-in-space-for-keeping-animals-safe, not ink-based-tool-for-writing, there was enough going on in the context that the students had some ideas about how to draw different pens, check if they fit the farmer’s specifications, and try to fit the frogs into the pens.
Students’ Concrete Action |
Teachers’ Organizing Response |
Student’s Concrete Response |
Mention 36 meters of fence | Great, that’s one of the requirements the farmer has | Check their guesses against the 36 meters of fence constraint |
Use guess and check and drawing pictures to try to find different shaped rectangles that would use 36 feet of fencing; it’s taking a while. | Organize the guesses that worked into a chart with the columns Length and Width | Immediately generate all of the other missing fence shapes that work, and confirm they had them all. No one explicitly mentioned that L + W = 18, but it was clear from the speed of their mental math they were using some version of that pattern. |
Mention that each frog needs one square meter | Diagnose student understanding by asking, “how many frogs do you think will fit in one of your pens?” | Make guesses using reasoning that shows they aren’t making sense of the area the frogs take up: 36 frogs or 9 frogs (each square meter uses 4 of the meters of perimeter). |
Assume that 36 meters of fencing means 36 frogs will fit in each pen | Invite students to use a drawing to show how many frogs will fit in a pen | Suddenly blurt out, “I can just multiply these! 6 rows and 12 columns of frogs is 72 frogs!” and even “that’s just the area!” One student who filled her 3×15 pen with lots of small squares (over 100) suddenly said, “I did it this way but I wasn’t supposed to. It should be 45 frogs but I drew the boxes too small. All I had to do was multiply.” |
The Common Core Math Practice encourages students to decontextualize and contextualize. Sometimes that means that as teachers, we need to get out of the way and stop helping students decontextualize. Instead we can support them to:
- determine if their thinking makes sense in the story
- organize their thinking to help them see patterns
- ask students to make conjectures (guess) and check their thinking based on the context
- encourage multiple representations for the same problem
When we let students be in charge of decontextualizing and making use of contexts, sometimes they surprise us!
PS — If you want to be surprised like I was at how often we do all the contextualizing and decontextualizing for students, watch Annie Fetter blow you away with 5 minutes jammed with examples of teachers “helping” students by thinking for them!
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