# Teaching Problems: Teaching to Deliberately Connect Content Across Lessons

This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.

This chapter of Lampert’s book has been the most interesting to me so far, but also the most complex. There is no three-step approach to connecting content across lessons. Instead, it’s a great deal of thinking about reoccurring representations, connecting contexts to content, using language purposefully, and formative assessment. I don’t know how I could summarize the chapter as a whole, but Lampert does investigate one domain that I had not thought much about before.

Lampert poses a problem to students:

a knot is one nautical mile per hour

She spends the class discussing the first problem, which students struggled with, and uses this diagram, which she calls a “journey line”, to relate the context and the mathematics.

Looking at student work after the class, she notes that only a few students make assertions about the second problem, and only one student, Charlotte, tries to figure out precisely how long it would take the Mimi to go 60 nautical miles at 8 knots. This problem is significantly more complex because the numbers don’t divide evenly. Lampert’s students are still struggling with the their knowledge of division, but Lampert knows she wants to move into teaching about remainders in a future lesson, and uses this as an opportunity to think about how to structure that teaching.

Here is Charlotte’s explanation for part two of the problem:

Charlotte’s interpretation of the remainder is wrong, but that’s not Lamperts focus. She writes:

Using the problem context to give meaning to the remainder is something I would want everyone in the class to be disposed and able to do (200).

This isn’t something I saw when I first read Charlotte’s work. I think it’s an important thing to notice because it is a focus not on the how, the mechanics of a computation or procedure, but instead a why, a habit of making sense of mathematics and relating mathematics to the world. It’s a change in perspective from looking at Charlotte’s reasoning as lacking because her answer is wrong, to looking at her reasoning as useful because it is attempting to make sense.

But more than using this piece of work as an evaluation of Charlotte’s thinking, it provides a “way in” to future mathematics.

Examining the mathematics involved in Charlotte’s assertion led me to think that I should pose problems that would engage students with these ideas. Although Charlotte may have been ahead of everyone else in confronting the problem of giving meaning to the remainder, I could use her work to clue me in to the kinds of things other students might do and what they should be able to do. The challenge for me would be to structure the learning environment so that neither Charlotte nor anyone else would be satisfied with an assertion about the remainder like the one Charlotte ventured without returning to the constraints of the problem situation to see if it fits.

Here, I expected Lampert to launch into a problem the next day to explore this mathematical territory. Instead, the class spends two days working on problems where a remainder is not necessary, so that students can continue to study division and its relationship to a context — dividing 135 by 5, and 180 by 6. Then, on the third day, she poses a problem about a rate. She uses the same distance, 180 nautical miles, as the day before, but changes the speed to 8 knots so that the division will not “come out even”. This problem was not entirely successful. Several students offered interpretations of the remainder, but the class ended before anyone could offer a coherent explanation for why the remainder of 4 represents half an hour.

The next day, Lampert chooses a similar problem, but instead chooses 20 nautical miles and 6 knots, reducing the algorithmic load on students and providing an opportunity to focus more mental effort on the meaning of the remainder. Lampert prompts students to figure out how long it will take for the ship to go one mile, and to use a diagram. This allows her to launch into teaching about what that remainder means, and to relate the hour to parts of the hour, and to do the unit conversion necessary to move between hours and minutes.

This is a bit of an anticlimactic end to a fascinating series of instructional decisions, but I learned several important things from seeing Lampert’s thinking. First, she is enormously patient, and content for students to leave class, perhaps for several days in a row, unable to solve a problem using a certain piece of mathematics. She does this because she has a laser-focus on sense-making, creating situations where students can believe that math makes sense, and they are capable of figuring out problems by making connections with what they already know. Finally, she launches this entire series of problems with an observation from student work, and uses that work to guide her planning for future lessons and frame her thinking about how other students are likely to interpret a new idea.

I’ve never done this level of thinking in my teaching, and it’s a bit intimidating, but also seems like an enormously valuable way to both examine student thinking, and use that thinking to inform future planning. It also highlights the complexity of connecting content across multiple lessons. Fun stuff to get better at.

## 6 thoughts on “Teaching Problems: Teaching to Deliberately Connect Content Across Lessons”

1. Michael Paul Goldenberg

I’ve actually written about the next problem Lampert posed (that you mention) and what another student wrote in response to it (though I think it came a few weeks later when the class revisited the issue of what to do with the “remainder”). I would need to review the paper or the student notebooks to get the particulars, but I can recall the gist of it if you’re interested.

Meanwhile, I’m struck here by what seems to be something crucial missing in the response. Rather than try to calculate the exact time, I’m puzzled that she didn’t simply state that a faster rate for a fixed distance by definition takes less time. This tells me that reasoning proportionally here is not yet within her grasp (though it will be a major focus of middle school math into which she’d be entering the following year (these are 5th-graders in 1989, well before the Common Core). And that sort of reasoning is often beyond the grasp of high school kids and adults: if
d =r*t (distance – rate * time), then for constant d, what happens to the other variables as one of them goes up? As one of them goes down?

These are “big” questions that I am not able to recall if the kids got into with Lampert in a 5th grade setting (way too much data and I looked at a tiny sampling of it 20 years ago as a graduate student on her project). Now I wonder why, if she didn’t try to elicit some sort of generalization along those lines, she didn’t. State math framework for Michigan in the late ’80s? I wish I could ask her.

2. howardat58

I see serious confusion in the way the kids are expected to deal with a measurement situation using whole number arithmetic. The problem is screaming out for fractions, decimals or rational numbers (call them what you like). remainders are a feature of whole number arithmetic, and whole number are COUNTING numbers. In this problem there is no remainder. The time taken is seven and a half hours. The end.

1. Michael Paul Goldenberg

Howard, I spent more than two years with the data (which are incredibly detailed and of multiple kinds, including video) from Lampert’s 1989-90 5th-grade math classroom. I wouldn’t presume to make the sort of broad statement you just served up even after that much familiarity. I find it hard to understand how someone who has effectively spent ZERO time with that data can presume to understand what’s going on here. You make a statement from the mathematical perspective of an adult. These are a very diverse group of 5th graders. And since I’ve seen where this lengthy arc of problems on time, speed, and distance goes (I still have a notebook with the problems of the day used for the entire school year), I can state that fractions, decimals, or rational numbers are precisely where things are headed (among other places). Do you know where in the school year this particular problem occurs, or doesn’t that matter? Are you intimately familiar with typical 5th-grade scope & sequence in math in Michigan in 1989-90? No? I didn’t think so.

FInally, if you think that teaching mathematics is tantamount to posing a computation problem and solving it (or having the kids solve it) as quickly and easily as possible and then moving immediately to something else, there are countless children who are benefitting greatly from the fact that you don’t teach elementary mathematics. Keep up the good work!

3. howardat58

“Finally…” – I don’t.

I do think that if the problem was supposed to get the kids to think about remainders (assuming that they hadn’t thought about them before) then it’s the wrong problem. Quote “..she notes that only a few students make assertions about the second problem”.