When I first saw this image, I have to admit, it didn’t match what I had been calling a Number Talk in my classroom. Having started my journey with Number Talks with Parrish’s book, I saw it as a string of problems with a specific strategy as the focus of the string that got progressively more difficult (which sounds more like a string in reading this slide). As I started creating my own and making variations to them over the past couple of years, I have simply started calling them Math Routines….it was just easier than trying to make things fit into a certain framework in my brain. After going back and forth about terminology, I started looking at these structures more in the sense of purpose than the name and I am finding it very interesting. Please keep in mind as you read, this is completely from my experience with Math Routines in the classroom and I find all of these talks so incredibly valuable!

First comparison: Single problem vs string of problems. In my experience, I think these two designs have a substantial difference in purpose. When I give one problem, I am going for one of two things: a variety of strategies to see where students are in their thinking OR connections/comparisons of multiple strategies. Personally, I like the variety of strategies before and after we have investigated different ideas that could impact their thinking. For both the students and myself, it shows growth and I can use what I find to help me in planning future routines. Connecting and comparing takes a bit longer and, for me, I don’t get as many strategies out because we focus on connecting and comparing only a few for time’s sake.

Second comparison: Difference in having a model and context. I don’t give a model or context unless a student brings one up in a explanation. If it comes from them, I go with it, if it doesn’t, I don’t write it up there. If there is a specific representation I am hoping comes up because we have been doing work with it during our math class, but doesn’t, I do have them do a quick journal response. I will ask them something such as, “How was our number talk similar to or different from our work in math class this week?” Then I can read their journals and have some students share the next day about the representation connection they made. I also have had students create contexts after we finish a number talk with a prompt such as, “Write a context that you think fits one of the problems in the string. How does the context change for another problem in the string?” For example if the string has “16 x 4” followed by “16 x 8” I am hoping to see connections between the two problems in the context.

Nothing to compare in the final piece, multiple strategies discussed in both!

In the end, students are talking math with a mathematical purpose so they are win/wins in my book, but I am curious to hear what others think around the purposes of different formats of these routines! Would love to hear other ideas so comment away!

-Kristin

Leah WolvertonYou might take a look at another book, Intentional Talk by Elham Kazemi and Allison Hintz. They differentiate different types of number talk activities and identify the purpose of each. Quick read…we used it in my district for a book talk and it fostered some great discussions!

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Maya QuinnSince you wrote that you would “love to hear other ideas,” I leave a comment with no assurance of its relevance.

Zeroth, a pedantic quibble with the accompanying tweet: Really ‘sets’ is not the best word to describe groups of problems that have been developed in a ‘sequence’. The former term refers to a collection, and, though they may be numbered, the problems are often done in the order most accessible to students. On the other hand, a sequence of problems suggest that they be done in a particular order.

Okay: First, if one encourages multiple strategies, then it is a bit tough to nail down what constitutes a “single problem.” More generally, rich tasks often have multiple components (sometimes in the ‘set’ form: i.e., without an intended sequencing). Is such a task considered a “single problem”? I don’t really know. (I could count the number of question marks, but I do not think this would be a reliable measure.)

Second, I am curious about what is meant by a “model.” Is it a model in the sense of modeling student thinking/conceptions? Is it modeling in the sense of mathematical modeling and CCSS-M? I cannot glean this from a single slide, though I am sure the answer is Somewhere Out There. (Actually, I am not quite sure what is meant by a “context,” either: Is this context in terms of, e.g., a real-world context? Or does it simply mean contextualized within the ordered sequence of problems? Or something entirely different?)

Third, I am a believer in discussing multiple strategies. However, I would have guessed that enacting this component would differ between the two approaches: In particular, if I am to use a sequence of problems (and I do, incidentally, use an ordered sequence in class!) then I not only want to elicit multiple strategies, but I want to see the multiple strategies evolve in ways that fit in with the sequence: Are algorithms emerging organically? Are lower-level strategies (e.g., guessing-and-checking) rearing their heads despite the articulation of higher-level strategies? Crucially, the multiple strategies discussed for a single problem may affect the wording for that problem; but the multiple strategies discussed in the context of an ordered sequence of problems may affect the wording for that problem, the wording for previous and subsequent problems, and the order of the sequence. (All of these points may have been broached in the actual talk; I do not expect the uploaded slide was intended to stand alone!)

Finally, for a dose of concreteness and actual mathematics: For those of us who sequence problems carefully, is there language to describe the tangents or interludes that can provide additional support for developing understanding? At the moment, I am teaching a summer course for 10 preservice elementary school teachers, and we are working through a sequence of problems on topics related to proportional thinking. Yesterday, 2 of the students were not able to make it to the class, which lasts 170 minutes. And so I deviated from the sequence to ask a real-world, proportional-thinking contextualized in-our-class problem: If we have typically spent 170 minutes for 10 students, then how long might we spend with just 8 students? Indeed, this gave students a chance to warm-up and think proportionally: They set up the proportions one might expect, 170:10 = N:8, and solved using a variety of methods (about three or four). Since they seemed engaged (perhaps the notion of ending class early was appealing!) I decided to deviate even further: Really our class is 170 minutes for 11 people — when a few faces countenanced confusion at this remark, I gently reminded them that I, too, am a person! If instead of class time and # of students we think about class time and # of people, then what will this effect be on the adjusted class length? One student quickly asked if they would just be equal (no effect at all); I only noted that they had already found the previous proportional class length, N, and wondered what would happen in this new setup: Would the adjusted class length be equal? Shorter? Longer? And then I left them to their own devices.

Again, the students were engaged with the task, and ultimately resolved it using proportional reasoning. (The setup of 170:11 = M:9 is decidedly more difficult to deal with; however, they were able to explain why M > N: despite an initial intuition that M would be less than N, and without computing M explicitly.)

Now: There was plenty of mathematical talk; multiple strategies were used; and the “single problem” (?) of comparing outcomes in class length with respect to # of students and then # of people was neither pre-planned nor intended to fit into an ordered sequence. It was truly improvised around the day’s attendance, but supported the general topic as well as the problems that preceded and succeeded it. At the moment, I would describe this as a “warm-up interlude”; but I also suspect that it enriched the problem sequence. Maybe a simple question to ask is, in the first comparison, does “developed as a sequence” include the tasks developed inside of the classroom?

(Sorry for the verbosity!)

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