A recent conversation between two New York City math educators, Carol Mosesson-Teig and Kara Imm
Kara,
I have been thinking a lot about the strings that you modeled last week for the city-wide math content seminar and thought I’d share some of my thoughts.
Throughout my history with Math in the City, context was hugely important and a constant topic of conversation. It was understood that context helped students mathematize concepts and therefore to understand them at a deeper level. We learned how difficult it was to come up with appropriate contexts as Cathy Fosnot was very clear to talk about the difference between realistic context and realized contexts which helped us as teachers understand what we were trying to accomplish with our students.
But, when we talked about the string work, we talked about the mathematical big ideas we were trying to help the students construct. The context of the strings was bare number and that in this work, we were helping the students develop number sense, becoming flexible thinkers, computation solvers by seeing relationships in number. It was amazing (and fun) work, and work that I continue to discuss today with every administrator, math coach and teacher with whom I work.
Fast forward several years, your presentation of strings at the content seminar was very different from what we had discussed. Your incorporation of a context — and your decision to stay within it for the whole string — onto the middle school and high school string, was new.
As a 4th grade teacher, I remember having my students explain the relationship of doubling and halving in two division problems (e.g., 24 ÷ 6 = 4 and 24 ÷ 3 = 8). In the class, there was an agreed upon sharing context of brownies at a party — if there were 6 people sharing 24 brownies, each person would get 4 brownies. But, if half as many people needed to share the same amount of brownies, then each person would get double the number, or 8 brownies, which was followed up with a representation on an open array.
For me it was one (in hindsight) very simplistic context and it wasn’t carried out for the full string. Now, I am seeing this in a very different light, I think a context (and not the same one always) could be extremely helpful in helping to build sense-making for the students.
— Carol
Carol,
Thanks for your e-mail.
You’ve got me really thinking about what I believe about context and numeracy routines. When I think back to my education at Math in the City, I think I understood contexts to be believable or imaginable situations that kids could mathematize — places where kids had bought into the “story” and allowed us as teachers to build the mathematics we wanted them to explore. Contexts for me were part of messy multi-day investigations.
My early learning on strings, however, rarely included context with the exception of money and the clock (which could be considered both contexts and models). Mostly strings were bare number problems that allowed us to make sense of mathematical relationships or big ideas, and take up new strategies.
What you said to me last week was that some math educators believe that in strings “the number was the context.” That really struck me. I encounter so many kids — on the rug in Kindergarten all the way up to seniors repeating Integrated Algebra — for whom number isn’t a useful context on its own. Just looking at the numbers isn’t enough, for them. They never got to construct the kinds of big ideas and generalizations that you and I tried to do with our students. So when working with kids like this, I have found that having a believable or imaginable story around which to reason about number makes all the difference. Road trips with my dad, packs and sticks of gum, Ferris wheels that have broken loose and are rolling to the next town — whatever it takes to help kids “enter” into the math. I’ve come to really value the role that story can play, particularly when it serves to make math interesting, intriguing and accessible to kids.
There’s also the question of how the context gets taken up by kids. Earlier, when leading money model strings for fractions I used to say nothing about money and somehow hope that students would offer money as a context for operating with fractions. Instead, I got a lot of kids who would offer the algorithm over and over again. So I stopped hoping and instead situated the string right in the context — I used a narrative to guide the string. Now when I lead a string I recognize my choices, which are informed by who my students are and what they need:
Over time I can let go of the story aspect of the context because the class has made sense of the relationships between fractions and money. Later I want kids to use context on their own — choosing, for example, whether a fraction problem like 3/4 + 1/6 is easier to think about on a clock, using money as a model, or perhaps with no context at all.
Thanks for getting us to think about these issues…..
— Kara
Hi Kara,
At the macro level, context makes strings, like all mathematical ideas, less abstract and more tangible, by providing “real-world” connections. This is at the heart of recognizing who students are and what they need.
Indeed, it seems to me now that situating strings in believable — or even fantastic — contexts will ultimately enable more students to realize that all of the contexts boil back down to the same essential mathematics (no matter how big or small the numbers). Many contextual examples should be provided before inviting students to construct their own contexts, so that they have a useful, more developed tool-kit from upon which to draw when working independently. Otherwise there is the danger that it will become more of a creative writing endeavor, rather than an aid in developing computational (and ultimately problem solving) skills.
After hearing a variety of situations involving 3rds, 6ths, or 12ths, students will realize that a clock might help them to visualize that context, but if exposed to 10ths and 100ths, maybe money or a football field would give them a better way to contextualize the problem. Exposing students to a variety of contexts, will give them a better background to recognize and understand when to use which contexts to help them solve problems, but also to understand the underlying mathematical relationships inherent in all of these contextualized situations.
— Carol
Number Strings 
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