This post is by reader and strings enthusiast Laura Bofferding, Assistant Professor, Purdue University
I was first introduced to number strings by Jennifer DiBrienza (one of the teachers highlighted in Young Mathematicians at Work: Constructing Number Sense, Addition and Subtraction by Fosnot and Dolk) when we both worked as teaching assistants for an elementary mathematics methods course. Captivated by their complexity and ability to hook students of all ages, I began to use number strings myself with teacher candidates. Now, as an assistant professor at Purdue University, I routinely explore them with my undergraduate mathematics methods students and require them to try a number string in their practicum classrooms. As I have looked for resources and talked with colleagues about this practice over the past few years, I’ve noticed an increased focus on mathematics instructional routines that people refer to as math talks, number talks, number strings, math strings, cluster problems, and problem strings. Some of these things are not like the others…
Although these terms of often used interchangeably, there are specific differences among some of these practices. Math talk is the most general term that refers to a mathematical discussion when students share their strategies and reasoning around a mathematics problem. Teachers use “math talk moves” such as revoicing, restating, and probing (Chapin et al., 2009) to facilitate the conversation and, through exploring the mathematics that arises, deepen students’ understanding of mathematics.
A more specific kind of math talk is a number talk. Also called strategy sharing (see Lampert et al., 2010), this refers to a practice whereby the teacher picks a single computational problem (or dot frame image) for the class to explore with the purpose of eliciting multiple strategies for solving the problem (or determining the total number of dots). Number talks grew in popularity recently with the publication of the book Number Talks (2010) by Sherry Parrish, although the examples she provides are more like number strings. However, Kathy Richardson and her colleague Ruth Parker may have first developed number talks in their work with teachers in the 1990’s. Their DVD Thinking with Numbers: Number Talks (2010) gave a vivid image of what these routines look like and sound like with students. Richardson describes a number talk as:
A short, ongoing daily routine that provides students with meaningful ongoing practice with computation…..a powerful tool for helping students develop computational fluency because the expectation is that they will use number relationships and the structures of numbers to add, subtract, multiply and divide.
In the more recent interpretation of a number talk, as described by Parrish, the teacher ideally uses representations to help illustrate and clarify the strategies, so that the class develops a better understanding of the strategies. Beyond this, a focus of number talks is delving into the presented strategies to decide in which cases one strategy is more efficient than another and/or when a particular strategy might not be helpful. Of course, problem choice is extremely important. The problem 6 + 8 might engender rich conversation in first grade (e.g., counting all versus counting on, starting with 6 versus 8, ways of making a ten and adding on) but would likely be less productive in third grade.
By contrast, number strings, can be seen as a series of number talks strung together — a more specific and elaborated form of a number talk with an additional focus on mathematical models. Also developed through work with teachers in the 1990s, number strings were first formally introduced in 2001 by Cathy Fosnot and Maarten Dolk. They were described in their series Young Mathematicians at Work and later became codified for teachers as part of the Contexts for Learning curriculum, published in 2007.
Whereas number talks could also be classified as sharing strategies, number strings are more complicated to classify. If I had to try, I might suggest comparing and connecting strategies and problems. Unlike number talks, number strings consist of a sequence of computational problems (or a series of dot patterns or quick images), with problems chosen so that a particular strategy is likely to emerge from students (e.g., compensation, doubling & halving) or big idea is likely to be discussed (e.g., commutative property, distributive property) (see DiBrienza & Shevell, 1998; Fosnot & Dolk, 2001). Number strings are carefully crafted and often follow a pattern in which a few helper problems are presented to support students to solve a challenge problem. For example, solving 10 x 6 (helper) and then 7 x 6 (helper) should help students solve 17 x 6 (challenge). After a few of sets of helper-then-challenge problems, it is not uncommon for a number string to end with a challenge problem with no helper problem preceding it. The removal of the scaffolded helper problem is designed to help students to transfer strategies in smart ways.
Further, central to number strings is the use of models (e.g., empty number lines, arrays, ratio tables) as a way to not only understand and analyze the strategies presented but also to make connections among the strategies. As with number talks, it is likely that conversations will naturally arise about how students chose their strategy and whether it was an efficient one; however, a greater emphasis is placed on exploring relationships between values and problems.
The term cluster problem is familiar to teachers using the TERC Investigations in Number, Data and Space curriculum. As with number strings, cluster problems are a series of problems consisting of easier problems that help students solve a more complex problem. The main difference is that students often solve these problems as part of written work — although they may be discussed later — and often must indicate how they used the helper problems to solve the more complex problems..
Finally, the term problem string was coined by Pamela Weber Harris whose recent work Building Powerful Numeracy for Middle and High School Students (2011) was a direct outgrowth of number strings work by Fosnot and Dolk. Though the terminology changed from number to problem string, the routine is used in an ideologically similar way. Harris’s work extends the number string work to middle and high school and beyond. This allows her to develop the notion of algebra and geometry strings, as well as pure computation strings.
Although these routines are slightly different, they all share a focus on children’s mental math strategies. Further, these routines address the communication and connections standards (NCTM, 2000) as well as the Common Core practice standards, supporting students’ learning in rich ways. However, as we continue to research the benefits of each of these routines, it is important that, as a field, we use specific vocabulary to distinguish them from each other so that we honor their differences noted above.
Chapin, S. H., O’Connor, C., & Anderson, N. C. (2009). Classroom discussions: Using math talk to help students learn (2nd edition). Sausalito, CA: Math Solutions.
DiBrienza, J. & Shevell, G. (1998). Number strings: Developing computational efficiency in a constructivist classroom. The Constructivist, 13(2), 21 –25.
Fosnot, C. T. & Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division. Portsmouth, NH: Heinemann.
Fosnot, C. T. & Dolk, M. (2001). Young mathematicians at work: Constructing number sense, addition and subtraction. Portsmouth, NH: Heinemann.
Harris, P. W. (2011) Building Powerful Numeracy for Middle and High School Students. Portsmouth, NH: Heinemann.
Lampert, M., Beasley, H., Ghousseini, H., Kazemi, E., & Franke, M. (2010). Using designed instructional activities to enable novices to manage ambitious mathematics teaching. In M.K. Stein & L. Kucan (Eds.), Instructional explanations in the disciplines (129-141). New York, Springer.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.