The session will explore multiple lines of evidence indicating that repeated addition of equal groups of discrete objects cannot effectively support the development of students’ understanding of multiplication and multiplicative relationships. In particular, repeated addition cannot support the learning of mathematical and scientific ideas that dominate the middle grades (5–8), including area and volume measurement; ratio, rate, similarity, and proportional reasoning; linear functions; and many scientific concepts, e.g., force, work, torque, and density. The session will draw on multiple perspectives, including those of professional mathematicians and educational researchers, and theoretical perspectives. Educational perspectives include analyses of curricular materials and of students’ work in varied multiplicative contexts. Parallels between current curricular and teaching practice (that focuses on repeated addition) and the treatment of multiplication and multiplicative reasoning in the Common Core Standards will be drawn. The symposium organizer will introduce the session and moderate the discussion. There will be no discussant. The session does not squarely address one of the three priority areas.

 The session is educationally significant because current curricula and teaching practice take repeated addition of equal groups of discrete quantity as a sufficient initial foundation for learning multiplicative relationships.

 Introduction: Jack Smith, Michigan State University (3 minutes)

 A Mathematician’s Concerns; Keith Devlin, Stanford University (~12 minutes)

This initial presentation will recount the author’s arguments for the problems that result from associating the multiplication operation with repeated addition. These arguments have been previously presented in his columns (Devlin’s Angle) on the Mathematics Association of America’s website between 2008 and 2011. The core argument is that multiplication must be presented, from the early years, as separate and distinct operation from addition. The fact that repeated addition and multiplication return the same numerical result in some situations is not ground for presenting multiplication as a kind of addition. The presentation will trace the problems of learning more advanced mathematics that result from thinking of multiplication as repeated addition. It will also include the author’s sense of more appropriate initial models for multiplication.

 Splits & Splitting: A Non-Additive Psychological Foundation for Understanding Multiplication; Jere Confrey, North Carolina State University (~12 minutes)

One entry into student reasoning in multiplicative structures is through fair sharing and the development of equipartitioning/splitting. The research reported in this presentation explores different paths in the evolution of division, multiplication and ratio. Fair sharing of discrete collections and continuous wholes serves as the foundation of partitive division but depends on what makes students believe they have formed equal groups and how they name and justify those groups. A project goal is to clarify the development of fair sharing. The research has also identified the idea of reassembly—how the parts are put together and how the whole relates to the parts, e.g., “times as much” or “times as many.” Careful examination of student thinking in these contexts has identified more elements of early multiplicative reasoning. These include the composition of splits, factor-based compensation, and covariation as a precursor to ratio as students work with multiple wholes and move towards the generalization, N wholes shared among P people results in N/P wholes per person.

 The Multiplicative Sleight of Hand: The Curricular Treatment of the Area Formula for Rectangles; Jack Smith, Michigan State University (~12 minutes)

The area of rectangles is fundamental to area measurement for all two-dimensional shapes. On one hand, rectangles are decomposed and recomposed into other basic shapes (triangles, parallelograms); on the other, rectangular area is where area methods shift from counting squares to multiplying lengths. Once this threshold is crossed, determining the areas of simple and complex shapes becomes primarily an application of area formulas. This presentation will report a detailed analysis of three elementary written curricula’s treatment of area measurement and how the formula for the area of rectangles (length x width) is developed. All three move from counting all enclosed squares, to counting composite units of squares (rows and columns), to multiplying lengths. But the third step is not explained by the first two: How two orthogonal lengths multiplicatively compose area (a count of squares) remains conceptually mysterious. The presentation will focus on particular lessons in each curriculum where the formula is introduced and motivated. Connections to the treatment of area measurement in the Common Core Standards will be drawn.

 Middle School Students’ Understanding of Cartesian Product Problems: A Context for Learning to Square Quantities; Erik Tillema, Indiana University at Indianapolis (~12 minutes)

This presentation will report research on the cognitive operations (i.e., mental actions) that middle grade students used to solve two types of multiplicative problems: Cartesian product and repeated groups. The presenter will argue that Cartesian product problems can support a meaning of multiplication for squaring a quantity in ways that repeated groups problems cannot. So Cartesian product problems can serve as an entry point for developing a meaning of multiplication for raising quantities to whole number powers—the power meaning of multiplication. In doing so, the presentation will address prior research findings that students have difficulty differentiating between a linear operator meaning of multiplication (e.g., doubling, tripling, etc.) and a power meaning of multiplication (squaring, cubing, etc.).

 Developing multiplication from a Dayvdov quantity perspective; Barbara Dougherty, University of Missouri (~12 minutes)

From the Davydov-Elkonin (D-E) perspective, all elementary mathematics can be developed from the foundation of measurement and relationships between continuous  quantities (Dougherty, 2008). This presentation will show how the measurement of continuous quantities can support the development of multiplication concepts. In D-E approach to elementary mathematics, the notion of a unit, the core of measurement, is the basis for the development of multiplicative relationships. The iteration of identical units and the construction of composite units support the view of multiplication as repeated addition. However, limitations arise in this approach as they do with repeated addition using discrete quantity. To develop richer understandings of multiplication, students also need to explore other models, such as area measurement, length measurement, arrays, combinations, and dilations. These models move from unit iteration to relating multiplicatively the size of units to the original referent quantity.

 Question & answer and audience dialogue (~27 minutes)