DRAFT-DO NOT CITE Designing, Teaching, and Learning 19

Designing, Teaching, and Learning with Multiple Representations:

A Case from Two-Digit Multiplication

Andrew Izsák

Abstract

This article presents analyses of one set of instructional materials for two-digit multiplication, implementation of those materials in one heterogeneous 4th-grade classroom, and case-studies of student problem-solving from that classroom. The instructional materials were designed to support conceptual understanding of two-digit multiplication by modeling areas of rectangles with an expanded numeric method. A focus on using representational features to accomplish problem-solving goals coordinates the analyses of design, instruction, and student solutions. The coordinated analyses examine classroom teaching and learning by comparing how taken-as-shared classroom practices and individual students accomplished similar problem-solving goals. The analyses lead to results about designing instructional materials with multiple representations, opportunities and challenges when teaching with such materials, and learning two-digit multiplication through a modeling approach.

DRAFT COPY DO NOT CITE

Designing, Teaching, and Learning with Multiple Representations:

A Case from Two-Digit Multiplication

Those involved in mathematics education agree that students' often low performance on a wide range of mathematics tasks poses serious challenges for educational practice and research. One approach to increasing students’ understandings of mathematics has been to develop curricula that use situations–often from the surrounding world–as contexts for initial sense-making out of which problem-solving strategies, computation methods, and conceptual understandings emerge. Such approaches require students to develop understandings of problem contexts and solution strategies simultaneously. Mathematics educators are still trying to understand opportunities and challenges that such approaches present to curriculum writers, teachers, and students. This study analyzes such opportunities and challenges by using rectangular areas as initial contexts in which to develop two-digit multiplication methods.

Multi-digit multiplication is an important, though underrepresented, area of research for two reasons. First, extending multiplication from single- to multi-digit numbers requires two mathematical coordinations that do not arise when extending addition or subtraction. The first coordination is between magnitudes of factors and magnitudes of products. Some single-digit products are of the same magnitude as both factors (e.g., 2 x 3 = 6), but related products differ in magnitude from at least one factor (e.g., 2 x 30 = 60 and 20 x 30 = 600). Students must learn when and how magnitudes of products differ from magnitudes of factors. The second coordination is between expanded forms for factors (e.g., 20 + 8) and the distributive property. Efficient multiplication methods that generalize to arbitrary numbers of digits rely on multiplying each term in the expanded form for one factor by all terms in the expanded form for the second, whether or not expanded forms are made explicit. Accomplishing such coordinations is likely to involve rich cognitive phenomena. Furthermore, if students understand the distributive property in the context of whole-number multiplication, they will be better prepared to apply the property in other domains such as fractions and algebra.

The second reason that research on multi-digit multiplication is important is that U.S. students have performed poorly on such items in international studies. Stigler, Lee, and Stevenson (1990) reported that only 54% of U.S. 5th-grade students in "traditional" courses could solve 45 x 26 correctly. Fourth-grade students in the present study performed better than 5th-grade students in the Stigler et al. study: 88% solved 45 x 26 correctly (Izsák & Fuson, 2000). The coordinated analyses of design, instruction, and learning reported here are an attempt to understand this result and its implications for other instructional approaches and classrooms.

Overview

The article first introduces the two-digit multiplication materials and the perspective on problem solving used in the present study. The article then reviews relevant research on multiplication, external representations, and classroom teaching and learning. Next, the article traces the evolution of whole-class strategies in one 4th-grade classroom over the course of the entire two-digit multiplication unit. The analysis of whole-class strategies reveals opportunities and challenges for teaching and learning multi-digit multiplication with rectangular area models. The article then compares whole-class strategies to strategies used by a cross-section of students during interviews to solve similar problems. The analysis of student solutions illuminates which aspects of whole-class strategies students appropriated and student difficulties that did not surface in whole-class discussions. The article closes with implications for revising the multiplication materials and, more generally, for designing, teaching, and learning with multiple representations.

The Children's Math Worlds Two-Digit Multiplication Unit

The present study was conducted in the context of the Children's Math Worlds project (CMW), an on-going project that develops instructional materials for elementary school mathematics and conducts research on teaching and learning as teachers use those materials in their classrooms. A main objective of CMW is to make the goals of the Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics, 1989) and the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000) accessible to urban and suburban students and teachers. Relevant to this study are standards about number and operations, geometry, representation, measurement, and problem solving.

The CMW two-digit multiplication unit builds on a preceding unit that develops connections among single-digit multiplication, repeated addition, arrays of unit squares, and areas of rectangles. The single-digit materials include rectangles with drawn unit squares and ask, "What is the total number?" To take one example, students can calculate the area of a 3 by 5 rectangle by counting individual unit squares, adding groups of 3 or 5 unit squares, or recalling 3 x 5 = 15. The materials also treat units for lengths and areas (e.g., centimeters and square centimeters). The two-digit multiplication unit begins with students’ existing connections among multiplication, repeated addition, arrays of unit squares, and areas of rectangles developed in the single-digit unit and scaffolds coordination of magnitudes of partial products, expanded forms for factors, and the distributive property. The next two sections describe the sequence of problems and external representations used in the materials and the perspective on external representations and problem solving used in this study.

The Sequence of Problems and Representations

The CMW two-digit multiplication unit begins with factors between 1 and 19 and continues using unit squares representations that were introduced in the single-digit multiplication unit (see Figure 1a). Students discover that effective single-digit strategies, like those described above, become inefficient with larger factors. This motivates the need for new, more efficient strategies.

The CMW materials scaffold new strategies with two additional area representations and a numeric method that break apart factors and products into smaller, easier-to-handle pieces. 100s/10s/1s representations break apart lengths (factors) into a ten and ones and areas (products) into groups of 100, 10, and individual unit squares (see Figure 1b). Groups that are 10 unit squares wide and long are called 100 squares. In preparation for larger factors, the materials also introduce quadrants representations (see Figure 1c). Unit squares and 100s/10s/1s representations are

Figure 1. The unit squares, 100s/10s/1s, and quadrants representations for 13 x 14 and the expanded algorithm.

drawn to the same scale, but quadrants representations are not. Quadrants representations are intended as sketches to be used with numbers of any size. Finally, the materials introduce the expanded algorithm (see Figure 1d), each line of which corresponds to one region in quadrants representations. The expanded algorithm generalizes to any number of digits and can collapse to the traditional U.S. method, but represents partial products and multi-digit addition more explicitly.

After developing methods for multiplying numbers between 1 and 19, the CMW materials continue with problems in which factors are at least 20 but products remain under 1,000. Figure 2 shows the unit squares, 100s/10s/1s, and quadrants representations for the problem 28 x 34. When factors are 20 and above, 100s/10s/1s and quadrants representations break apart factors in different ways. 100s/10s/1s

Figure 2. The unit squares, 100s/10s/1s, and quadrants representations for 28 x 34 and the expanded algorithm.

representations contain multiple 100 squares and help students determine magnitudes of partial products (the first coordination discussed above), while quadrants representations scaffold connections between expanded forms and the distributive property (the second coordination discussed above). The unit concludes with problems in which products are greater than 1,000. As the unit progresses, first unit squares and then 100s/10s/1s representations are dropped both for reasons of scale and to transition to numeric methods.

Other two-digit multiplication materials (e.g., Manfre, Moser, Lobato, & Morrow, 1992) have used area representations and numeric methods similar to the expanded algorithm. The CMW materials are distinguished by the coordinated use of all three area representations described above and the expanded algorithm. Three main goals of the unit are for students (a) to use 100s/10s/1s representations to coordinate magnitudes of factors and products; (b) to use quadrants representations to coordinate understandings of the distributive property, expanded forms for factors, and some version of the expanded algorithm; and ultimately (c) to use, understand, and explain two-digit multiplication methods without the aid of area representations. The unit provides particular representations and a target computation method, but also encourages teachers to elicit, discuss, and build upon students' problem-solving strategies. The intent is for teachers to help students understand the area representations and master numeric methods that generalize to larger factors.

Representational Features and Problem-Solving Strategies

Table 1 uses the example of 28 x 34 to summarize the features of each area representation and the expanded algorithm that correspond to factors and products (see Figure 2). Problems that include all three area representations are the most complex. For such problems, factors are represented in several ways numerically and geometrically; products are represented first as questions, then as areas broken into various parts, and finally as sums of four partial products.

Connecting representational features that correspond to the same factor or product, though necessary, is insufficient for solving problems with the CMW materials. Teachers and students must also have strategies for using representational features, like those in Table 1, to accomplish problem-solving goals. In particular, teachers and students must have strategies for accomplishing the following sequence of goals: (1) find groups of (sometimes imaginary) unit squares in unit squares, 100s/10s/1s, and quadrants representations; (2) determine areas of regions found in (1); and (3) find final products by adding areas found in (2). By connecting features across representations and developing strategies for accomplishing goals (1)-(3), students can construct solution methods that coordinate magnitudes of partial products, expanded forms for factors, and the distributive property.

Table 1

Features of CMW Representations that Correspond to Factors and Products for 28 x 34 = 952

Rep-
resentations / Factors
Numeric
Geometric / Products
Numeric
Geometric
Initial
Problem / Word problem describes an equal groups, array, or rectangular area situation with numbers (e.g., 28, 34).
or
Numeric problem (e.g., 28, 34). / The question.
The answer to be found.
Unit
Squares / Numeric labels along top and left hand side (e.g., 28, 34).
Unit lengths along perimeter.
Unit squares adjacent to perimeter. / Total number of unit squares.
Total area of unit squares.
100s/10s/1s / Numeric labels along top and left hand side (e.g., 28, 34).
Factors decomposed into single 10s and 1s along top and left hand side (e.g., 10 + 10 + 8, 10 + 10 + 10 + 4).
Lengths along perimeter (tens x tens).
Numbers of columns or rows (tens x ones).
Unit squares adjacent to perimeter (ones x ones). / Sum of 100s, 10s, and 1s.
Total area of regions, only one of which renders unit squares explicitly (ones x ones).
Quadrants / Numeric labels along top and left hand side (e.g., 28, 34).
Factors in expanded form (e.g., 20 + 8,
30 + 4).
Lengths along perimeter, but not to scale. / Sum of 4 partial products.
Sum of 4 areas none of which are drawn to scale.
Expanded
Algorithm / Numeric labels (e.g., 28, 34).
Factors in expanded form (e.g., 20 + 8,
30 + 4). / Sum of 4 partial products.

Three observations about strategies for using representational features will be key to my analysis of teaching and learning with the CMW two-digit multiplication unit. First, unit squares representations afford a range of strategies for determining areas. Figure 3 shows four. Strategy (a) relies on counting by 10s, while strategies (b), (c), and (d) rely on multiplication of dimensions inscribed in three different ways. These four strategies are distinct because, as shading indicates, each relies on different representational features.

Figure 3. Strategies for determining the area of a 4 by 10 unit squares representation: (a) counting groups of 10 unit squares as in "10, 20, 30, 40," (b) multiplying the number of unit line segments along the top and side, (c) multiplying the number of unit squares along the top and side, and (d) multiplying the numeric labels.

Second, as the CMW materials first drop unit squares and then 100s/10s/1s representations, the remaining area representations contain fewer and fewer features upon which to build strategies. Thus teachers and students who rely on unit squares for initial strategies have to refine or replace those strategies as the unit progresses. By including representations with fewer and fewer features, the instructional materials can scaffold convergence toward multiplication strategies that coordinate magnitudes of partial products, expanded forms for factors, and the distributive property.