ZOOMING IN ON THE LEARNING PROGRESSION Of NUMBERS AND OPERATIONS
Introducing students to the wonderful patterns in numbers and the structures of multiplication and division through interesting problems can be an engaging way to develop an appreciation for mathematics. There are multiple interpretations and structures for operations, such as addition, subtraction, multiplication, and division, and building these structures in interrelated and deep ways will help build mental models for students as they build on mathematical ideas in later grades. The authors from Putting Essential Understanding in Practice, Multiplication, and Division state that,
“The way in which you teach a foundational concept or skill has an impact on the way in which students will interact with and learn later related content. For example, the types of representations that you include in your introduction of multiplication and division are the ones that your students will use to evaluate other representations and ideas in later grades.” (Dougherty, Lannin, Chval, & Jones, 2013, p. 5)
A great way to develop meaning for operations is by using mathematical representations to model and interpret practical situations. Posing “math happenings” in class allows students to see how mathematics is all around them. They can develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models. In addition, using story problems allows students to be exposed to division contexts that involve partitive and measurement models. The use of area models to represent the distributive property in multiplication and can later be used to multiply fractions using the area model and also when learning algebra and multiplying two binomials using the area model and algebra tiles.
The Principles and Standards of School Mathematics (2000) introduced by the NCTM placed central importance on understanding and sensemaking in learning mathematics; for example, the Learning Principle states, “Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge (p. 20).” This calls for traditional basics such as multiplication facts, to be used in conjunction with new basics such as reasoning, learning with understanding, and problem solving. It is important to support the latter by incorporating new basics such as providing pictorial connections in an elementary classroom to help understand the process of multiplication better.
An alternative approach to memorization is an inquiry-based approach, which focuses on understanding and explaining the meaning for operations. To develop fact fluency, memorization and practice taught along with alternative approaches, such as pictorial or graphical or hands-on approaches, help students understand multiplication in a much more efficient fashion. The following sections discuss some ways students can learn facts and procedures with understanding by thinking about facts they already know (derived facts) and applying some useful properties to establish a connection between abstract and concrete multiplication through conceptual understanding. In addition, we can motivate students to learn multiplication through a discovery learning process using pictorial representations and manipulatives.
According to the learning progression document for operations and algebraic reasoning,
“Students will rely on [understanding the meaning and properties of multiplication and division and on finding products of single-digit multiplying and related quotients] for years to come as they learn to multiply and divide with multi-digit whole number and to add, subtract, multiply and divide with fractions and with decimals.” (From The Learning Progression documents at http://math.arizona.edu/~ime/progressions/, K, Counting and Cardinality; K-5, Operations and Algebraic Thinking, p. 22)
The important multiplication structures include using (a) equal groups of objects, (b) arrays of objects, and (c) multiplicative comparisons. For the equal group interpretation for multiplication, one of the factors refers to the number of objects in a group and the other is a multiplier that indicates the number of groups. For example, three cookies on a plate (number of objects in one group) and four plates (multiplier that indicates four groups). An array of objects is represented in rows and columns in a rectangular shape, and students are typically used to seeing this in the real world with chairs setup in the theater, displays at the grocery store, or just in organized arrangements of objects. A great literature connection is One Hundred Hungry Ants. Hundred is a nice composite number that has many configurations that students can display; 5 by 20 array, 10 by 10 array, 25 by 4 array, and 50 by 2 array and their commutative pairs.
Text Box 4.2 One Hundred Hungry Ants by Elinor J. Pinczes
(Multiplication interpreted as an Array Model)
Ants marched in 4 rows with 25 ants in each row. How many ants were there?
In the book Remainder of One, the main character Joe has to solve a problem where he has to think of an array so that he is not left out.
Text Box 4.3 Remainder of One by Elinor J. Pinczes
Relining the 25 bugs in his squadron from two lines to three lines to four lines, until inspiration and good math thinking result in five lines of five, and Joe fits in at last.
The third interpretation for multiplication, that is called multiplicative comparison, can be somewhat more sophisticated than the two previous structures of equal groups and array of objects because it introduces the notion of proportional reasoning and scale factor. In addition, it is a foundational mathematics concept to understanding a ratio as a multiplicative comparison of quantities.
According to the Developing Essential Understanding of Multiplication and Division for Teaching Mathematics in Grades 3-5 by Otto, Caldwell, Lubinski, & Hancock (2012), an important interpretation for multiplication is the scalar process.
Essential Understanding 1a
In the multiplicative expression, A x B, A can be defined as a scaling factor.
Multiplication is a scalar process involving two quantities, with one quantity—the multiplier—serving as a scaling factor and specifying how the operation resizes, or rescales, the other quantity—the multiplicative unit. The rescaled result is the product of the multiplication.
In the fourth-grade CCSS (CCSSM, 2010) for mathematics, multiplication is introduced as a comparison:
Operations and Algebraic Thinking
- 4.OA Use the four operations with whole numbers to solve problems.
- 1. Interpret a multiplication equation as a comparison, for example, interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
- 2. Multiply or divide to solve word problems involving multiplicative comparison, for example, using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (CCSS-M)
A great literature link to discuss the multiplicative comparison is the book called If you Hopped Like a Frog by David Schwartz. In this book, Schwartz uses proportional reasoning and amazing pictures to help students see what would happen if they had the same abilities as amazing animals. If you hopped like a frog, he says, you could jump from home plate to first base in a single bound. Since frogs are able to jump 20 times their body length, how far would a human being jump?
Text Box 4.4 If I Can Hop like a Frog by David M. Schwartz
(Multiplication interpreted as a Multiplicative Comparison Model)
Frogs are champion jumpers. A 3-inch frog can hop 60 inches. That means the frog is jumping 20 times its body length. If you hopped like a frog, How far could you hop?
Using mathematics literature is also great way to provide a meaningful context to the distinct meanings of partitive and measurement/quotitive division.
Text Box 4.5 The Doorbell Rang by Pat Hutchins
Fair Share is a common division structure familiar to even young children. In the case of The Doorbell Rang, Mother bakes 12 cookies and first she shares with 2 children, then the doorbell rings and 4, 6, and then 12 children join them having to share the cookies fairly.
For the partitive division interpretation, students are working on making equal groups, or sharing and distributing an amount to known number of groups. In the case of The Doorbell Rang, the children are partitioning the amount of cookies (dividend) into the number of children at the house (the number of groups indicated by the divisor) and then needing to find the number of cookies each child can have (items in each of the groups).
The measurement model is also known as the quotitive division interpretation. In a measurement division problem, students know how many should go in each group, and they have to find out how many groups there will be in all. In the case of the scenario in divide and ride, they know the number of children who want to get on rides at the amusement park and are encountered situations with 2-seater, 3-seater, and 4-seater rides.
Text Box 4.6 Divide and Ride by Stuart Murphy
The measurement model is illustrated in the book Divide and Ride where 11 Children go to the park and get into 2-seater and 3-seater rides. They need to model measuring off a quantity to find how many groups/sets can sit in the ride.
These two books provide a great springboard to discuss the important distinctions between different interpretations for division: partitive (fair share division) and quoti- tive (measurement division).