The Importance of Quick Images and Multiplicative Thinking

Nina Smith

Math Teacher grades 3-5 and Math Specialist grade K-8

Marcy Open School, Minneapolis Public Schools

MCTM Region 5 Director

Faye B. Clark and Constance Kamii report “Multiplicative thinking was found early (45% of second graders demonstrated some multiplicative thinking) and developed slowly (only about 48% of fifth graders demonstrated consistently solid multiplicative thinking)”. So how can we support students to develop this type of thinking?

Clark and Kamii write that in order to think multiplicatively students must be able to ‘think simultaneously about units of one and units of more than one (composite units)’ (Clark & Kamii, 1996). So students are moving from counting to coordinating counts of counts. This is a shift from thinking additively to multiplicatively.

As educators we usually begin by introducing the idea of multiplication through repeated addition, equal groups, arrays and equal jumps on the number line. Talking about multiplication through these examples helps to anchor the concept of multiplication but also keeps the discussion around addition. As students become comfortable with the concept of multiplication we also want to provide experiences where they are encouraged to see and talk about ‘composite units’ as they develop mental images. If students can visualize clearly how the numbers they are multiplying are related, they can develop flexible, efficient, and accurate strategies to build multiplicative thinking.

One way we can support students to develop multiplicative thinking is by using Quick Images (subitizing). Quick images are a number sense routine that helps students to immediately recognize a collection of objects as a single unit or sets of single units (composite units). Simultaneously visualizing units of one and units of more than one is an important element in developing fluency with multiplication and moving students from being additive thinkers to thinking multiplicatively.

An example of a quick image is show below. It provides an opportunity to describe a product in different ways. For example, this image might be described as 6 x 4 (6 rows of 4 dots), 8 x 3 (8 groups of vertical rows of 3 dots each), 2 x 12 (2 groups, each with 12 dots), or even 2 x 3 x 4 (2 groups, each with 3 rows of 4 dots.)

When using a subitizing routine with students, show the image quickly (2-3 seconds) and then cover it up. This is to encourage students to see sets of units (multiplicative) instead of counting by each unit (additive). Then show the image quickly one more time. Ask the students the questions below. The questions you ask are essential to build the understanding of a collection of objects as a single unit.

• How many do you see?
• How do you know?
• How did you see it?

In discussion use the image to highlight how the students saw the groups and record their thoughts using correct mathematical notation. This can help students make connections among the different ways they can describe a product.

Using Quick Images to break problems into parts becomes an essential skill students can use as a foundation to solve more complex problems. It encourages students to derive unknown results from know facts and flexible apply the properties of multiplication and division (eg. Commutative, associative, distributive, inverse) in problem solving. By exposing students to quick images we are providing an opportunity for students to use a mental image to develop multiplicative thinking.

Attached is a PowerPoint with several Quick Images to try with your students.

 

1) Clark, F.B & Kamii, C. (1996). Identification of Multiplicative Thinking In Children In Grades 1-5. Journal for Research in Mathematical Education, 27(1), 41-51