The focus of this unit is the introduction and early development of multiplication. By making use of realistic contexts, the unit invites students to find ways to mathematize their lived worlds with grouping structures. The unit uses many contexts: inside the grocery store; postage stamps; city buildings, windows, and buses; tiled patios; a baker’s trays; and sticker pages. Initially, formal multiplication notation is not the focus; efficient grouping is, as students are encouraged to make groups (and groups of groups) to find efficient ways to deal with repeated addition and determine totals.
Traditionally, multiplication has been introduced by representing quantities of things that come in groups (such as four wheels on a car or twelve eggs in a dozen) with the formal multiplication symbol (X). For example, the total number of wheels on five cars would be represented as 5X4. The multiplication tables are then emphasized and learned as lists of unrelated facts, usually with flash cards. Although this traditional method connects multiplication to repeated addition, the connection is fleeting, and for many students multiplication very soon becomes a world of it’s own, existing parallel and unrelated to subtraction and addition. In this new world, students use a symbol that can look to them like the addition symbol turned on it’s side, yet it has a very different meaning, and they need the multiplication tables to figure out the answers.
Multiplication introduced this way can be very difficult for young learners to understand. They have just learned to count large quantities and now they are expected to count groups of objects using the same words-words they must now use to count the individual objects in the group. Making the group a unit to count-unitizing it-is a major developmental shift in perspective. Students are able to make this shift, but only if reasoning with groups serves a purpose. Human beings cognitively structure their reality as a way to understand it. We make groups, and groups of groups. We even have the innate ability to perceive two, three, and maybe four items in a group as a whole without needing to count or establish one-to-one correspondence- an ability known as subitizing.
With respect to multiplication, the purpose of making groups, and groups of groups, is to find the total in an efficient way. Multiplication for students, starts with skip-counting and repeated addition, but structuring the situation leads naturally to strategies such as doubling, using partial products or benchmarks (ie. five-times and ten-times), and doubling and halving. This unit is designed to build on students’ natural ability to group and to develop efficient strategies for repeated addition. It prepares students to learn the multiplication tables not as lists of isolated facts to be memorized, but as number facts that can be related to each other in a multitude of ways.
You will be seeing your children continue to use the number line as a model because it stimulates a powerful mental representation of numbers and number operations that encourages students to become cognitively involved in their actions. You will also see them using using repeated addition, skip-counting, doubling, partial products, five-times, ten-times, and doubling and halving as strategies for solving problems. All of these strategies and this model will be used to encourage them to construct the meaning behind unitizing, the combinative property, the distributive property, place value patterns that occur in multiplication, and composition and decomposition of groups of groups. (excerpted from Cathy Fosnot’s Groceries, Stamps and Measuring Strips)
In order to best support your children at home it is best to allow them to do the work they are doing independently. You can then ask them questions about the strategies they are using and why. Encourage them to apply these strategies in real-life situations that occur (ie. doubling or halving a recipe, counting large quantities, calculating totals of groceries, estimating length, etc.) If you need suggestions of activities to do or if you have thought of a suggestion for other parents to do, please contact me. I’d like to create a list of real-life contexts for each unit to support students’ math thinking at home.
Thanks for your continuing support,