Course Guide - USAID Teacher Education Project

they move from the “real” to the “representational” by putting dots into the Ten Frames torepresent the 12 items.This is a key opportunity to highlight that when children “bridge a decade” as in 5 + 7 = 12there is a developmental sequence that occurs in chilren’s thinking when they are givenequation containing numbers and symbols: 1) a CGI type story context that explains theequation ("There were 5 ducks in a pond; there were 7 ducks on the shore. How many ducksin all?"), 2) the physical (the beans), 3) the representational (placing dots in Ten Frames),and 4) once again the symbolic (5 + 7 = 12).c) More importantly, this sequence of how youngsters think about, understand, and learnmathematics will be repeated not only in Numbers and Operations, but in other areas ofmaths, especially algebra: 1) narrative, 2) physical, 3) representational (which will laterinclude tables and graphs), and finally 4) the symbolic.d) After children become familiar with the one-dimensional linear model of the number linefrom 0 through 20, they can begin to transfer this visualization to the two-dimensionalHundred Chart, which is simply 10 number line segments reorganized into a stacked, morecompact, manageable format.e) Children learn not only what they are taught, but what they "see" in their classroom. Thisis why it is important to have a number line from 0 through 100 (and later, from -20 to 120)prominently displayed in the room. The same is true for a Hundred Chart. These aremathematical references that children can use on a daily basis to solve maths problems. Butmore importantly, visuals seen day-after-day will help develop children's "number sense,"transforming those visual models into intellectual models that will become a child's internalmental reference for understanding and working with mathematics.3. What is essential to know or do in class?a) Introduce the base-10 number system for whole numbers, linking this to the concept ofdecomposing two-digit numbers such as 12 into 10s with additional units.b) Provide students with three visual models to help them think about how children canunderstand place value: the number line, Ten Frames, and the Hundred Chart.c) Relate each of the above to children’s thinking.d) Mention that the base-10 number system does not relate only to whole numbers, but that itextends to negative numbers and decimals, both of which will be discussed later in theNumber and Operations unit.4. Class Activitiesa) To introduce the concept of place value in our base-10 number system, distribute copies ofthe Ten Frames and a handful of beans. Explain the Ten Frame model, and then have studentswork in pairs to model the equation 5 + 7 = 12. After they have done this have a classdiscussion to discover how students used the Ten Frames to solve the problem.Anticipate that some student may have placed 5 beans on one Frame, 7 beans on another, andthen moved 5 of the beans from the second frame onto the first, which resulted in one Framewith 10 beans and another with 2.Other students may have used the counting principle: Placing 5 beans on the first Frame, thentaking 7 beans and distributing 5 of them onto the first Frame until it was full. At that point