Math 647 Unit 1: Fundamental Notions of Operations

Section 2 : Counting and CGI (Cognitively Guided Instruction): ways to understand multiplication and division and how children learn them.

Goal: The goals this week are for you to

Know the CGI problem types for multiplication and division; be able to identify the problem type of a given word problem, and write word problems of a given type.

The most important way of thinking about multplication is as repeated addition: several groups, each having the same number of objects. Working with rates, prices and multiplicative comparisons are important variations on this concept. (We will look at the array and area model for multiplication next week). There are two basic ways of turning this around as a division problems: in one you know the number of groups, but not the number in each group (the partition model), and in the other you know how many in each group, but not how many groups (the measurement or repeated subtraction model). Your goal for this "day" is to be able to identify what type a problem is, and to write problems of each type.

Videos to watch:

About Multiplication

Partitive and Measurement Division part A

Division part B

To read:

Children's Mathematics chapter 4

My summary notes on problem types


Assn 5: Write a problem for each of the problem types in the table on page 48 of Children's Mathematics May 30

Know how how a child who is at the direct modeling stage of understanding would solve a multiplication or division problem.

Videos to watch:

You may already have a good idea about how children model multiplication and division problems, and you may not need all of these resources:

At a minimum watch: On the CD Childrens Strategies that came with the book Children's Mathematics, watch the Multiplication and Division Direct Modeling clips on Measurement and Partitive division (two of them). Also, on the CD Classroom Episodes (also in the Children's Mathematics book), watch all of the Kindergarten clip.

Also available if you want to listen to me are:

Modeling Multiplication

Modeling Partitive Division

Modeling Measurement Division


You should read chapter 4 of Children's Mathematics, but I think you just read it. I don't have anything else new for you to read.


Assn 6: Make a video of you solving these problems by direct modeling. Click here for the problems; click here for more info on the assignment June 1

Quiz: Take one or both of the quizzes: "CGI Multiplication and Division problem types quiz 1" and "CGI Multiplication and Division problem types Quiz 2". These quizzes are set up so that the computer will automatically grade them (and you will get the higher of the two scores). For each problem, you will be identifying whether it is Multiplication, Measurement Division or Partition Division. You will also be identifying whether it is Grouping, Rate, Price, or Multiplicative Comparison. I have not figured out how to get the computer to assign partial credit for these problems, however, at your request I can go in and regrade one or the other of the quizzes and assign partial credit where appropriate. I will not do this unless you request it, so if you want me to look at it, send me an e-mail request. Please treat these as closed book quizzes, and do your studying first! June 1