Math 647 Unit 1: Fundamental Notions of Operations

Section 3 : CGI (Cognitively Guided Instruction): ways to understand the four basic operations and how children learn them.

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Goal: The goals this week are for you to

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

First get out your Children's Mathematics book, and turn to page 12. All the problem types you need to learn and understand are in this summary chart. We're going to refer back to these ways of thinking about addition and subtraction for a lot of this course, so this is vocabulary you will continue to use after this unit; you especially need to know going forward the differences between the basic problem types, which are the four on the left (the result/whole/difference unknown problems).

I also have made a blank problem types grid that you can print to take notes on. You will use another copy of this grid for your first homework assignment

Videos to watch:

Intro to CGI

Join problems

Separate problems

Part-Part-Whole problems

Compare problems

Readings: read chapter 2 of Childrens Mathematics


Assn 11: In a copy of the blank problem types grid, make up your own problems, one of each type, and type it into the grid (upload to D2L) June 9

On your own practice: For each of these problems, figure out what the problem type is, then check your answers

Assn 12: To do after you have done the previous assignments, take the 2 quizzes: "CGI Addition and Subtraction probelm types quiz 1" and "CGI Addition and Subtraction probelm types quiz 1" on D2L . Your score will be the higher of the your score on the two quizzes. Please treat these as closed book quizzes, and do your studying first! June 9

Know how children progress in their ability to solve addition and subtraction problems of various types, and be able to show how a child who is at the direct modeling stage would solve problems of the types that can be readily direct modeled

We're concentrating on direct modeling first: what problems can be solved by acting them out with manipulatives. You should know that the easiest problem types to solve by direct modeling are Join Result Unknown and Separate result unknown; slightly harder are Part-Part-Whole Whole Unknown, and Join Change Unknown, and the next step up is Compare Difference Unknown. Those are the types of problems you are responsible for knowing how to solve by direct modeling.

Videos to watch

My overview

My walkthrough of direct modeling (though if the video is too hard to follow, you can skip this one)

My explanation of counting strategies

My discussion of the figure on page 31 in Children's Mathematics

My discussion of flexible strategies (which uses bar diagrams to show connections between problem types)

On the CD Childrens Strategies that came with the book Children's Mathematics, watch the Addition and Subtraction Direct Modeling clips (there are 3 of them), and the clips showing counting strategies (there are 4 of them)

Michael (who is not good at direct modeling) figuring out how to solve a separate problem by direct modeling.


Chapter 3 in Children's Mathematics


Assn 13: Write about problem type difficulties after watching this video due June 11

Assn 14: Make a video of you solving problems by direct modeling June11

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 15: Write a problem for each of the problem types in the table on page 48 of Children's Mathematics June 14

Assn 16: 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 14

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 17: 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 16

Assn 18: After watching the video of the Kindergarten class on the Classroom Episodes CD, write an analysis of how this lesson in this kindergarten class fits in with the process standards. You do not have to address every process standard--choose 2-3 that are particularly relevant to this clip, and discuss what children do that fits with those standards (upload to D2L drop box.) June 16