Math 647 Unit 1: Fundamental Notions of Operations

Section 1 : Counting and CGI (Cognitively Guided Instruction): ways to understand the addition and subtraction and how children learn them.

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Goal: The goals this week are for you to

Stages and understandings for counting:

Videos to watch:

Watch the first half of my video on early number concepts--paying attention to the stuff about counting. We'll come back to subitizing in week 3, so we'll re-watch the video then, paying attention to the stuff about subitizing: Early number concepts

Watch this clip of my son counting, and read my comments about counting

Assignment 1: Math Buddies which is # 3 in the Teaching Math Video Library K-4 at

Answer these questions (type your answers in, and upload your assignment to the appropriate drop box in D2L)
May 23

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 next 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 2: 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) May 23

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

Quiz: 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! due May 24

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 5: Write about problem type difficulties after watching this video due May 25

Assn 6: Make a video of you solving problems by direct modeling May 28