Math 647 Unit 1: Fundamental Notions of Operations

Section 2 : Counting and CGI (Cognitively Guided Instruction): ways to understand the addition and subtraction and how children learn them.

New computer skills needed: Help files
Answering a problem in SMART Notebook (assn 7)
If you are using a PC for your class work this summer, I recommend installing Smart Notebook 11 following these instructions. If you are using a Mac, you will need to use Smart Notebook Express.
Getting started with SMART Notebook 11
You will need to download and install a free screen recording software. I recommend using the free version of Jing

Here is a getting started with Jing video I made a few years back

Here's a longer video that shows how to use Jing and Smart Notebook Express (also linked later on this page)

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

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

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 (first edition) or 14 (second edition). 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 (Join, Separate, Part-Part-Whole and Compare).

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


AssignmentAssn 6: 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)

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

Assignment Assn 7 (CGI types quiz): Take the online re-takeable quiz. I'd like you to practice doing this as a closed book quiz (Please treat this as a closed book quiz, and do your studying first!) but this is a quiz that randomly generates a new quiz each time, so you may retake it as many times as you want prior to taking the Unit 1 test. When you take the unit 1 test, I will retrieve your scores on this quiz, and you will get the highest score of all of your attempts. After taking the quiz, you may find it useful to compare your results to this list of common errors and check out the other resources for fixing those errors.

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(first edition) or 41 (second edition) in Children's Mathematics

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

First Edition: 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 addition and subtraction counting strategies (there are 4 of them). If you didn't get a CD with your book, please let me know.

Second Edition: Follow the links on pages 19 and 20 to watch the direct modeling examples 3.1-3.5

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 (for these assignments, you'll only need the pages on direct modeling, but we'll be thinking about the rest of the chapter in the next section)


Assignment Assn 8: Write about problem type difficulties after watching this video due

Assignment Assn 9: Make a video of you solving problems by direct modeling