# Math Relearning/Progressions/K-5 Counting and Cardinality, Operations and Algebraic Thinking

These are my comments on the Common Core Math Progressions document Counting & Cardinality and Operations & Algebraic Thinking: Grades K-5 (PDF).

## Contents

## Counting and Cardinality

There's a lot of overlap between this section and what I was going to say, but each of us said things the other didn't.

If I wanted to define math concepts in terms of their physical meaning, I'd need to separate them from the teaching techniques that are meant to aid the mind in handling the operations (managing the memory load, etc.).

On the opposite end, I'd like to try to understand what these concepts mean on the most abstract level. That would let me say with certainty what the concepts mean so I can be sure I'm relating them correctly. But I don't know if it's possible to get away from the models and physical representations. Can the mind really think in pure abstractions? Does every model for a math concept (eg, the number line) distort it in some way?

Unlike my other sources, this doc explains how comparison relates to counting.

## Operations and Algebraic Thinking

### Overview of Grades K–2

Table 1 was confusing to me for some reason, or at least it took me longer than I expected to read through it and translate the words to the equations. In some cases I didn't know what the difference was between the situations. They made some fine distinctions. But maybe it's clearer to me now. Still, it's worth analyzing the differences between these situations.

One question I have with these situations is when you'd ever not know the unknown. That should be part of any word problem if you want the student to be more engaged with it.

### Kindergarten

"Mathematize" - Yes, everything has a quantitative aspect. I imagine putting on glasses with a filter that highlights the mathematical features of an object or situation.

### Grade 1

"Put together ... Addend Unknown" - The situations reveal different aspects of the nature of addition/subtraction.

"algebraic perspective" - I thought this was out of place at first because I didn't learn algebra till junior high, but surely younger kids can understand it.

"Linking equations" - I want to catalog these representations and think about what I can learn from them about problem solving in general.

New to me (NTM): "where the total is" - This is already very useful to me.

"decomposing one addend" - I really like the idea, but it starts to get hard to remember all the numbers.

These conceptual math procedures can still be done rotely, so you have to connect the concepts to the different procedures. I suppose the evidence of conceptual thinking comes out in solving new problems and in explaining your reasons for selecting a procedure.

### Grade 2

With two-step problems, math is starting to feel hard, if I'm doing it in my head. Too many numbers to remember. But part of problem solving is simplifying the problem, breaking it up, or transforming it in some other way so it's more manageable, so maybe that's the case here.

### Summary of K–2 Operations and Algebraic Thinking

"within 100" - I guess I'll have to wait till the curriculum to find out about this, but it seems like this would suggest learning some other algorithms. It seems strange that the progressions would go into such detail and then skip over this part. Actually I think it's in NBT.

"all sums of two-digit numbers from memory" - Isn't that thousands of facts?

### Grade 3

"no general strategies" - Explore what this means exactly.

The distinction between equal groups and arrays seems very fine.

The 5+n chart is confusing.

### Grade 4

I've noticed that problem with "more than" language in multiplication comparisons.

NTM: Differences in remainder problems.

NTM: Looking for pair reversals when factoring. Maybe I learned that.

Milgram points out that CC doesn't cover prime factorization. I've seen people insert it a couple of different places.

In his TEDx talk at USU (https://tedx.usu.edu/portfolio-items/david-brown/) David Brown points out that you don't necessarily know how a pattern will continue, so a better test question is to explain how each of several next numbers would continue it. I like this idea and want to be on the lookout for this kind of reasoning about how math works and what it's about, which is at a deeper level than even some of these CC documents.

### Grade 5

p. 32 - Students "write expressions to express a calculation." They also examine the relationships between numerical patterns they make up. I'd like to explore the idea of all the mathematical operations as explorations of the relationships among numbers.

### Connections to NF and NBT in Grades 3 through 5

Why do we group only these four operations into arithmetic?

### Where the Operations and Algebraic Thinking Progression is heading

p. 34 - NTM: "Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?" {6.EE.5}

pp. 34-35 - NTM: Situation equation vs solution equation.

## Appendix. Methods used for solving single-digit addition and subtraction problems

### Level 1. Direct Modeling by Counting All or Taking Away.

### Level 2. Counting On.

This taking away method is like the counting up method for giving change.

p. 36 - Why is counting down "difficult and error-prone"?

When I start to feel bored or lost as I'm reading, it helps to remind myself of the basic nature of what I'm studying. I like to think of math (at least at this level) as the relationships between numbers on the number line as revealed by our operations on them.

### Level 3. Convert to an Easier Equivalent Problem.

What makes subtracting finding an unknown addend? I know the answer is obvious at one level, but I want to think more about it.

What makes transformations between representations possible?