This summer is moving by quickly and thoughts of the upcoming school year are in view. I’m in the process of preparing materials and the first few lessons. This year my math sections are larger than before and it changes some of the ways in which I prepare. While researching a few ideas I came across a Tweet from Jamie.

Every elem T must read this book. It’s on my list of top 5 can’t live without resources. #elemmathchat pic.twitter.com/d7VZB9I5QW

— Jamie Duncan (@jamiedunc3) July 15, 2016

I’ve heard of Kathy Richardson and her role in the math education world, but haven’t delved too deep into what she’s written. This book interested me mainly because of the use of Kathy’s critical learning phases. As students progress in school they visit different stages of mathematical understanding. It’s not always linear, but these stages tend to follow each other. I wanted to learn more about this process.

So the book arrived yesterday and I was able reading through the first section. Kathy states that children learn number concepts in different learning phrases. She coined the phrase critical learning phases to describe the stages that kids pass on their way to making meaning of the math they encounter. The first section of the book focuses on understanding counting. On first glance I thought this would be very basic. Advertised and delivered. It’s basic, but also intriguing and gave me a few takeaways. After reading this section I started to draw parallels to how my own students make sense of numbers. I then wanted to reach out to my colleagues to direct them to this section. Understanding counting, although basic, is a foundational skill that is built upon. My paraphrased version of Kathy’s learning phases are below.

**Counting objects**

- Counts with 1:1 correspondence
- Knows “how many” after counting
- Counts out a specific amount
- Spontaneously adjusts estimates while counting to make a better estimate

**Knowing one more/one less**

- Knows one more and one less in a sequence without relying on counting out
- Notices if counting pattern doesn’t make sense

**Counting objects by groups**

- Counts by groups
- Knows quantity stays same when counted by different sized-groups

**Using symbols**

- Uses numerals to describe amount counted. Connects symbols to amount counted.

As I read through this I started looking through my school’s teaching materials for grades K-3. Some of the materials follow a linear progression while others tend to favor spiraling. I have a few takeaways after reviewing this first section of the book and looking at my district’s materials.

- If counting is important than students will start to see why keeping track/organizing numbers is important. If it doesn’t matter students won’t care whether they come up with a different amount when a specific amount exists.

- Being able to “spontaneously adjust estimates while counting to make a closer estimate” is a skill that’s found as students progress through the elementary grades. This plays a major role in measurement as students are asked to estimate using non-customary tools. I also see a connection to Estimation180 as students might use proportional reasoning after initially counting to find an estimate.

- Moving from 1:1 correspondence counting to group counting can be confusing for students. Understanding that counting by 2s, 5s, or 10s is a strategy that’s more efficient and that may be learned in time. Students might not initially come to this conclusion and replace 1:1 with 2:1, 5:1, or 10:1 if they don’t understand the reasoning and if they’re taught in the same period of time. For example, I find this evident when there’s a disconnect between counting by 10s and grouping numbers into 10s and then counting the groups.