## Understanding Number Relationships

Today I was able to dig a bit deeper into Kathy Richardson’s book. The first chapter was related to counting and critical phases that are needed as students develop numeracy skills.  The second chapter focuses on number relationships. In order for students to compare numbers they need to be able to distinguish between larger and smaller. Once at this stage students can recognize that numbers are found within numbers.  For example, eight is found within 10. When comparing numbers students generally start to identify differences between the original number and one.

Richardson states that being able to change a number by counting on or adding to a group is a Critical Learning Phase.  Counting on or adding to a group of numbers is a strategy students use when comparing numbers. I believe primary students might use this strategy to find out how many blocks are in both stacks below.

Assume that each block is the same size.  Now, how do you think primary students would count these two different stacks?  The strategy that they may use to solve this can tell  more about their understanding.  Are they counting each block individually or using the first stack to count on to find the second stack? While comparing numbers younger students often count each block individually.  The model below shows a different strategy.

In this case the student has taken the five and built upon it to find four more.  The five and four are nine.  This student didn’t count each stack individually.

I find this interesting as it may apply to other areas of mathematics.  After reading this I started to think of how increasing the complexity could apply to fraction concepts.  Specifically, I thought of how theses blocks and fractions are similar:

If the green stack is one whole what is the second stack’s value?  How would your students solve this?  In the example students may identify that each block is 1/5.  When looking at the parts on the right they might start off at 5/5 and add to that particular block line. Fractions can lead to confusion with a non-linear scale being present.  This is especially the case if students are always seeing 1:1 ratio when counting objects.

I thought a number line might be a better representations for a fraction problem. Richardson notes that number lines are only symbolic relationships.  She also states that when students use number lines they’re most likely not thinking of quantities, but more so using the line to find the solution. They’re using it as a tool to count on to find a solution.  Number lines are used frequently at the early elementary levels so this is something I’m going to keep in mind for the new school year.

## Learning Number Concepts

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.

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.