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.