When I was working in the school district office, we spent significant time putting together a plan that would meet the needs of a range of learners. Where there were almost an endless number of resources from which to choose for Reading, there were very few for Math. We looked at many options but struggled to put together anything as comprehensive as what we could offer for Reading. It was during this struggle that I was introduced to the work of Kathy Richardson, who is one of the leading math educators in the country.
As I studied Kathy’s work, I came to understand that the very foundational concepts of number—counting, for example—were much more complex than I had given them credit for. As I became familiar with the Critical Learning Phases that Kathy identified, I realized just how important it is for students to build their conceptual understanding of number relationships.
What I learned about counting was eye-opening for someone with a background in secondary math. When a student really knows what it means to count, they understand one-to-one correspondence, but they also can keep track while counting an unorganized pile of counters. If they recount, they notice when the result is a different number and respond to this difference. After they count, they retain the number that they just counted.
Kathy identifies nine indicators in the Counting Objects learning phase. Once I understood the intricacies of learning to count, my next question was how I could know when students had developed this level of understanding with counting.
Assessing for Understanding
The Assessing Math Concepts assessments help teachers know exactly where students are in their understanding of critical number concepts. The Counting assessment identifies a student’s instructional level—Ready to Apply, Needs Practice, or Needs Instruction—through four different counting tasks. The first task is simply having the student count an unorganized pile of counters. While the result is important, perhaps more important is the process.
While the students count, the teacher is watching and listening. What is their estimate? Do they line up the counters, look at them, point, or physically move them as they count? Are they able to keep track as they count, and do they do it easily? How do they react to their estimate? Do they remember how many they counted?
The observations are recorded and become part of the body of evidence that will determine each student’s working level.
Once we have determined a working level, Kathy also developed an array of instructional activities that build conceptual understanding of these key math concepts. These activities are found in the Developing Number Concepts series; counting is in Book 1. If I want to focus on one-to-one correspondence, I might choose to do the Creations activity. In this small-group activity, students will build connecting-cube creations that match the pictures on a card. Students must build their creation so it matches exactly, helping them learn this critical concept.
There are over 100 other activities that teachers can use to build understanding of these critical number concepts. The results of the assessment guides teachers in the selection of the activities. When used together, these resources can help ensure a solid foundation for students to build on as they continue their study of mathematics.