Conceptual Understanding versus Procedural Fluency through Skip Counting

Contributed by Lynne Taylor

How many in this community experienced a focus on procedural fluency in school? And what did our teacher training focus on? I knew instinctively as a student that if I could understand something conceptually, then I got it and could use it. I no longer stressed during tests, because I knew that I could recreate a formula since I understood the concept behind it. However, most of my teachers stressed procedures instead, which led to anxiety come test day. During my teacher training, I knew that I wanted to focus on having students understand concepts conceptually. This became my quest.

I was thrilled when I landed my first teaching position as a Kindergarten teacher. Here was my chance to teach conceptually and get students off to a great start. But alas, I fell into the procedural fluency trap with skip counting. Every day, my students built onto the number line that counted how many days we had been in school, which is a common practice in Kindergarten and first-grade classrooms. Students had the chance to see the number written out, and we practiced counting from 1 to whatever day we were on. One of our standards was for students to be able to count to 100, so this practice fit in perfectly. Each student took a turn to be our leader for the week and to point to each number while the class said the number out loud. They became proficient at this exercise, and every student could rote count to 100 before the end of the year.  

I coded the numbers for counting by 5s and 10s. Then, as we approached the higher numbers, and counting by 1s every day was becoming too long of a task, I shared the coding method and how we could count by 10s instead. We began counting by 10s on days that ended with a multiple of 10 and by 5s on days ending with a multiple of 5. The coding helped them know which number to go to next; however, they were not building conceptual understanding of skip counting through this method—they were simply using procedural fluency to do it.

enVisionmath2.0 helps build conceptual understanding beginning in Kindergarten when students develop number sense for counting objects to 20 and eventually to 100. Without this basic understanding of numbers and a one-to-one correspondence, skip counting is simply a rote exercise that they learn to do. They do not connect it with actual counting or even know why they are counting this way.

Students learn to use basic ten-frames in Kindergarten to organize their counters—first to compare groups introduced in Lesson 4-1 with the English Language Learners (ELL) suggestion. This is a useful tool for everyone, not only your ELLs, since it helps them see the concept more concretely and helps with the one-to-one pairing of comparing two numbers. The ten-frames then appear as homework problems for that night. The focus is on understanding number sense before teaching skip counting. The ten-frames set the stage for skip counting. Students see the arrangements on the ten-frames and learn that one row is 5 squares and the whole frame is 10 squares.

Students then move on to using place value blocks and connecting cubes in Kindergarten to compose and decompose numbers from 11–20—once again setting the stage for skip counting to come formally in second grade. In first grade, they further their work with place value to 90. When they count how many tens blocks they used to create a number, they learn place value concretely and skip counting informally. Once they have learned the foundations of number sense, they will be ready to use skip counting as a faster way of counting instead of counting every number. Teachers can use coins and their values to introduce skip counting in second grade.Money is a real-world item that students have seen and will use or are using already. Topic 9 moves them into skip counting with numbers by 5s, 10s and 100s—again setting the stage for multiplication in third grade.

Part of understanding multiplication is seeing the connection between skip counting and the sets of objects that students count. Again, teachers want students to become fluent with multiplication, but simply learning to skip count by rote does not ensure that they understand multiplication. When students recognize that multiplication is about equal groups of objects and they connect skip counting to those groups of objects, they have conceptual understanding of multiplication and skip counting itself.

Check out this interview  with Nita Copley about conceptual development. Conceptual development is the base on which all the areas of math proficiency are built. If students do not understand a concept, they do not know how to use it or build upon it and end up with misconceptions that can go undetected for months or years. Let’s make sure we’re helping our students build procedural fluency by helping them build their conceptual understanding first.

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