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Teachers are Lifelong Students

Aha! Moment

My favorite part of being on the Eureka Math team is our culture of continuous learning. The team readily considers and discusses thoughtful questions, which leaves room to be a learner as well as a teacher. Working in a culture that assumes we all have a lot to learn means that I am not afraid to ask questions — I ask a lot of questions.

Recently, I saw an online Rekenrek demonstration video created by a well-meaning educator. As I watched something felt odd and it took me a minute to identify the reason. The video is shot from a student’s perspective; students are moving beads to the right as they counted while the rest of the beads are stored on the left.

Note that we want students to see 4, shown on top right.

“Aha!” I thought, “The beads are moving the wrong way.” Then I stopped to think about why that felt wrong. Was I being too rigid in my thinking? We want students to think flexibly about numbers. Time to ask my teammates with fluency expertise!

Here’s what the team had to say about why we always move the Rekenrek beads from right to left from the student perspective:

Note that we want students to see 4, as shown on the top left.

The answer to this question lies in the number line and English print conventions (reading left to right). We most often see the number line (or number path in GPK-1) increasing in quantity from left to right. This follows the print conventions children internalize in their early literacy work.

Students and teachers tend to generalize the left-to-right directionality to the Rekenrek. Look at the top images. Where does your eye land first? Probably the top left. As a result, the quantity shown in the first image may be misinterpreted as 16 (6 on top and 10 on bottom). The second image is easier to see and count the four red beads because it corresponds to English print conventions.

When we are doing Rekenrek work, our purpose is often to quickly recognize and manipulate numbers. The fastest, easiest way for students to do that is by seeing the beads they should count in the top left of the Rekenrek. The point isn’t that the top image doesn’t show 4 — it does. It just isn’t the most efficient configuration for students.

Directionality also has implications for negative numbers in middle school. When we teach negative numbers in Eureka Math, we introduce them as opposites of positive numbers. More specifically, a number’s opposite is one that is positioned at exactly the opposite distance from zero. Most often, negative numbers are positioned to the left of zero and positive numbers to the right. We move left on the number line to find a positive number’s opposite (or negative number) and we move to the right in order to determine which numbers are greater. Is that the only possibility? No. We also use the vertical number line. However, you’ll be hard pressed to find number lines that have positive numbers increasing to the left and negative numbers decreasing to the right.

Implications for Practice

I find that it is easy to confuse the direction beads are moving when I hold the Rekenrek in front of me because my view is the opposite of what my students see. I worked through this by deliberately practicing Rekenrek exercises in front of a mirror. Better yet — find another inquisitive mind and practice in front of one another! The feedback you get from 5 minutes of practice with a colleague can make a significant difference in clarity and pacing in the classroom.

This post is authored by Kate McGill Austin a Eureka Math writer for PreK and Grade K.