Are there any dangers in training your students in the “strategy” of counting-on? After reading Dr. Les Steffe’s work, I would argue it is harmful. He calls counting-on a non-teachable scheme. This means that if you want counting-on to be meaningful for students you can present situations that would promote their construction of counting-on, but not before they are showing signs they are ready to make sense of it.
One way a teacher could support students to make sense of problems and persevere in solving them (SMP1) for tasks in which they could count-on is to attend to the mathematics of students and look for evidence that they are ready to construct counting-on. In the unit construction team’s last blog, my colleague Elin began our conversation around this presentation from the recent Palm Springs California Mathematics Council conference. Today I will write about the video we used of Chloe to set the stage for the presentation. Next week you will hear from Grace about the videos we used to highlight some of the progression that leads to meaningful counting-on in students.
Chloe was a 2nd grade student who struggled to solve problems when she didn’t have enough fingers to use in counting. This was an indicator for us that she needed visible things to count, which is the first stage of a counting scheme. Children do not meaningfully count-on until the 3rd stage of development. So, when we noticed her counting-on, we hypothesized that it was something she memorized. We looked for evidence to confirm or deny our hypothesis. In the following video you will see Chloe explain her counting-on behavior. Watch and see what she does.
Based on what she did in this video, it would be logical to think she was possibly in the 3rd stage of development. As I stated earlier, we had some experience and were skeptical. It is important when gathering data about the mathematics of students to look for multiple pieces of evidence to confirm a hypothesis. Let me explain how she solved another problem prior to this one.
The task: Adding 6 hidden marbles and 3 visible marbles
Chloe didn’t think she could solve the problem. It took prompting by the teacher before Chloe attempted the problem.
Chloe simultaneously lifts a finger pattern for 6 and says, “1, 2, 3” while lifting three fingers sequentially.
Teacher: “How did you know it was 9?”
Chloe: “Because I had nine fingers.”
Teacher: “9 what?”
Now watch this next video clip of Chloe continuing the interview.
Looking at all of the evidence we collected, my colleagues and I argue/infer that counting-on at this point was an empty procedure for Chloe. Chloe’s empty procedure made it possible for her to “solve” problems without making sense of a rather complex situation. She was just following a road map. This circumvented her meaningful development of number.
Stay tuned for Grace’s explanation of the other videos we used at the conference of exemplars on a meaningful journey for a child to count-on.