Read Michael’s essay, “Missing Factors: On Learning What You Don’t Know.”

Read Part 1: Counting with Celine, about a student I worked with for five years.

It was bittersweet to reflect back on my years with Celine as I wrote Part 1. For one, she had become a constant in my professional life, and I miss her now that she’s at the high school. She came back to visit recently, but unfortunately I had already left for the day. She sent me an e-mail to let me know, and also scrawled a cryptic “CELINE WAS HERE” on the window in my room. Current students rightfully assume she’s a legend.

I am both proud of our successes, and puzzled by our defeats. Celine and I tried so, so hard to get her to think additively and multiplicatively. Why did she latch on to major ideas about linearity and not to 5 + 9?

(1) Breaking Through the Counting Barrier

Is the breakthrough in thinking — moving past counting strategies — cited in that article really the critical point? That matches my experiences, but is it universal? I feel like I need to read more research.

(2) So… what?

What do we do next? Where do we begin? What are some high leverage learning experiences? There are no interventions that “work” for all students.

In Michael’s essay, Rachel is still working on this, but sounds like her interest is waning. She wants to keep up with current topics, and, I imagine, she wants to feel like she’s working on something that looks challenging like her classwork does. It is hard to argue that something is truly challenging when the student already has a (grossly inefficient) strategy for it. It may be challenging for the student to use the more sophisticated strategy, but not the problem in and of itself.

(3) How do we motivate students to adopt a new strategy for something they can already solve? …or something they recognize is grade levels “below” them?

This is a general question I have about intervention.

If students do not feel motivated to work on 7 + 4 for the fifth year in a row, how do we address this gap in strategy? We know it’s important. The students know it’s important. The students may even feel a sense of urgency, but they may also feel embarrassed.

Our work with Celine using the Transition to Algebra curriculum felt the most respectful for both teachers and student; it felt rigorous and more relevant to grade level work, and still allowed us to spend some time discussing derivational strategies. That said, Celine never did fully develop them.

Students that repeat a course in high school, or even a grade level, do not necessarily improve their performance. In fact, some students do worse because they do not work as hard when they perceive that they already know something. Does this sometimes happen in intervention? How can we stop it from happening? How do we keep minds engaged?

Also: does this mean we should be focusing even more effort into K-2 and early numeracy, so that we don’t get “repeat customers” working on fluency in later years? But what about these kids I see in K-2 that do not make the same progress as their classmates… that are still struggling to even “count on” at the end of second grade?

(4) If every students had a full neuropsychological battery done — ha — would there be a red flag that tells us when to change approaches to fluency work?

Would that red flag also result in a diagnosis of dyscalculia? …or is there just a cognitive marker that says “it’s okay to have this student memorize. The derivational strategies may cause them to hate everything about math class.”

As with much of teaching, none of these questions feel resolved. We just continue to work, and to refine, and to question.