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.

As Celine’s sixth grade math teacher, I too think about Celine and students like her and wonder whether she’ll/they’ll be okay. Even though Celine has many, many advantages (highly capable in non-arithmetic realms, well resourced school system, educated and affluent family), I am not convinced she will be. Unless what you mean by okay is that she’ll go through life not really understanding her mortgage, 401k plan, or how to compute a tip at a restaurant. I supposed those things don’t really matter, in that I don’t really understand how a transistor works; I am still using this computer. The difference is that I am not angry at or afraid of my computer.

One thing that stuck me about this thread is the contrast between Michael’s school where Rachel was grouped in what I presume is the not-fast paced class. (Michael, does your school really sectioning by ability as early as grade 4? or are they just fortunate enough to have a math department teacher teaching all fourth graders), and Jenna’s annoyance that Celine taken out of her general ed math class in 8th grade.

While I think the particulars of her situation (three different math teachers) were horrible, when I had her in sixth grade, I often felt she would have been better served with a different cohort of students along side. I certainly don’t relish having to decide which students get into that fast-paced class. However, I do know that students learn mathematics at dramatically different rates, and that very often (at least in our part of the country) that difference is magnified by participation in extra-curricular (not literally, of course, since the enrichment and school-based curricula are entirely duplicative) math programs. Celine was in the same classroom as students who spent 2x as much time doing math as the rest of the class, who also learned more quickly, and who had already previously studied sixth grade math topics. The question I have is how much her disposition to learning math and achievement would have been different if she’d spent her elementary years, especially K-2, learning arithmetic and math facts in a way and at a pace that worked and her middle school years using something like Transition to Algebra from the beginning, rather than just the last few months of 8th grade.

I think the real problem is what tends to happen when a teacher who isn’t part of a math department teaches a special-education math class, especially in middle school. There is no reason that a special education or a general ed course that is slower-paced shouldn’t feel respectful and rigorous to students. If we went slower for the students who needed slower, students wouldn’t have to repeat a same course. They may not get to all the grade level standards in a particular year, but they would get further along in the long run with a much better understanding, as compared to what happens now when they are being dragged along by a pacing guide and the very real pressures that teachers feel to “cover” the standards and also respect the pace of learning of the rest of the class.

Will she be okay? Well sure she will. My immigrant mom is *okay*, even though she can’t write a coherent email in English. But official correspondence gets her in a tizzy, and I worry that Celine will feel the same way about mathematics well into adulthood.

Also, Michael, have you looked at Chris Woodin’s work? (http://www.woodinmath.com/courses/course-d)

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My school sections by ability as early as Grade

3, though all students get a math department teacher starting that year as well.The only nice thing about our sectioning by ability is that we don’t have like six gradations of tracking, and it’s only roughly by ability. We try to track by fast- and regular-pace, and while this is sort of a euphemism it’s also sort of not, and there is a legitimate effort to keep the tracks porous. Any by high school, kids can choose the track that they want to be in.

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Interesting! How do you determine student placement? …anecdotal records from teachers? (That’s probably sufficient if it’s about pace. It would be really difficult to quantify that.) Do you get any parent push back?

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First of all, I am excited that you responded! It is nice to get the perspective of not just a colleague I trust, but also someone that knew this particular child. She hasn’t been the most challenging student I have had, but she is one that lingers in my mind. Maybe because I feel like I could have done better? That she should have gone to the high school more prepared? It bothers me.

I worry less about whether students like Celine will be able to understand their mortgages. As long as she understands that she needs to pay it, and that low interest numbers are better than high ones, she will be able to muddle through. I am more worried about your second point: the anger and fear. I worry that her anxiety around mathematics will bite her at inopportune times, and shut her out of opportunities.

When I first met her, Celine was adamant that she wanted to be a scientist. By 8th grade, she told me that she did not want to do anything in the sciences. “I think I will just be… a filmmaker or something. There’s no math, right?”

I worry that she will miss out on college opportunities and jobs. I worry that this impacts her self worth.

I do think that some kids should not be placed in their general education math class. The major reason for my reluctance to pull students from the general education setting is because I do not like to make the decision to veer kids from the path towards high school graduation. Our district’s current structure does not serve this goal: instead, classes are typically taught by non-math certified teachers lacking both guidance and a curriculum.

It sounds like Michael’s school teaches 4th grade curriculum to both groups, just at a faster pace for one. For many of our students presently taught math solely in the special education setting, they are no longer being taught anything approaching grade level standards. Honestly, I think we only addressed a handful of 8th grade standards with Celine’s group last year. She had to be placed in the lowest possible level of mathematics at the high school, and I am not convinced she belongs there. I pushed for her to be in a co-taught Algebra I class instead.

I don’t disagree, especially since one of her closest friends at the time poisoned Celine’s attitude. Any remnants of positivity Celine felt towards math class disappeared. And, undeniably, it is difficult to teach a heterogeneous math class. This is exacerbated by the growing (epidemic-like) student participation in outside math classes.

I have the same question! I wish we could have a do-over! According to some notes left behind for me, Celine worked with our previous math specialist when she was in 1st grade and 2nd grade. (It looks like she did not participate in remediation groups in 3rd grade.) An excerpt from her first grade report:

This makes no sense to me. So in first grade she was able to use “near doubles” strategies? Like… independently? It’s not clear what K-2 was like at all.

I wish we had used Transition to Algebra with her in 6 – 8. (Although this should have been feasible; I asked the district to purchase it for me when she was in 6th grade, and I was turned down.) That said, Celine dropped out of the morning group in 7th grade because it was “way too hard.” (We had finished Do the Math Now, and I was designing my own materials. Oops.)

If we went slower for the students who needed slower, students wouldn’t have to repeat a same course. They may not get to all the grade level standards in a particular year, but they would get further along in the long run with a much better understanding, as compared to what happens now when they are being dragged along by a pacing guide and the very real pressures that teachers feel to “cover” the standards and also respect the pace of learning of the rest of the class.

I agree! There are some standards that I feel are less urgent, and I would gladly sacrifice them in service of a deeper —or actual – understanding of other standards. There must be schools out there doing this better than we are. I would love to see what it looks like in action.

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