I read Michael Pershan’s essay, “Missing Factors: On Learning What You Don’t Know,” twice. He described a familiar experience: when students seem stuck with operational fluency, in spite of our most vigorous efforts. The struggle with math fact fluency creates a ripple effect through other mathematical understandings. Students and teachers are left frustrated. I had a handful of students that fit this description as a classroom teacher, and now, as a math specialist working with a school of 600+ students, I see even more.
Michael cites a study from a 1986 paper on “learning with technology.” This quote haunts me.
Even after as many as 70 session on the computer, children who came to the activity using counting strategies to solve basic facts left the activity using the same counting strategies. (Bransford et. al, 1986)
Oh. So does this mean there’s a mathematical altitude students need to hit in order to be successful with derivational strategies and fluency? That fits with what I have seen in upper elementary and middle school students. What about my students that haven’t hit it yet – or that we aren’t pushing to hit that point because we are so focused on a million other things?
This all reminded me of my former student, “Celine.”
(And now: in which I “borrow” Michael’s formatting to respond with my own story…)
I. The Story of Celine
I met Celine as a shy 4th grader. I taught a model lesson for her teacher that used pattern blocks to explore changing the size of the unit. (e.g. if a yellow hexagon = 1, what is the value of the green triangle? If the yellow hexagon = ½, NOW what is the value of the green triangle? Etc.) The classroom buzzed as students posed their own challenges to classmates. Celine looked at me blankly. When I crouched next to her desk to learn more about her thinking, she whispered, “why… I mean… what… umm, can I just go to the bathroom?”
I worked with Celine again in 5th grade, and 6th, and 7th… She became a permanent fixture in my pull out intervention groups. I enjoyed having her in them, despite her vocal hatred for math. (And she was increasingly vocal over the years.) Celine has a precocious wit about her. She loves science, and is curious about the adult world. When we had the opportunity to work one on one, I could easily engage her in mathematical conversations. Five or ten minutes into our discussion of down payments on a house, or a graph predicting the population of honey bees, and she would catch on to my ruse: “you tricked me into talking about math! How dare you!” she would say with a dramatic gasp.
Celine has the distinction of teaching me more about math learning than perhaps any other student I have encountered in the last 11 years. She expressed herself orally with strength and clarity, which gave me a window into her experience. She let me know exactly what she’s thinking – when she’s making connections, and, most importantly, when she’s frustrated, and why.
II. Do The Math, Now: Celine’s Experience in Morning Math Group
In sixth grade, Celine joined a group I taught before school – all girls needing to work on whole number computation. Marilyn Burns’ Do the Math Now served as my primary curricular resource. The first four units we did were about multiplication fluency (unit 2), multi-digit multiplication (unit 3), connecting multiplication and division (unit 4), and using place value strategies to divide (unit 5).
Below are the scores from Celine’s pre and post assessments for each 3-week unit (in blue) compared with the average score for the other students in the group (in red). I could write a lot about the pros and cons of these sorts of assessments, but for now it is most salient to point out that everyone else made fairly consistent progress with this measure, and Celine dropped off at Unit 4: Connecting Multiplication and Division. Unit 5: Using Place Value to Divide was even worse.
I thought that Celine’s scores dropped off around Unit 4 due to engagement. By that point, she hated her regular 6th grade math class with such passion that she started to view all numbers as demonic. She adorned them with horns and devilish anti-math quotes she collected from memes online. Never had a student’s hatred for math make me simultaneously smile.
Now – three years later – I wonder if Celine was not checking out solely because of some anti-math sentiments, but because of her frustration with her reliance on counting strategies. It’s one thing to work on multiplication with counting strategies, and it’s another to work on division. The world must have felt backwards.
Celine understood how to use place value to multiply larger numbers, but to calculate 30 x 60 she first had to figure out 3 x 6 – for which she counted, keeping track of her place with her fingers.
“1, 2, 3, 4, 5, 6.” She recorded a 6 on her paper.
“7, 8, 9, 10, 11, 12.” She wrote the 12 next to the 6.
“13, 14, 15, 16, 17, 18.” She wrote the 18 next to the 12.
Her written record gave the illusion of skip counting, but she was most certainly counting by 1s.
Sometimes, she would use her previous work to help her: “I need 5 x 6. I already did 3 x 6, so I can start from there.”
And then she would count by 1s.
III. “Long live the finger strategies.”
Celine hated the problem strings we did as a group – the same ones that helped Bianca and Daysha and Kylee and Mia blossom. “What’s so good about math in my head, anyway. Fingers are underrated,” Celine told me. “Long live my finger strategies.” (Jo Boaler agrees.)
Long live her finger strategies, indeed. This reminded me of that quote from Michael’s essay, about the students who remain stuck on counting strategies. I am all for the use of fingers to develop some perceptual memory, but what if it is indicative of a student who is stuck on counting? What is this student’s motivation to develop more sophisticated strategies?
It did not seem to matter that Celine and I were working on derivative strategies during group time, or that her family paid for an expensive private tutor to work on this gap outside of school, or that her fifth and sixth grade classroom teachers had volunteered countless hours of their own time outside of school to work with her. Celine stayed true to her counting strategies.
“You know, I get that this is holding me back,” she once confessed. “But I don’t know how to do magic like everyone else in the group. 6×9 doesn’t just live in my head.” I valued her candor.
I met with Celine for a few sessions outside of our group to focus on making arrays. I had her draw 2 x 5 on giant grid paper.
“2×5… 2×5… oh, that’s 5 and then 5, and 5 + 5 is 10.”
Then I had her turn it into 4 x 5, and describe what she noticed. We then turned it into 8×5, and did the same thing. We connected them to problems and problem strings.
“You know, I get these patterns,” Celine told me after 2 sessions. “But how am I supposed to remember to use them? And if I don’t know 3×5, I have to figure it out. And by the time I have figured it out, I don’t remember that I want to use that for 6 x 5.”
Michael described this, as well. In order to derive a fact, there has to be something to derive it from.
IV. The Neuropsychological Evaluation
When Celine was in 7th grade, she had a full neuropsych evaluation done, both by our school and a preeminent local children’s hospital. Celine was doing well in her other subjects, so what made math such a challenge?
I felt pretty confident that Celine had dyscalculia. I started reading a bit about it, and even purchased a book called “The Dyscalculia Assessment.” (…but didn’t get around to purchasing the even more useful books about what to do with a student that HAS Dyscalculia.)
I think that her working memory is better than these scores would indicate, but still. That’s quite a discrepancy – to demonstrate 3rd percentile scores in working memory and 84th percentile scores in processing speed.
Compare this with her reading scores:
A few other scores struck me, including low scores in rapid digit naming and sequencing. Celine had some cognitive reasons to need some extra help and attention, but I remained firm with Celine: she could do this.
V. Enter: NZ Maths Continuum
The neuropsych report was interesting, but did not offer any suggestions for where to go next. Towards the end of 7th grade, I assessed Celine using the Numeracy Project Assessment (NumPA) from Book 2 of the NZ Numeracy Project (nzmaths.co.nz). The assessment helps determine a student’s “strategy stage” (based on the mental processes students use to estimate answers and solve operational problems with numbers) and “knowledge stage” (based on the key skills students leverage into learning).
From NZ Numeracy Development Project, Book 1, p. 1:
There is also a continuum (in Book 1) that describes the indicators for three primary categories: addition/subtraction, multiplication/division, and proportions/ratios.
Celine’s skills were routinely higher than her strategies, almost all of which seemed stagnant at “counting from one by imaging” or “advanced counting.”
I showed Celine where this fell on the continuum. “Your addition strategies are mostly here,” I told her, indicating Stages 3 & 4. Celine agreed that the descriptions matched her thinking.
I pointed to stage 8: advanced proportional. “We keep trying to get you to think like this. I know you can do it. You are going to get there! But maybe, instead of pushing you to think like that right now, we should push for stage 5. That way, we aren’t skipping important steps for you. What do you think?”
She read over the description for stage 5: early additive thinking.
“Okay. I can do that.”
VI. Trying Something New
I left for 4 months of maternity leave. While I was away, Celine summarily failed out of 8th grade math. Her classroom teacher recommended that she and another student (with a very different profile) receive their math instruction outside of the general education classroom, and so I returned to find her spending 50 minutes per day doing math in a special education classroom in the basement. Her attitude towards math hadn’t improved. If she ever had a desire to develop fluency with “math facts,” it was now gone.
I was also informed that I would be teaching Celine’s math group 2x/week, alternating with another math teacher and a special educator. Poor Celine wasn’t even going to have the same math teacher every single day.
Honestly, I was livid; I didn’t think Celine and her mathematical partner-in-crime should have been removed from the general education classroom at all.
To develop some semblance of a cohesive experience between teachers, we used the Transition to Algebra curriculum with as much fidelity as possible. I like a lot of things about Transition to Algebra. I like that builds off strong visual representations (area models, number lines) that focus on the structure of the mathematics, and that the understandings are often uncovered through puzzling and problem solving. I was able to use the mental math warm ups to target some specific skills for Celine.
Most importantly for Celine, it felt and looked challenging. This wasn’t a typical intervention curriculum, with procedures broken down into barely decipherable parts. Students had to think. Students had to make sense of things.
In a later unit, we used “mystery number” puzzles to understand how to solve expressions. Here, early on in that unit, Celine wrote her own.
As much as Celine truly wanted to move from counting on to more additive strategies, working hard to calculate 6 + 7 felt insulting. She knew it was a problem students far below her chronological age had mastered. It made her feel stupid. Meanwhile, doing well in Transitions to Algebra made Celine feel capable. It included algebraic variables, just like the work her peers were doing in the general education 8th grade class.
Celine went from working on basic multiplication to multiplying polynomials.
Because of our separate setting, I was able to draw out the problems to work on some of those earlier skills. How do we solve 4×4? I was mindful about how frequently I slowed down a problem to put the spotlight on computation, knowing of Celine’s working memory weakness. Sometimes she just referred back to a multiplication chart or used a calculator.
Over those last few months of eighth grade, Celine made considerable progress both with her math and her mathematical disposition. She started to make connections between different ideas consistently, something she had always done but only in fervent spurts. In particular, she enjoyed our Transition to Algebra unit on linear equations.
…but I still saw her solving 5 + 9 by counting her fingers — from one.