Teaching Rachel: On Learning What You Don’t Know


I. Rachel

Rachel was a student in my fourth grade class. At the start of the year, she showed traits that I worry a great deal about. She was a quiet girl who didn’t raise her hand at even the safest questions I lobbed. (“What is your favorite number?”) Was she hoping to learn from the margins of the classroom? Did she have trouble making friends? Was she lost in mathematics, scared?

This was just my second time teaching math to fourth graders. The year before I had taught what’s called a “fast-paced” class at my school. Most of those fast-paced students hadn’t needed much help with single-digit multiplication, which was the first topic we started fourth grade with. I watched Rachel’s multiplication closely, but did not know what to be watching for.

Over the first few weeks of the year, I asked my class to solve problems in context: 8 cows have 4 legs each; 13 triangles have how many sides? I led my students in number talks — I posed problems and asked for their mental strategies. I named their mental strategies — I named them after students, like “Tori’s Doubling” — and watched most students make big, important leaps forward. I showed my class rectangular arrays of various sizes and asked them how many things they saw in them.

We estimated. We played the Factor Game, the Product Game, and the Array Game. I assigned worksheets, quizzes and challenging problems that pushed Rachel’s classmates beyond where they had previously gone.

It took me a shockingly long time to notice that Rachel wasn’t making much progress.

When I began thinking again about Rachel, I searched through old photos of student work from that year. I found two photos from that October, and both give a good sense of how Rachel handled multiplication at that point in the year.

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The first photo contains just three problem and their solutions. The work is on a small whiteboard, the only surface on which I could get Rachel to reliably write down her thoughts for much of the year. At the top of the board she wrote “8 x 8 = 12,” then “9 x 7 = 65,” and finally “14 x 13 = 22.” While her answer for 9 x 7 was wrong, in a way she sort of got close. (She knows it’s an odd number in the sixties.) More alarming are the other two. “8 x 8 = 12” isn’t just wrong — the answer is in the wrong realm of number. Likewise “14 x 13 = 22.” These equations only begin to work if she isn’t thinking of multiplication at all, and instead is thinking of them as addition.

If 8 x 8 should really be seen, in Rachel’s work, as 8 + 8, then the mistaken sum of 12 would be typical for her. Rachel would often solve problems like 4 + 5 or 6 + 3 by counting on with her fingers. When not using fingers, she was often counting on silently in her head. (Subtle movements of her mouth suggested this, and questioning confirmed it.) After embarking on addition problems whose sums crossed ten (like 8 + 8 or 14 + 13) she would fall quiet for several seconds and then announce her answer — very often it was off by a few.

It seemed to me that addition was hard for Rachel, though she didn’t see it that way. “I’m good at adding,” she told me, “but I’m bad at multiplying.”

Given all this, it might seem that the meaning of multiplication was missing for Rachel; maybe she couldn’t made sense of it. But I found a second photo of Rachel’s work from about the same time in the school year. This second photo reminds me that, while all of this was going on, Rachel did have fairly solid ways to interpret multiplication problems.


In this picture, Rachel was tasked with finding 8 x 6; she found that it makes 48. This board is hard to read, but I think I know what happened. She first drew groups of eight small marks. This is good — understanding multiplication as saying something about groups is a solid way to make sense of the operation. Rachel would have had to sum up all the tallies she made. Many other times I saw her count and recount marks to try to arrive at some stable result and she would have done something similar here, counting one-by-one through each group of eight. As best I can tell, though, she only counted the first 24 tallies one-by-one. After that she decided to add on eight tallies at a time, which is why her board shows 8 + 24. This addition would not be easy for her. Maybe, somewhere in this, I’m helping her — that could explain the quick move from 24 to 8 x 6, perhaps after I encouraged her to give Toni’s doubling strategy a shot.

Though a mess, her board is full of good math. Her board also shows a good approach for 6 x 12 — adding six 10s and then six 2s. (Though between these problems she writes 31 x 21 = 52.)

It makes sense that there is good math here, because Rachel is not stupid. She is smart. She is also funny, with a wry, intelligent sarcasm. Her most obvious talents are artistic. She would sometimes show me sculptures and toys she made, and they impressed me. (Her father is a professional designer who makes furniture as well as toys. She once gave me a wooden robot of his as a gift.)

Rachel also showed mathematical talent. This became clear — to my surprise, honestly — any time our class deviated away from numbers. When we studied statistics, she raised her hand often to argue why the mean, median or mode (or something entirely different) was the most appropriate measure for the data distribution at hand. She was usually right. Games and strategy were also a strong point — she could out-maneuver other students, and had incisive things to say about which move was best, and why.

As the year went on, Rachel had scattered successes with multiplication. I remember a certain problem — maybe 12 x 13 or something like that — that she quickly interpreted as 12 groups of 13, and drew a diagram that allowed her to tally up the product, slowly, oh so slowly, and while accuracy only visited her work ocassionally this time it did, and it was wonderful.

Rachel could make sense of multiplication on a conceptual level. And she was getting lots of chances to add or multiply single-digit numbers in class — I was careful to include that for her. I planned most of my lessons with Rachel in mind. Each day I would search for activities that were right for the class and would also give Rachel solid ground to traverse, some place on which she could build. I wanted her to grow into arithmetic, and I wanted her to come to trust my teaching. Instead, the year was slipping away and I was still watching Rachel completely fall apart when faced with multiplication.

II. When Deriving Leads to Remembering…

That first group of fourth graders I taught — the “fast-paced” kids — were, on the whole, quite mathematically talented. They enjoyed tough math; longer worksheets with bigger numbers sounded like a fun time to many of them. Those students would get antsy if we spent too long on a topic, wondering what we would study next. Many would tell you they were good at math.

I was curious to dig through my old files and pictures to see how I had approached multiplication with this class. In a folder labeled “Week 1” I found a single-digit multiplication assessment that I must have given my class. Alas, I couldn’t find any of their results. But in the folder marked “Week 2” all the files are images of large arrays of objects — there’s a picture of over four-hundred boats — that I had asked students to mentally estimate with multiplication. Those kids must have done pretty well on that first assessment!

I was leery of focusing too much on explicit practice of the facts, and it didn’t seem as if the kids needed that sort of work. I didn’t want to focus on the facts; instead I asked kids to mentally derive products, and we talked a great deal about our strategies. More than anything else, I didn’t want my students to memorize their facts, and I didn’t ask them to.

I brought up “memorizing”; this is as good a place as any to say what I meant by it. A basic distinction is between our capacity to derive a fact through reasoning and our capacity to recall that fact directly from memory. There are some things that I just know — that my middle name is “Brandon” and that two and two makes four. Other things, I figure out — that my middle name has seven letters, and that twenty-seven two times makes fifty-four.

What does it mean to “memorize”? You might say (back in the 1990s) “I have his phone number memorized.” By this you’d mean that you can recall the number directly from memory — you just remember it. But you also might say “I learned his number by memorizing it” in which case you’re saying something about how you learned it. You’re saying, roughly, “I studied the number and then tried to recite the phone number from memory, and I kept on doing this until I got it right, and when I forgot it I practiced it again, and now it’s not like I use some mnemonic to remember the number it just spills out of my head.”

Memorizing can refer to this how-I-learned as well as describe what-I-learned. When I said “I didn’t want my students to memorize their facts” I was talking about the how — I didn’t want them to practice remembering the facts. Instead, I wanted them to practice deriving them in cleverer and more efficient ways. Eventually those derivations would become second-habit, and then memorization (just knowing them) would happen all on its own.

I know many middle and high school math teachers who are indifferent as to whether students can recall facts directly or not: As long as you can fluently derive it, who cares? I was this sort of teacher when I taught my fast-paced class. Fluent derivation seemed to be all that mattered, and more often than not it resulted in a student’s ability to recall facts from memory anyway. This worked well for nearly every student in my class except for a girl named May.

May was the “Rachel” of my fast-paced class, though she is nearly the opposite of Rachel in every dimension of personality. Where Rachel is cool and absurd, May is goofy and really, really out-there. I can’t remember all the costumes she used to come into class with — tiaras, wings, face-paint. Where Rachel seemed to hide in the corners of my classroom, May put herself front-and-center, though rarely in mathematical ways. Rachel’s papers came to me blank, but May drew vibrant portraits of mermaids and bakeries on top of her math work.

Here is what I wrote about May for her mid-year report card:

May struggles with multiplication and division, but has made significant improvements over the course of the year. I’ve seen May use a widening range of multiplication strategies to reason herself toward the solution to a calculation problem, rather than relying merely on counting and recall. I’m excited to watch May’s calculation skills continue to improve as the year continues and we offer her more opportunities to practice them. In particular, I’ll be emphasizing the development of efficient mental techniques for doing arithmetic.

My report to May’s parents represents my most optimistic take. In more pessimistic moments, I felt fear that she was falling through the cracks. A focus on derivation was working for her classmates, but she was going to get left behind.

I tend to stew on my failures in the classroom, so I would have thought of May often while working with Rachel. Working with May hadn’t given me any techniques for helping Rachel learn to multiply. What May taught me most of all was to doubt in a new way: I no longer was at all certain that a focus on derivation would help all of my students.

III. …And When It Doesn’t

By December, Rachel was calculating 6 + 4 accurately. I wanted to spend another unit on multiplication and division with the class, but felt I had to push it off until Rachel’s addition and subtraction was ready for it. “I really like stacking,” she told me, though use of the standard algorithm nearly guaranteed that her answer would be off by a thousand or more. Even so, her skills were progressing. I was thrilled that she was putting thoughts on paper and not just on whiteboards. I took her move to this more permanent medium as a sign that she was feeling more comfortable with the class and with numbers, even if (in another picture I found) she still sometimes found that 6 + 5 made 10.


It was around this time that I waded into research on learning disabled students. This began as a job — someone asked me to read and summarize a bunch of stuff for them. (Not as sketchy as it sounds.) And while my school hadn’t told me that Rachel had a learning disability (“happily, math is an area of strength for Rachel” her report said), I kept on seeing connections between her struggles and my reading.

Ted Hasselbring is a researcher at Vanderbilt. (From his academic website: “Over the past thirty years, Dr. Ted S. Hasselbring has conducted research on the use of technology for enhancing learning in students with mild disabilities.”) I read several papers by him, including one where he asked learning disabled students to play an arcade-style learning game. A math problem appears on the screen wrapped up in, I don’t know, a rocket ship or something, and you have to shoot it out of the sky except the only way to shoot it is by answering the problem correctly. Hasselbring hoped the game would help his students come to recall more addition facts from memory. That wasn’t what happened, though:

[Hasselbring and co.] found that this form of drill and practice had no effect on developing fluency in learning handicapped students unless those students had already dropped their attachment to a counting strategy. 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)

Their solution was to tweak the game so that an answer appeared after a short amount of time, so that the student couldn’t fall back on the counting strategy. I wasn’t sure how this would go over with Rachel, but her problem with addition, I thought, was precisely the one these researchers had identified. She counted, and was committed to counting. All she was learning from me was better and better ways of counting.

The work of these researchers also helped me understand my experiences with May. May had many more strategies for deriving products than Rachel. May could double and use nearby facts to infer new ones pretty well. Despite all the times she practiced these strategies, though, she didn’t come to commit these facts to memory as I supposed she would. Why hadn’t she?

May was not learning handicapped, but there is a parallel to what Hasselbring had found in his study. He had found that students who were fluent in a derivation strategy (counting on) did not come to have these addition problems committed to long-term memory. For all their practice, they had only seen improvements in their use of counting on, one-by-one to find a sum. This was similar to what I saw in May, even though she was practicing more sophisticated strategies and working on multiplication.

How could it be that practice didn’t lead to remembering? It seems perverse. Practice, in general, makes perfect. If you’re practicing 4 x 3 or 6 + 5 over and over again, shouldn’t you come to remember it?

Practice does make perfect, but care is needed in specifying what exactly it is we’re practicing. I find it helpful to think that you get better at what you practice — but only what you practice. If you do a nice job practicing deriving facts, your ability to derive will improve. To get better at recalling facts, in contrast, we need to practice remembering them.

Recently, I was thinking about a geometry problem. The problem involved a right triangle whose height, when drawn, divides the whole triangle into scaled-down copies of each other. There is a name and formula summarizing this theorem that I’ve never been able to remember, though I’ve derived the result many times by first principles. In light of the learning disability literature, though, maybe it’s because I derive the result that I am unable to simply recall it. I have never given myself opportunities to try to recall the fact.

We help students come to recall facts through recall practice. From other papers I read, I developed a picture of what this might look like. As I understood it, a person’s memory grasps a fact weakly at first. When we encounter something new, the mind does not yet know its value. But we can teach our minds to value it. Each time we recall a fact, the fact proves its value and our mind’s grasp on it becomes stronger. (The grip grows especially strong around those things we’ve nearly forgotten, as if the possibility of loss makes the mind hold the fact even closer. Cognitive scientists call this “the spacing effect.”)

To strongly remember a multiplication fact, we begin by planting a barely-there memory in our minds, e.g. by studying a card that says 3 x 7 = 21. If this was all we did, the memory would quickly fade. Instead, we immediately draw this memory forth e.g. by removing the card and trying to remember what 3 times 7 makes. We then turn our attention elsewhere, causing ourselves to nearly forget the fact. The perfect moment for practicing our recall is when the memory is still-there-but-fuzzy. We continue this process until the fact has formed deep roots in our mind.

That spring, I tried to figure out what this approach to practice would look like with my students in class. Borrowing another idea from what I’d read, I wrote a series of multiplication problems on the board. I asked my students to answer the questions on the left side of a blank piece of paper. I asked students to raise their hands and share their answers to these questions, and once that was done I asked them to study their answers. Finally, I asked everyone to fold their pages over so that the answers (but not the questions) were covered, and to quiz themselves again, trying to remember the answers.

This was all fine, but I didn’t repeat the experiment. The whole routine took too much time, and the questions I wanted Rachel to practice were too easy to be worthwhile for the rest of the class. I still didn’t have a solution.

The students that animated all this inquiry were Rachel and May, both of whom I came to think were not well-served by the derivation work my classes were doing. And I now felt as if I understood why their practice in class wasn’t leading to their coming to recall more facts. This explanation, though, presented a new puzzle: why had the derivation approach worked for so many of my other students?

Perhaps the story goes like this: for my faster-paced students, coming to recall a fact from memory still depends on recall practice. It’s just that once they were able to fluently derive their facts, they suddenly find themselves afforded many more opportunities to practice recalling them.

Suppose a student has just derived 9 x 4. If they’re confident and successful, they might have an opportunity to share that solution with the class — I might ask them to share their solution, and they might have a moment where they ask themselves, “wait, what was 9 x 4 again?” This is recall practice. Or, maybe, they are working on a larger problem in which 9 x 4 is merely a step, and their later work calls on them to remember the product of 9 x 4. They derive it, and then turn back to the problem and ask themselves, “what was 9 x 4?” Or perhaps, while working on a large set of multiplication problems, a student derives 9 x 4 and is then asked to derive 90 x 4. They ask themselves: what is 9 x 4?

That strategy work can support recall was in line with other papers that I read. John Woodward, for one, randomly assigned some learning-disabled students to work on their multiplication via timed-drill, and others to use a combination of timed-drill and strategy practice. Both groups made nice progress (and both reported good feelings about the work), but in some ways the combination of strategy with drill was more potent and long-lasting (Woodward, 2006).

If students can practice their recall during derivation work, then students could come to have their facts memorized as a result of strategy practice. All this could happen naturally, without a need for explicit recall practice.

The possibility of this happening for a kid depends crucially on their degree of fluency with derivations confidence, efficiency, a certain mental speed. Without it, the natural opportunities to learn disappear. The student doesn’t finish computing the problem and so they aren’t able to remember the solution when they contribute to the discussion. The student spends all their time working on 9 x 4, and doesn’t get nearly as many chances to work on problems like 29 x 4. The student relies on addition for 90 x 4, which might be faster for the child but doesn’t give them a chance to practice remembering 9 x 4.

Rachel needed to become fluent — in addition, multiplication, whatever — if she was going to learn more facts via derivation in class. I had tried to encourage her to use a wider variety of strategies, but she still relied on counting for addition and repeated adding for multiplication. Why was she having so much trouble becoming fluent in derivation strategies?

IV. Helping Rachel

I wonder what Rachel’s math would have been like if, in her early years of school, her teachers had helped her learn more efficient mental strategies for adding and subtracting numbers. Maybe then she would have had the opportunities to recall facts that my “fast-paced” students had, and she wouldn’t have come into my fourth grade class with so much trouble with numbers.

The issue is that fluency in deriving facts is nearly always built on a foundation of directly recalled facts.

Fluency always depends on recall. Here is a derivation of 6 x 8 that I have seen many times: first, I realized that 2 x 8 is 16. Then I doubled that, and got 32 for 4 x 8, and then added another 16. That makes 48. Notice, though, the several times this student usually relies on things she directly knew. She often directly recalls that 2 x 8 is 16, or even that 16 + 16 = 32.

When a student attempts to derive 6 x 8 using such a strategy without directly recalling any facts, things get very messy. The students ends up wound up in loops of derivations, recursively spiralling into new problems. It’s not long until a student loses track of what’s in their head: To figure out 2 x 8 I counted by 2 eight times. Then to double that I did 10 + 10 and 6 + 6, that makes 20 and 12…

Rachel didn’t have any of these facts to support her multiplication derivations. She often used her fingers to answer single-digit addition questions, and for sums as small as 12 she often derived the answer incorrectly. Fluency depends on fluency, fluency depends on recall, which itself depends on fluency, and the whole thing was a huge tangle of not-know and can’t-do for Rachel.

It was nearly April when I decided that Rachel needed to be able to directly recall more basic math facts.  She continued to get problems such as 6 + 5 wrong, and my attempts to teach her addition strategies in class had failed entirely. Having more facts on recall would help her immensely, I believed.

Rachel finished out fourth grade math as well as she could, struggling whenever we worked on multiplication or division (or addition and subtraction), coming out from the shadows when attention turned to geometry, games or statistics. Fractions were fine, until it came to talk about equivalence and fraction addition, skills that call on multiplication and single-digit addition far more than I’d ever before noticed. I wished her a good summer, feeling both that I’d done her well and that in numerous other ways I had not.

At my school, all fifth graders take a once-a-week class to practice their numeracy skills. I taught this class, and often would get “repeat customers” from fourth grade. Rachel was on my roster — I had another chance.

I talked to Rachel’s fifth grade math teacher — kids call her Ms. H — often. We agreed: she was a smart girl whose experiences in class would improve drastically if she just knew more of her single-digit addition. She couldn’t master strategies for 4 x 8 because 16 + 16 took her too long to compute; 16 + 16 took too long to compute because she counted one-by-one to solve 6 + 6; she didn’t remember strategies for 6 + 6 because she didn’t know 6 + 5 and the leap from 5 + 5 was a bit much for her; 5 + 5 she knew well. Recalling even a few more addition facts would give her skills a shot in the arm.

Ms. H and I began a small-scale campaign to spread this message to the adults in Rachel’s life. The parents were helpful and receptive. Rachel already had a tutor who helped with all her academic subjects — they suggested the three of us coordinated our efforts. But the tutor was insistent: in their time together they needed to focus on other subjects and on helping Rachel with the grade-level math work. The tutor even agreed: Rachel definitely needed to work on her addition facts! But there was no time for the two of them to focus on them during their meetings.

It was once again April by the three of us sat down together — the tutor, the teachers, and now with the administration. We found a compromise: Ms. H and I would do it. She and I would each meet with Rachel once a week during lunch to practice addition.

At my first meeting with Rachel, she was tentative. I took out twenty index cards and placed them between us on the table. I took the first card and wrote an addition problem on it: 5 + 5. I asked Rachel to think about this in her head, and to share the answer and how she thought about it, when she was ready. I waited as long as she wanted to.

I asked her to write the correct answers on the back of each note card. We talked about strategies. If I found a problem that was hard for her, I’d make the next three easy. I wanted her to answer most of the questions correctly while practicing, another idea I’d pinched from my reading.

I sometimes asked her to write “strategy hints” at the bottom of the card. For 5 + 4 — which she counted by ones to determine — I asked her to write “4+4” as a hint, as she just knew what 4 + 4 made. I wanted to combine derivation and recall practice (ala Woodward) to cut a new path through addition for her.

Rachel and I met six times, hardly enough to start seeing a major difference. I was seeing progress, though. Rachel was becoming more confident with mental addition like 13 + 15, and there were things that she was coming to “just know,” like 7 + 8. I was seeing more engagement in our once-a-week class. I went to Ms. H to get her perspective on Rachel’s progress. She was also working on facts with Rachel also in her weekly sessions, and she saw Rachel in class each day. She too saw improvements in Rachel’s thinking and fact knowledge. Rachel was more willing to try problems involving addition in class, and she was caught using a good strategy or two. Finally, we were making progress with Rachel.

V. Conclusion: Now, Sixth Grade

That was that. Summer came, and once again Rachel and I said goodbye to each other. (That was when she gave me that little robot.)

This year, Rachel is in sixth grade. How is she doing?

Rachel’s parents asked Ms. H to continue to provide out of class support with math. Though I think about her all the time, I’m out of the picture. When I see Rachel in the halls, I’ve said hello and asked her how things were going, but she didn’t mention anything about math. I recently asked Ms. H how Rachel was doing, mathematically. Ms. H sighed.

“The conceptual stuff is down. It’s all the other stuff that slays her,” she said. I asked what she has been working on with Rachel. Ms. H said she was helping Rachel with her classwork — order of operations — but still trying to help Rachel improve her basic fact knowledge.

I ask her how the basic fact work is going. “I don’t know what good I’m doing, other than keeping it fresh,” she said. To help me get the picture, she told me a story. She had asked Rachel to roll a few six-sided dice and to sum their faces, to practice mental addition. She had been hoping that Rachel would group all the fives together. But Rachel didn’t do this — she instead took all the dice and organized them to make sixes and fours. It seemed to Ms. H that Rachel still didn’t know that five and five make ten. (Though, apparently, she does know what six and four make.)

Rachel wants to work on her classwork with Ms. H, not the basic facts. But Ms. H thinks the facts are important, and sends her home with a multiplication table to practice on each week. The most optimistic thing she’s willing to say is that progress goes slowly for Rachel.

And what next? After sixth grade comes seventh, then eighth and ninth, two more years after that and then she’s done with math for good, if she wants.

My teaching has changed since I taught Rachel. I don’t always have a students like her in my classes, but can I afford to wait two months to realize the seriousness of the issues a student is having? I don’t want to let down kids like May and Rachel. So I ask my classes to make little deck of cards to help them practice especially useful facts — like the deck I asked Rachel to make. I don’t want to ruin math, so I don’t time them (or at least I don’t tell them when I’m paying attention to their speed).

It’s not that I’ve decided that it’s best for kids to learn via drill over derivation. But fluency depends on some amount of recall, and fluency itself is needed to come to know facts via derivation. How many facts? How much practice? Will their teacher next year expect them to have their facts memorized? I don’t know, so I search for ways to protect my kids in the face of what I don’t (and can’t) know.

There’s a possibility I’ll have a chance to teach Rachel again. I’ve taught her older brother for the past two years, in eighth and ninth grades. Like Rachel, he has a fantastic artistic eye and a lot of mathematical talent. He’s seen a great deal of success in my classes, and apparently he had troubles in elementary school. Maybe a nice surprise is waiting for Rachel in middle school math.

Then again, maybe not. All of which leaves me with complicated feelings. This world — of teaching elementary, middle and high school students at the same school — is still fairly new to me. When I taught only high school students, their mathematical past was an mystery that was only partly filled in by hints from parents or administrators. The parents are totally clueless I was sometimes told, or his middle school was very weak, they basically didn’t study math. (Some private schools make very interesting academic choices.) There was a sort of abstract sense in which I took responsibility for the success or failure for these kids in math, but I knew that I was being put in a tough position.

That’s no longer an emotional option for me. This year, my elementary classes are third and fourth grade. There are a number of tricky kids. Levi gets so anxious that he often refuses to try a problem. I didn’t realize until just last week that Sofia struggles with multiplication. There’s also Lucy and Stella, both clever but both struggling. Things don’t seem dire for any of these children. They’ll all turn out fine, I think.

But who can be sure? There are so many things that I don’t know about my students, and the things I learn often come too late. So my students have their little notecards, and they use them to practice remembering their facts, just in case.


Bransford, J. D., Goin, L. I., Hasselbring, T. S., Kinzer, C. K., Sherwood, R. D., & Williams, S. M. (1986). Learning with technology: Theoretical and empirical perspectives. Peabody Journal of Education, 64(1), 5-26.

Woodward, J. (2006). Developing automaticity in multiplication facts: Integrating strategy instruction with timed practice drills. Learning Disability Quarterly, 29(4), 269-289.


24 thoughts on “Teaching Rachel: On Learning What You Don’t Know

  1. Your timing is exquisite. This past semester I had several students who were/are stuck in ‘counting,’ even needing instruction in how to do that (they counted the starting number as adding ‘one’). My experience concurs with yours: not having the basic fluency seriously impedes developing other fluencies.

    I also wonder whether it impedes with being able to deconstruct numbers and understand the “parts and wholes” relationship. I’ve never taught little guys but I’m finding lots of lessons where they spend tons of time with ten frames and doing subitizing. I wonder if some of my folks either didn’t get that, or weren’t developmentally ready for it — but got there… only once you get to middle school, it’s all about the symbols. I have taught at a school for students with learning difficulties and *everything* was done Concrete-Representational-Abstract… lots of manipulatives in math. I’ve also had classes in using manips for older kids and w/ more advanced math but it hasn’t been part of my actual job.

    So! I’ve decided based on experience & research that I want to make a mini-course in the Canvas LMS that includes the visual-to-symbols bridge so that memorizing 4 + 3 = 7 has more meaning than reciting the last word in a poem in a completely foreign language.

    I haven’t done a *hard* dig to see what’s been done w/ teaching older students / adults math w/ C-R-A (often expressed as “real life”)… usually that whole population is dismissed. Unfortunately, funders and sponsors want to see good, quick results — so they build programs for the other huge swath of humans who have a better chance of providing that.

    Ted Hasselbring has done a *lot* of good for folks with LDs. Unlike too many researchers, he actually questions things. He designed a reading program and then at a conference said that he had been completely floored by the fact that while the program’s focus on comprehension helped many students, there were to his shock many students who still weren’t fluent decoders. (I wished he’d called me ;P ) So… he designed another program.

    Back in 1998 I was an ed tech major and my grad assistant job was managing our educational software library. It bugged me that the math educational software was all “shoot the right answer,” and I didn’t see how that was going to help my guys. Glad to see somebody officially getting that ‘out there.’ Alas, it hasn’t really changed the software!

    Now I’m over to Canvas where the work in progress is at https://canvas.instructure.com/courses/1051168 — but I’m not sure who can get to that. (I have a “free for teachers” account.) I eventually want to have our whole “transitions” class adapted to trucking career up there…


  2. (Forgot to mention that as with your experience the students in question understood concepts … and also that ALEKS has helped a fair amount b/c it provides individualized practice and our course designer has spent lots of good time and effort picking the practice problems that help.)


  3. I read your post and ached for Rachel, as well as for your teaching struggle. There have been times in my teaching experience where I’ve had students bring me to my pedagogical knees. Many years ago, I was the closing speaker at the annual Asilomar math conference in California and I gave a talk about the group of students in every one of my classes (I was teaching middle school then and saw lots of kids each day) who I knew I wasn’t serving and who I didn’t know how to serve. I was trying to open up the conversation about how we might better meet the needs of students like these.

    The talk was a complete downer and many found it terribly depressing. Some criticized me for “giving up” on some kids, some offered me suggestions for improving my instruction, and others told me that I had pinpointed the same problem they were struggling with.

    In retrospect, the talk wasn’t the best choice for closing a conference. I’ve never given that talk again, but I think it’s an important issue, even more critical as students get older.

    Most recently, I’ve had conversations with a fourth grader, now a fifth grader. These weren’t in a school setting―a friend was concerned about his granddaughter, Z, and I invited them to come for a visit. This was in December 2015. I talked with Z first about things that she enjoys. She likes taking photos and posts regularly on Instagram, she told me. (She and I now follow each other on Instagram and I think her photos are marvelous.) With her permission, I conducted a one-on-one interview with her so that together we could learn about the source of her math difficulty.

    Z was a bit shy in the situation, which isn’t surprising. But she’s friendly, kind, gentle, with a quick smile, and was willing to give the interview a try. At the time, she was going to an intervention teacher daily when her class was learning math. The intervention teacher mostly helped Z with the homework, dealing with the same content that her classmates were learning. The class was working on long division. It seemed sort of OK with Z to be with the intervention teacher, except that she’d prefer to be with her friends.

    In the interview, I began by focusing on whole number addition and subtraction, interested in learning what she did understand and could do, and if there were gaps. As I’ve experienced with other students, Z’s responses gave a sort-of Swiss cheese profile―a mixture of holes and understanding. I didn’t move past addition and subtraction. Then Z, her grandfather, and I went out for lunch and had a fine time. Here’s the message I sent to Z’s mother and grandfather after this meeting:

    “I just spent a delightful afternoon with Z. She’s lovely (as I’m sure you know). I interviewed her about basic math concepts and skills, focusing on adding and subtracting and on having her reason mentally and explain her thinking. To me, mental reasoning is the kind of effort that can have the greatest payoff for students, with the goal that paper and pencil become tools for students to help keep track of thinking, not to reproduce procedures. It’s the thinking that’s most important.

    It’s clear to me that Z’s foundation with addition and subtraction is wobbly, though some of her knowledge is solid. Z understands the basic idea of place value, and uses the inverse relationship between addition and subtraction when solving problems (i.e., she uses addition and subtraction interchangeably when solving problems like 14 – 8 or 13 + ? = 20). She doesn’t have automatic recall of all number facts through 18 and relies on counting up, using her fingers to keep track. That said, she is fluent with combinations of two numbers that add to 10, but has to count for others. She can apply strategies for adding such as using doubles (e.g., to add 6 + 7, she knew that 7 + 7 = 13, so 6 + 7 had to be 13) and using known facts when adding numbers that involve multiples of 10 (e.g., thinking about 10 – 8 when solving mentally 100 – 98). She also demonstrated splitting numbers into place value parts to add and subtract, and using 10 or a multiple of 10 as a benchmark, but she didn’t use these strategies consistently. With contextual problems (i.e., word problems), she understands the intent of the problem and can “mathematize” situations, but then struggles when dealing with the needed calculations.

    I suspect that this summary of my interview is too much teacher-ese language, but in essence, Z needs a good deal more experience with the basics of addition and subtraction. My goal would be for her to be fluent not only with the basics of adding one-digit numbers, but also knowing for any two-digit number how much more is needed for a sum of 100, and to be able to add any two-digit numbers mentally. She can add some (e.g., 70 + 50) but is often uncertain and lacks confidence.”

    Basically, all of the questions I asked Z were based on Grade 2 standards. Her go-to, whenever possible, was to count by 1s. When I prompted her, she was able to decompose and explain how to go to the nearest 10 when adding numbers like 18 + 7 (18 + 2 is 20, 20 + 5 = 25), but she’d rather count up. When numbers were too large for counting by 1s, her choice was to rely on a paper-and-pencil algorithm to figure out answers, not always correctly.

    And she was working on long division???

    A month or so later, her grandfather sent me two poems Z had written and read in an evening program with her fellow fourth graders. The poems were beautiful. I wrote Z a note telling her what I especially liked about them. I ache. Here is a girl who is delightful and full of exuberance in many ways (as I see from her Instagram photos), loves to play with language (as her poetry revealed), but she has a serious problem with math.

    I think that Z has a kind of math learning block that makes me believe that dyscalculia is a real disability.

    Back to Rachel. How similar do you think Rachel and Z are? And what do you think about dyscalculia? As you described with Rachel, there are glimmers of understanding with Z. Also, she has learned to use some algorithms that she has practiced, but I don’t think practice leads to or supports understanding. (You wrote a bit about the role of practice when you described May.)

    Z is now a fifth grader, and the decision from the school and family was for her to stay with her regular class for math. I can understand that as friends are so important at this age. I’m trying to figure out a way to be helpful.

    Liked by 1 person

  4. I have two or three seventh graders that still resort to repeated addition. They will multiply if I insist, but addition is their security blanket. I have resisted recall practice (a la Dr. Boaler) but you make a persuasive argument that a little might go a long way. I have also taught these students in sixth grade, and so I feel twice the burden to send them into eighth grade prepared for Algebra 1. Having said that, I know that the Algebra 1 teacher just hands them a calculator if they struggle to calculate by hand, so I wonder how much good I’m really doing. (Still, you can’t factor a trinomial with a TI-83, can you?)


  5. Moving my comments from Twitter here…

    In my experience (which has only been a couple of years) working across K-5, I felt myself wondering what experiences K-2 could have helped students like Rachel or May? My heart ached for them and for the struggle you and SO many other teachers face with students just like them mathematically.

    Being involved with the RTI process in my school has been frustrating, but eye-opening for this very reason. Our school and district data supports the fact that many students who get the extra support of RTI by being in “Tier 2 or 3” in grades K-1 move out of the tier because something “clicks.” We could talk about what makes things click at a later time:) But if a student was in a Tier at the end of 2nd grade, they never move out of the tier and always end up in the extra support group. When I taught RTI in 5th grade to students who I would put in the same group as Rachel or May, I struggled because what they NEEDED was a 2nd grade idea but I needed to try and weave that into some on-grade content because I wanted them to feel success in their math class.

    This makes me wonder, if what every 4th or 5th grader needs is a 1st or 2nd grade idea, what could we do better there? I have seen so many students who can appear successful in K-2 because they can “do” things. They can add, identify combinations of 5, subtract, identify shapes, etc….however there is so much thinking that should be behind that “doing” that I wonder how we can do a better job with that? What good is being able to identify combinations of 10 if we never give students an opportunity to USE that understanding? Another example, the number line is a wonderful tool for students in K-2, however if “doing” things on the number becomes the focus to get an answer instead of “thinking” about what it is about that number line that makes it such a powerful representation, I can’t see students building on that understanding in future years.

    Not sure if that made much sense and I do agree with Marilyn there could be a dyscalculia thing going on here, but it just happens so often it leaves me wondering if we need to move even further back in the elementary years to see what happens.



    1. I agree that it helps to have 4th/5th graders revisit ideas from 1st/2nd grades. Or experience ideas from those grades in a new way. I’m working with a class of 5th graders now that I began working with as 4th graders last year. When I assessed them at the beginning of the year, they were woefully behind. So we began our focus for math instruction on addition and subtraction (using Do The Math Addition/Subtraction modules A and B, which I think you’re familiar with). We didn’t attend to the 4th grade curriculum. It felt like a risky thing to do, but the only thing that made sense for meeting THEIR needs, not the demands of the standards. Now they bring a foundation and attitude toward their grade-level work that never would have happened without this.

      Liked by 1 person

  6. Thank you, Michael. This was incredibly helpful to read.

    My experiences in working with students with learning issues in math have led me to believe more or less that typical math learners are alike; math learners with learning issues struggle in their own way. A student who I am teaching this year is having an issue I had somewhat seen before, but never to this degree. In her case, it seems that either derivation or recall work fine, but only in the very short term (within a day of working on a topic). I haven’t been able to get her to keep understanding going for longer than this very short time frame. It’s incredibly discouraging for her and very confusing for me. Does she understand a given concept or not? Does she need even more practice? Should the practice be more concentrated? More spaced? Should we really be working on remediating concepts from many years back with which she’s shaky? She’s in high school so the number of concepts/ideas/procedures in a given course grows ever larger and the number of prior skills that she needs to “remediate” continues to grow too.

    Your insight that deriving depends on recall rings very true for me — or else you get into an infinite spiral of derivation. At some point, derivation needs to become recall so that the next idea can be derived on top of that. But what happens to students with these types of holes in their understanding/recall as they progress through the curriculum? Watching my student try to derive becomes a series of painful realizations that she’s reaching back into her memory and there’s nothing there to stand on for many, many levels back.

    The other part is of course how much identity connects to learning. For her, an opportunity to do math is a chance to see herself as a failure. She sees her learning of math as a series of tricks against her own learning, which doesn’t seem to cooperate and do the things that other kids seem to be able to do easily. We’ve been focusing on identifying aspects of mathematical thinking in which she excels and feels proud of herself, but it’s been a challenge to have her build a positive self-image from these alone. She needs to build up some successes in order to see herself more positively, but she needs to see herself more positively in order to get these successes.

    Would love and any all thoughts on how these issues play out for high school students!


  7. We are so focused on answer getting in mathematics. It is our [US?] cultural expectation that mathematics has an answer and doing mathematics means getting an answer. For students, this starts so early in their formal education.

    We have an overemphasis on the procedural aspect of doing mathematics so that we can get to the answer the most efficient, quickest way. That often is rewarded.

    How do we support teachers and parents to see the “beauty” in mathematics (see https://mathyawp.wordpress.com/2017/01/08/mathematics-for-human-flourishing/#comments) is in mathematical thinking and communicating that thinking to others?

    I wonder what would happen if teachers and parents started giving the answers to mathematics problems to students, right off the bat. Rather of hiding answers, we state the answers up front.

    Instead we ask students to give possible ways to get that answer. We highlight different approaches students have to arriving at the same (given answer). We compare and contrast the approaches and strategies. We ask students to make sense of the problems and the different arguments. While it may throw people for a loop to begin with, this puts emphasis on mathematical reasoning and communication.


  8. I’ve been teaching mainly math, to mainly 5th grade students, for ten years now. About 5 years in I got a master’s in math education and then NBCT in EA Math and a high school math endorsement. I have fully bought in to teaching computation through thinking strategies and have deemphasized the memorization math facts. I haven’t done a timed fact test in years. But whatever it is that I am doing with arithmetic computation, it doesn’t always work for everyone. There are those kids each year that don’t seem to “pick it up.”

    About two or three years ago I had a student who made me question the pedagogies that I held so close to my heart… for that student… for those students we have each year. I would never stop teaching students how to take apart numbers and put them back together with meaning and flexibility, but I came to the realization that I wasn’t doing some of my students any favors by offering them flexibility when they didn’t have a basic facility with numbers. I came to believe in a middle road for fact fluency similar to what I think Michael is talking about. You need some footing in recalled facts to compose and work with larger products and sums. Somewhere we need to spend the time required for our students to gain familiarity with some of the basic patterns. When they have the basic patterns down, then fluid, flexible and efficient mathematics has a foundation: deeper patterns can be developed, connections can be recognized. These students need some prior knowledge with numbers to hang things on.


  9. This is a wonderful, heartbreaking essay, Michael. I love all the nuance and questioning, and your comments reflect how much thinking you provoked in your readers. Thank you for sharing it.


  10. I’ve been coming back to this essay and the comments over and over again for the past few days; I feel that Michael really pointed at my biggest struggle with my teaching.

    I found that my challenging kids push my thinking and make me question my pedagogical choices a lot. I taught grades 3 and 4, and it is easier to identify the gaps in the earlier grades but it doesn’t seem to be easier to fill them. I worked with a student who would count from 1 for any addition problem for the first few months of the year. Who would check how many little unit cubes are in base 10 ten blocks rod no matter how many times we used them. Who needs counters to equally share sets of less than 10 objects. I see the progress but it is not a fast one.

    Then I think about the age and wonder that if I just keep going with multiple representations and more experiences things will click eventually even though they didn’t click yet. In the meantime, he is falling behind and I have to eventually get to multiplication and division and I don’t know what to do to help this student succeed.

    I wonder and worry that if I rush and push for recall before flexibility with numbers I will muddle the waters even more. I am trying to balance both, but I am not certain I’m succeeding; it’s a day to day battle. This prior knowledge with numbers is not an easy thing for everyone to acquire.


  11. Beautiful comments across the board. Here are some thoughts I had while reading your comments.

    I read your post and ached for Rachel.

    I noticed this sort of bluntly emotional language several times in the comments. It made me think about how teacherly this sort of concern for kids is, and how central it actually is to our work.

    With someone like Rachel — or Marilyn’s Z, or many of the other students mentioned above — the thing that tugs at our heartstrings isn’t fear of their futures. (As Joe Schwartz has pointed out, kids mostly turn out fine.) Knowing how to multiply isn’t make-or-break for a kid’s adult happiness, I’d guess.

    So what breaks our hearts? It seems to me it’s the kids’ experiences in school itself that kill us. People outside the classroom don’t always get this. They only see kids as living in the future, with or without certain jobs or capacities. We, on the other hand, have a deep appreciation for the present lives of children. Making kids feel dumb, isolated or sad is what we worry about, and it breaks our hearts.

    Is it important for kids to be able to multiply fluently? That’s a great question that I don’t think at all about. Teaching well matters because the kids are here, and that’s reason enough.

    I think that Z has a kind of math learning block that makes me believe that dyscalculia is a real disability.

    I don’t know anything about dyscalculia. I do have a few — pretty uninformed — thoughts. Here they are:

    * The purpose of diagnosis is treatment.
    * The diagnosis of a learning condition that has no treatment is of no value to the “patient.”
    * Or else the purpose of the diagnosis is for the purposes of research, or to give the “patient” different expectations for their futures.

    So: how does a diagnosis of dyscalculia help us teach the student? if Rachel has dyscalculia, what does that mean? how would a diagnosis of dyscalculia make a student feel, if it isn’t treatable?

    These are all live questions for me. I’d love to know more about learning disabilities in general and this one in particular.

    I taught grades 3 and 4, and it is easier to identify the gaps in the earlier grades but it doesn’t seem to be easier to fill them.

    This came through in so many of our comments. It might be surprising to some high school teachers to hear that in Grade 3 teachers are already worried about these gaps!

    There are no easy answers here. In fact, there might be no answers, from the perspective on an individual teacher. Kristin brings up RtI, and it might be that missing knowledge is only resolvable at the school-wide level. (Which is maybe the level that’s structurally responsible for creating these issues via grade promotion.)

    There’s only one trick that I’ve ever found to help. If I can arrange one-on-one time with a student in my class, I almost never address their past knowledge. Instead, I use that time to preview the next lesson, or sometimes the next two. The way I figure, that’s an investment that yields benefits over and above our time together, since I’m making sure that they learn something from that next class or two. At least we’ll have that.

    Could I have used this trick with Rachel? If I had been able to arrange a once-a-week meeting when she was in my 4th Grade class I could have tried. Maybe, if I knew that we were going to work on 35 x 6 in class that day, I could have shown her a closely related problem. Maybe I would have previewed 5 x 6 for her, so that it would be fresh in her mind. Or maybe I could have practiced 35 + 35, so that she could use addition to handle the problem.

    I have a struggling student now. I’ve now managed to arrange a weekly meeting with him. (Thanks to Rachel, I was quite on top of this.) Each week I show him what we’re going to be doing that afternoon in class. I give him closely related examples, I preview the important language and concepts. And each week he has a truly phenomenal class following our session, participating and getting to be smart in public and writing things on his own.

    And then the next day is a bit worse, the day following worse still, and by the end of the week he’s lost in math again. That’s when we have our next meeting.

    Liked by 1 person

  12. “The answer is all, even when it is wrong.”

    So sad, and so unmathematical.

    Addition and subtraction go hand in hand. If I calculate 7 + 9 and I get 24 then what? It’s wrong. So I have another go and get 38? It is more sensible to UNDO the problem and go for 24 – 9. Result probably not 15, and even less likely to be 16. No amount of “check your working” is really any benefit. Same goes for multiplication and division.


  13. A question from the science perspective is: How do young students solve multi-step derivation problems if constituent math facts are not recallable “with automaticity” first?

    Cognitive scientists use a “working memory/long-term memory” model to explain how the brain solves math problems.

    Cognitive scientists Alloway and Gathercole (2006) describes working memory “as a mental workspace that can be flexibly used to support everyday cognitive activities that require both processing and storage, such as mental arithmetic. However, the capacity of working memory is limited, and the imposition of either excess storage or processing demands in the course of an on-going cognitive activity will lead to catastrophic loss of information from this temporary memory system.”

    If at any step in problem solving, the “chunks” of information that are the goal, data, and middle-step answers exceed working memory capacity, chunks are lost and confusion tends to result.

    Nelson Cowan (2001, 2010) found an average working memory capacity for non-recallable information of about 3 to 5 chunks of information in young adults, with lower capacity for fourth graders and lower still for first graders (near 2 chunks).

    Kirscher, Sweller, and Clark (2006) write that working memory is “very limited in duration and capacity” for new information, such as the data presented in a problem, but “when dealing with previously learned information stored in long-term memory, these limitations disappear.”

    Daniel Willingham writes that the “lack of space in working memory is a fundamental bottleneck of human cognition.” To work around cognitive limits when solving math problems, he advises facts and procedures used frequently “must be learned to the point of automaticity so that they no longer consume working memory space. Only then will the student be able to bypass the bottleneck imposed by working memory and move on to higher levels of competence.” (2004, 2010)

    Given the above, don’t math facts need to be quickly recallable from long-term memory BEFORE young students can carry out multi-step derivations?


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