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.
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.