Tag Archives: math facts

Memorization or Learning?

I originally started to write a post on a memorization technique I’m using for the unit circle, and went looking for representative jeremiads both pro and con. Instead, I found Ben Orlin’s piece When Memorization Gets in the Way of Learning (from five years back):


…which is the opposite of a standard, boring piece and serves as a good counterpoint to explain some recent shifts in my pedagogy.

It’s a good piece. In many ways, the debate about memorization runs parallel to the zombie problem–students regurgitate facts without understanding. Ben’s against that. Me, too. Ben says that testing requirements create tensions between authentic learning and manageable tests; I have various means of ensuring my students understand the math rather than just hork it up like furballs of unknown origin, so am less concerned on that point.

But I don’t agree with this sentiment as much as I probably did a decade ago: Memorizing a list of prepositions isn’t half as useful as knowing what role a preposition plays in the language. 

Not in math, anyway.


A couple years ago, after I’d taught trigonometry two or three times, I suddenly noticed that at the end of the year, my students were very fuzzy on their unit circle knowledge. (It’s no coincidence that Ben’s article and my observations are both focused on trigonometry, a branch of math with a significant fact base.) When working trig equations, they’d factor something like the equation above, use the Zero Product Property, solve for sin(x)…and then stop.

“You’re not done,” I’d point out. You’ve only solved for sin(x). What is the value of x?”

Shrug. No recognition. My tests are cumulative. Many students showed significant recall of concepts. They were using ratios to solve complex applications; they were sketching angles on the coordinate plane–both concepts we hadn’t revisited in months. They could sketch the unit circle from memory and eventually figure out the answer. But they had no automatic memories of the unit circle working backwards and forwards, even though I had emphasized the importance of memorizing it.

Upsetting, particularly at the end of the year. The name of the class is Trigonometry, after all. Solving for sin(x) requires not one tiny bit of trig. It’s all algebra. Trigonometry enters the picture when you ask yourself what angle, in radians or degrees, has a y to r ratio of 1 to 2.

The sine of π/2 is not among [the important things to memorize]. It’s a fact that matters only insofar as it connects to other ideas. To learn it in isolation is like learning the sentence “Hamlet kills Claudius” without the faintest idea of who either gentleman is–or, for what matter, of what “kill” means.

Well, okay, but….if a student in a Trig class can’t work a basic equation without a cheat sheet, what exactly has he learned? He already knew the algebra. Does the same standard hold for SOHCAHTOA, or can I still assume the student has successfully learned something if he needs a memory aid to remember what triangle sides constitute the sine ratio? What else can be on the cheat sheet: the Pythagorean Theorem? The ratios of the special rights?

Ben describes memorization as learning an isolated fact through deliberate effort, either through raw rehearsal or mnemonics, both of which he believes are mere substitutions for authentic learning. He argues for building knowledge through repeated use.

Sure. But that road is a hard one. And as Ben knows much better than I, the more advanced math gets, the more complex and numerous the steps get. Most students won’t even bother. Those who care about their grades but not the learning will take the easier, if meaningless route of raw rehearsal.

So how do you stop students from either checking out or taking the wrong road to zombiedom?

I’ve never told my students that memorization was irrelevant, but rather that I had a pretty small list of essential facts. Like Ben, I think useful memorization comes with repeated use and understanding. But what if repeated use isn’t happening in part because of the pause that occurs when memory should kick in?

So I’ve started to focus in on essential facts and encouraged them to memorize with understanding. Not rote memorization. But some math topics do have a fact base, or even just a long procedural sequence, that represent a significant cognitive load, and what is memorization but a way of relieving that load?

The trick lies in making the memorization mean something. So, for example, when I teach the structure of a parabolas, I first give the kids a chance to understand the structure through brief discovery. Then we go through the steps to graph a parabola in standard form. Then I repeat. And repeat. And repeat. And repeat. So by the time of the first quiz, any student who blanks out, I say “Rate of Change?” and they reflexively look for the b parameter and divide by 2. Most of them have already written the sequence on their page. The memorization of the sequence allows them repeated practice.

But it’s not mindless memorization, either. Ask them what I mean by “Rate of Change”, they’d say “the slope between the y-intercept and the vertex”. They don’t know all the details of the proof, but they understand the basics.

I take the same approach in parent function transformations, after realizing that a third of any class had drawn parent functions for days without ever bothering to associate one graph’s shape with an equation. So I trained them to create “stick figures” of each graph:stickfigures

I drew this freehand in Powerpoint, but it’s about the same degree of sloppiness that I encourage for stick figures. They aren’t meant to be perfect. They’re just memory spurs. Since I began using them a year ago, all my students can produce the stick figures and remind themselves what graph to draw. They know that each of the functions is committed on a line (to various degrees). Most of them understand, (some only vaguely), why a reciprocal function has asymptotes and why square root functions go in only one direction.

So did they learn, or did they memorize?

I haven’t changed my views on conceptual learning. I believe “why” is essential. I’m not power pointing my way through procedures. I am just realizing, with more experience, that many of my students won’t be able to use facts and procedures without being forced to memorize, and it is through that memorization that they become fluid enough to become capable of repeated use.

Like Ben, I think a zombie student with no idea that cosine is a ratio, but knows that cos(0) = 1, has failed to learn math. I just don’t think that student is any worse than one who looks at you blankly and has no answer at all. And addressing the needs of both these students may, in fact, be more memorization. Both types of students are avoiding authentic understanding. It’s our job to help them find it.

So I’ll give an example of that in my next post.

Math fluency

My Math Support class, for students who haven’t yet passed the state graduation test, is the most challenging of my preps. In many ways, though, the class offers the dream scenario for any math teacher who longs to focus on fundamentals.

I owe no allegiance to a curriculum. I’m not teaching arithmetic in and around an algebra course; arithmetic and a tiptoe into algebra is all the test requires. I only have 18 students (16 boys) in a 90 minute class, so I have tons of time to work one on one. While the kids probably wouldn’t strike the average observer as motivated, they are juniors and seniors who want to pass the test, so by their internal standards, motivation is high. Many (but not all) of the kids are acknowledged classroom challenges at the school. However, this school’s notion of a serious classroom challenge is something around the 30% mark of the students I taught for the last two years, so my basement has moved way, way up the stairs.

So I have a small class, a meaningful curriculum, motivated kids with low abilities, and, for that population, no significant management challenges. I was, and am, enthusiastic about the opportunity. However, please take renewed notice of the blog name. I am not under the impression that these students have merely been waiting for The Messiah, after years of suffering through false prophets (aka bad teachers). I was eager to see which of my assumptions played out, and which didn’t, and I wanted to test, anecdotally at least, some commonly held wisdoms that hadn’t, in my limited experience, borne out.

For example, I have long suspected that the received wisdom about math fluency has holes in it:

Educators and cognitive scientists agree that the ability to recall basic math facts fluently is necessary for students to attain higher-order math skills. Grover Whitehurst, the Director of the Institute for Educational Sciences (IES), noted this research during the launch of the federal Math Summit in 2003: “Cognitive psychologists have discovered that humans have fixed limits on the attention and memory that can be used to solve problems. One way around these limits is to have certain components of a task become so routine and over-learned that they become automatic.”

The implication for mathematics is that some of the sub-processes, particularly basic facts, need to be developed to the point that they are done automatically. If this fluent retrieval does not develop then the development of higher-order mathematics skills — such as multiple-digit addition and subtraction, long division, and fractions — may be severely impaired. Indeed, studies have found that lack of math fact retrieval can impede participation in math class discussions, successful mathematics problem-solving, and even the development of everyday life skills. And rapid math-fact retrieval has been shown to be a strong predictor of performance on mathematics achievement tests.

I used to accept this as a given until seven years ago, when I ran into my first kid who knew his math facts cold but couldn’t solve 2x + 7 = 11, unless I asked him what number I could multiply by two and add seven in order to get 11 and got the correct response almost before I finished the sentence. By that time, I’d already met a few 600+ SAT students who growled in frustration and reached for the calculator when it came to knowing 6 x 9. I’ve also tutored a dozen or more ISEE/SSAT (private school test) fifth and sixth grade students who went to precious little snowflake schools and knew none of their math facts with any fluency yet easily mastered fractions, ratios, and solving for unknowns and scored in the top 90% of a highly skilled population.

I’ve long since abandoned the notion that fluency might be necessary, but not sufficient, given the last group. Kids who can abstract can cope without fluency. What’s troubling is that fluency might be irrelevant.

None of this means we shouldn’t emphasize fluency. But plenty of solid math students don’t have fluency and—here is the important part—many exceptionally weak math students have strong fact fluency.

Every week, I get an extra 20 minutes with each of my classes. In Math Support, I use this time for drill competitions. The kids pair up and get a selection of MDAS flash cards. I set the timer and holler “GO!” First kid holds up cards for the second kid and go through the cards as fast as they can—correct answers in one pile, missed in the other. I stress that the “miss” is determined in 2-3 seconds for most kids (more on that in a minute). If the kid hesitates, it’s a miss.

I originally set the timer for 2 minutes, but all but two of the kids get through a whole pile of 30 cards in one minute, so I dropped it down to a minute.

The kids’ fluency falls into one of these zones:

  • High: I mean, 7×12, 6×9, 7×8 high. 121/11, 96/12 high. 7+9 and 15-8 high. No hesitation, no pauses. The five students in this group all struggle with abstractions, although two of them have solid arithmetic competency and excellent estimation skills. The rest struggle in every area of math. All of them test poorly, all are seniors.
  • Solid: Fluent except the usual suspects: higher 12s, the cross sections of 7, 8, and 9 and a few hard to remember addition/subtraction facts. Many of these kids have told me that this activity is improving their recall of their problem facts. All of my overall strongest students are in this category, the rest are average. Seven in total.
  • Weak: Say about 50% mastery. Four students, not noticeably different otherwise from the “average” students in the solid category. I haven’t yet noticed any improvement, but they’d likely take longer.
  • Non-existent: I have two kids who can’t quickly recall their 2 multiplication facts, struggle with basic addition. Clearly some sort of memorization issues. These two are given 6 seconds per card before it’s counted as a miss.

One of the two students in the non-existent zone is, hands down, the strongest procedural algebra student in the class. She can solve multi-step equations and identify linear equations from a graph. I have explained fractions and ratios to her on several occasions, and it all escapes her instantly. So no fluency, no proportional thinking, but algebra procedures and linear equations. If she can operate by rote, she’s fine. I haven’t checked yet, but I’d bet she can master the quadratic formula (with a calculator) more easily than factoring binomials. My strongest overall students, while not as solid on algebra procedures, are much stronger at proportional thinking, more capable of thinking abstractly, and are all in either geometry or algebra II. (Why yes, you can get to algebra II without passing the state math graduation test. Happens constantly.)

All of my students easily manage multiple digit addition and subtraction. A few of them are completely unfamiliar with long division. Fractions are a struggle for most of them. All but a few understand and use distribution. Combination of like terms, not so much. They all do quite well simplifying exponential expressions and have a solid grasp of scientific notation.

What does this mean? Beats me.

Assertion: Students who are categorically failing in math are almost certainly not doing so because of math fluency. They may or may not be fluent, but fluency is not the condition holding them back.

Tentative hypothesis: The rationale for math fluency (quoted above) does hold for many students who are moving through the math curriculum without ever achieving genuine proficiency, who would certainly be able to learn and hold onto more information if they weren’t spending so much of their time trying to remember what 6 x 3 is, particularly in algebra.

So go ahead and drill. Just remember that the kids it will help the most aren’t the ones you’re worried about, and many of the ones you’re worried about won’t need the drill.