Tag Archives: arithmetic

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.