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In the standards (3.OA.7) it states, “Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations. By the end of grade 3, know from memory all products of two one-digit numbers.” How do you measure “fluency” without timing the students? Also, if you do time students on these facts, then what time constraints and/or amount of problems are appropriate. We as teachers still have to give grades. HELP!!!
I would say that fluently refers to how you do a calculation and know from memory means being able to produce an answer without having to do a calculation. As a third grader if I can multiply 13 X 7 by reasoning that 10 x 7 is 70 and 3 x 7 is 21 so 13 x 7 = 91, and I can apply that strategy consistently and accurately (without hesitating and fumbling through it), then I would argue that I am fluent. I get that timed tests are easy to give and easy to score, but I don’t get how a timed test would give you much information on the strategies the students are using and whether they can use those strategies efficiently and accurately (fluently).
And as far as the knowing from memory, I guess you could use a timed test to measure that, but my concern is the overemphasis on timed tests that might send the unintended message that somehow math is about speed. I’d rather that kids get the message that math is a creative endeavor first, and then learn lessons along the way that efficiency matters also.
As an alternative to timed tests, I would submit that listening to a kid solve a problem involving the “know from memory” facts would tell you everything you need to know about how much they know from memory and how much they have to compute.
I’m sure Bill will add his own opinion again if necessary, but there was some talk of fluency and “knowing from memory” back in this discussion: http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/#comment-1428
In the standards, “fluently” means “fast and accurate.” So assessing fluency is always going to involve some observation of how long it takes a student to do a calculation. As abienek points out, that observation could be made by the teacher listening to a student solve a problem; or, it could also be made by a timed test. In talking to teachers I’ve found that timed tests are a very controversial topic: some people love them, others hate them. I don’t have a strong opinion either way. I do think that if you use timed tests you should try to make the fun and competitive, not scary and stress-inducing. That’s possible in athletics, so it should be possible in math. I also think they are probably not necessary in a classroom where students are explaining their solutions a lot; you can probably tell pretty well from that who is fluent and who is not.
Do the standards require “automaticity?” How would that be different from fluency? I ask in part because in the 2008 NMAP Report, on page 4-5, cognitive experts Geary, Renya, Siegler, Embertson, and Boykin wrote,
“At all ages, there are several ways to improve the functional capacity of working memory. The most central of these is the achievement of automaticity, that is, the fast, implicit, and automatic retrieval of a fact or a procedure from long-term memory.”
A key word there is “implicit:” Being able to do the right thing instinctively, without necessarily being able to explain why. In solving scientific calculations, because of limits on the duration of novel elements held in working memory during processing, automaticity is vital. How is fluency different? Is implicit recall of an appropriate procedure what students and instructors should aim for under the standards?
In my view “fluent” means “fast and accurate” and I don’t see a meaningful distinction between this and “automatic.” But, as you point out, people do make quite suble distinctions between all these words. I think a lot of these distinctions are more meaningful for assessment and curriculum then they are for standards themselves. The standards require fluency; implementers of the standards will be making more fine-grained decisions about how to teach and how to measure it.
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