(Mathematical) Fluency

Ken Davis, mathematics consultant at Wisconsin's Department of Public Instruction, contributed today's post.

The impetus for me writing this actually didn't come from a mathematical situation, but from a reading one.  I was having a discussion with one of my children about reading.  His concern was that he was reading slowly, especially when compared to others in his class. I'm not sure what data he was pulling in order to come to this conclusion or what he was using to measure how he stacked up to his peers, but my fear was that he was equating his reading ability to how fast he could read.  

I calmed his anxiety by telling him that his dad was not a fast reader either.  In order for me to read and read well, I have to use a highlighter or pen and track as I am reading.  I move across and down the page a little slower, but I understand or comprehend everything I read and seldom do I have to reread.  I asked him, "How do you feel as you are reading? Do you understand what you are reading?" He assured me that he is understanding what he is reading, but just feels as if he is going slow.  

It's ironic that about at this same time one of the chat groups that I belong to began to discuss mathematical fluency and the fact that mathematical fluency doesn't equal speed. In fact, many in this group spoke out against the idea of using timed-tests and memorization to teach mathematics because many students equate speed with being a "good" math student.

This can be damaging for students in two ways. First, it is discouraging for the student who is unable to reach the target of 20 problems in 60 seconds or whatever the objective of the timed-test may be. Some students are just not going to be able to work at a faster speed.  I turned 50 last year, (scary thought in and of itself) and still would not consider myself a fast reader, but do I have command of the English language?  I would like to think that I do. Second, are we setting up the student later for failure when they can answer the 20 problems in 60 seconds, but have no understanding of how they got the answer? Consider this scenario. We teach a student that to multiply by ten, simply add a zero on the end of a number. We give this student a timed-test that includes multiplication by 10 and the students gets them all correct in the specified amount of time.  The student (and parent) now have the sense that the student can multiply by ten.  Now, the student encounters this problem 3.14 times 10. The answer is not 3.140, but the student has no understanding of what it means to multiply by ten because they were not taught the concept of multiplying by ten, but rather a quick method of memorizing that did not teach number sense.

In an article from 2012, Linda Gojak, then National Council for Teachers of Mathematics (NCTM) president, pointed out that in Principles and Standards for School Mathematics  it states that “Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships” (p. 152).  The article doesn't mention that fluency is equivalent to speed.

In another article, Jo Boaler, Professor of Mathematics Education at Stanford states: "Mathematics facts are important but the memorization of math facts through times table repetition, practice and timed testing is unnecessary and damaging...It is useful to hold some math facts in memory. I don’t stop and think about the answer to 8 plus 4, because I know that math fact. But I learned math facts through using them in different mathematical situations, not by practicing them and being tested on them."  

For me, if we do equate fluency with speed, then how do we measure it? 20 problems in 1 minute or 45 problems in 2 minutes? How do we place time on fact fluency and say that "this" rate is proficient?  Let's begin by teaching students an understanding of mathematics; let's build their number sense.  Isn't this really what's important? Students should truly understand and appreciate the mathematics and are not just concentrate on being the fastest in the class.