Ok, here’s the deal:
I feel like the way most of us currently assess and view mathematical fluency and ability (timed tests) is complete garbage.
Now that I have that out of the way, let me explain myself.
I took German in high school. I can’t speak German, but I can recognize words and phrases, and I can usually tell if someone is speaking German. I can spout off phrases and words that I’ve committed to memory, but I wouldn’t consider myself even close to being fluent in German.
Math fluency is a lot like learning to speak a foreign language. Simply committing a series of facts or procedural steps to memory is no more a sign of math fluency (or numeracy) than my ability to speak German.
In English, I can carry an intelligible (perhaps intelligent even) conversation and think of many ways to express myself so that I will be understood by others. I can also understand what is being shared with me in the same language when it’s communicated in a variety of contexts, accents, dialects, and some colloquialisms or slang. If I’m communicating with someone who has a lesser command of the English language, I employ more simplistic words and phrases. Similarly, I can dip into a deep well of rich vocabulary when conversing with someone who is well-versed in my native tongue. Quite clearly, I am fluent in English.
Likewise, when individuals are fluent with math, they can speak about concepts in an intelligible manner and communicate their understanding in a variety of contexts for a wide range of audiences. Those with an understanding of the properties of numbers and the operations performed on those numbers are able to choose from multiple pathways to solve problems. People who are mathematically fluent are not limited to any one singular method or algorithm but are able to choose an efficient strategy based on the context of the problem.
Another important consideration is that speed is a byproduct of fluency. Too often, we focus on the speed with which an individual can solve a mathematical task or recall a fact from memory, mistaking it as the telltale marker of true mathematical ability or fact fluency. Yes, a reasonable rate of speed is desirable, but oftentimes, the cart is put before the horse; fluency begets speed. While I was in the Marine Corps, we had a phrase, “Slow is smooth. Smooth is fast.” Once you understand a concept and can carry it out accurately and proficiently, speed can be increased without additional stress or missteps taking place. We ought to work on building speed from conceptual understanding and repeated practice over time.
Some might assume that if we can just get a student to memorize their facts or the steps to a procedure, increasing the speed with which they solve problems, they’ll make sense of it later. NCTM (National Council of Teachers of Mathematics), however, would disagree.
Research suggests that once students have memorized and practiced procedures that they do not understand, they have less motivation to understand their meaning or the reasoning behind them.NCTM Standards and Positions: Procedural Fluency in Mathematics
Just as wise individuals in meaningful conversations often pause to measure the weight of their words before letting them loose from their lips, or a skilled writer allows themselves to wait briefly before putting pen to paper, being sure of the same, so also those who are fluent mathematically may take a brief moment before being sure they have calculated correctly.
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.NCTM. Principles and Standards for School Mathematics, p. 152
Clearly, there is much more to being fluent in mathematics than computing quickly. Simply committing a series of math facts or procedural steps to memory does not make one mathematically fluent. Mathematically fluent individuals, like those fluent in a language, possess the ability to think flexibly and communicate their reasoning effectively, in a variety of contexts, for wide ranges of audiences.
I feel we need to assess students for true understanding and not just speed. Are they able to communicate their understanding in an intelligible manner? Can they use effective and efficient strategies to arrive at correct solutions? Are they limited to using one method or strategy to solve a particular type of problem, or do they have options? Let’s not mistake memorization of facts and procedures or speed as a true measure of mathematical ability of fluency.
What’s you take on math fluency? How are you assessing fluency in your classroom or with your student(s)?
Nice article. I use Cathy Fosnot’s work, including her number strings almost daily. Her assessment app has something called “2 pen” assessments which are great assessments for fluency. Have you used her work?
Thanks, Emily! I recently purchased Conferring with Young Mathematicians at Work, but I haven’t got around to reading it yet. Is this where students use two colors and switch colors after time runs out? If so, I’ve done that in the past.