As she finishes coloring the last cell, the tutoring student is beaming as brightly as her multiplication fact chart.
She has proven her mastery of multiplication facts through card games, dice rolls, verbal quizzing, and good old fashioned worksheets. She already knew her 1s, 2s, and 10s three weeks ago. Since then she has learned the rest with various strategies. Most of the time she built on what she knew, and a few facts she just had to memorize. She did it, and nobody can take that away from her.
As her tutor, after I give her a crisp high five. Then I find myself reveling in the fact that the chart represents more than facts. Unlike a classroom teacher who needs to attend to many children at once, I was with her every step of the way in her journey to fluency. I was able to reinforce the underlying concepts of multiplication all along the way.
I am confident that genuine understanding underlies her ability to recall these facts quickly and accurately. This will serve her well as she uses these facts to solve complex problems. And when she begins learning algebra, she’ll be able to seamlessly transfer and expand her understanding of mathematical operations.
Conceptual understanding isn’t only crucial for students who struggle. As a middle and high school math teacher, I saw too many students hit a wall when they got to algebra. People praised them early in their math careers for their quick calculation skills or memorization abilities. But they didn’t have deep understanding.
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| Multiplication chart showing solid multiplication facts. |
Without deep conceptual understanding and flexibility, their fluency was like a house of cards that fell apart when they were asked to generalize arithmetic into algebra. It was tragic to watch these students’ math identities shift so abruptly in a negative direction. Many of them never recovered their confidence or love of the subject.
Mathematical fluency is crucial
All state math standards are based on the Common Core State Standards for Mathematics (CCSSM). Mathematical fluency is “more or less the same as when someone is said to be fluent in a foreign language. To be fluent is to flow: Fluent isn’t halting, stumbling, or reversing oneself” (CCSSM, p. 8).
We want our students to speak the language of math. And for too many students, early in their mathematics learning journey they lose the belief that they are native speakers of math. If early math facts don’t come easily, if they find themselves slow or stumbling as compared to their peers, they feel like outsiders.
And that alienation causes them to lean away from math, making fluency harder to come by. Thus starts the cycle of a negative math identity.
When a student can’t recall math facts quickly and accurately, it taxes their working memory. And so they have less mental resources for multi-step problems. Approaching complex problems with less resources leads to more struggle, and the cycle continues.
We need to break the cycle.
Go slow to go fast
The irony of fluency is that although it may be defined in shorthand as “quickly and accurately” (CCSSM, p. 8), it’s a big mistake to try to get students to fluency too quickly.
The Common Core standards note this tension: “Fluency is not meant to come at the expense of understanding but is an outcome of a progression of learning and sufficient thoughtful practice. It is important to provide the conceptual building blocks that develop understanding in tandem with skill along the way to fluency” (CCSSM, p. 12).
I frequently encounter parents who are baffled by school math. They wonder why students don’t learn their math facts more quickly. They can’t understand why students are playing with physical manipulatives or drawing elaborate representations rather than doing problem sets. Just teach them the algorithms and move on!
Teachers deliberately delay the teaching of algorithms for very good reasons. A quick history lesson on math education is useful here.
A bit of history about fluency vs. understanding in math education
The interplay between conceptual knowledge and fluency was a central focus of the 2000 landmark publication by the National Council of Teachers of Mathematics (NCTM), Principles and Standards of School Mathematics:
"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).
NCTM reacted to a dawning realization that American students tended to have flimsy conceptual understanding. This ultimately hampered their ability to be innovative problem solvers (and thus jeopardized American socio-political dominance). The “mile wide inch deep” nature of math curriculum caused incoherence. The math education community openly acknowledged that math skills and topics should not be taught in isolation. NCTM sought to bring coherence and introduced the idea of processes such as communication and making connections.
Around that time, the National Research Council published Adding it Up: Helping Children Learn Mathematics, where the strand model of mathematical proficiency was born:
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| Intertwined strands of proficiency from Adding It Up, 2001 |
The strand model literally interweaves the different aspects of fluency. All are equally important and need to be learned alongside each other.
Yet two of the strands were specifically called out for infighting: “Procedural fluency and conceptual understanding are often seen as competing for attention in school mathematics. But pitting skill against understanding creates a false dichotomy” (Adding it Up, p. 122).
We may have an expansive view of proficiency, but this key battle continued.
Tutoring uniquely supports full mathematical proficiency
Procedures or concepts? For decades, this fundamental question of the nature of school mathematics has been at the heart of efforts to improve American math education.
The Common Core cited both NCTM’s standards and NRC’s Adding it Up, in the introduction to the Standards for Mathematical Practice. These practice standards work in tandem with the content standards to define what a well rounded math student needs to know and be able to do.
Although the Common Core set out to bring coherence, to strike the right balance between procedures and concepts, there are some big obstacles that remain. Standardized testing looms large here, since these assessments are much better at testing procedures than the other strands of proficiency. Culturally, we still tend to equate speed with math skill, which overemphasizes fluency from an early age and often starts the negative cycle. And with so many students per class, it is almost impossible for teachers to interweave the strands perfectly for every student.
As a tutor, I have the luxury of avoiding those structural obstacles.
I can continuously bring conceptual understanding into our fluency work in creative ways. I can ask students to reflect or explain at any moment, leveraging metacognition to connect to established concepts. As students work through longer problems, I can “catch” their fluency weaknesses and do some targeted work right there on the spot to fill gaps. And without a classfull of peers, there is no competition getting in the way of our focus. We can celebrate successes without any comparisons creeping in.
This is why I wish every student could have a math tutor. Sure, tutoring is demonstrably powerful. But it is particularly perfect for building fluency while avoiding the house of card pitfall of weak conceptual understanding.
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