What is Fluency?

August 25, 2019

Mathematical fluency is easier to recognize than define.

It lies at the intersection of conceptual understanding and basic skills, but this flimsy descriptor undervalues its essence and fails to encompass its existential boundaries.

Fluent students are good at arithmetic, but haven’t necessarily automatized computational facts.  Instead, they possess a more holistic skill set that allows them to efficiently arrive at answers through multiple pathways, which they effortlessly select and toggle between.  Using sequences and foundational understanding, the fluent student constantly applies their number sense to connect the simple and complex, helping them confidently solve problems that they may or may not have seen before.  Nine plus five helps them answer 89 + 5 and rounding 67 to the nearest ten leads them to correctly round 3,967 to the nearest ten and 9,670 to the nearest hundred.

Similar to an eloquent writer, fluent mathematicians possess relative strengths and weaknesses within the subject.  Paul Theroux might be America’s greatest living author, but he doesn’t know every English word and his travel writing is far more renowned than his fiction.  Likewise, the scope of mathematics is too vast for any individual to master all of its regions.  What separates mathematically fluent and inarticulate students is that the latter expends unnecessary brain energy on implied knowledge and calculations while the former does not.

Like an efficiently run business or well-officiated athletic contest, fluency is often more noticeable in its absence than presence.  It is not a zone students pass into, and is hard to quantify or assess.  Instead, it is an amalgamation of skills, understandings, patterns, and sequences that – when mastered – fuse into an elegant mathematical language, allowing better access to the intricate discipline.

In the past, elementary school teachers only addressed a sliver of mathematical fluency.  Focused on skill automatization, instructors gave students fact sets to memorize at home.  Then, flash card games and computer-generated worksheets served as in-class catalysts to drive practice.  The teachers’ intentions were likely good, but their chosen tools lacked depth and nuance, favoring children with a high capacity to memorize, while leaving others to unnecessarily struggle.  The Math Brain myth and students hating the subject became unfortunate consequences of this approach. 

Traditional methods to build fact memorization restrict the boundaries on what students are capable of understanding and articulating.  Children in turn rarely learn to appreciate the subject’s beauty and fail to realize their mathematical potential.  Addressing basic skill practice through memorization tasks is like learning a second language by studying vocabulary lists.  Devoid of meaning, connection, and context, practice becomes laborious leading students to not understand – or like – math.

Recalling frustrating math experiences from their own schooling, many teachers choose to eliminate fluency from their lessons.  Although their frustrations might be justified, these educators merely eliminate one problem by replacing it with a bigger one.  They are right to try and create deeper, more meaningful mathematical experiences for their students, but wrong to dismiss the necessity of skill work.  This Throwing out the baby with the bathwater approach creates a new, but different hierarchical reality in elementary math classes.  Students continue to believe in the Math Brain myth but the source leading them to this conclusion is now flexible thinking and problem solving “abilities” instead of rapid-fire computation.

Doing mathematics without number sense is like strumming a guitar without knowing chords or playing soccer without fundamental ball skills.  Creativity is severely hampered and expertise is impossible.  To expand their potential in any discipline students need language proficiency and a teacher to help them acquire it.  The expert instructor knows this and starts every math class with fluency practice.

The tools and methods to deliver fluency differ from teacher to teacher, but the practitioners are bound by a few immutable characteristics:

  • 25-35% of their math block is devoted to fluency.

  • They begin class with student activity rather than passivity.

  • They facilitate practice using as few words as possible.

  • Their problem sequences start simple and gradually get harder, but always end with every student succeeding.

If mathematical eloquence provides students access to more complexity within the discipline, then teachers need to spend significant time developing its language.  Building number sense rarely occurs outside of math class, so fluency practice requires a significant time commitment – no less than 15 minutes of each one-hour class period.  Anything less makes the ensuing problem solving, lessons, and independent practice inaccessible.

Acquiring language requires focused concentration and intensive practice.  This only occurs when learners are actively practicing.  Responding to teacher prompts is the expectation, opting out is defiance, and cooperative passivity is non-existent.  Regularly solving six or more problems/minute, students experience math as a verb rather than a subject.

Teachers stimulate participation through succinct directives that produce frequent student responses, aiming for children to talk and/or write as much or more than they do.  This requires them to facilitate drills with hyper-brevity – a practice of saying all that needs to be said, but nothing else.

The phrase What’s three times five? is better than asking What is the answer to three times five? because it conveys the same question using half as many words.

Stating Three times five with a questioning intonation is an improvement on the latter.

Saying Three-five while crossing arms in a multiplication symbol is better still.

Teachers who sustain clear, concise prompts drastically increase student practice, and are more pleasing to the ear.

Although delivery modes vary, all fluency work follows a simple to complex trajectory that - in the end - circles back to something easy. This is accomplished through carefully sequenced problems that gradually become harder.

The master teacher orchestrates their drills by slowly stretching their students’ collective fluency comfort, but exiting the exercise before the class’ energy and enthusiasm fractures.Every activity concludes with an easy problem, so that each students’ last memory is success (see examples on right).

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