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Academic Spotlight: Rigor in the Fundamentals of Math Instruction

Yo-Yo Ma is recognized as one of the most talented and accomplished musicians in the world. One could surmise that a master like him no longer needs to practice like he once may have during his years at Juilliard and Harvard–but as he once noted in an interview: “What all string players have in common is that if we don’t play for a while, we actually start from ground zero.”

Even after decades of success and accolades, Yo-Yo Ma still goes back to the fundamentals of scales, chords, and arpeggios. Those repetitive exercises continue to build his muscle memory and hone his musical precision and fluency.

In mathematics instruction, there is a well-intentioned tendency to skip fundamentals in pursuit of what feels more rigorous—a rush to advance to more challenging problems and scenarios. Too often, students struggle unproductively because their fundamental skills are weak.

We believe that a foundation for rigor is laid through mastery of fundamentals. This principle is at the core of our math instruction. It’s not just being able to apply procedures with accuracy and automaticity, but also nimbly, with an awareness of why one might choose a particular strategy to begin with. With this in mind, our teachers tailored their remote instruction with the additional emphasis on procedural fluency.

The National Council of Teachers of Mathematics advises: To develop procedural fluency, students need experience in integrating concepts and procedures and building on familiar procedures as they create their own informal strategies and procedures. That is to say, students are not limited to utilizing mathematical strategies in isolation but rather encouraged to use them in tandem with one another.

For example:

A common third grade task requires students to find an equivalent fraction–like “2 thirds is equal to ____ twelfths.”  One common solution here is to multiply the denominator and numerator by a whole number.

A student with procedural fluency will know how to solve this problem in different ways: setting the problem up as a tape diagram, using a number line as guidance, or using fraction bars. The goal is that as the complexity of problems progress, they can pick and choose the most efficient and effective procedures for solving.

Here are examples of how two third grade students from our own St. Mark The Evangelist were able to take different fundamental strategies learned in previous lessons and grades and apply them to current fraction problems.

Student A

Student B

These lucky students are under the tutelage of third grade teacher Kelly Quinn, who knew that encouraging students to recall previously-learned procedures and apply them in conjunction with newly-learned procedures such as fraction bars would create a more rigorous math lesson, one that values solutions more than completion.

In addition to the application of strategies, Kelly also asks students to explain their thinking in writing, incorporating a modality often overlooked in math, but one that carries additional weight as a check for understanding in a virtual learning format.

To build this kind of procedural fluency effectively, teachers must acknowledge that the process of students understanding the most effective strategies to use for each type of problem will involve some trial and error.  The temptation is to place the guard rails on too soon and shift the students to a strategy that we know will work. However, our best work is allowing students to essentially play different mathematical chords a few times until they find the tune.

The incorporation of procedural fluency in the classroom is not just an exercise for students in efficiency; it offers a similar benefit to teachers.  It is an assessment opportunity for teachers, in that students’ work samples can reveal their misconceptions, allowing teachers to reflect on their delivery of a given strategy. This valuable information guides the teacher’s planning for upcoming lessons.

Towards the end of the aforementioned interview with Yo-Yo Ma, the musician shared the following: “And now I practice because I’ve experienced so much love that you practice out of loving a phrase, loving motivic change, loving a structure or harmony change or the way a sound can get to something.”

The work that our teachers do to ground student work in the fundamentals and provide opportunities for procedural fluency are the first steps in building a love for math, one that will be nurtured through practice and cultivated year after year.

John Bacsik is the Director of Professional Development and Nehemie Villarceau is the Talent and Academic Manager for Partnership Schools.