PRECALCULUS: Transformations of Functions Speed Dating

I know I am going to take some grief for this statement, but I have learned it to be the truth. There are some things that, once discovered, you simply need to memorize — and practice. This is about mental (and often physical) motor skills. Becoming fluent at some things requires more or less practice for different people, but for most of us, it does require something.

I come by this knowledge honestly.

I have played the piano since I was three or four. In growing as a pianist, there are conceptual learnings, procedural learnings, and contextual learnings. Technique, concepts, and contexts. To play well, you need first to become proficient and fluent technically. That is why there are so many established sets of technical figure exercises: Hanon, Czerny, and many others. The same is true for the violin (Schradieck, Kreutzer, Hrimaly) and every other instrument under the sun. If you want to be capable of playing the advanced works in your instrument's repertoire, your fingers and body need to know how to flow over the keys or strings in every known and familiar situation (scales, arpeggios, finger-crossings) so that you can summon them automatically in your pursuit of unknown and unfamiliar territory in the literature of your instrument.

But working your way through all of the possible finger movements will only make you a master technician. Doing only finger exercises will not give you a very deep understanding of music. It simply builds the finger and muscle memory you will need to face different and difficult technical situations and requirements as you dive deeper and deeper into the piano repertoire.

In other words, technical proficiency is necessary, but not sufficient to become a strong and capable musician. Without technique, you cannot progress.

Beyond technique, there are concepts to master — harmony, counterpoint, and composition, as well as music history. If you don't understand the basics of scales and harmonies and chord progressions in Western music, you will lack the basic musicianship that is required to make sense of the piano repertoire. You need to understand how melodic lines or voices can be woven together using harmonies and rhythms and chord progressions to build a piece of music with coherent and reproducible grammar and syntax that can be both encoded and decoded by others. Without the foundational concepts, your attempts to interpret and play the repertoire will be incoherent.

Finally, there are contexts — historical, cultural, interpretive. The ultimate context is your instrument's repertoire. Music, like mathematics, is a cultural act. As a musical learner, you need to be mentored into the repertoire. You need to experience other musicians' interpretations and experiences to learn what competent, coherent, and ultimately subtle musical communication sounds, feels, and looks like. And while you are doing this, you also need to be able to explore the repertoire for yourself so that you can find your own way.

It seems to me that this is a similar situation to what we face in mathematics education. Technical proficiency is boring but somewhat mindless. If you had to listen to anyone (including yourself) only playing Hanon's same 60 exercises day in and day out, you would undoubtedly lose your mind. As music, the technical patterns are boring — up and down, back and forth, crossing and uncrossing, stretching and shifting. But they're necessary to develop a foundation of muscle memory and motor skills, as well as the habits of mind and of practice you will need as you gain proficiency and advance to building the finer and finer skills of musicianship.

So there was a need for teachers to create pieces that are very rich musically but very restricted technically so that they are accessible to new learners. This is why Bach, for example, wrote the pieces in his Anna Magdalena Notebook. They created an on-ramp for his new wife to be included in the family's deep musical conversations. She needed a low barrier-to-entry, high-ceiling on-ramp to sophisticated music. These are also the motivation behind Bach's Two- and Three-Part Inventions. Bach's life as a composer was inextricably bound up in his life as a teacher, and these accessible works are the "rich problems" of piano education. They are accessible pieces that even the greatest virtuosi find beautiful.

This is why I love Glenn Gould's recordings of Bach's Two- and Three-Part Inventions. They are the music of his childhood, but from an advanced standpoint. The recordings are filled with joy and delight. To the dismay of critics, he often hums along with himself as he plays. The critics find this annoying. Personally, I find it inspiring. If Glenn Gould can still delight in playing these pieces, then so can I. It gives me great permission. The recordings and the playing are magical.

In my view, this is how we should be looking at the balance we need to strike in math education. There are technique and concepts and repertoire that learners need. Without strong technique, your understanding of concepts will be shallow. Without context/repertoire, your understanding and practice of mathematics will be joyless and without wonder.

The long-range purpose
of our practice was clearly
stated on the board

So this is the basis from which I used Kate Nowak's Speed Dating structure to solidify my Precalculus students' early understanding and integration of transformations of functions on Friday. As I keep reminding them, the parent functions and their graphs, as well as the transformations of these basic function graphs, are the essential vocabulary development work for calculus. This is the Hanon and Czerny mindset shift on which we are focusing this year: elementary things that we consider from an advanced standpoint. The order of operations becomes more sophisticated. "Groupings" replace "parentheses" in your thinking about where to start. Groupings, I warn them, are masters of disguise. Sometimes they show up as parentheses, but much of the time they show up wearing a moustache or another costume. They might show up in the form of the absolute value bars or the fraction bar. Don't be fooled, I warn them. Keep your focus on whether you are dealing with transformations of inputs or outputs.

We started simply, using only a single function du jour. On Friday, that was the square root of x. We also restricted our investigation to horizontal and vertical shifts as well as reflections across the x- and y-axis. We were considering the impact on the graphs of basic parent functions as we operated on either the input or the output of the function. We did not multiply by any value other than –1 to start. The Day 1 problem cards are here.  The Day 2 problem cards are also available now (formatted by the amazing Meg Craig - thank you!).

On Friday, each of my 36 students started out with his or her own problem card that contained two related transformations — a shift and a reflection. We organized the speed dating structure, moved our backpacks along the two empty walls, and established our rules of movement (the students along the window side of each row travel, while the hallway-side students stay where they are). If you run into problems, ask the expert in the room on that problem. He or she is sitting directly across from you.

36 precalc students in a state of flow

Trade, analyze, investigate, sketch, discuss. For forty minutes, my Precalculus students lost themselves in analyzing, investigating, sketching, and discussing functional operations on inputs and outputs. Every two minutes, my iPhone timer would go off and I would call out, "Shift!" And all 36 students would trade back their problem cards, while half of them stood up and moved one seat to their right. Then I would reset the timer and they would lose their ego-selves in each new immersion. I loved the hush that fell over the room after everybody settled down into the next round of problems. I eavesdropped on moving and purposeful conversation about inputs and outputs of functions, shifting and reflecting, as they worked collaboratively to help each other and to help themselves attain proficiency. The preciousness of each minute of mathematical conversation was not lost on me.

And I tell you, it was glorious.