Music, math, learning and immediate gratification

Have you ever wanted to play guitar? I’ve been reflecting on my own learning lately, particularly after attending a workshop last night by one of my all-time favorite guitarists, Chris Proctor.

I started off in a fairly traditional manner, learning basic chords and strumming. This was in the context of a high school music class where every student either took beginning recorder or guitar lessons.  I enjoyed it so much that I wanted to continue after the course was over. Well, I knew this teacher had private students, and I knew which instruction book he was using (having seen his students carrying it around), so I picked up a copy of Solo Guitar Playing by Frederick Noad and dove in.

I didn’t think much of it at the time, but this book took a very different approach from the one my teacher had taken. In class, we had started with left hand positions for strumming chords – you lock down your left hand at various positions on the fretboard and then strike all of the strings simultaneously to produce harmonies. Noad’s book, by contrast, began with single-voice melodies – you learn the pitches of the open strings and start playing simple 3 note exercises on each string, ultimately combing strings to have a greater range of pitches. Well into the book we encounter the two-note chord – striking a base note with the thumb while the fingers play a melody. It wasn’t until much later – after a year’s worth of study – that I encountered anything that looked like a “traditional” guitar chord. By then it was slowly dawning on me that the longer pieces with names (pieces other than short exercises) were all composed in the 18th century.

Eventually I came to realize that there were (at least) two pathways to learning the guitar. One can start with the chords and work up to adding melodic embellishment and finally full melody with harmony. This is the path of the folks/blues/rock guitarist. The classical path reverses the order – one begins with simple melodies, builds up to simple two-note harmonies, and eventually arrive at full melodies and chords.

Both learning paths can take us to the same place – being facile both melodically and harmonically. I would argue, however, that beginning with chords favors the short-term feedback and gratification needed to get through the frustrations of first learning an instrument. With a good teacher and some guidance, one can play recognizable music in the first one hour lesson. The “fun” and reward of playing is available almost immediately.

Taking up a classical method requires a tolerance of delayed gratification. It’s not very “musical” to play three note ditties over and over while memorizing standard music notation and locating fingers on the frets. Mistakes (particularly buzzing strings) sound much more acute when playing single notes. If I had not already been learning some chords and “real” music to keep me amused, I might not have had the patience to stick with the classical instruction path.

Every since my epiphany around the dual approach to learning guitar (I think I first had this insight while in college) I’ve been paying attention to other “dual paths” to learning.  Nowadays I sometimes treat my own continuing statistics education as a dual-path approach. On the one hand we have the formal theory of conditional distributions described by density functions and how they interact. On the other is a purely computational/simulation approach – let’s try doing something a gazillion times and observing the distribution of the outcome. When I’m feeling a bit shaky on my theory, I often simulate a problem computationally to confirm that the results conform to my expectations.

In formal education I’ve recently come across two initiative by the Carnegie Foundation for the Advancement of Teaching: Quantway and Statway. These are two “developmental” math courses for community college (i.e, “remedial” courses for those who are not yet ready to cover college level mathematics). One of the key features is that they allow students to engage in real mathematics by focusing on reasoning and applied statistics, developing the formalisms (the parts students generally get stuck on) along the way. This is analogous to how people learn folk guitar – begin by making real music, however simple, and then pick up the formalisms (tonic and dominant chord theory, for example) in the context of making music. I’m paying close attention to an evaluation of these programs, and am curious whether this approach helps struggling students over the hump.