I've been thinking a lot about why I hate the distinction that gets made in math education between a so-called "conceptual emphasis" and a so-called "skills emphasis."
Maybe it's because I grew up with a conservatory model of musical performance education, but after 10+ years of keeping quiet, I'm ready to propose a different distinction.
In a conservatory model of musical performance learning, there are multiple different dimensions from which a learner approaches their learning. The basic dimensions of focus are:
Repertoire: There is repertoire for your instrument or ensemble (the composed or improviseable pieces of music which the student is learning to play and/or perform. There are compositions for every level of player: as a very young child, I played pieces from Bach's Anna Magdalena Notebook that were appropriate for beginners because the technique they required was accessible to my level of technical and interpretive skill. More accomplished players might work on pieces from the Two- and Three-Part Inventions and even up to the Goldberg Variations and beyond.
Technique: A student also learns how to do focused regular work on technique, without which one lacks sufficient skill to support the playing of the repertoire. A steady diet of technical work (scales, arpeggios, Hanon exercises, etc) would probably lead to world wars because taken alone, these are boring. But they are the skills out of which we build our playing and understanding. No one wants to hear me play the Goldberg Variations. My technical skills are just not there. However, they are quite sufficient to not cause a casual listener pain while I work on Two-Part Inventions.
Music History & Theory: We study the context of the music we play because it gives us important insight into the composer's thinking. Understanding contemporary preferences, styles, beliefs, and historical context enable us to make some better sense of the pieces we play.
All three of these strands help us to make meaning & sense as we play, perform, practice, teach, and learn together. These are not the extent of everything we do in a conservatory model of teaching and learning, but they feel like a reasonable basis for comparison.
My Essential Question is: Why do we not take this kind of woven approach in math ed?
Modeling and problem-solving are the heart of our "repertoire." I want my students to be able to think about how to solve problems which may be "real-world" or not, but they are authentically problems that require thinking, activation, and transfer of prior learning in a new or novel way.
In music education, technique is the difference between musical performance and music appreciation. If I'm taking time to further develop technical skills that will enable me to access (i.e., to perform) more complex compositions, then am I not weaving together both conceptual and procedural forms of learning?
And if I'm simply listening and understanding musical compositions as a listener, then I am definitely accessing the concepts to be sure. But I am not at that moment engaged in the struggle of problem-solving/mastering/practicing/performing that composition. It's not that this posture in this moment doesn't inform my generative/productive performance/practice/learning as a performer. But I'm not acting in the role of performer in that moment. I'm acting in a receptive capacity.
By the same token, I have many times in my life had a major light-bulb moment while doing warm-ups like Hanon and realizing that THIS piece of technique can be used to improve my performance of THAT compositional passage. This is an authentic moment in which practicing skills can lead to true conceptual and performative insight.
So when I read Dan's latest blog post about how everything is modeling, that feels as true to me as saying that everything is based on skill-building. The kinds of mathematical thinking I want my students to be able to access includes powerful, flexible productive/generative mathematical modeling as well as sensitive and receptive mathematical listening/reading. I don't want my students to JUST unthinkingly repeat skill practice, but I also know that without a deep and flexible number sense and other forms of fluency, they will be cut off from the kinds of problem-solving I value most for them.
This is the question of "access" that I truly wrestle with.
I'm wondering if anybody else thinks of the nexus of conceptual understanding and procedural fluency in a related way.
