The point of this post is that there is much, much more to education than the "skills vs. concepts" frame. The skills vs. concepts framework is somewhat useful in determine the composition of a lesson or homework assignment. Perhaps more importantly using a skills-vs-concepts lens can lead to a perspective of teaching that can be limiting and potentially harmful.
Let's look at this from the baseball perspective. It would be completely absurd to talk to a baseball player and say we can either make you better at the fundamentals (throwing, catching, hitting) OR (exclusively) you can get better at game situations (thinking of the outs, where to throw the ball, when to tag or run). Good ballplayers combine concepts, skills, and more to *playing* the game.
Yet another example comes from music and the arts. Musicians and artists practice skills and learn theory as well. They do not, however, consider these as competing goals. Skills and theory are developed simultaneously to support a vision or idea of expression. Technical skill, theory, interpretation, vision, passion, all those things that make us human go into it.
The limitations of the skills vs. concepts framework is realized when we consider the point of school. If we look only at skills vs. concepts, we might obscure from our vision the notion of studying a subject with our whole being, in a way that the work is educative. It's one thing to add a new skill or learn a new concept. It's a whole other plane of existence when doing something changes the way you think. When students work together to create their own valid understanding and vision of how and why things work, rather than appealing to an external Math Authority (the back of the book or the teacher), then something fundamentally different and beautiful has occurred. Students see that they can do scientific inquiry for themselves and make sense of complex ideas. They learn that truth is established through logic and reason, and not by edict by a Math Authority. They see themselves as participants of the activities that lead to discovery.
It is completely understandable why the skills vs. concepts choice is commonly discussed. When one plans to tell students what they need to know, there exists a competition for space in the lecture notes. Linear teaching is linear.
Learning, however, is nonlinear.
A different way to think about the whole enterprise of preparing for class is to think first about what it means for a student to master a particular topic. Then the great teaching challenge is to find math problems that do the telling and provide contexts for posing and answering questions that solve the problems. Skills can be taught in the context of the larger problem-solving scheme as well as through a "minilesson" mode of instruction, where the skills development is intentional (more on minilessons in a future post).
A different way to think about the whole enterprise of preparing for class is to think first about what it means for a student to master a particular topic. Then the great teaching challenge is to find math problems that do the telling and provide contexts for posing and answering questions that solve the problems. Skills can be taught in the context of the larger problem-solving scheme as well as through a "minilesson" mode of instruction, where the skills development is intentional (more on minilessons in a future post).
Research also suggests that skills do not necessarily come at the expense of developing effective thinking. When students construct knowledge, not only is it good for them in terms of their overall education, even at the level of skills and concepts, students keep the skills and understand the concept better, while retaining it longer. It's a win-win.
Heid provides evidence that taking a radical approach to teaching calculus essentially resulted in nearly the same level of procedural skills, but with much better conceptual understanding. Resequencing Skills and Concepts in Applied Calculus Using the Computer as a Tool
In another article focused on Differential Equations, the IBL classes maintained the same level of skills compared to the non-IBL group, and retained their knowledge better one year later. (Kwon, O. N., Rasmussen, C., & Allen, K. (2005). Students’ retention of knowledge and skills in differential equations. School Science and Mathematics, 105(5), 227-239.)
Rasmussen, Kwon, Allen, Marrongelle, and Burtch found that in an IBL Differential Equations course, the students performed the same on procedural skills as students in a non-IBL, and scored significantly higher on items testing for conceptual understanding (Rasmussen, C., Kwon, O., Allen, K., Marrongelle, K., & Burtch, M. (2006). Capitalizing on advances in mathematics and K-12 mathematics education in undergraduate mathematics: An inquiry-oriented approach to differential equations. Asia Pacific Education Review, 7, 85-93.)
Last thought: "Skills vs. Concepts" as a viewpoint is a projection (a la vector spaces) of a vast, complex space of teaching and learning into two dimensions. It's clearly a limited framework, just as Macroeconomics is not captured merely by "Guns vs. Butter."
Rasmussen, Kwon, Allen, Marrongelle, and Burtch found that in an IBL Differential Equations course, the students performed the same on procedural skills as students in a non-IBL, and scored significantly higher on items testing for conceptual understanding (Rasmussen, C., Kwon, O., Allen, K., Marrongelle, K., & Burtch, M. (2006). Capitalizing on advances in mathematics and K-12 mathematics education in undergraduate mathematics: An inquiry-oriented approach to differential equations. Asia Pacific Education Review, 7, 85-93.)
Last thought: "Skills vs. Concepts" as a viewpoint is a projection (a la vector spaces) of a vast, complex space of teaching and learning into two dimensions. It's clearly a limited framework, just as Macroeconomics is not captured merely by "Guns vs. Butter."
