Last week, I spent several days with faculty who are working with the Carnegie Foundation’s Pathways – Statway and Quantway, at their National Forum (summer institute). I continue to be impressed with the quality of these professionals; Carnegie is fortunate to have them involved. One comment from a faculty member has been stuck in my thinking. In the context of Quantway, this faculty member said:
Everything in this course has to be practical. The math students see has to be practical.
I recognize that there is a high probability of head nodding and agreement with this sentiment among people reading this post. Can we … is it reasonable or desirable … to shift from a ‘nothing in this course is practical’ to ‘everything in this course is practical’ position?
First of all, we need to recognize that ‘practical’ is a matter of perception, communication, and culture. Our students will not see the same ‘practicality’ that we do. For example, if we have a series of material looking at the cost of buying a car including operating and finance, many students will definitely not see this as practical. The majority of my students are not able to consider this situation in their real life now, nor for several years; for some, they can not even imagine having a real choice to make about a car. What we often mean is that math needs to be contextualized, not practical — context is a simpler matter to establish than practical.
Secondly, the ‘practical’ or ‘contextual’ emphasis reminds me of the old school approach to low-performing math students: If a student was not doing well in math, put them in an applied math course (business math, shop math, personal finance), as a way of being polite about lowered expectations. I realize that many of our students are initially happy with the lowered expectations of ‘practical math’; however, this approach does not honor their real intelligence, nor does it recognize the capacities in our students to understand good mathematics just because it is enjoyable to do so.
More important than these two points is the learning implications of ‘practical math’. I’ve been reading theories of learning and research testing these theories … for close to 40 years now. Nothing in the theory suggests that learning in a practical context is better than learning without the context; without deliberate steps to decontextualize the learning, the practical approach often inhibits general understanding and transfer of learning to new situations. I do not believe that ‘all is practical’ is a desirable approach to learning mathematics.
However, context and practicality can be very motivating. Motivation is the most elemental problem in developmental mathematics. Therefore, it is reasonable to provide considerably more context for students than the traditional developmental math courses with its ‘train problems’. I also would add that most students are motivated by learning mathematics with understanding when they can see the connections; true, our students need some extra support for this process, and it conflicts with the approach emphasized with them in the past (primarily memorization without understanding).
I have summarized my view on the ‘practical’ issue with this statement:
I will always include some useless and beautiful mathematics in all of my math classes.
Education is about expanding potentials and creating new capacities; practical learning is the domain of ‘training’ (which is also critical … but it is not education). I encourage all of us to help our students learn mathematics in different ways: sometimes practical, sometimes in a context, sometimes imaginative, and sometimes logical extensions. The mix of these ingredients might reasonably shift as a student progresses; developmental math courses might be more practical than pre-calculus. Diverse learning is better than limited learning. Diverse learning respects the intelligence of our students and maintains high expectations for all students.
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