Within our mathematics community, much of our recent efforts have been directed at presenting students mathematics related to problems (contexts) that are likely to be important to them. Some curricular work is limited to the mathematics for which such a context can be presented. Although relevant context is helpful, we lose something important when the context becomes more important than ‘mathematics’.

A related movement is the ‘guided pathways’ (see http://www.aacc.nche.edu/Resources/aaccprograms/pathways/Pages/ProjectInformation.aspx) which has a goal of aligning mathematics with the intended major, a guideline based on research showing improved completion when this is done. The guideline is being applied to both college-level and developmental course work.

In some ways, this makes sense … Mathematics has always had roots deep in practicality.

However, I see two failures resulting from these approaches:

- Mathematics is not always practical when ideas are developed or discovered.
- General education seeks to go beyond the parochial.

In the American culture of 2016, we seem to validate the notion that “I only have to care about things that impact me directly.” When we honestly tell students that this mathematics is important even though we are not showing a context for it, we should be able to expect students to honor the statement. In many ways, learning mathematics without context is a good training program for employment … I suspect that the majority of workers work in a job with little innate value to them, in which they need to honor a supervisor’s statement that doing a job a certain way is important.

The role of general education has been both integral to higher education and marginalized in higher education. The values of ‘different perspectives’ and ‘modes of thought’ represent the building of capacity in a society to think about difficult problems without resorting to slogans and over-simplifications. When general education works, it is a beautiful thing. This type of rising to a higher level of problem solving can not occur when the classroom is limited to the shared current concerns of those present.

If we truly believe that students are well-served by allowing them to focus on their own interests and concerns, sure … let’s limit their mathematics to contexts that they can understand at the time.

I think that limitation is a dis-service to students (and is not respecting mathematics as a set of disciplines). Sure, we can have lots of fun when students are enthusiastic about our work in class. Do they have any better notion of what ‘mathematics’ is? Did the experience result in anything more than a few concepts that are applied in concrete ways?

Our courses should always contain significant elements of what I call “beautiful and useless mathematics”. “Beautiful” refers to the aspects of mathematics which appeal to mathematicians … which can vary from person to person, and from one domain to another. “Useless” refers to the ideas being developed in an abstract way without knowing if there will ever be any practical use.

One example of such ‘beautiful and useless mathematics’ would be functions which have a rate of change equal to the function. The number e is not immediately reasonable when we deal with concrete multiplicative change. We can contrive some contexts where the base e can be used, though most of these are more accessible to students using a percent growth rate (or decay). The use of e for the function, and for the rate of change in the function, is a thing of beauty.

In some ways, this post boils down to this statement:

Don’t sanitize any mathematics course to the point where all artistic merit is destroyed.

Although this post relates to a recent post on ‘where STEM students come from’, I think the idea is valuable for every student who walks in to a math class. We are not mathematicians because it is practical (though it is); we are mathematicians because there was something that attracted us. Our students deserve to see at least a small corner of the wonderful canvas called ‘mathematics’.

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