Guest Post: Bernd Schroeder on preparing students, creating texts

The following is a guest post from Bernd Schroeder, the current academic director and program chair of Mathematics and Statistics at Louisiana Tech University.  His specialties include discrete mathematics, harmonic analysis, and probability theory. He has about 20 years of teaching experience and wrote a few titles including Fundamentals of Mathematics: An Introduction to Proofs, Logic, Sets, and Numbers (2010) and A Workbook for Differential Equations (2009).

Below, he talks about the difficult task of providing both STEM and non-STEM students with the skills they need to succeed in this increasingly analytic workplace.  He says that he does not to have all of the answers but wanted to share some observations with fellow authors.

This is his first of two posts.  Check back next week for a piece on how to teach analysis to the varied levels of those seeking this skill set.

If you have any comments or thoughts on this topic, feel free to reply to the post below.

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How to Prepare Mathematics Majors and STEM Undergraduates for Jobs and for Work with Non-Mathematicians?

Labor statistics suggest that there is an impending shortage of STEM talent but not at the Ph.D. level. Therefore, the mathematical community needs to (re)focus on students whose final degree will be a bachelor’s or a master’s degree. Moreover, we need to focus on students whose degree will not be in mathematics. This is a substantial task even if your program’s focus is already wider than the stereotype in which “the courses get you ready for your Mathematics Ph.D. qualifying exams.”

So how can mathematics and the job relevant “mathematics abilities” (see page 57 of this document) be made “more accessible” without “watering down” the students’ preparation?

For some student populations, changes in the first two years may well turn out to be minimal.

For engineering, physics, and mathematics majors, there does not seem to be a replacement for calculus. Multivariable calculus is the mathematical basis for the theories of fields and flows, which are central to subjects like Physics, Electrical Engineering, Mechanical Engineering, and more.

For mathematics majors, the case can be made that calculus is the “applied version” of mathematical analysis. When I surveyed the AMS subject classification, in my judgment, I found that analysis is the indispensable foundation for 38 of the 62 branches of mathematics. In addition, another 9 branches were  closely related to analysis. This means that less than one fourth (15/62) of all branches of mathematics might get by without analysis. If you go through the same exercise, I’m quite confident that your count will be similar to mine. For many disciplines, the question is not if to teach calculus, but how. Similarly, at a more advanced level, the question is not if to teach analysis, but how

Although the populations above are sizable, for other students, changes are possible, maybe even desirable.

A calculus prerequisite for Discrete Mathematics helps assure that students have “mathematical maturity.” But questions arise like:

  • What parts of an introductory Discrete Mathematics class truly need calculus content?
  • Can these parts be replaced with content that is similarly beneficial?
  • Would it be sacrilege to consider a hypothetical computer science graduate who has not had calculus?

My only constraint in this regard is that I would want this graduate to have developed/trained certain overall cognitive abilities (ability to concentrate, ability to think about a problem in different ways, ability to correctly follow a procedure, deductive reasoning, etc.) to the same level as would have been gained through calculus.

The situation gets muddier in disciplines that do not require the (whole) calculus sequence and need STEM skills, such as business and biology. One reason STEM majors are considered useful is because they can analyze data. However, data analysis appears to be fundamentally different from proving theorems. For example, I proofread a proof of the Central Limit Theorem as a graduate student, but I only developed an understanding of the Central Limit Theorem when I wrote this simulation.

So how can we authors help with the changes that will either be made by the mathematical community or that someone else will make for us?

First of all, we face a paradox that affects something publishers care about: sales. Take a calculus book as an example. If you write a calculus book that is similar to the standard texts in the field (10 years ago, sequences and series was always Chapter 8), then why would people buy your book? On the other hand, if you write a text that is very different from the standard texts, will people dare adopt your book?

Personally, I have no interest in writing a book that’s already been written by someone else. The above remarks on future needs also indicate that there is little to be gained from incremental changes. That is, unless you write an incrementally changed text that sells millions of copies, in which case at least one person has gained substantially: you; and (without sarcasm or envy) congratulations.  The market leaders are market leaders because their products are good. Specific users will always find certain things that they wish would be different, but the texts satisfy the needs of many quite well.

So here is the hard questions are when you write something that radically departs from the canonical setup:

What do you include?  What do you drop and what will be the effect?

For the above examples, we immediately obtain some specific questions:

Can you design a reasonably deep Discrete Mathematics book that does not need to touch upon calculus (definitely) and would students who have not been vetted by calculus respond to the presentation (There is a lack of data available)?

Can you teach the deep data analysis skills needed in the working world without touching upon the theory of continuous distributions (probably, as long as you’re okay using tables and computers and simply quoting results) and would students who have not been vetted by calculus respond well to the presentation (There is a lack of data available to me but a colleague told me about positive experiences with graduate students in biology)?

Finally, once you have answered these (and other) questions in a convincing fashion, how do you get other people to agree that your answer is convincing?

I have not tackled the questions above yet, but I would be interested in doing so. Discrete Mathematics and its connection to computer science should be well within my competence. However, I have never let a lack of education stop me from exploring other areas. Every book of mine has at least one chapter of which I knew little when I started writing. (Replies of the nature “All chapters read so poorly, which one is the one you had no clue about?” are not needed.  ) So, despite my shortcomings, data analysis would be interesting, too.

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