The "Mathematics Problem"

Our journal club for this semester is looking at the "mathematics problem" in engineering education. At least this is what we've been calling it, but this may be due to our own ability to define what "mathematics problem" really means. The idea arose from listening to faculty complain that students don't seem to know math, or at least seem unable to do the mathematics asked of them in engineering classes. Over here in engineering we tend to blame either the math faculty or--even more easily--the students' high school preparation. But this viewpoint has always seemed a little simplistic to me; i.e. it is somebody else's fault.

In looking at this issue, there is a surprising dearth of information on the connection between engineering and mathematics, considering how fundamental math is to engineering. Wendy, a doctoral candidate in the journal club, found a paper in the Journal of Research in Science Teaching (vol 54, p, 197, 2008) that looks at whether what students learn in mathematics courses transfers to chemistry courses.

This study devised two tests, one giving an algebra problem phrased in the context of chemistry and the same problem with all the chemistry content removed and written as a math problem. While the paper had it's weaknesses, the experiment was simple and the results were interesting:

• Students were able to transfer algebra procedural skills to chemistry problems. The authos reported they were surprised by the proficiency with which students did this.


• Some students' inability to transfer seemed to be more a result of inadequate understanding of mathematics rather than failures to transfer results to the context of chemistry.


• In contrast to their facility with algebra procedures, students seemed unable to draw graphs or use graphical information to solve problems. Students actual scores on this aspect of the test were lower and they showed significantly less confidence in their abilities.


• Finally the paper concludes, perhaps tenuously, that students have trouble transferring between graphical and algebraic thinking. In other words they can't use information from graphs to supplement what they learn from algebra or vice versa.
  

It should be noted that this study was performed in South Africa so the preparation of students there may be different than our own students here at OSU.

Two of these results support things I have repeatedly observed in my own classes of engineering seniors. First students aren't as comfortable using graphs as they should be at this point in their degree, and they certainly are not good at creating graphs. Second, the last of these conclusions is perhaps the most interesting to me from an educational research perspective. This paper hints at the fact that students seem to have a rote procedure for getting correct answers without being able to draw from multiple approaches. All of us that teach engineering education have certainly seen this in our classes. A common complaint is that many students don't think about procedures before they apply them.

I would argue that a sound conceptual understanding is vital to developing such metacognition. And now I've identified a gap in my own understanding of education. What are the best and most effective techniques for teaching conceptual understanding? Anybody out there have any good reference on this?

"...we cannot remain who we are"

The title of this post is taken from a short editorial in the Christian Science Monitor (linked in the title) from the president of Ohio State University, Gordon Gee. In the editorial he speaks about the need to use these days of financial uncertainty to fundamentally reform higher education. He posits that in a very global world the structure of higher education is too restrictive to really let us prepare graduates for the very (strange and wonderful) | (dangerous and terrifying) world in which we find ourselves.

While not deep in the sense of referencing research or statistics on education, the editorial did home in on something I personally have come to believe over the last few years- the university's problems are more structural than pedagogical. In the way the editorial was phrased, President Gee sees that a narrow focus on disciplinary knowledge can't prepare students for working in the world. Students collaborate naturally; course structures which inhibit sharing knowledge and experience no longer mimic life. In this he is right but I don't think he goes far enough...

Our focus at universities has historically been to address the need for students to master some part of the shared knowledge of the human race. Knowledge must be personal to be useful, so we see our goal as transmitting through the mouths of faculty what is known about a discipline to our students, who we hopefully develop enough of a relationship with to make learning somewhat personal. There is really nothing wrong with this approach, a lot of research supports the fact that experts have a lot of relevant knowledge in their heads where it can be accessed easily and quickly when they are confronted with a problem. And passing this on to non-experts is a good thing. The problem comes from how we transmit this knowledge and, perhaps more subtly, the need for us to grade. To assign "meaningful" grades we believe we need to give rigorous tests and homework assignments. However the very rigor of the grading schemes we use are, I think, their downfall. The knowledge I personally have found most useful in my career is the knowing of ways of doing things. Such knowing is general, flexible, adaptable, and very very hard to teach. Such tacit knowledge is difficult to measure using an exam; I still can't do this well despite years of tinkering with exams.

And this is the message I took from President Gee's article. At universities we trade effort for grades with the hope that learning occurs somehow in the effort. But the efforts needed to develop effective and learned citizens for today's society don't mesh well with what schools have historically done. We can't remain who we are if we are to claim that the education we offer is worth the price we now charge.