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?