I’m at the American Society for Engineering Education Annual Conference right now through Thursday, not presenting this time but keeping the plates spinning as Mathematics Division program chair. This morning’s technical session featured a very interesting talk from Kathy Harper of the Ohio State University. Kathy’s talk, “First Steps in Strengthening the Connections Between Mathematics and Engineering”, was representative of all the talks in this session, but hers focused on a particular set of interesting data: What engineering faculty perceive as the most important mathematics topics for their areas, and the level of competence at which they perceive students to be functioning in those topics.

In Kathy’s study, 77 engineering faculty at OSU responded to a survey that asked them to rate the importance of various mathematical topics on a 5-point scale, with 5 being the most important. Here are the top 5 specific mathematics skills as rated by faculty with the average rating in parentheses:

- Evaluating solutions/checking work (4.5)
- Being familiar with units and dimensions (4.4)
- Knowing how to create and interpret graphs (4.4)
- Performing algebraic manipulations (4.3)
- Knowing how to convey and interpret engineering relationships through mathematical expressions (4.3)

The faculty were then asked to rate skills on the basis of how prepared or competent students were in those skills when the students entered into their courses, which are typically post-calculus. Here are the top 5 results (i.e. the five areas in which students are best prepared) also rated on a scale of 1 to 5 with 5 being the top level of preparation:

- Performing algebraic manipulations (3.3)
- Knowing how to create and interpret graphs (3.2)
- Being familiar with units and dimensions (3.1)
- Knowing when to integrate or differentiate (2.9)
- Using parameters/symbols, rather than numerical values, in analysis (2.9)

I find several takeaways in the data here.

First: It’s interesting to see what *isn’t* listed in the top 5 math skills list. Those include “Formulating mathematical models”, “knowing when to differentiate or integrate”, and “working with multivariable problems”, all of which in some form or another are staples of most university calculus courses. I don’t think anybody is saying that these skills shouldn’t be taught or even de-emphasized, but the fact is that they didn’t rise to the top of the engineering faculty’s list, and this seems revealing.

Second: What *did* rise to the top are *not* always staples of a university calculus course, certainly not the super-traditional style of course that’s offered frequently at universities and in many high school calculus courses. There, the entire point of the course is algebraic manipulations – to be able to do as many as possible, as quickly as possible and with as few mistakes as possible. But this is not what engineering faculty say is most important. They didn’t say it was *unimportant* either – just not *singularly* important, and just one part of a well-balanced skill set.

Third: The overlap of the “important skills” list with the “student preparation” list is noticeably incomplete. In particular, the top-rated item on the importance list – being able to check one’s work – came in at number 8 on the preparation list. And the “Knowing how to convey and interpret engineering relationships” item was at number 7 on that list. So there is a gap between some of the most important skills engineering faculty want students to have on the one hand, and how skilled those students actually are on the other.

So it seems like the data here are saying something important: That some of the skills that engineering faculty perceive as most important are those skills for which student preparation is lacking the most; and conversely, some of the skills in which student preparation is highest are skills which matter the least in the grand scheme of things for engineers. This speaks to a misalignment between how we teach and what we teach in calculus and what students actually need if they are going to be engineers – and one could make the case that it’s not just engineers who might find such skills important.

Let me focus on the top-rated “importance” item, being able to validate solutions, for a minute. I think you would be hard pressed to find a university calculus course that intentionally focuses on some of these top-rated importance items, especially self-validation. Intentionality is very important here because self-validation is a *skill* that has to be built, not just an activity to be done. We math teachers often approach this by telling students to look up their answers in the back of the book. But this is a form of validation that doesn’t scale – what about problems from outside the book? Or problems with no clear right or wrong answer? Instead, we ought to be teaching students how to check to see if their work is correct and their solutions valid using methods that scale to any problem at all, and to an extent equal to or even greater than some of the mathematical topics that we’ve always considered “canonical” in a calculus course. (I’ll let you debate what those topics are, in the comments.)

How might we do that? In the Calculus course that I just finished teaching in our 6-week Spring term, I experimented with making self-validation one of the foremost learning goals of the course. I think I may have more to say about this in another post, but for me and my class, getting self-validation to be perceived as important *among the students* was hard work and required a culture change. Most students are used to validating their work by using the back of the book or the teacher as an oracle or a black box. They ask: *Is this right? Am I on the right track?* The first step toward getting the culture to change was to avoid giving direct answers to these kinds of questions. I’d respond: *Talk me through what you’ve done so far. Why did you do this? What’s the basis for that step? Where did this assumption come from? What did Wolfram|Alpha say was the derivative of that function? What did the graph look like on Geogebra and does it corroborate your answer?* And so on.

Note that the technological tools of the course were key for this self-validation process, and we embraced those tools warmly throughout. Did students universally appreciate this approach? Of course not. You’ll get students who believe their tuition pays for my direct answers to their questions, no matter what the question, and whose primary concern is the grade in the course. But I think many students did, and the data from Kathy’s talk at least gives me something substantive along those lines to share with students in engineering, a demographic that makes up about half of the students in our calculus courses.