Sometimes when I read or hear discussions of innovation or change in teaching mathematics or other STEM disciplines, whether it’s me or somebody else doing the discussing, inevitably there’s the following response:

*What do we need all that change for? After all, calculus [or whatever] hasn’t changed that much in 400 years, has it?*

I’m not a historian of mathematics, so I can’t say how much calculus has or hasn’t changed since the times of Newton and Leibniz or even Euler. But I *can* say that **the context in which calculus is situated has changed — utterly**. And it’s those changes that surround calculus that are forcing the teaching of calculus (any many other STEM subjects) to change –radically.

What are those changes?

First, the **practical problems** that need to be solved and the **methods** used to solve them have changed. Not too long ago, practical problems could be neatly compartmentalized and solved using a very small palette of methods. I know some things about those problems from my Dad, who was an electrical engineer for 40 years and was with NASA during the Gemini and Apollo projects. The kinds of problems he’d get were: *Design a circuit board for use in the navigational system of the space capsule*. While this was a difficult problem that needed trained specialists, it was unambiguous and could be solved with more or less a subset of the average undergraduate electrical engineering curriculum content, plus human ingenuity. And for the most part, the math was done by hand and on slide rules (with a smattering of newfangled mechanical calculators) and the design was done with stuff from a lab — in other words, standard methods and tools for engineers.

Now, however, problems are completely different and cannot be so easily encapsulated. I can again pull an example from my Dad’s work history. During the last decade of his career, the Houston Oilers NFL franchise moved to Tennessee. Dad was employed by the Nashville Electric Service and the problem he was handed was: *Design the power grid for the new Oilers stadium.* This problem has some similarities with designing the navigational circuitry for a space capsule, but there are major differences as well because this was a *civic* project as well as a technical one. *How do we make the power supply lines work with the existing road and building configurations? What about surrounding businesses and the impact that the design will have on them? How do we make Bud Adams happy with what we’ve done?* The problem quickly overruns any simple categorization, and it required that Dad not only use skills other than those he learned in his (very rigorous!) EE curriculum at Texas Tech University, but also to learn new skills on the fly and to work with other non-engineers who have more in the way of those skills than he had. Also, the methods use to solve the problem were radically different. You can’t design a power grid that large using hand tools; you have to use computers, and computers need alternative representations of the models underlying the design. And the methods themselves lead to new problems.

So it is with calculus or almost any STEM discipline these days. Students today will not go on to work with simple, cleanly-defined, well-posed problems that fit neatly into a box. Nor will they be always doing things by hand; they will be using technology to solve problems, and this requires both a different way of representing the models (for calculus, think “functions”) they use and the flexibility to anticipate the problems that the methods themselves create. This is not what Newton or Leibniz had in mind, but it is the way things are. Our teaching must therefore change to give students a fighting chance at solving these problems, by emphasizing multiple representations of functions, multiple methods for solution of problems, and attention to the problems created by the methods. And of course, we also must focus on teaching problem-solving itself and on the ability to acquire new skills and information independently, because if so much has changed between 1965 and 1995, we can expect about the same amount of change in progressively shorter time spans in the future.

Also, the **people who solve these problems**, and **what we know about how those people learn**, have changed. It seems undeniable that college students are different than they were even 20 years ago, much less 200 years ago. Although they may not be natively fluent in the use of technology, they are certainly steeped in technology, and technology is a primary means for how they interact with the rest of the world. Students in college today bring a different set of values, a different cultural context, and a different outlook to their lives and how they learn. This executive summary of research done by the Pew Research Foundation goes into detail on the characteristics of the Millenial generation, and the full report (PDF, 1.3 Mb) — in addition to our own experiences — highlights the differences in this generation versus previous ones. These folks are not the same people we taught in 1995; we therefore cannot expect to teach them in the same way and expect equal or better results.

We also know a lot more now about how people in general, and Millenials in particular, learn things than we did just a few years ago. We are gradually, but also rapidly, realizing through rigorous education research that there are other methods of teaching out there besides lecture and that these methods work better than lecture does in many situations. Instructors are honing the research findings into usable tools through innovative classroom practices that yield statistically verifiable improvements over more traditional ways of teaching. Even traditional modes of teaching are finding willing and helpful partners in various technological tools that lend themselves well to classroom use and student learning. And that technology is improving in cost, accessibility, and performance at an exponential pace, to the point where it just doesn’t make sense not to use it or think about ways teaching can be improved through its use.

Finally, and perhaps at the root of the first two, the **culture** in which these problems, methods, people, and even the mathematics itself is situated has changed. Technology drives much of this culture. Millenials are highly connected to each other and the world around them and have little patience — for better or worse — for the usual linear, abstracted, and (let’s face it) *slow* ways in which calculus and other STEM subjects are usually presented. The countercultural force that tends to discourage kids from getting into STEM disciplines early on is probably stronger today than it has ever been, and it seems foolish to try to fight that force with the way STEM disciplines have been presented to students in the past.

Millenials are interested to a (perhaps) surprising degree in making the world a better place, which means they are a lot more interested in solving problems and helping people than they are with epsilon-delta definitions and deriving integrals from summation rules. The globalized economy and highly-connected world in which we all live has made almost every problem worth solving multidisciplinary. There is a much higher premium now placed on getting a list of viable solutions to a problem within a brief time span, as opposed to a single, perfectly right answer within an unlimited time span (or in the time span of a timed exam).

Even mathematics itself has a different sort of culture now than it did even just ten years ago. We are seeing the emergence of massively collaborative mathematical research via social media, the rise of computational proofs from controversy to standard practice, and computational science taking a central role among the important scientific questions of our time. Calculus may not have changed much but its role in the larger mathematical enterprise has evolved, just in the last 10-15 years.

In short, everything that lends itself to the creation of meaning in the world today — that is, today’s *culture* — has changed from what it used to be. Even the things that remain essentially unchanged from their previous states, like calculus, must fit into a context that has changed.

All this change presents challenges and opportunities for STEM educators. It’s challenging to go back to calculus, and other STEM disciplines, and think about things like: *What are the essential elements of this subject that really need to be taught, as opposed to just the topics we really like? What new facets or topics need to be factored in? What’s the best way to factor those in, so that students are really prepared to function in the world past college?* And, maybe most importantly, * How do we know our students are really prepared?* There’s a temptation to burrow back in to what worked for us, when faced with such daunting challenges, but that really doesn’t help students much — nor does it tap into the possibilities of making our subjects, and our students, richer.

Excellent discussion of multi-variable dynamics of ‘modern’ problems requiring student learning. However, must still start with basics? Then with coaching, hands-on teaching, launch next generations of learners to manage change/transformation. Such a fluid world of teaching and learning now. Thanks for post.

Your last statement really sums it up. But the folks who are “burrowing back to what worked for them” are not reading this (or any other) progressive blog. Are not using even the simplest of technologies. Have no desire to even listen to the conversation. It’s just a lot simpler to lecture on Examples 1 through 5, in that order, and then assign #1-79, odds. Any day. Any math class. Any semester.

Add to that the increasing number of part time faculty who teach business math and math for liberal arts students. These folks do not have the time or incentive to prep a really a good course for an audience that needs to be quantitatively literate.

How do we get these new challenges across to those who want to stay in their comfort zone?

Reva, that’s an extremely good and important question. My take on the answer is: You have to be a change agent where you are, and the best way to do this is to teach your own classes such that the outcomes driven by the changes I’ve described above are more attractive than the status quo. I think that if you can show that your own students — who are probably no different than the students of the change-resistant colleagues you’ve mentioned — learn better, are more engaged, and are happier by using technology, problem-based learning, and so on than they are with straight lecture, then colleagues will notice. Then you can gently share ideas and practices with them to give them simple, low-cost steps to including these things in their own courses.

A lot of faculty aren’t simply recalcitrant. They’ve just made a cost-benefit analysis of changing their teaching and decided that the change isn’t worth the cost. If somebody can show them that it is worth it, then my experience is that those colleagues are more willing to change than they might first appear.

Having taught Grades 1 to graduate school, change(s) often a result of administrative will/policy with appropriate incentives. In education we jokingly discuss how K-3 are most innovative hotbeds of teaching and post-secondary the most entrenched. Difficult to assess without ‘hard’ data and some courses lend themselves to straight-out lecture-style teaching, particularly if fortunate to have prof with adequate knowledge/experience to share. This sharing of the ‘mind’ can be amazing. Yet we fall back to RAs and TAs being given responsibility to ‘teach’ and this I see as a demise of education but we Ph.D.s are to pursue research dollars and time crunches keep us from really instructing those next generations. Before I retired I spoke with old (I mean old) faculty who confirmed my suspicions that in their time the primary emphasis was instruction/teaching and there was no expectation of publish-paper-of-the-week or go-get-money. If there is to be return to teaching-as-educating then faculty and administrative support will be one of the methods of advocacy.

Excellent post that could go for pretty much any other subject in addition to calculus and math.

Context and technology may change, but our minds are really the same as minds have been for the last several years (last few centuries, maybe?). Traditional ways of teaching are still what is what humans need to use, and then learn newer methods separately before integrating the traditional with the newer technologies.

Realize, back just a few years ago, when we studied College Algebra, or Trigonometry, or Calculus, we really had to LEARN the material, learn to draw graphs on paper, learn to understand problem solving processes, perform steps, and we were often permitted to use a scientific calculator. So much technology creates its own problems – mainly, first learning it all the while struggling to learn and understand Calculs, College Algebra, and perform proofs. I say, bring in the technology at a time when the student has first reasonably learned the course content of the Math courses in order that they do not become overloaded with both. Hey, realize again that just about 25 years ago, most students in Math and Science did not yet commonly have graphing calculators