Call me crazy, but undergraduate courses should be qualitatively different from their high school antecedents. One key change? At the college level, students should be moving toward *understanding* things instead of just *knowing* them. How can we introduce and reinforce the difference?

In my freshman-level world history class, I challenge learners in the first week with an unusual exercise. Pretend you’re a student and see if you can do it.

With 15 learners in attendance this semester, I asked how many of them knew the quadratic formula. Hands shot up: 12 out of the 15 (80 percent) claimed to. I then distributed a blank sheet of paper to the 12 with instructions to write out the formula anonymously. After a minute or so, the results came in: only four had it right. Many of their attempts contained elements of the formula but were lacking or incorrect. I also got the Pythagorean theorem a couple of times (*a ^{2} + b^{2} = c^{2}*), as well as the equation for a straight line (

*y = mx + b*).

Still, four out of 15 (27 percent) isn’t bad for a history course. The successful ones were rightfully proud of themselves, and some congratulatory high-fives were exchanged.

To those four who wrote the formula correctly, I then posed a follow-up: “What is the quadratic formula good for? That is, in what circumstances would we use it?”

Now I got puzzled looks and silence. Nobody knew. I prodded them a bit. “Do you use it any time you see *x* in a problem?” They really couldn’t say. I finally revealed to them that the formula derives from a quadratic equation in the form of *ax ^{2} + bx + c = 0*. Some quiet nods followed, but there was clear disappointment in most of their expressions.

Are you thinking the exercise is disingenuous, that quizzing a history class on algebra just sets people up for failure? Perhaps. We don’t retain knowledge and skills if we don’t actively use them, and maybe these students had encountered the formula years earlier. Daniel Willingham points to specific evidence that math abilities taper off over time, although those who learn more advanced mathematics seem *never* to forget algebra. (As someone who earned a math degree decades ago before turning to history, I can personally attest to this.) Still, there was a palpable disconnect between what many of my students thought they knew and what they actually knew.

But I wanted to give them one last chance. “You know when the formula is used,” I said. “Now, can you derive it from a general quadratic equation?”

Some eyes brightened a bit. “You mean, we just have to solve *ax ^{2} + bx + c = 0 *for

*x*?” asked one. Yes, I assured her. “Piece of cake,” she announced. Two of the successful four set off, while I ventured back into historical issues with the rest of the class.

After a few minutes of furious scribbling, the pair of problem-solvers had clearly given up. “What happened?” I asked. Both had hit an insurmountable roadblock. For the second time, their hopes were dashed.

Now, think about what had just transpired. A clear majority of the class was confident in its recall of mathematical arcana, but that knowledge turned out to be either faulty or incomplete for most of them. A decent minority remembered the formula, but now didn’t know what it was for, let alone how one could derive it.

In other words, those latter students had dutifully committed to memory an important piece of mathematical language, and they were able to reproduce it on command. But it’s questionable whether they ever understood it, much less its origins and reason for being. These are Ken Bain’s so-called “strategic learners,” who outwardly show great command of course material, but perhaps never understand it in any deep sense. And all too often, our classes don’t require them to do so.

For my purposes, the exercise serves as a wake-up call to learners: This class will be different. It will require you to think in altogether different ways, and it will assess you not on how well you absorb and replicate material, but on your ability to arrive at logical conclusions based on disparate, oftentimes conflicting evidence from and about the past. Trying to feign your way through it is akin to passing off the Pythagorean theorem as the quadratic formula. It’s a non-starter.

There’s a second lesson I want to get across early in the semester. Many things are much tougher and more complex than they appear, so students need to keep their confidence in check. There are myriad studies showing that weaker learners tend to be the surest of their abilities (see for example the works by Chew and Girash below). My own surveys of history students indicate tremendous confidence in such difficult matters as analyzing primary sources and sorting through historians’ variant interpretations of the past. These are challenging issues for experienced professionals, let alone undergraduates. And indeed, learning artifacts all too often betray students’ overconfidence and naivety when they try to rely on faulty or insufficient knowledge and skills in an altogether different environment.

This becomes the basis for our first unit in the course, which tackles a seemingly easy question: Why did early civilizations originate along riverways? Many students think they know the answer: People need water to drink in order to survive. But that’s a truism, not an explanation—there’s that overconfidence and naivety again. But even if they learn the answers, there’s nothing saying students really understand them. They could faithfully reproduce justifications without knowing how or why, just as they’d done in the quadratic formula exercise.

The follow-up historical problem we grapple with is even more daunting, namely: Given a set of documents and artifacts from the distant past, can we *derive* the necessity of riverways for early complex societies?

This is an epistemological question, getting at the heart of how we know what we claim to know about the past. It calls for training in historical interpretation that’s unfamiliar, even what Sam Wineburg calls “unnatural.” It’s to historical understanding what deriving the quadratic formula is to math students. It’s hard and it may not be a whole lot of fun, but it’s the difference between knowing *what* and *why*. And that should be the sort of goal we can all agree on.

Oh, lest I forget, here’s that pesky quadratic formula. Did you get it right?

**References
**Bain, Ken,

*What the Best College Teachers Do*(Harvard University Press, 2004).

Burkholder, Pete, “Metacognitive Roadblocks: How Students’ Perceived Knowledge and Abilities May Hinder Performance in Undergraduate History Courses,” *American Historical Association Tuning Project Report* (2015).

Chew, Stephen, “Helping Students Get the Most out of Studying,” in *Applying Science of Learning in Education*, eds. Victor Benassi et al. (Society for the Teaching of Psychology, 2014), 215-23.

Girash, John, “Metacognition and Instruction,” in *Applying Science of Learning in Education*, eds. Victor Benassi et al. (Society for the Teaching of Psychology, 2014), 152-68.

Willingham, Daniel, *Why Don’t Students Like School?* (Jossey-Bass, 2009).

Wineburg, Sam, *Historical Thinking and Other Unnatural Acts* (Temple University Press, 2001).

*Pete Burkholder is professor of history at Fairleigh Dickinson University, and he serves on the editorial board of *The Teaching Professor*.*