# The Infinite Cycle of Gladwell’s David and Goliath

I recently finished reading Malcolm Gladwell’s David and Goliath: Underdogs, Misfits, and the Art of Battling Giants. The book is like most Gladwell books. It has a central thesis, and then interweaves studies and anecdotes to make the case. In this one, the thesis is fairly obvious: sometimes things we think of as disadvantages have hidden advantages and sometimes things we think of as advantages have hidden disadvantages.

The opening story makes the case from the Biblical story of David and Goliath. Read it for more details, but roughly he says that Goliath’s giant strength was a hidden disadvantage because it made him slow. David’s shepherding was a hidden advantage because it made him good with a sling. It looks like the underdog won that fight, but it was really Goliath who was at a disadvantage the whole time.

The main case I want to focus on is the chapter on education, since that is something I’ve talked a lot about here. The case he makes is both interesting and poses what I see as a big problem for the thesis. There is an infinite cycle of hidden advantages/disadvantages that makes it hard to tell if the apparent (dis)advantages are anything but a wash.

Gladwell tells the story of a girl who loves science. She does so well in school and is so motivated that she gets accepted to Brown University. Everyone thinks of an Ivy League education as being full of advantages. It’s hard to think of any way in which there would be a hidden disadvantage that wouldn’t be present in someplace like Small State College (sorry, I don’t remember what her actual “safety school” was).

It turns out that she ended up feeling like a complete inadequate failure despite being reasonably good. The people around her were so amazing that she got impostor syndrome and quit science. If she had gone to Small State College, she would have felt amazing, gotten a 4.0, and become a scientist like she wanted.

It turns out we have quite a bit of data on this subject, and this is a general trend. Gladwell then goes on to make just about the most compelling case against affirmative action I’ve ever heard. He points out that letting a minority into a college that they otherwise wouldn’t have gotten into is not an advantage. It’s a disadvantage. Instead of excelling at a smaller school and getting the degree they want, they’ll end up demoralized and quit.

At this point, I want to reiterate that this has nothing to do with actual ability. It is entirely a perception thing. Gladwell is not claiming the student can’t handle the work or some nonsense. The student might even end up an A student. But even the A students at these top schools quit STEM majors because they perceive themselves to be not good enough.

Gladwell implies that this hidden disadvantage is bad enough that the girl at Brown should have gone to Small State College. But if we take Gladwell’s thesis to heart, there’s an obvious hidden advantage within the hidden disadvantage. Girl at Brown was learning valuable lessons by coping with (perceived) failure that she wouldn’t have learned at Small State College.

It seems kind of insane to shelter yourself like this. Becoming good at something always means failing along the way. If girl at Brown had been a sheltered snowflake at Small State College and graduated with her 4.0 never being challenged, that seems like a hidden disadvantage within the hidden advantage of going to the “bad” school. The better plan is to go to the good school, feel like you suck at everything, and then have counselors to help students get over their perceived inadequacies.

As a thought experiment, would you rather have a surgeon who was a B student at the top med school in the country, constantly understanding their limitations, constantly challenged to get better, or the A student at nowhere college who was never challenged and now has an inflated sense of how good they are? The answer is really easy.

This gets us to the main issue I have with the thesis of the book. If every advantage has a hidden disadvantage and vice-versa, this creates an infinite cycle. We may as well throw up our hands and say the interactions of advantages and disadvantages is too complicated to ever tell if anyone is at a true (dis)advantage. I don’t think this is a fatal flaw for Gladwell’s thesis, but I do wish it had been addressed.

# Issues with Stewart

The old faithful Calculus text of Stewart has provided me with countless headaches this quarter. I must confess that my calc class is really quite bright. These problems do not occur in your standard freshmen calc class.

The Catch-22:
Stewart presents the definition of continuity at a point. A function is not continuous at “a” if it is not defined at “a.” Got that? Now he goes on to say a function is “continuous” if it is continuous at every point in its domain. This means that the function $f(x)=\begin{cases} 1 \ \text{if} \ x>0 \\ -1 \ \text{if} \ x<0\end{cases}$ is continuous. Fine and dandy. Now the bright student pipes up and asks, how can it be continuous since we know it isn’t continuous at 0? Excellent question! A function can be continuous even if it isn’t continuous at every point. Wait! What? You just said that was the definition. No, no. I said a function is continuous if it is continuous at every point in its domain.

Etc. Etc. This has been the source of questions nearly everyday since this was covered. In fact, now that we are well into differentiation and things like “a function that is differentiable is continuous” has provided a new source of attempting to get the students to keep all the examples straight of differentiable not continuous, not continuous, not in the domain, is a function continuous at a vertical asymptote, etc.

My fix for the problem is quite simple. Let’s go slightly more general and define continuous on a set. This sounds horrific from the veteran professor’s point of view that can’t even get the students to understand continuous at a point or on its domain, but hear me out. If we start there, then have them do drill exercises such as is f continuous on (1,2) what about [1,2], what about R, what about etc. Then this concept of thinking in terms of sets is not really any more abstract than what they already do. The drill exercises will make sure they understand the concept. Now they can note that “at a point” really means, on the set {a}. Also, we now just specify continuous on its domain, and they realize that this is also a set. I’m not saying we should give them the Dirichlet function and ask whether it is continuous on the rationals or something. But intervals and unions of intervals are things they can do.

Why this fixes the problem: the term “continuous” will no longer be used in an ambiguous sense. Right now the source of confusion seems to come solely from the fact that it hasn’t been drilled into them that secretly you can’t say the word continuous without meaning “on a set.”

The catch-22: As a textbook writer, Stewart most certainly wanted to adopt a convention that would mean he could drop the phrase “on its domain,” (which I concede is explicitly stated, but what freshmen reads their text). It would be rather annoying to both read and write that phrase all the time, especially when it is so natural a convention (what would one even mean by saying continuous and including points that aren’t even in the domain of f?!?). There are also many students struggling with the concrete side of things, so introducing “continuous on a given set” is bound to completely lose those students. We should definitely consider the flip-side, though. This text is meant for a freshmen calc class, and so if drilling this concept and explicitly writing out something that isn’t very clear (to them) is going to help in understanding, then I think it is worth it.

Any fellow calc teachers think this is a good/bad idea? I typed this in a heated frenzy and have been thinking about it for quite some time, so it might not be crystal clear what my complaint is.