The "Calculus Rush"

Some of you may have heard of this, some of you may have not. The “Calculus Rush” is a reference to when students (particularly those in high-school) “rush” to the next math class – that is, instead of actively developing their problem-solving skills with what they already know, they spread themselves thin by learning new material. Several articles have been written about this which you can find by googling for or “the rush to calculus” or some similar alliteration. This does not apply to every student and/or school, but it’s foolhardy to deny that this is a problem in education in general.

Calculus, by far, receives the most negative end of the pro-acceleration accusations (hence why it’s called the Calculus Rush and not the Geometry Rush). Even though this applies to a lot of math subjects whether it be algebra, geometry, precalculus, calculus, linear algebra, etc. Calculus in particular receives all the attention is for a variety of reasons:

  • Calculus, in theory, is the subject that transitions most students from a purely computational-based environment (another issue entirely) to a more conceptual and abstract one. While one certainly doesn’t need to study Spivak and bash rigor out of everything, it’s pretty important to intuitively understand why the stuff in calculus works to be able to use it effectively.
  • Another related reason to the one above is that most high-schools do not offer any other higher-level mathematics courses other than calculus.
  • For most high-schools, calculus is typically offered as AP courses (AB and BC) and is the highest-level mathematics available. As a result, calculus is perceived as the hallmark of mathematical achievement.
  • Many students (and universities too, sadly) believe taking calculus indicates significant talent; therefore, it is required to be competitive.

Now here’s where people make the mistake: I am not saying it is bad to learn calculus. Calculus is really, really, really powerful as both a tool and how it changes your mindset of mathematical reasoning. What I am saying is that it is bad when a student learns high-level material without sufficient preparation.

You might disagree, after all, should we not be pushing mathematically gifted students ahead as quickly as possible? Hell no. That’s a road with a high possibility of disaster. What you’d be suggesting is that if we have any prodigy pianist, we should immediately let them play very difficult piano pieces with only 3 to 4 years of experience. Neither I nor my piano teachers would ever let such a thing happen: in the case of piano, it’s extremely important to get a solid foundation before you let someone play difficult pieces. It’s only when they are ready do you let them play those pieces. This also applies to mathematics and moving ahead without a sufficient foundation is often unforgiving. But how is mathematics unforgiving? Well, there are two particular flavors: math majors and other STEM majors.

In the case of STEM majors other than mathematics, the curriculum can be pretty unforgiving. At a university level, students are expected to know the basics of mathematics (especially if you’re taking say, vector calculus) and use said basics for more complicated material and problem-solving – in the case of the engineering math curriculum (differential equations, calculus, vector calculus, linear algebra, etc.), students should have their algebra and geometry skills hands down solid. If they don’t, the likelihood of unnecessary struggle is pretty high.

In the case of mathematics majors, the curriculum required for a mathematics degree is VERY unforgiving to those who lack a solid foundation and in-depth knowledge of material. What’s worse is that this often doesn’t hit hard until students take a course in Real Analysis, often took in the second sophomore semester or first junior semester. Real analysis involves a lot of deep abstract thinking and is completely proof and axiomic-based. Those without a solid foundation will obviously be at a disadvantage and honestly, screwed as they’re already well into their university studies and have to go through hell.

Some root-causes

Okay, maybe you’re still not convinced. If this is such a problem, then schools should have fixed it already, right? Again, no. There are so many flaws with the education system that it’d be faster to list what’s not flawed. But then why do these issues persist? Well, there are a few prevailing reasons:

  • One of the worst causes is parents. It’s natural to want to see your child succeed, be smart, and make a boatload of cash. Unfortunately, parents also have a tendency to overestimate their child abilities and overly-encourage (if not force) them into subjects that they aren’t sufficiently prepared for.
  • Education is also really outdated. Most curriculums (even calculus ones) are constructed in a way that would be appropriate for a time when computers didn’t exist. Routine computation-based math problems can be done far faster on my LEGO Mindstorms EV3 brick than students could ever compute. Humans excel at novel problem-solving compared to computers, computers excel at computing things we already know how to do.
  • The standard curriculum in algebra, geometry, and precalculus is not rigorous enough. In particular, the Common Core standards lack proofs (not those two-column “proofs”). Being able to communicate and prove why you are correct about your result is one of the many important things mathematics is about.
  • Many universities and students perceive taking calculus as an elite qualification.

Solutions?

Admittedly, not much can be done about these points. It’s pretty hard if not impossible to make all parents and students aware that accelerating can easily go horribly wrong and it’s pretty hard to force all schools and universities to change their views on allowing students who shouldn’t be taking calculus into a calculus class. Unfortunately, it’s simply an issue that goes on and can’t really be fixed quickly. However, for those students willing to try to avoid the stigma, there’s a couple of options:

  • Get a very solid grasp on algebra, geometry, and trigonometry and using them as problem-solving tools. Both are needed to fully appreciate calculus. A great place to practice on some non-routine and challenging problems is through the Art of Problem Solving’s Alcumus system (disclaimer: I work for AoPS).
  • Learn discrete mathematics (combinatorics, number theory, graph theory, sets, logic, etc.). Discrete math is a very good way to delve yourself into some very challenging problems which rely heavily upon understanding compared to your typical plug-and-chug. It’s also a really good place to begin writing proofs.

Hopefully, I was able to persuade you enough to consider another view on the calculus fever. Again, I’m not saying that it’s bad to learn calculus, but it’s bad to venture on without sufficient preparation. Everyone learns at a different pace and a different way, when they are ready to learn something, then they should learn it.

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