The High School Gap

See you on the other side.

DISCLAIMER

While I’ve observed this phenomenon independently early on, I cannot take credit for being the first to point it out. I first heard the version I will be elaborating on in Mar del Plata, July 2012. The diagrams, further explanation, and tips below, however, are my original work.

I am not (yet) an expert in mathematics education, and the explanations and diagrams below come from my personal observations and not from any formal study. If you want to subject this phenomenon to research, however, do consider e-mailing me. I’d be happy to help!

What it is
I think this issue is more important for students who plan to focus on sit-down exam questions (e.g. Olympiads). A similar high school gap should exist for students who specialize in fast-paced oral contests, but I believe it is less of a problem in that field.

To start off, I made a little diagram :

NOTE 1
I think it's very subjective to quantify the common definition of skill level, so I'm taking it here to mean the maximum level of mathematical complexity/abstraction that the individual is comfortable with. Quiz bee style questions should usually require a level in the middle of the gap, so the challenge there is not to get to that level, but to be fast and accurate in it.

What is bothering, is that gap of despair -- the insidious phenomenon that slows virtually every student down upon entering high school. Sometimes a victim never really recovers, and regardless of his/her previous grade school performance, by the time (s)he enters university, (s)he’s indistinguishable from a math major who never engaged in competition at all (who has a less pronounced experience with the gap, having gone through the regular educational ladder). I can identify two large factors contributing to this gap, namely (i) an demanding learning curve, and (ii) a totally new class of questions.

The Learning Curve

The first point is clear — high school students have to start with questions ranging from intermediate-level (some of which are easier than grade school contest problems), and in four years, work their way up to the often intractable (pre-olympiad to olympiad). Indeed, victims don’t ‘suffocate’ from the absence of recognition in the gap, rather they ‘choke’ at the great jump in workload. The gap is not empty space.There’s a smorgasbord of resources, so much that it’s a real problem to figure out where to start, and how quickly to advance.

And Now for Something Completely Different

I can illustrate the second point better with examples. These are two very easy problems, one in the grade school style, and the other in the high school style:

A GRADE SCHOOL MENSURATION EXERCISE

Points $D$ and $E$ are on sides $AB$ and $AC$ respectively of acute triangle $ABC$, so that $CD$ and $BE$ intersect at $F$. The areas of triangles $DEF$, $DFB$, and $FEC$ are 4, 8, and 5 respectively. Find the area of triangle $ADE$.

A HIGH SCHOOL GEOMETRY PROBLEM
In triangle $ABC$, the altitudes from $A$ and $B$ intersect at $H$. Take $J$ to be the point so that $BC$ is the perpendicular bisector of the segment $HJ$. Show that there is a circle $\Gamma$ that passes through the four points $A$, $B$, $C$, and $J$.

Let’s look at the differences that set these two apart:

• While the grade school problem is replete with given, the high school problem didn’t give any numerical information at all (except if maybe you consider ‘bisector’ to be numerical).
• The grade school problem is very specific — it covers only one possible case, and there are no generalizations made.
• Grade schoolers rarely have to draw their own figures; high schoolers draw their own figures, and have to make sure all configurations (see note 2)  have been covered.
• The most glaring difference: the grade school problem asks for an answer, while the high school problem asks for a proof — an explanation. Fresh from a world that demanded only numerical results, the new high schooler must suddenly learn to reason, argue, and communicate.

NOTE 2
Some problems intentionally describe the situation very vaguely, leaving multiple possible figures (say, certain points may not always be ordered the same way, or a point can be in the exterior instead of the interior of a triangle, etc). The naive solver would draw and solve only one of the possible figures (usually the trivial one), and get few or none of the marks. There are tools (eg directed angles mod $180^\circ$) built specifically to avoid this kind of pest, and solve all configurations in one fell swoop. In this problem, $H$ can be inside or outside the triangle.

Many of these differences still exist outside geometry problems. It’s no small wonder, then, that many good grade school mathletes suddenly disappear come high school (or perhaps excel exclusively in short-answer oral contests). There’s so much to learn by way of technique, style, and rigor that very few manage to keep improving steadily.

How to survive

Begin early.
I think the one best way to avoid disappearing during the gap is to prepare early. That means training in certain techniques and skills long before they become important (i.e. before they are needed in contest questions). For example, it’s becoming more and more of a necessity to start doing algebra in Grade 4-5. Unlike the Calculus trap, there is no Algebra trap — it’s best to get those symbolic skills ready before the going gets tough. The same goes for angle chasing and basic trigonometry, which should probably be picked up by Year 1 or 2 of high school.

Moreover, I cannot emphasize how important it is to start learning how to do proofs — paragraph style explanations that are concise, don’t overuse words like “clear” or “obvious”, and are humanly readable.

Read books.
I really think it says something for a young student to plow through a book, and not just training worksheets — it sort of develops a mental resilience to digest large chunks of text (see note 3) . For young mathletes I would passionately recommend The Art of Problem Solving. It caters to young and old readers alike, and gives a solid foundation to the requisite mathematics involved in later years. Mathematical Excursions is also a great text, but the difficulty level may appeal more to students in the middle of high school. They don’t even have to be competition math textbooks: I remember my introduction to axiomatic Euclidean geometry was kindled by an old edition of Michael Serra’s Discovering Geometry.

NOTE 3
This is one of the many skills a mathlete can easily develop that translates to actual collegiate and workplace efficiency. It may not be high literature, but mathematical writing, even at high school Olympiad levels, can be extremely information-dense and demands as much focus from the reader as would close reading of a passage, or analysis of legalese.
At any rate, it feels more like an achievement to have finished a book compared to a bunch of worksheets and problems sets of the same length.

Be aware of difficulty levels.
The scariest part of the gap is the distance students have to cover by way of complexity, and also the immense, unorganized variety of material to work on. I’ve found it vital to have a good sense of pace, to select and work on resources that are just beyond one’s comfort zone, so one neither becomes too demoralized (if the problems are too hard) nor lulls oneself into a false sense of security (if the problems are too easy). Of course, this is easier said than done, and I think it does take some experience (and many spoiled problems) to get a feel of whether a problem is just right.

I also think the gap is a test of who can learn to be patient with a problem, for unlike grade school problems, the harder high school problems (especially the Olympiad questions) can take much, much longer to crack.

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Any thoughts, opinions, or firsthand experience of the High School Gap you'd like to share? Shoot a comment below!

Image Credit:
"South Kaibab Trail Hiker", NPS, Michael Quinn. Public Domain.