Solution Avenues

Teaching as map-making

Each test prep student has a mental map of the test. The map has locations (specific questions/answers), but it also has roads (solution paths). Teachers help students improve these maps. But you don’t want to help them build and maintain a bunch of isolated gravel roads that they’ll only use once. Nor do you want to include lots of disconnected destinations. You want to help them to build avenues, powerful solution paths that intersect with lots of other paths. Then they’ll be able to navigate from one destination to another on their own.

How would you teach this question? What is 40% of 80?

You could just multiply .4 by 80. I prefer a slower approach: set up a proportion, cross-multiply, solve. You may prefer the faster method, or a different one entirely. It doesn’t matter. What does matter is how your explanation here relates to all your other explanations, and to the abilities of your student. What do they already know? How does your explanation connect to that knowledge? Will your explanation help them on other problem types as well?

I like my slower method because it connects with other methods students use all the time. Proportions are useful for ratios, fractions, basic probability, circle wedges, and many other questions. If they understand how proportions work, then they don’t need to build a new road on the edge of town for this question type – they can just add a side street to their avenue.

Luckily, the town itself does not change much — there is a lot of repetition on these standardized tests. As I mentioned in a previous post (Score Tiers), roughly 90% of the material on a student’s actual test should be familiar from previous practice tests. But each student’s mental map of that town is unique, with half-built roads, abandoned avenues, and missing destinations. And students only have so much time: it’s impossible to build an identical, complete map for every student.

Therefore, it’s not just the student who is building this mental map — the teacher is building that same map! Something like “Stephanie’s mental map of the ACT” exists in the mind of the tutor, who is referencing it at every moment of every session, deciding which roads to build, which routes to suggest, and which destinations to skip entirely.

Consider this question, a version of which has appeared on 5 of the last 26 ACTs:

There’s a very fast solution: Area = (.5)(a)(b)(sin(c)). But I won’t teach it this way to all students. If their scores are too low, I won’t cover it at all. If they are very comfortable with special right triangles (which appear on the top 75 list) and have trouble with trig functions, I might explain it with special right triangles instead. And if it’s an advanced student who learned this formula at some point, I’ll just teach the formula. Meanwhile, other tutors might teach a calculator-based method. And that could be the right approach too! For many tutors and students, the calculator is a very powerful way to solve all kinds of questions; for others, it’s a last resort. It all depends on the map you’ve built together.