Math and science problems fall into four categories: Easy, Medium, Ugly, and Hard.
Easy Problems are ones you can solve with no difficulty in a short time. An example from Algebra I might be:
The problems that come at the beginning of each group of problems in a textbook are usually Easy Problems. If you had a pulse during class when they were explained, you can probably do them without hesitation.
Medium Problems are the ones in the middle of each group of problems in a textbook. An example from Algebra I might be:
They may take a few more steps to solve than an Easy Problem. If your brain was engaged during the class when they were explained, you can probably do them without too much difficulty – perhaps after some practice. Most problems on most quizzes and tests I have seen are Easy and Medium problems.
Ugly Problems are ones that will take a while to solve. They involve many steps. They may require you to use concepts from multiple chapters. They are problems which, when you first set eyes on them, lead you to pull out two or three fresh pieces of paper and make sure your eraser still has some life in it. Once you have mastered a topic, an Ugly problem will typically lead you to think something like: “I see how to start this, I don’t know how many steps it will take to solve, I have no idea how much time this problem is going to take to solve, I don’t really want to tackle this ugly thing, but I’m not getting any younger so I had might as well tackle this. Oh, and I had better work extra neatly and keep organized, or else it could take even longer”. An example from Algebra I might be:
Once you solve an Ugly problem you will probably experience a profound sense of satisfaction, and your self-confidence will have gone up by several notches.
High School students are seldom assigned Hard Problems. Hard Problems usually involve one or more things you don’t know anything about. Nothing was said about them in class. There is a brief reference to something that seems related in your textbook, but no explanation. A trip to a really good library is needed… or perhaps it’s time to buy someone who took this course last year a pizza so you can ask them many questions. Hours and hours of research and study are ahead of you before you can even begin to try to solve this type of problem. Hard Problems are typically assigned only in college or graduate school, and usually only in courses taken predominantly by people majoring in the subject. Hard Problems arise in everyday life more often than you might think though, so perhaps more High School teachers should assign some (with fair warning given and a suitable support structure in place, of course).
Most math students perceive problems “relatively”. If the hardest problem you have ever done was an Easy Problem, anything else will look pretty intimidating to you. If a test has only Easy and Medium Problems, just as your homework did, then the test will feel “hard” to you because it has problems similar to the hardest ones you have done.
Therefore, I urge math and science students and teachers to seek out Ugly Problems as part of the process of mastering a topic. If you can solve Ugly Problems efficiently and confidently, then you know the material well. Furthermore, most of the problems on the test or quiz will probably look and feel easy to you – because in comparison to your experience with Ugly Problems they are… well… simple.