So I've told my 500's student that getting a 700 on the SAT will be "a lot of work." But what should that work consist of? I have some activities we could do and some problems we could work, but we need to do the activities and work the problems over a long period of time so that she can actually internalize the lessons. Unfortunately, working one-on-one with me for that period of time would be prohibitively expensive for a lot of people. Is there a cheaper alternative? Can someone do it on her own?
I have been using materials from Art of Problem Solving with a handful of elementary school students, and I am struck with the fact that they place more emphasis on the number sense than they do the algorithms. They walk students through steps toward comprehension that focus on the meaning and assume that the algorithm will come on its own. That process works best for the advanced kids, but that is their target market. Still, I thought they might have something useful for the high school student who has learned the algorithms, but would like to retro-actively work on the number sense.
Introducing Alcumus. Alcumus is a free online problem bank. Once you register, it will give you a math problem. Get that right, and you will get a more difficult problem. Complete enough problems in a row, and you will move on to the next topic. It is somewhat similar to the practice modules on Khan academy with a couple of notable differences: First, you will be given an explained solution even if your answer was correct! In fact, to get anything out of this exercise, you need to carefully read every explanation to see if there was a different, more intuitive (as opposed to algorithmic) method of solving the problem. There are a few videos to watch for more instruction, but Art of Problem Solving believes in a problem-first approach. There are also references to chapters in Art of Problem Solving math text books. (The books can be a bit pricey. If you have the means, buy a couple of copies and donate one to your school library.)
Try to see if these methods lead to being able to solve complicated-looking problems in your head. (It goes without saying that you should NOT be using a calculator.) You will be led through addition, subtraction, the distributive property - simple stuff, but there are lessons here for how to think about these problems differently. How to use your head instead of that hand-held machine you have been using as a crutch.
Try it! I'd love to hear how it works out!