Lesson Five: Rely less on your textbook

One of the most important things I have realized during my math blog is how frequently my math book leads me astray and into dangerous territory. What could Andrew possibly be talking about? I thought math books were perfect!

Example problems!

Initially I thought example problems were key in helping me understand the material. Looking back, I notice how much they have hindered me. In the math book I have been working through, they introduce new topics with a paragraph or two. Then, you are asked to solve a series of basic problems covering the material. These problems are carefully chosen, as each one teaches the student a bit more about the topic being covered. Then more advanced material relating to the topic is introduced through a series of new problems. Instead of the book solving "key" problems for me, I must first try and struggle through them.

On a personal note, I have found it is much more satisfying to understand the implications of a certain theorem or concept on my own than having my book explain it to me. Using this method, the mathematics student is challenged every step of the way, forced to think through unknowns and apply new ideas to problems.

I found this to be much different than my own math homework, where I get to do a few problems practicing X formula/method, which is easily found in the example problems. I got really good at glancing at example problems and being able to think only in terms of these 5-6 key problems per chapter.

An example of the effects of narrowing a student's mindset:

On my chapter five calculus test (optimization, critical points, etc.) there was an optimization problem relating to minimizing the cost to make a cube. This was a standard optimization problem. However Mr. Markey, the sly dog that he is, added cost as an X factor, something the book hadn't done. We were fine at minimizing the meters of fence needed, but once price was introduced my class struggled. Few people got this problem right, even though all that was necessary was multiplying the various lengths (bottom, sides, etc.) by their respective prices before optimizing. This, however, was a twist on the example problem. Sadly, because we had been conditioned to solve problems of an almost identical nature to the example problem in the textbook, this trivial problem became a great challenge.

While just one example, I think it highlights the dangers of relying too much on a book to provide typical problems. I recently flipped through my calculus book (I only do this a couple times a week =P ) and tried to find problems that were more like the problem solving problems I have been doing recently. It turns out the best problems in the entire section were at the end, as these problems were not cut and dry and instead combined ideas from the section with previous sections/chapters. Furhtermore, these problems were not under the header of "Use X theorem" or "Use Y method". This forces students to think critically and rely on all four steps of problem solving more.

Sorry for the rant-like nature of this post, it was just something that I noticed that really frustrated me.

Cliffnotes: Find problems that are related to the topic you are studying/learning about, but yet where the solution is not easily accessible. If you know how to solve the problem within 5-10 seconds, chances are it's too easy for you! Challenge yourself and you will do better in math, I promise!