Gaining Mathematical Maturity Today I would like to talk about mathematical maturity and how it relates to your performance on standardized tests such as the SAT, ACT, and GRE. Although there is no single agreed upon definition of mathematical maturity, I would define it as “one’s ability to analyze, understand, and communicate mathematics.” There is no strict measure of mathematical maturity, but let’s examine what various levels of mathematical maturity might look like: Level 1: These students can do basic high school math when taught in a classroom setting. Students at level 1 can solve basic algebra and geometry problems by following the procedures that were taught to them. If such a student were asked, “Why does this work?” they will not be able to verbalize a satisfactory answer. Most high school students are at this level. Level 2: These students can learn basic mathematics without any outside help. Students at level 2 can read an algebra or precalculus book on their own and can probably explain basic mathematics to other students. If shown how to do a problem, they will be able to solve problems that are similar using the same strategy. These students would have no problem in a Calculus course. Level 3: These students can learn advanced mathematics on their own. Students at level 3 can not only do problems, but can also explain why the methods they are using work. They can often solve problems in several different ways and develop their own techniques for solving problems. These students will be able to solve problems of a type that they have never seen before as long as they know the appropriate definitions and relevant background material. These “levels” are just here to illustrate a point. It is often hard to pinpoint the exact level of a student. Some students naturally have a higher level of mathematical maturity than others. For example, students taking AP Calculus usually have a higher level of mathematical maturity than a student in remedial algebra. As such, the AP Calc student is more likely to do better in math on the standardized tests. Simply knowing more mathematics is not enough to guarantee a higher level of mathematical maturity. An AP Calculus student does not have a higher level of mathematical maturity simply because they know more math. It is because they have been attempting problems that require a higher level of thinking. The good news is that mathematical maturity can be increased. Doing so will greatly improve your problem solving ability as well as increase your math score on the SAT. Here are a few ways you can increase your level of mathematical maturity. Do SAT/ACT/GRE math problems. If you can currently only solve Level 1 geometry problems, and a month from now you can solve Level 2 geometry problems, then guess what—you have increased your level of mathematical maturity. So keep doing SAT (or ACT, etc.) problems a level above your current ability until the day of the test. If you have only about 3 months before your test, then this is the method you should employ most. Learn some mathematics by yourself. This may seem contrary to what has already been stated—but it is not. Knowing more mathematics will probably not raise your score very much, especially if the math is hand fed to you. But the process of learning mathematics will increase your mathematical maturity. What kind of math should you learn? It does not matter! Pick something you find the most enjoyable, and try to learn it on your own. If you get stuck, then ask for help—but developing the skill to learn math yourself will definitely increase your mathematical maturity. This strategy is more long term and you should consider it if you have 6 months or more before your standardized test. If the idea of learning advanced math on your own seems scary to you, you may want to check out either Pure Mathematics for Beginners or Set Theory for Beginners. These books are perfect for anyone just starting out in theoretical math. Attempt difficult math problems. There are a lot of challenging math problems out there that only require the math that you already know. There are math magazines (yes, I know these sound like a lot of fun!), websites, and books full of problems. Pick a few and struggle with them. Just trying difficult problems and developing strategies will increase your mathematical maturity, even if you never solve a single one on your own. Also, solving problems using the advanced methods that are described in the many books I have written will boost your level of mathematical maturity. I can still remember the day I got back my first real analysis homework as a first year graduate student at Rutgers University. I received a 62. This is by far the lowest grade I had ever received in mathematics. The worst part was that I had spent many hours on this homework, and I thought that it was completely correct. I remember approaching my teacher explaining that I never received such a low math grade, begging for him to tell me what to do. But he did not have an answer for me. I was on my own. That whole first year of graduate school was a struggle. I wound up with B’s in most of my classes despite working harder than I had ever worked in my life. I was unprepared for how difficult graduate school was, especially after receiving a 4.0 Math GPA as an undergraduate with very little effort. This was quite disheartening and I thought that there was no way I would be able to do mathematical research and get my Ph.D. I just was not smart enough. But an interesting thing happened during my second year. Apparently my hard work paid off. All of a sudden nothing seemed difficult anymore — I was back at the top of my class. Despite the fact that I was taking even harder classes than the previous year I was breezing through them with very little effort. All of a sudden I was able to read advanced graduate texts on my own and do the problems at the end of each section without any help from others. What happened? Did I all of a sudden accumulate a massive amount of mathematical knowledge. The answer is no. In fact, this would not have helped. The mathematical language in my new classes was actually quite different from what I had learned the previous year. What happened is that by struggling so hard the year before with little to no help from others, I managed to increase my level of mathematical maturity. Things that I did not stand a chance of understanding just a year earlier now made perfect sense. Other students were now coming to me for help, and I was able to help every single one of them. To think just a year earlier I was considering dropping out because it was just too hard. What a shame that would have been. My level of mathematical maturity had to take several more jumps before I would graduate. Mathematical research is highly sophisticated and requires a clever mind that is willing to think in unconventional ways. The one thing that my years at graduate school have taught me is that anyone is capable of doing anything that they set their mind to. Just don’t give up and you will persevere! P.S. If you are preparing for a standardized test, you might want to take a look at the Get 800 collection of test prep books. Comments comments