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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:

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.

Pure Mathematics for Beginners

Set Theory for Beginners
 

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.

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