Mathematical Maturity and its Applications
to Standardized Test Preparation

What is mathematical maturity?

Before defining mathematical maturity, I should probably mention that there is no single universally accepted definition. If you ask two different mathematicians, “what is mathematical maturity?” you will probably get two different answers. One way that I like to define mathematical maturity is as the ability to analyze, understand, and communicate mathematics.

When I tell someone who I have just met that I am a mathematician, I will often get the response “I’m not a math person.” My personal belief is that there is no such thing as “math people” and “non-math people.” Each of us is just walking around with a different level of mathematical maturity. These “non-math” people simply have not had enough of the right kind of exposure to mathematics.

How can you determine your “level” of mathematical maturity?

Just like there is no single definition of mathematical maturity, there are not really well defined “levels” of mathematical maturity. You’ll never hear someone say “I have a level 2.7 mathematical maturity. What’s yours?” However, we can still describe what certain levels might look like.

For example, you have your typical high school student who can plug numbers into formulas and follow procedures to solve basic problems in subjects like algebra and geometry. Many of these students would not be able to explain where these formulas come from or why various procedures work to give you the correct answer.

A student at a slightly higher level might be able to learn some math on his or her own without the help of a teacher or tutor. They might also be able to apply techniques they have learned to problems somewhat similar to ones they have already seen.

Then you have students that can solve problems that they have not seen before by making connections to mathematics that they have already seen. These students can often solve problems in several different ways and can explain their methods to other students in an effective manner.

Is it possible to improve mathematical maturity?

It certainly is! No matter what level someone is at, it is always possible to improve that level further. I have improved my own level of mathematical maturity many times throughout my life. My most significant experience with mathematical maturity occurred when I began graduate school.

I remember handing in my very first homework assignment in graduate school. It was for my Real Analysis course. I was confident that I got everything right. Imagine my shock when I got my homework back with a grade of 62 out of 100. Not only did I get many of the answers wrong, but I didn’t even have the mathematical maturity necessary to realize that I didn’t understand what I was doing.

Throughout the rest of that first year of graduate school, I worked harder than I had ever worked in my life. Nonetheless, I could not get higher than a B in any class. As a student who received only A’s in my math classes as an undergraduate, this was devastating to me. I was actually thinking about dropping out of the PhD program.

Instead, I worked really hard the next summer to fill in all my gaps in knowledge and to work on really difficult math problems. I also sought out the smartest students in my classes to work with. What I lacked in knowledge and experience, I made up for with hard work.

Going into my second year of graduate school, I was terrified. I was about to take classes that were even more difficult than any class I had taken the previous year. However, something amazing happened. I found the classes to be very easy. I was now back to getting perfect scores on all my assignments and tests. Instead of asking other students for help, students were coming to me for help. Not only did I understand all the material, but I was able to explain it to others. Moreover, these classes were completely different from the classes I took the first year. This was when I realized that mathematical maturity was a real thing and more importantly, that it was possible to improve it.

How does mathematical maturity relate to standardized test scores?

For this section, my explanations will be in terms of the SAT, but anything I explain applies to just about any standardized test.

I once believed that every student had a maximum potential SAT math score that he or she could attain. For example, maybe a student is currently scoring 400 and with the right preparation, they can get up to around a 600. Therefore, they practice easy to medium difficulty questions and over a period of a few months, they work their way up to a 600.

What I have since realized is that mathematical maturity is the key to getting that “maximum” score even higher. In other words, a student’s “maximum potential” can be improved by increasing their level of mathematical maturity.

Just to clarify, an increased level of mathematical maturity will increase only the potential SAT score, not the actual score. To increase the actual score, one needs to prepare specifically for the SAT.

For example, let us take two students with current SAT scores of 400 and a maximum potential of 600 (we can pretend that these students are clones of each other – identical in every way). Suppose that one student spends a summer learning some Abstract Algebra, while the other does not do any math during that summer.

Now, suppose that both students begin preparing for the SAT at the same time for about three months. If they both attain their potential, the student that did nothing will get a 600. The student that studied Abstract Algebra, however, might get a 700 or even an 800. His maximum potential increased over the summer because he was working on improving his level of mathematical maturity.

What are some specific actions someone can take to increase their level of mathematical maturity?

Before providing specific methods, I think that it is important that I stress the fact that mathematical maturity cannot be improved overnight. The process of increasing one’s level of mathematical maturity requires patience and hard work. That said, here are three of my favorite methods:

• Learn some real mathematics. Pick up an advanced math book in a subject you are unfamiliar with and just start reading the chapters and working on the problems at the end of each section. Maybe try an Abstract Algebra book or book in Set Theory. I have actually written several pure math books on various topics designed to ease you in to the world of pure mathematics, while increasing your level of mathematical maturity (you can click on each image below to take you to the book’s Amazon page)

 

 

 

• Work on some challenging math problems in the subjects you already know. Look for problems from math competitions such as the Mathcounts exam (at a lower level) or the Putnam exam (at a higher level). You can also find challenging math problems in math magazines and other publications. Many such problems require only the most basic mathematics, and yet, they are still quite difficult. Struggle with some of these problems, even if you cannot get a single solution. The struggle will make you mathematically stronger.

• When preparing for a standardized test, go a bit “deeper.” In other words, instead of just answering a question, think a little more about what the question is asking. Try to solve each problem in more than one way. Think about why someone would ask this question. How does this question relate to other questions? Try to make connections between seemingly unrelated questions. As one specific example, questions about arithmetic sequences are often quite similar to questions involving linear equations (see the following post for details: https://satprepget800.com/2016/06/01/arithmetic-sequences-linear-equations-sat-act-prep/ )

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