October 2, 2015

First-Year Mathematics: More Methods, Less Madness

Just in time for your looming first math exams, here are some expert tips from math professors and graduate students on how to approach your work and improve your understanding of math:

Making Connections, Moving Past Procedures.  Let's face it: a monkey can plug numbers into a calculator; that's not your goal. You want to understand what you're doing.

The biggest difficulty is that students are often much more focused on a "plug and chug" learning that demonstrates they can put numbers into a formula and calculate an answer. But, hitting the right buttons on the calculator does NOT mean that you fully understand the concept and when/how it is used. That type of learning can be useful for quizzes, but does not prepare you for the conceptual understanding and applying concepts that is required on exams.

First-year Rutgers students have the advantage of taking courses with highly-esteemed mathematicians knowledgeable about the applications of content as well as the logic behind mathematical algorithms and procedures.  A conceptual understanding of mathematics allows one to make sense of the WHY's and HOW's of mathematical procedures for a more interconnected knowledge base of the discipline.  

Effective studying does require some advanced planning. Many mathematics instructors will include a list of suggested and required homework problems to complete in their syllabi. Take a look at the amount and complexity of homework problems that are assigned for the course. Use this list to gauge your competency with the material PRIOR to lecture and recitations so you will be ready to pose your most pressing struggles with the material. 

PRE-READING the section before attending the lecture helps you to follow along and will either give you more confidence or will begin to signal areas of difficulty.  Next, complete as many problems from a particular section as you can before preparing for the next section.  Keep note of the questions that you can't answer on your own and try to look them up online or in the solutions manual. 
EXAM REVIEW is essential to patching any holes in your understanding before moving forward. Review the exam and all problems that you got wrong; meet with your TA to review any errors that you don’t fully understand. The only thing worse than losing points for a mistake on the first exam is losing points on subsequent exams for the same mistake!

Getting Help. Once you’ve made a real effort,  go to office hours! If you're still not getting it, you will be able to ask questions about specific problems or ideas, and get more effective help from the TA. Even if you feel confident in your approach, office hours are a terrific opportunity to hear other students’ questions and deepen your understanding of mathematical content beyond what can be addressed in an 80-minute class period. Meet with the TA, professor, and/or tutor at the campus learning center prior to an examination (and throughout the semester).  Remember that tutoring is not just to learn concepts that you don't understand; it's also for really mastering material and seeing connections between concepts. It's about gaining conceptual – versus rote – knowledge.

For exam preparations, find out the structure of the exam in advance.  Is it proof-based? Calculator, no calculator, or half and half? These are valid questions. "What's on the exam?" is not. Practice problems under actual exam conditions.  Also, know when and where your exams are! Some exams are given in the usual classroom and at the usual class time. And others, generally called common-hour exams, are not in the usual time and place, so you need to pay particular attention to those details. After all, you can only be successful at a test that you actually take.

Additionally, the mathematics department posts samples of past examinations. Search the "CourseMaterials" section of the Math website to find review sheets and sample exams. YouTube or Khan Academy can be used for additional support, particularly in peer study groups. These sites may provide a new way of "seeing" a problem or concept with which you're struggling in understanding.

Keep in mind that math is a truly cumulative subject. With strong understanding of how basic concepts are interconnected, you can build higher levels and form a conceptual understanding of mathematics, but with a weak foundation, you'll struggle in your work as you move higher. 

Thanks to John Kerrigan, Alice Seneres, and Luis Leyva for the words of wisdom!