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Math Study Skills

I came across this excellent article about Math Study Skills.  The information is very useful.


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Tips for Overcoming Math Test Anxiety

Math test anxiety can be described as overwhelming, anxious feelings when preparing for and taking math tests.  Students who are affected by math test anxiety “freeze up” the moment they sit down to take a math test, even though they know and understand the material very well.  Many of my students come to me with concerns about taking math tests (a necessary part of the college education process).  I usually ask a few questions about why they feel anxious, their study habits, their eating habits, and their feelings about math in general.  This helps me to understand why they feel anxious about taking math tests.  Sometimes I get some understanding of the root of the anxiety, but most of the time, it’s just math test anxiety.  I give my students the usual spiel and send them on their way.  Most of the students who take my advice come back later and thank me.

Below I listed some of the tips I share with my students.  They are very basic, but helpful.  I hope you, your child, or student can benefit from these tips.  Please feel free to share other tips, as I’m sure I missed some!!!

1.      Acknowledge you are affected by math test anxiety

The first step in getting help in any area of life is admitting you need help.  You are probably affected by math test anxiety if you get excellent grades on homework assignments, understand the teacher, take great notes, study effectively for class, but still perform poorly on math tests.  Some symptoms may include sweaty palms, a racing heartbeat, uncontrollable shaking, etc.  GET HELP!!  Read about test anxiety, in general, and apply the tips to your math test anxiety.

2.      Write about your math test anxiety

I read this great article about writing down your concerns just before taking a test.  I took this information and expanded it a little further.  I advise my students to write about their math test anxiety the moment they come to me with their concerns.  I usually ask them how long they’ve noticed the anxiety and what they do to ease the anxiety.  Then I suggest they write about their feelings about math tests just to get it out in the open so they can move forward.  My students usually express relief after completing this exercise.

3.      Share your feelings with your teacher or professor, they may be able to help ease your concerns

It’s important to share these concerns with your teacher.  Most teachers are aware of test anxiety and have tips for their students to prepare for tests mentally, so they are not overwhelmed with anxiety.  Even if your teacher is not helpful, you should feel some relief by merely discussing your concerns.

4.      Become familiar and comfortable with the test site (classroom, lecture hall, etc.)

Most students study in their living room, bedroom, dining room, or some other place of comfort.  This makes it easier to relax and learn.  However, when test time comes, students are out of their comfort zone and have difficulty relaxing.  This allows test anxiety to set in.  It is best to study in the room where the test will be administered.  This room should become your comfort zone.  You should read your notes, complete assignments, and study for tests in the same room (or a very similar room) where the test will take place.  You will be comfortable with your surroundings and you will be able to relax during the test.

5.      Relax before taking a math test

Avoid the urge to study up to the minute before the test.  If you study effectively, you will learn the material and it will come back to you when you take the test.  You need a few moments to wind down and relax before you take your test.  When possible, use the night before or morning of the test for light review (formulas, examples, etc.).  You must allow yourself time to relax and recover from all the studying over the days leading up to the test.  You also need time to process material you’ve worked so hard to learn and retain.

6.      Write formulas on the test before you begin

When you sit down to take your test, take a few minutes to write down formulas, examples, equations, and other study aids before you begin your test.  Most students who are affected by test anxiety tend to forget most of what they learned the moment they begin to work on the first problem.  My students often refer to this as a “brain freeze” or “black out.”  To avoid having a moment of forgetfulness, write out everything you need to know to perform well on your test before you begin working through the problems.  This will alleviate some anxiety during the test.

These tips may be helpful for most subjects, but I know they are effective for overcoming math test anxiety.  I hope they will be useful for you!!!

What other tips do you have for students who are affected by test anxiety?


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The Teacher is Nervous

It’s that time again.  That time when I’m preparing for the 30 or so students eager to get it over with: my class, precalculus.  Every semester for the past two years, I say this is my last semester teaching and I say it with conviction.  It’s not the students, the administration, the workload, or the working conditions.  It’s that nervous feeling, anxiety if you will.  Yes, I am a college instructor and I get nervous and experience intense anxiety on the first day of class, every semester.

My heart races, my palms get sweaty, my voice quivers…  I always give the credit to some random epiphany: this isn’t for me, it was only temporary, I should be doing something else, and whatever other excuses I can conjure up to justify wanting to quit so I don’t have to have a quasi-anxiety attack.

Then I go into the first class, take a few deep breaths, and begin to speak.  Yes, my voice still quivers for about 3 – 5 minutes.  After a while, I calm down, get into the lesson, and go into my “zone.”  Last semester during office hours one of my students told me I go into a zone when I’m teaching.  She said it’s interesting to watch, because I look as though no one else is in the room.  I know that feeling and I can imagine that look.  I feel it when it happens.

While I’m teaching, I become absorbed as new ways of teaching the same material emerges out of nowhere.  Every now and then I pull back, look around the room, and ask if there are any questions.  I can feel the intensity of their stares, so I smile, realizing they are more nervous than I am.  I remember that I am the instructor, they are the students.  I must be calm and confident or they will retreat.  I search my memory for a story that will make me seem human again.  Something always surfaces.  I share the story, they lighten up, and I go back to teaching.

It’s a little after 6 am and I’ve been awake since about 5 am.  I’m, probably anxious about the 30 or so students who will sit in my class next week waiting for that “aha moment” that happens later in the semester.  Some will sit there hoping I will take it easy on them, but of course I can’t.  It’s summer session and we only have 6 weeks to cram 14 weeks of material into their minds.  But they asked for it so here I go.

Writing this blog actually helped me relax a little, but I know this is temporary.  In about a week, I will be nervous, anxious, and ready to declare that this will be my last semester teaching, again.  Then this process will start over again 7 weeks later.


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Preparing for Calculus

Calculus is a specified field of mathematics.  Limits, derivatives, integrals, and infinite series are all applied to various types of functions (i.e., linear, quadratic, exponential, trigonometric, etc.).  They each depend on the understanding of basic arithmetic as well as specific concepts of other branches of secondary level mathematics such as algebra, geometry, and trigonometry.  Prior to taking a calculus course students should develop specific skills in each of these areas of study.

Below, I briefly discuss key skills and concepts required from three branches of mathematics (algebra, geometry, and trigonometry) to prepare students for calculus.  I discuss this material as it relates to differentiation, a fundamental component of calculus.

Algebra (Functions)

Differentiation converts formulas into other formulas.  These formulas are functions.  Algebra is the study of functions.  Since functions are the object of differentiation, it is imperative that students entering into a course of study in calculus have a thorough understanding of functions.

Many secondary students can follow instructions to substitute a value into an equation, but the study of functions will provide a greater understanding of why the method of substitution works and why a relationship exists between the parameters.

Geometry (Graphs)

The result of differentiation is a derivative.  Derivatives are used to calculate the slope of a line tangent to a point on the graph of a function.  A student can find an estimated value using the slope of a line secant to the same graph, but calculus makes it possible to find the exact value.  Students must have a thorough understanding of graphs, slopes, and functions to understand the use of derivatives with respect to graphs of functions.

Trigonometry (Angles, Periods of revolution)

Trigonometry is the study of trigonometric functions.  Trigonometric functions are used to calculate measures involving triangles, periods of revolution, etc.  During the course of a calculus class students will encounter problems involving trigonometric functions which are useful for explaining periodicity.  Students studying calculus should understand trigonometric functions, their properties, and their graphs.

As discussed above, students preparing to take calculus should have a specific set of skills relating to calculus content.  The major subjects required for the development of these skills, and the understanding of the related concepts, are algebra, geometry, and trigonometry.  More specifically, students need to have a thorough understanding of functions, graphs, slopes, and trigonometric functions, to name a few, in order to effectively prepare to enter into a calculus course.

Although students have the benefit of the use of technology to solve many problems, they must have an understanding of the relationship between the functions used, their graphs, and the derivative so they can understand the output generated from the use of technology.  Simply plugging values into a technological tool does not provide this level of understanding.  In most cases, students will not have access to technological tools in college level math courses (see “Calculators Not Allowed” on this blog site), so it’s best to learn and understand the concepts without these aids.

Do you have any other ideas about concepts students should learn prior to taking a calculus course?


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COMPLEX NUMBERS …in brief

I don’t remember exactly when I learned about imaginary/complex numbers, but I do remember feeling excited about this “new” math; well, it was new to me.  I easily grasped the idea of complex numbers, but my classmates struggled with their meaning and purpose.  For me, it was easy; I treated i like a variable when performing operations on the complex numbers.  However, I understood very well the value of i as the square root of negative one.  I didn’t understand why it was so difficult for others to comprehend complex numbers.  Even now, some of my students do not “get it.” 

When approaching the topic, I often ask my students to calculate the square root of negative one.  They look at me puzzled and say “one” or “negative one.”  I then ask “what is one squared” or “negative one squared” and they say “one.”  I repeat this cycle of questioning until they finally ask “what is the square root of negative one?”  Ah-ha!!  Now we’re getting somewhere.  This is usually how I introduce my students to complex numbers.  It never fails; my students almost always say “that’s stupid” or “what’s the point?”  After I allow them to vent their frustrations, I explain that i is short for “imaginary” (the name given by René Descartes in 1637) and represents the value of the square root of negative one.  I tell them the letter was assigned by a mathematician since the exact value is unknown; that i simply represents the value of the square root of negative one.  They buy it (well some of them), open their minds to it, and learn enough to get through their lessons.  In the meantime, I sigh with relief that they don’t ask additional questions. 

For years, I’d wanted  research i and complex numbers so I could give a more accurate account of their beginnings, history, and existence.  Finally, the day came that I actually learned a brief history of these obscure numbers known as “complex numbers,” a term given by Carl Friedrich Gauss in 1831.  I’ll spare you the details.

As with many topics in mathematics, complex numbers became useful, and necessary.    Therefore, it was important for mathematicians to carefully analyze, define, introduce, and explain complex numbers.  This did not happen without opposition.  Some mathematicians argued against the usefulness of complex numbers.  Even Leonhard Euler did not care for the idea of these “impossible” numbers, but realized their necessity.  In spite of the arguments stacked against complex numbers, the quest to validate them did not end.  Many mathematicians have worked with complex numbers and found uses for them. 

Delving any deeper into the world of complex numbers would require extensive research into other fields, which time does not allow.  Maybe one day I will have the resources to discover all the truly amazing characteristics of this number system.  As with many mathematical concepts, it took many centuries and multiple mathematicians to finally come up with yet another set of numbers.  I can’t help but wonder if there are any other number systems that have not yet surfaced.

NOTE:  Some of the information shared here was inspired by “Math through the Ages: A Gentle History for Teachers and Others,” written by William P. Berlinghoff and Fernando Q. Gouvêa.  A must read for Math Educators!