A big problem that students run into,
especially in the more difficult classes, is test questions that go beyond the homework and require quick, creative thinking
under pressure. Over the years I've gotten pretty good at anticipating what the test questions might be for a given course.
Once I've made certain that a student is up to speed with the homework, we start working on likely test questions. Many students
have reported to me that this has been a life saver.
In many classes, key concepts are presented too quickly, too abstractly
or without enough explanation (with the teacher sometimes just repeating what's in the book). Sometimes the book isn't much
help either, since it will provide examples of how to solve the easier problems, but give no examples of how to approach the
harder ones. I have found that if I go over the concepts before they come up in the classroom, it allows the student to better
absorb the material as it is presented by the teacher, and relieves a common source of anxiety.
One of the complaints I hear most often from kids goes something
like this: "I understand the material. I know I do, because I can do the homework. The thing is, I do really badly on the
tests." The kid, and often the parents as well, conclude from this that the problem must have something to do with test taking
skills. In my experience, this little problem is hardly ever due to a lack of test taking skills. In most cases, it really does have to do with not understanding the material very well.
If a student does not understand the underlying ideas in math,
that student often resorts to rote memorization as an alternative method for getting through math classes. This "memorization"
approach can more or less work all the way from the first grade up through pre-algebra.
As far as math goes, there simply aren't that many things to either learn or memorize during those years. So if the student
finds the concepts hard to grasp, they figure out early on that good grades can still be obtained by resorting to rote memorization.
In fact, when such a student says that they "understand" the math homework, what they often mean is that they have sufficiently
memorized the necessary formulas and procedures.
The trouble usually begins around the time the student gets to
algebra, because at that point, the number of things that one must memorize in order to survive the class, begins to increase
dramatically. If the student, ever since the first grade, has been mostly relying on their basic understanding of the concepts,
the switch to the more rapid pace and the more sophisticated ideas encountered in high school, is a graceful one. However,
if the student has long since abandoned the attempt to understand concepts, and has been using rote learning as their basic
math "tool," they will find that that tool is no longer equal to the task; there are simply too many things to memorize. This
basic problem first begins to show up on the tests, not on the homework. Why? Because as the student does his or her homework,
they have access to the book, so if they don't remember the correct procedure for solving a certain kind of problem, they
can just follow the steps as outlined in the book (or in their own class notes). This doesn't require much understanding;
all you have to be able to do is follow steps. However, on the tests, without
the book to prompt them, some students have only a very fuzzy idea as to which procedure should be used when. They also might
not perfectly remember the procedures themselves, to say nothing of having no understanding of why they work at all.
The real problem with taking the approach of rote memorization
of words, formulas and procedures, is that in so doing, you are not really learning math at all. I am not saying that one
never has to memorize stuff in a math class, it is just that rote memorization shouldn't be your main tool as you approach
the material. The main tool needs to be understanding the ideas, seeing how they fit together. It is somewhat like the difference
between memorizing a poem and actually understanding it. It might be a good thing to memorize the poem, but that is only the
beginning. Imagine a student taking a literature class who simply memorizes a couple of poems, and thinks that this is going
to be an adequate preparation for the upcoming test. Our poor student might be
in for a rude shock when the test questions all turn out to be something along the lines of "compare and contrast these two
poems in terms of what they are saying about the role of the family in modern society." If the student had only memorized
the poems, but given no thought to their meaning, he or she might have no idea how to answer such a question. A somewhat similar
dynamic can exist for students taking a math test. I sometimes find that students do not understand what the test questions
are even asking for.
The problem I have described above does not have a quick or easy
solution. Or maybe I should say that if it does, I don't have it. When I work with a student who comes to me believing that he or she simply needs to be taught some "test taking skills," I know that I am faced with
a rather daunting task: I have to convince the student to try to reorient their entire approach to math. Basically, I have
to convince them to think more, and memorize less. Almost always, this takes
time and considerable effort, but I can honestly say that I usually get good results, provided the student is at least somewhat
willing to make this kind of change.
Finally, many students have reported to me over the years that
their teachers just don't seem to "get it" that the basic common-sense logic behind some of the complicated-sounding ideas
or theorems in math can seem extremely non-obvious to somebody seeing them for the first time. As one kid described her math
teacher's attitude: "Maybe it was never hard for her, or she just learned this
stuff so long ago that she doesn't remember what it was like to not understand it."
One advantage I have as a teacher, is that while I get a big kick out of math, it has not always been easy for me,
and I definitely do remember what it was like to not understand things that seem to me today to be beyond obvious.
Please feel free contact me with any questions sarahmathtutor@gmail.com or call me at (925) 930-9356.
