[RUME] Blanking on Tests

[Metronym]

  Andy brings up an important topic.  Students can "blank on tests" for two
reasons.  The first is probably the more common - they have watched you do
the problems, they can follow the model to do the homework, they can retain
the model long enough to accomplish a quiz but they do not retain nor do
they know the material.  The second is a more serious reason.  They may
suffer from math anxiety.  I will adress both problems.  The "cure" is
similar in both cases.
  In the case of the first problem, the student simply lacks concentrated
practice.  Often, the student voices, "I understood everything you did in
class, but I just blanked when I did the homework, quiz, etc."  Many of our
students come to college without study skills for mathematics.  High schools
very typically teach on short term topics.  High schools textbooks further
this problem.  It is the it-must-be-Tuesday-because-we-are-doing-logarithms
connundrum.  First of all, students better retain material if it is always
joined with other content.  On the college level, we are not particularly
teaching algebra, logarithms, trigonometry, or whatever in our approach to
Calculus.  However, the student must have those tools polished and at the
ready each day of study.  Placing tools within context often helps that
problem.  Secondly, in some high schools, the student grows accustomed to
learning a skill for the test and then dropping it in favor of the next
skill.  They do not learn how to study comprehensively.  They learn one
trick and then are tested on one trick.
  I recommend that all of my students, approximately 4 days before a
preliminary examination (7 days for a midterm or final) read their notes,
highlighting any detail that does not immediately bore them with its
familiarity.  They are then asked to transcribe each of the highlighted
topics onto a piece of paper with sufficient detail to understand each facet
of concern.  Needless to say, this is the appropriate time for the student
to conference with the instructor if additional understanding is needed.  In
the meantime, they are asked to formulate their own sample test:  10-15
questions (depending on the material) that are culled from their notes,
homework and/or quizzes.  These problems should be the ones that offer them,
individually, the most challenge and/or trouble.  I will return to this
strand of thought after a few comments about math anxiety.
  Math anxiety is a pervasive problem.  There are multiple research papers
that explore this very real difficulty.  Suffice it to say, students who
suffer with this are not learning-disabled but rather, at some point,
subject to poor teaching for their particular learning style.  The typical
story is about the second grade teacher who decides to present an enrichment
activity to her class.  She challenges them with three rows of three dots,
comprising a nine-dot square and asks them if they can connect all nine dots
with four straight line segments without ever lifting their pencil.  The
next day, the children rreturn and one child, let's say Johnny, proudly
announces he can do it.  He is invited to the board where nine dots are
awaiting his efforts.  After successfully connecting the dots, he is praised
in front of his classmates for his cleverness.  In the meantime, Susie, in
the back of the room, raises her hand and says she can connect the dots, but
she did it a different way.  The teacher, perfectly aware there is only one
way for solution (the first problem) invites Susie to the front of the room
for failure in front of her classmates.  The teacher redraws the nine dots
on the board, but Susie says she cannot do it on the board and aks for the
paper easel to demonstrate her solution.  Faced with the nine dots now
available for her on the paper, she folds the paper, and carefully connects
the dots.  The teacher says that Susie is wrong.  Susie objects, stating
that the teacher never said "you can't fold the paper."  Susie returns to
her seat, having learned a very valuable lesson.  Stifle creativity and wait
for the "right" answer.  Otherwise, you can end up humiliated in front of
your friends.
  If you pair that sort of experience with one similar to my following
observation, you observe math anxiety in the student.  During a two-year
hiatus from my graduate work, I taught in a high school.  A new hire
appeared on the scene, famous for her discipline and keeping students after
school.  One afternoon, needing to use the computer that was available in
the back of the classroom, I observed this new hire lecturing her
imprisoned-after-hours students.  she wrote a third degree polynomial on the
board, setting it to zero.  She moved the constant to the other side of the
equation, factored an 'x' out of the remaining terms, factored the resulting
quadratic and then set each of those factors equal to zero.  One of the
prisoners was awake enough to query how zero could equal that isolated
constant on the other side of the equation.  At this point, I expected her
to erase her work, apologize (or mutter that covering phrase "I just wanted
to see if you were paying attention...") and correct her work.  She did not.
After erasing the final answer, but leaving the entire development on the
board, she said, "Well, those answers are wrong but it is your job to find a
number such that (sic) 'x' times 'x + 5' times 'x ­ 2' equals 7."  I sat in
the back of the room thinking, "Even God could not do that.  That's the
reason we have the rational root theorem."  Two days later, her class of 29
pre-calculus students all failed her exam.  And she had just mounted a
horrendous achievement.  Left in her wake were 29 high school juniors and
seniors who honestly believed that they could not do math - that it made no
sense.
  Query those students that you may suspect are suffering from math anxiety.
In my own experience, every single one of them will tell you of the time
that math no longer was rewarding, logical or sensible.  Convinced they
cannot succeed, they suffer terrible anxiety under testing situations even
in a new venue, with excellent instruction.
  As I mentioned, in both cases, ask them to go through their notes,
codifying all topics of interest for study as well as forming a sample test
based on the problems they find the most challenging.  Each day, prior to
the exam, the list of topics should shrink in size but not in number, until
the night before the test, they have a small notecard that brings immediate
recall of each area.  In the  meantime, they should take their sample test
at least twice daily.  For those students with math anxiety, they should
take the test five times in a row the night before the exam.  They will hate
doing this, because it seems like busy work, but the repetition will imbue
them with a rhythm, a drumbeat of performance, that will tide them over when
faced with the actual exam.
  In addition, I warn them that if, during the examination procedure, they
start to feel that "bubble" of anxiety rising in their throats, they are to
put their pencil down, look at the clock or some other inanimate object,
take a deep chest breath or two and relax their shoulders and neck before
resuming the test. 
  It is a shame that these students are paralyzed to act during an
examination due to past poor instruction, but we must deal with our
students, one at a time, as they come to us to effect meaningful learning.
  For those students who simply do not know how to study, the techniques
mentioned will give them a structure for studying and improve their
performance.
  Some students are so badly injured that they may need to be reminded often
about these techniques with requisite reassurance in order to improve their
performance.  Those students who do not need this attention are not
necessarily brighter or more capable.  They simply have the good luck of
past instruction that was appropriate and meaningful for them.  Let us know
how your students fare!

Regards,
Alden
Alden L. Monberg, PhD
Applied Mathematics and Coastal Processes
Maine Maritime Academy
amonberg at mma.edu
http://www.monbergmath.com