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The Math Debate:
To Drill It or To Explore It?
by Bill Lauritzen
Drilling math, proponents claim, is topdown, "backtobasics,"
nononsense.
Drilling math might best be characterized by these words: drill,
abstraction, memorize, practice, review, instruct, teach, indoctrinate,
exercise, discipline, onlyonerightway, by rote, by lecture,
read about, by procedure, listen.
In this math, in its extreme form, the students work individually,
and are spoonfed formulas, rules, theorems, procedures, etc.,
to memorize, drill, drill, memorize, drill, drill, drill (all
completely disconnected from the real world), until they are stuffed
completely full, so that they can then simply regurgitate this
information on national tests such as the SATs, so, theoretically,
that they can go to the collegeoftheirchoice, and will thus
not end up as homeless bums or bagladies.
I have received this method and I have taught this method. I received
it in elementary school, middle school, and high school, while
the US was trying desperately to catch up with the "Sputnik
science" of the Russians. I scored in the 700s on the math
portion of the SATs, and went to the collegeofmyparentschoice.
I also received more nononsense math while I was a cadet at the
US Air Force Academy, in Colorado Springs, where I graduated near
the top 1% of my class. I went to Purdue, another engineering
school, and took more math courses in the same way, getting straight
"A"s for my master's degree.
At the end of all this drilling and memorizing, not only did I
hate math, but I hated school, and never wanted to spend another
day in a classroom.
Only when I turned 30, did I discover that I could explore math,
that I could invent and discover things on my own in math,
and this discovery delighted me and has stayed with me. (Recently,
I was the annual guest speaker at the Dozenal Society in New York.
Also recently, a school in Russia had a Versatile Number Day in
which they studied my publication on these numbers.) So when I
decided to study education, by working in it as a substitute
teacher, as opposed to reading about it in a university,
I chose to teach math.
I "drilled" the students on math when I first started
substitute teaching. However, I knew this could not be the best
method, and I was always trying to come up with better methods.
(When I subbed at Garfield High for two months in 1982, I sometimes
went by Jamie Escalante's class to overhear what he was saying.)
The drillers claim that the explorers are fuzzy thinkers (not
to be confused with the book, Fuzzy Logic), that simply
confuse kids, and lower test scores. Drillers also claim that
explorers neglect to teach fundamentals, especially by allowing
calculators into the classrooms.
Where I currently substitute here in Glendale, I sometimes see
classes who have been overdrilled on math. I look into their
glassy eyes, and there is no spark of curiosity. Sometimes there
is a silent resentment. Oh, they are usually quite competent
at solving the problems, as long as I don't change anything
from the way it was presented to them in the text.
Drilling math might raise test scores artificially, like steroids
can give strength, but what about the side effects? Like hating
math. (Despite what they say to please their parents and teachers;
I'm talking about what they tell each other.)
The "explorers" claim that students can be taught to
explore math in meaningful situations, so that they can then come
up with the rules governing it on their own. The National Council
of Mathematics Teachers (NCMT) endorses this approach. (Its standards
are now available online.)
This method might best be characterized by these words: experiment,
understand, develop, realism, guide, model, question, discover,
focused play, manyrightways, by grasping, by touch and manipulation,
by invention, by guess and check, by experimentation, by application.
Explorers claim that drillers practice "drill and kill"
math. Probably because it kills the spirit of exploration and
creativity that is natural in every Homo sapian.
Let me give an example of how each side might teach one simple
fact.
Fact: The three angles of a triangle add up to 180 degrees.
To Drill Math: the instructor might simply write this on
the board, and then point out where in the book it also says this.
Then he would do an example problem. He would sketch a triangle
(with say, 90 degrees, 30 degrees, and x degrees) and show how
you can get the missing angle, by adding 90 and 30, and subtracting
the result from 180, to get 60 degrees for the unknown angle.
Then he would have the students do about 20 or 30 example problems
from the book like this. Later, they would be tested.
To Explore Math: the instructor might first introduce a
broad problem from the real world in which angles and triangles
are used, depending on the level of the class, such as finding
the height of a tree. He would give every student a protractor
(a device for measuring an angle; something that was never given to me throughout my entire education from Kindergarten through
master's degree.)After the students knew how to comfortably use the protractors to measure and draw angles (something that
could take several days), the instructor might ask them
to draw three different triangles on construction paper. Then
he would have them measure the angles, and add them for each triangle.
They might do all this in groups of four, though this is not necessary
and, as in the real world, they might help each other out when
they were having difficulty.
They might get 179 for one triangle and 181 for another and 180
for another.
The students would be allowed to use calculators in this approach,
for this takes the drudgery out of the math in more lengthy computations,
and allows more time for explorations such as this. Calculators
can never replace understanding, however, calculators are what
people use in the workplace (probably even by those who would
tell us that they should be removed from the classroom).
The instructor would ask the students if there was a pattern?
Yes, they might answer, all the angles of the triangles add up
to around 180 degrees. They might be then asked to formally state
this as a rule.
If I were teaching this class, I might ask the students to try
to draw a triangle with 120 degrees or one with 200 degrees. Of
course, this is impossible, but instructive. I might also have
the students discuss whether it were ever possible to have a triangle
in the real world of exactly 180 degrees. (The answer is
no. A fact that drillers might never find out.) Then each group
might present their results to the rest of the class. The work
of each student might be kept in a portfolio for grading. All
this would be with the real world view of eventually figuring
out the height of the tree.
Sometimes the exploration might be "integrated," or
combined, with another subject like science, English, or even
history. This is how you find math in the real world. Mixed with
other things.
These explorers here are learning about the world around them
in a concrete manner. They look at it, measure it, look for
patterns in it, describe those patterns, and apply those patterns
in a new situation. How is that for fundamentals? You
can't get more backtobasic.
They are allowed to think and see for themselves.
There is nothing "fuzzy" about this approach. It is
rock bottom, downtoearth, rigorous, demanding, and challenging.
Any discussion by the students should be strictly about math.
Principles and fundamentals are uncovered, discussed, formalized,
and applied.
I have seen several teachers apply this program well, and one teacher extremely well. When it's applied correctly, it can be a remarkable thing.
However, I have also
seen this program totally misapplied by teachers.
For example, I've seen teachers read the solution to a problem
in the teacher's guide (none of us knew how to do many of the
problems as we were mostly trained by the "drill and kill"
method), but somehow expect the student to figure it out, without
the teacher's guide. These teachers should have modeled the correct solution for the students (given them the darn answers), even if it took several months of doing this before the students
(and teachers) caught on.
Another thing these teachers didn't know was this: if you understand
a problem, then you can usually solve a problem. I would sometimes
spend a lot of time getting the students to understand the
problem, and then they would usually come up with the correct
answer on their own.
No matter which way you teach, you have to keep the class moving.
So exploring math has gotten a bad name in some places where the
teachers were just not ready for it, or were not trained right.
I can't say that every math "exploration program"
or book is good. I have primarily used the Interactive Mathematics
Program (IMP) that was developed by Berkeley and the National
Science Foundation. Overall, it is a brilliant program, at times
the work of genius, with only a few minor bugs.
In actual fact, there
is a broad spectrum with abstract drilling on one side
and realistic exploring on the other side. Many teachers
lie somewhere in the middle. They drill their students, but also
attempt to get them to understand why.
With abstract drilling, you can cover a lot of material. You can
sometimes boost up test scores. With realistic exploring, you
teach how to think, but you cover less material. "Drill and
kill" math is better than no math education at all. At times,
I lean toward drilling when I haven't completely thought through
how to explore something.
Exploring math takes more money, more time, and more qualified
teachers. It also may cause teacher burnout, as it sometimes requires
more time and energy.
However, pure drilling math sort of reminds me of junk
food. It fills you up, but you may feel sick afterwards, and you
won't get the nutrients you need for the long haul.
In other words, to just drill math and only drill math is a sort
of a imitation of a more natural, realistic, and very ancient
education. A natural education using ruler, protractor, compass,
and sextant. When geometry still meant geometry, or "earthmeasuring"
and not "theoremmemorizing." A realistic education
that built the great pyramids, Stonehenge, the great cathedrals,
and now the geodesic dome, the modern observatories, and the space
telescope. An ancient education which nurtured the urge, designed
through millions of years of trial and error evolution, to use
the eyes, hands, and brain.
Can we afford to explore math? Or should we just spoon feed formulas
and hope for the best? Or should we try for something in the middle,
in which the teacher explains why? These are some of the
questions that face parents, teachers, administrators, and legislators.
William Lauritzen has published
several articles on science, math, and education.
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