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[1] Teaching PhilosophyErica Slate Young
U. S. Military Academy
I have always known I wanted to be a teacher. As a high school student, I had
not yet decided what I wanted to teach, but I did know I wanted to teach
something. In my undergraduate program, I sought out opportunities to teach
and work with students. I modeled my teaching style after teachers I had in
the past whom I admired. Mostly I admired the teachers who were well-organized
and gave good notes; the teachers whom, if I just listened in class and took
careful notes, I would have everything I needed to do well on tests. This
method worked for me, I believed it would work for my students as
well.
Once I began taking formal teacher education classes, and learning about the
Constructivist philosophy, I started to see that maybe everyone did not learn math the same way I did. I bought into the notion that
everyone constructs their own knowledge, however, I still saw my role as the
teacher as a transmitter of that knowledge. If I was clear enough and worked
through enough examples my students would construct their math knowledge
correctly, assimilating everything I said, exactly as I said it. I always
tried to be responsive to student questions in class as well as understand
their individual needs, so I believed I had a student-centered classroom. Even
though I was focused on the students, their role in the classroom was still
largely passive.
Over the past few years, I have started approaching my lessons differently. I
know that each student constructs his or her knowledge differently based on
his or her experiences. As a mathematics teacher, I am responsible for
creating those experiences for my students in such a way that the students can
make meaningful constructions for themselves. For most students, this means
they need to be actively engaged with the material rather than passively
waiting for me to impart wisdom to them. I have come to see that a particular
student's success in my course has very little to do with what I say in class,
and a lot to do with what I have them do on their own.
Novice learners (many college freshmen still fall in this category) need more
structure in their courses. I look at it as being similar to how many of us
learned to ride a bike when we were children. We had training wheels until we
got used to the mechanics of how a bike worked -- how to pedal, how to steer
and how to stop. We then progressed to trying to ride without the training
wheels, but with someone that ran behind us to make sure we did not fall. Then
we progressed to riding on our own with no safeguards. Some people were
naturals, but most of us fell at least once before we got the hang of it.
Similarly, I try to structure classroom activities that scaffold the student
learning appropriately. I start with tasks that are closely linked to prior
math knowledge, and then advance to more complex concepts. For example, a
common topic in college algebra is solving a system of equations. With
students working in teams of two, I begin with a simple example -- graphically
approximate, using the Trace key on their calculator, the solution of the
system
I also try to establish a safe environment in my classroom. Students are free
to ask questions and make mistakes without fear of embarrassment. Each day, I
try to have several problems for the students to work through on their own and
I frequently call on students to work through problems at the board or discuss
their solution strategies. If a student does not know how to solve the
problem, I have them start with what they know and then elicit help from the
rest of the class on what they do not know. I try to emphasize the value in
learning from one's mistakes.
Regarding assessments, I believe in using multiple measures to evaluate the
level of student learning. Typically, I assign problem sets for graded
homework, give short quizzes fairly frequently, assign several major projects
and then, of course, give mid-term exams and a final exam.
In the homework problems sets and in the projects, I try to give the students
exposure to solving more complex and sometimes ill-defined problems. The
projects also have a writing component. The students are assigned a problem
(usually one that requires some assumptions be made and involves multiple
mathematical tasks) which they must solve and then explain their solution in a
formal report. I generally use quizzes to assess how the students do when they
have a time constraint and no outside resources. I find that giving frequent
quizzes helps the students prepare for mid-term exams by making them face what
they can do on their own during a timed event.
When I design an exam, I include some questions that are very straightforward.
These questions are similar to homework problems and can assess a student's
understanding of the basic concepts from the course. I also include some
questions that are designed to assess more complex reasoning skills. These
questions require the students to apply the skills and concepts covered in
class to answer unfamiliar problems. I have found that designing exams this
way helps me assess which students truly have a good understanding of the
material versus the ones who have just learned the basic skills.
After each mid-term exam, I also require my students to do test corrections on
any problems they missed. This is not for points back on the test, but just as
an additional homework assignment. However, I do not have them simply do the
corrections. I also insist that they include for each problem a short
paragraph reflecting on why they did not receive full credit. I find that this
helps my students become more reflective on their work in general. They are
also less likely to make the same mistake again. [2] Beginning Class Activities
These two "Beginning Class Activities," are designed to be done by two-person
teams. Conclude the activity with two or three teams presenting their work to
the class and explaining their reasoning. a. Determine two linear equations whose graphs intersect at the point (3, 4) with one line having slope three and the other line having slope negative two. b. Determine two parallel lines, one pass- ing through the point (2, 1) and the other passing through the point (-3, 3).
[3] Query: Debt
According to the U.S. Census Bureau the current population of the U.S. is
303,162,770 and rising at the rate of 2,425,850 per year. The U.S. national
debt is currently $9,000,000,000,000 and rising at the rate of $
500,000,000,000 per year. a. What is the current per capita (per person) national debt? b. Is the per capita national debt rising
or falling? At what rate? [4] Asymptotes
For each of the following, state if the function has a vertical asymptote
only, a horizontal asymptote only, both a vertical and a horizontal or no
asymptote. If the function has one or more asymptotes, give an example and
state the asymptote(s). If the function has no asymptote, explain why it does
not have an asymptote. a. An exponential function. b. A logarithmic function. c. A power function with a positive exponent. d. A power function with a negative
exponent. [5] Query: Oil-Gasoline Prices
Although both oil and gasoline prices fluctuate, their prices are linked and
the price trends are increasing. Assume that 50% of the price of gasoline is
the cost of the oil used in the refining process for the gasoline. Assume also
that the price of gasoline is $3.20 per gallon. If the price of oil declines
10%, how much should the price of gasoline decline? [6] PizzaThe proprietor of Central Pizza in York, PA estimates that a 16 inch pepperoni pizza that sells for $8 contains 4 ounces of pepperoni, 7 ounces of sauce, 13 ounces of cheese, and 20 ounces of crust. a. Display this information using a pie chart. showing the weight percentage of each ingredient. b. What would be a reasonable price to pay for a single slice whose surface area measured 25 square inches?
Explain your reasoning. [7] Challenger Explosion
Twelve years ago this month the space shuttle, Challenger,
experienced a catastrophic explosion shortly after launch killing all the
astronauts on board. The resulting investigation headed by William Rogers, a
past Secretary of State, concluded that the failure of an 0-ring was the cause
of the accident.
The space shuttle system consists of the orbiter, the external liquid fuel
tank, and two solid rocket boosters, all joined together at launch time. The
solid rocket boosters responsible for giving the initial thrust necessary to
launch the space shuttle into orbit are approximately 150 feet in length and
12 feet in diameter. They are manufactured and shipped in sections and
assembled at the launch sight. 0-rings, primary and secondary, are used to
seal the joints between the sections. The day before the fateful launch,
concerns were raised about the effectiveness of the 0-rings in cold weather.
The concern was whether or not the cold weather would make the 0-rings too
stiff to seal properly. The temperature forecast for launch time was
31 The following data on 0-ring performance comes from the inspection of the jettisoned solid fuel rockets that were recovered from the ocean on 23 of the previous 24 launches.
There was no indication of damage to 0-rings on the remaining 16 launches. The
temperatures
(
Form a scatter plot of this data and then, based on quadratic and exponential
models, predict the 0-ring damage for launch at
31 [8] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
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