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[1] VCU's Instructor's GuideYvette Stepanian
Virginia Commonwealth University
(This Instructor's Guide is given to the instructors of the Contemporary
College Algebra course at Virginia Commonwealth University at the beginning of
the semester.)
This course is very different from a traditional course both in content,
pedagogy, and student expectations. Students as well as instructors need to
frequently review and discuss the course goals.
The primary goal of this text is to empower students to become
exploratory learners, not to master a list of algebraic rules. The structure of the course and text is designed with this goal in mind. We want our students to be fully participating in their learning process, each time they come to
class. As an instructor, here is some guidance to always have in
mind. 1. Frequently review and discuss the goals of the course. The students are not used to this teaching approach; constantly remind them, particularly during the first two weeks, on the importance of this approach: - Why is it that we are not "just" giving the right answer? - Why is it that students have to think as a group about different ways to solve a problem? - How is it possible that there is no answer, or that there are many possible answers, or that the answer we found does not make sense? . . .
Students need to learn that the mathematics used in real life is not a magic
trick; it cannot give one perfect, unique solutions to all problems. The
student remains the one who will have to make the best decision.
2. Repeatedly remind and encourage the students to read and study
the text with a pencil: the text was written for it to be
personalized. This is something the students do not do naturally. It might be
useful to personalize your own copy before class and show it to your
students.
3. Practically, you should not lecture for more than 15 minutes;
the course is about learning through problem solving and critical thinking,
not about mimicking repeatedly the instructor's examples. This can feel
uncomfortable at first for you as well as for your students who are usually
never asked to think by themselves and who expect you "to do all the
work."
4. The course is designed so that students work in small groups
regularly and hand in graded worksheets (or graded Class Activities)
weekly. They learn to manage a group, they learn to make compromises, they
learn to rely on each other's knowledge, and they learn to be responsible
while working in a group with people of different motivation.
5. Rather than just giving students an answer when they feel stuck, you should
encourage them to talk to each other to try to explain the
problem to each other. Stronger students could be paired with weaker ones.
Students need to be reminded of the benefits of such an approach; as they talk
to each other they stimulate their communication skills, they are more likely
to feel comfortable expressing their misunderstandings, . . .
6. Spending a whole session on one exercise is fine; always remember that your
goal is not to give an answer, but is to make students think, talk,
write about the problem. The schedules you are given are designed with
this in mind. If you are too much ahead of the schedule, then it means that
either you are lecturing too much or you are not letting your students be
stuck enough!
7. As the semester goes on, the students learn about different approaches for
solving a problem (algebraic, graphic, . . .). They need to be reminded of
these possibilities; your goal is to make them feel comfortable, to
develop their personal confidence as problem solvers, to let them
know that there is not only one way to do it.
8. In this course the calculator is used expansively/daily so
students learn the process of analyzing a real situation: (1) collecting data,
(2) plotting data, (3) fitting an appropriate curve to the plotted data, (4)
making predictions, (5) obtaining insights into the situation. 9. Expect your students to show some resistance to this teaching approach; after all: - They have rarely been left stuck on a problem, - they have been asked to think critically about an answer to a math problem, - They have rarely been asked to think and discuss new situations without mimicking a model exercise, - They have rarely been asked to use technology as an important part of their problem solving, - And more importantly, they have rarely been asked to rely on their own knowledge as much as on their peers to discuss a problem and solve it. [2] Joint Mathematics Meeting
San Antonio, TX, Jan. 12-15, 2006
The Joint Mathematics Meetings includes the American Mathematical Society
(AMS), the Mathematical Association of America (MAA), the Association of Women
in Mathematics (AWM), the National Association of Mathematicians (NAM), the
Association of Symbolic Logic (ASL), and sessions by the Society for
Industrial and Applied Mathematics (SIAM). A rich spectrum of talks,
minicourses, panel sessions, poster sessions, contributed papers, committee
meetings, and, of course, numerous hallway discussions will make the Meetings
memorable to several thousand mathematicians.
Interest in refocusing college algebra will be high at the Meetings. The
following sessions will be of particular interest to those interested in
college algebra:
Friday, 8:00 - 9:55 am (Room 217B, Convention Center) MAA Session on
Courses Below Calculus: A Continuing Focus (Aimee Ellington, who
conducted a research study of the effect of a modeling-based college algebra
on student achievement will speak at 8:00 am.)
Friday 2:30 - 3:50 pm (Room 205, Convention Center) What Business Looks
For in New Hires. A panel session of four business speakers discussing
what business related skills/experiences are important in college algebra
level courses. Sunday 2:30 - 3:50 pm (Room 217A, Convention Center) Reunion of Participants in Refocused College Algebra Programs. Bill Haver (Virginia Commonwealth University) and Laurette Foster (Prairie View A&M University) will lead the session. All persons who are interested or involved in renewing college algebra courses are encouraged to participate by sharing their experiences, plans, and dreams. [3] Soda Preference
(This activity is designed for the first week of classes, possibly over the
first weekend. It would serve well as a guided discovery for Section 2.5
(Circle Properties and Pie Charts). The purpose is to initiate group work and
to emphasize working with data.)
Campus vending machines are usually limited to dispensing fewer than six
varieties of soda. How does a vending company decide which varieties to offer?
This activity provides an insight into how such decisions may be made. The
activity consists of three stages.
Stage 1. (in class) Divide the class into groups (e.g., 3-person groups). The
groups are then given 10 minutes of class time to organize themselves. In
particular, each group determines a procedure for determining the soda
preference for thirty people, including themselves. Each group member is to be
involved in collecting the data and the procedure adopted should avoid having
any person surveyed by more than one group. Stage 2. (out of class) Each group a. Collects data b. Consolidates the data collected and displays it in both a bar graph and a pie chart. c. Prepares a class presentation, including a discussion of which display (bar graph or the pie chart) is better at conveying the information on soda preference and a discussion of how reliable the results
are for making predictions. Stage 3. (in class) Group presentations. [4] Proportional Reasoning
Proportional reasoning occurs throughout mathematics and is an important
aspect of problem-solving. We are all familiar with the statement
"corresponding sides of similar triangles are proportional" and use it in
solving problems of ladders sliding down the side of a barn or determining the
length of a shadow when walking away from a lamp post. Formulas in Section 2.5
for circle properties such as measures of central angles, areas of sectors,
and length of arc are the result of proportionality reasoning. For example,
the length of a subtended arc of a circle is proportional to the corresponding
subtended central angle (arc length =
Two quantities,
Another important category of problems involving proportionality reasoning are
those involving inflation, change in prices over time. For example in 1975, a
gallon of gasoline cost 56.7 cents. How does that price when adjusted for
inflation compare to the price in 2005 when the price was $1.60 per
gallon? In order to obtain the inflation adjusted price, one assumes that the
proportional increase in gas prices is the same as the proportional increase
in the Consumer Price Index (CPI) over the same time span. That is
or
(2005 Jan. Gas Price)
=
Consulting the CPI table in www.bls.gov gives
(2005 Jan. Gas Price)
In Cornwall, NY, the price of gasoline in January 2005 was approximately $1.75
per gallon. Thus gasoline was a "better buy" in 2005 than it was in 1975 as
its rate of increase over this 30 year period was less than the inflation
rate.
The proportionality constant
indicates that the rate of inflation over this 30 year period was
366%.
Another way to make a comparison on gas prices over the period 1975-2005 would
be to compute the 1975 price in "2005 dollars." That is, solve the
equation
for the 1975 Jan. Gas Price (in 2005 dollars)
(1975 Jan. Gas Price)
[5] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
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