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[1] Developing Mathematical Intuition
There is no "Book" on how to develop mathematical intuition. In fact, the
phrase mathematical intuition is probably not well defined.
Nevertheless, most mathematicians would probably say that their mathematical
intuition is an important aspect in their understanding and ability to do
mathematics. This suggests that college algebra courses should address the
challenge of developing student's mathematical intuition. What can be
done?
Experience doing mathematics is certainly a critical component in developing
one's mathematical maturity. Thus given a one semester course that is probably
the terminal mathematics course for most of our students, we are faced with
the question: What types of student experiences should we emphasize in order
to maximize the development of mathematical intuition? Is the drill and skill emphasized in the past
best? Or, is it best to emphasis modeling experiences? Or, is it best
to emphasize discovery experiences? Or, is it best to . . .
. ?
Many of the experiences designed to help students develop into exploratory
learners also serve to develop students' mathematical intuitions. Here are a
few queries whose discussions could (hopefully) enhance a student's
mathematical intuition.
a. Which of the following choices : (i) 2 gallons, (ii) 4 gallons, (iii) 6
gallons, (iv) 8 gallons is the best approximation of the amount of water in
one cubic foot? (Hint: Think of how many half gallon milk cartons are needed
to form a cubic foot block.)
b. Is there a relationship between the length of the boundary of an enclosed
two-dimensional region and the area of the enclosed region? Hint: Experiment
with a circle or triangle or rectangle.
c. What does the highway sign 6% grade indicate?
d. What would be the angle of elevation of a road if its grade was
100%?
e. What, if anything, is common to the meanings of the terms pitch (as used by
a roofing carpenter), grade (as used by a highway engineer), "slope" (as used
by a skier), slant (as used with a slide in playground equipment), ramp
incline (as used by movers)? [2] Fundamental Skills (Elementary Algebra)(Please refer to the article on Fundamental Skills in the January issue of this Newsletter.)
This is short activity to begin a Monday morning class. Divide the class into
groups of two and then have each group do the following: For each of the five
equations, state if it is True or False and then give an example that supports
your answer. (That is, assign integer values to "
a.
b.
c.
d.
e.
Readers are invited to send their lists of fundamental skills to this
Newsletter along with suggestions for teaching fundamental
skills. [3] Class Quickies
The following four Class Quickies are examples of short (quick) exercises that
can be given to begin a class on a participatory mode or to reengage students
who have or are drifting off.
a. Which of the following functions best models the graph?
b. Sketch the graph of
c. A line is drawn through the points (-7,2) and (5,8) and another line is
described by the function
d. This weekend you plan to spend several hours studying. If you plan to spend
0.15 of your time reading an English play,
[4] Temperature Distribution in a Rod
(Instructor: This in-class activity involves modeling with a system of
equations. Solving the system offers opportunities to explore different
approaches. The follow-on questions offer nice opportunities for students to
explain their reasoning to the class.)
Consider five points A, B, C, D, E ordered from left to right on an unevenly
heated rod. Let points A and E be the endpoints of the rod and points B, C,
and D be interior points. The temperature at point A is maintained at 0
degrees (lowest temperature on the rod) and the temperature at point E is
maintained at 40 degrees (the highest temperature on the rod). Assume for each
interior point that the temperature is the average of the temperatures at
their adjacent points. (Note that A and C are adjacent points to B.) Model the
temperature distribution with a system of equations and then solve for the
temperatures at points B, C, and D. Follow-on Questions:
1. Does the location of the interior points make a difference? Explain your
reasoning.
2. Sketch the graph of a temperature function for the rod assuming that the
points B, C, and D are clustered near point A.
3. Sketch the graph of a temperature function for the rod assuming that the
points B, C, and D are clustered near point E.
4. Sketch the graph of a temperature function for the rod assuming that the
points B, C, and D are evenly distributed between points A and
E. [5] PREP Workshop Revitalizing College Algebra
(PREP is an acronym for PRofessional Enhancement Programs of the
Mathematical Association of America (MAA). The programs are sponsored by the
MAA and funded through a grant from the National Science
Foundation.)
Revitalizing College Algebra Bill McCallum, Don Small, Bill Haver June 18 - 21, 2007 University of Arizona Tucson, AZ Co-sponsored by the Institute for Mathematics and Education Registration Fee: $300 by May 7, $400
after May 7.
Priority will be given to departments that send at least two participants. The
registration fee will be reduced to $200 for the second, third, and fourth
participant from a department registering by May 7.
A growing number of colleges and universities are modifying their college
algebra courses to focus on mathematical modeling. Students address problems
presented as real world situations by creating and interpreting mathematical
models. Solutions to the problems are formulated, validated, and analyzed
using mental, paper-and-pencil, algebraic, and technology-based techniques as
appropriate. MAA's Committee on Curriculum Renewal Across the First Two Years
(CRAFTY) recommends that all College Algebra courses be organized in this
manner. The recently approved CRAFTY College Algebra Guidelines describe the
features of such a course. Complementing the modeling approach is an approach
aimed at making symbols more meaningful for students by emphasizing structure
and purpose in expressions, formulas and equations.
The workshop is designed to help faculty from departments who have plans to
renew their algebra curriculum. During the course of the workshop,
participants will: Explore a number of College Algebra
topics from a modeling based perspective; Explore problems that help students interpret the structure of symbolic
representations;
Learn about a number of possible texts; Consider testing and assessment issuers by reviewing tests and assignments that
have been used at other institutions; Review group projects that have been
assigned to students Consider issues of training part time instructors or graduate students who have not experienced courses of this
nature; Discuss challenges involved in refocus-
ing college algebra courses.
Participants will be provided with some reading before the workshop and will
be asked to share their experiences and provide mutual support as they revise
their college algebra courses.
For more information on the Institute for Mathematics and Education, visit
http://ime.math.arizona.edu. [6] Two Geometrical Problems
a. Determine the area enclosed by the polygon pictured below.
b. Dilation. Informally dilations are projections of figures in the plane from the perspective of the origin of a coordinate system.
Formally, a dilation centered at the origin with scale factor r maps
the point
How does a dilation centered at the origin with scale factor r affect the area
of a triangle? In particular, what is the relation between the area of
triangle ABC and triangle AB'C' in the preceding example? If a dilation centered at the origin with scale factor r is applied to a triangle of area 10, what is the area of the resulting triangle? Does it matter if all of the vertices of the original triangle are different from the origin? Explain your reasoning.
[7] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
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