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400-Level Advanced Algebra (Writing- Intensive)
WRITING AND REASONING
I think writing is more a matter of
clear thinking. And if students can think clearly enough to write
these proofs correctly and, where possible, write simply, then that's
going to help them in other kinds of writing as well. This is a good
exercise in ordering your thoughts, eliminating unnecessary baggage,
and getting things down in a clear and concise way. — Professor
Christopher Allday
Formal mathematical proofs . . . teach
you to structure your thinking process into a stream of relevant data,
ignoring whatever is irrelevant. — Student
COURSE GOALS
The major objective is to foster
student understanding of abstract algebra and the ability to use
the method of reasoning required to prove theorems and explain
solutions to abstract, mathematical problems. As students become
increasingly skilled in thinking clearly and ordering their
thoughts, they should gain greater aptitude in writing clearly and
concisely.
WRITING ACTIVITY
FIVE MATH PROOF ASSIGNMENTS
Every week, students are assigned
homework exercises in which they write proofs of various math
principles. For each assignment, students must write out proofs to
answer a set of mathematical problems such as the following:
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Writing was the ONLY way to learn
the material. . . Neither writing nor mathematics is a
"spectator sport." Moreover, in writing proofs, you
learn and practice not only mathematics, but also the principles
of logical argument; and the discipline of these communications
skills is widely APPLICABLE. --Student |
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Proofs of the solutions to such
problems are to be logical arguments written in the English
language, much of which is later replaced by mathematical symbols.
Here is an example of part of a proof:
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to complete the set of proofs for each assignment. The instructor
does not collect the first two assignments, but goes over them in
class, writing answers in proof-form on the chalkboard. Students
compare those answers to their own, asking for clarification when
needed. The next assignment is graded. All the assignments require
logical organization.
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I would encourage students to form
study groups to help one another on non-graded assignments. This
exchange of ideas, I think, would reinforce the instruction. [To
encourage more class participation] I might suggest that from time
to time a portion of a lecture period be devoted to a brief
assignment of current material and the class be divided into
groups, each tackling the same or different problems and then
sharing results with the entire class.--Student |
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PURPOSE: Writing proofs of
theorems or other statements allows students the opportunity to
practice logical thinking and document logical arguments. Students
also gain proficiency in the language of abstract mathematical
proofs and gain greater understanding of the methods and level of
abstraction necessary in abstract algebra.
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Aside from the core material, I
strengthened my skills at analytic thought and logical argument. I
also practiced research and review skills in order to complete the
assignments.--Student |
RELATED ACTIVITY
FOUR TAKE-HOME EXAMS
Take-home exams require students to
answer approximately ten problems in written proof form, which
results in six or more pages for each exam.
PURPOSE: Take-home exams
allow students enough time to thoroughly contemplate algebraic
problems and write them out in neat and logical form. Students
thus demonstrate that they understand the concepts, format, and
reasoning used in abstract algebra and that they are proficient in
logical analytical thinking. |
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Professor Allday comments on his class (excerpts from an
interview):
The first
learning goal is to understand the material: abstract algebra, the
method of reasoning, the style of proving things in a logical and
orderly fashion. This is all very new to the students. There's
very little computation involved compared to reasoning.
I think the more advanced the
material gets, the more likely it is that there are different
approaches. Usually in the more elementary stages there is the
simplest approach; maybe there is also a more circuitous approach
to the problem. I encourage the students to find the simplest one,
and sometimes I even take points off if they've presented a
solution that is roundabout but not necessarily wrong. And if the
reasoning is correct, that's good; but if the reasoning is correct
and the simplest possible, that's even better. And I might give
ten out of ten for a correct and simple solution and reasoning,
and eight out of ten for correct and unnecessarily long reasoning.
By example, I try to teach them a
writing strategy [and the writing style specific to math]. I write
a tremendous number of proofs on the board and they also read
proofs in the book, and I hope the combination of the two will
give them an idea of how to do it.
To some extent it seems odd that a
math course is a writing-intensive course. In a sense, I would
encourage them to write as little as possible, tell them that it's
a thinking-intensive course rather than a writing-intensive
course, and emphasize finding the shortest, most concise proof. On
the other hand, ordering their thoughts, getting their thoughts
and reasons clear in their heads, and putting them on paper are
good experiences for many other kinds of expository writing or
reasoned writing. |
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