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Q: What problems in mathematics education are addressed by C&M courses?
A: As Sue C. Heffelfinger,M.D. put it: "My husband and I suffered through many years of dull college math, I for one didn't learn anything about calculus until I took physical chemistry." Like Dr. Heffelfinger,when most people reflect on the math courses they took in college or high school, they get a hazy look on their face. Those who have been through a conventional mathematics class are often hard-pressed to explain what the course was about. They remember math was hard but often have little memory of what they actually learned.
Often these classes presented students with collections of "skills" which they were to memorize. The problem was that many folks forgot these skills shortly after they completed the class. The skills had to be memorized because they meant little to the students' intellects.
C&M courses address this problem using modern technology to concentrate on
the ideas and concepts of mathematics treating the "skills" as ancillary
but sometimes important. Former C&M students can rekindle the skills when
they need them because they have the conceptual background to understand
what the "skills" try to do.
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Q: What C&M courses are available the Ohio State campus and in CROSU Internet
Distance Education?
A:
A selection of five courses are available to students at the Ohio State University campus or qualified high school students within the state of Ohio:
For a complete description of the courses, go to the courses page.
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Q: What is C&M courseware?
A:
Imagine a magic textbook in which every example can be rekeyed and rerun to
do as many additional examples as the student wants and in any way the
student wants. Imagine a magic text book in which the student is free to
run unlimited experiments, add notes and check his/her progress. Imagine a
magic textbook the student can enter and interact with creatively any time
the student wants. Imagine a magic text book that supports graphics in
nearly 200 colors which can be applied by the student anytime the student
wants. C&M courseware provides this and more.
In C&M courses, active student interaction with C&M courseware replaces
introductory lectures.
The results:
One C&M student put it this way:
"C&M makes mathematics come alive to me. It's no longer a dry tedious
subject but one I can actually explore and play with. It's fun."
Another C&M student said:
"I've been studying math for years and doing very well at it, but I never
knew what I was studying until you got me on this computer so I could see
it."
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Q: What types of students should consider C&M courses?
A:
Students who:
These courses have rescued many students who were either turned off by math
or felt they could not learn math.
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Q: What types of students should NOT consider C&M courses?
A:
Students who:
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Q: Are C&M students likely to run into trouble when they take
follow-up
engineering and math courses in which the professor bans calculators and
computers?
A:
Follow-up studies done by the UIUC College of Engineering at
Illinois indicate that former C&M students are fully competitive in these
courses.
Many C&M students have gone onto graduate school in engineering and science
(Illinois, Cal Tech, MIT, Cal-Berkeley, Texas, Minnesota, etc). Others are
in medical schools. Many have excellent jobs in the business world ( Arthur
Anderson, Intel, Caterpillar, Pixar, Ford, Motorola, Wolfram Research, U.S. Navy,
etc). Others are teaching in high school.
Many C&M students have opted to go on to minor in mathematics thanks to
their early C&M experience.
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Q: Must students have experience with computers and the Mathematica
software before they can take a C&M class?
A:
Not at all.
No previous Mathematica training is required or expected.
Students learn to use the Mathematica code gradually as they progress
through the courses.
The overwhelming majority (85% as measured by survey) of new C&M students
with no previous Mathematica experience report they are comfortable within
two weeks.
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Q: How about the role of copying, pasting and editing Mathematica code?
A:
The C&M computer-based text is full of successful Mathematica code which
can be copied, pasted and edited by the student anytime the student wants
to. This fact minimizes the need to learn lots of Mathematica code and
makes C&M courseware easier to navigate than possible with a calculator.
But the role of copying and pasting play a much larger role than this.
Copying and pasting is a high order skill. To use it well, the student must
know when and what to copy, what to edit and how the results will be
interpreted. This intellectually heavy activity separates the math of C&M
from the programming of Mathematica. The outcome: Good copying, pasting and
editing reinforces mathematical understanding.
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Q: Is C&M a course in computer programming or is it a course in mathematics?
A:
Mathematics.
In C&M courses, the computer is only a tool. Students are encouraged to do pencil-and-paper calculuations while they are working on the computer assignments.
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Q: Is it true that C&M students do no hand work with paper and pencil?
A:
No. C&MStudents do hand work with paper and pencil, just not as much as traditional calculus courses. C&MStudents are required to do hand-written assignments on usually a weekly basis. Typical C&M tests are also pencil and paper-based. However, many of the test problems are more conceptually based and do not require intensive hand calculations. C&M
students are urged to use the calculational device most appropriate for the
problem. Sometimes hand calculations are the most appropriate calculational
device.
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Q: Can students enter and be successful in a C&M course after taking a traditional calculus course?
A:
Yes.
In fact at Ohio State about 50-60% of the students in each of these courses
consist of newcomers. Newcomers are generally paired up with an experienced C&M student to help the newcomer make the transition into the class.
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Q: How much personal attention do C&M students get?
A:
Almost as much as they want.
Attached to each C&M campus class is an faculty or graduate instructor and an undergraduate
class assistant. The C&M class meetings are in labs in which
the instructors and class assistants work one-on-one with their students.
To get a consultation during lab time, all a student has to do it to raise
a hand. Instructors and class assistants are also available to help students outside of class hours. Everyone gets to know each other in a way impossible in the
lecture model of teaching; frequently the friendships built in C&M extend
way beyond C&M itself.
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Q: Do C&M courses prepare for life beyond the classroom?
A:
Many former students report back that C&M experience is a big plus in their
careers.
For example, former C&M student Brad Winemiller is a Render TD for Pixar
Animations Studios in Richmond, CA. Brad puts it this way: "I wouldn't be
doing this stuff if it weren't for those classes. It's all math and working
together with other people."
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Q: Why do C&M students work in teams?
A:
Experience has shown that C&M students learn more when they work in teams
of two or three. Students working alone seem to miss a lot of things that
teams tend not to miss. C&M labs tend to be noisy places; the sound is the
sound of learning.
When folks talk together about math (or about anything else), they learn
together and they learn more.
This is one reason modern businesses usually assign work to teams and not
to individuals.
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Q: What employment possibilities are there for C&M students?
How does a C&M student qualify for employment?
What advantages do C&M students get from these jobs?
A:
C&M students can become:
Hiring is based on performance in previous C&M courses and ability to deal
with students.
The advantages you get:
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Q: Are C&M courses easier than corresponding standard math courses?
A:
Some students have said that C&M is in fact easier than standard math
courses. But the reasons for this view are complex.
First, C&M students are spared much (not all) of the tedious pencil and
paper calculation found in other courses. This in itself makes C&M courses
less laborious than other courses.
Second, as all mathematicians know, mathematics is more easily understood
when it rests on a conceptual background rather than a memorized
background. C&M students have the conceptual background needed to make math
easier than it would be otherwise.
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Q: Are C&M courses harder than corresponding standard math courses?
A:
Some students have said that C&M is in fact harder than standard math
courses. Some reassons:
First, C&M is more intellectually demanding and richer conceptually than
other courses.
Second, most C&M students put in more hours than traditional students.
Student homework is not optional in C&M courses.
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Q: Do many C&M students ask:"What's this stuff good for?"
A:
"What's this stuff good for?" is an FAQ in standard mathematics courses..
This is not unexpected because many standard math courses present math for
math's sake.
But this question is rarely asked by C&M students. The reason is that in
C&M courseware each new idea is accompanied by a solid application
chosen to promote student interest and to put the new idea in a concrete
context many students need in order to learn.
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Q: Are there many women in C&M courses?
A:
C&M student demographics tend to mirror the demographics of the majors of
the students in the courses. C&M classes with heavy engineering enrollment
tend to have fewer women than C&M classes with heavy life science
enrollment.
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Q: Why do some students, professors and advisors argue against C&M?
A:
Lots of students, professors and advisors have excelled in their studies of
traditionally taught mathematics courses. They feel good about the old ways
and often have the idea that everyone must learn the way they did.
What they forget is that many others did not excel and were turned off by
the old ways.
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Q: Some folks have claimed calculators have nearly ruined grade school and
high school math education. Won't the use of computers in C&M ruin math
education completely?
A:
Misuse of technology is a big problem in many mathematics classes - often
with disastrous results. The problem is that the course texts were not
written with technology in mind. Instead technology is injected as a layer
in a text whose orginal edition is likely to have been written years ago -
before calculators and computers were on the scene.
The C&M courses are the first math courses to be written from the start
with the computer in mind. The computer is woven throughout the courses and
is often used to introduce new ideas visually so that students can play and
experiment with new idea before dealing with the new terminology. The idea
is understanding - not button pressing. As Richard Hamming once put it:
"The goal of computing is insight, not numbers."
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Q: How does the mathematical content of C&M courses measure up against the
mathematical content of standard courses?
A:
C&M courses drop some of the parts of the standard courses that a former
President of the American Mathematical Society calls "inert."
C&M courses replace the inert matter with fresh (mostly 20th Century) ideas
such as:
In addition C&M courses include healthy doses of complex numbers and give
plenty of experience with the complex exponential function.
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Q: Are the standard courses more "theoretical" than C&M courses?
A:
No. In many standard courses, the underlying ideas are in the book or in
the lecture but not engaged by students. C&M students are fully engaged in
underlying ideas.
Dr. Francis Sullivan, Director of the Computing Laboratory of the Institute
for Defense Analysis says this about C&M calculus courses:
"For me the wonderful thing about C&M is that it concentrates on what
calculus is about."
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Q: When it comes to hand calculations, how do C&M students compare?
A:
Hand calculations are a part of all C&M courses - so much so that the 1995
Park-Traverse study concluded that C&M calculus students can do hand
calculations (derivatives, integrals etc.) on a par with students in
corresponding standard math courses.
The same study also concluded that C&M students have a stronger conceptual
background than students in corresponding standard courses. C&M students
know that math is much more than a collection of rote hand calculations to
be memorized.
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Q: Are repeated hand calculations necessary for understanding?
A:
In the past, many teachers believed that repeated hand calculations were
the path to understanding and in fact were necessary for understanding.
This belief has been exploded in recent research by David Tall, Stanislas
Dehaene and others. Their research indicates that repeated hand
calculations sometimes actually interfere with conceptual understanding.
To get the idea, try counting a group of objects while maintaining a
conversation about the unifying characteristics of those objects.
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Q: How do the graphics in C&M compare with the graphics in other courses?
A:
C&M Mathematica graphics make calculator plots seem primitive.
And in C&M, you get more vivid colorful graphics than found in any printed
text. Also, in C&M you are not limited to the graphics chosen for a printed
text. As a C&M student said: " I've been studying math for years and doing
very well on it. But I never knew what I was studying until you got me on
this computer and I could see it."
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Q: Why is C&M written in conversational English rather than in the usual
style of standard math text books?
A:
C&M is written the way actual American people talk to each other and not in
the stilted prose associated with most conventional math texts. Many C&M
students have reported that after their C&M experience, they were able to
pick up and read conventional math books - mainly because of the conceptual
grounding they obtained in C&M.
Special thanks to C&M at University of Illinois Urbana-Champaign for their assistance in answering these questions.
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