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What`s Wrong with College Algebra?

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What`s Wrong with College Algebra?
What’s Wrong With
College Algebra?
Sheldon P. Gordon
[email protected]
College Algebra and Precalculus
In 2000, between 1,000,000 and 2,000,000
students took college algebra and
precalculus courses
At most schools, these are the bread-andbutter courses – huge numbers of students
and very cheap to run (staffed usually by
TA’s or part-time faculty).
College Algebra and Precalculus
The focus in most of these courses is on
preparing the students for calculus.
But only a very small percentage ever go
on to start calculus.
Enrollment Flows
Based on several studies of enrollment flows from
college algebra to calculus:
• Only about 5% of the students who take college
algebra courses ever start Calculus I
• The typical DFW rate in college algebra is
typically well above 50%
• Virtually none of the students who pass college
algebra courses ever start Calculus III
• Perhaps 30-40% of the students who pass
precalculus courses ever start Calculus I
Why Students Take These Courses
Required by other departments
• Satisfy general education requirements
• To prepare for calculus
• For the love of mathematics
Four Special Invited Conferences
•
Rethinking the Preparation for Calculus
• Reforming College Algebra
• CRAFTY Curriculum Foundations Project
• Forum on Quantitative Literacy
Common Recommendations
“College Algebra” courses should be real-world
problem based:
Every topic should be introduced through a
real-world problem and then the
mathematics necessary to solve the problem
is developed.
Common Recommendations
A primary emphasis in “College Algebra” should
be Mathematical Modeling:
– transforming a real-world problem into
mathematics using linear, exponential and power
functions, systems of equations, graphing, or
difference equations
– using the model to answer problems in context
– interpreting the results and changing the model
if needed.
Common Recommendations
“College Algebra” courses should emphasize
communication skills: reading, writing,
presenting, and listening.
These skills are needed on the job and for
effective citizenship as well as in academia
“College Algebra” courses should emphasize small
group projects involving inquiry and inference.
Common Recommendations
“College Algebra” courses should make
appropriate use of technology to enhance
conceptual understanding, visualization, and
inquiry, as well as for computation
“College Algebra” courses should be student
centered rather than instructor centered
pedagogy: they should include hands-on
activities rather than all lecture
Important Volumes
• CUPM Curriculum Guide: Undergraduate Programs and
Courses in the Mathematical Sciences, MAA Reports.
• Ganter, Susan and Bill Barker, Eds.,
A Collective Vision:
Voices of the Partner Disciplines, MAA Reports.
• Madison, Bernie and Lynn Steen, Eds., Quantitative Literacy:
Why Numeracy Matters for Schools and Colleges, National
Council on Education and the Disciplines, Princeton
• Baxter-Hastings, Nancy, Sheldon Gordon, Florence Gordon and
Jack Narayan, Eds., A Fresh Start for Collegiate
Mathematics: Rethinking the Courses Below Calculus, MAA Notes.
CUPM Curriculum Guide 2004
• All students, those for whom the (introductory
mathematics) course is terminal and those for whom it
serves as a springboard, need to learn to think
effectively, quantitatively and logically.
• Students must learn with understanding, focusing
on relatively few concepts but treating them in depth.
Treating ideas in depth includes presenting each
concept from multiple points of view and in
progressively more sophisticated contexts.
CUPM Curriculum Guide 2004
• A study of these (disciplinary) reports and the
textbooks and curricula of courses in other disciplines
shows that the algorithmic skills that are the focus of
computational college algebra courses are much less
important than understanding the underlying
concepts.
• Students who are preparing to study calculus need
to develop conceptual understanding as well as
computational skills.
AMATYC Crossroads Standards
In general, emphasis on the meaning and use of
mathematical ideas must increase, and attention to rote
manipulation must decrease.
•Faculty should include fewer topics but cover them in
greater depth, with greater understanding, and with
more flexibility. Such an approach will enable students
to adapt to new situations.
•Areas that should receive increased attention include
the conceptual understanding of mathematical ideas.
NCTM Standards
These recommendations are clearly very much
in the same spirit as the recommendations in
NCTM’s Principles and Standards for School
Mathematics.
If implemented at the college level, they would
establish a smooth transition between school and
college mathematics.
Some Conclusions
We cannot simply concentrate on teaching the
mathematical techniques that the students need. It is as
least as important to stress conceptual understanding
and the meaning of the mathematics.
We can accomplish this by using a combination of
realistic and conceptual examples, homework problems,
and test problems that force students to think and
explain, not just manipulate symbols.
If we fail to do this, we are not adequately preparing
our students for successive mathematics courses, for
courses in other disciplines, and for using mathematics
on the job and throughout their lives.
Further Implications
If we focus only on manipulative skills
without developing
conceptual understanding,
we will produce nothing more than
students who are only
Imperfect Organic Clones
of a TI-89
Curriculum Foundations Project
A series of 11 workshops with leading
educators from 17 quantitative
disciplines to inform the mathematics
community of the current mathematical
needs of each discipline.
The results are summarized in the MAA
Reports volume: A Collective Vision:
Voices of the Partner Disciplines, edited
by Susan Ganter and Bill Barker.
Curriculum Foundations Project
The discussions and recommendations of
virtually every other discipline focused
on the mathematics below calculus.
Even such mathematically intense fields
as physics offer far more non-calculus
based courses than calculus-based
courses.
The Common Themes
• Strong emphasis on problem solving
• Strong emphasis on realistic applications
and mathematical modeling
• Conceptual understanding is more
important that skill development
• Development of critical thinking and
reasoning skills is essential
The Common Themes
• Use of technology, especially spreadsheets
• Development of communication skills
(written and oral)
• Greater emphasis on probability and
statistics
• Greater cooperation between mathematics
and the other disciplines
Some Implications
Although the number of college students
taking calculus is holding steady, the
percentage of students taking college
calculus is dropping, since overall college
enrollment has been rising rapidly.
But the number of students taking calculus
in high school already exceeds the number
taking it in college. It is growing at 8%.
Some Implications
Few, if any, math departments can exist
based solely on offerings for math and
related majors. Whether we like it or not,
mathematics is a service department at
almost all institutions.
And college algebra and related courses
exist almost exclusively to serve the needs of
other disciplines.
Some Implications
If we fail to offer courses that meet the
needs of the students in the other
disciplines, those departments will
increasingly drop the requirements for
math courses. This is already starting to
happen in engineering.
Math departments may well end up offering
little beyond developmental algebra courses
that serve little purpose.
What is Quantitative Literacy?
Quantitative literacy (QL), or numeracy, is the
knowledge and habits of mind needed to
understand and use quantitative measures and
inferences necessary to function as a responsible
citizen, productive worker, and discerning
consumer.
QL describes the quantitative reasoning capabilities
required of citizens in today's information age -- from
the QL Forum White Paper
QL and the Mathematics Curriculum
The focus of the math curriculum is the
geometry-algebra-trigonometry-calculus
sequence.
•
•
•
•
•
In high school, the route to competitive colleges.
The sequence is linear and hurried.
No time to teach mathematics in contexts.
Courses are routes to somewhere else.
Other sequences are terminal and often second rate.
How does the US compare to other countries?
Many College Graduates Demonstrate
Weak Quantitative Literacy Skills
Grads:
2 Yr. Colleges
Grads:
4 Yr. Colleges
Level 5: High
5
13
Level 4
30
40
Level 3
44
40
Level 2
17
10
Level 1: Low
4
3
Source: USDOE, NCES, National Adult Literacy Survey, 1992, in Literacy in the Labor Force: Results from the NALS,
September 1999, p. 61.
The Forum on Quantitative Literacy
Participants were top-level representatives of
business, industry, and government, and very
few academics.
The discussions focused on the broad need of
every educated person to have a significantly
higher level of quantitative understanding and
skills.
The recommendations were amazingly similar to
those mentioned previously.
Where Does College Algebra Fit In?
QL is something that should permeate the entire
mathematics curriculum, so that every student
develops this knowledge and skills.
The one existing course that provides the best
opportunity to stress QL is college algebra:
• It has the largest enrollment
• It does not prepare or motivate large numbers
of students to go on to calculus
• It is taken to prepare students for courses in
other disciplines, and the themes of QL are the
mathematical topics needed in most other
disciplines today.
What Can Be Removed?
How many of you remember that there used to
be something called the Law of Tangents?
What happened to this universal law?
Did triangles stop obeying it?
Does anyone miss it?
What Can Be Removed?
• Descartes’ rule of signs
• The rational root theorem
• Synthetic division
• The Cotangent, Secant, and Cosecant
were needed for computational purposes;
Just learn and teach a new identity:
1  tan x 
2
1
cos2 x
How Important Are Rational Functions?
• In DE: To find closed-form solutions for several differential
equations, (usually done with CAS today, if at all)
• In Calculus II: Integration using partial fractions–often all four
exhaustive (and exhausting) cases
• In Calculus I: Differentiating rational functions
• In Precalculus: Emphasis on the behavior of all kinds of
rational functions and even partial fraction decompositions
• In College Algebra: Addition, subtraction, multiplication,
division and especially reduction of complex fractional
expressions
In each course, it is the topic that separates the men from the
boys! But, can you name any realistic applications that involve
rational functions? Why do we need them in excess?
Realistic Applications and
Mathematical Modeling
• Real-world data enables the integration of data
analysis concepts with the development of
mathematical concepts and methods
• Realistic applications illustrate that data arise in a
variety of contexts
• Realistic applications and genuine data can
increase students’ interest in and motivation for
studying mathematics
• Realistic applications link the mathematics to
what students see in and need to know for other
courses in other disciplines.
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