ALTERNATE LEARNING ENVIRONMENT HELPS STUDENTS EXCEL IN CALCULUS
A PEDAGOGICAL ANALYSIS
by Jerome Dancis
Executive summary: "The overemphasis on testing, skill
development, and fact level content ... [in high school has]
inhibited [student] interest in learning, motivation, ability to
work with and enjoy ideas, use creativity and attain satisfaction
from an educational experience." In math, this non-intellectual
approach comes from math textbooks which "never offer intellectual
challenges, or chances to build confidence and problem-solving
skills." This emphasis on testing, skill development, and fact
level content results in entire curricula which do not include the
training of pupils on how "to read critically, synthesize
information, interact effectively with both peers and instructors
in academic settings, and participate actively in discussions" and
to "express a logical, coherent line of argumentation or to follow
another's argument". It results in students going off to college
with study habits which are ineffective and inefficient.
The highly motivated, hard working, smart black students at
U.C. Berkeley were unsuccessful in Calculus (average grade 1.5
(D+)). Uri Treisman developed a program in which minority students
excelled and thrived. Training in an alternate learning
environment on (i) how to learn in general and (ii)how to learn
cooperatively facilitated students replacing their skill-oriented
attitudes and their ineffective study habits with effective
learning habits. Some of the crucial pillars of this alternate
learning environment are: (i) group learning on (ii) interesting
hard problems with (iii) informal interaction with an instructor
rich in mathematical expertise, together with (iv) a support
system for the students. There is (v) much correcting without
grading and (vi) peer discussion, also (vii) major emphasis on
explanations and understanding, also (viii) minor emphasis on
skills and memorization of facts. It was not just groups of
students doing textbook problems overseen by someone with moderate
knowledge of mathematics. In 1983-84, the alternate learning
environment enabled the black students to achieve an average grade
of B- (2.6) in calculus in spite of their math S.A.T. average of
500. For comparison, the average for white students without this
training was 1.9 (C) and they had a math S.A.T. average of 640.
The good learning habits developed in the workshops carried over
to success in later courses (without workshops) and to a
significantly higher rate of graduation especially in math based
fields . Most likely similar programs in most subjects and at
many educational levels would result in considerably raising the
achievement level of both black and white students. It is time to
organize such programs for all students.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
At the University of California at Berkeley, most of the
white students from blue collar and farm families and most of the
black students were not succeeding in the calculus course for
engineering and science students. In 1973-75, the average grade
in Calculus 1A for Engineering and Science students for black
students was 1.5 on the usual scale of 1.0 = D and 2.0 = C.
Calculus for Engineers is the traditional 2 year college calculus
course for freshmen and sophomores who will study math, physics,
and engineering (even fire protection engineering). Students need
to understand calculus on an A or B level in order to be
comfortable with the calculus aspects of later math, physics and
engineering courses; C is not sufficient. After only 11 weeks of
college, their low grades in Math 1A dashed the academic
aspirations of many black students at U.C. Berkeley. Various
remedial programs were tried without success.
Enter Uri Treisman. Over several years, he developed an
alternate learning environment for training students how to learn
both individually and cooperatively. Black participants in his
alternate learning workshops exceeded the achievement levels of
those white students whom he had not trained. Another result was
a significantly higher rate of graduation, especially in math
based fields, for the "workshop" black students over the rate for
earlier black students with very similar backgrounds and very
similar prior academic achievement. For the development of this
program, Newsweek, in 1989, named him one of three American
educators "on the leading edge of innovation" and in 1992, he was
named a MacArthur fellow.
How did Uri Treisman devise his winning strategy when a
wide range of traditional, remedial programs had failed to help
math non-achievers succeed? First, 400 math and science faculty
members were polled on the question: what were the reasons for
black studentsı low grades? The "off-the-cuff" faculty
responses were the usual "blame the victim" type reasons: the
students who flunk are (i) stupid, (ii) lazy, (iii) they do
not study much and (iv) they do not know high school mathematics,
etc. (Of course the faculty barely know the students.) Then, by
living in the student dorms for 18 months, Uri (and 5 cohorts)
observed 20 black students. He found that all the stereotypes
were wrong. These black students at U.C. Berkeley were smart,
highly motivated, and very hard-working; some had been
valedictorians. Many of these students had even studied calculus
in high school. They followed the "guidelines" by doing two hours
of study for each class period and by doing the assigned work. He
was the first white person to visit some of their homes; in doing
so, he found that each of them had at least one parent who
strongly encouraged their academic pursuits and urged them to go
to college. The majority of these students had a parent who was a
teacher.
Since these characteristics presuppose success, why were
these black students doing poorly in calculus instead of
excelling? Treisman discovered that
(i) these black students had good "study" habits but
ineffective learning strategies.
(ii) social isolation and other non-academic factors had
significant negative impact on their learning;
(iii) and many students attempted an ambitious academic
load while working many hours each week.
The Berkeley program includes important non-instructional
support and it makes a real effort to persuade incoming freshmen
to reduce their academic and work loads to a reasonable amount.
This article will concentrate on the pedagogical aspects.
A partial pedagogical analysis of high school education and how
well it prepares students for college
Many freshmen use the same strategies that earned them
success in high school. Pseudo-good "study" habits often includes
too much time spent memorizing too many formulas and
prescriptions for calculations and doing lots of computations with
an emphasis on getting the correct answer. These types of ³good²
study habits are usually rewarded by success in high school, but
they can be a trap in a college calculus class. As we will
note, ineffective study habits of this kind are commonly
inculcated in both black and white students in most high schools
by (i) the overemphasis on testing, skill development and fact
content and (ii) the underemphasis on understanding, reading
critically and synthesizing information
Also the skill-based instruction (which is popular in many
high schools) leaves the students stymied when confronted with a
problem that is slightly different from the ones they have been
trained to do. That solving a large number of conventional
exercises is largely ineffectual and turns misconceptions into
"bad" habits has been observed by Owen and Sweller (JRME Vol. 20)
and by Dr. Bernice Kastner of U.MD. (at Towson) resp.. In
contrast, good learning habits include trying to gain
understanding and trying to develop facility with the important
ideas and techniques of the subject.
At the request of the Prince Georgeıs county public school
system (in Maryland), a series of discussions were held by college
faculty members on the topic: What contributes to the failure of
college freshmen? The participants in the initial discussions
were faculty members (who have much connection with freshmen) from
10 departments of my university, together with high school
teachers and college freshmen). The main conclusion (Ref. #1) was
that: "The overemphasis on testing, skill development, and
fact level content, etc. [in high school] seems to have
inhibited [both white and black student] interest in
learning, motivation, ability to work with and enjoy
ideas, use creativity and attain satisfaction from an
educational experience." In a later discussion among college
faculty members (mostly from departments of speech and
communication), it was noted that: "Entering college freshmen
appear severely limited in their ability to read
critically, synthesize information, interact effectively
with both peers and instructors in academic settings, and
participate actively in discussions." (Ref. #7) This is a
natural consequence of these activities not being included in the
curriculum of most school systems.
Neither black nor white students read their math
textbooks. Only foreign students read the math textbooks, and
they do so, Uri Treisman discovered, to improve their English!
That there is little value in reading school textbooks was
documented in Harriet Tyson-Bernstein's book: The Textbook Fiasco;
A Conspiracy of Good Intentions. In Ref. #5, Prof. Davis of
Worcester Polytechnic Institute wrote: "The [high school
precalculus textbook] ... is no more mathematics than the noise
made by trained seals is music. But the trained seal approach
abounds in textbooks and in classrooms. It never provides a
foundation of fundamental ideas ... . It never offers
intellectual challenges, or chances to build confidence
and problem-solving skills." Of course students are able to
solve those math problems for which the text has supplied
cookbook-type directions, but there is little training in serious
problem-solving skills, that is solving problems that are
different than the examples worked out in the textbook. In this
way high school textbooks "set-up" students with a wrong
orientation for college.
High schools and colleges together provide 8 years of
education to many students. But there is little communication
between them. High schools and colleges are quite different types
of institutions, with different cultures, goals and views of
education. Traditionally, most high school graduates went on to
become clerks, secretaries or technicians; they did not become
college graduates. As is appropriate for the training of clerks,
secretaries and technicians, the emphasis (in many high schools
today) is still on acquiring skills. By contrast, in colleges the
emphasis is on acquiring understanding. This mismatch of goals
results in a difficult transition from high school to college for
many college freshmen. While the specific math, English skills
etc. that students learn in high school are crucial for college
work, the skill-oriented attitudes about learning are
counterproductive in college.
An excellent assessment of U.S. School Mathematics is
contained in the national report on the second international
mathematics study sponsored by the International Association for
the Evaluation of Educational Achievement (Ref. 3). Its main
conclusion was that the public school math curriculum is a
repetitive, fragmented, underachieving curriculum. There
is a "cause and domino effect" here, namely:
(i) Many textbooks use mainly a skill development approach
to learning.
(ii) Teachers model class instruction on the textbooks.
(iii) Students model their studying on their textbooks and
on their skill-development classroom instruction .
(iv) Students do calculations in a skill based manner
using many memorized formulas which results in much of
the mathematics being easily confused and easily
forgotten over the summer.
(v) So much is confused and forgotten over the summer
that it necessitates
(a) the Grade 6 arithmetic textbook starting with
addition of integers.
(b) the reteaching of much of Algebra 1 in
Algebra II
(c) the reteaching of much of Algebra 11/ Trig in
Precalculus
(d) the reteaching of Algebra 1 and Algebra II
to large numbers of college students. For example:
About 500 students retook Algebra 1 and another
1000 retook Algebra 2 at the University of
Maryland in 1990. (Having successfully completed
Algebra II (in high school) is a requirement of
admission to U MD)
(e) the reteaching of calculus in college to an
estimated 94% of the high school calculus students
(of 1982). (Ref. #9.)
(f) large numbers of students enter college
calculus classes with only moderate knowledge of
algebra and trigonometry; often their knowledge
includes many misconceptions. A major
contributory cause to the high failure rate in
calculus classes is that "fluency" not merely
moderate knowledge of algebra and trigonometry is
needed for success.
(vi) The large amount of class time devoted to
reteaching old topics greatly reduces the amount of
time available for new topics. The content of the
Algebra II/ Trig - Precalculus sequence has been cut
by a third since I went to school in the 1950's. This
results in an underachieving curriculum and our
children being taught far less than they are capable
of learning. If effective teaching methods were used
the Algebra II/ Trig course would become superfluous
since all its material is also taught in Algebra I
and Precalculus.
To be fair to the high school teachers, we note this
comparison: A community college instructor of Algebra II,
Trigonometry and Precalculus spends 12 hours per week in the
classroom teaching. A high school teacher of Algebra II,
Trigonometry and Precalculus may spend almost 23 hours per week
in the classroom teaching. Thus community college instructors
have almost an additional 9 hours each week for preparation.
High school teachers are not allotted the preparation time
needed to prepare exciting non-skill based lessons.
In reaction to the ineffectiveness of precollege
mathematics instruction, the National Council of Teachers of
Mathematics (NCTM) is recommending a new national math curriculum
(outline) called the "Standards". These NCTM Standards call for a
major deemphasis on skill based learning and rote practice
together with a new major emphasis on students gaining knowledge
and understanding.
In 1987, in reaction to the ineffectiveness of math
instruction, the Maryland State Dept. of Education was wise
enough to issue the following Official goals for math
instruction for the State of Maryland:
Goal #1 is that the students will "develop an
appreciation of and a positive attitude toward
mathematics".
Goal #2 is that the students will develop an
understanding of mathematics: concepts, properties and
processes.
Goal #3 is that the students will "acquire
mathematical facts and skills".
Goal #4 is that the students will "develop the
ability to express and interpret mathematical ideas and
relationships".
Goal #5 is that the students will "develop the
mathematical reasoning ability required in problem
solving and decision-making situations".
Goal #6 is that the students will develop the ability
"to apply mathematics in personal, societal, technological,
scientific and career settings".
Unfortunately, it is probably a rare school (even in
Maryland) that has started to implement these goals.
Prof. Perry of Harvard Univ. (Ref. #2) has found that
successful adaptation to college includes successive major
reorientations of the student's approach to learning. It
is a wise college that organizes its freshmen math courses in a
manner which facilitates the students reorienting their approach
to learning as they replace their skill-oriented attitudes and
their ineffective study habits with effective learning habits.
This is easy to say, but hard to do, especially with large
lectures. Uri Treisman's group learning workshops do this. Prof.
Clarence Stephens did this so well first at Morgan State
University (a traditionally black university) and then at the
State University of New York (SUNY) at Potsdam, that he was
honored by the governors of MD and N.Y. The students at SUNY at
Potsdam feel so good about math that 20% of the bachelors degrees
awarded there are in math compared to less than 1% nationwide
(This 1% is down from 5% in 1980). Unlike at most colleges,
remedial math courses are not taught at SUNY at Potsdam.
Basically many students study mathematics like this: They
skip the reading material and just start doing problems. If they
can't do a problem, they turn a few pages back in their texts in
search of sample problems to mimic, a practice which worked
wonderfully well with the skills-oriented high school texts. But
this strategy only half works for college calculus courses, with
their greater emphasis on understanding.
Back to Uri Treisman
Uri Treisman's team also intensively observed 20 first-
generation Chinese-Americans students (who grew up speaking
Cantonese) as they studied in the student dorms, and thus he could
compare their behavior patterns with those of both black and white
students. Most of these first generation Chinese-Americans
students used a different learning strategy: they regularly spent
two to three hours (per day) studying by themselves, and then
would come together in groups to critique each other's work and to
work on difficult problems. Separately, Professor Richard J.
Light found that participation in student organized study groups
was crucial for thriving at Harvard University (Ref. #11).
Uri Treisman then suggested to the black students that they
form study groups, but they felt that "one is supposed to be self-
reliant". This was also a reason why they did not make use of
the available free tutoring. As this is the American spirit of
rugged individualism many white students share this attitude.
Another reason is that many college freshmen do not consider
their fellow students to be a useful resource (Ref. #2) Dr.
Shearn of the U. MD Studies Skill Center has noted the antagonism
of freshmen students (in an elementary math class at U. MD.)
toward participating in study groups. As one student told her
"Juniors and seniors work together all the time, but you can't
expect freshmen to do it!" Dr. Shearn credits the compulsory
group work both for the fact that one class of students scored
especially high on the "uniform" final exam and that these same
students gave their instructor an especially low evaluation. If
an instructor does not "fully program" the students on how to
solve every problem (the expectation of the many students at
Perry's Position #1 (Ref. #2)) because the instructor wants the
students to learn to figure some things out for themselves, then
it is "natural" for the students to claim (on the teacher
evaluation sheets) that the instructor is not explaining the
material clearly. Times they are a changing. The recent
introduction of group learning in high schools has resulted in 36%
of the incoming college freshmen at my campus, looking forward to
forming study groups. (Ref. #8.) Dr. Hight observed that many
students, in remedial math classes at U MD, were study some
subjects with other students, but not math.
Uri Treisman noted that "I had come to question the
efficacy of individualized tutoring, self-paced
instruction, and short courses aimed at the development of
study skills - the traditional pedagogical arsenal of special
programs for minorities. I questioned also the wisdom of these
programs' remedial underpinnings: the focus on minority students'
weaknesses rather than on their strengths." He also observed the
high level of black students suspicion to programs
specifically aimed at helping them. Uri Treisman wanted a
program where minority students would excel and thrive not
merely survive. He believes students should both enjoy
mathematics and feel sufficiently self-confident about their math
ability that they will do extra mathematics problems for "sport".
Over several years he developed an alternate learning
environment for training college students how to learn. The core
of his program was twice weekly honors-type calculus workshops
where students were trained how to learn by participating in an
effective and efficient learning environment. (The workshops
which were composed 75% of blacks and Hispanics were in addition
to the traditional "review" or "quiz" sections.) He tried
aggressively to recruit all the black freshmen to join his honors-
type workshops. Uri Treisman noted that some educators
"considered the unabashed elitism of the new program and its
willingness to enroll seemingly unprepared minority students in
difficult mathematics courses to be unwise, if not downright
irresponsible".
As would be natural for an honors type program, he gave to
these potentially flunking students problems which were as hard
and harder than those given to the rest of the students. No easy
problems. To quote Treisman: "Not remediation; students hate
remediation." The students were asked to solve as many problems
as possible individually. But when a student got stuck, instead
of sitting there frustrated, he or she was to join with one to
four other students to work on the harder problems. When this
team could not solve the problem, they would then request help
from the instructor supervising the workshop. In this way, the
students spent half the time solving problems individually and
half the time cooperatively. Timely intervention by the
instructors saved the students from floundering, from much
frustration and from going on wild goose chases.
Speaking. The students were also required to explain
their solutions to their peers. This is especially important
since speaking is not one of the pillars (along with reading,
writing and arithmetic) of the American education system. In the
discussion held by college faculty members mostly from departments
of speech and communication, (Ref. #7) on why freshmen have
difficulties succeeding in college it was noted that:
"Freshman have great difficulty in expressing a logical,
coherent line of argumentation or in following another's
argument." Unfortunately for American students, elocution is
completely absent from their education. Clearer understanding is
obtained during the "give and take" discussions when a students
has to explain and defend his/her answers to others. Uri
Treisman summed up a basic premise underlying the workshops:
"Through the regular practice of testing their ideas on
others, students will develop the skills of self criticism
essential not only for the development of mathematical
sophistication, but for all intellectual growth."
Group learning is a non-graded, low stress, nurturing
learning system which can expose and then deal with the variety
of misconceptions about mathematics that students have collected
during previous mathematics classes. Group work enables the
student to work through more problems in a limited time period.
An instructor (with expertise in mathematics) will see to it that
the students avoid time-wasting inefficient approaches as well as
dead end approaches to problems. Therefore the amount and variety
of learning that occurs in a two hour workshop is usually far more
than what normally occurs in two hours of individual study.
Results in calculus at U.C. Berkeley. (1983-84)
For black students whose S.A.T. math scores were 550 or
higher:
Average grade %earning As, Bs % earning Ds, Fs
Workshops (22 students) 2.8 71 0
Non workshop 1.7 24 30
............................................................
Average S.A.T.math scores Average grade
Workshop black students 500 2.6
Non-workshop white students 640 1.9
Also, the group of black students whose S.A.T. math scores
were 460 or lower but participated in the workshops had a higher
percentage of A's and B's (30%) than the non-workshop black
students (24%) whose S.A.T. math scores were 550 or higher. The
D's and F's rates for non-workshop white and first generation
Chinese-Americans students were 15% and 10% resp.
Graduation and persistence rates among black students who had
taken calculus at U. C. Berkeley
SAT-Math Historical Workshop Non-Workshop
group participants participants
1973-77 1978-79 1978-79
200-460 36% 58% 33%
470-540 47% 59% 48%
550-800 48% 77% 49%
This chart counts the percentage (numbers) of black students who
had taken calculus at U. C. Berkeley and then went on to graduate
or were stilling enrolled at U.C. Berkeley in Fall of 1985.
(Ref. # 10.)
It took the ruse of an honors-type program to get the black
students to learn math together, as some of their Asian peers did
regularly on an informal basis. Once the black students changed
their learning strategy, in a effective, efficient learning
environment, they did very well, and not only in calculus, but
also in later courses. Then many white and Asian students demanded
to be admitted to this largely minority program! The workshops
were then expanded to include more non-minority students.
The moral underlying both Uri Treisman's results and those
of the Underachieving Curriculum (Ref. #3) is that American school
systems should abandon their practice of setting low goals. The
traditional teaching till "mastery" method with its too-easily
achievable goals is a prescription for frustration and boredom, as
well as stress caused by perfectionism. It is a play-it-overly-
safe approach to writing a curriculum. The natural result is our
underachieving precollege math curriculum; This is documented in
Ref. #3. Learning new things is always fun. Solving hard
problems raises self-esteem. The endless repetition of teaching
to mastery is drudgery, and takes time better given to expanding
the curriculum. Students cannot learn what is not taught. If
college courses were taught at the high school learning pace, it
would take students ten years to acquire a college education!
There is a further moral: Minorities are setting their
sights too low when they demand that the math achievement of their
children be raised merely to the current underachieving level of
American white children. In contrast, Uri Treisman tried to
change D and F students into A and B students not merely into C
students. While he did not fully achieve this goal, 2/3rds of the
black students in his program earned grades of A's and B's which
was significantly better than their white peers without group
learning.
Of course, organizing these workshops and the increased
instructor time costs money. At Berkeley, it may have taken a
faculty member three hours to prepare the problem set for
a single two-hour workshop. Important non-instructional
support is also included in the Berkeley program. While the cost
per student in these workshops is higher; the cost per student who
learns the material is much lower. The workshops are
especially cost effective when they results in students
completing engineering degrees instead of dropping out of college.
It would be highly cost effective for government to fund Treisman-
type workshops in all freshmen math courses. Doing this would
reduce the drop-out rate, reduce math anxiety and increase math
self-confidence. It would result in students being much more
comfortable with the mathematical aspects of all their courses.
This would enable many students to choose their majors freely
without first crossing off desired majors because of math
requirements.
There is an accounting question here of form versus
substance. Colleges usually allocate monies to a department
(for teaching) based on the numbers of students enrolled in its
courses not on the numbers who actually learn the material. The
workshops at Berkeley were part of a (separate) Professional
Development Program. (Also the faculty at U.C. Berkeley has
donated $50,000 to the program.) The funding basically needs to
come from state legislatures and the general college budget.
The pedagogical problem reexamined. Of course, this
raises a major pedagogical question: if last yearıs students had
considerable difficulty with the standard homework, how can giving
this yearıs students even harder problems help them? The usual
assumption is that they should start with easier problems and
slowly work their way up to the hard ones.
The answer: first of all, in doing interesting hard
problems students get practice in all the skills they need to
review. For example, doing short division exercises gives
students plenty of practice with both subtraction and
multiplication. Uri Treisman organized his workshops so that the
students would be learning much algebra and trigonometry along
with the calculus; he did this indirectly by purposefully
assigning calculus problems whose solutions required "mastery"
knowledge of algebra and trigonometry. In the course of trying to
do hard problems, the sources of the student's difficulties become
clear, and the instructor can address the weaknesses.
The second part of the answer is that by working together
in teams, the students help each other over all sorts of rough
spots, as they fill in each other's gaps, correct each other's
misconceptions and share their knowledge. Thus, four students
working together can and will solve many problems that they would
all give up on if working individually. In supervised group
work, the instructor gives much semi-personal attention to the
students in the form of "one-on-four" instruction along with
individual and class instruction. The workshops at Berkeley had
20 or fewer students; with classes of 25-30 students, an
instructor cannot run from one group to another fast enough to
provide the needed one-on-four instruction. I can't emphasize
enough how important it is that students get help with their
mathematical misconceptions in ungraded situations.
While I strongly believe that all seatwork should be team
seatwork, this by itself is not a panacea. Group work is being
treated as the current fad in some school systems. Sometimes this
means having students work together in small groups on straight-
forward textbook-type problems while receiving occasional guidance
from the instructor. Technically this is group learning and is
much better than individual seatwork, but this type of group
learning bears little substantive resemblance to Uri Treisman's
program.
Knowing how to work effectively with one's peers is an
important job and learning skill. Of course, students should be
taught not to make personal remarks when another student makes a
mistake. They should say "the mathematics is incorrect" and
correct it, not "you are stupid". Ethnographer Dr. Asera, who
works with Treisman, has noted; "False praise is never given [by
the workshop instructor]. Minority students are especially
sensitive to the underlying message when receiving inappropriate
praise for a simple task." (Ref.#4)
A college does not simply copy Treisman's program; each
college must organize a variation of this program that is
appropriate for its own environment. Currently 30 or more
universities in the U.S. are experimenting with variations of this
program. The most successful is the one at the University of
Texas where a Treisman type program resulted in the average score
of minority students becoming 20 points higher (on the common math
dept. final exam) than the average score of the white students
(who were not participating in the program). During the
implementation of a Treisman type program at the University of
Texas, Treisman was giving advice via telephone for more than a
hour each week. Striking results were also obtained in
experimental Calculus workshops at the University of Illinoisı
Chicago campus and at the City College of New York (CCNY). See
Ref. #6.
References
1. Report by Prof. Jim Greenberg, Director of Office of Outreach
Programs [to public school systems], Univ. of Maryland. (May
26, 1986).
2. William G. Perry, Jr., Cognitive and Ethical Growth: The
Making of Meaning
3. Curtis C. McKnight et.al. The Underachieving Curriculum -
Assessing U.S. School Mathematics from an International
Perspective, A national report on the second international
mathematics study sponsored by the International Association for
the Evaluation of Educational Achievement (1987).
4. Rose Asera, The Mathematics Workshop [at Berkeley]: A
description.
5. Paul Davis, Teaching mathematics and Training seals, SIAM News
(the newsletter of the [professional] Society for Industrial and
Applied Math.), (March 1987) page 7.
6. Rose Asera, The professional development program and a study in
adaption, Undergraduate Mathematics Education (UME) Trends
(1990)
7. John L. Brown, Curriculum Dialogues, Prince Georges County
Public School System, Upper Marlboro, Maryland (1987) Page 6.
8. In/sights - Research and information about UMCP students,
Spring 1991:#2
9. Don Small (Colby College) et al, Report of the CUPM Panel on
Calculus Articulation. CUPM = Committee on the Undergraduate
Program in Mathematics of the Math Assoc. of Amer.
10. Uri Treisman & R.E. Fullilove (1990) Mathematics achievement
among African American undergraduates at the University of
California, Berkeley: An evaluation of the mathematics workshop
program. Journal of Negro Education 59 (3): 463-478.
11. Richard J. Light, Harvard Assessment, Kennedy School of
Government, Harvard University, Cambridge, MA 02138
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Jerome Dancis is an associate professor of mathematics at
the University of Maryland, College Park, MD, 20742-4015.
Dr. Uri Treisman was director of the Professional
Development Program, Univ. of Cal., Berkeley, CA. He wrote "A
study of the Mathematics Performance of Black Students at the
Univ. of Cal., Berkeley" (1985). He is currently at the Univ. of
Texas at Austin.