Gregory F. Bachelis, Ph.D.
Professor of Mathematics,
Wayne State University
Detroit, Michigan [1]
This
report, and the six months I spent working on the underlying survey, is
dedicated to the students, past and present, of Andover and Lahser High
Schools.
I
would like to the thank several parents and others in the West Bloomfield and
Bloomfield Hills School districts, and in particular Mark Schwartz, Ph.D., for
their support and encouragement during the duration of this project
This
survey was conducted to compare how high school students with differing high
school math programs do in college. I
conducted the survey in my capacity as a Professor of Mathematics at Wayne
State University, where for 27 years I have taught and done research in
mathematics, and more recently also in mathematics education and theoretical
computer science. I conducted this
survey as a public service to parents, students and others in the Bloomfield
Hills School District, the West Bloomfield School District (where I live) and
other districts where new programs have been introduced which have caused
concerns among community members. It
was also done as a service to the mathematical community, whose two main organizations[2]
are closely following the evolution of these new math curricula and the effect
they are having on incoming college students.
I
sent the survey questionnaire (see below) along with a covering letter bearing
the letterhead of the Department of Mathematics of Wayne State University. I gave my office phone number for people to
call if they had any questions or concerns.
There was also a stamped return envelope, addressed to my office, with
which to return the survey. None of the
above made it a “Wayne State Survey," nor did I imply in any of my
communications with the people being surveyed that this was anything but my own
research project. Nor was any
implication given that Wayne State would endorse any of the research’s
conclusions.
In
addition, the research was not funded by Wayne State University. The stamps on the return envelopes were paid
for by interested parents; I did all of the work connected with the survey --
clerical, follow-up phone calls and e-mail messages, data transcription and the
like -- and I was not given release time from my usual duties in order to accomplish
all this.
The
report that follows is preliminary in nature. Further analysis needs to be done
on the data, and this is currently being done by professors at another
university. However, the report does include all the comments made by
all the respondents, with accompanying information supplied by them -- GPA’s,
SAT and/or ACT scores, college attended, college major, etc.-- in order to give
the comments a context. Since I
promised confidentiality to the respondents, I have blurred the contextual information,
in a manner explained later, so as to preserve their anonymity. I have also, in a few cases, paraphrased or
deleted portions of the comments in order to protect the identity of the
respondent. In each case this is
clearly indicated. None of the above
actions detract in any material way from the information that they provided.
In
closing, I wish to reiterate that this research project was performed by myself
as an individual faculty member of Wayne State University, conducting a survey
on what I considered to be a matter of public interest within the area of my
professional expertise.
The Bloomfield Hills School District
(BHSD) is located in Oakland County, Michigan.
It is a 25-square-mile area, which is comprised of virtually the entire
city of Bloomfield Hills, most of Bloomfield Township, a large portion of
eastern West Bloomfield Township, and a small part of the city of Troy. The district has two high schools, Andover
and Lahser, and three middle schools, West Hills, Bloomfield Hills and East
Hills. The middle schools all house
grades six through eight. West Hills
feeds Andover, East Hills feeds Lahser, and Bloomfield Hills Middle School
feeds both high schools.
In the fall of 1993, Andover High
School began what would be a four-year phase-out of its (non-accelerated) “traditional”
math program, which had been as follows:
·
Ninth
grade --- Algebra 1
·
Tenth
Grade -- Geometry
·
Eleventh
Grade --Algebra 2
· Twelfth Grade - Pre-Calculus[3]
In its place the Core-Plus Mathematics
Project (CPMP or Core-Plus) was installed; it is an integrated math program
using modeling, simulation and cooperative learning, which makes extensive use
of graphing calculators. Core-Plus was
phased in on a year-by-year basis, so that by the 1996-97 school year it was
the exclusive math program at Andover, with the exception of AP Calculus and AP
Statistics.[4] The latter courses are typically taken in
the twelfth grade by “accelerated students,” by which I mean those who take
Algebra 1 or Core 1 before the ninth grade.
“Integrated”
or “Reform Math” refers at the high school level[5]
to constructing the curriculum out of four “strands”:
·
Algebra
and Functions,
·
Probability
and Statistics,
·
Geometry
& Trigonometry,
·
Discrete
Mathematics.
These are woven together for a three or four-year
curriculum, rather than being taught as separate courses.
Core-Plus was introduced at Andover in 1993 as a pilot
project, which means that this was the first time it was used anywhere in an
actual classroom setting. According to
Professor Harold Schoen, evaluation Director of Core-Plus, “The pilot test was designed mainly to
provide feedback to the authors from teachers and students concerning what
worked well, what did not and what improvements were needed.”[6] In 1994 the field testing of Core 1 began
in 36 high schools in Michigan and around the country. Core-Plus was originally intended as a
three-year curriculum. However,
according to Marcia Weinhold, the Outreach Coordinator of Core-Plus, “During
their senior year, [the non-accelerated Andover] students studied three
prototype units for a possible fourth-year course that was envisioned by
Core-Plus. Thus, the curriculum these
students pursued was not a complete four-year curriculum.”[7] The field testing for Core 4 started in the
fall of 1998.
A
substantial portion of the Andover Class of 1997 consisted of students who had
been accelerated in math, and hence had not taken Core-Plus. These students as
a rule had taken Algebra 1 during the 1992-93 school year, while still in
middle school. So when this group
arrived at Andover in 1993, they rode the last wave of the traditional math
sequence while the non-accelerated students rode the first wave of Core-Plus. Lahser, the other high school, stayed
“traditional.” Core-Plus was introduced
at the middle school level, so that accelerated students destined for Andover
could take it in the eighth grade, in (I believe) January, 1994.
So,
in June 1997, the first class having completed four years of Core-Plus
graduated Andover High School. Subsequently,
reports of some of these graduates having difficulties on the math placement
exams at the University of Michigan - Ann Arbor (UMAA) and Michigan State
University (MSU) began to surface. On October 28, 1997, a joint meeting of the
Bloomfield Hills and West Bloomfield School Boards was held to discuss reform
math. After the featured speaker had finished
the main part of his presentation, several parents and others took to the floor
to express their thoughts and concerns about Core-Plus and about how this first
graduating class was doing in college.
I live in the West Bloomfield School District (WBSD), which had started
introducing Core-Plus, and phasing out their existing math program, in 1995.[8]
I had
participated in several meetings of my school board during the preceding year,
at which Core-Plus was discussed, and I attended this joint meeting. After attending a number of additional
meetings organized by parents, in both BHSD and WBSD, concerning the impact of
Core-Plus, and mindful of the intense scrutiny the mathematical community is
giving the evolution of such reform math programs, I decided to do a survey of
the 1997 graduates of Andover, in order to determine their opinions about
Core-Plus and to get as complete as possible a picture of their mathematical
experiences since graduation.[9]
I
surveyed the entire Andover class of 1997; the reason being that I did not know
a priori who had been accelerated and
who had not. Also, with the accelerated
students I could study how well students with a traditional high school math
background do in Reform Calculus courses such as “Harvard Calculus,”[10]
The latter is the flagship calculus course of UMAA; it is also taught at MSU,
but on a more limited basis.[11] One of the main claims of Core-Plus and
other programs of its type is that they are a better preparation for Reform
Calculus courses than the more traditional curricula.[12]
The survey commenced in late
April of 1998 and concluded in mid-September, as far as any activity on my part
soliciting responses. The covering
letter and survey questionnaire are given below in compressed form. (Most blank
lines and some lines for answers have been deleted.) The original questionnaire consisted of three pages plus the
optional section.
******************************************************************************************
|
College of Science |
Department of Mathematics Detroit, Michigan 48202 (313) 577-2479 (313) 577-7596 FAX |
April 22, 1998
Dear __________________,
I am conducting a survey of 1997 graduates of several high schools, including yours, who entered college in the summer or fall of 1997. We would appreciate your cooperation in this effort to evaluate how high school math programs are preparing students for college level mathematics. The results of this survey will be used for independent research regarding high school math curricula and individual names will be kept confidential. We are asking you for a few minutes of your time to complete the enclosed questionnaire and then to return it in the enclosed envelope. Please feel free to call me at my office at Wayne State at 313-577-3178, or to send e-mail to greg@math.wayne.edu, if you have any questions or concerns.
Thank you for your cooperation,
Gregory Bachelis, Ph.D.
Professor
********************************************************************************************
MATH SURVEY
1. High School graduated from in 199___ ________________________________High School
2. High School GPA ______ Honors or Awards__________________________________
3. Academic interests in high school 1.______________________2._______________
3. 4. 5.________________
4. Scores on SAT Math ________Verbal ________ PSAT Math______ Verbal ____
ACT ________ PLAN _______ Other (specify) ______________________
5. Did you take any Advanced Placement tests? Yes No . If so, please specify.
Subject Score Year Subject Score Year
________________ _______ _____ ________________ ______ _____
6. Did you take any ACT or SAT prep courses? Yes
No If so, please tell from whom
and give dates.:
________________________________________________________________________
7. Math courses taken in High School: (Please fill in the appropriate box with the grade(s) received.)
|
Year |
Alg
1 |
Alg
2 |
Geom |
Trig |
Adv
Alg |
Pre
Calc |
AB Calc |
BC
Calc |
Core
Plus 1 |
Core
Plus 2 |
Core
Plus 3 |
Core
Plus 4 |
Other
(specify) |
School
Name |
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Fresh |
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Soph |
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Junior |
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Senior |
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8. Did you receive any math tutoring while in high school? (besides SAT or ACT prep, if any) Yes No. If so, please give details on the next page. (Indicate if tutoring was private, provided in school, or by a commercial organization.)
I received tutoring in:
Subject _____________________ Year ____________ From ________________
9. Did you participate in any summer math programs during the years you were in high school?
If so, please specify subject, year, and who sponsored them.
______________________________________________________________________________
10. Did you enter college after graduating from high school? Yes No. If no, or if you haven’t taken any math courses or placement exams in college, please skip to question 16. Otherwise, please continue with questions 11 - 15.
11. Please specify any college math courses taken during the summer of 1997, or the 1997-98 academic year.
|
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Math Course |
College/University |
Grade |
Text * |
|
Summer
‘97 |
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Fall
‘97 |
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Winter
‘98 |
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*Please identify the text by listing the author or first author, if there are several. (e.g. Stewart, Thomas, Finney, Hughes-Hallett, Stein, Anton, Ostebee, Ellis, Edwards, Swokowski, Varberg, Larson, Dick, Wattenberg)
12. Did you take a math placement exam in college? Yes No . If so, please tell where taken, the nature of the exam and your score and/or in what course you were placed.
______________________________________________________________________________
13. Have you sought any math tutoring in college? Yes No If so, please give subject(s) tutored in and reason(s) for seeking tutoring.
______________________________________________________________________________
14. What is your intended major? _____________________________________________
15. Please answer the following two questions, when applicable, on a scale from 5 to 1.
a) Math courses I had in high school, other than calculus (if taken), helped me with my
college math courses (circle one)
5 4 3 2
1
very much somewhat not at all
b) Calculus I took in high school helped me with my college math courses. (circle one)
5 4 3 2 1 (Does not apply)
very much somewhat not at all
16. Please give any additional comments you wish to make concerning your math experiences in high school or college. ______________________________________________________________________________
______________________________________________________________________________
Thank you for your cooperation. There is an optional section on the next page. When you have completed this questionnaire, please return it in the enclosed envelope to
Professor Gregory Bachelis
Department of Mathematics
Wayne State University
Detroit, MI 48202
******************************************************************************
OPTIONAL SECTION
NAME ___________________________ AGE _________ SEX : M F
HOME ADDRESS________________________________________________________
COLLEGE ADDRESS (if different from above) _________________________________
May we contact you to obtain any further comments? Yes No
TELEPHONE NUMBER(S) ________________________________________________
E-MAIL ADDRESS __________________________________
******************************************************************************
In
the eighteen months prior to my decision to conduct the survey, I had spoken
out about Core-Plus. I was skeptical of
the claim being made that it was suitable for all students, and I was critical
of the fact that it was being implemented at certain schools to the exclusion
of almost the entire previously existing math courses; the latter was the case
at Andover and at West Bloomfield High School, the sole high school in
WBSD. I felt qualified to speak out,
since I have been a mathematics professor for over 30 years, and I have taught,
at one time or another, all of the subjects that are included in these new integrated
curricula. I felt that it was
impossible to do these subjects justice by cramming them into a three or four
year high school curriculum
However,
I do not feel that my voicing criticisms and concerns disqualified me from
conducting the survey. Certainly I am
capable of wearing different hats, and people doing surveys are entitled to
have opinions on the subject under study.
The key point is whether the survey is conducted in a fair and impartial
manner. In this regard, note that the
tone of the letter and questionnaire is quite neutral. There are no “loaded” questions. My follow-up phone calls and e-mail
messages, encouraging people to respond, were also quite neutral in tone. I
knew none of those surveyed beforehand. Also, only a few of them attend Wayne
State, which has about 17,000 undergraduates.
For these reasons there was no pressure on them to respond. In any case, the respondents to this survey
were not analogous to a jury being picked prior to a trial. They had had the
course, so to speak.
In
surveys of this type, there is a lot of
“noise” that needs to be filtered out.
If the Core-Plus graduates were the treatment group, then who to use as
controls? It would not have been fair
to the Core-Plus group to use the accelerated Andover students as
controls. Virtually all of the latter
had taken calculus in the 12th grade, and it had been determined that they were
among the better math students in their class - by exams, grades or other means
- or else they wouldn’t have been accelerated in the first place[13].
I
decided to survey the 1997 graduates of Lahser, the other high school in the
district, which had stayed traditional.
I could then use the non-accelerated Lahser students, or perhaps the
Lahser students who didn’t take Calculus in the 12th grade (which includes the
non-accelerated ones), as controls.
With the remaining Lahser students I could study the same question as
with the accelerated Andover students, concerning Traditional vs. Reform
curricula. I could also compare these
Lahser students with the accelerated Andover students, pre-Core-Plus.
I
believe my choice of control groups was a reasonable one. The populations of the
two high schools are similar socioeconomically, they are in the same school
district, and one of the middle schools even feeds both of them. In 1993, when Core-Plus was introduced at
Andover, students in the district could in fact choose which high school to
attend. I have only anecdotal evidence
as to what effect, if any, the advent of Core-Plus had on traffic between
Andover and Lahser, or out of the public system entirely to private schools
such as Cranbrook Academy or Detroit Country Day School, or to parochial
schools such as Marian High School or Brother Rice High School. In 1997, choice of high schools in the
District was ended for the time being because of an imbalance in favor of
Lahser.[14]
In 1997, there were 228 graduates of Andover High School,
and 258 of Lahser High School.[15] I determined the size of the population
being surveyed and response rate of each high school as follows:
·
1997:
228 total graduates
·
one
exchange student -- returned home
·
at
least four other students had left the country
·
probable
valid addresses for all but one of the remaining students
Therefore
student population size = 228 - 1 - 4 - 1 =
222
·
112
total replies
Response
rate = 112 total replies/222
population size = 50%
·
1997:
258 total graduates
·
one
exchange student -- returned home
·
probable
valid addresses for all but six of the remaining students
Therefore
student population size = 258 - 1 - 6 =
251
· 75 total replies
Response rate = 75 total replies/251 population size = 30%.
In
late April and early May the questionnaire and covering letter were mailed to
all 1997 graduates from both high schools.
A stamped envelope with my return address at Wayne State was
included. The initial mailing was
followed up by phone calls or e-mail messages (when e-mail addresses could be
determined). These follow-up contacts
were made by me, so that there would be consistency in the messages being sent
by e-mail or left on phone answering machines, and in the phone conversations.
Based on these contacts, a second questionnaire was often dispatched, as its
predecessor had been misplaced or discarded for sundry reasons. I have communicated in the ways indicated
above with over 80% of the 473 graduates (or in some cases, family members) who
were being surveyed.
One
can only speculate as to the reasons why some graduates did not respond to the
survey. The non-respondents might
include those who
·
for
one reason or another, never received the questionnaire;
·
were
apathetic, busy, etc.;
·
objected
to a survey in the first place;
·
didn’t
want to revisit high school issues.
I
wish to make the following three points.
1) I want to stress that I
was not trying to determine how well the various curricula were taught. This is certainly an issue, especially with
a radical new curriculum like Core-Plus.
I was simply trying to find out how the various curricula, as taught,
prepared the students for college math, their reaction to their high school
math experiences, and also how much extra help, such as tutoring, they sought.[16]
2) The information is only
that provided by the respondents, and has not been independently verified. They were promised confidentiality by me,
and I believe they made a good faith effort
to give accurate answers to the questions. Certainly a number of the comments were quite candid.
3) Many schools besides
Andover High School have phased in new math programs like Core-Plus, mainly in response to the
promulgation of the 1989 NCTM Standards[17],
although some of these schools have
also kept the traditional track, thus allowing for “choice” and for comparison
of the two curricula.
SOME STATISTICAL ANALYSIS
The Andover respondents fall
naturally into two groups:
·
I: non-accelerated (Core-Plus), and
·
II:
accelerated (virtually all of who took Calculus).
The Lahser respondents fall
naturally into three groups, since a lot of the accelerated students did not
take AP Calculus, although a number of them did take AP Statistics. These groups are
·
I:
non-accelerated,
·
II:
accelerated, no Calculus, and
· III: accelerated, with Calculus [18]
The answers to 15a) of Andover I are
compared to those of Lahser I and II in the following table. Recall that question 15 was:
15. Please answer the following two questions, when applicable, on a scale from 5 to 1.
a) Math courses I had in high school, other than calculus (if taken), helped me with my college math courses (circle one)
5 4 3 2 1 very much somewhat not at all
b) Calculus I took in high school helped me with my college math courses. (circle one)
5 4 3 2 1 (Does not apply) very much somewhat not at all
We have the following results.
Group Mean response Standard Deviation Number of Responses
Andover I 1.78 0.94 53
Lahser I 3.39 1.09 23
Lahser I&II 3.46 1.05 35 ________
Table I: Answers to 15a) for Andover I,
Lahser I, and Lahser I&II
Assuming linearity of the response
scale, this means that the Lahser students without calculus thought their high
school math was approximately twice as helpful with college math than the
Andover Core-Plus students did.
I should mention that a number of
respondents appear to have been confused about 15b) and to have thought that
the word “calculus” applied to college rather than high school. So, for example, all of the respondents in
the above three groups should have circled “Does not apply” in 15b), and this
was not the case. The answers to 15a)
and 15b) for Andover II and Lahser III are now compared.
______________________________________________________________________
Group Question Mean
response Standard Deviation Number of Responses
Andover II 15a) 3.81 1.13 36
Andover II 15b) 4.01 1.34 35
Lahser III 15a) 3.50 1.24 28
Lahser III 15b) 3.80 1.30 30
______________________________________________________________________
Table II: Answers to 15a) and 15b) for Andover II and
Lahser III
This means that all groups who had
high school calculus answered between 3.5 and 4 on average, when asked about
the helpfulness of high school calculus or of high school math before calculus,
and that Andover was slightly more generous than Lahser. Considering both
tables, we see that the average answer of each of the groups, except Andover
Core-Plus, was roughly 2 times more than Andover Core-Plus.
Further statistical analysis of the
data is needed, and this is currently being done by several professors at
Stanford University.
I have decided to report all the
comments verbatim, subject to the following protocol. I have corrected spelling errors and expanded abbreviations. In a few instances I have deleted a word or
phrase to protect the confidentiality of the respondent. These deletions are denoted “[...]”. In a few cases I have added a few words or
paraphrased. Such paraphrasing or
additions are enclosed in square brackets.
I have given the answer and
accompanying remarks to question 13 about math tutoring in college, when there
were remarks made that were worth noting, since this question did indeed
generate a lot of comments. I have also
included those parts of answers to question 12, which asks whether a math
placement exam was taken in college, that relate to the score received and the
resulting placement, since this is a matter of some controversy.[19] I have also included affirmative answers to
question 8 about math tutoring in high school, and excerpts from the answers
given, since this too has been a matter of some interest; to wit, did Core-Plus
generate more than the “usual” amount of math tutoring. In this regard, I have included affirmative
answers to question 9 about summer programs when they relate to summer school
as opposed to summer “math camps.” In
addition, I have given the answers to question 15a) for Andover I and Lahser
I&II and to questions 15 a) and b) for Andover II and Lahser III.
Besides the above, in order to give the comments more
context, I supply some information about the respondent, blurred somewhat so as
to preserve anonymity. This “blurring”
is accomplished as follows: High School GPA’s, and SAT and/or ACT scores, when
given, are reported in a certain range, rather than by exact value.
Colleges, except for UMAA (The
University of Michigan - Ann Arbor) and MSU (Michigan State University), are
reported by category, these being:
Michigan-Public,
Non-Michigan-Public,
Private,
Private-regional, and
Other (specialty schools, no college attended, or college unknown).
College majors are also reported by category, these being
Science (which includes
mathematics and psychology),
Engineering,
Business,
Education,
Nursing (which includes
medical technicians and physical therapy),
Design,
Fine & Performing Arts;