REFORM VS. TRADITIONAL MATH CURRICULA:*

 

 

 

PRELIMINARY REPORT ON A SURVEY

OF THE GRADUATING CLASSES OF 1997 OF

ANDOVER HIGH SCHOOL AND LAHSER HIGH SCHOOL,

Bloomfield Hills, Michigan,

CONCERNING THEIR HIGH SCHOOL MATH PROGRAMS AND HOW

WELL THESE PROGRAMS PREPARED THEM FOR COLLEGE MATH

 

Gregory F. Bachelis, Ph.D.

Professor of Mathematics,

 Wayne State University

Detroit, Michigan [1]

 

         

    

Dedication

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.

 

 

 

 

Acknowledgment

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

STATEMENT OF PURPOSE

 

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 Advent of Core-Plus in Bloomfield Hills

          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.

The Accelerated Students

          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.

Controversy over Core-Plus

          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.

Covering Letter and Survey Questionnaire

******************************************************************************************

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

 

 

Fresh

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Soph

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Junior

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Senior

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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.

 

 

Math Course

College/University

Grade

    Text *

Summer ‘97

 

 

 

 

Fall ‘97

 

 

 

 

Winter ‘98

 

 

 

 

 

 

 

 

 

   *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.

The Control Group

          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]    

Survey Mechanics, Population Size and Rate of Response 

          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:

Andover High School

·        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%

Lahser High School

·        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.

     SOME WORDS OF CAUTION                                                                                                              

          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.

The Comments

          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;

                   </