ZALMAN USISKIN

The UCSMP:Translating Grades7-12 MathematicsRecommendations

into RealityThe University of Chicago's School

Mathematics Project is developing curriculumto motivate middle-ability students.

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EDUCATONAL LEADERSHIP

I

In 1983 the Amoco Foundationgave the University of Chicagofunding for six years to improve

school mathematics in grades K-12.The result is the multifaceted Universi-ty of Chicago School Mathematics Pro-ject, aimed at preparing students in themiddle-ability range for the 1990s andbeyond. The project has since re-ceived additional funding from theCarnegie Corporation of New York,the National Science Foundation, andthe General Electric Foundation.

At the secondary level, the projecthas focused on designing and produc-ing usable and effective materials for acomplete mathematics curriculum forstudents in grades 7-12. This articleoutlines the curriculum developmentprocess and describes the motivationfor the major features of thecurriculum.

Recommendations and RealityAmong the many school reform re-ports published in 1983 were severalthat made specific suggestions regard-ing mathematics (National Commis-sion on Excellence in Education 1983and The College Board 1983). Sincethese recommendations largely mir-rored views of mathematics educatorsand mathematicians in reports pub-lished between 1975 and 1980 (NA-COME 1975, NIE 1975, NCSM 1977,NCTM and MAA 1977, NCTM 1980), theUniversity of Chicago project has notcreated its own set of recommenda-tions. Instead, it has attempted totranslate existing ones into the realityof classrooms and schools.

These recommendations respond totwo generally perceived problems inmathematics education in grades 7-12.The first is that high school graduatesare not learning enough mathematics.Specifically:

1. many students lack the mathemat-ics background necessary to succeedin college, on the job, or in dailyaffairs;

2. students do not get enough expe-rience with problems and questionsthat require some thought before an-swering; and

3. many students terminate theirstudy of mathematics too soon, notrealizing the importance of mathemat-ics in later schooling and in themarketplace.

These situations have given rise torecommendations to upgrade students

by increasing graduation require-ments and by improving students'problem-solving performance andtheir skills in arithmetic and algebra

Upgrading is not as simple as somereports and remedies suggest. For in-stance, in the past few years, in re-sponse to the first and third problems,many states and schools have in-creased requirements for secondaryschool graduation. Yet in 1983, beforemany of these increases went intoeffect, the one million seniors whotook the SAT had, on average, taken3.62 years of mathematics; by 1985 theaverage had become 3.68 (AdmissionsTesting Program of the College En-trance Examination Board 1983, 1985).Thus, increases in requirements affectmore non-college-bound studentsthan those who will attend college.Consequently, students with a recordof poorer academic performance arewinding up in courses intended toprovide preparation for college,courses in which they have littleinterest.

Furthermore, high school mathe-matics courses are characteristicallyamong the most severely gradedcourses with the highest failure rates.To respond to the second problem bygiving students harder questionswould seem only to exacerbate theexisting situation in which too manystudents experience failure.

The second general issue confront-

"The motivation [inthe project courses]often comesfrom real-worldapplications ofmathematics.Dealing withapplicationsprovides substantialand wide experiencein problem solving."

ed in recent reports is that the schoolmathematics curriculum has not keptup with changes in mathematics andthe ways mathematics is used in busi-ness, industry, and the marketplace.Specifically:

4. current mathematics curriculumshave not taken into account today'scalculator and computer technology;

5. students who do succeed in sec-ondary school mathematics are pre-pared for calculus but not for theother mathematics they will encounterin college;

6. statistical ideas are seen every-where, from newspapers to researchstudies, but are not found in most highschool mathematics curriculums;

7. the emergence of computer sci-ence has increased the importance ofa background in topics from discretemathematics, which is not usually ahigh school focus;

8. mathematics is applied to areasoutside the physical sciences as muchas inside, but these applications areseldom found in textbooks; and

9. estimation and approximationtechniques are important in all ofmathematics, from arithmetic on, buthave not been emphasized in thecurriculum.

These situations have led to recom-mendations to update the curriculum.Updating the curriculum is also noteasy to translate into reality, even if weignore the usual difficulties in imple-menting any new content. Time is theomnipresent enemy. Even with thepresent curriculum, the pace of pre-senting new mathematics content isfast for average students. The contentsuggested in problems 4-9 cannot becovered in a couple of weeks or in atextbook chapter. To cover the addi-tional topics in statistics and comput-ing currently recommended by theCollege Board (1985) would take upto a full year. Some help comes fromsuggestions to delete certain paper-and-pencil skills that can be donemore accurately and quickly by calcu-lators and computers (CBMS 1983,NCTM 1984), but learning to deal withthis technology takes time.

Finally, it is widely recognized thatstudents do not read mathematicsbooks and, thus, are not ready to learnmathematics on their own outside ofschool. This situation suggests that weneed to change the character of mathtextbooks.

DEcEMBER l986qsNuAiiy 1987 31

DECEFMBER 1986/JANUARY 1987 31

School Mathematics ProjectCurriculum, Grades 7-12The project's response to the problemof upgrading students is to developcourses for the average student-notfor those who are mathematically in-clined (as was the case with most ofthe new math curriculums of the late'S0s and early '60s)-nor for the low-est achievers (as was the case withmost of the back-to-basics curriculumsof the 70s). The target population forproject materials is perhaps best de-scribed as students who will graduatefrom high school but who are notadvanced placement students inmathematics.

Characteristics of all project courses(Table 1) reflect the target population.The motivation in them often comesfrom real-world applications of mathe-matics (motivations seldom found ei-ther in new math or back-to-basicstexts). Dealing with applications pro-vides substantial and wide experiencein problem solving. Skills are carefullyselected to complement the applica-tions and sequenced so that thoseneeded by a wider populace (e.g.,calculations of area and volume ingeometry) are introduced early andreviewed frequently, while thoseneeded simply for college-levelcourses (e.g., operations with rationalpolynomial expressions in algebra)are covered in a later year than iscustomary

Confronting the problem of time,project members searched for roomin the curriculum. We knew thateighth grade is a time at which manystudents are needlessly held back towait for algebra in the ninth grade.Furthermore, there is a substantial gapbetween the difficult eighth grade al-

gebra course offered to future ad-vanced placement students and thetypical course offered to most stu-dents. We hypothesized that manyeighth graders would be ready for anaverage algebra course, even moreready for an algebra course of the typewe might design, and still more readyif they were given a strong preparationin seventh grade. Studies of mathemat-ics performance in other countriesconfirm this view (McKnight et al.1985).

Our analyses of mathematics text-books in use in the U.S. support thiscontention. In typical seventh andeighth grade texts, an average of only20 percent of the first 200 pages con-tain any mathematics new to the stu-dent. (We counted anything as newcontent that was so indicated by pub-lishers-even computer exercises or"calculator corners" that many teach-ers skip.) Only 30 percent of the pagesof these entire books contain any newcontent. Thus, on the average, a teach-er who conscientiously follows thebook will introduce only one or twonew bits of content a week. In contrast,about 70 percent of the pages of atypical algebra course contain contentnew to the student that year, so alge-bra teachers are introducing new ma-terial three or four days out of five.Many ninth graders are destroyed notby the content of ninth grade algebra,but because this is their first encoun-ter with a mathematics course inwhich ideas are introduced at such abrisk pace. They do not possess thestudy habits necessary for success be-cause they have been lulled to sleep ingrades seven and eight.

We concluded that a curriculum totranslate today's recommendationsinto reality must begin at grade seven

at the latest, with algebra at gradeeight. This spreads out the learning oftraditional subject matter while offer-ing room for newer content (see Table2). Not all students are able to beginthis curriculum at grade seven; prelim-inary testing has shown the first-yearcourse to be appropriate for studentswho are at or above the achievementlevel of average seventh graders at thebeginning of that grade regardless ofage.

The general goal of the project'ssecondary curriculum is to preparehigh school graduates with the mathe-matics most relevant to all citizens andto all college majors. Paralleling thebreadth of mathematics found at thecollege level, the secondary curricu-lum includes substantial work withapplications of mathematics in allyears, statistics as an emphasis in thesecond and fifth years, computer ac-cess required in the fifth year andrecommended in all other years, andattention to those mathematical topicsbasic to an understanding of computerscience.

Today's mathematics textbooks contain very little reading. Students areforced to learn through the explana-tions of others, which goes against aprimary purpose of schooling, namely,to teach students to learn indepen-dently. In contrast, all lessons in pro-ject textbooks contain reading to re-late material from a lesson withprevious content, to introduce exam-ples, and to provide motivating infor-mation. Each lesson is followed byquestions specifically covering thatparticular lesson. Suggestions to teach-ers strongly recommend that they fre-quently ask students to read and an-swer questions before givingexplanations.

The Development ProcessTabr Currculum The process used to develop theseSecondary Curriculum materials has four major steps: plan-General Characteristics of Project Courses materials has four maor steps: plan-ning, writing a pilot, formative evalua-

To update content: * scientific calculators required and exploited in all years tion, and summative evaluation. Each* applications throughout step takes roughly a year (Table 3).* data handling, statistics, and probability given emphasis Project staff are involved rather obvi-* mathematics needed for computer science interspersed ously in each phase. For instance, they

To improve performance: * reading required attend professional meetings and visit* modified mastery learning sequence after each chapter school districts to obtain information* continual review within chapters, within years, and and guidance. Outlined here are thebetween years more formal ways in which school· multidimensional approach to understanding: skills, personnel (administrators, mathemat-properties, uses, and representations ics supervisors, and classroom teach__32ics supervisors, and classroom teach-32 EDUCnTIONAL LEADERSHIP

Speal CaApplied arithmetic, pre-algebra, and pre-geometryinterspersed

About 1/5 statistics interspersed, contrived wordproblems and complicated manipulations deemphasized

3 Geometry Measurement formulas, coordinates, transformations firstsemester; proof second semester

4 AdvancedAlgebra

5 Functions andStatistics withComputers

Graphic techniques and geometry emphasized;complicated manipulations deemphasizedContent interspersed; computer access required 1 hour/week for class; 1 more hour/week for students

6 Pre-College Manipulative skills, pre-calculus, and mathematicsMathematics needed for computers emphasized

ers) are involved.In the planning phase, each year we

convene a full-day meeting of 20-30school personnel to test whether ideasproposed by the project are reason-able and to come up with new strate-gies on particularly thorny issues. Todate we have held three meetings, onthe general themes of "Changing theCurriculum in Grades Seven andEight," "Changing Standards in SchoolAlgebra," and "Functions and Statistics

Course

1983-84

1984-85

1985-86

1986-87

Planning

TransitionMathematics

AlgebraAdvanced Algebra

GeometryFunctions andStatistics withComputers

Pre-CollegeMathematics

1987-88

in Secondary School Mathematics."These meetings generate important in-formation about sensitive points re-garding implementation. For example,administrators have told us to be boldbut to remember that our materialswill be judged by old tests, Teachershave told us that the best existingsoftware for doing manipulative alge-bra is not easy enough to use. At themost recent meeting, we were encour-aged to go ahead with a plan to re-

Table 3Development Schedule

First Draft/Pilot

TransitionMathematics

quire computer access by classes andstudents in our fifth-year course, butnot to chapge any content as a result Abonus of hese meetings is that almosteveryone who attends later seems will-ing to become pan of the testingprogram.

A curriculum project is judged by itsmaterials. Because materials can be nobetter than their authors, we havetaken great care to obtain the bestauthors available.i The algebra andadvanced algebra writing teams wereselected in a nationwide competitionadvertised in national newsletters. Allrespondents received an applicationthat required them to edit a givenlesson and to write a second lessonLWe brought finalists to the universityand evaluated them on their ability toplan with others and to write a lessonon the spot. It is noteworthy that 7 ofthe 11 members of these writing teamsare teachers or supervisors be-low the college level. Authors of theother texts are all experienced text-book authors. Each project editor is anexperienced secondary school mathe-matics teacher.

The wuiting/pilot phase is based onthe philosophy that, whenever possi-ble, all materials should be taught firstby their authors or surrogates of the

I _________________________

Revision/Formative Revision/Summative

TransitionMathematics

AlgebraAdvancedAlgebra

GeometryFunctions andStatistics withComputers

Pre-CollegeMathematics

1988-89

TransitionMathematics

AlgebraGeometry

Advanced AlgebraFunctions andStatistics withComputers

Pre-CollegeMathematics

1989-90

AlgebraGeometry

Advanced AlgebraFunctions andStatistics withComputers

Pre-CollegeMathematics

DECEMBER 1986/JANUASY 1987 33

Table 2Secondary Curriculum

Special Charaterhistia of Protect Courses

Year Course Name1 Transition

Mathematics

2 Algebra

DECEMBER 1986/JkNUARY 1987 33

An ef4hb grade student at bte Waft Dney Magnet Sc.ool uses a conrete modet of coordinateto piaue nmuUtiaion, an approach used in UCSMP Algebra.

authors. In this way, as many as possi-ble of those who revise the materialshave the benefit of firsthand classroomexperience. The actual logistics havediffered from course to course. Transi-tion Mathematics was written and thentaught by its author and one otherteacher. The first drafts of AdvancedAlgebra and Algebra were written dur-ing the summer, then edited and pilot-ed during the next school year. Thewriting of Geomeny and Functionsand Statistics with Computers beganabout four months before the begin-ning of the year in which they werepiloted.

After the pilot, the original authorsor editors revise the materials. Thencomes a formative phase, in whichmaterials are tried with "typical"teachers. Evaluators closely studythose teachers and their classes. Theymeet periodically at the university todiscuss materials, so schools partici-pating in this phase must be withincommuting distance of Chicago. Thepurpose of the formative phase is tolearn as much as possible for thesecond revision, which takes placeduring the following summer.

The summative evaluationsplanned for the first three years of thiscurriculum are major national studies.For example, the 1985-86 evaluation

of Transition Mathematics involvedover 2,000 students from 35 schools in10 states. In this stage, schools providenames of pairs of teachers willing toteach either project or their usual ma-terials; to avoid the "interested teach-er" variable, we randomly decide whoteaches which. In both formative andsummative evaluations, we interviewstudents, teachers, and administrators,observe classrooms, administer pre-and post-tests, and make results avail-able to the public.

At the time of the summative evalua-tion, our materials are available for

"We concluded thata curriculum totranslate today'srecommendationsinto reality mustbegin at grade sevenat the latest, withalgebra at gradeeight."

general use. We hold a conferenceeach fall at which users and prospec-tive users can find out the latest pro-ject news, share ideas, and commenton our work. The conference providesus with guidance and other informa-tion even from schools not in ourformal studies.

Alone But PromisingFrom the 1984-85 formative evalua-tion of Transition Mathematics, wefound that students using this text heldtheir own with arithmetic skills and faroutstripped their counterparts in read-iness to study either algebra or geom-etry. The formative evaluation teachersresponded positively to the materials,the reading in them, and the use ofcalculators, but they had the benefit ofbeing directly involved with us. As Iwrite this (September 1986), the testresults from the summative evaluationof Transition Mathematics are beingprocessed, and we expect to have datato report by early 1987. We arepleased that several districts in ourstudy have adopted these materials forlarge student populations.

Still, we are often asked if our workisn't pointless. What happened to allthose projects of the '60s? We respondthat virtually all commercial textbooksof the '60s are out of print. A projecttextbook should not be expected tohave a longer life than a commercialtextbook with a sales force behind it.Also, if the curriculum currently inschools were an experimental curricu-lum being tested, we would be forcedto pronounce it a failure for manystudents.

We think the project is affectingefforts in many states and on manyfronts to improve the mathematicscurriculum. Our materials demon-strate that some of the recommenda-tions of national commissions can beimplemented. Project test items havebeen sought by states and school dis-tricts who wish to evaluate students onnewer content. A number of majorpublishers have sent representativesto our meetings (we use their input,too), and before long we hope to seesome effect of our work in commer-cially published materials.

When we began our work in 1983,we expected to learn of other projectsof the same size and scope; to date weare alone. We feel pressured to per-form the seemingly impossible: to de-

EDUCATIONAL LEADERSHIP

velop a curriculum that is immediatelyimplementable, fully uses the latestwidely available technology, and pre-pares students for success in the fu-ture. We hope that the University ofChicago School Mathematics Projectwill spawn other large-scale efforts toimprove the mathematics curriculumin schools. Mathematics is too impor-tant today to the average citizen whomust deal with numerical informationand to the person who must use math-ematical techniques on the job to al-low them to leave school with insuffi-cient preparation. ]

i. Course leaders includeJohn McCon-nell (Glenbrook South High School, Glen-view, Illinois); Arthur Coxford (Universityof Michigan); Sharon Senk (Syracuse Uni-versity); James Schultz (Ohio State Univer-sity); and Rheta Rubenstein (RenaissanceHigh School, Detroit, Michigan)

References

Admissions Testing Program of the CollegeBoard. National Report on College-Bound Seniors, 1983, 1985 New York:

College Entrance Examination Board,1983, 1985.

The College Board Academic Prqdae/ onfor College: Wbat Students Need to Knowand Be Able to Do. New York: TheCollege Examination Board, 1983.

The College Board. Academic Preparationfor College. New York: College EntranceExamination Board, 1985.

Conference Board of the Mathematical Sci-ences. The Mathemaical Sciences Cur-riculum K-12: What Is Still Fundamntaland What Is Not? Report to the NationalScience Board Commission on Pre-Col-lege Education in Mathematics, Science,and Technology. Washington, D.C.:CBMS, 1983.

McKnight, Curtis C., Kenneth J. Travers, FJoe Crosswhite, and Jane Swafford."Eighth-Grade Mathematics in U.SSchools: A Report from the Second In-temational Mathematics Study." Arftbme-tic Teacber 32 (April 1985): 20-26.

National Advisory Committee on Mathe-matical Education (NACOME). Oterewand Analvysis of Scbool Matbematics:Grades K-12. Reston, Va: National Coun-cil of Teachers of Mathematics, 1975.

National Commission on Excellence inEducation. A Nation at Risk: The Impera-tie for Educational Reform Washing-

ton, D.C.: U.S. Department of Educaion,1983.

National Council of Supervisors of Mathe-matics. "Position Paper on Basic Mathe-matical Skills." Distributed to membersJanuary 1977. Reprinted in Matbe-matiCs Teaober 71 (February 1978):148-152.

National Council of Teachers of Mathema-ics. An Agenda for Aaon. Rest, Vaa:NCTM, 1980.

National Council of Teachers of Mathemat-ics. Co uting and Maematics, editedby James Fey. Reston, Va: NCIM, 1984.

National Council of Teachers of Mathemat-ics and the Mathematical Assoation ofAmerica. Recommendatos for thePreparation of Higb ScidoolSidnsforCollege Mathemati Courss. Reston,Va., and Washington, D.C.: NCTM andMAA, 1977

National Institute of Education. The NIE(Eudci Ohio) Conference on BasicMatematical SWis and Lem img. Vols.I and 2. Washington, D.C.: NIE, 1975.

ZaLman Usikiln is Professor of Educa-tion, Department of Education, Universityof Chicago. 5835 S. Kimbark. Chicago, IL6063'

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DEcEMBrR 1986/JANUARY 1987 35

Copyright © 1986 by the Association for Supervision and Curriculum Development. All rights reserved.