DOCUMENT RESUME

ED 389 887 CE 070 372

AUTHOR Brodsky, Stanley M.

fITLE Tech-Prep Mathematics in New York State.

INSTITUTION City Univ. of New York, N.Y. Center for AdvancedStudy in Education.

SPONS ACENCY New York State Education Dept., Albany. Bureau ofPostsecondary Grants Administration.

REPORT NO CASE-10-95PUB DATE Sep 95CONTRACT VATEA8080-95-0005NOTE 107p.

PUB TYPE Reports Research/Technical (143)

EDRS PRICE MF01/PC05 Plus Postage.

DESCRIPTORS College Preparation; Integrated Curriculum;*Mathematics Curriculum; *Mathematics Instruction;Secondary Education; State Curriculum Guides; *TechPrep

IDENTIFIERS *New York

ABSTRACTA study sought to determine the extent to which

mathematics content in tech prep programs in New York state matchedor differed from the content of the three Regents college preparatory

courses. Data were gathered through questionnaires received from a

sample of 24 of the state's 30 tech prep consortia providinginformation about 27 curricula. Some of the results included the

following: (1) almost 60 percent of the tech-prep mathematicscurricula reported using only applied courses and no collegepreparatory sequential mathematics courses; (2) only two mathematicscurricula were integrated into other subjects rather than being givenas separate courses; (3) 20 percent of the tech prep programsincluded either 2 or all 3 of the Regents college preparatorymathematics courses; (4) the curricula typically covered just over 40

percent of the topics in the Regents curriculum; (5) agreement with

State Framework Math Statements was substantial; and (6) anticipatedchanges to be brought about by various legislation and technologychanges will have an impact on the curricula of the tech prepschools. (The report includes 31 tables and figures representing thetopics covered in the various mathematics courses in the tech prepschools and in the mathematics framework.) (KC)

Reproductions supplied by EDRS are the best that can be madefrom the original document.

********).......%).***************************************************

TECH-PREP MATHEMATICSIN lsIEW YORK STATE

Stanley M. Brodsky, Ph.D., P.E.Project Director

U S DEPARTMENT OF EDUCATIONry . ,. 1,,I,11,c lir.ea, alEDUCATIONAL RESOURCES INFORMATION

IVCENTER ;ERIC)

This docum eM has been reproduCed asI ec. el ye d ham the poison or organilatirmoriginating it

0 Mins changes have been made tOimprove reproducliOn quality

Paints cif von, or opinions stald In thtsdocument do not necessarily reprer.entofficial OEDI position or policy

"PERMISSION TO REPRODUCE THISMATERIAL HAS BEEN GRANTED BY

V.,/ "'I

TO THE EDUCATiONAL RESOURCESINFORMATION CENTER (ERIC)"

This study was supported as part of the activities of the Tech-Prep TechnicalAssistance Center at the Center for Advanced Study In Education at the City Universityof New York Graduate School by a grant from the Bureau of Post-Secondary GrantsAdministration of the New York State Education Department, VATEA No. 8080-95-0005.

September 1995

CASE #10-95

I3EST COPY AVAILABLE

TABLE OF CONTENTS

Page

Table Of Contents ii

List Of Tables & Figures iii

Acknowledgements

Executive Summary vi

Tech-Prep Mathematics In New York State 1

Background 1

The Study 2

Assumptions 4

Findings 5

Coverage Of Topics From Sequential Mathematics Courses 6

Coverage Of Topics By Subject Area 7

Time Allotment Comparison: Tech-Prep vs. Traditional Courses 8

Curriculum Analysis By Subject Areas & Topics 10

Additional Topics Includei In Tech-Prep Programs 11

Instructional Materials 12

Responses To Framework Mathematics Statements 12

:Iiscussion 14

Tables & Figures 17 -- 62

LIST OF TABLES & FIGURESPage

Table 1 Tech-Prep Mathematics Curriculum Category 17

Table 2A Category A Tech-Prep Curricula With No Sequential Courses: N = 16 17

Topics In Sequential Courses Covered In Tech-Prep Mathematics

Curricula By Sequential Course

Table 28 Category B Tech-Prep Curricula With One Sequential Course: N = 5 18

Topics In Sequential Courses Covered In Tech-Prep Mathematics

Curricula By Sequential Course

Table 2C Category C Tech-Prep Curricula With Two Sequential Courses: N = 2 18

Topics In Sequential Courses Covered In Tech-Prep Mathematics

Curricula By Sequential Course

Figure 1 % Median Number Of Topics From Sequential Math Courses Which

Are Covered In Tech-Prep Curricula By Category & Sequential Course 19

Table 3 Number Of Topics In Sequential Math Courses By Subject Area 20

Table 4 Topics In Sequential Courses Covered In Tech-Prep Mathematics 21

Curricula By Subject Area

Table 5A Category A Tech-Prep Curricula With No Sequential Courses: N = 16 22

Time Devoted To Topics From Sequential Math Courses In Tech-Prep

Mathematics Curricula By Subject Area & Sequential Course

Table 5B Category B -- Tech-Prep Curricula With One Sequential Course: N = 5 23

Time Devoted To Topics From Sequential Math Courses In Tech-Prep

Mathematics Curricula By Subject Area & Sequential Course

Table 5C Category C -- Tech-Prep Curricula With Two Sequential Courses: N = 2 24

Time Devoted To Topics From Sequential Math Courses In Tech-Prep

Mathematics Curricula By Subject Area

Figure 2 % Median Time Used For Topics In Tech-Prep Curricula Compared To 25

Sequential Math Course "Standard" Time By Category & Sequential

Cou rse

Table 6 Logic Topics: Topics In Sequential Math Courses Covered In Tech-Prep 26

Math Curricula

Table 7 Number Sense & Numeration Concepts Topics: Topics In Sequential 27

Math Courses Covered In Tech-Prep Math Curricula

Table 8 Operations On Number Topics: Topics In Sequential Math Courses 28

Covered In Tech-Prep Math Curricula

Table 9 Geometry Topics: Topics In Sequential Math Courses Covered In Tech- 29

Prep Math Curricula(Continued)

LIST OF TABLES & FIGURES (Continued)Page

Table 10 Measurement Topics: Topics In Sequential Math Courses Covered In 33

Tech-Prep Math Curricula

Table 11 Probability & Statistics Topics: Topics In Sequential Math Courses 34

Covered In Tech-Prep Math Curricula

Table 12 Algebra Topics: Topics In Sequential Math Courses Covered In Tech- 36

Prep Math CurriculaTable 13 Trigonometry Topics: Topics In Sequential Math Courses Covered In 40

Tech-Prep Math CurriculaTable 14 Other Topics: Topics In Sequential Math Courses Covered In Tech-Prep 42

Math CurriculaTable 15 Topics From Sequential Math Courses Which Were Most Often And 43

Least Often Covered In 16 Category A Tech-Prep Math Curricula

Table 16 Additional Topks Covered In Tech-Prep Math Curricula As Reported 48

By Respondent;

Table 17 Instructional Materials In Tech-Prep Math Curricula As Reported By 50

Respondents

Table 18 Percent Agreement With New York State Framework Mathematics 51

Statements By Category And By Subject AreaFigure 3 Percentage Of Topics In Sequential Math Courses Which Are Covered 52

In A Tech-Prep Curriculum vs. Percentage Agreement With StateFramework Math Statements For Each Tech-Prep Curriculum

Table 19 Responses To Framework Mathematics Statements On Logic 53

Table 20 Responses To Framework Mathematics Statements On Number Sense & 54

Numeration ConceptsTable 21 Responses To Framework Mathematics Statements On Operations On 55

Number

Table 22 Responses To Framework Mathematics Statements On Geometry 56

Table 23 Responses To Framework Mathematics Statements On Measurement 57

Table 24 Responses To Framework Mathematics Statements On Probability & 58

Stat istics

Table 25 Responses To Framework Mathematics Statements On Algebra 60

Table 26 Responses To Framework Mathematics Statements On Trigonometry 61

Table 27 State Framework Mathematics Statements With Agreement By 62

50 % Or Fewer Of The Respondents

ACKNOWLEDGEMENTS

This study was triggered by an idea which Lee Traver of the New York StateEducation Department suggested and was endorsed by the Tech-Prep Technical Assistance

Center Advisory Committee consisting of Tech-Prep consortium personnel: Anne Gawkins,New York City Technical College; Kevin Hodne, SUNY College of Technology at Delhi;

Marilyn Kelly, Oneida-Madison BOCES; Constance Spohn, Capital District Consortium &Two-Year College Development Center at SUNY Albany; Suzy Tankersley, OnondagaCommunity College; and Kathleen Thompson, Eastern Suffolk BOCES. The plan andinitial draft of the questionnaires were guided by Dr. Bert Flugman, Director of the Centerfor Advanced Study in Education at the CUNY Graduate School and critiqued by BenLindeman of the New York State Education Department. James Donsbach and David

Martire, both of the New York State Education Department were consistently supportive ofthis effort. Two graduate students, Sherri Horner and Frederick Nassauer, Jr. providedexcellent direct assistance in data analysis, table construction and necessary research. Eachof those mentioned above had a substantial and positive effect on this study and I amgrateful to them all.

The Tech-Prep Project Coordinators of the 30 consortia in New York Statefacilitated this project by generally favoring the methodology, seeking and identifying

mathematics professionals from within their consortia to participate and respond to thequestionnaires, and providing follow-up when requested to do so. While not everyconsortium participated, those that did not participate did discuss their situations with me.

These folks are among the most dedicated professionals I have had the pleasure of workingwith. The mathematics faculty respondents provided their professional expertise to thisstudy and I salute them all for a job well done.

The study benefitted greatly from the work of several mathematics experts. I amgrateful to Dr. Arthur Kramer of New York City Technical College and Irvin Barnett ofBushwick High School. Susan Zakaluk, formerly Head of the Central Mathematics Unit atthe New York City Board of Education, helped greatly by providing her technical expertiseand advise as well as a comprehensive review of the draft. Helpful reviews of the draftreport were provided by Bernard McInerney (State Eduration Department), Jack Fabozzi(Two Year College Development Center), Kevin Hodne (SUNY Delhi), Linda Halligan

(Oneida-Madison Tech-Prep Consortium) and Richard Lukaschek (Bayshore High School,Eastern Suffolk Tech-Prep Consortium).

S. M. Brodsky, September 1995

TECH-PREP MATHEMATICS IN NEW YORK STATE

EXECUTIVE SUMMARY

This study was designed to determine the extent to which mathematics content in

Tech-Prep programs matched or differed from the content of the three Regents college

preparatory courses. In addition, the applicability to Tech-Prep mathematics curricula

of statements in "Framework for Mathematics, Science & Technology" was measured.

Secondary school mathematics was studied in a sample of Tech-Prep programs in

New York State. Useful responses were received from 24 of the State's 30 Tech-Prep

consortia providing information about 27 curricula on two extensive questionnaires and

a brief follow-up questionnaire.

One questionnaire listed 317 mathematics topics taken from the Regents college

preparatory curriculum series of three courses, with 89 topics from Sequential Math

Course I, and 114 topics each from Sequential Math Courses II and III. Respondents

were asked to indicate whether a topic was not included in the Tech-Prep program

(NO), was optional or the respondent was uncertain (?), or was included (YES). For

items answered YES, an estimate of the clock hours devoted to that topic was asked for.

A second questionnaire listed 92 statements taken from the "Framework for

Mathematics, Science & Technology" which is in draft form and was distributed for

discussion by the New York State Education Department in mid-1994. These statements

are grouped into 8 broad subject areas. While the State document provides statements

at three levels, elementary, intermediate and commencement, only the latter two were

included in this questionnaire.

The following are among the more interesting results.

Almost 60 percent of Tech-Prep mathematics curricula reported using only

applied courses and no college preparatory Sequential Math Courses.

Typical applied course titles were Tech-Prep Math and Applied Math. Only two

math curricula were integrated into other subjects rather than being given as separate

math courses.

Another one-fifth of the Tech-Prep curricula reported including Sequential Math

vi

Course I along with additional applied mathematics.

The remaining one-fifth included either two or all three of the Regents college

preparatory math courses.

The curricula with no Sequential Math Courses exhibited a wide range in the

percentage of topics covered with an overall median of just over 40 percent. The typical

curriculum included 65 of the 89 topics from Sequential Course I and another 65 out of

228 topics from the other two Sequential Courses, totaling 130 topics. Most of these

applied courses avoided proofs and derivations, but devoted more hours to the topics

than traditionally allotted. In many cases, these curricula covered additional math

topics, not included in the Sequential Courses, as well as other topics relevant to the

technical curriculum.

The curricula which included Sequential Math Course I, also tended to include a

larger number of topics from the other two Sequential Courses, with median coverage of

over half of the Sequential Course II topics and two-fifths of the Sequential III topics.

While these curricula also showed a rather wide range, the typical curriculum of this

type was clearly closer to the Regents course topical series (overall median coverage of

about 60 percent) than those with no Sequential Courses.

Mathematics topics were also grouped into the following subject areas which

mirrored the groupings of the statements in the State Mathematics Frameworks.

Logic topics

Number Sense & Numeration Concepts topics

Operations on Number topics

Geometry topics

Measurement topics

Probability & Statistics topics

Algebra topics

Trigonometry topics

Other topics not included above

For the curricula with no Sequential Courses, the three subject areas with the

lowest median percentage coverage were Logic (25%), Geometry and Trigonometry

(31% each). The three areas with the highest median percentage coverage for these

vii

curricula were Measurement (75%), Numeration Concepts (69%) and Operations with

Number (66%). However, the four subject areas with the largest number of topics

covered in all groups were Algebra, Geometry, Trigonometry and Probability &

Statistics which accounted for almost 80 % of the topics. Similar median percentage

and numerical trends were noted for the curricula with one Sequential Math Course.

One or more curricula with no Sequential Courses included 299 of the 317 topics

listed and required approximately 25 percent more class time than was judged to be

used in Sequential Courses for these same topics. Semester class time for a Sequential

Math Course is traditionally 120 clock-hours, but Tech-Prep applied math courses

ranged from 120 to 200 clock-hours with a median of about 140 hours.

Curriculum developers can use the subject area Tables 6 through 14 to find, for

each of the 317 topics, the number of Tech-Prep curricula with no Sequential Math

Courses which include that topic and the median number of hours reported for each

compared with a set of "standard" hours. In addition, for these same curricula, Table

15 lists by subject area those topics which are included in 75 % or more curricula and

25 % or fewer curricula; i.e. the most often and least often used topics in Tech Prep.

Agreement with State Framework Math Statements was substantial with an

overall median of 87 percent for the curricula with no Sequential Courses and 91

percent for those with one Sequential Course. The three subject areas with the lowest

percentage agreement for those with no Sequential Courses Vs1:1',. Logic (44%),

Probability & Statistics (66%) and Geometry (77%). All other subject areas had median

agreement percentages from 96% to 100%. Evidently, those involved with applied

curricula may agree with many of the broader Framework statements even though

coverage of specific topics may vary.

Readers who are interested in the detailed responses to the State Framework

Math Statements will find that information in subject area TaHes 19 through 26. In

addition, Table 27 lists those Framework Math Statements which received 50 % or less

agreement from respondents.

Several anticipated changes in Tech-Prep curricula are outlined in the Discussion

section. In addition, that section also treats the issue of curriculum decision rules which

may result in diverse mathematical curricular content.

viii

TECH-PREP MATHEMATICS IN NEW YORK STATE

BACKGROUND

Tech-Prep is a national program as part of the Vocational & Applied Technology

Education Act (VATEA) which encourages secondary and post-secondary institutions to

form consortia to create applied curricula which smoothly connect the 11th and 12th

grades through an associate degree in career fields. The legislation also allows

equivalent apprenticeship programs in lieu of associate degrees but the programs in this

state have focused on associate degree goals and beyond. The program fosters close

working relationships between high school and college faculty and counselors including

joint training, planning and curriculum development. Tech-Prep normally includes

development of workplace readiness skills, career education and exploration, initial

stages of workplace learning, educational preparation for rigorous associate degree

technical programs, and often opportunities for advanced standing through attending

college level work prior to completing high school.

The New York State Education Department launched Tech-Prep as part of the

newly reauthorized VATEA in 1991 when seven consortia were funded. In subsequent

years, an additional seven, then fourteen and finally two consortia were approved giving

a total of 30 operating consortia, geographically distributed across the state. In

addition, two Tech-Prep Technical Assistance Centers have been supported, one at the

Two-Year College Development Center at the State University of New York (SUNY) in

Albany, and the other at the Center for Advanced Study in Education (CASE) at the

City University of New York (CUNY) Graduate School in New York City.

The New York State Education Department was interested in assessing how

applied curricula in core subject areas compared with content in college preparatory

courses. A Technical Assistance Center Advisory Committee consisting of Tech-Prep

Consortium Project Directors endorsed the idea and recommended starting with

mathematics. The CASE Technical Assistance Center was asked to carry out this study.

1

THE STUDY

Each of the 30 Tech-Prep Consortium Project Directors was asked to select an

experienced mathematics person who was familiar with their Tech-Prep curriculum to

participate in the study. Those identified were sent two extensive questionnaires and

detailed instructions for completing them. In addition, the author spoke to each of the

potential respondents as well as the 30 Project Directors to ensure that the materials

were understood and that the math person would be able to respond in a timely

manner. A total -A 25 respondents from 24 of the state's 30 (80%) Tech-Prep consortia

provided usefv; information about 27 different Tech-Prep curricula. The remaining

consortia did not participate either because there was no available respondent or

because their math curriculum was not sufficiently developed to provide complete

responses to the questionnaires.

One questionnaire consisted of a listing of math topics from the three Regents

college preparatory courses in the state. These courses are officially designated as a

"Three Year Sequence for High School Mathematics, Courses I, II & III". In this

study, these courses will be called "Sequential Math Courses I, II and III" or variations

of that terminology. The topics in the questionnaire were listed by Sequential Course

with 89 topics associated with Sequential Math Course I, and 114 topics for each

Sequential Math Course II and III; a total of 317 items. For each topic, the respondent

was to indicate by check mark if the topic was not included in the Tech-Prep math

curriculum (NO), if the topic was either optional or the respondent was uncertain about

it (?), or if the topic was included in the Tech-Prep curriculum (YES). For each YES

response, the number of clock-hours used for that topic in the curriculum was to be

estimated to the nearest quarter hour.

The list of topics was taken from the 1988 New York City revision of Sequential

Courses 1, 2 and 3, which detailed lesson topics numbering 120, 120 and 130,

respectively. This source was selected over the New York State syllabi for these courses

because the latter were subdivided into many more topical items. Although the state

2

syllabi were more recent, use of the New York City version limited the length of the

content questionnaire while covering essentially the same material.

The topics were listed on the questionnaire in the order they appeared in the

source noted above. After the questionnaires were distributed to the various consortia

in early February 1995, three mathematics experts provided a classification of these

items into 9 subject areas:

Logic

Number Sense & Numeration Concepts

Operations on Number

Geometry

Measurement

Probability & Statistics

Algebra

Trigonometry

Other

Responses to the questionnaire were subdivided and analyzed by Sequential Math

Course from which the item was drawn and by subject area classification.

An expect in secondary mathematics assigned hours normally needed to cover

each of the topics in each Sequential Math Course with instructional hours totaling 120.

These hours represented a math course meeting for 180 instructional days in 40 minute

periods each day. These hours were designated as "standard" hours and were used for

comparison with those reported by the respondents.

The items in the topics questionnaire were arranged in three sections. Items 1

through 89 were from Sequential Math Course I, items 90 through 203 were from

Sequential Math Course II, and the last group from Sequential Math Course III were

items 204 through 317. To simplify the respondent's task, those whose Tech-Prep

curricula included any of the Sequential Math Courses were instructed to skip the

corresponding section of the questionnaire. Thus, the four respondents who reported

that their Tech-Prep curricula included all three Sequential Math Courses did not need

3

to give information for any of the 317 topical items.

The second questionnaire incorporated 92 statements from "Framework for

Mathematics, Science and Technology" issued by the New York State Education

Department as a draft for discussion. The statements, as issued, are given in three

levels; elementary, intermediate and commencement. Statements from the latter two

levels were included in the questionnaire as being appropriate to secondary grades 9

through 12.

The Framework statements were organized, in the publication noted above, into 8

subject areas:

Logic

Number Sense & Numeration Concepts

Operat;ons on Number

Geometry

Measurement

Probability & Statistics

Algebra

Trigonometry.

The items in the topics questionnaire were classified into the same set of subject areas

as the Framework statements after the topics questionnaire was distributed.

Respondents were asked to indicate for each statement if it applied to their

curriculum (YES), if it did not apply to their curriculum (NO), or if they were

uncertain (?).

A brief follow-up questionnaire solicited information about the number of

separate mathematics courses in their curricula, the course titles, the time devoted to

eacii course, and for integrated math, the subjects with which it is included.

ASSUMPTIONS

This study is delimited by the following assumptions.

All respondents were math professionals who were involved in one or more Tech-

4

Prep curricula and completed the questionnaire using their best professional judgement.

The topics listed in the questionnaire accurately describe the essential content of

the Sequential Math Courses I, II and III.

All topics in the questionnaire can be treated as though they are numerically

equal in importance to facilitate quantification. The question of interest here is the

comparison of the extent of coverage of the content of Sequential Mathematics courses.

For this purpose, no consideration of the relative importance was made, and they were

not weighted but rather treated as equal.

All topics listed in the questionnaire for a given Sequential Math Course were

included when such courses were taught in their entirety as part of a Tech-Prep

curriculum.

The "standard" hours assigned to topics in Sequential Math Courses in this study

accurately reflect the time distribution in such courses generally and these hours apply

when such courses are taught in their entirety as part of a Tech-Prep curriculum.

All Framework statements can be treated as though they are numerically equal in

importance to facilitate quantification, rather than being weighted in some way.

Both the intermediate and commencement levels of the State Framework

statements are applicable to secondary mathematics curricula.

FINDINGS

The 27 Tech-Prep curricula reported fall into four different categories as shown

in Table 1. About 60 percent of the curricula (Category A) do not require any

Sequential Math Courses, but include applied mathematics instruction. Another 18.5

percent (Category B) included Sequential Math Course I, usually with additional applied

mathematics instruction, while the remaining curricula included either two (Category C)

or three (Category D) Sequential Math Courses.

The respondents reported that applied mathematics instruction was most often

offered in separate courses with a variety of titles, the most common of which were

Tech-Prep Math and Applied Math. Only two programs had the math integrated into

5

other subjects rather than M separate math courses. Approximately half of the

curricula use materials developed by the Center for Occupational Research &

Development (CORD) of Waco, Texas either as the primary instructional material or as

a supplement to texts or teacher-made material. Most of the others rely on textbooks,

including all of the Category C & D, most of the Category B and several of Category A

curricula. A list of the instructional materials reported, including the titles of the 40

CORD units, is shown in Table 17.

COVERAGE OF TOPICS FROM SEQUENTIAL MATHEMATICS COURSES

Table 2A shows the range and median of the number of topics from the

Sequential Math Courses which are covered in the 16 Category A Tech-Prep

mathematics curricula; those with no Sequential Math Courses. Also given is the

median of the percentages of topics covered which is determined by dividing the median

number of topics covered by the total number of topics in that group. See the example

in the footnote to Table 2A which shows that the median Tech-Prep curriculum in

Category A covered about three-quarters of the Sequential Course I topics. This table

shows the extremely broad range of topical coverage within this category. Note that

while a median of three-quarters of the topics from Sequential Course I are covered in

Tech-Prep curricula, this measure drops to one-third for Sequential Course II topics

and to less than one-quarter for Sequential Course III topics. Overall, a median of

about two-fifths of the topics are covered. Thus, the typical Category A curriculum

includes about 65 topirs from Sequential Course I, and another 65 topics from

Sequential Courses II and III, totaling 130 topics from college preparatory courses.

The corresponding data for Categories B and C are shown in Table 2B and 2C. The

median curriculum with one Sequential Math Course (Category B) appears to

incorporate about three-fifths of the college preparatory topics, while those with two

Sequential Math Courses (Category C) show almost 90 percent coverage. Note that is

was assumed that all topics listed for Sequential Math Courses were covered in those

Tech-Prep curricula which included specific Sequential Courses. These latter two

categories (B and C) include very small numbers of programs, 5 and 2, respectively,

6

which suggests the probability that these data may be unreliable. Category D curricula

were assumed to cover all 317 topics in the three Sequential Math Courses which they

include and therefore no separate table for Category D topics was needed. Variations

from the assumption of full topical coverage when Sequential Math Courses are given as

part of Tech-Prep curricula may occur in some cases but were not measured. A

comparison of the median data from Tables 2A, 2B and 2C is shown graphically in

Figure 1. Category D curricula are represented by tars which reach the 100 percent

level. Category C data start at 100 percent for Courses I and II and then drops to 68.4

percent for Course III. Category B data start at 100 percent for Course I and then

drops to 52.6 and 36.0 percent. While tech-prep math curricula with no Sequential

Courses cover only a fraction of the topics in the college preparatory courses, the tech-

prep math curricula cover additional topics in math and other areas relevant to the

technical curriculum; see Table 16.

Coverage of Topics by Subject Area

The 317 topics from the three Sequential Math Courses have been subdivided

into nine subject area classifications as shown in Table 3. In order of total number of

topics, Algebra and Geometry predominate with more than half of the topics (91 and

87, respectively), followed by Trigonometry (39), Probability & Statistics (34), Logic and

Operations on Number (20 each), Number Sense & Numeration Concepts (11), and

Measurement (7). Eight topics were classified as Other.

The range, median numbers, median percentages and size ranks of subject area

topics covered in Tech-Prep curricula for Categories A, B and C are shown in Table 4.

The rank order of all 317 topics is somewhat different than the rank order of topics

covered in Tech-Prep curricula. While Algebra and Geometry continue to dominate as

1st and 2nd ranks, the remaining size rankings in Category A are different than for

Sequential Courses. Probability & Statistics, Number Sense & Numeration Concepts,

and Measurement all moved up one rank position to 3rd, 6th & 7th,

respectively, while Logic dropped back more than two positions to 8th and

Trigonometry moved one position down to 4th. Rankings for Category A and B were

7

similar except that in Category B Logic was down one-half position to 6th and

Numeration and Measurement were tied for 7th. Category C which included two

Sequential Courses matched the Sequential Course rankings with one small exception.

TIME ALLOTMENT COMPARISON: TECH-PREP VS. TRADITIONAL COURSES

The median number of hours devoted to Tech-Prep topics in each subject area

compared to the "standard" hours for the same topics is shown for Category A curricula

in Table 5A. In this table and the two which follow, the number of topics listed for

each subject area are those which are included in one or more Tech-Prep curricula of

the corresponding category. Thus, if a topic in the questionnaire is not represented in

any Category A curriculum, it is not included in this table and neither are its

"standard" hours or Tech-Prep hours. Furthermore, responses which indicated that a

topic was included in the Tech-Prep curriculum but provided no time estimate were

included in the No. TP TOPICS column but were not involved in either "standard"

hours or medians. These Tech-Prep programs appear to spend more hours than the

"standard" hours for each group of topics. The percentage of increased time in Table

5A for each Sequential Course group of topics is +22.6 % for I, +33.8 % for II and

+16.2 % for III, with an overall increment of +23.7%. This last number can be seen

in the TOTAL line under % TP MEDN, shown as 123.7. The percentage hourly

increments are calculated from the TOTAL line by dividing TP MEDN by STAND.

Certain Tech-Prep subject areas appear to involve from 70 to 100 percent more time

than Sequential Courses; Number Sense & Numeration Concepts (+100 %), and

Operations on Number and Measurement (+70 % each), although taken together these

three subject areas represent relatively few topics. Ten percent fewer median hours

than "standard" are devoted to Probability & Statistics by Category A curricula which

may imply somewhat briefer coverage with less problem solving time than is given

proportionately to other topics. The three mathematics subject :reas to which these

Tech-Prep programs appear to devote the heaviest time commitments are Algebra,

Geometry and Trigonometry. These topics together account for 66 percent of total

Tech-Prep math course time, and exceed Sequential Course "standard" hours by about

8

one-fifth for Algebra and Geometry, and by about one-third for Trigonometry.

Table 5B for Category B curricula, shows the time devoted to Tech-Prep topics

compared to "standard" hours. Note that the assumed time for topics in Sequential

Math Course I, which these Tech-Prep Category B curricula include, is equal to the

"standard" time. Therefore, the hours listed for Sequential Course I under the heading

TP MEDN are the same as those listed under STAND, as are both column totals.

However, the Tech-Prep median time allocations for Sequential Courses II & HI topics

which total 135.9 and 112.6 hours, respectively, exceed the "standard" hours of 109.8

and 100.8 by +23.8 % and + 11.7 %, respectively, with an overall increment of + 11.5

%. This difference is half as much as for Category A programs and may be attributed

to the inclusion of Sequential Math Course I in Category B programs which eliminates

the time differential for the first 89 topics. Note that the largest increment is for the

Sequential Course II topics, which was true of Category A as well. For these five

Category B curricula, the largest subject matter hourly median increment was in Logic

(+31.9 %), followed by Trigonometry (+19.7 %) and Geometry (+17 %). The major

time allocations were for Algebra, Geometry and Probability & Statistics which together

accounted for almost 70 percent of the hours in the median Tech-Prep math curriculum.

Table 5C shows only the overall data for two Category C curricula which include

Sequential Courses I & II in the Tech-Prep math program. The hourly allocations for

the two Sequential Course topics were, once again, assumed to be the "standard" hours.

Therefore, the only effect on the overall increment comes from the Sequential Course

III topical hours. With an overall increment of +11.9 %, topics for III contributed a

surprising +36.5 % more time than Sequential Course III topics. This may be partially

explained because most of the Trigonometry topics appear in the III group of topics

which show a median increment of +58.3 % over "standard" hours. Trigonometry

combined with Algebra and Geometry command almost 70 percent of the median Tech-

Prep hours for this category.

Figure 2 shows graphically the relations between the median percentage time

allocations for topics in all four categories of Tech-Prep curricula. Category A & B

9

curricula appear to follow somewhat similar patterns with higher incremental hours

devoted to Sequential Course II topics. One would have expected the increment for

Category C in group III topics to have been closer to those of the other two categories;

i.e. somewhere near +10 to +15 %. Instead, an increment of almost triple the expected

number was found. While this may be partially explained as noted above, these data

come from just two programs which were the only Category C curricula reported by

respondents in this study. Therefore, this result may not be typical of the group.

CURRICULUM ANALYSIS BY SUBJECT AREAS AND TOPICS

In order to provide Tech-Prep mathematics curriculum developers with a

maximum of useful information, the 317 topics used in this study are arrayed by subject

area ;4,3 Tables 6 through 14. Individuals may differ with the classification of topics into

the subject areas shown, but the data for each topic represent the responses from math

professionals who were unaware of the subject area classification when they responded.

The participants in the study and the 30 Tech-Prep Project Directors in New

York State should have a copy of the topics questionnaive and those who responded

would have copies of their own completed questionnaires. To relate the following nine

tables to a copy of a completed questionnaire, it will be useful to number the items from

1 to 317, consecutively. There is no need to number a blank topics questionnaire nor

does one need a blank questionnaire since all 317 topics are identified in Tables 6 -- 14.

These nine tables reflect the data for the 16 curricula in Category A, that is,

those Tech-Prep math curricula with no Sequential Math Courses. Tech-Prep personnel

whose curricula fall into Categories B or C might simply use the less detailed data in

Tables 4, 58 and 5C, or examine the appropriate group topics from II and or III in

Tables 6 through 14. Each of these nine tables gives for each topic in one subject area,

the number of curricula and the percentage of the 16 programs for which respondents

checked YES, indicating that the topic was being covered in their Tech-Prep math

program. In addition, the assigned "standard" hours are given for each topic, the range

of Tech-Prep hours reported for each topic and the median Tech-Prep hours for each

topic. Since not all respondents who checked YES for a topic provided time estimates,

10

the median hours were determined from those reporting the time, which may have been

fewer curricula than the number of YES responses.

Tech-Prep math study participants and consortium Project Directors who have

their completed questionnaires may want to compare their msponses with these tables

for several possible purposes, for example:

To see how their program compares with the statewide patterns

To examine whether to include or delete a series of topics based on what othershave done

To adjust their time allocations for certain topics or subject areas

To broaden the coverage of their program, where warranted

To include or delete Sequential Math Courses, or

To provide a basis for seeking Regents credit recognition for applied mathcourses via the variance process.

Others who did not participate in this study might simply work from the dataprovided herein to accomplish some of the same objectives mentioned above.

Table 15 provides a listing of the topics used by 75 percent or more of the

Category A curricula (12 or more of the 16) and the topics used by 25 percent or less ofthe Category A curricula (4 or fewer of the 16) of the 317 used in this study. In

viewing this table, one should keep in mind that more than 100 of the topics used in

Tech-Prep curricula are not shown on Table 15. The largest contribution to the topicsin this table which are used less often comes from topics involving proofs or derivations,

which many Tech-Prep math curricula minimize to allow for emphasis on applications

and additional topics related to specialized mathematics or a technical field.

ADDITIONAL TOPICS INCLUDED IN TECH-PREP PROGRAMS

Table 16 lists some of the additional topics used in Tech-Prep math curricula.Not all respondents provided information about such topics, but one would suspect thatmost Tech-Prep math programs do include some additional instructional topics. Thelargest group of items in this table were in mathematics. Although a few were actually

in the questionnaire list, some represented either advanced or specialized math topics.

11

In the technology group, items were related to manufacturing, drafting, surveying,

electronics, and health careers. A number of items dealt with applications of computers

and calculators, while others focused on measurements and instrumentation.

INSTRUCTIONAL MATERIALS

Table 17 lists some of the instructional materials which were reported. The most

widely used material is from the Center for Occupational Research & Development

(CORD) which has been adopted in whole or part in 13 of the 27 curricula in this study.

This material is organized into 40 units, as shown in Table 17, 36 of which were

available at the beginning of the 1994-1995 academic year. The units come with a

variety of back-up media and lab instruction. There are enough units to support three

applied math courses and several of the curricula have done precisely that.

Eleven of the curricula reported using particular textbooks. Specific texts were

reported used by 4 Category A, 2 Category B, 2 Category C and 3 Category D

curricula. Many of the textbooks listed in Table 17 were either supplemented by other

materials or were themselves supplementary. There does not appear to be very many

texts which would directly support Tech-Prep programs, perhaps because of the variety

of activities which are normally included in such programs as well as the tendency to

avoid single source and print-only instructional materials.

RESPONSES TO FRAMEWORK MATHEMATICS STATEMENTS

Table 18 shows the extent of the reported agreement of these Tech-Prep curricula

with the Framework statements. The median percentage agreement with statements

exceeded 85 percent over all 27 programs. The statements were classified by subject

area in the Framework document and were taken from the intermediate and

commencement levels, as noted earlier. Each statement was treated numerically

equally. The statements in the five subject areas of Algebra, Trigonometry, Number

Sense & Numeration Concepts, Operations on Number, and Measurement produced

strong agreement from the typical (median) respondent with either complete agreement

or nearly so. The statements for the subject area of Logic showed a median agreement

of only 50 percent. Curricula with Sequential Math Courses typically showed

substantial agreement with the Framework statements. In general, the respondents

were much more likely to check YES for a Framework statement, indicating that the

statement applied to their curriculum, than for a Sequential Math Course topic, which

would have indicated that the topic was part of their curriculum. This finding would

imply that the Framework mathematics statements apply broadly to Tech-Prep

programs with notable individual exceptions.

The dispersion of Tech-Prep math curricula reported when measured by both the

percentage of Sequential Math Course topics covered and the percentage agreement with

the Framework statements is shown graphically in Figure 3. Note that those programs

which include all three Sequential Math Courses come close to perfect matches with the

Frameworks while they were assumed to match the college preparatory topics. Many of

the Category A curricula cluster around the median data for both of these variables,

although a few show lower matching percentages on both variables. While there were

only two Category C curricula reported, they both showed somewhat lower Framework

agreements than the Category B programs and many of the Category A programs.

Tables 19 through 26 show the responses to each of the Framework statements,

with each table covering one of the eight subject areas. Within each table, the responses

from Category A curricula (those with no Sequential Math Courses) are given separately

and the other categories (B, C & D) are grouped. In each set of subject area data, the

number of responses and the percentage of the total (for that group) which agreed that

the statement was appropriate to the Tech-Prep math curriculum (YES), the number

that disagreed (NO), and the number of uncertain responses (?) are given for each

statement. The statements shown in these tables have been substantially abbreviated in

order to limit each to one line in the tables. The statements which were given in the

questionnaire were much more elaborate and represented the published Framework

statements closely. If one is interested in the applicability of any of the Framework

statements to Tech-Prep mathematks curricula, these eight tables will provide that

information.

Table 27 shows selected Framework statements which received 50 percent or

13

24

fewer YES responses from either the 16 Category A curricula or the 11 group of

responses from Categories B, C & D combined. The latter grouping represents all those

programs with at least one Sequential Math Course. Of the 12 statements listed which

received 50 percent or fewer YES responses, Logic and Geometry each had 4, although

one of the Probability & Statistics statements was the only entry in the Category BCD

column. Several statements seemed to cause respondents some uncertainty with

numbers of responses (grouped for all 27 ABCD curricula) ranging from 4 to as high as

7. These are listed in the No. ? column in Table 27. It is likely that some statements

may have caused confusion and perhaps need revision, although the uncertainty about

"Validate formulas used to compute measurements" is surprising.

DISCUSSION

A good deal of previously unmeasured information has been gathered from

knowledgeable mathematics professionals regarding the nature and content of Tech-Prep

mathematics curricula in New York State. While these data are believed to be accurate

for the spring 1995 semester, it is clear that several changes are in process giving these

programs a dynamic element. Among these anticipated changes are:

The new School-To-Work Opportunities Act which 'NM impact Tech-Prep

curricula in several predictable and other unpredictable ways. A number of the New

York State Tech-Prep consortia are involved in School-To-Work partnerships which

may change the way they operate. There will undoubtedly be an increase in workplace

learning which might include some elements of the mathematics curricula.

The increasing use of calculators will impact the way in which certain topics are

taught. This year, scientific calculators are to be used in Regents exams. One expects

that there will be changes in the "standard" time distributions and the relative

importance of topics.

In New York City, the Board of Education has approved a change which will

phase in a requirement that all students take the Sequential Math Courses. This corr ing

fall (1995) the requirement for Sequential Math Course I will be implemented and the

14

others will follow in successive years. Currently, two of the five New York City Tech-

Prep consortia include Sequential Math Courses (Category D; all three Sequential

Courses) and shortly they will be joined by the other three.

Approximately half of the respondents reported using math instructional

materials produced by the Center for Occupational Research & Development (CORD)

either solely or as supplements to texts or other materials. Several respondents noted

that they were awaiting one or more of the new units which CORD had under

development at the time they responded to this study. The units mentioned specifically

were Logic, Coordinate Geometry, and Transformations, all of which showed limited

current inclusion in the math curricula, particularly in the Category A (no Sequential

Math Courses) programs.

Several Tech-Prep programs are in their 4th or 5th years of operation and are

planning to institutionalize their programs or have done so in large measure. Some

other consortia may find it imperative to institutionalize their current programs due to

reduced or eliminated external funding even if they may be at early stages. There may

be consortia which are planning a major expansion to meet the mandate in the newer

legislation to serve all students. Still others may terminate their Tech-Prep articulation

agreements and shut down the programs.

Those mathematics faculty in Tech-Prep or similar programs who make use of

the detailed tables of mathematics topics, particularly Tables 6 through 14, as well as

those who re-examine the number and timing of their math courses may decide to

consider changes in their programs as a result of this report. To the extent that this

happens, the report may be thought to have a useful impact.

The wide range of mathematics topics which are revealed in this study may be

attributable to the nature of decision rules related to curriculum planning for career

paths for associate degree graduates in business, allied health, engineering technology

and industrial technology fields. Associate degree programs in some of these fields are

governed by specialized accreditation requirements which influence many curriculum

decisions. Other fields may be shown not to be math dependent and curricula for these

15

areas often place emphasis elsewhere in the curriculum and lessen the mathematics

demand. Some Tech-Prep curriculum planners insist on the strongest mathematics

content for their program espousing the view that everyone should take college

preparatory mathematics, regardless of their occupational goals. It appears that all of

these decision rules have operated to some extent in the Tech-Prep programs sampled in

this study.

The New York State Education Department which promulgated the Framework

statements in draft form for discussion may find the tables which deal with the

Framework statements of some use in considering future editions of the "Framework for

Mathematics, Science and Technology". The pertinent tables are Tables 19 through 26

and Table 27 in particular which shows several of the statements which appear to

deviate from what the respondents felt were valid for their Tech-Prep math curricula.

There were a few statements noted in Table 27 which indicated some uncertainty on the

part of a cluster of respondents which may be sufficient cause to re-examine those

statements.

The cooperative response which this study received from the Project Directors

and the mathematics faculties across the state is an indication that Tech-Prep is

important to these practitioners, and by extension to the students of New York State.

Tech-Prep has typically generated significant enthusiasm from all concerned and has

been responsible for a long list of innovative approaches to secondar:,. and community

college education. While this study has focused narrowly on mathematics courses and

their content, one should be aware of the joint curriculum planning in all subject areas

by secondary and post-secondary faculty working together -- itself an important

innovation. These efforts involve visits by students, faculty and counselors to each

other's institutions as well as a good deal of bridging into college courses by secondary

students prior to their graduation. While the mathematics respondents seemed to

indicate relatively little computer activity in math courses by students, there is extensive

computer use throughout most of the T-ch-Prep programs, most often in the

communications, science and technology components.

16

TABLE 1

TECH-PREP MATHEMATICS CURRICULUM CATEGORY

CURRICULUM NO. OF PERCENT OFCATEGORY CURRICULA CURRICULA

A 16 59.3

B 5 18.5

C 2 7.5

D 4 14.8

TOTAL 27 100

A = Curricula with no Sequential Math Courses; All Math in Applied Courses

B = Curricula with Sequential Math Course I and other Applied Math

C = Curricula with Sequential Math Courses I & II and other Applied Math

D = Curricula with Sequential Math Courses I, II & III

TABLE 2A

CATEGORY A -- TECH-PREP CURRICULA WITH NO SEQUENTIAL COURSES(N 16)

TOPICS IN SEQUENTIAL COURSES COVERED INTECH-PREP MATHEMATICS CURRICULA BY SEQUENTIAL COURSE

SEQUENTIAL No. TOPICS COVEREDCOURSE RANGE MEDIAN % MEDIAN*

I 21 -- 88 67.5 75.8

II 11 -- 76 38.5 33.8

III 2 -- 96 26 22.8

TOTAL 37 -- 251 130 41.0

* % MEDIAN = Median No. of Topics / Total No. of Topics in the Sequential Course

e.g. For Course I, % MEDIAN = 67.5 / 89 = .758 or 75.d%

17

2 I,

TABLE 2B

CATEGORY B TECH-PREP CURRICULA WITH ONE SEQUENTIAL COURSE= 5)

TOPICS IN SEQUENTIAL COURSES COVERED INTECH-PREP MATHEMATICS CURRICULA BY SEQUENTIAL COURSE

SEQUENTIAL No. TOPICS COVEREDCOURSE RANGE MEDIAN % MEDIAN*

I 89 89 100

II 26 -- 88 60 52.6

III 24 -- 87 41 36.0

TOTAL 149 -- 264 190 59.9

* See Footnote to Table 2A

TABLE 2C

CATEGORY C TECH-PREP CURRICULA WITH TWO SEQUENTIAL COURSES(N = 2)

TOPICS IN SEQUENTIAL COURSES COVERED INTECH-PREP MATHEMATICS CURRICULA BY SEQUENTIAL COURSE

SEQUENTIAL No. COVEREDCOURSE MEDIAN % MEDIAN*

I 89 100

II 114 100

III 78 68.4

TOTAL 280.5 88.5

* See Footnote to Table 2A

18

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NUMBER OF TOPICS IN SEQUENTIAL MATH COURSESBY SUBJECT AREA

NUMBER OF TOPICSSEQUENTIAL COURSE

SUBJECT 1 II III TOTAL % of TOT RANK

LOG 7 12 1 20 6.3 5.5

NUM 5 ---- 6 11 3.5 7

OPS 13 1 6 20 6.3 5.5

GEO 16 52 19 87 27.4 2

MEA 2 3 2 7 2.2 8

PRB 15 9 10 34 10.7 4

ALG 29 31 31 91 28.7 1

TRG ---- 6 33 39 12.3 3

OTHER 2 6 8 2.5 ----

TOTAL 89 114 114 317 100

LOG = Logic TopicsNUM = Number Sense & Numeration Concepts TopicsOPS = Operations on Number TopicsGEO = Geometry TopicsMEA = Measurement TopicsPRB = Probability & Statistics TopicsALG = Algebra TopicsTRG = Trigonometry TopicsOTHER = Topics Not Included Above

20

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148

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the

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oble

ms

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GE

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07G

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etri

c T

rans

form

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ns w

ith M

athe

mtc

l Sys

tms

308

Glid

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9Is

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rics

310

Sim

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rans

form

atio

nsPR

B26

Wha

t are

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etw

n L

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roba

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0Pe

rmut

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ns w

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epet

ition

s19

4A

pply

ing

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bina

tions

to C

ount

ing

195

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lyin

g C

ombi

natio

ns to

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met

ry19

6U

sing

Com

bina

tions

to S

olve

Pro

babi

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6T

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s in

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lity

Prob

lem

s29

8Pr

obab

ility

of

an E

vent

in S

eque

nce

of I

ndep

Tri

als

299

Use

Bin

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l The

orem

to S

olve

Pro

babi

lity

Prob

lms

300

Exp

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ent!

Dat

a to

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ify

Prob

abili

ties

301

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ion

AL

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raph

s &

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ts7

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s10

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dditi

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ve P

rope

rtie

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btra

ctio

n Pr

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ty16

9E

quat

ion

of a

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us17

2Fi

ndin

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quat

ions

of

Loc

us in

the

Coo

rdin

ate

Plan

e17

4In

ters

ectio

n of

Loc

i in

the

Coo

rdin

ate

Plan

e18

0C

ompl

etin

g th

e Sq

uare

181

Der

ivin

g th

e Q

uadr

atic

For

mul

a18

2N

atur

e of

the

Roo

ts o

f a

Qua

drat

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quat

ion

183

Rel

Bet

wn

Coe

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ient

& R

oots

of

Qua

drat

ic E

quat

184

Find

ing

Poin

ts o

f In

ters

ectio

n of

a P

arab

ola

& L

ine

185

Solv

e O

ther

Qua

drat

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inea

r Sy

stem

s G

raph

ical

ly18

6So

lvin

g Q

uadr

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ear

Syst

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ebra

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ly19

8Id

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203

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227

Sum

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278

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quat

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279

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lyin

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blem

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wth

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61Fi

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gent

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262

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gore

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263

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met

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titie

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quat

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266

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282

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aw o

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nes

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287

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quen

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thm

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8 2

TABLE 16

ADDITIONAL TOPICS COVERED IN TECH-PREP MATH CURRICULAAS REPORTED BY RESPONDENTS

SUBJECT TOPIC

MATHEMATICS 3 Equations, 3 Unknowns (Listed by 1 A, 1 B)Matrices (1 A)Surface Area (1 A)3 Dimensional Shapes (1 A, 1 B)Application of Statistics (1 A)Designing a Statistical Study (1 A, 1 B)Applicatiors of Formulae (1 A)Inverse Proportions (1 A)*Ratios & Proportions (1 A)[Topic 137]*2 Equations, 2 Unknowns (1 B)[Topics 78, 81]Topics in Jr. High School Math (1 A)Binary Numbers (1 A)*Systems of Inequality -- Graphically (1 B)[Topic 801Exponents (1 B)Constructing Models in Geometry & Trigonometry (1 B)*Area of Any Triangle (1 C)[Topic 67]Piece-wise Function (1 C)

COMPUTERS &CALCULATORS

Computer Spreadsheets for Statistics (1 A)Computer Spreadsheets -- General (1 A)Using Scientific Calculators (1 A, 1 B, 1 C)Computer Graphics (1 A)Using Computer Graphics to Solve Practical Problems (1 A)Using Scientific Calculator for Linear Regression, Linear

Programming, Quadratic Progression (1 C)

MEASUREMENT Converting Measurements (1 A)SYSTEMS & Using Vernier & Micrometer Calipers (1 A)INSTRUMENTATION Using Stopwatches, Rulers & Line Levels (1 A)

Metric & English Measurement Systems (1 A)English -- Metric Conversion (1 B)

COMMUNICATION &CAREER EDUCATION

Oral & Written Presentations (1 A)Communication (1 B)Career Exploration (1 A, 1 B)

(Continued)

48

83

TABLE 16 (Continued)

ADDITIONAL TOPICS COVERED IN TECHPREP MATH CURRICULAAS REPORTED BY RESPONDENTS

SUBJECT TOPIC

TECHNOLOGY Precision, Accuracy and Tolerances (1 A, 1 B)Drawing Detailed Scale Drawings (1 A, 1 B)Quality Assurance & Process Control (1 A)Outdoor Surveying (1 A)Outdoor Design Mapping (1 A)Using Drawing Instruments (1 A)AND/OR Gates in Electronics (1 A)Euler Circuits (1 A)Survey Taking & Data Analysis in Health Careers (1 A)Measurements in Health Careers (I A)

MISCELLANEOUS Preparation for RCT Examination (1 A)Managing Personal Finances (1 A)Map Coloring (1 A)Graph Coloring (1 A)

* Denotes Topics Reported as Additional Which Were Listed in Questionnaire

49

84

TABLE 17

INSTRUCTIONAL MATERIALS IN TECH-PREP MATH CURRICULAAS REPORTED BY RESPONDENTS

TEXTBOOKSAlgebra 2. Larson, R., Kano ld, T & Stiff, L. Heath. Lexington, MA. 1993.

Curriculum k Evaluation Standards for School Mathematics. Addenda Series. Grades 9-12.

National Council of Teachers of Mathematics. Reston, VA. 1995.

Elementary Algebra for College Students. Angel, A. Prentice-Hall. 1991.

Integrated Math Course I. Dressler, I. & Keenan, E. Amsco School Publishing, Inc. New

York. 1980.

Integrated Math Course II. Keenan, E. & Dressler, L Amsco. New York. 1980.

Integrated Math Course HI. Keenan, E. & Ganter, A. Amsco. New York. 1980

Integrated Mathematics Course I. Kelly, D., Atkinson, P., & Alexander, B. McDougal, Littell

& Co. Evanston, IL. 1991.

Integrated Mathematics Course II. Kelly, B., Atkinson, P., & Alexander, B. McDougal, Linen

& Co. Evanston, IL. 1991.

Math Connections. Gardena, F. Houghton-Mifflin, 1992.

Mathematics: A Topical Approach. Bumby, D. & Klutch, R. C. E. Merrill Publishing Co.

Columbus, OH. 1986.

Mathmatters. Lynch, C. & Olmstead, E. South Western Publishing Co. Cincinnati, OH. 1993.

Practical Problems in Mathematics for Electronics Technicians. Herman, S. & Sullivan, R.

Delmar Publishing Co. Albany. 1995.

CENTER FOR OCCUPATIONAL RESEARCH & DEVELOPMENT MATERIALS

Dividing the 40 Units of CORD Applied Mathematics into One-Year Courses.

9th Grade (15 units)UnitA. Getting to Know Your CalculatorB. Naming Numbers in Different

WaysC. Fmding Answers with Your

Calculator1. Learning Problem-solv-ing

Techniques2. Estimating Answers3. Measurint in English and Metric

Units4. Using Graphs. Clans, and Tables5. Dealing with Data6. Working with Lines and Angles7. Working with Shapes in Two

Dimensions8. Working with Shapes in Three

Dimensions -

9. Using Ratios and Proportions10. Working with Scale Drawings11. Using Signsd Numbezs and .

Vectors12. Using Scientific flotation

106 Grade (13 units)Unit

13. Precision, Accuracy, and Tolerance14. Solving Problems with Powers and

Roots15. Using Formulas to Solve Pioblems16. Solving Problems That Involve -

Linear Equations17. Graphing Data18. Solving Problems That Involve

Nonlinear Fquations19. Working with Statistics"c0. Working with Probabilides23. Factoring24. Patterns and Functions25. Quadratics26. Systems of Equauons27. Inequalities

lltn Grade (12 units)Unit

21. Using Right-triangle Relationships22- Using Trigonometric Functions28. Geometry in the Workplace 129. Geometry in the Workplace 230. Solving Problems with Computer

Spreadsheets31. Solving Problems with Computer

Graphics32. Quality Assurance and Process

Control I ,

33. Quality Assurance and ProcessControl 2

34. Logic (new unit)35. Spatial Viivalization (new unit)36. Coordinate Geometry (new unit)37. Transformations (new unit)

-ED

Applied EDEBvt

matics,,A CCM/DIU 1/4 AVP0A04 TO MOAT= Pmall.MA ICS

e50

TABLE 18

PERCENT AGREEMENT WITHNEW YORK STATE FRAMEWORK MATHEMATICS STATEMENTS

BY CATEGORY AND BY SUBJECT AREA

No. OFFRAMEWK

SUBJECT STATEMNTS % RANGE % MEDN % RANGE % MEDN % MEDN

PERCENT AGREEMENTA

(N = 16) (N = 5)ABCD

(N = 27)

LOG 8 0 -- 100 43.8 0 -- 100 75.0 50.0

NUM 8 62.5 -- 100 100 75.0 -- 100 100 100

OPS 11 63.6 -- 100 100 81.8 -- 100 100 100

GEO 13 23.1 -- 100 76.9 30.8 -- 100 92.3

MEA 12 25.0 -- 100 95.8 25.0 -- 100 91.7

PRB 22 0.0 -- 100 65.9 0.0 -- 95.5 86.4 86.4

ALG 13 46.2 -- 100 96.1 92.3 -- 100 92.3

TRG 5 0.0 -- 100 100 0.0 -- 100 100 100

TOTAL 92 37.0 -- 98.9 86.5 38.0 -- 95.7 90.6 85.9

Category A = Tech-Prep curricula with no standard sequential coursesCategory B = Tech-Prep curricula which include Sequential Course ICategory C = Tech-Prep curricula which include Sequential Courses I & IICategory D = Tech-Prep curricula which include Sequential Courses I, II & HI

LOG = Logic StatementsNUM = Number Sense & Numeral ion Concepts StatementsOPS = Operations on Number StatementsGEO = Geometry StatementsMEA = Measurement StatementsPRB = Probability & Statistics StatementsALG = Algebra StatementsTRG = Trigonometry Statements

51

86

FIGURE 3

PERCENTAGE OF TOPICS IN SEQUENTIAL MATH COURSES

WHICH ARE COVERED IN A TECH-PREP CURRICULUM

VS.

PERCENTAGE AGREEMENT WITH STATE FRAMEWORK MATH STATEMENTS

FOR EACH TECH-PREP CURRICULUM

lod

90

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D = Curriculum with ThreeSequential Courses

A

A

r-/ 0 20 30 4:0 io 610 710 8'0 90 /00

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100

00

1487

.50

29

81.8

11

1381

.31

211

100

00

732

552

12

162

59

114

21

5

93

'FA

BL

E 2

2

RE

SPO

NSE

S T

O F

RA

ME

WO

RK

MA

TH

EM

AT

ICS

STA

TE

ME

NT

S

ON

GE

OM

ET

RY

LE

VE

LFR

AM

EW

OR

K S

TA

TE

ME

NT

INT

ER

ME

D E

xplo

re w

ays

in w

hich

geo

met

ry is

use

d in

the

real

wor

ld

Use

tran

sfor

mat

ions

to s

ee p

rese

rvat

ion

of s

ize

&/o

r sh

ape

Cla

ssif

y 2-

& 3

-dim

ensi

onal

fig

ures

on

part

icul

ar a

ttrib

utes

Con

stru

ct g

eom

etri

c co

nclu

sion

s us

ing

logi

cal r

easo

ning

Use

gem

idea

s to

ana

lyze

pro

bs in

volv

ing

geom

con

cept

scr

iSu

b-T

otal

CO

NIM

EN

Rel

ate

geom

etri

c pr

inci

ples

to r

eal-

wor

ld p

heno

men

a

Show

und

rstn

dg o

f tr

ansf

orm

atio

ns:

cong

ruen

ce, s

imila

rity

Ana

lyze

pat

tern

s in

2 &

3 d

imen

sion

s

Con

stru

ct c

onvi

ncin

g ar

gum

ents

usi

ng g

eom

etri

cco

ncep

ts

App

ly g

eom

etri

c id

eas

in th

e so

lutio

n of

pro

blem

s

From

ass

umpt

ns, d

educ

epr

ops

of &

rel

tn b

etw

n ge

om f

igs

Mak

e co

ord

repr

esen

tatn

s of

geo

met

ric

figu

res

&co

ncep

ts

Ded

uce

prop

s of

fig

ures

usi

ng tr

ansf

rmat

ns &

coo

rdin

ates

Sub-

Tot

al

TO

TA

L

94

CA

TE

GO

RY

AY

ES

NO

?N

o.%

No.

No.

CA

TE

GO

RIE

S B

CD

YE

SN

ON

o.%

No.

No.

1610

00

010

90.9

10

956

.36

29

81.8

20

1593

.81

08

72.7

12

425

.010

28

72.7

12

1062

.54

210

90.9

10

5421

645

64

1487

.52

010

90.9

10

956

.35

29

81.8

11

1275

.03

17

63.6

13

850

.08

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63.6

13

1381

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1275

.04

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1275

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63.6

40

425

.011

17

63.6

40

8440

471

1510

138

6110

116

2114

TA

BL

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RE

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NSE

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RA

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WO

RK

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STA

TE

ME

NT

S

ON

ME

ASU

RE

ME

NT

LE

VE

LFR

AM

EW

OR

K S

TA

TE

ME

NT

INT

ER

ME

D K

now

mea

sure

men

ts c

an b

e to

spe

cifi

c de

gree

of

accu

racy

Sele

ct a

ppro

pria

te to

ols

to m

easu

re d

egre

e ac

cura

cy d

esir

ed

Mea

sure

& c

ompu

te in

bot

h E

nglis

h &

met

ric

syst

ems

Info

rmal

ly d

eriv

e an

d us

e fo

rmul

as in

mea

sure

men

t act

ivs

Solv

e w

ide

arra

y of

pro

blem

s us

ing

mea

sure

men

t con

cept

s

Com

pute

qua

ntiti

es: a

rea

& v

olum

e us

ing

stan

dard

uni

ts

Sub-

Tot

al

CO

MM

EN

Rel

mea

srm

t pre

cisi

on w

ith c

alc

accu

racy

usi

ng m

easu

rem

ts

Use

inst

rum

nts

& p

roce

dure

s to

mak

e in

dire

ct m

easu

rem

ts

Use

dim

ensi

onal

ana

lysi

s in

pro

blem

s in

volv

ing

mea

sure

s

Val

idat

e fo

rmul

as u

sed

to c

ompu

te m

easu

rem

ents

Solv

e w

ide

arra

y of

pro

bs in

mat

h, s

cien

ce &

tech

cur

ric

Com

pare

rel

s of

per

imtr

, are

a, v

ol o

f va

riab

le s

ize

figu

res

Sub-

Tot

al

TO

TA

L

96

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TE

GO

RY

AY

ES

NO

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o.%

No.

No.

CA

TE

GO

RIE

S B

CD

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SN

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%N

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1593

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110

90.9

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1593

.81

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1593

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1593

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90.9

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1610

00

011

100

00

913

259

7

1381

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105

166

16I

1011

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117

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9 7

TA

BL

E 2

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RE

SPO

NSE

S T

O F

RA

ME

WO

RK

MA

TH

EM

AT

ICS

STA

TE

ME

NT

S

ON

PR

OB

AB

ILIT

Y &

ST

AT

IST

ICS

LE

VE

LFR

AM

EW

OR

K S

TA

TE

ME

NT

INT

ER

ME

D D

istin

guis

h be

twee

n em

piri

cal &

theo

retic

al p

roba

bilit

ies

Dev

ise/

cond

uct e

xper

imen

ts/s

imul

atns

to f

ind

prob

abili

ties

Use

pro

bab

mod

el to

com

pare

res

ults

with

mat

h ex

pect

atns

Mak

e pr

edic

tions

App

reci

ate

the

perv

asiv

e us

e of

pro

babi

lity

in r

eal w

orld

Col

lect

, org

aniz

e, d

escr

ibe,

inte

rpre

t gro

uped

& in

div

data

Use

sam

plin

g in

sta

tistic

al s

tudi

es

Use

mea

sure

s of

cen

tral

tend

enci

es to

ana

lyze

dat

a

Ext

rapo

late

info

fro

m d

ata

set i

n nu

mer

icor

gra

phic

for

m

Sub-

Tot

al

(Con

tinue

d)

98

CA

TE

GO

RY

AY

ES

NO

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o.%

No.

No.

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TE

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CD

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SN

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1062

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09

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956

.37

010

90.9

10

1062

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08

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1168

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09

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1062

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09

81.8

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1381

.33

010

90.9

10

1381

.33

010

90.9

10

1487

.52

010

90.9

10

1487

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19

81.8

11

1104

113

9184

99

TA

BL

E 2

4 (C

ontin

ued)

RE

SPO

NSE

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O F

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WO

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MA

TIW

MA

TIC

S ST

AT

EM

EN

TS

ON

PR

OB

AB

ILIT

Y &

ST

AT

IST

ICS

LE

VE

LFR

AM

EW

OR

K S

TA

TE

ME

NT

CO

MM

EN

Use

pro

babi

l;ty

to r

epre

sent

& s

olve

cha

nce

prob

lem

s

Cre

ate

& in

terp

ret d

iscr

ete

prob

abili

ty d

istr

ibut

ions

Und

erst

and

the

conc

ept &

use

of

rand

omne

ss

Use

com

pute

r &

oth

er s

imul

atio

ns to

est

imat

e pr

obab

ilitie

s

Und

erst

and

a no

rmal

dis

trib

utio

n

Solv

e pr

oble

ms

invo

lvin

g pr

obab

ility

con

cept

s

Kno

w th

at a

s ex

peri

mnt

l n in

crse

s pr

obab

nea

rs th

eore

tic!

Des

ign/

cond

uct s

tatis

tcl s

tudi

es: i

nter

pret

/com

mun

ic r

esul

ts

Dra

w in

fere

nces

fro

m d

ata

in c

hart

s, ta

bles

and

gra

phs

Dif

fere

ntia

te b

etw

een

norm

ally

dis

trib

uted

& s

kew

ed d

ata

App

reci

ate

stat

istic

al m

etho

ds a

s m

eans

of

deci

sion

mak

ing

Det

erm

ine

mea

sure

s of

dis

pers

ion

for

a gi

ven

set o

f da

ta

Use

com

pute

r so

ftw

are

to m

odel

dat

a fr

om th

e re

al w

orld

Sub-

Tot

al

TO

TA

L

160

CA

TE

GO

RY

AC

AT

EG

OR

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BC

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ES

NO

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NO

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No.

No.

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o.N

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1062

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09

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956

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07

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31

956

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010

90.9

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531

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24

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1275

.04

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90.9

10

1062

.56

08

72.7

12

850

.07

17

63.6

22

1168

.85

09

81.8

11

1487

.52

010

90.9

10

1062

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08

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21

1168

.83

210

90.9

10

1168

.85

010

90.9

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850

.07

16

54.5

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128

746

108

19IS

232

113

719

228

21

101.

TA

BL

E 2

5

RE

SPO

NSE

S T

O F

RA

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WO

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MA

TH

EM

AT

ICA

L S

TA

TE

ME

NT

S

ON

AL

GE

BR

A

LE

VE

LFR

AM

EW

OR

K S

TA

TE

ME

NT

INT

ER

ME

D M

odel

sol

utio

n of

sim

ple

equa

tions

usi

ng c

oncr

ete

mat

eria

ls

Gra

ph li

near

rel

atio

nshi

ps

Dev

elop

pro

cedu

res

for

com

putin

g w

ith in

tege

rs

Use

var

iabl

es

Wri

te &

sol

ve e

quat

ions

Com

pare

dir

ect &

indi

rect

var

iatio

n

Sele

ct/u

se a

ppro

p te

chno

logi

es to

inve

stig

alg

ebra

ic c

once

pts

Sub-

Tot

al

CO

MM

EN

Use

alg

ebra

ic &

gra

phic

tech

niqu

es in

sol

ving

equ

atio

ns

Com

pare

/con

tras

t dir

/indi

r va

riat

ion,

incl

udg

use

of g

raph

s

Exp

lore

fun

ctio

ns th

at d

epic

t rea

l wor

ld p

heno

men

a

Use

alg

ebra

ic te

chni

ques

in th

e so

lutio

n of

pro

blem

s

Rel

ate

alge

brai

c id

eas

to c

oord

inat

e ge

omet

ry

Sele

ct/u

se a

ppro

p te

chno

logi

es to

sol

ve a

lgeb

raic

pro

blem

Sub-

Tot

al

TO

TA

L

lo

CA

TE

GO

RY

AC

AT

EG

OR

IES

BC

DY

ES

NO

L.

YFS

NO

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o.%

No.

No.

No.

foN

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o.

1381

.31

29

81.8

11

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011

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NT

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ON

TR

IGO

NO

ME

TR

Y

LE

VE

LFR

AM

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OR

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TA

TE

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1NT

ER

ME

D I

nves

tigat

e re

latn

ship

s am

ong

the

side

s of

sim

ilar

tria

ngle

s

Exp

lore

rel

atns

hps

with

sim

ilar

tria

ngle

s in

pro

blem

sol

ving

Sub-

Tot

al

C(M

IME

NR

el c

onsi

stnt

rat

ios

in s

imila

r ri

ght t

rian

gles

to tr

ig f

untio

ns

App

ly tr

igon

omet

ry to

pro

blem

s w

ith r

ight

tria

ngle

s

Use

cal

cula

tors

to g

et tr

igm

mm

etri

c fu

nctio

ns o

f gi

ven

angl

e

Sub-

Tot

al

TO

TA

L

104

CA

TE

GO

RY

AY

ES

NO

?N

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No.

No.

CA

TE

GO

RIE

S 11

CD

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1487

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399

030

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6614

050

50

105

SIJB

JEC

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LO

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STA

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AM

EW

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WO

RK

ST

AT

EM

EN

T

Con

stru

ct s

impl

e ar

gum

ents

usi

ng la

ws

of lo

gic

Use

geo

m id

eas

to a

naly

ze p

robs

invo

lvin

g ge

om c

once

pts

Ded

uce

prop

s of

fig

ures

usi

ng tr

ansf

rmat

ns &

coo

rdin

ates

& ju

dge

valid

ity o

f lo

gica

l arg

umen

ts

Flse

com

pute

r &

oth

er s

imul

atio

ns to

est

imat

e pr

obab

ilitie

s

Rec

ogni

ze &

app

ly in

duct

ive

reas

onin

g

Rec

ogni

ze &

app

ly d

educ

tive

reas

onin

g

Nla

ke &

eva

luat

e m

ath

conj

ectu

res

& a

rgum

ents

Con

stru

ct c

onvi

ncin

g ar

gum

ents

usi

ng g

e(-

netr

ic c

once

pts

Val

idat

e fo

rmul

as u

sed

to c

ompu

te m

easu

rem

ents

Kno

w th

at a

s ex

peri

mnt

l n in

crse

s pr

obab

nea

rs th

eore

ticl

Ilse

com

pute

r so

ftw

are

to m

odel

dat

a fr

om th

e re

al w

orld

Ilse

con

cret

e m

ater

ials

for

ops

with

fra

ctns

, dec

hns,

inte

gs

Ana

lyze

pat

tern

s in

2 &

3 d

imen

sion

s

CA

TE

GO

RIE

SA

% Y

ES

BC

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YE

SA

BC

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o. ?

18.8

25.0

4

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31.3

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.

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.

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GL

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ber

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Geo

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ry le

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Res

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106

107