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York - Seneca Institute for Mathematics, Science and Technology
Education
COLLEGE MATHEMATICS PROJECT 2011
FINAL REPORT
For the
Ontario Ministry of Education
and the
Ontario Ministry of Training, Colleges and Universities
Graham Orpwood, Laurel Schollen, Gillian Leek,
Pina Marinelli-Henriques, Hassan Assiri
© Seneca College of Applied Arts and Technology
2012
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COLLEGE MATHEMATICS PROJECT
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Preface
Once again, we are pleased to present the annual report of the College Mathematics Project (CMP). In
doing so, the CMP team acknowledges the critical work of many individuals and groups whose
contributions have enabled the project to operate throughout the year. These include:
the Ministry of Education and the Ministry of Training, Colleges and Universities for their
ongoing support, both financial and professional;
the CMP Steering Committee, the college Vice-Presidents, Academic and the CMP College Leads,
who have ensured that CMP had the data with which to conduct its research;
the School/College/Work Initiative (SCWI), whose Regional Planning Teams have organised
forums at which CMP research has been shared with school and college educators;
Seneca College’s Information Technology Services (ITS) department, especially John Meskes,
Mehrdad Ziaei and Mohsen Rezayatmand, who have enabled the CMP data to be assembled,
analysed and displayed.
Thank you to all of these individuals and groups and also to the many others who have made
contributions to the CMP this year.
Le présent document est également disponible en français au site
http://collegemathproject.senecac.on.ca
Margaret Sinclair
As this report was going to press, our dear friend and colleague, Margaret Sinclair,
passed away. Margaret was an Associate Professor of Mathematics Education at
York University and co-Director of the York/Seneca Institute for Mathematics,
Science and Technology Education (YSIMSTE). She was a key member of the CMP
team in its early years and her critical but constructive contributions will be missed.
This 2011 final report of the College Mathematics Project is dedicated to her
memory.
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Table of Contents Preface ............................................................................................................................................ 2
Executive Summary ......................................................................................................................... 4
Chapter 1: Introduction to the College Mathematics Project ....................................................... 8
Chapter 2: Mathematics Achievement ........................................................................................ 20
Chapter 3: College Mathematics in Context ................................................................................ 31
Chapter 4: Provincial Forum ........................................................................................................ 57
Chapter 5: Conclusions ................................................................................................................ 64
Appendix A .................................................................................................................................... 74
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Executive Summary
The College Mathematics Project (CMP) is a collaborative program of research and deliberation
concerning mathematics achievement of first-year college students in Ontario. Its goals are:
To analyse the mathematics achievement of first-semester college students, particularly in relation to their secondary school mathematics backgrounds;
To deliberate with members of both college and school communities about ways to increase student success in college mathematics.
CMP 2011 included all 24 colleges and 72 district school boards in all regions of the province. It was
funded by the Ministry of Education and the Ministry of Training, Colleges and Universities, and led by a
team of researchers from the York-Seneca Institute for Mathematics, Science and Technology Education
(YSIMSTE) based at Seneca College.
The CMP employs the overall methodology of deliberative inquiry, in which research into the current
situation is linked to deliberations among stakeholders over appropriate courses of future action. The
CMP 2011 research analysed the secondary school and college records of almost 95,000 students who
enrolled in all college program areas in fall 2010. Of these, over 35,000 took a first-semester
mathematics course and the research focused on their achievement in these courses, relating this to a
variety of factors, including the choice of mathematics courses taken at secondary school. Because this
was the third year in which all 24 colleges participated, we have been able to identify preliminary trends
emerging in our data,
Highlights of the CMP 2011 research include the following:
67.6% of students achieved good grades (A, B or C) in first-semester college mathematics, while
32.4% were considered to be “at risk” (having received a D or F or having withdrawn from the
course). This percentage (of good grades) represents very little change from the previous two
years.
Of the Very Recent Ontario Graduates – those who have taken the most recently revised
mathematics curriculum in secondary school – 61.6% of the males and 68.7% of the females
achieved good grades.
Second Career students achieved better than other students: 80.0% of males and 84.5% of
females achieved good grades.
Graduates of French-language school boards attend English- and French-language colleges in
approximately equal numbers; the mathematics achievement of students attending French-
language colleges has increased since last year while that of students attending English-language
colleges has shown very little change.
Older students, particularly females, achieve significantly better than younger students: for
example, 77.6% of males and 85.5% of females aged 30-39 obtained good grades.
Patterns of achievement analysed according to secondary school mathematics pathways follow
similar patterns to those found in the past:
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o Many more students came to college with MAP4C than MCT4C in Grade 12 but their achievement was lower overall. The same applies to MBF3C and MCF3M in Grade 11.
o However students with high marks in MAP4C do well in college mathematics; 76.8% of those with 80% and over in MAP4C obtained good grades in college.
o Over 3,000 students in our sample took no mathematics after Grade 11; only 50.2% of those who took MBF3C as a terminal mathematics course achieved good grades in college mathematics.
o The numbers of students transferring from Grade 10 Applied Mathematics to MCF3M is increasing each year from 0 (in 2007) to 381 (in 2008) to 806 (in 2009) to 1,213 (in 2010). 63.9% of the 2010 students go on to obtain good grades in college mathematics.
Systematic qualitative research was undertaken this year into the content of first semester
mathematics courses, particularly those of a foundational or preparatory nature. This was set in
the context of curriculum case studies corresponding to the complete mathematical experience
of students taking selected programs in Business and Technology.
o The case studies illustrated how mathematics is taught both as standalone courses and as embedded into specialist courses within each program (such as accounting or electrical theory).
o They also demonstrated how, at different colleges, the curriculum of programs focused on the same occupation may differ, while still being based on the same provincial program standard and aimed at the achievement of the same learning outcomes.
o CMP analysis of a systematic selection of foundation program mathematics courses from across the province showed that they share a common emphasis on the numeracy skills required for college diploma programs and the occupations for which these programs prepare students.
o Analysis of diploma level mathematics courses showed a greater proportion of program-specific mathematics topics but also a strong continued emphasis on numeracy skills.
o The numeracy skills most frequently encountered in first-semester college mathematics courses were also mapped onto the provincial curriculum for Grades 1-8 and 9-12 mathematics. This analysis showed that the grades at which these key numeracy skills were most frequently taught were Grades 6, 7 and 8.
The CMP 2011 project included a Provincial Forum, hosted by Seneca College in October 2011, where
themes of a provincial policy nature were discussed. Half of the day comprised the discussion of two
major themes of the CMP: Mathematics Education for the 21st Century Economy and Student Success in
Secondary-Postsecondary Transition. Also included in the forum was an outline of CMP 2011 research
results and a presentation from the Conseil supérieur de l’Éducation du Québec on their research in this
area. Finally, forum participants had the opportunity to deliberate over three sets of questions arising
from the research. The forum is described in more detail in the report and individual presentations are
available on the CMP web site.
The CMP report concludes with a discussion of two themes that emerged from the research and
deliberation this year. The first of these is Numeracy and in support of this theme, we suggest that a
provincial numeracy strategy is required. Such a strategy would aim to increase levels of numeracy
among Ontario secondary school graduates through a multi-faceted program of action, which could
include some or all of the following ideas:
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The implementation of a Grade 10 numeracy test having the same status as the existing Ontario
Secondary School Literacy Test (OSSLT) in lieu of the present Grade 9 assessment of
mathematics.
Involving employers, college faculty, and parents in identifying the numeracy skills in greatest
need of support.
Increased efforts, through both pre-service and in-service teacher education, to support teachers’
mathematical skills and understanding and to eradicate negative stereotypes associated with
mathematics.
Considering the use of a numeracy test for teacher candidates, such as has recently been
introduced (along with tests in literacy and information and communication technology-ICT) in
England & Wales, starting in 2012.
Research into ways in which Junior/Intermediate teachers with a mathematics teaching
qualification are currently being deployed by school boards, and the impact of such deployment
on achievement.
Development of sample instructional materials to support the teaching of numeracy across the
curriculum (for example in social studies as well as in science and mathematics).
Colleges and universities have a role to play here also. Until innumeracy among incoming students is
eradicated, there will be a continuing need for developmental courses of the variety that have grown up
over the past few years. Colleges can support schools’ efforts at developing higher levels of numeracy if
they could work together in a number of related areas:
To develop a common numeracy assessment tool to be used by all colleges as part of their
admission and placement process for all incoming college students – perhaps with a technology
version and a business version – based on a numeracy framework approved by both the college
system and the Ministry of Education. This assessment would be consistent with, but not the
same as, the Grade 10 numeracy test since it would be designed specifically for students
applying to college programs in technology and business.
To reframe program admission and placement requirements to take into account students
achievement on the common numeracy assessment.
To develop a system-wide college numeracy course (again, perhaps in a technology version and a
business version) for students whose scores on the numeracy assessment show that they need
such a developmental course.
To share both the assessment framework and course information with elementary and
secondary schools so that teachers at earlier levels understand better the expectations of the
college system of students entering into vocational diploma and certificate programs.
To use the CMP data collection system to collect students’ numeracy assessment scores and to
provide feedback to school boards and schools on the (aggregate) achievement of their
graduates.
The second theme of this year’s final report is called College Knowledge and is focused on the varieties
of knowledge and skill that students require to be successful at the postsecondary level of education.
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While the CMP has neither the resources nor the research evidence on which to support specific
recommendations, we believe that our evidence supports the need for further work in this area. We
therefore invite the education community – both at governmental and local levels – to discuss possible
solutions and, specifically, that:
The Ministry of Education and the Ministry of Training, Colleges and Universities set up an expert
panel to study the assessment of students at the interface of secondary and postsecondary
education and to recommend possible policies and practices that could ensure that students are
adequately prepared for postsecondary education.
The School/College/Work Initiative be asked (and resourced) to expand the range of mechanisms
for facilitating students’ successful transitions from school to college as well as maintaining its
ongoing support for dual credits and forums.
Colleges, Universities and School Boards work together at the local level to develop joint
programs aimed at providing all students who intend to go on to postsecondary education
sufficient college knowledge to maximise their chances of success.
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Chapter 1: Introduction to the College Mathematics Project
The College Mathematics Project (CMP) is now seven years old. It grew from a small pilot study of 500
students in three schools of technology at Seneca College, in which the data was analysed on an Excel
spreadsheet, to become an annual project tracking almost 100,000 students in all 24 Ontario colleges.
Its roots can be traced to meetings of the Ontario College Heads of Technology, where concerns about
the levels of student success (or lack of it) in mathematics had been an ongoing topic of discussion
informed by anecdotal evidence of the participants. At that time also, the Ministry of Education had
invited feedback on its (then) new mathematics curriculum and responding to this invitation focused the
group’s thinking on what would constitute the most appropriate Grade 12 preparation for college
mathematics. From there, informal discussions developed into deliberations involving a wider range of
stakeholders – Ministries, Colleges, School Boards, mathematics teachers in schools and colleges, school
and college administrators, and students themselves – and these deliberations have been informed by
systematic research into the mathematics achievement of every college student participating in a
program that includes a first semester mathematics course.
The goals of the College Mathematics Project reflect these two aspects of its work:
To analyse the mathematics achievement of first-semester college students, particularly in relation to their secondary school mathematics backgrounds;
To deliberate with members of both college and school communities about ways to increase student success in college mathematics.
While these goals have remained the same, the scope of the research questions has broadened during
the past seven years. At the beginning, as outlined above, it seemed natural to consider just the
students’ immediate past mathematical experience as preparation for college mathematics. The initial
research conducted by CMP focused on the choice of Grade 12 mathematics courses taken by students.
Soon, however, deliberations suggested that this was not enough and that the scope should be
broadened to include the choice of mathematics courses from Grades 9 to 12. Yet further deliberations
led to an examination of the levels of students’ achievement within their Grade 12 courses, the age and
gender of the students, and an increasing range of factors for which the CMP had access to data. Finally,
this year, the CMP research has begun to explore qualitatively the substantive aspects of first year
college mathematics courses and to relate these both to the mathematics curriculum in elementary and
secondary schools and to the curricula of typical college programs for which they formed a foundation.
Overall, the range of data collected and analysed through the CMP has enabled the scope of
deliberations to expand and broaden correspondingly.
Finally, and perhaps most significantly, the deliberative context, within which the central problem –
student success in college mathematics – has been considered, has broadened and deepened. From
seeing the issue as mainly matters of secondary school course selection and adjustment of college
admission requirements, we came to recognize that these were parts of a more complex set of issues
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related to students’ transition from secondary to postsecondary education, involving different
curriculum goals, different ways of thinking about teaching and learning, different expectations for
students’ learning skills and attitudes. In addition, local deliberations among college and secondary
school educators at the regional level have led to discussion of issues of provincial policy and ultimately,
this year, to a full deliberative forum at the provincial level. Here, some of the questions about why so
many students are at risk in college mathematics are now being considered in the context of broader
questions about a society where, it was claimed, the lack of numeracy skills appears to be quite
acceptable, even as a corresponding lack of literacy skills is not. This report therefore goes further than
its predecessors in terms of the scope of its concluding themes and recommendations. It is hoped that
the CMP will be seen not only as a “college mathematics project” but also as a contribution to the
growing body of policy studies relating to the relationship between elementary/secondary education, on
the one hand, and postsecondary education, on the other.
CMP Methodology
The methodology used for the CMP is called “Deliberative Inquiry”. It is a cyclical methodology (see
Figure 1) designed for integrating research with deliberations about future courses of action. The
questions for inquiry (2) are derived from deliberations about the problems of practice (1) and
deliberations are based on questions (5) emerging from the research. The cycle of inquiry and
deliberation continues, drawing ideas from existing theory (3), generating recommendations for practice
(6), and contributing further ideas to theory (4).
Figure 1.Deliberative Inquiry - the CMP Methodology
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This methodology is responsible for the continuing interplay between research and deliberations which
has resulted in the growing complexity of the issues being addressed by the CMP over the past seven
years which is described above.
The CMP is directed by a steering committee comprised of representatives of the supporting Ministries,
and provincial organizations related to colleges and secondary schools1. This committee met three
times during 2011 and once in early 2012: its first meeting in January approved the research questions
for the CMP to address during 2011; the second (in June) focused on plans for a provincial forum to be
held in the fall; the October meeting discussed the preliminary results of the CMP 2011 research; and
the final meeting (in January 2012) reviewed the first draft of this report.
Researchers from the York/Seneca Institute for Mathematics, Science and Technology Education
(YSIMSTE) based at Seneca College conduct the CMP research, supported by technical staff of Seneca
College’s Information Technology Services department. All data used in the CMP is obtained from
participating colleges. The principal data sources include students’ secondary school transcripts as
provided to colleges from the Ontario College Application Service (OCAS) and students’ first semester
grades in mathematics courses. These files are combined and student identifiers are then removed in
order to ensure student anonymity2. Finally, the data is validated by each college prior to its being
mounted on a web-based database, from which more specific analyses can be made. Data reports are
presented in a manner and sequence consistent with the CMP research questions. Following data
collection and analysis, a provincial policy forum was held in October and this is described in more detail
in chapter 4 of this report.
CMP 2011 Research Questions
The research questions addressed by the CMP this year developed the same four areas of interest in
CMP 2010 – information about the participants, distribution of grades in first semester college math, the
relationship between college achievement and secondary school mathematics background, and the
relationship between students’ first semester math achievement and their math achievement at the
school boards (and secondary schools) from which they came. In addition, this year, the CMP
introduced a new qualitative research strand exploring the content of first year college mathematics
courses and their relationships with the content of both the elementary and secondary school
mathematics curriculum and that of selected college programs. The results of the quantitative research
are summarized in chapter 2 and those of the qualitative research in chapter 3 of this report.
The questions, which relate to students taking first semester college mathematics in Fall 2010, are:
1The members of the CMP Steering Committee along with the “CMP leads” from each College are listed on the project’s web
site (http://collegemathproject.senecac.on.ca). The CMP project team acknowledges with thanks the contributions of all members of this committee. 2CMP has a policy on data confidentiality available on its web site (http://collegemathproject.senecac.on.ca) and the research
methodology has been given ethics reviews by participating colleges.
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A. PARTICIPATION
A1. What are the numbers of students in our sample, by college, gender, and program cluster?
(with course type and age filters)
A1f. As A1 but for graduates of French language boards only (also by region).
A2. What are the numbers of students under the age of 23 (December 31, 2010) and who are
graduates of Ontario secondary schools (ROGs3) by college, gender, and program cluster? (with course
type and age filters)
A3. What are the numbers of students with a Grade 11 mathematics course taken in fall semester
2006 or later, by college, gender and program cluster (VROGs)? (with course type and age filters)
A4. What are the numbers of students (ROGs, non-ROGS and VROGs) enrolled in all math courses, in
college-level math courses, and in preparatory math courses, by college, gender and program cluster?
(with course type, gender and age filters)
A5. What are the overall numbers of students (ROGs only) from each school board by college,
gender and program cluster?
A6. What are the overall numbers of “second career” students in our sample, by college, gender,
and program cluster (with course type and age filters).
B. COLLEGE MATH ACHIEVEMENT
B1. What is the mathematics grade distribution for ROGs, non-ROGs and VROGs, by college, gender,
and program cluster? (With age, gender and course filters)
B2. What are the % of students achieving a “good passing grade” (A, B, C) and “at risk” (D, F, W) for
ROGs, non-ROGs and VROGs by college, gender, and program cluster?(With age, gender and course
filters)
B2f. As B2 but for graduates of French language boards only (also by region)
B3. What are the % of students achieving a “good passing grade” (A, B, C) and “at risk” (D, F, W) for
“second-career” students (with age, gender, and course filters).
C. SECONDARY SCHOOL MATHEMATICS BACKGROUNDS
C1. What are the numbers of students taking each secondary school mathematics pathway
(including MCF3M-MCR3U, MFM2P-MPM2D, and MFM1P-MPM1D) and what % of these achieve good
3 These and other groups of students are defined on page 15.
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grades or are at risk in college? (with filters for student type <ROGs and VROGs> and college course
type).
C1f. As C1 but for students from French language boards only.
C2. For both ROGs and VROGs following a pathway culminating in MCT4C, MAP4C, or a 12U course,
how do students’ Grade 9, 10, 11 &12 math marks compare with their college math marks? (with filters
for age, gender, school course and college course type).
C3. What is the profile (gender, college, program cluster) of students following selected
pathways?(with filters for student type <ROGs and VROGs> and college course type)
D. COLLEGE ACHIEVEMENTS BY SCHOOL BOARD AND SCHOOL
D1. What are the % of students with a “good passing grade” (A, B, C) and “at risk” (D, F and W) from
each district school board? – also broken down by secondary school? (with filters for student type
<ROGs and VROGs> and college course type)
D2. What are the % of students enrolled in college-level courses and preparatory courses from each
district school board? (with filters for student type <ROGs and VROGs> and college course type)
E. COLLEGE FOUNDATION PROGRAMS
E1. Program Identification: Which programs at each college are included as “foundation programs”?
E2. Comparative analysis of mathematics courses and assessments: secondary, college foundational,
and college diploma-level: What are the topics and emphases in each course and in college placement
assessments?
E3. Student selection processes: How are students selected for foundation courses or preparatory
mathematics courses?
E4. Participation: How many students take foundation programs or preparatory mathematics
courses within diploma-level programs?
E5. Achievement: What is the mathematics achievement of students taking foundation programs
and preparatory mathematics courses?
E6: Pathways: Which (secondary school mathematics) pathways are followed by student’s
foundation programs and preparatory mathematics courses in college?
This report includes the highlights of research results relating to these questions from a provincial
perspective. More detailed results, particularly as they relate to specific colleges and school boards, can
be found on the CMP interactive database, which is accessible from the CMP web site4.
4http://collegemathproject.senecac.on.ca/cmp/en/research.php
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COLLEGE MATHEMATICS PROJECT
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College Programs and Grading Policies
As in past years, the CMP has reviewed all college programs as part of its data collection process. The
project includes all full time Ontario College Certificate, Ontario College Diploma and Ontario College
Advanced Diploma programs. In 2006/07 these programs represented more than 90% of the system
registrants5. College bachelor degree, apprenticeship, and graduate certificate programs are excluded
from the study. Once the list of programs from each college is collected, they are classified according to
the program cluster system used in previous iterations of the project. The CMP uses Ministry of
Training, Colleges and Universities (MTCU) program codes to organise all programs into four major
clusters, each of which is subdivided into sub-clusters, as shown in Table 1. This clustering ensures
comparability of the aggregate analysis across colleges and also affords researchers opportunities to
“drill down” further into the data to investigate achievement at the sub-cluster and program level. The
results of classifying all of the college programs according to these clusters and sub-clusters are shown
on the CMP web site6.
Table 1. CMP System of Program Clusters
Major Cluster Sub-clusters Sample Program
Applied Arts (AA) Applied Arts Human Services Health Services Hospitality & Tourism
Broadcasting-Radio Early Childhood Education Practical Nursing Hotel and Restaurant Management
Business (B) Accounting & Finance Business Administration & Management Office Administration
Business –Accounting Business – Human Resources Office Administration - Legal
General (G) General Arts & Science Pre-Business Pre-Health Pre-Technology
General Arts & Science Business Foundations Pre-Health Science Technology Foundations
Technology (T) Applied Science Computer Construction Electrical Mechanical
Chemical Laboratory Technology Computer Engineering Technician Civil Engineering Technology Electronics Engineering Technician Mechanical Engineering Technology - Automation
5King, A.J.C. et al.“Who Doesn’t Go to Post-Secondary Education?”- Final Report of Findings, Colleges Ontario, 2009.
6 Program cluster and college grading policy information is available for review at the CMP website:
http://collegemathproject.senecac.on.ca/cmp/links.php
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We have noted in earlier reports that, since all colleges have their own grading systems, the College
Mathematics Project has developed – for the purposes of aggregating achievement data across multiple
colleges – its own simplified system of grades, to which grades from all college data sets are
transformed. The final report of the CMP 2007 study contained a full discussion of this issue7 and is not
repeated here.
The CMP grading system is shown in Table 2 and the detailed comparison of this system with that of
each participating college is also available on the CMP web site. In addition, CMP has found from earlier
studies that a D grade in first semester mathematics is often followed by a student dropping out or
changing programs. We therefore classify D grades along with F and W, as evidence that students are
“at risk” of not completing their chosen program.
Table 2.
CMP Grading System
Good Grades
A(includes A+ and A-) 80% - 100%
B (includes B+ and B-) 70% - 79%
C (includes C+ and C-) 60% - 69%
P (used for courses with Pass/Fail grades)
At Risk
D (includes D+ and D-) 50% - 59%
F under 50%
W withdrawal
Mathematics Courses and College Programs
The programs selected for detailed study in the CMP all have mathematics scheduled in the first
semester of the college curriculum. While most students entering college take the regular curriculum of
their chosen program, an increasing number of students in all colleges are taking alternate programs in
order to prepare them for their desired occupation-focused program8. In some cases, students are
required to take these foundational programs; in others they are advised to take them following
assessments – usually in mathematics and language – taken after admission; in still other cases, students
choose to take a foundational program (such as a pre-technology, pre-business, or pre-health science
program) because they are not yet decided which career direction to take. In some colleges, particularly
in the Greater Toronto Area, preparatory mathematics courses are used where skills assessment tests
7Laurel Schollen et al. College Mathematics Project 2007: Final Report. (Toronto: Seneca College of Applied Arts & Technology,
2008), pp 10-13. 8 Data supporting this is presented in Table 6 on page 17.
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suggests that students would benefit from such courses. More detailed discussion of these foundational
programs and preparatory courses is contained in chapter 3 of this report.
Students
CMP 2011 analyses data for students who undertook first-semester mathematics in fall 2010.
Occasionally the first mathematics course in a college program occurs in the second or a later semester.
The CMP does not capture data for such programs. Among all these students, the CMP has defined a
variety of sub-groups, most of which appear in the analyses contained in this report. Readers should
therefore note the following definitions.
Recent Ontario Graduates (ROGs): Students under the age of 23 (as of December 31, 2010) and having
an Ontario Secondary School Diploma (OSSD). These students are those whose secondary school
mathematics backgrounds are analysed. Others – “non-ROGs” – include older students or those whose
secondary education is from outside Ontario. While we still have data for ROGs and non-ROGs available
in the CMP database, the analyses displayed in this year’s backgrounder do not show them separately.
Very Recent Ontario Graduates (VROGs): ROGs whose most recent Grade 11 mathematics course is
recorded since September 1 2006. This group of students is assumed to have followed the most
recently revised Ontario mathematics curriculum. It should be noted that this definition has been
modified this year to be more inclusive; the data from past years, used here to identify trends, has
therefore been updated to reflect this new definition and may not therefore correspond exactly to that
reported in previous CMP reports.
Direct Entry Students: VROGs whose OSSD was awarded since January 1 2010.
Second Career9 Students: Students designated as such on their college admission documentation.
CMP 2010 Student Cohort
The student cohort that has been studied in CMP 2011 entered college in the fall of 2010. This section
of the report contains a description of this cohort in terms of program cluster, gender, age, and
participation in first semester mathematics courses. The tables also include comparisons with the fall
2008 and fall 2009 cohorts as reported in the last two year’s CMP reports10.
Enrolment by Program Cluster and Gender
Table 3 shows the distribution of all students in all colleges analysed by program cluster in each of the
past three years. After a sharp increase in enrolments in Fall 2009, there has only been a modest
9 Second Career is a program sponsored by Ontario’s Ministry of Training, Colleges and Universities designed to assist laid off
workers. The program provides skills training in high demand occupations and financial support during retraining. 10
Graham Orpwood et al. College Mathematics Project 2009: Final Report (Toronto: Seneca College of Applied Arts & Technology, 2010), pp. 18-21.
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growth this year and we see that enrolment in business programs actually decreased somewhat in 2010,
compared with 2009. The number of college programs in both business and technology also declined
somewhat in 2010.
Table 3. Programs and Enrolments, Fall 2008, 2009 and 2010
Fall 2008 Fall 2009 Fall 2010 Changes 2008-10
Cluster Programs Enrolment Programs Enrolment Programs Enrolment Programs Enrolment
Applied Arts 936 40,828 1,041 46,405 981 46,978 4.8% 15.1%
Business 387 13,680 421 14,468 386 13,640 -0.3% -0.3%
General 191 10,331 220 11,482 201 11,586 5.2% 12.1%
Technology 782 19,157 877 21,792 778 22,447 -0.5% 17.2%
TOTAL 2,296 83,996 2,559 94,147 2,346 94,651 2.2% 12.7%
Table 4 shows the numbers of programs having first semester mathematics courses with the enrolments
in these programs. While the largest percentage increase in mathematics enrolment is in Applied Arts
programs, the actual numbers involved here are small. The increases in mathematics enrolments in
General programs reflect the increase in the number of Foundation programs (noted in last year’s report
and shown later in Table 6). Mathematics enrolments in Technology programs continue to increase
though at a slower rate in 2010 than in 2009.
Table 4. Mathematics Enrolment by Program Cluster: Fall 2008, 2009 and 2010
Fall 2008 Fall 2009 Fall 2010 Change 2008-10
Cluster Programs Enrolment Programs Enrolment Programs Enrolment Programs Enrolment
Applied Arts 65 2,625 76 2,871 98 3,618 50.8% 37.8%
Business 269 9,678 293 10,100 285 9,137 5.9% -5.6%
General 92 5,336 118 6,204 126 6,574 37.0% 23.2%
Technology 620 14,167 691 16,115 598 16,071 -3.5% 13.4%
TOTAL 1,046 31,806 1,178 35,290 1,107 35,400 5.8% 11.3%
When mathematics enrolments are analysed by gender (as in Table 5), we can see that in Business, the
decreased numbers result almost entirely from fewer females enrolling in Business programs in 2010. By
contrast, in Technology programs, the increases in female enrolment in 2010 are greater than the
increases in male enrolment. As we have noted in previous years, numbers analysed by gender do not
total 100% because some students do not declare a gender.
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Table 5. Mathematics Enrolment by Gender: Fall 2008, 2009 and 2010
Fall 2008 Fall 2009 Fall 2010 Change 2008-10
Cluster M F M F M F M F
Applied Arts 890 1,715 1,064 1,800 1,257 2,322 41.2% 35.4%
Business 4,877 4,510 5,316 4,767 4,883 4,223 0.1% -6.4%
General 2,086 3,218 2,485 3,692 2,549 3,994 22.2% 24.1%
Technology 12,160 2,061 13,815 2,264 13,602 2,373 11.9% 15.1%
TOTAL 20,013 11,504 22,680 12,523 22,291 12,912 11.4% 12.2%
Enrolment by Type of Program and Mathematics Course
While the majority of college students enter diploma or certificate programs leading to occupational
qualifications, enrolments in foundation programs and preparatory mathematics courses are also on the
increase. This is because colleges seek to enhance student success by providing additional support and
preparation where this appears to be needed or where students are uncertain as to which occupational
program is the most appropriate for them. Table 6 shows how pre-diploma programs in Health,
Technology and Business have grown both in number and enrolment over the past three years. In
addition, seven colleges (principally in the GTA) offer preparatory courses in mathematics as additions to
the regular program curriculum and enrolments in these are also shown in Table 6. The foundation
programs are all included in other analyses in the General program cluster and so enrolments in the
preparatory courses are shown for the Applied Arts, Business and Technology programs only.
Table 6. Mathematics Enrolment in Foundations Programs and Preparatory Courses: Fall 2008, 2009 and 2010
Fall 2008 Fall 2009 Fall 2010 Change 2008-10
Pre-Diploma Programs Programs Enrolment Programs Enrolment Programs Enrolment Programs Enrolment
Pre-Health 21 2,143 23 2,568 29 2,692 38.1% 25.6%
Pre-Technology 13 415 16 483 20 471 53.8% 13.5%
Pre-Business 8 434 12 514 13 478 62.5% 10.1%
General Arts & Science 25 1,419 37 1,835 49 2,110 96.0% 48.7%
Sub-totals 67 4,411 88 5,400 111 5,751 65.7% 30.4%
Preparatory courses
2,506
2,712
2,552
1.8%
TOTAL
6,917
8,112
8,303
20.0%
% of total math enrolment 21.7%
23.0%
23.5%
Overall, there has been a 65% increase in the number of Foundation programs in the past three years
and a 30% increase in the numbers of students enrolled in these programs. If the preparatory course
enrolments are added in (as shown in Table 6), we can see that in 2010, nearly one-quarter of all
incoming students are taking these programs or courses, a 20% increase over 2008.
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Table 7 takes this analysis one step further and summarizes the changes in mathematics enrolments in
three types of college courses: mathematics courses in the first semester of foundation programs;
preparatory mathematics courses taken in the first semester of regular diploma programs; and regular
first semester college-level mathematics courses. The analysis also compares enrolments of all students
with enrolments by Very Recent Ontario Graduates (VROGs). This shows that, while overall enrolment
in foundational and preparatory mathematics has increased from 21.7% in 2008 to 23.5% in 2010 (a
change of 1.8 percentage points), VROG enrolment increases in these courses is 2½ times higher (from
22.4% in 2008 to 27.1% in 2010) – a change of 4.7 percentage points.
Table 7. Mathematics Enrolment by Student Type & Program/Course Type, Fall 2008, 2009 and 2010
2008 2009 2010 Change 2008-10
Course Type All Ss VROGs All Ss VROGs All Ss VROGs All Ss VROGs
Foundations Programs 4,411 1,578 5,400 2,531 5,751 3,642
Preparatory Math 2,506 1,096 2,712 1,502 2,552 1,650
College-level Math 24,889 9,241 27,178 12,646 27,046 14,268
TOTAL 31,806 11,915 35,290 16,679 35,349 19,560
% College-level 78.3% 77.6% 77.0% 75.8% 76.5% 72.9% -1.8% -4.7%
% Fndn. + Prep. 21.7% 22.4% 23.0% 24.2% 23.5% 27.1% +1.8% +4.7%
Note: ‘All Ss’ refers to all students and ‘Fndn. + Prep’ to foundation programs and preparatory courses.
Enrolment of French-Language Students
While the CMP database does not include records of students’ mother tongue, it does include
information relating to the (Ontario) school board from which they graduated. For the past two years,
CMP research has examined the college enrolment and mathematics achievement of the graduates of
Ontario’s French-language school boards. While many of these enroll in programs at the province’s two
French-language colleges, many also participate in programs at the 22 English-language colleges. In this
report therefore, we use the term “French-language students” to refer to the graduates of Ontario
French-language school boards. We recognize that the French-language colleges (particularly La Cité
collégiale) enroll other French-language students (from Québec, for example) but these are not included
in the following analyses.
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Table 8. First Semester French-language Student Enrolment in French- and English-Language Colleges, Fall 2008, 2009 and 2010
Fall 2008 Fall 2009 Fall 2010 Change 2008-2010
Cluster F E T F E T F E T F E T
Applied Arts 780 481 1,261 779 602 1,381 877 522 1,399 12.4% 8.5% 10.9%
Business 158 147 305 174 152 326 165 129 294 4.4% -12.2% -3.6%
General 113 115 228 121 140 261 108 143 251 -4.4% 24.3% 10.1%
Technology 362 321 683 407 351 758 492 337 829 35.9% 5.0% 21.4%
TOTAL 1,413 1,064 2,477 1,481 1,245 2,726 1,642 1,131 2,773 16.2% 6.3% 11.9%
Table 8 shows the enrolment of first-semester French-language students in English- and French-
language colleges over the past three years. It should be noted that while the overall increase in
enrolment (11.9%) is very close to the overall provincial increase, a much larger proportion of this
increase (16.2%) is in the French-language colleges compared with 6.3% in the English-language
colleges. Table 9 shows the corresponding changes in mathematics enrolment and this disproportional
increase in favour of the French-language colleges is even more pronounced, owing to an over 80%
increase in enrolment in Technology mathematics.
Table 9. French-language Mathematics Enrolment in French- and English-Language Colleges, Fall 2008, 2009 and 2010
Fall 2008 Fall 2009 Fall 2010 Change 2008-2010
Cluster F E T F E T F E T F E T
Applied Arts n/a 34 34 n/a 44 44 94 39 133 n/a 14.7% n/a
Business 94 78 172 106 88 194 103 86 189 9.6% 10.3% 9.9%
General 64 65 129 86 85 171 62 70 132 -3.1% 7.7% 2.3%
Technology 209 259 468 278 273 551 381 262 643 82.3% 1.2% 37.4%
TOTAL 367 436 803 470 490 960 640 457 1,097 74.4% 4.8% 36.6%
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Chapter 2: Mathematics Achievement
The central feature of the College Mathematics Project’s research program since the outset has been
the identification of the level of mathematics achievement of first semester college students11 and
Figure 2 shows the grade distribution of students in the fall 2010 cohort. A chart similar to Figure 2 has
appeared in every CMP report and its general shape has remained almost constant over the years, with
a bimodal distribution showing peaks at the A grade and F grade levels. Last year we noted a generally
superior achievement level shown by females relative to males and this can be seen again in the fall
2010 data.
Figure 2. Mathematics Grade Distribution, Fall 2010 (n=35,431)
As in past years, for all the remaining analyses, we designate the grades A, B, C, and P as “good grades”
and the grades D, F, and W as “at risk” – meaning that a student with these grades is at risk of not
completing his or her chosen program.
Figure 3 shows students’ mathematics achievement in this manner for each program cluster. It also
shows how little these have changed over the past three years. Achievement by students in applied arts
programs have improved somewhat but the overall numbers there are relatively small (as shown in
Table 4) particularly compared with business and technology where achievement has been virtually
static. Considering the efforts that have been made over the past three years by both secondary schools
and colleges to enhance student success this apparent lack of improvement is somewhat disappointing.
11
Unless otherwise stated, the achievement data in this chapter refer to the mathematics achievement of all first semester students.
A B C P D F W
Males 30.4% 16.8% 14.9% 2.7% 10.7% 20.8% 3.7%
Females 36.5% 17.6% 13.4% 5.0% 8.8% 15.3% 3.4%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
Males Females
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It suggests that either we have not yet identified those factors that most influence mathematics
achievement in college programs or that these contributing factors are very resistant to change. At the
very least, it suggests that the ultimate goals of the CMP – to improve college mathematics achievement
significantly – have yet to be reached.
Figure 3. Mathematics Achievement (% Good Grades) by Program Cluster, Fall 2008, 2009 and 2010.
Figure 4.Mathematics Achievement by Gender and Student Type, Fall 2010.
All Programs Applied Arts Business General Technology
Fall 2008 67.0% 74.8% 64.8% 65.2% 67.8%
Fall 2009 68.6% 77.1% 67.2% 64.7% 69.5%
Fall 2010 67.6% 78.1% 64.2% 65.7% 68.0%
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
90.0%
Fall 2008 Fall 2009 Fall 2010
M F M F M F M F
All Students(n=35,203)
VROGs (n=19,795) 2nd Career (n=936)Direct Entry
(n=7,339)
Good Grades 64.8% 72.5% 61.6% 68.7% 80.0% 84.5% 62.6% 70.2%
At Risk 35.2% 27.5% 38.4% 31.3% 20.0% 15.5% 37.4% 29.8%
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
90.0%
Good Grades At Risk
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Figure 4 shows the results of the analyses of mathematics achievement for the Fall 2010 cohort by gender
and student type. It shows once again that females outperform males in all groups. It also shows
relatively small differences among the Direct Entry students, the Very Recent Ontario Graduates, and all
students, but a clearly higher achievement by the 2nd career students. We noted this last year and it is
also reflected in the age-based analysis (see Figure 7).
Figure 5.Mathematics Achievement by Student Type and Course Type, Fall 2010.
Figure 6. Mathematics Achievement of French-language Students by Language of Instruction, Fall
2008, 2009, and 2010.
C(n=32,370)
P(n=3,110)
C(n=17,910)
P(n=2,000)
C(n=891)
P(n=52)
C(n=6,616)
P(n=747)
All Students(n=35,480)
VROGs (n=19,910) 2nd Career (n=943)Direct Entry
(n=7,363)
Good Grades 68.1% 62.7% 64.1% 59.5% 81.5% 82.7% 65.3% 64.1%
At Risk 31.9% 37.3% 35.9% 40.6% 18.5% 17.3% 34.7% 35.9%
0.0%10.0%20.0%30.0%40.0%50.0%60.0%70.0%80.0%90.0%
Good Grades At Risk
French English French English French English
Fall 2008 Fall 2009 Fall 2010
Good Grades 74.4% 70.4% 72.1% 69.6% 77.8% 69.8%
At Risk 25.6% 29.6% 27.9% 30.4% 22.2% 30.2%
0.0%10.0%20.0%30.0%40.0%50.0%60.0%70.0%80.0%90.0%
Good Grades At Risk
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Achievement by the same four groups of students can also be analysed from the perspective of the
types of mathematics course being taken at college. In Figure 5, courses at the college level are
designated C and those of a preparatory nature P. Once again there are relatively small differences
among the groups with the exception of the 2nd Career students. It is interesting to note that even when
students take a preparatory mathematics course, smaller proportions achieve good grades.
French-language students attend both English and French language colleges as noted in chapter 1.
Figure 6 shows the mathematics achievement of the past three years’ cohorts analysed by the language
of instruction. In each year, students attending French language colleges have achieved higher grades
but the proportions have not varied much over the three years. Readers are cautioned not to make too
much of the inter-college comparisons since each college has a different mix of programs. Nonetheless,
given that the proportion of French-language students attending French-language colleges has increased
over the past three years (as shown in Tables 8 and 9 in chapter 1), those colleges can be encouraged to
note that the levels of mathematics achievement of these students have also been maintained.
The CMP 2009 report first analysed the mathematics achievement data by age and showed the
existence of a relationship between students’ ages/genders and their mathematics achievement. This
relationship is shown once again for the Fall 2010 cohort in Figure 7 and Table 10. Once again we see
that students in their 30s and 40s out-perform both younger and older students and that females
outperform males in all age groups. In last year’s report we noted an unusually lower level of
achievement by males in their 30s and in the light of this year’s data which is similar to that of the Fall
2008 cohort, we assume that last year’s results were anomalous, though it should also be noted that
males and females are not equally represented in all program areas.
Figure 7.Mathematics Achievement by Gender and Age, Fall 2010.
under 23(n=25,822)
23-29(n=5,758)
30-39(n=2,022)
40-49(n=1,082)
50 & over(n=519)
TOTAL(N=35,203)
Males 61.8% 71.8% 77.6% 76.2% 71.9% 64.8%
Females 68.8% 80.5% 85.5% 83.0% 76.6% 72.5%
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
90.0%
Males Females
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Table 10. Mathematics Achievement by Age and Gender Math All Students Males Females
Age Enrolment %GG %AR GG AR GG AR
Under 17 77 57.1% 42.9% 63.3% 36.7% 46.4% 53.6%
17 93 72.0% 28.0% 77.8% 22.2% 64.1% 35.9%
18 7,409 66.4% 33.6% 65.0% 35.0% 68.6% 31.4%
19 8,216 62.2% 37.8% 59.8% 40.2% 67.1% 32.9%
20 4,792 62.1% 37.9% 59.0% 41.0% 67.9% 32.1%
21 3,059 65.0% 35.0% 61.3% 38.7% 72.2% 27.8%
22 2,176 68.9% 31.1% 65.9% 34.1% 73.9% 26.1%
Under 23 25,822 64.3% 35.7% 61.8% 38.2% 68.8% 31.2%
23 1,641 72.2% 27.8% 69.9% 30.1% 75.8% 24.2%
24 1,218 74.4% 25.6% 70.4% 29.6% 81.4% 18.6%
25 871 76.8% 23.2% 74.0% 26.0% 81.5% 18.5%
26 715 76.6% 23.4% 74.1% 25.9% 81.2% 18.8%
27 538 79.0% 21.0% 73.1% 26.9% 87.6% 12.4%
28 452 77.0% 23.0% 73.6% 26.4% 81.3% 18.7%
29 323 75.2% 24.8% 70.5% 29.5% 82.9% 17.1%
23-29 5,758 75.1% 24.9% 71.8% 28.2% 80.5% 19.5%
30 336 80.7% 19.3% 76.2% 23.8% 86.1% 13.9%
31 270 80.4% 19.6% 76.8% 23.2% 84.9% 15.1%
32 225 80.0% 20.0% 77.4% 22.6% 84.1% 15.9%
33 184 84.2% 15.8% 77.1% 22.9% 93.7% 6.3%
34 194 84.5% 15.5% 80.7% 19.3% 89.4% 10.6%
35 181 81.8% 18.2% 83.9% 16.1% 79.5% 20.5%
36 178 78.7% 21.3% 73.3% 26.7% 85.7% 14.3%
37 145 85.5% 14.5% 83.1% 16.9% 88.2% 11.8%
38 155 81.9% 18.1% 79.8% 20.2% 84.8% 15.2%
39 154 73.4% 26.6% 70.0% 30.0% 78.1% 21.9%
30-39 2,022 81.1% 18.9% 77.6% 22.4% 85.5% 14.5%
40 140 82.1% 17.9% 81.1% 18.9% 83.3% 16.7%
41 143 78.3% 21.7% 73.3% 26.7% 83.8% 16.2%
42 130 78.5% 21.5% 75.7% 24.3% 82.1% 17.9%
43 104 80.8% 19.2% 74.6% 25.4% 88.9% 11.1%
44 102 77.5% 22.5% 71.0% 29.0% 87.5% 12.5%
45 101 86.1% 13.9% 86.0% 14.0% 86.3% 13.7%
46 108 75.9% 24.1% 76.3% 23.7% 75.0% 25.0%
47 74 79.7% 20.3% 77.3% 22.7% 83.3% 16.7%
48 92 68.5% 31.5% 67.9% 32.1% 69.2% 30.8%
49 88 83.0% 17.0% 80.0% 20.0% 86.8% 13.2%
40-49 1,082 79.1% 20.9% 76.2% 23.8% 83.0% 17.0%
50 & over 519 73.6% 26.4% 71.9% 28.1% 76.6% 23.4%
TOTAL 35,203 67.6% 32.4% 64.8% 35.2% 72.5% 27.5%
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Secondary School Mathematics Pathways
From the earliest reports of the CMP, we have been interested in exploring how students’ college
mathematics achievement might be influenced by their choice of secondary school mathematics
courses. Grade 12 courses were the first we looked at and there has been a fairly consistent pattern to
be seen here as shown in Figure 812. The results continue to show that:
The largest proportion of students who continue to college mathematics take MAP4C as their
highest level mathematics course in secondary school and that only about 58% of them obtain
good grades in college mathematics.
The course designed to prepare students for college technology programs (MCT4C) is taken by
fewer than 10% of all students, although their subsequent achievement in college mathematics
is better (with about 68% obtaining good grades);
Over one quarter of college mathematics students have taken one or more 12U mathematics
course in secondary school and these students obtain significantly higher levels of success in
college mathematics (nearly 80% with good grades).
Figure 8. Achievement of VROGs with different Grade 12 mathematics courses, Fall 2008, 2009 and 2010
12
Data in this section of the report refers to students’ highest level Grade 12 mathematics course. For example: a student who has taken MAP4C and no other Grade 12 mathematics course would be included in the MAP4C group; a student who has taken only MCT4C or both MAP4C and MCT4C would be included in the MCT4C group; and a student who has taken either of these and also a grade 12U course would be included in the “any 12U” group. This convention applies throughout this section unless otherwise specified.
Fall 2008(n=5,664)
53.8%
Fall 2009(n=7,089)
46.6%
Fall 2010(n=8,332)
45.5%
Fall 2008(n=1,170)
11.1%
Fall 2009(n=1,343)
8.8%
Fall 2010(n=1,567)
8.6%
Fall 2008(n=2,779)
26.4%
Fall 2009(n=4,347)
28.5%
Fall 2010(n=5,136)
28.0%
MAP4C MCT4C any 12U
Good Grades 57.7% 58.2% 58.2% 68.3% 69.2% 67.8% 79.6% 78.4% 78.3%
At Risk 42.3% 41.8% 41.8% 31.7% 30.8% 32.2% 20.4% 21.6% 21.7%
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
90.0%
Good Grades At Risk
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This analysis is based on records of students having obtained particular Grade 12 mathematics credits,
regardless of the marks obtained in the course. Figure 8 thus shows the average college mathematics
achievement of all students who have taken different Grade 12 mathematics courses. Figures 9 and 10
takes this analysis one stage further and compares the college mathematics achievement of students
with varying levels of achievement in each of MAP4C (Figure 9) and MCT4C (Figure 10).
Figure 9. College Mathematics Achievement by Level of Achievement in MAP4C, Fall 2010
Figure 10. College Mathematics Achievement by Level of Achievement in MCT4C, Fall 2010
Overall(N=10,743)
<50%(n=619)
50-59%(n=1,968)
60-69%(n=2,548)
70-79%(n=2,804)
80% & over(n=2,804)
Good Grades 58.7% 43.9% 41.9% 51.5% 62.2% 76.8%
At Risk 41.3% 56.1% 58.1% 48.5% 37.8% 23.2%
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
90.0%
Good Grades At Risk
Overall(N=2,877)
<50%(n=242)
50-59%(n=582)
60-69%(n=598)
70-79%(n=746)
80% & over(n=709)
Good Grades 69.7% 55.4% 57.2% 64.0% 74.8% 84.1%
At Risk 30.3% 44.6% 42.8% 36.0% 25.2% 15.9%
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
90.0%
Good Grades At Risk
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The steep gradient of college mathematics achievement in both of these graphs show that merely
obtaining the credit is of less importance in predicting subsequent achievement than the level of
achievement in each course. In the case of MAP4C, it can be seen that 80% or higher is needed in order
to be confident of success in college mathematics but over half of those who come to college
mathematics with MAP4C as their highest level mathematics course have obtained less than 70% in that
course and fewer than half of these obtain good grades in college mathematics.
Figure 11. Achievement of VROGs with different combinations of Grade 11 and Grade 12
mathematics courses, Fall 2008, 2009 and 2010
Figure 11 furthers the analysis shown in Figure 8 by showing the college mathematics achievement of
those who have taken the most commonly encountered combination of Grade 11 and Grade 12
mathematics courses. The MBF3C and MAP4C combination is the most common of all, taken by close to
one-third of all students, barely half of whom go on to obtain good grades in college mathematics. The
patterns of achievement of those taking MCF3M/MCT4C and MCR3U/any 12U have also remained very
steady through the three cohorts now included.
Current Ministry policy requires that students obtain three mathematics credits (one of which must be
at the Grade 11 or 12 level) in order to graduate with an Ontario Secondary School Diploma (OSSD).
When we examined the numbers of those in college mathematics whose last secondary school
mathematics course was at the Grade 11 level, we were surprised to find the numbers of college
mathematics students taking this option to be so high. Figure 12 shows that over 3,000 students are
now in this group, representing an increase from about 13% of all students in 2008 to 18.8% in 2010.
Not only have these students received less mathematics by way of preparation for college, there could
Fall 2008(n=3,732)
35.4%
Fall 2009(n=4,945)
32.5%
Fall 2010(n=5,724)
31.2%
Fall 2008(n=645)
6.1%
Fall 2009(n=871)
5.7%
Fall 2010(n=1042)
5.7%
Fall 2008(n=1,918)
18.2%
Fall 2009(n=2,935)
19.3%
Fall 2010(n=3,515)
19.2%
MBF3C + MAP4C MCF3M + MCT4C MCR3U + any 12U
Good Grades 52.5% 54.8% 55.3% 68.7% 68.5% 68.0% 83.8% 83.1% 81.1%
At Risk 47.5% 45.2% 44.7% 31.3% 31.5% 32.0% 16.2% 16.9% 18.9%
0.0%
10.0%
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30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
90.0%
Good Grades At Risk
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be a significant time gap between the last school mathematics course and the first college course13. Not surprisingly, therefore, their
achievement at college is not strong. Those who take MBF3C as their final secondary school mathematics course make up the largest group and
barely half of these obtain good grades in college mathematics.
Figure 12. Achievement of VROGs with Grade 11 terminal mathematics courses, Fall 2008, 2009 and 2010
13
For students entering college directly following secondary school, this could be up to a year and a half; for those taking a break between school and college, it could be much longer.
Fall 2008(n/a)
Fall 2009(n=183)
1.2%
Fall 2010(n=235)
1.3%
Fall 2008(n=724)
6.9%
Fall 2009(n=1,072)
7.0%
Fall 2010(n=1,508)
8.2%
Fall 2008(n=452)
4.3%
Fall 2009(n=760)
5.0%
Fall 2010(n=966)
5.3%
Fall 2008(n=154)
1.5%
Fall 2009(n=327)
2.1%
Fall 2010(n=365)
2.0%
MEL3E MBF3C MCF3M MCR3U
Good Grades 39.9% 38.7% 38.7% 46.6% 50.2% 56.0% 57.9% 57.8% 71.4% 70.6% 69.6%
At Risk 60.1% 61.3% 61.3% 53.4% 49.8% 44.0% 42.1% 42.2% 28.6% 29.4% 30.4%
0.0%
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20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
Good Grades At Risk
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The CMP has reviewed each year the achievement in college mathematics of students who have taken either the Academic or Applied
mathematics courses in Grades 9 and 10 and Figure 13 shows the results of this analysis for the past three years.
Figure 13. Achievement of VROGs with Grades 9 and 10 Applied and Academic courses in Mathematics, Fall 2008, 2009 and 2010
Readers will note the small but steady increase in both the numbers of students with Applied mathematics in Grade 9 and 10 and their
achievement levels in college mathematics. This is partly due to the implementation of the (2007) revised mathematics curriculum for Grades 11
and 12 which permitted a pathway from Grade 10 applied mathematics to the MCF3M course in Grade 11. Figure 14 shows the growth in
numbers of students taking this pathway from 3.6% in 2008 to 6.6% in 2010 and the growth in achievement from 59.1% with good grades (in
2008) to 63.9% in 2010. These results provide evidence for the value of this particular revision, at least for those students who go on to take
mathematics in college.
Fall 2008 (n=2,392)22.7%
Fall 2009 (n=3,424)22.5%
Fall 2010 (n=4,634)25.3%
Fall 2008 (n=4,456)42.3%
Fall 2009 (n=5,288)34.7%
Fall 2010 (n=7,429)40.6%
Grades 9 & 10 Applied Grades 9 & 10 Academic
Good Grades 47.3% 50.2% 51.9% 74.4% 76.1% 74.7%
At Risk 52.7% 49.8% 48.1% 25.6% 23.9% 25.3%
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
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Figure 14. Achievement of students with MFM2P and MCF3M, Fall 2008, 2009, and 2010
Table 11 shows, for Fall 2010 only, the numbers of students taking other combinations of mathematics
courses which were included in this year’s research questions as outlined in chapter 1.
Table 11. Numbers and Achievement of Students with Selected Mathematics Pathways, Fall 2010
n Good Grades
At Risk
MCF3M-MCR3U 418 66.5% 33.5%
MFM2P-MPM2D 28 46.4% 53.6%
MFM1P-MPM1D 17 64.7% 35.3%
Further Analysis of CMP Data
Analyses shown in this chapter focuses on province-wide data. Using the interactive CMP Database,
educators from all 24 colleges and 72 school boards in Ontario are able to pursue these and other
analyses based on data pertaining to their own institutions. Access to this data is free and requires only
that users be authorised by a College Vice-President Academic (in the case of college users) or a
Superintendent of Program (in the case of school board users). The details for obtaining authorisation
along with related policies and procedures are available at the CMP web site.
Fall 2008 (n=381) 3.6% Fall 2009 (n=806) 5.3% Fall 2010 (n=1,213) 6.6%
Good Grades 59.1% 62.0% 63.9%
At Risk 40.9% 38.0% 36.1%
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
Good Grades At Risk
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Chapter 3: College Mathematics in Context
Curricula at the college level are not simply an extension of curricula at elementary and secondary
schools into specific program areas. They are not set out in Ministry policy documents for province wide
application but are separately developed by each of the 24 colleges. This sometimes leads to confusion,
even frustration, on the part of members of the elementary and secondary educational community, who
are accustomed to statements of provincial curriculum policy for each subject, grade level and course.
Indeed, it is sometimes (but mistakenly) thought that there is no overarching curriculum policy
framework operating at the college level at all.
In fact, all college programs in Ontario must meet standards set out by the Ministry of Training, Colleges
and Universities. The Ontario College Program Standards14 identify vocational learning outcomes,
essential employability skills, and the general education requirements for most college programs. In the
words of the introduction to each standard, “collectively, these elements outline the essential skills that
a student must demonstrate in order to graduate from [a] program”15. In other words, each college
offering a given program is free to build its own curriculum as long as it can demonstrate and verify that
its graduates meet the learning outcomes in the corresponding program standard. So while the
endpoint of a program is consistent from one college to another, the journey to that endpoint (including
where students start – the admission requirements) may vary16. In this respect, college programs can be
seen to be quite distinct from secondary school programs and understanding the place of mathematics
in a given program requires this perspective.
The present chapter is divided into three sections. In the first, we illustrate the place of mathematics in
college programs through a set of case studies of programs in business and technology from a number
of typical colleges. These will take the reader from the provincial program standards “back” through the
curricula where mathematics is embedded in program-specific applied courses to the standalone
mathematics courses in first semester that have been the primary focus of CMP study over the years.
The second section of the chapter examines these courses in more detail – this time, from all 24 colleges
– to analyse the mathematics topics and skills most frequently taught at this point. Both regular
diploma/certificate-level courses and preparatory or foundational mathematics courses are analysed in
this way. The final section of the chapter maps those topics most frequently encountered in first
semester foundational or preparatory college mathematics courses onto the framework of topics found
in elementary and secondary mathematics curricula.
14
http://www.tcu.gov.on.ca/pepg/audiences/colleges/progstan/index.html 15
For example: Business Administration – Accounting Program Standard. (Toronto: Ministry of Training, Colleges and Universities, 2009) p. 1. (emphasis added). All program standards are freely available from the internet at web site in footnote 14. 16
Individual colleges may also add local learning outcomes to reflect specific local needs and/or interests.
.
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Case Studies of Mathematics in College Programs
One business program and two technology programs were selected for the study: Business
Administration – Accounting, a three year program (MTCU code 60100), Electrical Engineering
Technician, a two year program (MTCU code 55613) and Electrical Engineering Technology, a three year
program (MTCU code 65613). These programs are offered by a broad selection of colleges across
Ontario and between them represent the spectrum of programs in Business and Technology.
The CMP team then selected nine colleges offering these programs to ensure as diverse a
representation as possible, including French language, northern, large urban and smaller colleges. Four
colleges were selected for the business study and five others for the technology study – three of which
offer 3- and 2-year programs, one offering only the 3 year program, and one only the 2 year program.
Components of the study for each program and each college included: student admission requirements,
the program curriculum to identify courses with embedded mathematics and areas of emphasis, and a
topical analysis of all standalone mathematics courses in the program. Program information was
collected from college web sites and, where necessary, phone calls were made to the CMP college lead
or the Ontario College Mathematics Council representative to obtain the required information.
A framework for analysis was developed using the program learning outcomes for the published
program standard as a reference. Each college was presented with its individual program analysis with
admission requirements, a semester-by-semester representation of the curriculum including course
names, course codes, course descriptions, designation as standalone mathematics course or course with
embedded mathematics, topical analysis, emphasis, and mapping to the published program learning
outcome for validation. Finally, a comparative (cross-college) analysis of validated cases was prepared.
Business Administration – Accounting
Program Standard
The basis for all college programs in Business Administration – Accounting is the program standard17. As
noted earlier, this comprises several parts, of which the program– specific part to be analysed here is a
list of vocational learning outcomes, as shown below.
17
Business Administration – Accounting Program Standard. MTCU 60100 (Toronto: Ministry of Training, Colleges and Universities, 2009).
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In the standard itself, each of these is further elaborated into elements of performance (see next page
for an example of these for outcome #8).
The program is aligned with the requirements set forth by the Certified General Accountants (CGA) of
Ontario, which grants transfer credits to graduates with a grade of “B” or better in each course. The
number of transfer credits is dependent upon the specific college program and ranges between 10 and
14 credits.
Curriculum Analysis
Most of the mathematics taught in this program in all four colleges is actually embedded in the
curriculum, in this case within accounting, financial management, micro and macroeconomics, auditing
and taxation courses. It is within these courses that 6 of the 8 program learning outcomes are met (at
Synopsis of the Vocational Learning Outcomes – Business Administration – Accounting Program The graduate has reliably demonstrated the ability to: record financial transactions in compliance with Canadian Generally Accepted Accounting Principles
for sole proprietorships, partnerships, private enterprises, publicly accountable enterprises and
non– profit organizations.
1. prepare and present financial statements, reports and other documents in compliance with
Canadian Generally Accepted Accounting Principles for sole proprietorships, partnerships and private enterprises.
2. contribute to recurring decision– making by applying fundamental management accounting
concepts.
3. prepare individuals’ income tax returns and basic tax planning in compliance with relevant legislation and regulations.
4. analyze organizational structures, the interdependence of functional areas, and the impact
those relationships can have on financial performance.
5. analyze, within a Canadian context, the impact of economic variables, legislation, ethics, technological advances and the environment on an organization’s operations.
6. outline the elements of an organization’s internal control system and risk management.
7. contribute to recurring decision– making by applying fundamental financial management
concepts.
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least in part) and there is consistency between the four colleges in terms of which learning outcomes
are addressed (see Table 12). The mathematical topics in these courses are entirely in the area of
arithmetic.
Table 12 Courses with Embedded Mathematics
College # Courses with Mathematics
# of Courses with Embedded
Mathematics
Topical Analysis Program Learning Outcomes Mapping
A 14 13 Arithmetic 1, 2, 3, 4, 7, 8
B 17 15 Arithmetic 1, 2, 3, 4, 7, 8
C 16 14 Arithmetic 1, 2, 3, 4, 7, 8
D 15 12 Arithmetic 1, 2, 3, 4, 7, 8
In addition to the embedded mathematics, all programs offer at least one and as many as three
standalone mathematics courses. These courses map against program learning outcome #8 in the
program standard whose “elements of the performance” are elaborated below18.
18
Op. cit. p. 13.
The graduate has reliably demonstrated the ability to
contribute to strategic decision making by applying advanced financial management concepts.
Elements of the Performance
calculate and interpret financial ratios
calculate present and future values of financial instruments
calculate, analyze and evaluate, past and present data to prepare estimates and forecast trends
apply capital budgeting methods such as Net Present Value (NPV), cost– benefit analysis, payback period and internal rate of return, and evaluate investment opportunities explain and apply discount rates
identify, analyze and evaluate various sources of financing including leasing, debt and equity
collect, organize and interpret statistical data related to an organization’s operations utilize the appropriate software to produce reports and other documents related to financial management
apply risk management analysis to generate information for decision– making and the creation of financial strategies
evaluate the financial implications of changes to components of working capital
prepare budgets and statements of cash flow
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The decision to teach these in one course or several courses is made at the college level taking into
account the length and structure of courses in the college (which vary across the system from 42 to 84
hours) as well as the perceived needs of the students in their region. A summary of the standalone
mathematics courses in the four colleges in this case study is shown in Table 13. All four colleges teach
statistics in these courses, college B also identifies arithmetic, and college D actually specifies the topics
of ratios, proportion and percentages.
Table 13. Standalone Mathematics Courses
College # Courses with Mathematics
# of Stand Alone Mathematics
Courses
Topical Analysis Program Learning Outcome Mapping
A 14 1 Statistics 8
B 17 2 Arithmetic, Statistics 8
C 16 2 Statistics 8
D 15 3 Ratios, proportion, percent, statistics
8
Admission Requirements
The admission requirements for each of the four colleges are summarized in Table 14. All four colleges
require that applicants possess an Ontario Secondary School Diploma (OSSD) or the equivalent; in some
cases, specific equivalencies are stated. All colleges accept students under the Mature Applicant basis.
Eligibility is determined based on the outcome of an assessment process. Only one college states that,
regardless of status, the mature applicant must possess the academic credits specific to the program
and that applicants “may be allowed” to write a test in lieu of having the required credits.
Mathematics requirements vary across the 4 colleges. Only one college requires a Grade 12
Mathematics credit at the college preparation level (i.e. MAP4C or MCT4C), while two others identify
Grade 12 Mathematics or a Grade 11 university/college or university credit. Only one college accepts
students with a Grade 11 College preparation (MBF3C) credit as their highest level of secondary school
mathematics. Students who lack the expected mathematics background may be counseled or required
to take an alternative program (such as a pre– Business program).
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Table 14. Admission Requirements for Business Administration – Accounting Programs – 2011/12
Admission Criterion
College A College B College C College D
Diploma OSSD or Equivalent OSSD or equivalent with senior credits at the C, M or U level or appropriate credits from Academic and Career Entrance
OSSD or Equivalent Academic and Career Entrance (ACE) certificate General Education Development (GED) certificate
OSSD with a majority of senior credits at the C,U, or M level or Mature student status (age 19 or older)
Language Grade 12 English (ENG4C or equivalent)
Grade 12 English; ENG4(C) or ENG4(U)
Mathematics MAP4C, MCR3U, MCF3M or equivalent
Grade 11 College preparation Mathematics (MBF3C)
Grade 12 Mathematics (MAP4C or equivalent)
Grade 12 Math (C,U) or Grade 11 Math (U,M)
Mature Student Status (age 19 or older and without a high school diploma)
Eligibility determined by academic achievement testing
Successful completion of Mature Student Assessment
Eligibility may be determined by academic achievement testing
Academic credits in Mathematics, English. Mature Status applicants may be allowed to write the Mature Student Entrance Test in lieu of having these Ontario Secondary school credits or their equivalent.
Other Applicants will be selected on the basis of their proficiency in English and mathematics in the event that the number of applicants exceeds the number of places.
Applicants will be selected on the basis of their proficiency in English and mathematics in the event that the number of applicants exceeds the number of places.
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Electrical Engineering Technician and Technology
Program Standard
The basis for program development in the areas of Electrical Engineering Technology and Technician is
once again the Program Standards. Synopses of the Vocational Learning Outcomes for each of these are
shown on this and the next pages.
Synopsis of the Vocational Learning Outcomes – Electrical Engineering Technology Program
The graduate has reliably demonstrated the ability to
1. analyze, interpret, and produce electrical and electronics drawings, technical reports including other related documents and graphics.
2. analyze and solve complex technical problems related to electrical systems by applying mathematics and science principles.
3. design, use, verify, and maintain instrumentation equipment and systems. 4. design, assemble, test, modify, maintain and commission electrical equipment and systems
to fulfill requirements and specifications under the supervision of a qualified person. 5. commission and troubleshoot static and rotating electrical machines and associated
control systems under the supervision of a qualified person. 6. design, assemble, analyze, and troubleshoot electrical and electronic circuits, components,
equipment and systems under the supervision of a qualified person. 7. design, install, analyze, assemble and troubleshoot control systems under the supervision
of a qualified person. 8. use computer skills and tools to solve a range of electrical related problems. 9. create, conduct and recommend modifications to quality assurance procedures under the
supervision of a qualified person. 10. prepare reports and maintain records and documentation systems. 11. design, install, test, commission and troubleshoot telecommunication systems under the
supervision of a qualified person. 12. apply and monitor health and safety standards and best practices to workplaces. 13. perform and monitor tasks in accordance with relevant legislation, policies, procedures,
standards, regulations, and ethical principles. 14. configure installation and apply electrical cabling requirements and system grounding and
bonding requirements for a variety of applications under the supervision of a qualified person.
15. design, commission, test and troubleshoot electrical power systems under the supervision of a qualified person.
16. select and recommend electrical equipment, systems and components to fulfil the requirements and specifications under the supervision of a qualified person.
17. apply project management principles to contribute to the planning, implementation, and evaluation of projects.
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While, at first glance, the two program standards appear similar, there are subtle but important
differences between the 3-year trained technologist and the 2-year trained technician. For example in
outcome #2, the technician is expected to “analyze and solve routine technical problems,” the
technologist can “analyze and solve complex technical problems” (emphasis added). And in outcome #3,
the technician will “use, verify and maintain instrumentation equipment and systems”; the technologist
is also expected to be able to “design” such equipment and systems.
Synopsis of the Vocational Learning Outcomes – Electrical Engineering Technician The graduate has reliably demonstrated the ability to
1. interpret and produce electrical and electronics drawings including other related documents and graphics.
2. analyze and solve routine technical problems related to electrical systems by applying mathematics and science principles.
3. use, verify, and maintain instrumentation equipment and systems. 4. assemble, test, modify and maintain electrical circuits and equipment to fulfil requirements
and specifications under the supervision of a qualified person. 5. install and troubleshoot static and rotating electrical machines and associated control
systems under the supervision of a qualified person. 6. verify acceptable functionality and apply troubleshooting techniques for electrical and
electronic circuits, components, equipment, and systems under the supervision of a qualified person.
7. analyze, assemble and troubleshoot control systems under the supervision of a qualified person.
8. use computer skills and tools to solve routine electrical related problems. 9. assist in creating and conducting quality assurance procedures under the supervision of a
qualified person. 10. prepare and maintain records and documentation systems. 11. install, test and troubleshoot telecommunication systems under the supervision of a
qualified person. 12. apply health and safety standards and best practices to workplaces. 13. perform tasks in accordance with relevant legislation, policies, procedures, standards,
regulations, and ethical principles. 14. configure installation and apply electrical cabling requirements and system grounding and
bonding requirements for a variety of applications under the supervision of a qualified person.
15. assist in commissioning, testing and troubleshooting electrical power systems under the supervision of a qualified person.
16. select electrical equipment, systems and components to fulfill the requirements and specifications under the supervision of a qualified person.
17. apply project management principles to assist in the implementation of projects.
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While the Ontario Association of Certified Engineering Technicians and Technologists (OACETT)
recognizes applied science and engineering technology graduates from all Ontario colleges for purposes
of individual certification as Certified Engineering Technologists (C.E.T.) or Certified Engineering
Technicians (C. Tech), only one of the colleges in this case study (College H) has obtained national
accreditation for its electrical engineering technology program through the Canadian Technology
Accreditation Board (CTAB) of the Canadian Council of Technicians and Technology. CTAB accredits
programs against the National Technology Benchmarks, which includes a common mathematics
benchmark for at the technician and technology levels as well as mathematics related benchmarks
specific to the discipline.
Curriculum Analysis
The programs of the five colleges were analysed in a similar manner to those in the Business case study.
Courses that included embedded mathematics were identified and analysed and standalone
mathematics courses were similarly analysed. Table 15 shows that all technology programs have at
least 4 courses in mathematics. One of the programs (College E) has 7 mathematics courses in the three
years of study. This may be related to the local situation of this college which serves boards from which
very few students have taken MCT4C and most enter college with MAP4C as their final mathematics
course. As well courses at the colleges vary in length from 42 to 84 hours; so that while some colleges
may have fewer mathematics courses in their curriculum, the hours of instruction may be equal to
colleges with more mathematics courses.
Table 15. Standalone Mathematics Courses – Electrical Engineering Technology
College # Courses with
Mathematics
# of Standalone Mathematics
Courses
Topical Analysis Program Learning Outcome Mapping
E 23 7 Algebra, Trigonometry, The Straight Line, Vectors, Quadratic Equations, Logarithmic Functions, Calculus, Statistics & Probability
2,8
F 9 4 Algebra, Trigonometry Linear Equations, Exponential & Logarithmic Functions, Radicals and Complex Numbers, Logarithmic Functions
2
G 12 4 Algebra Trigonometry Linear Equations
2
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H 12 5 Algebra, Arithmetic, Trigonometry Exponential & Logarithmic Functions Exponents & Radicals, Vectors Oblique Triangles, Quadratic Equations, Statistics & Probability, Linear Equations
2
All of the mathematics courses map against program learning outcome #2 for the technology program
standard as shown here.
Table 16. Courses with Embedded Mathematics – Electrical Engineering Technology
College
# of Courses with
Mathematics
# of Courses with Embedded
Mathematics
Topical Analysis Program Learning Outcomes Mapping
E 23 16 Algebra, Trigonometry Calculus, Arithmetic Number Systems, Vectors Electrical Theory & Applications
1,2,3,4,5,6,7,8,9, 10,11,12,13,14,15, 16,17
F 9 5 Algebra, Electrical Theory & Applications, Arithmetic
2,16,17
2. The graduate has reliably demonstrated the ability to analyze and solve complex technical problems related to electrical systems by applying mathematics and science principles. Elements of the Performance
apply mathematical and scientific concepts and analysis (e.g. differential calculus, algebra and trigonometry) as part of the design process;
apply advanced mathematical theory and scientific analysis to troubleshoot, maintain, and test electrical circuits, equipment, systems, and subsystems;
evaluate and quantify complex technical problems and formulate alternative solutions;
use statistical measures to analyze and solve technical problems;
represent graphically and analyze experimental data;
develop mathematical and graphical models to test alternatives and deduce optimal solutions;
apply Laplace Transform and Fourier Transform and their applications to circuit analysis, behaviour and design;
perform conversion in and among number systems such as hexadecimal, decimal, octal, binary, and binary-coded decimal.
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G 16 8 Arithmetic, Trigonometry Algebra, Electrical Theory & Applications
2,3,4,5,12,16,17
H 12 7 Algebra, Electrical Theory & Applications, Vectors
2,16,17
Table 16 shows the courses in which mathematics is embedded and the topics contained in them.
Algebra, trigonometry and linear equations are the common topics taught in all colleges. The
application of mathematics is embedded in the curriculum. It is within these courses that at least 3 of
the 17 program learning outcomes are met at least in part. These outcomes are related electrical
measurements, electrical fundamentals, control systems, digital fundamentals, project courses, etc.
The analysis of the technician mathematics curriculum presented in Tables 17 and 18 shows similarities
to that of the technology curriculum. There is variation in the number of standalone mathematics
courses (ranging from 1 course to 4 courses). As noted before the number of hours per week of
instruction may differ from college to college and within an institution, for example a first mathematics
course may have more hours of instruction than subsequent courses. All of the standalone mathematics
courses map against program learning outcome #2 for the technician program standard.
2. The graduate has reliably demonstrated the ability to analyze and solve routine technical problems related to electrical systems by applying mathematics and science principles. Elements of the Performance
• use mathematical and scientific applications to solve routine technical problems (e.g., basic
algebra and trigonometry);
• apply accurately mathematical and scientific concepts to troubleshoot, maintain, and test
electrical circuits, equipment, systems, and subsystems;
• identify routine technical problems related to electrical systems and formulate alternative
solutions;
• use statistical measures to analyze and solve routine technical problems
• represent graphically experimental data;
• use appropriate software for calculations;
• interpret and support the results of calculations;
• perform conversion in and among number systems such as hexadecimal, decimal, octal,
binary, and binary-coded decimal.
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As was the case for the technology program, there is significant mathematics embedded in the
technician curriculum (Table 18); however there is no one program learning outcome other than #2 that
is identified by all sample colleges.
Table 17. Standalone Mathematics Courses – Electrical Engineering Technician
College # of Courses with
Mathematics
# of Standalone Mathematics
Courses
Topical Analysis Program Learning Outcome Mapping
E 12 4 Algebra, Trigonometry, The Straight Line, Calculus
2,8
F 4 1 Algebra, Trigonometry 2
G 9 2 Algebra, Trigonometry, 2
I 5 2 Algebra, Trigonometry, Statistics
2
Table 18. Courses with Embedded Mathematics – Electrical Engineering Technician
College
# of Courses with
Mathematics
# of Courses with
Embedded Mathematics
Topical Analysis Program Learning Outcomes Mapping
E 12 8 Algebra, Electrical Theory
1,2,3,4,5,6,7,9,11,12,13,15,16
F 4 3 Algebra, Electrical Theory & Applications, Arithmetic
2,16,17
G 9 7 Arithmetic, Trigonometry Algebra, Electrical Theory & Applications
2,3,4,5,12
I 5 3 Arithmetic, Trigonometry, Electrical Theory & Applications, Vectors
2,16,17
Admission Requirements
The published admission requirement for electrical engineering technician and technology programs for
the 2011/12 academic year is summarized in Table 19. All colleges require that applicants possess an
OSSD or equivalent such as an Academic Career Entrance or upgrading certificate, an Ontario College
Certificate in technology foundations, etc. All colleges accept students under the Mature Applicant basis
and eligibility may be determined based on the outcome of an assessment process. Two colleges state
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that regardless of status, the mature applicant must possess the academic credits specific to the
program. Mathematics admission requirements vary across the 4 colleges; only 1 college requires
MCT4C for admission while 2 indicate MCT4C is the preferred course. One college lists all Grade 12 C, M
or U courses as acceptable, however notes that meeting the minimum standard does not guarantee
admission. Admission requirements for colleges offering both the technician and technology programs
are the same due to the number of courses that are common to both programs, thus enabling ease of
transfer between programs.
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Table 19. Admission Requirements for Selected Electrical Engineering Technician and Technology Programs (for 2011/12 Academic Year)
Admission Criterion
College E College F College G College H College I
Programs Offered
Technician & Technology
Technician & Technology
Technician & Technology
Technology Technician
OSSD or Equivalent OSSD with courses C, U or M or O destinations
OSSD or equivalent or Mature student status (age 19 or older)
OSSD or equivalent or Mature student status The majority of Grade 11 and 12 courses must be college or university preparation level.
OSSD or equivalent. The majority of Grade 11 and 12 courses must be college or university preparation level.
English (French) Grade 12 English (C or U)
Grade 12 English (C or U)
Grade 12 English (ENG4C or 4U or equivalent.
Grade 12 English (ENG4C or ENG4U).
Grade 12 English (C or U)
Mathematics Grade 12 Mathematics (C, U) (MCT4C preferred; MAP4C is accepted with a minimum GPA of 60%
One of: MCT4C, MCR3U, MCF3M (with minimum grade of 65%. **Note-MAP4C was ill the minimum math requirement for the 2011/12 academic year.
Grade 12 Mathematics, C or U (MCT4C, MHF4U, MCV4U) or equivalent. Applicants with MAP4C or MDM4U will be considered only after writing the college Math admission test, min. grade cut off will apply
Grade 12 Math (MAP4C, MCT4C, MDM4U, MCB4U, MGA4U, MCV4U or MHF4U)
Grade 11 Math at the university or college level.
Mature Student Status (age 19 or older and without a high school diploma)
Mature students must undergo academic testing prior to admission.
Mature applicant with standing in the required courses and grades.
Eligibility may be determined by academic achievement testing.
Assessment by review of completed high school credits or through admission testing.
Mature applicants must provide proof of Gr. 12 English and Gr. 11 Math at the C or U level.
Other Grade 11 or Grade 12 Physics (C) or (U).
Required proficiency with word processing and spreadsheet applications.
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First Semester College Mathematics Courses
The first section of this analysis has provided illustrations from the fields of Business and Technology of
how college programs are based on provincially mandated program standards and how much of the
mathematics strand of the curriculum is embedded in program– specific courses. However, most
college programs in business and technology (including those illustrated here) also include one or more
standalone mathematics courses in the first or second semester. These form the basis for the courses
containing embedded mathematics that follow in later semester and mastery of the content of these
early courses is critical to the eventual achievement of the program outcomes.
The strategic importance of these early standalone mathematics courses provides the rationale for the
research into first semester students’ mathematics achievement, which has been the main focus of the
CMP since its inception. This year, however, we have extended our research into the content of these
first semester courses to enable us to understand better what students are expected to know and be
able to do, why too many students are failing or coming close to failure, and, most important of all, how
we can enhance their success.
First semester college mathematics courses are of two main types as noted in chapter 1. The first are
the standalone mathematics courses in regular vocational programs of the type we have illustrated in
the case studies. The second type are either mathematics courses contained in one-year pre-technology
or pre-business foundation programs or preparatory mathematics courses that are not part of any
program but which students may be advised to take if they do not appear ready to take the regular first–
semester mathematics course. Enrolments in each of these types of course were outlined in chapter 1
(Table 7) and student achievement in chapter 2 (Figure 5). The focus here is on the content of these two
types of course.
Numeracy Skills at College
The CMP research program collected samples of course outlines for each of these types of course in
both business and technology programs from each of the 24 colleges19 and analysed the topics they
contain. This analysis represented a more comprehensive version of the pilot study conducted last year
and described in the CMP 2010 final report20. From the pilot study, a framework of Numeracy Skills was
established which is used again this year to analyse the courses.
Table 20 shows the topic framework and its correspondence to the regular first semester mathematics
courses in both Technology and Business programs. A “Y” in any cell indicates that topics corresponding
to a particular numeracy skill (shown in the first column) were present in all first semester diploma
mathematics courses in either Technology or Business program clusters. Other topics, where there
were others, were of a program or cluster-specific nature.
19
Not every college had courses in each of these four categories. 20
G. Orpwood et al. College Mathematics Project 2010: Final report (Toronto: Seneca College, 2011) chapter 3.
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Table 20.
Numeracy Skills Framework in Relation to College Programs
Technology Cluster Business Cluster
Order of Operations Y Y
Fractions Y Y
Decimals Y Y
Percentages Y Y
Ratio and Proportion Y Y
Algebra Y Y
Exponents Y Y
Accuracy and Precision Y
Scientific Notation Y
Roots and Radicals Y Y
Analysis of Diploma Level Mathematics Courses
Each of the colleges provide samples of first semester mathematics courses from diploma– level
business and technology programs and these were analysed using this framework. In the 23 business
mathematics courses analysed, the proportion of topics on numeracy skills ranged from 0% to 100% of
all the topics, with a median of 28%, as shown in Figure 1521. A more detailed analysis of topics in each
course is in Appendix A. (This and each of the following three figures shows course identifiers on the
horizontal axis and the percentage of numeracy topics in the course on the vertical axis.)
21
One college does not offer Business mathematics in first semester.
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Figure 15: Proportion of Topics on Numeracy Skills in Diploma-level Business Courses at 19 Ontario Colleges
A similar analysis of 29 technology mathematics courses is shown in Figure 1622. Here, topics focused on numeracy skills comprise from 0% to
55% of all of the course topics with a median of 28%. Again, the full analysis is shown in Appendix A. Topics other than those involving
numeracy skills, such as logarithms, measurement, trigonometry and geometry are more specifically related to technology programs. Only two
courses allocate no time to Numeracy Skills. In one case, the college uses a placement test to stream students. In the other, only students with
MCT4C or MCR3U or an equivalent course are allowed entry.
22
Several colleges offer more than one first semester Technology mathematics course.
Average
A B C D E F G H I J K L M N O P Q R S
% ofTotal NS
32% 59% 25% 28% 53% 29% 100% 17% 38% 27% 18% 21% 0% 24% 29% 30% 32% 45% 33% 37%
0%
20%
40%
60%
80%
100%
120%
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Figure 16: Proportion of Topics on Numeracy Skills in Diploma-level Technology Courses at 23 Colleges
Analysis of Foundation Mathematics Courses
Pre-business mathematics courses from 11 colleges are included in this study23 and the summary analysis is shown in Figure 17. The complete
analysis is shown in Appendix A. The proportion of topics in these courses on numeracy skills ranges from 33% to 100% with a median of 76%.
This is significantly higher than is the case in diploma-level business mathematics shown in Figure 15 (where the median proportion was 28%).
The balance of these courses contains more cluster-specific topics such as Compound Interest, Descriptive Statistics and Graphing Linear
Equations. Accuracy in answers is critical in the business world, whether in accounting and break-even analysis or finance and prediction. The
ability to use these skills is necessary for students to be successful in their core courses.
23
Not all colleges offer a pre-Business program.
Average
A B C D E F G H I J K L M N O P Q R S T U V W
% ofTotal NS
29% 12% 39% 43% 50% 0% 14% 34% 20% 45% 33% 33% 45% 27% 20% 38% 26% 19% 21% 30% 38% 45% 6% 30%
0%
10%
20%
30%
40%
50%
60%
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Figure 17: Proportion of Topics on Numeracy Skills in Pre-Business Courses at 11 Colleges
In the case of pre-technology mathematics, 18 courses were analysed (see the summary analysis in
Figure 18)24. Once again, the full analysis is shown in Appendix A.
Figure 18: Proportion of Topics on Numeracy Skills in Pre-Technology Courses at 18 Colleges
24
Not all colleges offer pre-Technology programs
Average A B C D E F G H I J K
% ofTotal NS
63% 41% 33% 78% 84% 100% 55% 100% 76% 33% 50% 100%
0%
20%
40%
60%
80%
100%
120%
Average
A B C D E F G H I J K L M N O P Q R
% ofTotal NS
64% 41% 100%100% 61% 55% 48% 45% 100% 56% 61% 71% 38% 81% 61% 42% 91% 100% 50%
0%
20%
40%
60%
80%
100%
120%
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Here one can see once again the emphasis being given to numeracy skills with a range from 38% to
100% of the course topics and a median proportion of 61%, compared with 28% in the diploma level
technology mathematics courses as shown in Figure 16.
It is clear from these analyses that mastery of numeracy skills is of great importance for success in both
Business and Technology programs and, for this reason, students with limited skills in this area are
encouraged to take foundational or preparatory mathematics courses that place even greater stress on
these key topics. The final analysis in this section shows the frequency with which each of the specific
numeracy skills in the framework is featured in the college pre– business and pre– technology
mathematics courses.
Table 21. Frequency of Numeracy Skills in Foundation Mathematics Courses
Numeracy Skills Pre-Technology Mathematics (19
courses)
Pre-Business Mathematics (11
courses)
Order of Operations 19 (100%) 11 (100%)
Fractions 17 (89%) 11 (100%)
Decimals 14 (74%) 10 (91%)
Percentages 11 (58%) 9 (82%)
Ratio and Proportion 13 (68%) 9 (82%)
Algebra 19 (100%) 8 (73%)
Exponents 16 (84%) 6 (55%)
Accuracy and Precision 7 (37%) 0 (0%)
Scientific Notation 9 (47%) 0 (0%)
Roots and Radicals 2 (11%) 1 (9%)
The analysis in Table 21 shows which numeracy skills are considered to be the most critical in relation to
preparing for diploma level college mathematics in both business and technology programs. Not
surprisingly, perhaps, order of operations, fractions, decimals, percentages, and ratio and proportion
head both lists. These are also the skills that college faculty report as being lacking in those students
who are most at risk of not completing their college programs successfully. This leads us therefore to a
consideration of the place of those numeracy skills in the elementary and secondary school curriculum.
Mapping of Numeracy Skills to Topics in Grades 1-12 Mathematics Curriculum
Throughout the history of the CMP, special attention has been paid to students’ selection of secondary
school mathematics courses because it seemed reasonable to suppose that this selection might affect
achievement at college level mathematics. We have therefore paid less attention to students’
elementary school mathematics experience. (This has also been due to a lack of availability of student–
level data on this experience.) However, the CMP 2010 final report hypothesised that the topics of most
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importance to student success in college mathematics might not be those taught in Grade 12, or even in
Grade 11, but those found much earlier in the mathematics curriculum.
Accordingly, the qualitative research undertaken as part of CMP 2011 has mapped back to the
elementary and secondary school mathematics curriculum those topics found most frequently in first
semester Business and Technology mathematics courses at college.
Table 22 shows the grades at which eight numeracy skills of importance to all college mathematics
courses are most frequently encountered in the elementary and secondary mathematics curriculum.25
Table 22. General Numeracy Skills in the Elementary and Secondary School Mathematics Curriculum26
Topic Grades 1-8* Grades 9-12 mathematics courses
Order of Operations 6, 7, 8 9(MPM1D), 9(MFM1P)
Fractions 1, 2, 3, 4, 5, 6, 7, 8 9(MPM1D), 9(MFM1P), 10 (MFM2P)
Decimals 1, 2, 4, 5, 6, 7, 8 9(MPM1D), 9(MFM1P)
Percentages 6, 7, 8 9(MPM1D), 9(MFM1P)
Ratio and Proportion 5, 6, 7, 8 9(MPM1D), 9(MFM1P), 10(MPM2D),10(MFM2P) 11(MCR3U), 11(MCF3M),11(MBF3C) 12(MCT4C), 12MAP4C)
Algebra 1, 2, 3, 4, 5, 6, 7, 8 9(MPM1D), 9(MFM1P) 10(MPM2D), 10(MFM2P) 11(MCR3U), 11(MCF3M),11(MBF3C) 12(MCV4U), 12(MHF4U), 12(MCT4C),
Exponents 7, 8 9(MPM1D), 9(MFM1P) 10(MPM2D), 11(MCR3U), 11(MCF3M),11(MBF3C), 12(MHF4U), 12(MDM4U), 12(MCT4C), 12(MAP4C)
Roots and Radicals 7, 8 9(MPM1D), 9(MFM1P) 10(MPM2D), 11(MCR3U), 12(MAP4C)
*Grades at which a topic is specially emphasised are bolded
Some additional numeracy skills are particularly related to business mathematics at college and some to
mathematics for college technology. The mapping of these two groups onto the elementary and
secondary mathematics curriculum is shown in Tables 23 and 24 respectively.
25
These tables are only summaries of the findings. Appendix A shows the full spreadsheets. 26
These analyses are based on: The Ontario Curriculum, Grades 1-8, Mathematics (Toronto: Ministry of Education, 2005); The Ontario Curriculum, Grades 9-10, Mathematics (Toronto: Ministry of Education, 2006); and The Ontario Curriculum, Grades 11-12, Mathematics (Toronto: Ministry of Education, 2007).
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Table 23. Business Related Numeracy Skills in the Elementary and Secondary School Mathematics Curriculum
Topic Grades 1-8 Grades 9-12 mathematics courses
Compound Interest 11(MCR3U), 11(MCF3M), 11(MBF3C) 12MAP4C
Descriptive Statistics 1, 2, 3, 4, 5, 6, 7, 8 9(MPM1D), 9(MFM1P), 11(MBF3C), 12(MDM4U)
Graphing Linear Equations 7, 8 9(MPM1D), 9(MFM1P), 10(MPM2D), 10(MFM2P),
Table 24. Technology Related Numeracy Skills in the Elementary and Secondary School Mathematics Curriculum
Topic Grades 1-8 Grades 9-12 mathematics courses
Logarithms 12(MHF4U), 12(MCT4C)
Trigonometry 7 10(MPM2D) 11(MCR3U), 11(MCF3M), 11(MBF3C) 12(MHF4U), 12(MCT4C)
Measurement 1, 2, 3, 4, 5, 6, 7, 8 10(MFM2P), 12(MAP4C)
Geometry 1, 2, 3, 4, 5, 6, 7, 8 9(MPM1D), 9(MFM1P) 10(MFM2P), 11(MBF3C), 12(MCT4C), 12(MAP4C)
Since 1997, the Ontario mathematics curriculum has been standards based, in which concepts are
expressed in the form of curriculum expectations defined as “the knowledge and skills that students are
expected to acquire, demonstrate, and apply.27” A general principle on which the curriculum has been
developed has been that any given expectation appears explicitly at one grade level only. Thus while a
given numeracy topic may appear in this analysis in several grades, the fundamental concept may be
developed at a few grade levels only with subsequent appearance requiring only its use or application.
The topic of fractions provides a good example of this and the detailed analysis showing the explicit
treatment and subsequent applications or use of this key numeracy topic is set out in Table 25.
Table 25. Mathematics Curriculum Expectations in relation to Fractions, Grades 1– 12
Grade Curriculum Expectations Related to Fractions
Grade 1 – divide whole objects into parts and identify and describe, through investigation, equal– sized parts of the whole, using fractional names (e.g., halves; fourths or quarters).
Grade 2 – determine, through investigation using concrete materials, the relationship between the number of fractional parts of a whole and the size of the fractional parts (e.g., a paper plate divided into fourths has larger parts than a paper plate divided
27
The Ontario Curriculum, Grades 11 and 12, Mathematics (Toronto: Ministry of Education, 2007), p. 11.
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into eighths); – regroup fractional parts into wholes, using concrete materials (e.g., combine nine fourths to form two wholes and one fourth); – compare fractions using concrete materials, without using standard fractional notation (e.g., use fraction pieces to show that three fourths are bigger than one half, but smaller than one whole).
Grade 3 – divide whole objects and sets of objects into equal parts and identify the parts using fractional names (e.g., one half; three thirds; two fourths or two quarters), without using numbers in standard fractional notation.
Grade 4 – represent fractions using concrete materials, words and standard fractional notation, and explain the meaning of the denominator as the number of the fractional parts of a whole or a set, and the numerator as the number of fractional parts being considered; – compare and order fractions (i.e. halves, thirds, fourths, fifths, tenths) by considering the size and number of fractional parts; – demonstrate and explain the relationship between equivalent fractions, using concrete materials (e.g., fraction circles, fraction strips, pattern blocks) and drawings; – count forward by halves, thirds, fourths, and tenths to beyond one whole, using concrete materials and number lines; – determine and explain, through investigation, the relationship between fractions and decimals to tenths, using a variety of tools and strategies
Grade 5 – represent, compare, and order fractional amounts with like denominators, including proper and improper fractions and mixed numbers, using a variety of tools (e.g., fraction circles, Cuisenaire rods, number lines) and using standard fractional notation; – demonstrate and explain the concept of equivalent fractions, using concrete materials; – determine and explain, through investigation using concrete materials, drawings, and calculators, the relationship between fractions; – represent, using a common fraction, the probability that an event will occur in simple games and probability experiments.
Grade 6 – represent compare, and order fractional amounts with unlike denominators, including proper and improper fractions and mixed numbers, using a variety of tools and using standard fractional notation; – represent ratios found in real-life contexts, using concrete materials, drawings and standard fractional notation; – determine and explain, through investigation using concrete materials, drawings, and calculators, the relationships among fractions (i.e., with denominators of 2, 4, 5, 10, 20, 25, 50, and 100), decimal numbers, and percents.
Grade 7 – represent, compare, and order decimals to hundredths and fractions, using a variety of tools; – select and justify the most appropriate representation of a quantity (i.e., fraction, decimal, percent) for a given context; – divide whole numbers by simple fractions and by decimal numbers to hundredths, using concrete materials; – use a variety of mental strategies to solve problems involving the addition and subtraction of fractions and decimals;
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– add and subtract fractions with simple like and unlike denominators, using a variety of tools (e.g., fraction circles, Cuisenaire rods, drawings, calculators) and algorithms; – demonstrate, using concrete materials, the relationship between the repeated addition of fractions and the multiplication of that fraction by a whole number; – determine, through investigation, the relationships among fractions, decimals, percents, and ratios; – research and report on real-world applications of probabilities expressed in fraction, decimal, and percent form (e.g., lotteries, batting averages, weather forecasts, elections);
Grade 8 – represent, compare, and order rational numbers (i.e., positive and negative fractions and decimals to thousandths); – translate between equivalent forms of a number (i.e., decimals, fractions, percents); – use estimation when solving problems involving operations with whole numbers, decimals, percents, integers, and fractions, to help judge the reasonableness of a solution; – represent the multiplication and division of fractions, using a variety of tools and strategies; – solve problems involving addition, subtraction, multiplication, and division with simple fractions.
Grade 9 MPM1D (Grade 9 Academic) – solve first-degree equations, including equations with fractional coefficients, using a variety of tools and strategies; MFM1P (Grade 9 Applied) – solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms (e.g., calculating simple interest and sales tax; analysing data).
Grade 10 MFM2P (Grade 10 Applied) – solve first-degree equations involving one variable, including equations with fractional coefficients (e.g. using the balance analogy, computer algebra systems, paper and pencil).
Grade 11 (none) Grade 12 (none)
Careful study of this analysis shows that the concept of fractions is introduced and developed gradually
in Grades 1 through 3, with a major emphasis on the nature of fractions in Grade 4. Following
consolidation and elaboration of this in Grades 5 and 6, operations on and with fractions (such as
addition, subtraction, multiplication, and division) are taught in Grades 7 and 8. The secondary
mathematics curriculum only refers to fractions three times in Grades 9 and 10, all in the context of
solving problems and equations involving their use, and not at all in Grades 11 and 12. Clearly, there
are two grades (7 and 8) during which the key operations using fractions are expected to be learned.
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Table 26. Mathematics Curriculum Expectations in Relation to Percentages, Grades 1– 12
Grade Curriculum Expectations Related to Percentages
Grade 1 (none) Grade 2 (none) Grade 3 (none) Grade 4 (none) Grade 5 (none) Grade 6 – determine and explain, through investigation using concrete materials, drawings,
and calculators, the relationships among fractions (i.e., with denominators of 2, 4, 5, 10, 20, 25, 50, and 100), decimal numbers, and percents.
Grade 7 – select and justify the most appropriate representation of a quantity (i.e., fraction, decimal, percent) for a given context (e.g., “I would use a decimal for recording the length or mass of an object, and a fraction for part of an hour.”); – use estimation when solving problems involving operations with whole numbers, decimals, and percents, to help judge the reasonableness of a solution; – determine, through investigation, the relationships among fractions, decimals, percents, and ratios; – solve problems that involve determining whole number percents, using a variety of tools (e.g., base ten materials, paper and pencil, calculators); – research and report on real-world applications of probabilities expressed in fraction, decimal, and percent form (e.g., lotteries, batting averages, weather forecasts, elections).
Grade 8 – translate between equivalent forms of a number (i.e., decimals, fractions, percents); – solve problems involving percents expressed to one decimal place (e.g., 12.5%) and whole-number percents greater than 100 (e.g., 115%); – use estimation when solving problems involving operations with whole numbers, decimals, percents, integers, and fractions, to help judge the reasonableness of a solution; – solve problems involving percent that arise from real-life contexts (e.g., discount, sales tax, simple interest).
Grade 9 MPM1D (Grade 9 Academic) – solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate, and proportion. MFM1P (Grade 9 Applied) – solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms (e.g., calculating simple interest and sales tax; analysing data).
Grade 10 (none) Grade 11 (none) Grade 12 (none)
Analysis of one other key numeracy topic will help to illustrate the importance of these “middle years”
still further. Many faculty at the college level, when asked about topics they find that at-risk students
struggle with, identify percentages (along with fractions and ratio/proportion). Table 26 shows the
development of knowledge and skill in relation to percentages through the Ontario curriculum. Once
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again, we see the concept being developed and applied in Grades 6, 7, and 8 with some application in
Grade 9. Thereafter, the topic is not referred to again.
It would appear therefore that most of the key numeracy topics that have been shown to be critically
important in both college business and technology programs depend on their having been effectively
taught and learned in the senior years of elementary school rather than at some later point in secondary
school. Unfortunately, at this stage also, there are other factors that militate against ensuring that all
students master these key skills. First, the students themselves are at a difficult transition point at this
age, physically, mentally and emotionally and there are many reasons why individual students may not
be inclined to focus their minds on what may not seem to them at the time to be an important or
relevant topic. In addition, there is no provincial assessment covering mathematics in Grades 7 and 8
that might reassure the students, their parents, and others that these key numeracy skills have been
mastered. The Grade 9 EQAO mathematics assessment covers only the Grade 9 curriculum28 which, as
we have shown, does not emphasise some of the most important numeracy skills. Even then, it is often
difficult for elementary schools to know how their own graduates performed on the Grade 9
mathematics assessment and to receive feedback, in the way they can from the Grade 6 provincial
assessments. We shall reflect further on these issues in chapter 5 of this report.
28
Framework: Grade 9 Assessment of Mathematics (Toronto: EQAO, 2009).
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Chapter 4: Provincial Forum
Introduction
The College Mathematics Project (CMP) has been operating for six years – three with full province-wide
participation – on the basis of the Deliberative Inquiry model, in which research into the present
situation is blended with deliberations about ways to move forward. While the research has been
coordinated centrally, the deliberations have largely taken place regionally, through forums organised in
cooperation with School/College/Work Initiative (SCWI) regional planning teams.
Regional forums have drawn together teachers and administrators from school boards and colleges to
discuss local strategies for enhancing student success, and deliberations have therefore focused on
classroom issues (curriculum and instruction), guidance and student support issues, and institutional
issues (such as program admission requirements, professional development linkages, and dual credits).
Two of the strengths of these regional forums have been the large numbers of people engaged in them
(over one thousand over the past three years) and the consequent focus on local solutions to locally
identified needs. Sometimes, however, the problems identified regionally are symptomatic of broader
province-wide issues and while the CMP annual reports have attempted to identify these, there has
been little focused deliberation over them by people who are accustomed to taking a provincial, as
distinct from a local, perspective.
It was with this in mind that the College Mathematics Project hosted its first provincial forum in October
2011 where the focus of the deliberations was on themes of a provincial nature. Each theme was
addressed by a keynote speaker either from outside Ontario or outside education but with particular
relevant expertise. Each address was followed by a panel including two additional participants – one
from the college sector and one from the K-12 sector - each offering their distinct perspective on the
theme. Participants were invited to join in the discussion through an open question and answer session.
The CMP team presented the initial research of the CMP2011 study and engaged with participants in a
discussion about their work. Deliberative sessions at the end of the day provided participants with an
opportunity to explore questions that were focused on the same themes. The results of the
deliberations were captured and were used to inform the recommendations presented in Chapter 5.
Theme 1: Mathematics Education for the 21st Century Economy
The first theme was premised on the fact that schools and colleges prepare students for life and work in
a rapidly changing global economy where numeracy and other literacies are imperative to social and
economic growth. Participants were asked to consider the demands of this economic context in relation
to mathematical competence and general numeracy, and the implications for schools that must serve
the full range of students and colleges whose focus is on occupational preparation.
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Carol Wilding, CEO of the Toronto Board of Trade provided a business community perspective on
building future leaders, those who are not just educated to perform a job, but who are educated to be
contributors to society and the economy. The Board develops policies and strategies to ensure the
Toronto region remains competitive, not just locally, but globally. The availability of talent is key to the
region’s competitive edge therefore students need to be prepared for the changing and sometimes
uncertain landscape. A key question posed by the Board during the Vote Ontario 2011 campaign was “Is
Ontario making the transition to the knowledge economy fast enough to help the Toronto region?”
Infrastructure and education were identified as critical factors to making the transition. Businesses
looking to locate in the Toronto region require a number of resources to be in place, including ease of
transportation, availability of capital and quality educational institutions. Ms. Wilding identified a
number of challenges the region will face and emphasized the importance that we must prepare future
leaders to deal with these challenges. Interviews conducted with business leaders about what they look
for in employees identified a number of attributes, including discipline, determination, adaptability,
agility, creativity and innovation and most importantly passion about the field. In addition to these
attributes, Ms. Wilding identified a number of skills that are essential in today’s economy, including
critical thinking, communication (oral and written), and mathematics as being vital to an individual’s
success. Networking is also essential for future leaders and she noted that students should learn this
skill an early age.
The first panel participants were Gillian Leek (former Mathematics Coordinator, Mohawk College), Kathy
Kubota-Zarivnij (Principal, Mathematics, Science and Technology (MST) focused school, Toronto Catholic
District School Board), and Carol Wilding (CEO, Toronto Board of Trade). Discussion revolved around
industry expectations and how the education system prepares students to meet these expectations.
Panelists noted that educators must ensure students are able to communicate, perform basic arithmetic
skills and are able to conceptualize a question as there are different ways of understanding and solving a
problem. Students need to be able to listen, not simply to respond, but to question and challenge as this
fosters innovation and creativity. They will need to be able to work in teams, often with members of
differing perspectives. Ms. Kubota-Zarivinij noted that in the province there is a strategy underway to
change the demeanor of mathematics teaching with an emphasis on the developmental sequences that
are needed in mathematics. This transition can be difficult for teachers who are using traditional
methods of “telling” students about mathematics and for parents who may be more comfortable with
the traditional approach. This transition will not occur overnight and so it will be some time before the
results appear at the college level.
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CMP Research Highlights
Laurel Schollen and Graham Orpwood presented the preliminary research findings based on the fall
2010 data collection. These are presented in Chapter 2 of this report. The findings, along with the
keynote presentations and panel discussions provided the context for the group deliberation sessions.
Articulation, Alignment and Integration
Edith Brochu, Coordonnatrice de la Commission de l'enseignement collégial par intérim
at the Conseil supérieur de l’éducation of Quebec spoke about the Council’s work in understanding and
improving the transition from secondary school to college. The Council published a brief in 2010 which
identified three aspects of transition: articulation, alignment and integration. It was this conceptual
framework that was referenced in the CMP2010 report and used as a way to help think about Ontario’s
solutions to the same range of problems.
Articulation refers to the education system, its structures, and rules governing certification of studies at
one level and admission requirements at the next level. Alignment focuses on knowledge and
pedagogical and evaluation practices and is primarily the domain of teachers. Integration is the process
of “incorporating” the individual (i.e. the student) into a new environment (i.e. the college community).
The Council has recommended three courses of action to address articulation issues. First, in reference
to the implementation of the Quebec Education Reform in Secondary Schools, to monitor, on an
ongoing basis, the effects of the structural changes made as a result of the reforms on student
transition to and success in college, and undertake any modifications required at each respective level of
instruction. Second, to promote collaboration between secondary schools and colleges, and third to
support those who wish to continue their education.
The reform resulted in the need to align knowledge and practice of the Quebec Education program. A
greater emphasis was placed on competency. The result was the integration of subjects, broad areas of
learning and cross – curricular competencies. Some subjects were eliminated, others were added or
reconfigured. The reform did not prescribe pedagogical practices, however it did encourage teachers to
innovate their practice by embracing a student centric approach and collegiality with other teachers and
professionals. The Council has noted that implementation of curriculum is complex – there is the official
curriculum that is established by the Ministry, the “taught” curriculum, which is the curriculum that
teachers understand, interpret and share with their students, and there is the “acquired” curriculum –
what students actually learn. Colleges must address the changes brought about by the reform through
discussions with secondary school teachers and modification of college curriculum.
True integration occurs when the “whole” student is considered. The integration process must take into
account institutional, intellectual, social and vocational relationships between the student and the
institution, the curriculum, their peers and their educational goals. At college or university there is a
tendency to focus on intellectual integration, however all are essential if the student is to successfully
make the transition to college student culture. Figure 19 shows the framework of secondary to college
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transition and identifies the challenges associated the processes of integration, alignment and
articulation.
Figure 19. Framework of Secondary to College Transition29
Theme 2: Student success in secondary-post secondary transition
The second theme focused attention on the needs of students as they transition from secondary to
postsecondary education and the conditions required for them to be successful. Dr. Katherine Hughes,
Assistant Director at the Community College Research Center of Columbia University provided a
perspective from the United States. Her presentation, “Evidence and Innovations to Improve the Tough
Transition to College” identified problems with the “pipeline” to college, including challenges to high
school completion and attainment, challenges to transition from high school to college and challenges to
persistence and success in college. She noted that academic and non-academic factors contribute to the
situation. Public high school completion rates in the United States average 70.1%, however 14 states
have rates between 47.6 to 67.1%, another 12 have rates between 67.2 and 72.8%. The CCRC has also
determined that in 14 states there is a 26.3-38.6% chance of a 9th grader attending college; the overall
average in the U.S. is 41.8%. And the news is not good for students who make it to college. In 2007 the
29
This figure is reproduced by permission of the Conseil supérieur de l’Éducation du Québec
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average rate for graduation from a three year associate degree was 27.5%. Dr. Hughes noted that one
of the main problems is the lack of alignment of high school and college content and standards, resulting
in large numbers of new college students being placed into developmental education. Their research has
found that 60% of (U.S.) community college students and 25% of 4 year students are referred to
developmental education and these students are less likely to earn a credential. Cohorts of students
were tracked for three years through a series of developmental mathematics courses to a “gatekeeper”
algebra course. The research found that very few (6%) of students complete the gatekeeper course in
the stepwise manner that is recommended if they are referred to low levels of remediation. The result is
that colleges are helping students to exit before graduation.
Dr. Hughes suggested three points at which this problem could be addressed. First, prior to enrolling in
college, students should have the goal of avoiding developmental education. Second, at enrolment,
there needs to be improved assessment and placement. Finally there is a need to improve the format
and structure, curriculum and content and pedagogy of developmental education. Alignment of high
school exit standards with college entrance standards and interventions such as dual enrollment,
summer bridge programs and diagnostic early assessments with targeted interventions would assist
students in avoiding developmental education. Reforms to assessment and placement including use of
multiple measures, diagnostic tools and non-cognitive assessments and educating students about the
assessment process are needed. At the present time there is a lack of consistency in the assessments
used and cut scores for college readiness and the instruments provide little predictive value and
inadequate diagnostic information. Structural reforms focused on the organization of instructional time
and content, curricular reforms focused on refining content and pedagogical reforms focused on
changing teaching practice are essential to improving student success. Of these, Dr. Hughes noted that
pedagogical reform is the most difficult to enact, mandate and study.
Panelists Chris McGrath (Seneca College), Mary-Jean Gallagher (Ministry of Education), and Sina
Okhovat, a Seneca College student joined Dr. Hughes on a discussion about transition between
secondary and postsecondary. Panelists were asked to comment on the nonacademic factors affecting
success and how colleges are helping students to exit. Chris McGrath talked about transformational
education and serving the “whole” student. He noted that learning does not just happen in the
classroom; the student experience is the academic experience and in order to measure student success,
creative and complex ways of measuring the totality of the student need to be found. Mary Jean
Gallagher noted that these same ideas should inform the elementary and secondary school experience.
She noted that much of our educational programming is compartmentalized – they may know what we
taught them in a particular course, but when we put them into the situation where we expect them to
transfer and apply that knowledge some students struggle. Since direct application of knowledge is
expected of all college students, they would benefit from more opportunities to integrate and apply
their learning earlier in their school careers. Sina Okhavat noted that his pathway was not a direct one,
as a student new to Ontario he noted that his lack of English compromised his ability to learn, although
he was passed in early grades and only realized in senior secondary school years that he would have to
work hard to achieve good grades. He noted that it would have been more helpful for him to have
stayed behind his peer group in order to improve his English skills. Sina enrolled in college, but
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struggled and withdrew, enrolling in adult school to gain additional credits before re-enrolling in another
program. He noted there are supports in place to assist students, including the peer mentors, tutoring
and study groups as well as the availability of a math course that is structured over two semesters
instead of one semester, thus allowing students more time to learn the concepts.
The panelists discussed the non-academic factors affecting student persistence, including integration.
Dr. Hughes noted that not much research has been done on assessment of nonacademic factors for
success. In the U.S. there are significant challenges with engagement for 2 year colleges as compared
with 4 year schools, primarily due to the nature of the student body. Involvement in student life outside
of the classroom results in better outcomes in terms of engagement. Small, residential, 4 year, liberal
arts colleges have the highest scores on the National Survey of Student Engagement (NSSE). Chris
McGrath noted the traditional view of coming to campus, living on campus and meeting people – the
model for a full time student no longer holds true and that we must find ways to engage all students
through rethinking, redefining, and realigning engagement and how we think about this academically.
In Ontario, dual credit and Specialist High Skills Major programs have been implemented with great
success. Both programs are tools for engagement, in this case secondary school students who may be at
risk of not graduating. Students enrolled in a dual credit course have a 80% success rate in college
courses, which may be higher than the success rate for college students only. Mary-Jean Gallagher
noted the Ministry of Education recognizes the value of research to educational reform; a culture shift is
taking place in Ontario’s elementary and secondary schools and programs such as dual credit and
Specialist High Skills Majors are part of the re-culturing process. Finally, the panelists agreed that
success comes from connectedness – students need to see how their preparation is connected to what
they are/will be learning, to a caring adult, and to the program -when the student sees something that
is relevant to them, thus setting the tone for the deliberative sessions.
Deliberative Sessions
Forum participants were pre-arranged into 6 groups, including 1 francophone group, ensuring broad
representation. Each group was provided with questions focused on foundation programs and
preparatory mathematics courses, college mathematic achievement, and learning skills to deliberate.
1. Foundation Programs and Preparatory Mathematics Courses
The CMP research shows a 30% increase in the numbers of students taking college foundation programs
over the past three years and a 20% increase in the numbers taking preparatory mathematics courses
(where these are offered). These compare with an 11.3% increase in overall college enrolment over the
same period.
1. How do you interpret these changes?
CMP qualitative research shows that many of the mathematical skills and concepts taught in foundation
programs and preparatory courses were first taught in Grades 6 through 9.
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2. What are the implications of this finding for curriculum and assessment policy and practice in
both the K-12 system and the college system?
2. Mathematics Achievement in College
CMP research shows that mathematics achievement levels of first year college students have not
increased significantly over the past three years. In addition, EQAO results have shown relatively
modest increases in mathematics achievement at Grades 3, 6 and 9.
1. How do teaching and assessment methods used at both school and college levels affect student
success? What other factors are at play here?
This year we found that the proportion of students taking college mathematics after only three
mathematics courses in secondary school (the minimum required for secondary school graduation) has
increased over the past three years to over 18%.
2. Does this data, along with the growing importance of mathematics for many careers, suggest
that the number of mathematics courses required for graduation should be increased to four, or
what other policy changes might be considered?
3. Learning Skills
Participants at regional forums have often pointed out the importance of generic “learning skills” for
student success in postsecondary education (and also in employment). The CMP 2010 Final Report
references research from a variety of sources supporting this claim and the Ministry of Education and
the Ministry of Training, Colleges and Universities have policies stressing the importance of these skills.
1. What evidence is there that progress is being made at elementary, secondary, or postsecondary
levels to enhance these skills?
2. What additional changes to policy and/or practice at all levels are required?
3. How can students and their parents become more aware that while marks on academic courses
enable entry to postsecondary programs, learning skills can enable success at that level?
The resultant discussion was recorded by a member of each group and was reviewed by the CMP team
to inform the recommendations presented in Chapter 5 of this report. The summaries of the
deliberations are posted on the CMP website30.
30
http://collegemathproject.senecac.on.ca/cmp/en/forums.php.
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Chapter 5: Conclusions
The research and deliberations undertaken by the College Mathematics Project each year have resulted
in recommendations for immediate action to support improvements in student success. They have also
raised questions for further investigation in the ongoing research program. For example, deliberations
in previous years have questioned college participants about the content of their preparatory and
foundational mathematics courses and this has led the CMP to develop a new qualitative analysis of
these courses, as described in chapter 3.
But they have also led us to conceptualise the issues in new ways. For example, in last year’s report we
drew on the work of the Conseil Supérieur de l’Éducation and used their ‘framework of secondary-to-
college transition’ to discuss three ways in which student transitions should be supported31. In addition,
the problems uncovered by the CMP have often come to be seen as symptomatic of broader problems
faced by students in transitioning from secondary to postsecondary education. Our continuing
discussion of ‘learning skills’ featured in several of CMP’s recent reports and raised again in this year’s
forum is an instance of this. This year, the qualitative analysis of college foundational and preparatory
mathematics courses (as outlined in chapter 3) and the subsequent forum presentations and
deliberations (as described in chapter 4) have led us to think about the mathematics needs of students
in a new way. This is one of two themes emerging from the CMP research and deliberations, which are
the focus of this concluding chapter.
Mathematics or Numeracy?
The primary focus of the College Mathematics Project has always been on students’ achievement in
mathematics at college. And, insofar as elementary and secondary school mathematics is a preparation
for mathematics at postsecondary levels32, the CMP has been investigating aspects of that curriculum
also. Along the way, we have uncovered a variety of puzzling paradoxes. For example:
College admission requirements (at least for programs in which mathematics forms a significant
foundation) are almost always framed in terms of students’ Grade 11 and 12 mathematics
credits. Yet our research has shown that the relationship between high school mathematics
courses taken and college mathematics achievement is far from simple.
Figure 7 and Table 10 (in chapter 2) and similar displays in previous CMP reports show that
students who have been out of school for quite a long time (and whose secondary school
mathematics is a distant memory) regularly outperform those whose secondary school
mathematics experience is more recent.
31
A graphic representation of this framework was presented at the Provincial forum by Edith Brochu and is reproduced in chapter 4 of this report. 32
K-12 mathematics has a broader range of goals of course (see, for example, The Ontario Curriculum, Grades 11 and 12, Mathematics (Toronto: Ministry of Education, 2007).
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While early CMP reports suggested that the Grade 12 mathematics course taken by most college
bound students (MAP4C) might not be the most appropriate basis for college mathematics,
students who achieve high marks in that course achieve good grades in college mathematics.
Foundation programs and preparatory mathematics courses at the college level – designed to
meet students with a need for such courses – place less emphasis on Grade 11 and 12 course
topics than on topics and skills originally taught much earlier.
While school mathematics achievement has remained steady (as measured by provincial and
international assessments), concerns continue to be expressed by employers, parents and
academics about the mathematical abilities of Canadians33.
Even as mathematical skills are seen as critically important to Ontario’s transition to a
knowledge economy, many people are open, sometimes even boastful, of their lack of
mathematical ability.34
Participants at forums including the most recent provincial forum have helped us to think about these
paradoxes as has the broader literature on mathematics education. In particular we have been drawn
this year to think more about the distinction between mathematics and numeracy35 and to ask ourselves
if the key to student success at college is really their choosing the “right courses” at grades 11 and 12 (or
for colleges to try to influence those choices by adjusting admission requirements). The qualitative
research reported in chapter 3 suggests that, if something is lacking in students’ mathematical
preparation for college, it is their basic numeracy skills, rather than the more advanced mathematics
courses.
Numeracy has been defined in a variety of ways but all of them have certain common features:
It involves mathematical knowledge and skills;
Its purpose is the effective functioning in work and society;
In consequence, it specifically includes the ability to use the mathematical knowledge and skills
in concrete, real-world situations.
Lynn Arthur Steen, who has been a strong advocate for improved numeracy for many years, has written:
Numeracy is not the same as mathematics, nor is it an alternative to mathematics. Mathematics is abstract and Platonic, offering absolute truths about relations among ideal objects. Numeracy is concrete and contextual, offering contingent solutions to problems about real situations.
33
For example, in one ten-day period last year, the following news reports appeared: Amy Chung “Canada’s public schools doing a poor job teaching basic math skills” Postmedia News (www.vancouversun.com) September 21; Margaret Wente “Too many teachers can’t do math, let alone teach it” The Globe and Mail, September 29, 2011; Anne-Marie Tobin “Is your child struggling with math?” Canadian Press (www.therecord.com) September 30, 2011. 34
Both of these observations were made by panelists at the Provincial forum (see chapter 4). 35
In this report, we use the term ‘numeracy’ as being equivalent to ‘mathematical literacy’ (as in Leading Math Success: Mathematical Literacy, Grades 7-12 (Toronto: Ministry of Education, 2004) and in the OECD Programme for International Student Assessment, PISA) and ‘quantitative literacy’ (as commonly used in the United States).
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Whereas mathematics asks students to rise above context, quantitative literacy is anchored in the messy contexts of real life. Truly, today’s students need both mathematics and numeracy36.
This is entirely consistent with the report of the Expert Panel on Student Success in Ontario cited earlier, which points out that:
Mathematical literacy involves more than executing procedures. It implies a knowledge base and the competence and confidence to apply this knowledge in the practical world. A mathematically literate person can estimate; interpret data; solve day-to-day problems; reason in numerical, graphical, and geometric situations; and communicate using mathematics.37
In reflecting on the day-to-day uses of numeracy, Wade Ellis (himself a mathematics professor)
comments:
On any given day, for any one person, quantitative literacy may include reconciling a bank statement, analysing data to support or oppose a local government proposal, estimating how to split a lunch bill, debugging a program by working from assumptions to a logical conclusion, deciding which medical treatment to pursue based on statistical evidence, building a logical court case, or understanding the risks in investing for retirement.38
To these we could add a broad range of tasks drawn from the occupations for which college programs
prepare students, including: preparing correct dosage for a patient’s medication; making sense of a
company’s balance sheet; estimating the cost of rewiring a house; the list is endless. But the vast
majority of these tasks require not the relatively sophisticated mathematics of Grades 11 and 12 as
much as the thoughtful application of fundamental skills taught much earlier39.
This is the essence of numeracy: it involves both knowledge and skill combined in the performance of a
real world task. In particular, we note that numeracy is not the narrowly rule-bound or mindless
application of learned procedures as is the case, for example, with riding a bicycle or responding to a fire
alarm. The frequently encountered dichotomy of knowledge and skill is particularly unhelpful here. The
application of numeracy skills in context – such as is the case in all of these examples – requires both
theoretical and practical knowledge and acquiring and interpreting knowledge of the context are in
themselves critically useful skills. The acquiring of numeracy skills is therefore not something that can
be done – once for all – at a particular grade level. Rather, a person’s numeracy ability is continually
developed as new contexts appear in which the previously learned knowledge and skill is reapplied over
and over again. In this respect, numeracy is very like literacy. It is not something to be taught in Grade
5 or 6 and then assumed to be growing on its own but must be constantly supported and developed as a
student moves on.
36
Lyn Arthur Steen. ‘Mathematics and Numeracy: Two Literacies, One Language’ The Mathematics Educator (Journal of the Singapore Association of Mathematics Educators) 6:1 (2001) 10-16. 37
Leading Math Success: Mathematical Literacy, Grades 7-12 (Toronto: Ministry of Education, 2004) p. 10. 38
Wade Ellis. “Numerical Common Sense for All” p. 63 39
A new study from the UK (Celia Hoyles, Richard Noss, Phillip Kent & Arthur Bakker. Improving Mathematics at Work: The Need for Techno-Mathematical Literacies. Abingdon: Routledge, 2010) identifies the need for what they call “techno-mathematical literacies” – the ability to understand and use the forms of mathematics pervasive though often implicit in the modern workplace.
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Yet most authors agree that, traditionally, school mathematics has focused more on the abstract
mathematical concepts and skills and less on the real world problems that call for the use of
contextualised mathematical knowledge and skill. Partly this is a reflection of the mathematics and
mathematical education that teachers themselves have received. Partly it is also the result of a
mathematics curriculum that is structured on the basis of the traditional canon which underlies the
mathematics curriculum worldwide40. And partly it is that numeracy is something that has been left to
the mathematics teachers, much as literacy was traditionally left to the language teachers. But if
numeracy is essentially a context-based rather than a discipline-based set of abilities, then it should be
the responsibility of all educators wherever the use of numbers or other mathematical concepts are
encountered in the context of a student’s experience. Numeracy can be developed in the context of
music, art and drama, through history and geography, even sometimes in the study of literature as well
as through mathematics and science subjects.
Traditional approaches to mathematics leave many students cold. Even in countries where
mathematical achievement is high such as Japan and Singapore, there is a negative attitude towards
mathematics and students are loathe to use it in their daily lives.41 This is leading to serious
consideration in those countries of actually reducing the amount of mathematics in the school
curriculum. The EQAO mathematics assessments, in which improvements in achievement have been
less than had been hoped for, and the equally disappointing lack of growth in college mathematics
achievement (as reported through CMP research) lead one to wonder if too many Ontario students are
not sufficiently engaged by mathematics when the foundational concepts and skills are being taught,
because they do not see the use of these concepts and skills in context. If this is the case, it could lead
to their failure to grasp key concepts or master critical skills at that time and to their encountering
difficulties and frustration at later stages of their education.
As our forum discussions reminded us, Ontario cannot afford to lose out economically because its
citizens lack the necessary literacy or numeracy abilities. While literacy continues to be important – and
Ontario educators have worked very hard in recent years to increase the literacy levels of all students –
numeracy appears to have been given rather less attention. Concerns have been expressed at CMP
forums that, while in recent years educators have placed a high priority on literacy, there has not been
an equally high priority afforded to numeracy. Consequently, the announcement at the CMP provincial
forum by Mary-Jean Gallagher (Chief Student Achievement Officer for Ontario) that numeracy was going
to be a Ministry priority in the coming years was warmly welcomed by participants.
A recent Harvard study estimated that poor math skills in the United States could cost that country’s
economy $75 trillion over the next 80 years42. Eric Spiegel, a leading American CEO pointed out recently
that “over the past decade, STEM43 job openings grew three times faster than non-STEM jobs. STEM
40
Lynn Arthur Steen, Ibid. 41
Wade Ellis. Ibid. 42
Paul Peterson, Ludger Woessmann, Eric Hanushek, and Carlos Lastra-Anadon. Globally Challenged: Are US Students Ready to Compete? 2011 43
Science, Technology, Engineering and Mathematics.
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workers are expected to earn, on average, 26 per cent more than their non-STEM counterparts.” He
went on to point out that his concern was not just for scientists and engineers but also for sales and
marketing staff who also need a STEM background and are “fluent in the language of medicine, energy,
and high-tech manufacturing.”44 In Canada, while the numbers vary somewhat, the needs are similar, as
studies such as Rick Miner’s have shown45.
The economic benefits of improved numeracy are not for business and the national and provincial
economy alone; they are of direct benefit to the students themselves. Carnevale and Desrochers point
out that “the value of quantitative reasoning has surged at each of the great economic divides: in the
shift from agriculture to an industrial economy and most recently in the shift from an industrial to a
knowledge economy.”46 They claim that:
“mathematical ability is the best predictor of the growing wage advantages of increased
postsecondary educational attainment.
“improvements in mathematical skills account for at least half of the growing wage premium
among college-educated women…and
“although the wage premium has increased across all disciplines, it has increased primarily
among those who participated in curricula with stronger mathematical content, irrespective of
their occupation after graduation.47”
These authors go on to argue for what they call the “democratization of mathematics” which, they
insist, “does not mean dumbing down. It means making mathematics more accessible and responsive to
the needs of all students, citizens and workers.”48
In Ontario, as was demonstrated at the CMP provincial forum, we have recognised the need for this
broadening and democratising of mathematics. The future of both the community and its members
requires that Ontarians endorse the beliefs that:
everyone can learn mathematics;
everyday life and work requires that everyone should learn mathematics;
every secondary school graduate should be both literate and numerate.
With these beliefs in mind, the CMP proposes a series of recommendations. First, we suggest that a
provincial numeracy strategy is required. Such a strategy would aim to increase levels of numeracy
among Ontario secondary school students through a multi-faceted program of action. The actual
composition of such a strategy would require deliberation among a broad range of stakeholders but the
following ideas, most of which have been suggested before or are being implemented in other
jurisdictions, might be on the table for specific consideration.
44
Blog by Maureen Downey. http://blogs.ajc.com/get-schooled-blog/2011/12/05/ 45
Rick Miner. People without jobs – Jobs without People: Ontario’s Labour Market Future. (Toronto: Miner Management Consultants, 2010) 46
Anthony Carnevale & Donna Desrochers. “The Democratization of Mathematics.” In Quantitative Literacy: Why Numeracy Matters for Schools and Colleges (Princeton, NJ: National Council on Education and the Disciplines, 2003), p. 22. 47
Anthony Carnevale & Donna Desrochers. Ibid. 48
Anthony Carnevale & Donna Desrochers, Op. cit., p 29.
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The implementation of a Grade 10 numeracy test having the same status as the existing Ontario
Secondary School Literacy Test (OSSLT) in lieu of the present Grade 9 assessment of
mathematics.
Involving employers, college faculty, and parents in identifying the numeracy skills in greatest
need of support.
Increased efforts, through both pre-service and in-service teacher education, to support
teachers’ mathematical skills and understanding and to eradicate negative stereotypes
associated with mathematics.
Considering the use of a numeracy test for teacher candidates, such as has recently been
introduced (along with tests in literacy and information and communication technology-ICT) in
England & Wales, starting in 201249.
Research into ways in which Junior/Intermediate teachers with a mathematics teaching
qualification are currently being deployed by school boards, and the impact of such deployment
on achievement.
Development of sample instructional materials to support the teaching of numeracy across the
curriculum (for example in social studies as well as in science and mathematics).
A public awareness campaign to highlight the importance of numeracy both to individuals and
society as a whole.50
Colleges have a role to play here also. Until innumeracy among incoming students is eradicated, there
will be a continuing need for developmental courses of the variety that have grown up over the past few
years. Colleges can support schools’ efforts at developing higher levels of numeracy if they could work
together in a number of related areas:
To develop a common numeracy assessment tool to be used by all colleges as part of their
admission and placement process for all incoming college students – perhaps with a technology
version and a business version – based on a numeracy framework approved by both the college
system and the Ministry of Education. This assessment would be consistent with the suggested
Grade 10 numeracy test, though at a higher level since it would be designed specifically for
students applying to college programs in technology and business.
To reframe program admission and placement requirements to take into account students
achievement on the common numeracy assessment.
To develop a system-wide college numeracy course (again, perhaps in a technology version and
a business version) for students whose scores on the numeracy assessment show that they need
such a developmental course.
To share both the assessment framework and course information with elementary and
secondary schools so that teachers at earlier levels understand better the expectations of the
college system of students entering into vocational diploma and certificate programs.
49
Training and Development Agency for Schools. http://www.tda.gov.uk/trainee-teacher/qts-skills-tests.aspx. 50
Such as that recently launched in the UK by a new non-profit organization, National Numeracy, (http://www.nationalnumeracy.org.uk).
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To use the CMP data collection system to collect students’ numeracy assessment scores and to
provide feedback to school boards and schools on the (aggregate) achievement of their
graduates.
The implementation of both of these sets of initiatives will require a substantial effort on the part of
both school and college systems and the CMP recommends that, in the interests of the economic
benefits to be gained and the economic costs of failure, the Government of Ontario assign special
resources to the Ministry of Education and the Ministry of Training, Colleges and Universities for its
support.
College Knowledge
Katherine Hughes’ presentation at the provincial forum highlighted the distinction between what she
described as “academic” and “non-academic” factors that contribute to students’ success or lack of
success at college51. She described the lack of alignment between secondary and postsecondary courses
which we in the CMP have also been addressing – most recently through the discussion of numeracy
above. She also mentioned that the non-academic factors were equally important but significantly
under-researched. This led to a vigorous discussion among panellists about how to think further about
the needs for students to be better integrated into postsecondary education.
In the course of the discussion the term “college knowledge” was used to characterise the sorts of
knowledge that students need to be successful in college, apart from their content knowledge. As David
Conley (whose book of that title has popularised the phrase in the United States) has noted, college
knowledge or college readiness “continues to be defined primarily in terms of high school courses taken
and grades received, combined with scores on national tests.”52 Apart from the reference to national
test scores, it would appear that the same claim can generally be made of college readiness in Ontario.
Conley’s analysis of what makes a student ready for college provides a helpful starting point for
organising our consideration of how well we in Ontario are supporting students’ successful transitions to
postsecondary education. His research has led him to distinguish four categories of college knowledge:
Key Cognitive Strategies: Examples include analysis, interpretation, precision and accuracy,
problem solving and reasoning. Abilities such as these have been “consistently and emphatically
identified by those who teach entry-level college courses as being of equal or greater
importance than any specific content knowledge taught in high school.”53
51
In the American literature, the term ‘college’ is used to cover all types of postsecondary institution – both 2-year and 4-year colleges – whereas in Ontario we distinguish more sharply between colleges and universities. However we believe that the terms ‘college knowledge’ and ‘college readiness’ and the related research can be applied with equal appropriateness to both colleges and universities. 52
David Conley. Redefining College Readiness. Volume 5. (Eugene, OR: Educational Policy Improvement Center, 2011): 1. 53
David Conley, Ibid. Also: David Conley. “What Makes a Student College Ready?” Educational Leadership 66, number 2 (October 2008): 1-4. This section of this report draws from both of these documents.
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Key Content Knowledge: This refers to the “big ideas” of the subjects most related to a student’s
college program – of the sort that the CMP has focused its research on – but also other key
academic skills such as writing.
Key Self-Management Skills: These include skills and attitudes required for success not only in
college but also in life and work more generally – including study skills, time management,
awareness of one’s performance, persistence, and the ability to work in groups. There is a
strong link here to the learning skills identified by both the Ministry of Education and the
Ministry of Training Colleges and Universities and discussed at length in the final report of the
CMP last year54.
Key Knowledge About Postsecondary Education. This is important contextual knowledge about
colleges, about college programs, and about the admission requirements and other expectations
of colleges for their students. It also includes understanding about the costs associated with
going to college, and (as Conley puts it) “perhaps most important, understanding how the
culture of college is different from that of high school.”
The discussions at the provincial forum suggest that, while general awareness of this college knowledge
– all but one component of which are considered to be ‘non-academic’ – is growing, we still lack
systematic ways of ensuring that all students are provided with opportunities to acquire it. The
consequence of this situation is that some students (likely those most academically and socially
advantaged already) are likely to acquire sufficient college knowledge intuitively to be successful, while
less advantaged students – including aboriginal students, certain ethnic groups, and those from families
for whom postsecondary education is a new experience – are less likely to acquire it without support.
The ‘college knowledge’ issue must therefore be considered as one of equity and one that is going to
grow as the proportions of students attending postsecondary institutions grow.
The four categories of college knowledge described above can serve as organisers for consideration of
the mechanisms that need to be in place to ensure that all college students are well prepared not just to
be admitted to college but to enjoy success while there.
The first two categories have traditionally been built into the structure of elementary and secondary
school curricula both in mathematics and all other subjects. However, since Ontario has no external
examination system at the end of secondary school, nor an independent system for reviewing the
assessments undertaken at individual secondary schools, there is no way of knowing the extent to which
individual students have acquired either the ‘big ideas’ of individual subjects or key cognitive strategies
such as analysis and interpretation, reasoning, or problem solving. Postsecondary institutions are
therefore obliged to frame their admission requirements in terms of the limited information available to
them on the Ontario Student Transcript. This is not a problem for Ontario alone, of course, and there is
54
Ministry of Education. Growing Success (Toronto: Ministry of Education, 2010). Ministry of Training Colleges and Universities. http://www.tcu.gov.on.ca/eng/general/college/progstan/essential.html College Mathematics Project 2010: Final Report (Toronto: Seneca College, 2011) pp. 54-56.
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a growing research literature about innovative assessments of students at this stage of their education
in other jurisdictions.55
The third category of college knowledge – or ‘learning skills’ as they are known in Ontario – has been the
focus of CMP interest for several years and last year we identified the need for further research both on
the acquisition and assessment of these skills in general and the activities being undertaken by Ontario
schools and colleges to support students’ acquisition in particular. This year’s forum underlined this
need and we believe that until we understand more about how these skills can be taught, learned, and
assessed, we shall not make further progress.
The fourth category of college knowledge – knowledge about postsecondary education – is being
addressed whenever teachers from colleges and schools get together in forums, exchanges, visits, team
teaching situations and in other ways and the School/College/Work Initiative (SCWI) has done a lot to
advance this goal. However, it has been acknowledged by senior officials of SCWI that the very success
of some of its activities, notably the dual credit system, has commanded time and financial resources
such that the potential for its Regional Planning Teams to explore other aspects of supporting
transitions from school to college has been more limited56.
The problem of ensuring that students have sufficient college knowledge to be ready for postsecondary
education is a broad issue for which the College Mathematics Project does not have complete
information, sufficient resources or, indeed, the mandate to prescribe a comprehensive set of specific
responses. However, the CMP can identify what appears to be a general problem and invite the
education community – both at governmental and local levels – to discuss possible solutions. We
therefore recommend that:
The Ministry of Education and the Ministry of Training, Colleges and Universities set up an
expert panel to study the assessment of students at the interface of secondary and
postsecondary education and to recommend possible policies and practices that could ensure
that students are adequately prepared for postsecondary education.
The School/College/Work Initiative be asked (and resourced) to expand the range of
mechanisms for facilitating students’ successful transitions from school to college as well as
maintaining its ongoing support for dual credits and forums.
Colleges, Universities and School Boards work together at the local level to develop joint
programs aimed at providing all students who intend to go on to postsecondary education
sufficient college knowledge to maximise their chances of success.
55
An example that recently came to our attention is from the Australian Capital Territory (ACT) where reasoning tests developed at the Australian Council for Educational Research are used as a key external assessment for students intending to apply for postsecondary education. (D. McCurry. “Using external testing as an opportunity to learn: Making a teachable but non-toxic cross-curricular test.” Paper presented at the annual conference of the Association for Educational Assessment – Europe, Belfast NI, 10-12 November, 2011. 56
Private communication to CMP team.
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Further Research
This latest round of research and deliberation from the College Mathematics Project has, like all its
predecessors, raised as many new questions as it has answered. At the invitation of the Ministries of
Education and of Training, Colleges and Universities, the CMP team has proposed a continuing program
of research and deliberation for 2012 and beyond. We would welcome commentary on what we have
achieved to date and suggestions for further research as we move forward.
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Appendix A
Topics in College Mathematics Courses: Diploma Level Business Mathematics
College Order of
Operations Fractions Decimals Percentages Ratio &
Proportion Algebra Exponents Roots & Radicals
% of Total NS
A y y y
y y
59%
B y y y y y y 25%
C y
y y
y
28%
D y y y y y y Y 53%
E y y
y y y
29%
F y y y y y y y 100%
G
y
17%
H y y y y y y 38%
I y y y y y y y
27%
J y 18%
K y
y y
y
21%
L 0%
M
y
y
24%
N y y y y y y 29%
O y
y y y
30%
P y 32%
Q y y y y y y
45%
R y y y y y 33%
S y y y y y y y
37%
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Topics in College Mathematics Courses: Diploma Level Technology Mathematics
College Order of
Operations Fractions Decimals Percentages Ratio &
Proportions Algebra Exponents
Accuracy &
Precision Scientific Notation
Roots & Radicals
% of Total NS
A
y
y y
y
12%
B y y y y y y y 39%
C y y y y y y y
y
43%
D y y y y y y 50%
E
0%
F y y y y 14%
G y y y y y y
y y
34%
H y y y y y y 20%
I y y y y y y y
45%
J y y y y y y y y y 33%
K
y y
33%
L y y y y y y y y 45%
M y y
y y y y y
27%
N y y y y y y y y y y 20%
O y
y y y y y 38%
P y 26%
Q
y
y y
19%
R y y y 21%
S y y
y y y y
30%
T y y y y y y y y y y 38%
U y y y y y y y y y
45%
V y y 6%
W y y
y
y y y y y 30%
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Topics in College Mathematics Courses: Pre-Business Foundational Mathematics
College Course Hours
Order of Operations Fractions Decimals Percentages
Ratio & Proportion Algebra Exponents
Roots & Radicals
% of Total NS
A 56 y y
y y y y 41%
B 56 y y y y y 33%
C 45 y y y y y y
78%
D 60 y y y y y y y 84%
E 45 y y y y y y y
100%
F 42 y y y y y y 55%
G 42 y y y y y
100%
H 60 y y y y y y y 76%
I 42 y y y y y y y
33%
J 48 y y y y 50%
K 56 y y y y y
100%
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Topics in College Mathematics Courses: Pre-Technology Foundational Mathematics
COLLEGE Order of
Operations Fractions Decimals Percentages Ration &
Proportion Algebra Exponents
Accuracy &
Precision Scientific Notation
Roots & Radicals
% of Total NS
A y y
y y y
y y 41%
B y y y y y 100%
C y y y
y y
y
100%
D y y y y y y y y y 61%
E y y y y y y y y y
55%
F y y y y y 48%
G y y y y y y
45%
H y y y y y y 100%
I y y y y
y y
56%
J y y y y y y y 61%
K y y y y y y y y y
71%
L y y y y y 38%
M y y y y
y y
81%
N y y y y y 61%
O y y y y y y y y y
42%
P y y y y y y y y y 91%
Q y y y y y y y
y
100%
R y y y y 50%
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