Pathways Report:  Dead Ends and Wrong 
 
Turns on the Path Through Algebra 

 
 
 
 

Prepared for the Noyce Foundation 
April 4, 2010 
Steve Waterman 

 

 
 
 
ABSTRACT 
This report reflects on the implications of a study that examines the 
 
progression of students in several Bay Area school districts as they proceed 
from eighth to ninth grade math classes.  The findings indicate that students 
 
and their parents face a bewildering array of course titles, that many students 
– even those who are successful – repeat Algebra, that repeating Algebra is 
 
not certain to yield better grades in ninth grade, and that placement decisions 
are correlated to ethnicity and parent education, but not gender. 
 
 

 PATHWAYS REPORT:
DEAD ENDS AND WRONG TURNS ON THE PATH
THROUGH ALGEBRA
Conducted on Behalf of the Noyce Foundation
Steve Waterman

Introduction
As we began what was to be a long-term evaluation study of the Noyce Foundation’s
ambitious mathematics staff development program in the San Francisco Bay Area, we
discovered some troubling patterns. What we had believed was a straightforward
pathway from one mathematics course to the next turned out to be closer to a labyrinth.
With the completion of the original study, we have focused on some of the troubling
findings. This paper looks specifically at findings that appear to impede student
progress and provides some analysis and recommendations. The complete study is
presented in the Appendix.
As we reviewed the data we found that an alarming number of students were taking
Algebra I (standard freshman high school Algebra) for a second time in high school.
This situation was a common thread throughout the districts studied and for most of the
ethnic groups represented by the students. The more we probed, the more concerned
we became that we were witnessing one of two possibilities – that middle schools were
calling a class Algebra when they weren’t teaching Algebra, or that high schools were
not accepting eighth grade Algebra as a legitimate class. Not only were large numbers
of students retaking Algebra in ninth grade, a majority of students who had received
good grades in Algebra in eighth grade were retaking Algebra in ninth grade. Large
numbers of students who had “passed” objective assessments of Algebra were retained
in Algebra in ninth grade. Moreover, while it appeared that boys and girls were
promoted through Algebra at equal rates, progression was more uncertain for students
from some ethnicities than from others.
After reading this report, we are left with many questions, not the least of which is
whether we educators are, inadvertently and unnecessarily, creating a California at risk
of being unable to compete in an increasingly technological society – not because we
are teaching badly, but rather because we are needlessly holding many capable
students back from progressing through advanced mathematics in high school by
flunking them in Algebra, destroying their confidence, and leaving them to languish in a
mystifying morass of confusing course titles with bewildering and narrow exit gates.
The sin here, if such there is, is one of omission. There is no evidence that any school
district deliberately set out to confuse students and parents and hold them back.

Pathways Report, Steve Waterman, April 4, 2010, Page 1

 Likewise, there is no evidence that any district set out to hold back any students of color
or to advance one ethnic group rather than others. Rather, the movement to Algebra in
eighth grade seems to have run into a series of unwritten beliefs and rules.
Unfortunately, belief systems are intractable precisely because they are rarely
expressed. These belief systems include teacher beliefs about the ways students prove
they understand math, the ways students demonstrate they have the “right stuff” to
advance in math, how much math is needed, when topics should be introduced,
whether mathematics is linear, how important homework is, how much a teacher should
reach down, and what a teacher should do when a student fails to understand a concept.
In the hope that beliefs are susceptible to evidence, we have provided a set of
intervention points or strategies for changing the situation.
One might ask why anyone should worry about Algebra and whether it is offered in
eighth grade. We contend that Algebra is a key to entry into the world of advanced high
school classes. These are the classes where the environment is academic rather than
remedial, where expectations are set for college, where students are seen and begin to
see themselves as capable, where children whose first language is not English can still
catch the brass ring, where children from financially limited backgrounds are not held
back by the paucity of their outside-of-school experiences. Moreover, students who
succeed in Algebra in eighth grade and go on to succeed in Geometry in ninth grade
stand a good chance of passing advanced mathematics in high school – sparing them
from college calculus classes with hundreds of students moving at a fast pace while they
are adjusting to college life, and affording them entry into a range of STEM majors.
This report is organized so that the findings are followed by an analysis of pathways for
students and strategies/interventions/pathways readers can use to assist schools and
students to make more effective transitions into high school and thereby to increase the
pool of students leaving high school confident in their math ability and with advanced
mathematics courses under their belts. All of the findings should be considered
preliminary in nature as the sample of schools was not random and may not be
representative of the Bay Area or California. However, after discussions following early
workshops based on these findings, we believe they may reflect practices in many other
school districts.
A. We will begin with a brief summary of the intent of the Pathways Study and its
role within the First in Mathematics Consortium.
B. This will be followed by the major findings of the study. These are essential to
understand the context in which the analysis was made and the limits of the
scope of interpretation of the data
C. The findings will be followed by possible interventions
D. Finally, we will provide some suggested long-term interventions to ameliorate this
situation.

Pathways Report, Steve Waterman, April 4, 2010, Page 2

 A.

The First in Mathematics Consortium

The First in Mathematics Consortium was an investment by the Noyce Foundation in the
improvement of fourth through eighth grade mathematics instruction in a group of nine1
California school districts that demonstrated a commitment to meeting California’s goal
of having all eighth graders succeed in Algebra. The logical chain of the program began
with that commitment. The districts and the Foundation then teamed up to support
intensive, multi-faceted staff development for large numbers of teachers. The training
and commitment was vertical in that it included boards of education, teachers, principals
and superintendents. The Foundation and districts believed that if teachers learned and
tried out more effective strategies for assisting students to access more conceptual
mathematics, the students would be more successful on tests and in more rigorous
mathematics classes as they moved through junior high and into high schools.

Purpose of Study
The main purpose of this study was to establish a starting point for a long-term, outcome
or summative analysis of the impact of the First in Mathematics Consortium over the
period of time between July 2007 and June 2010. Using the eighth grade class in the
last, pre-treatment year as the control group, the full study will compare the success of
that group with the success of the first and second cohort of students who benefited from
the staff development program as they transitioned from middle to high school.
Thus, this study sought to determine dependent and independent variables of interest in
conducting the research, to develop a useful database system that would yield good,
pre-treatment baseline statistics, and to develop and report those statistics. Problem
statements for the study itself might be framed as follows:
1. Did teacher training result in more eighth graders being placed in
high school level Algebra in eighth grade?
2. Did teacher training result in more students completing high
school level Algebra by the end of eighth grade?
3. Because of the training, did more girls and non-white students
have access to and success in Algebra?
4. Did student success in eighth grade carry over into high school in
terms of math course placement and success?
To determine whether sufficient data was available to carry out the study, the
Foundation supported a mini-study in the summer and fall of 2008. The feasibility study
uncovered the possibility of confounding variables that had the potential of overwhelming
any long-term effects from any type of staff development program. The mini-study found
that placement in eighth grade and success in that placement did not seem to be closely
related to placement decisions or success in high school placements. As the study only
included 70 students from one district, the Foundation believed it was essential to
1

One district dropped out of the program. Its data are not included here.

Pathways Report, Steve Waterman, April 4, 2010, Page 3

 discover whether this pattern was widespread. Thus, this study, while identifying the
baseline data needed for a quasi-experimental design with a long-term evaluation, also
looked specifically at the issue of transition into high school math. The information
gleaned from these analyses may encourage middle school and high school teachers to
work more closely together in insuring that students are placed in the most appropriate
mathematics class in high school, and that high school teachers can then believe the
students are placed appropriately and will commit to their success.

Study Methodology
All nine participating districts were asked to provide several pieces of information about
the eighth graders in the 2006-07 school year. These included eighth grade placement
information and grades, eighth grade CST (California Standards Tests) and MARS
(Mathematics Assessment Resource Services) test results, ethnicity, and gender. High
school districts into which the eighth graders were zoned for matriculation were asked to
provide the placement, final grades, and CST results for the 2007-08 ninth grade. This
data was merged into the Noyce database with student names removed. The database
already included information about MARS scores and teacher variables.
Once assembled, the database was to be used for a number of analyses to provide a
baseline in a number of dimensions for an impact study. (The first group of students
whose teachers received intensive training, completed eighth grade in June 2008 and
ninth grade in June 2009. Their CST results for ninth grade are now available.)

B.

Major Findings

There are several major findings. Each has implications for further work with school
districts and implications for maximizing the impact of the staff development:


There is a large and confusing array of names for math classes in eighth
and ninth grades among the nine elementary schools, four high schools
and one unified school district in the treatment group.



Nearly 65% of the students who were placed in Algebra in eighth grade
were placed in the same level of Algebra or Honors Algebra in ninth grade.
(Note: All non-honors Algebra titles were merged into Algebra for this
analysis.)



Nearly half of the students who were successful in Algebra in the eighth
grade and who were placed again in Algebra in ninth grade were no more
successful in their second experience.



For the group of students who took Algebra in eighth grade, success on
either the MARS or CST assessments did not translate into Geometry
placement for most students.



There were indications that both placement decisions and final grades in
mathematics classes differed by ethnicity – that is, the opportunity to take
Algebra in eighth grade and the grade given in an Algebra class both
differed by ethnicity.

Pathways Report, Steve Waterman, April 4, 2010, Page 4

 Implications/Discussion
The findings from this study portray a mixed picture of progress toward increasing the
number of students successful in Algebra in eighth grade, and an even more mixed
picture for students of color and children whose parents have not completed college.
Alternative explanations include unintended consequences of the state push for STAR
testing success, poorly coordinated data and decision systems for high school
placement, unaligned performance expectations between middle and high schools,
unsuccessfully trained teachers, insufficient numbers of teachers with skills in both
content and teaching the content, teacher grading practices, pressure on middle school
teachers from high school teachers, belief systems of math teachers about the value and
need for Algebra in eighth grade, expectations of students that are biased by socioeconomic status or ethnicity, and lack of supervision by principals.
While papers could be written about each of these possibilities, here we are going to
assume that some or all of these are possible factors influencing success, and we are
going to look at low- or no-cost strategies various stakeholders can take to improve the
situation.

Finding 1 – Math Course Titles are Multiple and
Confusing
In the eight elementary school districts and their coordinating high school districts with
data in the study, there were 27 names for Algebra. We do not know whether students
and their parents understood what the course titles implied, and we were unable to find
clear descriptions on most of the websites. The potential problem with this situation is
that parents who are not sophisticated in the school verbiage would not know what the
course titles mean. This possibility combined with their general unfamiliarity with the
university system requirements and their lack of knowledge of the career possibilities
that open through advanced math and their fear of or trust in the professionals lead them
to accept teacher recommendations or their children’s choices. A partial fix for this
problem can be accomplished at little cost.
The first step in helping parents and students know what is happening is to label the
courses clearly. For example: Algebra 1 – This course is designed as a one-year
sequence of Algebra that will fulfill the California A-G high school requirement. Success
in this class will set a student on a path to completing calculus by the end of her/his
senior year. Student success in this class is determined by a final grade of ”C” or higher
or a score of “Proficient” or “Advanced” on the CST, or a score of “Meets Standards” or
“Exceeds Standards” on the MARS test. Successful students will be matriculated into
Geometry or Honors Geometry in ninth grade. Students who do not meet one or more
of these criteria will repeat Algebra 1 in ninth grade unless they successfully complete a
summer school intervention program.

Pathways Report, Steve Waterman, April 4, 2010, Page 5

 Finding 2 – Most Eighth Grade Algebra Students Do Not
Progress to Geometry in Grade Nine
Currently, placement in math classes in ninth grade does not clearly follow eighth grade
placement. In this way, math classes are unique in the eighth grade. Students enrolled
in English, science, and social studies classes in eighth grade are generally promoted to
the ninth grade versions of these classes. It is only in Algebra that students are routinely
retained. The study found that 65% of the students who were enrolled in one of the
Algebra classes in eighth grade were retained in Algebra in ninth grade. A failure rate of
this magnitude would not be tolerated in other academic fields.
Possible explanations for this include the following: The eight grade class is called
Algebra but is not really a high school level Algebra class; the eighth grade teachers do
not believe they can deliver a high school level class; the grading practices get in the
way of student accomplishment; students come to the class unprepared to succeed;
high school teachers undo the eighth grade teacher recommendations; a belief system
operates that holds that Algebra in eighth grade is either unnecessary or too early; a
belief system operates that some students who seem successful in eighth grade will fail
in ninth grade, or that they will be more successful after repeating the course; a belief
system that some students are not as deserving of advanced placement as others;
passive parental involvement; and confusion among teachers about which concepts are
core and how to measure competence in them. One possible solution is to look at the
timetable and events that lead to placement decisions and to identify key intervention
points. We do this after this section.
In an attempt to isolate the key variables at work, we then looked at those students who
showed success in eighth grade Algebra. Again, our assumption was that success in
eighth grade would automatically lead to advancement.

Finding 3 - 46% of the Eighth Grade Algebra Students
with Final Grades of ‘B-‘ or Above Were Retained in
Algebra or Dropped to “Math” Classes in Ninth Grade.
We assumed that students whose eighth grade Algebra teachers had given them
passing grades would surely have been promoted, as no one could argue that the
students had not participated in class, completed homework, or had passed exams. We
selected a grade of ‘B-‘ as a demonstration of success because most readers would
consider that grade as demonstrating more than basic understanding and performance
in a class. But with 46% of these students not progressing, it was clear that success by
common sense criteria – good grades – was not sufficient for movement to Geometry.
Even in this relatively small sample, the result of opening the Geometry classes to an
additional 247 students could have had a significant impact on the number of students
completing advanced mathematics courses by twelfth grade.
The implications are staggering if this sample is even mildly representative of the state of
California. If at every level of high school math the numbers of students doubled, the
need to recruit engineers from other countries could disappear.

Pathways Report, Steve Waterman, April 4, 2010, Page 6

 Finding 4 – 45% of Eighth Grade Algebra Students Who
“Met Standards” or “Exceeded Standards” on the MARS
Assessment and 44% of Students Who Scored
“Proficient” or “Advanced” on the CST Algebra Test
Were Retained in Algebra in the Ninth Grade.
If success under the usual criterion of good grades by the teacher was not a clear
pathway to Geometry in ninth grade, neither was success on two objective assessments
of Algebra given to eighth graders in these districts.
Retention is not harmless. One can only imagine the bafflement of a students who did
well on one or more of these three data points – grades, MARS, or CST, and then found
him/herself back in Algebra as a ninth grader. Perhaps a self-assured adult would
recover. Perhaps an insecure thirteen-year-old would form the belief that he or she is
incapable of succeeding in and understanding mathematics, stop taking math courses
as soon as possible in high school, and avoid math in college.
The CST Series were advertised to Californians as tests exemplifying “World Class
Standards.” Yet the system allows many students who pass these tests to be retained.
The combinations of Findings 2, 3, and 4 seem to remove many objective explanations
for the overall 65% failure rate in Algebra for this sample. These findings leave belief
systems and their implications as the most viable explanations for the findings. Finding
5 does nothing to dispel this conclusion.

Finding 5 - Placement Differs by Ethnicity and Parent
Education Level. Within Placement, Grades Differ by
Ethnicity and Parent Education Level.
Both of these variables – ethnicity and parent education level – were self-reported by
students in the sample on their CST tests. Thus, accuracy is not perfect.
It is not surprising that nearly all of the eighth graders in Geometry and Honors Algebra
in eighth grade reported that their parents had college or graduate degrees. Degrees
are correlated with family income and more understanding by parents of the value of
taking geometry in ninth grade.
Placement by ethnicity in eighth grade also varied. For example, while 56% of Filipino
students took Math or Pre-algebra in eighth grade, only 17.5% of Asians and 18.8% of
white students were enrolled in courses below Algebra. It is not clear whether these
differences were due to district practices about offering Algebra in eighth grade or
because of different treatment of the students. Obviously, students enrolled in classes
below Algebra in eighth grade had no access to Geometry in ninth grade. However,
even among the “successful” eighth grade Algebra students, placement in ninth grade
varied – with Asian students showing the highest percentages in Geometry or Advanced
Geometry (77%) and Filipino students with the lowest (40%), while Latino and white

Pathways Report, Steve Waterman, April 4, 2010, Page 7

 students were approximately equally successful at about 66% enrolled in Geometry.
Other ethnic groups samples were too small for inclusion.
Thus, while Tables 7 and 8 in the study show a potential differentiation related to
ethnicity, because the sample of students and school districts was not random, it is
impossible to generalize these findings. However, it is an indication that a problem may
exist. A replication study with a larger sample size would be needed to confirm the
problem. Still, it is an area that these and other school districts could monitor as a
potential problem.

C.

Pathway Timeline

The timeline below is an attempt to deconstruct the progress of students and provide
school districts and agencies with possible structural and programmatic intervention
points.
Seventh Grade
The seventh grade teacher determines class placements for eighth grade. Decisions
are based on student progress, the course program of the middle school, and possibly
by the availability of teachers. Students not recommended for Algebra in eighth grade
cannot get beyond Pre-calculus in high school without extraordinary efforts.
Eighth Grade – Refers to Students Enrolled in Algebra Only
Fall
Because the decisions about fall placement are often made in January, fall is the time for
students to gain confidence in Algebra and demonstrate understanding of fundamental
concepts.
January
In January, parents of eighth graders are mailed high school course catalogs and
enrollment materials. The students and their parents fill them out and they are signed by
the parent, student, and the eighth grade teacher advisor. In some schools, the eighth
grade teachers provide separate recommendations to the high school for placement in
math.
February
In February, the high school counselors visit the eighth graders in middle school to
review their program choices. Sometimes parents are involved, sometimes not.
Programs are modified at this time and finalized.

Pathways Report, Steve Waterman, April 4, 2010, Page 8

 March through May
Some high schools administer placement exams during the spring. Results of
placement tests sometimes override already set placements. The programs are then
entered into the high school program and classes are preliminarily set up.
Some districts administer and score MARS tests during this period. As the MARS
requires students to apply knowledge and skills in open-ended performance questions,
success in the MARS is a very good indicator of competence.
In May, students in California take the CST or STAR test in Algebra or eighth grade
math. This test, a traditionally formatted, multiple-choice test, nonetheless has proven to
be difficult for students. A relatively modest percentage of students who take the test
are successful.
June through August
Students are notified of their class schedules during this period – usually in August, just
before school starts.
CST results come back to the school districts in early August. Eighth grade
administrators could take one last look to be sure that any students who pass this
difficult test in Algebra be moved to Geometry in ninth grade.

Analysis and Interventions
Seventh Grade
Structural interventions can be made here in clarifying the best predictors for success in
Algebra, training teachers to provide more access paths to students in problem-solving,
setting up programs that support access for students to Algebra in eighth grade, and
clearly describing classes.
Eighth Grade – Algebra Classes Only
Fall
Interventions here include: (1) helping parents and students understand the value of
Algebra as a pathway to many college majors and careers, and why it is a value for
students to pass Algebra and move to Geometry in ninth grade; (2) providing early
interventions for students who begin to struggle. There is some research to indicate that
the most effective assistance for students in Algebra occurs immediately when a student
has trouble understanding a new concept. These interventions are perhaps most
effectively done by the classroom teachers, but coordinated tutoring can also work; (3)
some schools have found “core” programs can provide the time teachers need to help
students consolidate their learning before moving on. (Core programs are those in
which math and science are taught in a double period by one teacher. This enables the
teacher to have two periods available for science labs or for teaching difficult math
concepts.)

Pathways Report, Steve Waterman, April 4, 2010, Page 9

 Outside agencies could assist here with funding, volunteer tutors, etc.
January
In low-income communities, twelve- and thirteen-year-old students often rely on teachers
for help in math, while in affluent communities, students are more likely to rely on
parents for help. It is difficult to determine in January whether the concepts will kick in
for a student by the end of the year. Teachers sometimes worry about “setting their
students up for failure” in high school by recommending Geometry unless the student is
acing Algebra already. Thus, students are often convinced to take courses that will
“insure success.”
The institution could change this practice in several ways. First, high schools could
provide clear descriptions of ninth grade courses so parents and students understand
the implications of the courses they are selecting. Second, teachers could encourage
students to stretch, and could show them data that illustrates the students might do no
better with Algebra the second time around and should instead sign up for some extra
tutoring or study sessions. Third, the principal could request grades and placement
recommendations for eighth graders, then monitor student success on the MARS and
CST later in the year.
February
High school counselors sometimes appear to be schooled by math teachers to
discourage students from tough math courses. Counselors have been observed talking
students into lower level math classes on the rationale the students would have a better
start to their high school careers. Principals could work with counselors and high school
math teachers to change this practice. These discussions require considerable finesse
as high school math teachers can argue that middle school simply isn’t preparing
students for high school. If principals cannot get their teachers past this stance,
whatever the principal tries will fail for many students.
High school mathematics placement tests may or may not have any predictive validity.
The tests most commonly used may not have been given to a random sample of
students who were later placed without regard to the test results and then followed for
indications of success. The schools could use the placement results only to move
students up a level and never to move them down.
March through June
Usually while the high school is programming the students, the middle school has
completed its task and goes on with the chaos of spring and eighth grade promotion and
field trips. Instead, the eighth grade principal could monitor the progress of students
recommended for retention in Algebra and review the results of the MARS tests and
later report cards to possible recognize student growth or achievement based on more
independent measures. The principal could then notify the high school of recommended
placement changes before the high school program is in stone.

Pathways Report, Steve Waterman, April 4, 2010, Page 10

 August
In the first week of August, the last chance for change occurs with the release of the
CST results. This is the last chance for the elementary district to catch students who
have somehow managed to score Proficient on the CST while not having been
recommended for Geometry.

D.

Long Term Interventions

These structural interventions can increase the number of students who move on to
Geometry in ninth grade. They will pick up students who are capable but need
encouragement to challenge themselves, students who come to grasp Algebraic
concepts later in the school year, and students who were capable but were not
recognized as such by their teachers. Even a ten percent increase in numbers would
dramatically change the number of students completing high school prepared to enter
more technical fields in college
Reaching percentages of success approaching universality will require additional
interventions that will require extra efforts or money. Below are some suggestions.


School administrators might review the study findings in detail and help their
staffs understand the study’s implications and strategies for avoiding becoming a
school that fails half or more of its math students. In addition to the study, the
appendix includes some scenarios of real situations various players face in
moving students through math classes. Staff development sessions could use
these scenarios as discussion initiators.



While elementary trained teachers are often more appreciative of varying
strategies for approaching math solutions, their often limited understanding of
mathematics hinders their ability to adequately prepare students for abstract
concepts in math. Formal teacher training could help. On the other hand,
teachers who were math majors were taught in traditional ways that only a few
students understood and may not know alternative strategies for assisting
students to access new concepts. Both groups would benefit from programs
such as a Masters Degree in the Teaching of Mathematics – a program that
would blend knowledge with teaching strategies.



Schools could redirect some of their after-school tutoring funds from programs
such as the ASES Program in California to assisting Algebra students after
school.



Businesses could provide volunteers with math backgrounds to tutor, teach short
courses after school, monitor students’ online, provide summer work, etc.

Pathways Report, Steve Waterman, April 4, 2010, Page 11

 Moving Forward
Obviously, the findings presented here have to be considered preliminary. Replicating
the study in a broader context would provide leaders with a firmer foundation for
compelling change in their systems.
In the meantime, every school district might benefit from reviewing these findings and
timelines and comparing them to data and timelines within their own systems. The
following questions could focus this work: Is the system rational? Does it truly maximize
student success? Are unexamined assumptions impeding student progress? What
strategic interventions might improve the system?
Success will be measured in the increase in numbers and percentages of students from
all backgrounds who progress smoothly through Algebra in eighth grade and into
Geometry in ninth. The beginning of this process is appreciating the possibility that
nearly all students really do want to learn and that it might not be the “fault” of the
students who do not immediately grasp each new concept.

Acknowledgements:
David Foster, Executive Director of the Silicon Valley Mathematics Initiative for editing
and data retrieval
Sara Spiegel, Noyce Foundation for assembling the database
Dr. Ray Gamba, Professor of Mathematics, City College, and Michael Jang, Executive
Director of the Institute for Scientific Analysis for statistical assistance
Aida Gamba, Brisbane School District, for database help and editing
Penny Noyce, Trustee, Noyce Foundation for editing
Jon Wagner, Professor of Education, U.C. Davis

Steve Waterman, 1045 Contra Costa Drive, El Cerrito, CA 94530, email
stevewatermanca@aol.com. phone 510-525-7970, 510-301-0883.

Pathways Report, Steve Waterman, April 4, 2010, Page 12

 Appendix

Pathways Study  
Prepared for the Noyce Foundation 
January 20, 2010 
Steve Waterman 
 

ABSTRACT 
This Report examines the progression of students in several Bay Area 
School Districts as they proceed from eighth to ninth grade math classes.  
The findings indicate that students and their parents face a bewildering 
array of course titles, that many students repeat Algebra, that repeating 
Algebra is not certain to yield better grades in ninth grade, and that 
placement decisions are correlated to ethnicity and parent education, but 
not gender. 

 PATHWAYS  STUDY 
 

Conducted on Behalf of the Noyce Foundation 
 
 
 
 

Steve Waterman 
 
 
 

The First in Math Program 
 
The First in Math Program was an investment by the Noyce Foundation in the improvement of fourth 
through eighth grade mathematics instruction in a group of nine1 California school districts that 
demonstrated a commitment to meeting California’s goal of having all eighth graders succeed in 
Algebra.  The logical chain of the program began with that commitment.  The districts and the 
Foundation then teamed up to support intensive, multi‐faceted staff development for large numbers of 
teachers.  The training and commitment was vertical in that it included boards of education, teachers, 
principals and the superintendents.  The Foundation and districts believed that if teachers learned and 
tried out more effective strategies for assisting students to access more conceptual mathematics, the 
students would be more successful on tests and in more rigorous mathematics classes as they moved 
through junior high and into high schools.   
 
 
 

Purpose of Study 
 
The main purpose of this study was to establish a starting point for a long‐term, outcome or summative 
analysis of the impact of the First in Math Program over the period of time between July 2007 and June 
2010.   Using the eighth grade class in the last pre‐treatment year as the control group, the full study will 
compare the success of that group with the success of the first and second cohort of students who 
benefited from the staff development program as they transitioned from middle to high school. 
 
Thus, this study sought to determine dependent and independent variables of interest in conducting the 
research, to develop a useful database system that would yield good, pre‐treatment baseline statistics, 
and to develop and report those statistics.  Problem statements for the study itself might be framed as 
follows: 
1. Did teacher training result in more eighth graders being placed in high school level 
Algebra in eighth grade? 
1

One district dropped out of the program. Its data are not included here.

 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation       

1 

 2. Did teacher training result in more students completing high school level Algebra by 
the end of eighth grade? 
3. Because of the training, did more girls and non‐white students have access to and 
success in Algebra? 
4. Did student success in eighth grade carry over into high school in terms of math 
course placement and success? 
 
To determine whether sufficient data was available to carry out the study, the Foundation supported a 
mini‐study in the summer and fall of 2008.  The feasibility study uncovered the possibility of 
confounding variables that had the potential of overwhelming any long‐term effects from any type of 
staff development program.  The mini‐study found that placement in eighth grade and success in that 
placement did not seem to be closely related to placement decisions or success in high school 
placements.  As the study only included 70 students from one district, the Foundation believed it was 
essential to discover whether this pattern was widespread.  Thus, this study, while identifying the 
baseline data needed for a quasi‐experimental design with a long‐term evaluation, also looked 
specifically at the issue of transition into high school math.  The information gleaned from these 
analyses may encourage middle school and high school teachers to work more closely together in 
ensuring that students are placed in the most appropriate mathematics class in high school, and that 
high school teachers can then believe that students are placed appropriately and will commit to their 
success. 
 
 
 

Study Methodology 
 
All nine participating districts were asked to provide several pieces of information about the 2006‐07 
eighth graders.  These included eighth grade placement information and grades, eighth grade CST 
results, including results by ethnicity and gender.  High school districts into which the eighth graders 
were zoned for matriculation were asked to provide the placement, final grades, and CST results for the 
2007‐08 ninth grade.  This data was merged into the Noyce database with student names removed.  The 
database already included information about MARS scores and teacher variables.  
   
Once assembled, the database was to be used for a number of analyses to provide a multi‐dimensional 
baseline for an impact study.  (The first group of students whose teachers received intensive training, 
completed eighth grade in June 2008 and ninth grade in June 2009.  Their CST results for ninth grade are 
now available.) 
 
 
 

Findings 
 
There are several major findings.  Each has implications for further work with school districts and 
implications for maximizing the impact of the staff development: 

 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

2 

 

There is a large and confusing array of names for math classes in eighth and ninth grades among 
the nine elementary schools, four high schools and one unified school district in the treatment 
group. 

 


Placement in ninth grade math class is not always clearly related to prior placement in eighth 
grade math class in the way one would expect – a movement from Algebra to Geometry, for 
example.  Nearly 65% of the students who were placed in Algebra  in eighth grade were placed 
in the same level of Algebra or Honors Algebra in ninth grade. (Note:  All non‐honors Algebra 
titles were merged into Algebra for this analysis.  The term “Math” refers to general, non‐
Algebra, non‐Geometry or other higher mathematics courses. ) 

 


For the group of students who took Algebra in eighth grade, the best predictor of success in 
ninth grade, as defined by final grades, was the eighth grade MARS raw score. 

 


There were indications that grades and placement in math classes differ by ethnicity and parent 
education. 

 


Nearly half of the students who were successful in Algebra in the eighth grade and who were 
placed again in Algebra in ninth grade were no more successful in their second experience. 

 
 

Finding 1:  Thirteen (at least) Ways of Characterizing Math2 
 
School district websites were reviewed to try to ascertain the meaning of titles.  In most cases, these 
titles were not on the websites, or were confusing.  Titles are listed below: 
 

Mathematics – Pre‐Algebra, Math 8, Math 8‐1, Math 8‐H, Math Support, Math SDC, 
Math 6, Math 7/8, ELD Math, Math 8 I.L., Tutorial Math 
 

Eighth Grade Algebra – Algebra A, Algebra 1, Fund Algebra, Acc. Algebra, Algebra, 
Algebra Honors, Algebra 8, Algebra 8A, Concrete Algebra,  
 

Ninth Grade Algebra ‐ Algebra 1 (9th), Algebra 1 unit 6, Algebra 1A, Algebra 2, Algebra 
Concepts, Algebra 1A1B (9), Algebra A (9), Algebra B (9) (P), Algebra 1 (9) (P), Algebra 
Support, Fund Algebra A, Algebra 1 (8th), P‐Alg 2 Acc, P‐Algebra 1, P‐Algebra 2, P‐Algebra 
2‐H, P‐Seq Alg 1, P‐SeqAlg1/’Geom 
 

Geometry – Geometry (9) (P), Geometry, Geometry Acc. 
 
From conversations with teachers, we gathered that placement decisions vary greatly both across and 
within high school districts.  Many factors are involved:   
 

  Eighth grade teachers make recommendations early in the second semester, before external 
assessments have been completed;  
 

2

See poem by Wallace Stevens, “Thirteen Ways of Looking at a Blackbird”
 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

3 

   Grading practices (mastery determinations) of teachers vary; some rely more on tests, others 
rely more on homework, others include effort; 
 

  Students, with their parents’ help, sign up for high school classes in January of their eighth grade 
year;  
 

  High school counselors sometimes meet with eighth graders to confirm choices and provide 
further guidance in the spring;  
 
  High schools sometimes modify decisions based on CST performance; and  
 
  High schools and eighth grades sometimes give Algebra Placement tests.   
 
This study did not focus on the types of explanations given to students during placement meetings held 
during the spring of their eighth grade year.  Anecdotal comments from teachers indicate that at least 
sometimes, students are advised to re‐take Algebra in ninth grade to be sure they “master” all of its 
concepts.  High school teachers have been overheard castigating eighth grade teachers for placement 
recommendations the ninth grade teachers feel are too advanced for the students.  This may have 
frightened eighth grade teachers into being conservative in their placement recommendations. 
 
The number of course titles that include the word “Algebra” may confuse parents who are not 
sophisticated in unraveling the complexities of math class placement.  These parents may rely on the 
teachers and counselors to advise them. 
 
The various steps in the recommendation process almost never appear to encourage student placement 
in more academic classes, and may have been the single largest contributor to the re‐placement of 
students who take Algebra in eighth grade back into Algebra in ninth grade – regardless of student 
success on the state’s Algebra CST. 
 
 

Finding 2:  Ninth Grade Placement, a Function of Eighth Grade Class? 
 
The first data finding of the study deals with the way Algebra and general math classes are described by 
the schools and on transcripts.  This finding deals with student placement in ninth grade. 
 
As the chart below indicates, nearly all of the students who were in Math 8 classes (math or pre‐
Algebra) were placed in Algebra classes in ninth grade (Algebra two or three semesters are both 
considered Algebra here).   Nearly all of the students who took Geometry in eighth grade were placed in 
Algebra II as ninth graders.  Likewise, more than 80% of the students who were enrolled in Honors 
Algebra in eighth grade moved to either Geometry or Honors Geometry. Thus, for these three groups of 
students, the system provided predictable results.  
 
This was not the case for the largest subset of students, those taking Algebra as eighth graders.  
Unfortunately, the findings of the study generalize the finding of the mini‐study.  Slightly more than half 
of the students took Algebra in eighth grade.  Approximately 64% of those students were again placed in 
Algebra as ninth graders – or worse – were placed in pre‐Algebra or general Math as ninth graders.   
 
 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

4 

 One could then look at the results from two perspectives.  On the one hand, about 65% of 1653 
students advanced from one level of math to the next as they moved from eighth to ninth grade.  On the 
other hand, 35% did not advance.  Moreover, the failure rate (if defined as non‐advancement) was 
nearly 65% for students taking Algebra in the eighth grade. 
 
Table 1.  Placement in ninth grade mathematics courses by eighth grade class, expressed in percentages on first 
line, number of students on the second line.  Sample size: 1653, 57% of the total population.  For example, 1.9% of 
the students who took Math 8 were placed in math or pre‐Algebra in ninth grade.   
 

 
  Crosstab 

Math 
Algebra 
Hon Algebra Geometry 
Ninth Grade  Ninth Grade  Ninth Grade  Ninth Grade 
 
1.9% 
94.3% 
4.7% 
0% 
Math 8 & Pre‐algebra 
8 
461 
20 
 
4.2% 
58.1% 
1.9% 
32.5% 
Algebra 8 
37 
513 
17 
287 
 
10.8% 
1.3% 
48.5% 
0% 
25 
3 
112 
H Algebra 
 
2% 
98%* 
0% 
0% 
1 
49 
Geometry 
*This appears to have been Second Year Algebra 

Hon 
Geometry 
0% 
3.3% 
29 
39.4% 
91 

Total 
Number 
489 
883 
231 

0% 

50 

 
Table 2.  Compares students by district who were placed in Algebra or honors Algebra in eighth grade to their 
placement in ninth grade. (The percentage in the Algebra eighth grade column indicates the percent of students 
from that district with matched scores who were placed in Algebra in eighth grade.) 
 

Crosstab, 
district by 
Algebra 
Dist 1 

Algebra or  Hon Algebra 
Eight Grade 
# of Students 
182 (35.4%) 

 

Algebra or 
Hon Algebra 
Ninth Grade 
121 (66.5%) 

 

Math 
Ninth Grade 

Geometry 
Ninth Grade 

Hon 
Geometry 

61 (33.5%) 

 

135 (60%) 

90 (40.0%) 

 

Dist 2 

225(99.5%) 

Dist 3 

116 (50.6% 

 

27 (23.3%) 

60 (51.7%) 

29 (25%) 

Dist 4 

242 (89%) 

45 (16.4%) 

116 (42.4%) 

 

91 (33%) 

Dist 5 

93 (60.3% 

 

26 (28%) 

67 (72.0%) 

 

Dist 6 

146 (77.6%) 

 

104 (71.2%) 

42 (28.8%) 

 

Dist 7 

17 (54.8%) 

 

3 (17.6%) 

14 (82.4%) 

 

Dist 8 

38 (100%) 

 

27 (71.1%) 

11 (28.9%) 

 

 
Finding 3:  MARS performance as a Predictor of Ninth Grade Placement 
 
As the MARS test is given in early March each year, with scores available to the districts by early May, it 
provides the strongest independent data for teachers to use in placing or adjusting the placement 
recommendations for ninth grade.  One would predict that since the MARS is closely correlated with CST 
results, it would be a generally strong predictor of ninth grade placement. 
 
From the chart below, it is clear that the MARS performance levels were not good predictors of later 
placement.  Indeed, students were just as likely to be placed in Geometry in ninth grade if they did not 
meet standards as if they did.  It is possible that teachers do not check placement decisions after the 
 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

5 

 MARS results come back to the local districts. 
 
Table 3.  Compares student success on the MARS test administered in the spring of 2007 with the students’ later 
placement in math classes in the high schools.  There were 1410 valid matches for this analysis out of a total 
population of 2897 students.  MARS has four levels.  This analysis consolidated the analysis into two cells, those 
meeting or exceeding standards and those not meeting standards. 
 

MARS to Placement 

Math 
Algebra 
34 
440 
Met Standards 
4.4% 
56.4% 
16 
412 
Standards not Met 
2.5% 
65.4% 
Total 
50 
852 
*Includes two students placed in Trig 

Hon Alg 
48 
6.2% 
13 
2.1% 
61 

Geometry 
169 
21.7% 
151 
24% 
320 

Hon Geom 
87 
11.2% 
38 
6% 
125 

Total 
780* 
630 
1410 

 
Because only about 50% of the eighth graders were enrolled in Algebra class, that was the only group to 
take the Algebra version of the MARS in 2007.  Table 3a below summarizes the results for this group of 
students.  This analysis is critically important to examine because historically, nearly all of the students 
who scored either “Met Standards” or “Above Standards” on the MARS Algebra tests also scored at 
“Proficient” or “Advanced” on the CST.  Because the MARS proved to be a more difficult test than the 
CST and required greater understanding, and because the results come early enough in the year to 
change students’ schedules for high school, these results could change a student’s learning 
opportunities for the rest of her/his educational career. 
 
The data in this table has profound implications.  First, only a little over half of the students who met 
standards on this difficult test were promoted to Geometry.  These were students who understood 
Algebra well enough to use it to solve new word problems.  These students demonstrated competence 
in Algebra in the spring, early enough to have their programs changed.  But their programs were not 
changed.  If this percentage is representative of even a portion of the State of California, many 
thousands of students who demonstrated math capability, were effectively shut out of Calculus in high 
school – and all that completing such an AP class represents to their college and college major choices. 
 
Second, about half of the students who took the assessment did not meet standards.  These were 
students taking Algebra in eighth grade who were not succeeding by March of the school year.  A fifty 
percent failure rate would not be acceptable in other core academic subjects, but in Mathematics, 
apparently a failure rate of this magnitude is accepted. 
 
Table 3a.  Includes only students enrolled in Algebra in eighth grade in 2007.  Compares student success on the 
MARS test administered in the spring of 2007 with the students’ later placement in math classes in the high schools.  
There were 739 valid matches for this analysis out of a total population of 2897 students.  MARS has four levels.  
This analysis consolidated the results into two cells, those meeting or exceeding standards and those not meeting 
standards. 
 

MARS to Placement 
Met Standards 
Standards not Met 
Total 

Math 
9 
2.5% 
26 
6.9% 
35 

Algebra 
137 
38% 
307 
81% 
444 

Hon Alg 
17 
4.7% 
0 
 
17 

Geometry 
169 
46.9% 
45 
11.9% 
214 

Hon Geom 
28 
7.8% 
1 
0.3% 
29 

Total 
360 
379 
739 

 

 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

6 

 Finding 4:  Parent Education and Eighth Grade Placement 
 
As one of the purposes of the staff development program was to open the availability of higher level 
math classes to students from lower socio‐economic levels, the study looked at placement prior to 
training for baseline data.  By looking down the column labeled “Math” one can see that there is an 
increase in the percent of each group who took lower level math classes based on socio‐economic 
status.  It is important to note that this information was self reported in some school districts by the 
students.  Conversations with school secretaries in past years revealed that students have been hesitant 
to report their parents as not having at least a high school diploma – or students are simply unaware of 
their parent education level.  This tends to be less the case for students whose parents hold jobs that 
clearly require college or advanced degrees.  However, even assuming some uncertainty, the results 
show a clear bias for children of parents with college or graduate degrees to be placed in advanced math 
courses.  This is not at all unexpected and forms a good baseline for change as brought about by 
program effects. 
 
Table 4.  Compares parent education level to eighth grade placement in mathematics classes.  Parent Education is 
self‐reported and is taken from the students’ CST results.  Approximately 8% of the students who took the CST left 
this field blank. 
 

Parent Ed /  Eighth Grade Placement 

Parent Education Level 

Math 
Graduate 
School 

Count 
% within 

18.6% 

College 
Degree 

Count 

136 

165 

Geometry 

Total 

58 

39 

322 

51.2% 

18.0% 

12.1% 

100.0% 

284 

107 

8 

535 

20.0% 

1.5% 

100.0% 

41 

3 

347 

% within 

25.4% 

53.1% 

Some 
College 

Count 

117 

186 

% within 

33.7% 

53.6% 

High School 
Grad 

Count 

124 

128 

23 

0 

% within 

45.1% 

46.5% 

8.4% 

.0% 

44 

39 

1 

2 

86 

51.2% 

45.3% 

1.2% 

2.3% 

100.0% 

58 

57 

18 

1 

134 

43.3% 

42.5% 

13.4% 

.7% 

100.0% 

539 
31.7% 

859 
50.6% 

248 
14.6% 

53 
3.1% 

1699 
100.0% 

Not HS Grad 
Blank 

 

60 

Eighth Grade Placement 
Algebra 
Hon Alg 

  Total Count 
% of Total 

Count 
% within 
Count 
% within 

11.8% 

.9% 

100.0% 
275 
100.0% 

 
 

Finding 5:  Gender and Placement 
 
There has been an assumption that, as they progress through the grades, girls do not enroll as often as 
boys, either by choice or by prejudice, in advanced math classes.  If this phenomenon occurs generally, it 
did not occur in the target school districts during the 2006‐07 school year.  There were statistically 
insignificant differences in math placement between boys and girls overall.  Only in Geometry placement 
were more boys than girls enrolled.  See Table 5, below. 
 

 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

7 

 Table 5. Comparison of gender to eighth grade placement in mathematics classes. Gender is self‐reported and is 
taken from the students’ CST results.  Approximately 19% of the students who took the CST left this field blank. 
 

 

Eighth Grade Placement 
Pre Alg 
Algebra 
Hon Alg 

Math 

Count 
% within 
Count 
Male 
% within 
Count 
Missing 
% within 

0 
.0% 
0 
.0% 
13 
3.2% 

253 
31.3% 
286 
32.1% 
35 
8.6% 

0 
.0% 
0 
.0% 
91 
22.3% 

413 
51.1% 
447 
50.1% 
238 
58.3% 

124 
15.3% 
124 
13.9% 
29 
7.1% 

18 
2.2% 
35 
3.9% 
2 
.5% 

808 
100.0% 
892 
100.0% 
408 
100.0% 

Total Count 
% of Total 

13 
0.6% 

574 
27.2% 

91 
4.3% 

1098 
52.1% 

277 
13.1% 

55 
2.6% 

2108 
100.0% 

Female 

Geometry 

Total 

SDC 

 
 

The findings for ninth grade mirror those for eighth grade.  Again, the difference in placement between 
boys and girls was not statistically meaningful. 
 
Table 6.  Comparison of gender to ninth grade placement in mathematics classes. Gender is self‐reported and is 
taken from the students’ CST results.  Approximately 22% of the students who took the CST left this field blank. 
 

 
Female 
Male 
Missing 

Ninth Grade Placement 
Count 
% within 
Count 
% within 
Count 
% within 

Total Count 
% of Total 

Geometry 

Hon Geom 

Total 

Math 

Algebra 

Hon Alg 

Trig 

Calculus 

18 
2.5% 

416 
58.8% 

23 
3.2% 

190 
26.8% 

61 
8.6% 

0 
.0% 

0 
.0% 

708 
100% 

25 
3.1% 

504 
61.7% 

42 
5.1% 

188 
23.0% 

57 
7.0% 

1 
0.1% 

0 
.0% 

817 
100% 

28 
6.6% 

250 
59.1% 

6 
1.4% 

84 
19.9% 

45 
10.6% 

9 
2.1% 

1 
0.2% 

423 
100% 

71 
3.6% 

1170 
60.1% 

71 
3.6% 

462 
23.7% 

163 
8.4% 

10 
5.0% 

1 
0.1% 

1948 
100% 

 

 
 

 

Finding 6:  Mathematics Placement in FiMP Schools by Ethnicity 
 
Opening access to higher level mathematics classes to children from diverse ethnic backgrounds was 
one of the goals of the First in Math Program.  The table below clearly illustrates the differences in 
eighth grade placement for different ethnic groups, with white and Asian students being placed in more 
advanced classes much more often than students of other ethnicities.  More than forty percent of the 
students from most of the ethnic groups were placed in math/pre Algebra classes in eighth grade, while 
fewer than twenty percent of the white and Asian students were so placed.  In fact, for most of the 
minority groups, more than ninety percent of the students were placed in either Algebra or math 
classes, and aside from Filipino students, few non‐Asian minority students were placed in higher‐level 
math classes. 
 

 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

8 

 Table 7.  Comparison of ethnicity to eighth grade placement in mathematics classes. Ethnicity is self‐reported and is 
taken from the students’ CST results.   
 

Eighth Grade Math Placement (2006‐07 School Year) 
Ethnicity 

Math/ Pre Alg 
  N 

Algebra 

  % 

  N 

Hon Algebra 

  % 

  N 

 
Total 

Geometry 

  % 

  N 

  % 

   N 

        % 

Am Indian 

    7 

53.8% 

    6 

46.2% 

   0 

0% 

  0 

0% 

    13 

0.8% 

Asian 

  60 

17.5% 

181 

52.9% 

  62 

18.1% 

39 

11.4% 

  342 

20.4% 

Filipino 

183 

56.1% 

111 

34.0% 

  32 

9.8% 

  0 

0% 

  326 

19.5% 

Pac Islander 

  13 

44.8% 

  15 

51.7% 

    1 

3.4% 

  0 

0% 

    29 

1.7% 

Latino 

122 

43.0% 

144 

50.7% 

  18 

6.3% 

  0 

0% 

  284 

17.0% 

African Am 

  25 

43.9% 

  30 

52.6% 

    2 

3.5% 

  0 

0% 

    57 

3.4% 

White, non‐H 

117 

18.8% 

365 

58.6% 

130 

20.9% 

11 

1.8% 

  623 

37.2% 

Totals 

527 

31.5% 

852 

50.9% 

245 

14.6% 

50 

3.0% 

1674 

100.0% 

 
 

The placement pattern for this cohort of students in ninth grade followed a similar pattern to their 
placement in eighth grade.   While more than 80% of most of the minority groups were placed in 
Algebra in ninth grade, only 31% of the Asian students were so placed.  The proportion of white 
students who moved from Algebra in eighth grade to Geometry in ninth grade was not as large as that 
for Asian students. 
 
Table 8.  Comparison of ethnicity to ninth grade placement in mathematics classes. Ethnicity is self‐reported and is 
taken from the students’ CST results.   
 

 
Total 

Eighth Grade Math Placement (2006‐07 School Year) 
Ethnicity 
Am Indian 
Asian 
Filipino 
Pac Islander 
Latino 
African Am 
White, non‐H 
Totals 

Math/PreAlg 
     N 
  1 
  4 
  0 
  0 
  5 
  0 
  33 
  43 

      % 
8.3% 
1.3% 
0% 
0% 
2.1% 
0% 
5.5% 
2.9% 

Algebra 
 
 
 
 
 
 
 
 

     N 
10 
97 
189 
20 
197 
37 
358 
908 

    % 
83.3% 
30.8% 
70.3% 
90.9% 
81.4% 
82.2% 
60.1% 
60.5% 

Hon Algebra 
 
 
 
 
 
 
 
 

     N 
0 
51 
0 
0 
0 
0 
10 
61 

       % 
0% 
16.2% 
0% 
0% 
0% 
0% 
1.7% 
4.1% 

Geometry 
 
 
 
 
 
 
 
 

  N 
1 
163 
80 
2 
40 
8 
195 
489 

      % 
8.3% 
51.7% 
29.7% 
9.1% 
16.5% 
17.8% 
32.7% 
32.6% 

     N 
 
12 
  315 
  269 
 
22 
  242 
 
45 
  596 
  1501 

        % 
0.8%
21.0%
17.9%
1.5%
16.1%
3.0%
39.7%
100.0%

 
 
 

Finding 7:  Issues Related to Performance of Students who were Placed in Algebra 
Classes in Eighth Grade 
 
Because so many of the students who had been placed in Algebra in eighth grade, remained in Algebra 
in ninth grade, analyses were performed to try to determine variables that may have contributed to this 
static placement. 
 

Frequencies 8th Grade B‐ Grade and above....9th Grade Placement 
 
One focus was on students with grades of ‘B‐’ or better in eighth grade, on the assumption that teachers 
would have considered them to have “mastered” Algebra sufficiently to move on to Geometry in ninth 
 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

9 

 grade.  As can be seen, despite their eighth grade teachers’ determination that the students had earned 
grades of ‘B‐‘ or better, nearly 35% of the 993 students were placed either in Math or Algebra classes in 
ninth grade.  The reasons for this placement can only be speculated upon.   
 
Table 9.  Frequency distribution in ninth grade math classes for students enrolled in Algebra or Honors Algebra in 
eighth grade and who received final grades of “B‐“ or better in eighth grade. 
 

 
Valid 
  
  
  
  
  
Missing 
Total 

Math 
Algebra 
H Algebra 
Geometry 
H Geom 
Total 
System 

Frequency 
9
234
20
372
118
753
180
993

Percent 
1.0
25.1
2.1
39.9
12.6
80.7
19.3
100.0

Valid Percent 
1.2
31.1
2.7
49.4
15.7
100.0
 
 

Cumulative Percent 
1.2
32.3
34.9
84.3
100.0
 
 
 

 
 

The situation is somewhat more dramatic for students who were enrolled in Algebra classes in eighth 
grade.  Table 10, below, illustrates the transition for these students.  This chart shows that 46% of the 
students who were enrolled in Algebra classes in eighth grade repeated Algebra or were dropped to 
Math as ninth graders.  By pulling the students enrolled in Honors Algebra in eighth grade, the numbers 
of students enrolled in Geometry drop considerably.  Clearly, the Honors track of students makes the 
transition to ninth grade courses more successfully than those enrolled in straight Algebra classes in 
eighth grade. 
 
As not all of the sampled school districts offered Honors Algebra, the differences represented in these 
two charts may well under‐represent the differences in expectations for students.  But where Honors 
Algebra is offered, it offers a more predictable path to Geometry and Honors Geometry than enrollment 
in Algebra classes. 
 
Table 10.  Frequency distribution in ninth grade math classes for students enrolled in Algebra in eighth grade and 
who received final grades of “B‐“ or better in eighth grade. 
 

 
Valid 
  
  
  
  
  
Missing 
Total 

Math 
Algebra 
H Algebra 
Geometry 
H Geom 
Total 
System 

Frequency 
9
221
17
261
29
537
137
674

Percent 
1.3
32.8
2.5
38.7
4.3
79.7
20.3
100.0

Valid Percent 
1.7
41.2
3.2
48.6
5.4
100.0
 
 

Cumulative Percent 
1.7
42.8
46.0
94.6
100.0
 
 
 

 
As with placement overall, placement of students of various ethnicities differed for the subset of 
students with grades in eighth grade Algebra among the various ethnicities represented in the study.  As 
can be seen from Table 11 below, there were fewer Asian students placed in ninth grade Algebra or 
Advanced Algebra than other ethnic groups, Filipino students fared worst in the sorting process, and 
 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

10 

 Latino and non‐Latino white students  fared about the same.  
 
Because the different ethnicities were not represented proportionally across the various school districts, 
it is impossible to determine what impact, if any, the relationship between the high school districts and 
their respective feeder elementary districts played on class assignments. 
 
Table 11.  Frequency distribution in ninth grade math classes for students enrolled in Algebra in eighth grade and 
who received final grades of “B‐“ or better in eighth grade by Ethnicity.  (The numbers for African American, Native 
American and Pacific Islander Groups each had fewer than 25 students in this sample and are not included.) 
 

 
Math 
Algebra 
H Algebra 
Geometry 
H Geom 
Total Num 

Asian 
.5% 
14.3% 
8.2% 
60.7% 
16.3% 
196 

Latino 
0% 
32.6% 
0 
67.4% 
0 
92 

Filipino  
1.6% 
59% 
0 
36.1% 
3.3% 
61 

White 
2.6% 
37.3% 
0 
31% 
29.2% 
271 

Total Number 
9 
195 
16 
287 
113 
620  

 
 

Frequencies 8th Grade B‐ Grade and above … 9th Grade Algebra Grades 
 
The study then focused on students who had been successful in eighth grade Algebra and followed them 
through their final grades in ninth grade Algebra to determine whether they maintained their success 
level in ninth grade.  From the table below, one can see that 50.6% of the students who had been 
successful in Algebra as eighth graders but were placed again in Algebra as ninth graders, received the 
same or lower grades the second time around. 
 
Table 12.  Frequency distribution of final grades in Ninth Grade Algebra for students enrolled in Algebra in eighth 
grade and who received final grades of “B‐“ or better in eighth grade. 

 
Grade in Algebra in 
Ninth Grade 
F 
D‐ 
D 
D+ 
C‐ 
C 
C+ 
B‐ 
B 
B+ 
A‐ 
A 
A+ 
Total 

Frequency 
13
5
5
7
18
28
20
15
33
14
17
38
6
219

Percent 
5.9
2.3
2.3
3.2
8.2
12.8
9.1
6.8
15.1
6.4
7.8
17.4
2.7
100.0

Cumulative Percent 
5.9 
8.2 
10.5 
13.7 
21.9 
34.7 
43.8 
50.6 
65.7 
72.1 
79.9 
97.3 
100 
  

 
 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

11 

 CST Performance Level for Eighth Grade Algebra Students, Compared to Ninth Grade 
Placement 
 
The CST Algebra test was created to independently assess the success of California students in Algebra.  
Parents and citizens are told each year that scoring “proficient” or “advanced” on these tests means that 
students have mastered the skills needed to move on to the next level.  California’s standards are 
reputed to be among the highest in the U.S.  While teachers and administrators have long argued that 
these tests represent performance on a single day or in a single week, and that they do not measure all 
of the attributes students possess, those arguments have usually been made to rationalize lower test 
scores.  That is, administrators and teachers rarely have argued to parents that their child’s success on 
this test was only a fluke and that the child really did not master the material.  Thus, one would expect a 
one‐one correspondence between success on the CST and placement in the next level of class – here a 
passage from Algebra to Geometry. 
 
While administered before the school year ends, the CST, or STAR tests arrive back in school districts by 
early August, several weeks before the start of the next school year.  Thus, it is not too late for school 
districts to change the placement of students based on their success on the CST.   
 
It is then surprising that more than 60% of the students who had scored Proficient or Advanced in 
Algebra in Eighth Grade were again placed in algebra in ninth grade. 
 

Table 13.  Comparison of Ninth Grade placement in Math Classes as a Function of CST Performance levels in Eighth 
Grade for Students Taking Algebra as Eighth Graders 
 

 
Far Below 
Basic 
Below 
Basic 
Basic 
Proficient 
Advanced 
Missing 
Total Count 
% of Total 

Count 
% within 
Count 
% within 
Count 
% within 
Count 
% within 
Count 
% within 
Count 
% within 

Math 
1 
2.6% 
8 
6.2% 
16 
8.5% 
8 
3.2% 
0 
0% 
4 
2.4% 
37 
4.2% 

Algebra 
37 
97.4% 
117 
90.7% 
137 
72.9% 
109 
44.0% 
23 
20.5% 
90 
53.6% 
513 
58.1% 

Hon Alg 
0 
0% 
0 
0% 
0 
0% 
0 
0% 
17 
15.2% 
0 
0% 
17 
1.9% 

Geometry 
0 
0% 
4 
3.1% 
35 
18.6% 
124 
50.0% 
50 
44.6% 
74 
44.0% 
287 
32.5% 

Hon Geom 
0 
0% 
0 
0% 
0 
0% 
7 
2.8% 
22 
19.6% 
0 
0% 
29 
3.3% 

Total 
38 
100.0% 
129 
100.0% 
188 
100.0% 
248 
100.0% 
112 
100.0% 
168 
100.0% 
883 
100.0% 

 
Statistical Analyses to Determine the Best Predictors of Ninth Grade Placement for Students 
who had Taken Algebra as Eighth Graders 
 
Finally, the study attempted to determine which of four variables was the best predictor of placement in 
ninth grade.  As this analysis for the entire group of schools yielded significant within‐group variance, 
and as it had been determined that placement decisions seemed to follow a predicted pattern for 
students enrolled either in Math or Honors Algebra in eighth grade, these analyses focused only on 
those students who were placed in Algebra as eighth graders. 
 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

12 

 The analyses below indicate that the sum of the CST scores, grades and MARS results were able to 
explain about 45% of the variance in placement for students from this group who were passing from 
eighth to ninth grade.  This was statistically significant.  Moreover, the eighth grade grades, MARS scores 
and CST Scale scores were each statistically significant predictors of later placement.  MARS raw scores 
explained 35% of the variance.  MARS raw scores and CST scale scores were highly correlated, (.738).   
 
Table 14.  Regression Analysis, Grades, CST scores, MARS scores and CST performance levels as predictors of ninth 
grade placement 
Model Summary 
Adjusted R  Std. Error of 
R 
R Square 
Square 
the Estimate 
.674(a) 
.454 
.452 
.826 
a  Predictors: (Constant), Eighth Grade Class, MARS Raw, MARS Perf, CST Scale, CST Perf 
 
 
ANOVA(b) 
Model 
1 

Sum of 
Squares 
df 
Mean Square 
F 
779.215
5
155.843
228.514
938.412
1376
.682
 
1717.627
1381
 
 
a  Predictors: (Constant), Eighth Grade Class, MARS Raw, MARS Perf, CST Scale, CST Perf 
b  Dependent Variable: Ninth Grade Placement 
 
Coefficients(a) 
 Model 
Regression 
Residual 
Total 

 
 Model 
  

Unstandardized 
Coefficients 

B 
Std. Error 
.278 
.155
.250 
.023
.075 
.008
‐.349 
.069
.003 
.001
.017 
.053
a  Dependent Variable: Ninth Grade Placement 
 
(Constant) 
8th class 
MARS Raw 
MARS Perf 
CST Scale 
CST perf 

Standardized 
Coefficients 

t 

Beta 
 
.259
.605
‐.309
.210
.017

  
1.791
11.068
9.385
‐5.089
3.682
.310

 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

Sig. 
.000(a) 
  
  

Sig. 
  
.074 
.000 
.000 
.000 
.000 
.756 

13 

 Table 15.  Correlations between Eighth grade school performance as indicated by letter grades, MARS Scale Scores, 
MARS Performance, CST Scale Scores and CST performance levels with ninth grade placement. 
 

  
8th Gr. 2006‐07 

MARS Raw 

MARS performance 

CST Scale 

CST performance 

Math Placement 9th 

  
Pearson Correlation 
Sig. (2‐tailed) 
N 
Pearson Correlation 
Sig. (2‐tailed) 
N 
Pearson Correlation 
Sig. (2‐tailed) 
N 
Pearson Correlation 
Sig. (2‐tailed) 
N 
Pearson Correlation 
Sig. (2‐tailed) 
N 
Pearson Correlation 

8th Gr. 
2006‐07 

MARS 
Raw 

MARS 
Perform 

9th 
Placement

CST 
Perform 

CST Scale 

1 
. 
1072 
.442(**) 
.000 
839 
.450(**) 
.000 
839 
.516(**) 
.000 
808 
.442(**) 
.000 
812 
.521(**) 

.442(**) 
.000 
839 
1 
. 
862 
.943(**) 
.000 
862 
.738(**) 
.000 
830 
.642(**) 
.000 
834 
.588(**) 

.450(**) 
.000 
839 
.943(**) 
.000 
862 
1 
. 
862 
.706(**) 
.000 
830 
.621(**) 
.000 
834 
.510(**) 

.516(**) 
.000 
808 
.738(**) 
.000 
830 
.706(**) 
.000 
830 
1 
. 
830 
.933(**) 
.000 
830 
.566(**) 

.442(**) 
.000 
812 
.642(**) 
.000 
834 
.621(**) 
.000 
834 
.933(**) 
.000 
830 
1 
. 
834 
.484(**) 

.521(**)
.000
862
.588(**)
.000
739
.510(**)
.000
739
.566(**)
.000
715
.484(**)
.000
718
1

Sig. (2‐tailed) 

.000 

.000 

.000 

.000 

.000 

.

N 

862 

739 

739 

715 

718 

883

**  Correlation is significant at the 0.01 level (2‐tailed). 
 
 

Correlations for Students taking Algebra in Eighth Grade by Ethnic Group and Ninth Grade Placement 
 
The last table compares letter grades for the students as eighth graders, MARS scores and CST scores to 
ninth grade placement.  One would expect the correlations to be similar for the three independent 
variables.  This was the case for non‐Hispanic white students.  As can be seen, the correlation between 
final grades and ninth grade placement was much stronger for Asian students than for Filipino students, 
African American students, and Latino students.  For these two groups of students, test scores proved 
better correlates with ninth grade placement than teacher‐given grades.  These results are not 
conclusive; however, they deserve further research in that they could relate to teacher expectations and 
how they might differ as teachers work with students of different ethnicities. 
 
Table 16.  For eighth grade Algebra students only, correlations by race between final grade in eighth grade, MARS 
raw score, CST scale score and ninth grade placement. 
 

Ethnicity 
Asian 
Filipino 
Latino 
African Am 
White 

Number of 
Students 
163 
111 
96 
24 
313 

Grade to Placement 
0.504 
0.258 
0.444 
0.543 
0.452 

MARS Raw to 
Placement 
0.535 
0.450 
0.677 
0.606 
0.467 

CST Scale to Placement 
0.444 
0.515 
0.625 
0.640 
0.482 

 
 
 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

14 

 Final Thoughts/Recommendations 
 
This study provides a baseline for determining whether the staff development program supported by 
the Noyce Foundation has resulted in an increase in the access of eighth graders to Algebra content, and 
whether that increase will carry forward to success in terms of the grades the eighth grade math 
teachers give the students, the students’ CST success, and later placement and success in ninth grade.  
As expected, minority students and students coming from homes in which neither parent has graduated 
from college had less access to Algebra in eighth grade than their white and Asian and affluent peers. 
 
On the way to providing the baseline data, the discovery that many students placed in Algebra in eighth 
grade, and who were successful in that placement based on common sense indicators, were not moved 
to Geometry in ninth grade raises a significant educational issue for the elementary and high school 
districts to confront.  It appears that for many students, success on the usual indicators does not lead to 
the opportunity to move on with their math education.  It would appear that earlier efforts by 
elementary and high school teachers to bridge the placement/recommendation gap have failed. 
 
These findings lead to two recommendations to the Noyce Foundation: 
 
Recommendation 1:  Create a Forum for Reducing the Arbitrariness of Placements 
 
 
All of the work done so far by the elementary school districts in the collaborative can 
easily be undone by poorly conceived placement decisions by either or both of the 
eighth‐ or ninth‐grade teachers.  Moreover, there is reason to believe this situation is 
widespread in the state.  The Foundation has the resources to provide a credible forum 
for the discussion of expectations and standards as they are presently conceived by 
teachers at this critical juncture in children’s lives.  The harm possible to the self‐esteem 
of a child caused by being placed again in a class he/she has already passed is significant, 
and can at least partially explain the lag the United States experiences in producing 
graduates of high school and college with math acumen.  The Foundation can expose 
the problem and provide a venue for districts to solve it. 
 
Recommendation 2:  Continued, if Modified, Support to the First in Math Districts  
 
 
It is apparent from both the findings of this report and those of the surveys that 
changing the culture of mathematics thinking in middle schools and high schools is 
extremely difficult.  During the 2008‐09 year, it was clear that some of the districts were 
really moving ahead, others not.  In completing the three‐year cycle of support for those 
districts the Foundation judged to be true leaders, the Foundation would be able to 
establish a credible database to support the efficacy of having districts across the 
country re‐examine their belief systems. 
 
 
 
 
 
The author would like to acknowledge the contributions of the following individuals: 
 
 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

15 

 Aida Gamba for data base work, and editing and formatting this paper 
Sara Spiegal for assembling the data base 
Dr. Ray Gamba and Michale Jang for statistical analysis 
David Foster for advice and guidance 
 
 
 
Steve Waterman 
stevewatermanca@aol.com 
510‐525‐7970 

January 20, 2010 
 

 
Pathway Study by Steve Waterman, January 20, 2010, On Behalf of the Noyce Foundation 

16 

 Scenarios 
 
Principal 
 
After hearing this presentation, you have gone back to your school and checked on 
the recommendations your teachers are making to the students for high school 
placement.  (You now have 60% of your eighth graders taking Algebra. ) You have 
become aware that one of your Algebra teachers has only recommended a third of 
her students to Geometry in the ninth grade.  You have asked her the reason and she 
says, “This class is just a weak one.”  The average for your other Algebra teachers is 
60% being recommended to geometry. 
 
Midterm grades for Algebra are as follows: 
 
Teacher a – 70% ‘B’ or better 
Teacher b – 60% ‘B’ or better 
Teacher c – 40% ‘B’ or better 
 
You check with your English teachers and find out what they are doing.  They are 
recommending 25% to advanced English. 70% to college bound ninth grade English 
and 5% to remedial ninth grade English. 
 
What do you think is the right proportion of Algebra students to be recommended to 
geometry? 
 
What might you do between now and June to get to that percentage? 
 
 
Coach 
 
Because of your influence, all of the students in eighth grade are now taking Algebra.  
The eighth grade teachers are angry and frustrated.  They are giving failing grades 
to more than half of the students.  You and a few of your math teachers have begun 
meeting with the high school math teachers to smooth the transition for students.  
The high school teachers are loudly telling your teachers that they are 
recommending too many students to geometry and the students are not doing well.  
In fact, the teachers are saying that the Algebra ninth graders are not prepared for 
Algebra.  This is making your eighth grade math teachers feel they have made a big 
mistake in changing to an all‐Algebra eighth grade. 
 
What do you do? 
 
 
 

Scenarios 

 

1 

 Assistant Superintendent, Instruction 
 
Last year, having heard of the research conducted by the Noyce Foundation, you 
talked the superintendent into increasing the number of students in Algebra classes 
at the eighth grade.  Recently, parents have been trooping to your door to express 
their “concerns” that the addition of unprepared students in Algebra was causing 
teachers to water down the curriculums.  They fear this is holding their Stanford‐
bound children back.  They feel you should keep your liberal experiments for other 
people’s kids. 
 
You suspect that teachers and/or principals are propelling the parents with 
comments like – “There was nothing I could do, the central office told me to increase 
the number of sections,” – the principal, and “I realize we are going slowly.  I have to 
aim my instruction to the middle of the class, and now the middle of the class is 
lower,” – a teacher’s comment at the Back to School night. 
 
What do you do? 
 
 
Principal 
 
After leaving this meeting you requested an ethnic breakdown of enrollment in the 
eighth grade Algebra and Advanced Algebra classes.  You discovered a trend similar 
to that in the report given you at this meeting. 
 
You share the data with your teachers.  They reply, “These kids are simply less 
prepared than the Asian and white kids.  We are taking kids as they are, not as we 
might wish them to be.  We can promote more to Algebra next year, but either we 
will end up failing them or inflating grades for everyone.  What do you want from 
us? We teach over one hundred kids a day.  There is simply not enough time to bring 
these kids up to speed without holding back others.” 
 
What do you do? 
 
 
Coach 
  
The teachers in the middle school mathematics department have been reluctant to 
increase the number of students in Algebra 1.  As more students are placed in 
Algebra 1, a vocal few continue to complain about how they have to water down the 
curriculum and cannot get through the topics.  Their beliefs are reinforced by a 
couple of high school teachers, who claim that the high school teaches a more 
rigorous Algebra 1 program.  These middle school and high school teachers are 
arguing for a two‐year algebra 1 course sequence.  You are worried that the data in 
this report will be cherry‐picked to support their argument against allowing these 
students to take Algebra 1. 
Scenarios 

 

2 

   
How might you address this issue? 
 
 
Parent 
 
Your daughter was enrolled in Algebra in eighth grade, and struggled through it 
during the early part of the year, but seemed to gain confidence as the year ended.  
While her midterm grade was a ‘C’, her second semester grade was an ‘A‐‘.   You 
didn’t give placement much though during the summer, but were pleased when you 
say that she scored, “proficient” on the CAT.  As with most teens, you have to pry 
information about school from your daughter, so didn’t learn until the third week of 
school that she was again placed in Algebra this year.   
 
You have gone to the new teacher, but she has told you that it is too late to switch 
classes this semester.  The teacher goes on with comments about girls and math and 
that she thinks your daughter will gain by repeating the course because you she will 
be covering topics in more depth.   
 
You leave the meeting shaking, but have held your tongue because you fear 
retaliation on the part of the teacher.   
 
What steps can you take to help your daughter? 
 
What steps can you take to try to see that other kids don’t suffer the same fate? 

Scenarios 

 

3