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TEACHEREFFICACYINSECONDARYMATHEMATICS 1
TeacherEfficacyinSecondaryMathematics:FosteringConfidenceandFluency
KellyLynWilson
HighTechHighGraduateSchoolofEducation
AuthorNote
KellyLynWilsonisnowtheDirectorofInnovativeandEntrepreneurialProgramsatSevern
HighSchoolinAnnapolis,Maryland.Thisresearchwassupportedinpartbyafellowshipfrom
theWaltonFoundation.CorrespondenceconcerningthispapershouldbeaddressedtoKellyLyn
Wilson,SevernSchool,201WaterStreet,SevernaPark,MD,21146.
Contact:[email protected]
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TEACHEREFFICACYINSECONDARYMATHEMATICS 2
Abstract
This research focused on understanding what factors affected the
perception of efficacy in the
teaching and learning of mathematics in several progressive
secondary schools. Efficacy is the
belief in ones ability to produce the desired or intended
results. For teachers, this is the belief
the practices and structures they use and work in contribute to
student success. For students, this
is the belief they can use mathematics and are prepared for
college level work. Through surveys,
interviews, focus groups, observations and test score analyses,
several themes emerged that
influenced teachers and students sense of efficacy including
unclear expectations or vision of
the mathematics program. For teachers this also included the
need for more effective strategies
for reaching all learners in a classroom. This research
highlights the importance of defining an
institutions goals for secondary mathematics, and aligning
teacher preparedness and support
aroundthosegoals.
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TEACHEREFFICACYINSECONDARYMATHEMATICS 3
TeacherEfficacyinSecondaryMathematics:FosteringConfidenceandFluency
In these days of better, faster, more, it is all about the
numbers. Where are we ranked as a
nation in mathematics prowess in regards to other economically
developed countries? How many
students are preparing to study science and other technical
fields? What measures are being taken
to focus teachers on these goals? The setting of this research
was in progressive secondary
mathematics classrooms focused on projectbased learning with an
emphasis on openended
problems. The hypothesis was there would be higher than average
state test scores because the
methods used in these schools were focused on deeper learning.
Deeper learning, as defined by
the Hewlett Foundation (2013), states student learning should
contain the following aspects:
mastering content, thinking critically, collaborating,
communicating effectively, and developing
a growth mindset. Hence the anticipated results of using deeper
learning techniques would
equate with a deeper grasp of the concepts and procedures, which
would translate into higher test
scores. However, Figure 1 shows the percentage of students and
their proficiency in Algebra II
topics on the 2013 California STAR test (California Department
of Education). The comparison
between the setting school and the state averages was concerning
due to the high percentage
(74%) of students falling into the Below Basic and Far Below
Basic categories. Though the
setting school follows an integrated math approach, students in
traditional or integrated series
shouldbereachingequivalencyinknowledgeuponthecompletionofMath3orAlgebraII.
Other areas of concern existed
about mathematics instruction at the
setting. There were parent meetings with
the setting school director which focused
on the mathematics program and its ability
to prepare students for both college
entrance exams and college mathematics
courses. Students wishing to pursue a
higher level of mathematics were
supplementing their school learning with
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TEACHEREFFICACYINSECONDARYMATHEMATICS 4
community college classes. Additionally, initial conversations
with several math instructors
revealed a lack of understanding of the vision of the
mathematics program and feeling
unprepared to provide curriculum to the wide breadth of learners
in their classroom. The above
factors led to an inquiry process surrounding the teaching and
learning of mathematics in the
settingofaprogressivesecondaryschool.
Many questions surround the challenges and goals of learning
mathematics. There are
pedagogical questions surrounding teacher and classroom
practices. Do students need to be
drilled in the basics before being able to apply them to higher
level concepts? How can students
discover mathematical formulas without knowing the language of
mathematics? Mathematical
fluency, or the state of being able to understand and transfer
knowledge, may also be impacted
by institutional practices or the ways in which schools define
structures like student and teacher
schedules, inclusion decisions, daily schedules and other
factors. It is also important to identify
and understand the goals and design principles of a school in
relation to the structures it has
developed.
In an attempt to influence the content of what high school
graduates should know, state
leaders in government and education united to create standards
or a listing of academic goals for
students. However, these standards do not address the
institutional structures or pedagogy which
should be in place to ensure their successful attainment. These
new standards are also linked to
student outcomes (yes, back to numbers) in the form of new
standardized tests created by the
Smarter Balanced Assessment Consortium (SBAC) and Partnership
for Assessment of Readiness
for College and Careers (PARCC), along with upcoming changes to
the current college entrance
exams created by the College Board (SAT) and ACT organizations.
Whether individual teachers
or schools view these assessments as a valid predictor of
college success or preparedness, they
both will be judged by student performance on these exams from
rating boards,
colleges/universities and probably their harshest critics,
parents. This paper will not discuss the
merits or inferiority of the standards and related tests, but it
does hold the belief learning
outcomesneedtobemeasurableandattainable.
This research reviewed current theories in the pedagogy and the
learning of mathematical
knowledge. From there it examined teacher preparedness and
analyzed reasons why teachers
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TEACHEREFFICACYINSECONDARYMATHEMATICS 5
may have a gap in their sense of efficacy in the classroom. As
per long term selfefficacy
researcher Albert Bandura, Peoples beliefs about their
capabilities affect what they choose to
do, how much effort they mobilize, how long they will persevere
in the face of difficulties
(Bandura, 1994, p.1). The research investigated the practices
and data of a progressive,
constructivist school who incorporate a projectbased learning
(PBL) pedagogy. The goal was to
determine whether the practices being employed contributed to a
sense of efficacy in teachers
based on the current strategies provided by past and present
experts. Examples of effective
practices and recommendations for possible areas of improvement
were provided with the intent
of boosting the numbers: the numbers of students prepared for
college and careers, along with
thenumberteacherswhofeelpreparedtoteachthem.
Research Question: What practices and school structures create a
sense of efficacy in secondary
mathematicsteachersinordertodevelopmathematicalfluencyinstudents?
LiteratureReview
There is a call to arms in this nation surrounding the scores of
secondary students in
mathematics. According to the Programme for International
Student Assessments (PISA) latest
test results in 2012, the United States is 27th among the 34
OECD (Organization for Economic
Cooperation and Development) countries, and performed below
average in mathematics
(Gurria, 2014, p.1). President Obama stated in his address at
the Third Annual White House
Science Fair he is focused on creating an allhandsondeck
approach in areas like math and
we need to make this a priority to train an army of new teachers
in these subject areas (Educate
to Innovate, 2013). However, before we begin to analyze the
statistics and train teachers in the
latest methods to achieve student and national success, let us
first define how and what students
need to learn in order to know mathematics, or be mathematically
fluent. This will then be
followedbythoughtsonteacherpreparednesstofacilitatesuchlearning.
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TEACHEREFFICACYINSECONDARYMATHEMATICS 6
DefinitionofMathematicalFluency
Stanford University professor George Polya defines mathematical
knowledge as a
combination of demonstrative reasoning and plausible reasoning.
Demonstrative reasoning
is safe, beyond controversy, and final or a mastery of skills
plausible reasoning is hazardous,
controversial, and provisional (Polya, 1954, p. vvi). Plausible
reasoning is how math connects
to disciplines like business, science, etc. or making
connections to real world experiences. The
formulas and processes which students use to present their
solutions to an algebra problem are
demonstrative in nature but in order to know what theorems or
steps to take was the result of
plausible reasoning, or guessing. Polya further goes on to
reflect, a serious student of
mathematics...must learn demonstrative reasoning yet for real
success, he must also learn
plausible reasoning this is the kind of reasoning on which his
creative work will depend (1954,
p.vi).
Though Polyas thoughts were written in the 1950s, his research
is echoed by leading
mathematics education researchers today (there is still a course
Math 193: Polya Problem
Solving Seminar at Stanford). Guershon Harel, mathematics
professor and researcher from the
University of California San Diego (UCSD), has respectively,
very similar definitions of
demonstrative and plausible reasoning but uses the terms Ways of
Understanding (WoU) and
Ways of Thinking (WoT) (2008, p.8). He defines knowledge of
mathematics as a union of
thesetwosets.
Harel has also written many papers regarding the pedagogy of
mathematics including his
theory of DNR or DNRbased instruction for mathematics. The D, N,
and R respectively stand
for duality, necessity, and repeated reasoning (2008, p. 3).
Duality posits students only develop
ways of thinking when constructing ways of understanding, and
these ways of understanding are
determined by the ways of thinking they possess. In other words,
students gain insights into the
purpose of mathematics by investigating the formulas and
procedures of mathematics. The
necessity principle refers to the idea students must have an
intellectual need to learn. The last
principle, repeated reasoning, states students need to have
repeated experience or practice to
gather and retain the ways of understanding and thinking.
(Harel, 2008, pp.1921). Again, Harel
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TEACHEREFFICACYINSECONDARYMATHEMATICS 7
contends providing intellectual need, or utilizing humans
remarkable capacity to be puzzled
(2008a,p.488)mustbeofutmostimportance.
These researchers and many others have outlined the components
needed for deeper
learning in mathematics as ways of doing and knowing
mathematics, combined with an
intellectual need or purpose for the math being studied. Again,
a deeper learning emphasis is not
solely on mastering the content, but also on gaining the skills
needed to be able to use it, share it
and extrapolate the learning to new situations. Now we have a
basic grasp of how students
shouldlearn,theconversationmovesontowhattheyshouldlearn.
WhatistheImpactofStandards
In 2009, state leaders in government and education came together
to promote the
development of standards to ensure all students, regardless of
where they live, are graduating
high school prepared for college, career, and life (Common
Core). These standards are
promoted as providing demonstrative and plausible reasoning, or
the combination of acquiring
skills with realworld connections. The Common Core State
Standards (CCSS) were developed
by consulting leading experts, teachers and other standard
communities, such as the National
Council for Teachers of Mathematics (NCTM). In August of 2010,
the state of California
adoptedthesestandardsforallpubliceducationinstitutionsingradesK12.
However, there is swirling controversy surrounding these
standards and their potential
effects on learning mathematics. Opponents of the CCSS object
the standards are an imposition
of federal rights over a states rights and lack of evidence the
standards will meet the desired
goals of improvement (McDonnell & Weatherford, 2013, p.494).
Outcomes of the effects of the
CCSS are currently unknown as inaugural testing is commencing in
the 20142015 school year.
Teacher support for the initiative is also waning due to
perceived ties to teacher evaluation
systems and restriction of their freedom in the classroom. A
report from U.S. News & World
Reportprovidedsomestatisticsregardingteachersupport:
SupportershavetoutedasurveyconductedbyEducationNext,aneducationjournal,thatlastyearfound76percentofteachersweresupportiveofthestandards.Butinits2014poll,EducationNextfoundoppositionhadmorethantripled,from12percentin2013to40percentin2014.Now,just46percentofteacherssaytheysupportthestandards.(Bidwell,2014)
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TEACHEREFFICACYINSECONDARYMATHEMATICS 8
Progressive school educators may translate the idea of following
standards as an
encroachment to their ability to define curriculum. In attending
a recent conference with one of
the authors of the CCSS for Mathematics, Phil Daro stated the
standards were written as a guide
to what mathematical knowledge students should be able to
perform, demonstrate an
understanding of and transform. However, Daro clearly indicated
they were not a guide for
howtoteachtheseskillsandpracticesofthemind(MFASD,2014).
Other areas of concern exist surrounding the CCSS and students
with learning
disabilities, especially those diagnosed with mathematics
learning difficulties (MD). Powell et al.
researched students with MD and cites the research over the last
thirty years has indicated that
students with MD require explicit, systematic instruction (2013,
p.41). Explicit instruction
generally involves teacher demonstration of detailed stepbystep
instructions along with
independent practice. In further addressing the needs of MD
students, concern exists surrounding
the assessment programs which have been developed to coincide
with the CCSS, like the SBAC
and PARCC. Powell et al. concluded, schools may find it
necessary to use tracks or for
students with MD, the supplementary instruction required in RTI
(Response to Intervention) to
prepare students for Common Core assessments (2013, p. 46). The
idea of tracking is in
contradiction to some of the latest theories in the epistemology
of learning though this topic is
beyondthescopeofthisresearch.
StandardsandAssessments
Regardless of the controversy surrounding the standards, they
are the expected learning
outcomes for US students in fortyfour states. (Six states have
either not adopted or withdrawn
support of the standards, Academic Benchmarks). The standards
represent a shift from a more
traditional view of education as teachercentered delivery of
instruction to a more progressive
studentcentered approach to education. The SBAC and PARCC have
been developed as
comprehensive, technologybased assessment systems to measure
students' attainment of the
CCSS. Testing will commence in the 20142015 school year. The
National Center for Research
for Evaluation, Standards and Student Testing (CRESST) based at
UCLA asserts these
assessments, are likely to represent goals for deeper learning,
particularly those related to
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TEACHEREFFICACYINSECONDARYMATHEMATICS 9
mastering and being able to apply core academic content and
cognitive strategies for complex
thinking,communication,andproblemsolving(Herman&Linn,2013,p.4).
It is the contention of this research the CCSS and its
associated testing measures are
shifting towards a focus in deeper learning. Proceeding from
this basic understanding of how
students should learn and what they will be learning, what are
the current theories of how
educatorsshouldbefacilitatinglearning?
CurrentResearchonMathematicsPedagogy
Referring back to researcher Harel, he questioned whether
guidelines for instructors
should have been written alongside the standards to assist
educators with the transition (MFASD,
2014). Harel suggested educators are currently more focused on
ways of understanding and have
lacked providing means for obtaining ways of thinking, ...
without targeting ways of thinking,
students are unlikely to become independent thinkers when doing
mathematics (2008, p.13).
Progressive schools are leading the shift in educating students
in the ways of thinking about
mathematics but there may be some confusion or angst in
educators about the balance between
the more traditional methods of tell, show, practice and the
deeper learning model of
discover, explore, use. This unrest may also be heightened due
to a lack of professional
development in shifting to the CCSS. Educators are unclear about
where to focus their
instructional efforts, and many school leaders are overwhelmed
by trying to lead multiple, major
reform efforts and uncertain about where to direct professional
development (ASCD, 2012, p.
12).
During research to create an assessment of mathematics teachers
pedagogical content
knowledge, Hauk et al. (2010) defined the four components of a
professional understanding of a
discipline as having content, discourse, anticipatory and action
knowledge. Content knowledge is
the knowledge of topics, procedures and concepts and
substantiates the idea of teachers
possessing demonstrative reasoning (Hauk et al., 2010 Polya,
1954). Possessing discourse
knowledge allows one to inquire and communicate in mathematics.
The researchers defined
anticipatory knowledge as an awareness of, and responsiveness
to, the diverse ways in which
learners may engage with content, processes, and concepts (Hauk
et al., 2010, p.3). Action
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TEACHEREFFICACYINSECONDARYMATHEMATICS 10
knowledge is the ability to differentiate instruction based on
students needs and the ability to
enact the previous three components while teaching. It appears
whether one is a teacher or a
student, the possession of plausible reasoning or ways of
thinking about mathematics (Polya,
1954, Harel, 2008) is essential. An interesting result of their
research was that professional
development (their subjects participated in 80100 hours of PD
over the course of a year)
provided a significant improvement in knowledge, particularly
discourse knowledge (Hauk et al.,
2010, p. 14). So given this model for what teachers should know
and be able to do, what
practiceshelpfosterasenseofbeingabletoenactthemodel?
TeacherEfficacy
Anita Woolfolk Hoy, a preeminent researcher in teacher efficacy,
stated, Teachers who
set high goals, who persist, who try another strategy when one
approach is found wantingin
other words, teachers who have a high sense of efficacy and act
on itare more likely to have
students who learn (Shaughnessy, 2004). A teachers sense of
efficacy, or the perception of
having an effect on student learning, has been researched for
the last forty years. Hoy, and fellow
researcher Rhonda Spero, also suggests some of the most powerful
influences on the
development of teacher efficacy are mastery experiences during
student teaching and the
induction year (2005, p. 343). Thus, the first years of teaching
could be critical to the
longterm development of teacher efficacy. Not only could they be
critical, but it could be their
last formative experience unless meaningful professional
development is provided. Dylan
William, Emeritus Professor of Educational Assessment at the
University of Londons Institute
of Education, discusses this phenomenon, People make claims
about having 20 years
experience,buttheyreallyjusthaveoneyearsexperiencerepeated20times(Leslie,2015).
Teacher and education researcher Doug Lemov wrote a book, Teach
Like a Champion,
about effective teaching techniques. He spent years observing
teachers and capturing their best
moves. Lemovs claim is good teaching doesnt just happen, it
needs to be coached and
practiced. He equates it with witnessing seemingly effortless
excellence in a sport, but the made
free throw or golf shot is in fact the product of countless
hours of practice and analysis (Lesle,
2015). While some teachers may be natural educators, the
majority of us need to work at it.
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TEACHEREFFICACYINSECONDARYMATHEMATICS 11
Figure 7 shows the top experience of novice teachers for
developing their effectiveness as a
teacher was having access to a mentor. This study comes from the
National Network of State
Teachers of the Year (NNSTOY) and the
Center on Great Teachers and Leaders
(GTL Center) at American Institutes for
Research, which conducted surveys on
exemplary teachers on increasing teacher
effectiveness across their careers
(BehrstockSherrattetal.,2014).
The implications of teacher efficacy go
beyond individual student concerns. Eric Hanushek, a Stanford
educator and member of the
Hoover Institute, along with fellow researcher Steven Riskin,
have reviewed and conducted
studies on the impact of teacher effectiveness on the economy
and other policy matters. While
the statistics and calculations are above the scope of this
paper, their findings clearly state
teacher effectiveness has an impact on individual students
future earnings and cumulatively the
effect of replacing only 5% to 8% of the U.S.s most ineffective
teachers could quadruple our
gross domestic product (Hanushek & Rivkin, 2012). However,
they also concluded determining
thecharacteristicsofeffectiveteachersisanareaforcontinuedstudy.
EfficacyandInclusion
Teacher efficacy can also be impacted by the wide range of
student learning styles and
needs, including those of special education needs (SEN)
students. Special education needs (SEN)
students are generally supported jointly by teachers and other
SEN or inclusion personnel.
However, support of SEN students is primarily attributed to the
classroom teacher. Research
conducted on teachers attitudes towards inclusion, perceived
adequacy of support, and the
classroom learning environment found, Teachers attitudes towards
inclusion increased with
greaterperceivedadequacyofinternalandexternalsupport(Monsenetal.,2013,p.1).
Researchers Hobbs & Westling wrote about their experiences
in conducting a course
instructing both general and special educators in best practices
in inclusive education. One of the
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TEACHEREFFICACYINSECONDARYMATHEMATICS 12
main components of their course focused on building an emphasis
on cooperative learning and
team decision making (2002, p. 188). This idea of collaborative
effort between general and
special educators was found to be imperative for successful
inclusion to incur as found by
researchers Broderick and Vakil et al. as reported by Monsen et
al. (2013, p. 124). Hobbs &
Westling also discuss the need for general and special educators
to be trained as partners and
collaborators as a cooperative venture in the education of SEN
students (2002, p. 188). One of
the strategies Hobbs & Westling used to improve
collaboration was the use of in vivo or
reallife cases both types of educators brought to the class to
investigate. The participants of the
class stated the cases were an irreplaceable component to the
class (Hobbs & Westling, 2002,
p. 192). This area of learning for teachers around the needs of
SEN students is another
componenttoconsiderwhenevaluatingateacherssenseofindividualefficacy.
BacktoDeeperLearning
From early progressive educational theorist John Dewey to
current researchers Guershon
Harel and others, the answer to how deeper learning occurs has
not changed. Students and
teachers need to be engaged in lessons and assessments which
challenge them to such activities
as thinking critically, justifying their reasoning, and
communicating their findings. Deeper
learning organizations have shown improvement in students
learning of mathematics (at least
according to standardized tests). One of these organizations is
the Silicon Valley Mathematics
Initiative (SVMI). SVMIs work is in providing professional
development, establishing
contentfocused coaching in schools, and collaboratively
examining student work to inform
teachers of pupils understandings to foster teacher efficacy and
students deeper learning of
mathematics. After their first decade of work with teachers they
found, ...when teachers teach to
the big ideas, participate in ongoing contentbased professional
development, receive support in
the classroom from welltrained coaches, and use specific
assessment information to inform
instruction, their students will learn and achieve more (Foster
& Noyce, 2004, p. 11). The
questions remain as to why there is an ongoing necessity of
instructional coaches for
mathematics teachers? As the literature and my research has
uncovered, the following factors
havebeenshowntoaffecttheteachingandlearningofmathematics:
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TEACHEREFFICACYINSECONDARYMATHEMATICS 13
Degrees earned or the years of teaching experience do not
necessarily matter
whenformingteacherefficacy.
Teacherefficacydoeshaveanimpactonstudentlearning.
Teacher efficacy is best fostered by employing a community
approach in both
defining common mathematical practices, the use of mentors
and/or other
instructionalcoaches,andpeerlearning.
It is also important to consider the impact the school leader
has on teacher efficacy. Teachers
level of confidence about their ability to promote learning can
depend on past experiences or on
the school culture. Principals can help develop a sense of
efficacy for individual teachers and for
the entire school (Protheroe, 2008, p. 42). School leaders can
foster efficacy by providing
professional development, time for preparation, supporting
teachers in difficult student/parent
situationsandvaluingthemascurriculumdesigners.
Based on the review of literature spanning almost a century it
is the claim of this paper
for students to gain mathematical fluency they need to develop
both habits of thinking and
understanding. Students also need to practice those habits
immersed in environments which
provide realworld context and a need to learn. Teachers need a
sense of efficacy to be able to
foster this type of learning and facilitate the desired results
of students, parents, universities, and
futureemployers,regardlessofthesubjectarea.
Setting
The research in this paper was conducted at a group of
progressive charter secondary
schools in southern California utilizing a projectbased learning
approach. The schools are
grounded in the philosophies students learn deeply by being in a
fully inclusive environment and
by participating in authentic realworld experiences. Students
are not tracked by perceived ability
and teachers are respected as the designers of their curriculum.
Admission into the school is via a
lottery system based on zip codes in an attempt to model the
surrounding demographics and to
ensureequityofaccess.
The research was conducted around mathematics classrooms across
four of the five
secondary schools. There are no honors distinction of classes in
the 9th and 10th grade years.
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TEACHEREFFICACYINSECONDARYMATHEMATICS 14
There are student selfselected honors options in the 11th and
12th grade levels, but the honors
and nonhonors classes are still contained in one classroom. As
the schools are a part of
Californias public school network, they have adopted the CCSS of
Mathematics (CCSSM) as
the framework for their math content and skills requirements.
Teaching practices were observed
and catalogued along the spectrum from traditionalbased or
didactic instruction to openended,
experientialmethodologies.
Teachers are required to be state credentialed or be enrolled in
a valid credentialing
program with credential attainment within two years of
employment. Teacher backgrounds range
from those with degrees in education to experts or doctorates in
specific fields of business or
study. New teachers to the charter, regardless of their previous
experience, are required to attend
a ten day training program prior to the school year. All
teachers attend a weeklong preservice at
their given school campuses. Professional development occurs
throughout the year, primarily
three mornings a week, during the time period from 7:30am to
8:15am (though this and the types
ofmeetingsoractivitiesvarybycampus).
Specific demographics of the student
community consist of a student population
who are 37.6% Caucasian, 34.1% Latino,
13.7% Asian, 9.5% African American, 3%
American Indian, and 1.6% Pacific
Islanders(seeFigure2).
Figure 3 presents the percentage of the
student population who has special educational
needs (SEN) as 12.8%, free or reduced lunch
(FRL) as 38.2%, and only 3.8 % are designated as
Englishlanguagelearners(ELL).
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TEACHEREFFICACYINSECONDARYMATHEMATICS 15
MethodsMy original research question was to determine what
school structures and teaching
practices led students to obtain mathematical fluency, or
learning the language of math and to
use it effectively and transformatively. This question was
formed through preliminary
conversations with parents, teachers, students and looking at
numerical data, such as SAT and
state test scores. There seemed to be questions regarding the
college preparedness of some
students, specifically in mathematics. After initial rounds of
data gathering, the research question
evolved to one focused on school practices and structures which
promote a sense of teacher
efficacy in the teaching of secondary mathematics. Efficacy in
teaching is the ability of teachers
to produce the desired results they wish to see in their
students. Practices and structures are the
methods schools use to develop and support their teachers
examples include professional
developmentopportunities,instructionalcoaching,dailyschedules,andprepperiods,etc.
Data collection tools included: surveys, interviews, focus
groups, exit cards, journal
entries/field notes and math discipline meeting notes. Though
some quantitative data was
extracted from the teacher survey results, the remainder was
gathered by analyzing the schools
student survey data (YouthTruth), and standardized test scores,
including both state and college
entranceexams.
In order to provide context and uncover areas of strengths and
challenges, I surveyed
secondary math teachers across the organization. The questions
began with information on
professional background/history and current teaching
assignments. There were perception
questions on various mathematical teaching beliefs and practices
via a scaled ranking system.
Finally, there were openended questions ranging from the
definition of math projects to how an
allinclusivemodelaffectstheirteaching(seeAppendixA).
Next, I interviewed school directors (principals). The
interviews gathered background
and context information as to what roles and how long they had
been associated with the
community. From there, the interview transitioned to how and why
they created the structures
and supports for the mathematics program at their site. I used a
semistructured interview process
with the school directors to understand their vision and allow
for unknown developments to
evolve (Appendix B). The goal was to understand what practices
school directors employed with
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TEACHEREFFICACYINSECONDARYMATHEMATICS 16
the aim of being able to corroborate certain areas where
teachers were feeling supported and/or
areastofurtherexamineconcerns.
Meanwhile, I sent college advisors a short survey (Appendix C)
to gain their perceptions
of college preparedness, successes or challenges in the
application process, and the rigor
regarding course offerings, all in respect to mathematics. As
college advisors, they are aware of
challenges surrounding students in applying to and choosing
colleges. The advisors also work
with students and colleges through the Early Assessment Programs
(EAP) which determines
college placement for many students. My focus was on college
mathematics course placement
and percentages of students requiring remedial courses. As
teacher efficacy and student success
has been found to have a correlation (Hoy & Spero, 2005
Bandura, 1997), this data helps
providecontexttotheproblem.
The remainder of my qualitative data collection was focused on
the work done with a
math discipline (department) group. Data gathering techniques
included: meeting notes, journal
entries, exit cards and individual interviews of team members.
The interviews focused on their
background and preparation for teaching, challenges and rewards
surrounding their current work,
andidentifiedneedsforsupport/growthasaneducator.
The quantitative data collected included an analysis of
YouthTruth (student survey data)
to analyze student perception of the mathematics instruction and
structures, and their feelings of
college preparedness. Test scores, with a focus on the college
entrance exams of SAT and ACT,
were gathered and compared to state and national averages. The
goal of gathering and analyzing
all of this data is to provide school communities with a picture
of what structures and practices
helpfacilitateasenseofefficacyforsecondarymathematicsteachers.
DataCollection
Surveys(December,January)
The surveys provided a means for capturing teacher and college
advisor perceptions of
the work surrounding the instruction and preparedness of
students in mathematics. They
provided both qualitative and quantitative data with the hope of
developing key concepts and
themes around the highlights and areas for improvement. The
respective surveys were sent to
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 17
thirty teachers and five college advisors across five campuses.
The data for perception questions
was quantified in tables and graphs to determine trends. The
openended questions were coded to
capturekeywordsandtrendssupplementingbaselineinterviewquestions.
YouthTruth, a service which collects student perceptions on
schools and their learning,
surveyed the students of the setting schools. The results were
analyzed for trends in student
satisfactionsurroundingthemathematicsprogramandtheirperceptionsofteacherefficacy.
Interviews(January,March)
The interviews provided more in depth reporting and analysis of
director, instructional
coach and teacher thoughts regarding mathematics instruction.
Four director interviews were
focused on the reasoning for certain school structures, like
daily schedule, course definition and
support personnel. Two instructional coach interviews were
focused on what practices they felt
supported teachers and areas for continued teacher/instructional
growth. Five individual teacher
interviews were focused on their preparation for teaching
mathematics, perceptions and
reasoning for issues surrounding mathematics instruction and
student learning, and areas for
support. Partial transcripts of the interviews were coded and
triangulated with results from the
surveysandtheorytocreatefindings.
InquiryJournal,FieldNotes(OngoingDecemberMarch)
These notes were a compilation of my observations and
participation in meetings,
conversations and teacher and student interactions. They
provided insights into the methods
employed and other perceptions of the work of teaching and
learning mathematics. These notes
werereviewedandcodedwithkeywordstosupplementearlierresearch.
MathDisciplineMeetingNotes(BiWeeklyDecemberMarch)
These notes were a collection of participant activities and
discussions surrounding the
investigation and defining of mathematical practices at one of
the school sites. Teachers worked
together to brainstorm, rank and further define and reflect on
these practices. These notes were
analyzed for trends and real life examples of teachers creating
both individual and collective
sensesofefficacy.
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 18
Timeline
November2014 IRBsubmission
December2014 IRBapproval
Teachersurveysent(withonlineconsentquestion)
Ongoingjournal/fieldnotes
January2015
Mathdisciplinemeetingnotescommence(withparticipantsprovidingconsentviaonlineform)
Collegeadvisorsurveysent(withonlineconsentquestion)
Directorinterviews(withsignedconsentforms)
Instructionalcoachinterview(withsignedconsentforms)
Ongoingjournal/fieldnotes
February2015 Ongoingmathdisciplinemeetingnotes
Exitcardfollowingmathdisciplinemeeting
Ongoingjournal/fieldnotes
March&April2015 Ongoingmathdisciplinemeetingnotes
Ongoingjournal/fieldnotes GatheredtestscoreandYouthTruthdata
Individualteacherinterviews
Findings
Over the course of six months, I observed, participated, and
recorded events in math
classrooms and schools across the setting. I began my research
by surveying math teachers to
gauge their background and experiences with progressive math
education in an inclusive setting.
All students in a given grade level are placed into the math
course for their grade level with the
vision the teacher will provide access and challenge to all
students in the room. Only secondary
math instructors were polled with a response rate of 33% of the
population across five sites. In
combination with this activity, results from a student
perception survey, YouthTruth, was
analyzed to help determine students view of the mathematics
program. The results used were
from one of the five secondary schools with 87% of the students
responding. These results then
assisted in the collecting of information from school directors,
teachers and instructional coaches
viainterviews.
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 19
VisionandEfficacyTeachers
In trying to understand the mindsets of teachers, questions
surrounding the purpose of
mathematics were posed. As per the indicated research (Harel,
2008 Polya, 1954), teachers
understood the importance of thinking critically and providing a
foundation of math skills.
However the teacher responses to the purpose of these skills
varied with eight of the ten
responses focused more on college and exam preparation,
...because of the traditional exams
and courses in their near future...it is to prepare them for
those things. However, a few
responsesfocusedmoreontheexplorationofmathasdemonstratedbythisresponse:
I think math education should be allowing students to find their
identity, empowering them, allowing them to be autonomous with each
other, free from some (mathematical) authority through an inquiry
based system where they are collaborating, critically thinking and
making sense out of math (reality) with each other. I also feel
like they can meet standards and
developprocessesthroughthissystem.While teacher passion and
commitment to their students was evident in their responses,
there was confusion, and sometimes disagreement, as to what the
purpose of mathematics
education is across the respondents. This sentiment was echoed
in interview responses gathered
from a teacher support specialist or instructional coach at one
of the school sites. When asked
what the coach interpreted as challenges math teachers faced,
one response focused on the lack
of vision or clarity. Questions such as What are the
expectations of us as math educators in a
PBL school? and How does it look? were two themes which emerged.
This was echoed by
another instructional coachs
reflection on the question of vision.
People in math education are having
trouble progressing as quickly as the
other disciplines are for some reason,
because we all learned math this
way, and we like math this way, and
math worked for us this waybut
people arent walking around loving
math.
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 20
In correlation to the vision and purpose of mathematics
questions, the idea of efficacy, or
the belief one is effective, is a factor in the success of
teaching and learning in any subject area
(Hoy & Spero, 2005 Shaughnessy, 2004). One of the survey
questions posed to teachers was
whether they believed the schools structure for math was
adequately challenging all math
students. Figure 4 shows the results 72.7% of the respondents
either disagreed or were neutral
withthestatementindicatingahighlevelofdissatisfactionwiththecurrentstructure.
VisionandEfficacyStudents
This variety of expectations and dissatisfaction with the
mathematics program was also
evident in the student YouthTruth openended responses. Students
were asked to comment on
areas of strengths and improvements about their schools.
Students selected areas where they
interpreted the school needed improvement from various
categories, including the option of
Nothing. Of the 479 responses, 32% of the responses indicated
there were no areas for
improvement. Of the 68%, or 325 respondents, indicating an area
for improvement, 20% of those
comments included the keyword of math.
Figure 5 displays the comparison to other
discipline specific comments using the
keywords humanities and science
showing of the three disciplines, math
received87%oftheresponses.
There were many responses
regarding the math program not adequately
preparing students for exams and future math courses, but some
students appreciated the
emphasis on thinking about math. An 11th grade student shared,
For example, in math class, I
learn and practice learning HOW to think like a mathematician
and HOW to think out of the box
to solve problems by myself rather than being told how to solve
a problem and memorizing the
steps. However, the majority of comments focused on a desire for
more foundational work
which is reflected by this 10th grade student response, My
school's math curriculum works with
conceptual math, which is important, but learning can be very
confusing when a teacher starts
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TEACHEREFFICACYINSECONDARYMATHEMATICS 21
with the conceptual math and teaches solely with it. One really
needs procedural math as well.
This comment is corroborated by students who have supplemented
their learning through outside
resources, As a result of our inept math program, I've had to
work very hard at Community
College to make up for lost ground just to be prepared for a
four year college. Student
perceptionoftheprogramaffectsallthestakeholders,includingparents,teachersanddirectors.
Though the majority of comments focused on students feeling
unprepared or wanting
additional challenges, it could also be perceived as a
misunderstanding regarding what the
institution values. Students (and parents) may be unclear of the
design principles of deeper
learning and/or projectbased learning, specifically in terms of
math as evidenced by the
following student comment and others like it I want to learn
high school math from public
schools where they give you a lecture on how to do this math and
giving examples with the
class. This shared lack of clarity by teachers and students as
to the purpose and methods to
teachandlearnmathematicsisacontributingfactorintheefficacyoftheprogram.
InclusionandEfficacy
Another question posed to teachers dealt with the subject of
whether an allinclusive
model affected their teaching, and if so, how. Of the eleven
respondents, ten of them, or 91%,
said the model affected their teaching eight respondents felt it
was a difficult challenge while
three of them were either neutral or felt it challenged them to
be better teachers. Figure 6
providessomesampleresponses.
Figure 6: Responses from teacher survey question "Does having an
allinclusive model affect your teaching, and if so, how?"
Commentson
Challenges
Ifinditincrediblychallengingtoassistallstudentsattheircurrentlevelofmathematics.Yes,
I have to make sure to plan both interventions and extensions for
my units. It can be
verydifficulttobalanceandmakesureallneedsarebeingmetallthetime.Absolutely.
Theoretically it is a great idea. In practice, I am not equipped
with enough timeormanpowertoserviceallstudentsaswellastheydeserve.I
do not feel that I have enough time to modify, scaffold and
accommodate all learners in thissetting.
Commentson
Benefits
Yes,youhavetobeanevenbetterteacher.Youhavetofindtherightproblemsandstrategiestoprovidearigorousclassforallconstituents.Yes,
it makes the classroom richer and pushes me/us to a more open ended
approach and valuesalltypesofthinkingandapproachestoproblems.
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TEACHEREFFICACYINSECONDARYMATHEMATICS 22
While a few teachers expressed optimism with the current school
structures and the
allinclusive nature of the classrooms, the majority of teachers
did not. The teachers comments
regarding challenges were also substantiated by interview
responses from one of the instructional
coaches. Responses to the question of teacher challenges
included having mixed classrooms
and meeting the needs of all of our learners. If a teacher
doesnt feel they are helping all the
students in their classroom, combined with the student
perception of dissatisfaction with the
math program and individual teacher performance, than their
degree of confidence in the ability
todotheirjobhasbeencompromised.
The research conducted found these two main factors,
misunderstanding the goals of the
program and the ability to reach all learners in a classroom,
contribute to the perceived and
documented challenges of teachers and students. These problems,
though not officially stated as
such, are not new to the organization and there have been
attempts to lessen the effect on
teaching and learning by the various school sites. The following
section will attempt to provide a
samplingofthestrategiesemployed.
StrategiesWhichLeadtotheFormationofVision
During the course of my research I also embedded myself within
the mathematics
discipline group at one of the school sites. The group consisted
of seven teachers with three of
them being first year hires. Discipline meetings occurred
approximately every other week for
fortyfive minutes before classes began. Facilitation of the
meetings was on a rotational basis to
follow the design principle of teacherasdesigner without imposed
hierarchical structures.
However, this structure was not initially conducive to teacher
learning as there was no defined
vision or arc to the meetings. It was perceived no one teacher
wanted to step forward to define
thisvisionlestitbeperceivedtheyweresomehowsuperiortotheirpeers.
After dealing with parent complaints on the effectiveness of
teachers and the math
program, the school director and I brainstormed with the group
on what practices they believed
should be evident in all classrooms (see Appendix D for a
listing of those practices). This event
was followed by several meetings where the group individually
defined and defended the
meanings of these practices to reach a common definition for
each of them. This was then
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 23
followed by threemeeting arcs covering the various practices.
This processes of collectively
agreeing on what was important was a unifying factor for the
group. In an exit card response
after one of the meetings to What worked well for you today? the
response was I think the
structureandpositivedialoguewithapurpose(ExitCard).
Other school sites are taking advantage of additional
professional development meeting
times to selfselect into action groups to continue their
discussions and support of math learning.
This creates a weekly checkin with discipline members, as
opposed to a two to three week
cycle, which allows them to gain traction in advancing their
vision and improved practices.
Some practices, including the use of improvement science or
rapid cycles of measurable change
initiatives, have focused on improving lessons and classroom
management. I did not directly
observethesepracticesbutlearnedofthemthroughinterviews.
UsingInstructionalCoaches
Other school sites are employing the use of the aforementioned
teacher support
specialists or instructional coaches. These positions tend to be
temporary from year to year
depending upon available funding. Of the five secondary sites,
two of them have either a full or
parttime math instructional coaches with another site utilizing
a graduate student in the role. As
per one of the instructional coachs reflection, having a
dedicated resource to assist in planning
has been very helpful to teachers. Staff meetings are more big
picture, the coach relationship is
like having a mentor that is readily available. On a related
note, during followup interviews
with three teachers, all three mentioned mentors during their
first year was one of the most
helpfulsupportstructurestheyexperienced.
In delving deeper to understand how and why the school settings
are structured the way
they are, I interviewed the directors of four out of the five
secondary schools. Their leadership
experiences with the setting schools ranged from first year to
eight plus years, and in addition
they were all experienced educators. All the directors stated
mathematics instruction and learning
was an area for growth. In support of having instructional
coaches, one director stated Teachers
are really excited to have that consistent, ongoing support and
it is An amazing luxury
having a structure for dialogue as to what we want our math to
look like. Another director, who
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 24
uses two parttime teacher/coaches as management designees for
math does so in response to
not having a background in math and having a larger staff than
other sites. When directors were
asked whether other discipline groups have had a similar
structure of coaches, the answer was
no. This revelation seems seminal in understanding teacher
efficacy and will be discussed further
in the conclusion. Also, one director tied in the need for
students to develop a growth mindset
and overcome a lack of selfefficacy. Finally, multiple directors
spoke about the need to prepare
teachers for the transition from traditional teaching methods to
a more progressive approach and
the Common Core standards. A major source of funding for the
instructional coaches are paid
from Common Core transition grants. A further area for research
may to be to examine the
perceived effectiveness of the coaching based on their
qualifications and/or training in
implementingcommoncoremethodsandstandards.
There were many practices occurring across the school sites to
improve the instruction
and learning of mathematics. It is an area of ongoing
contemplation and research surrounding the
practices, methods and support structures to increase efficacy.
Whether it is student or teachers,
the idea of efficacy and having a mindset which allows for
growth and a positive experience
appearstobeanimportantfactorinaffectingmathinstructionandlearning.
Conclusions
This research focused on understanding what factors affected the
perception of efficacy
in the teaching and learning of mathematics in several
progressive secondary schools. Efficacy is
the belief in ones ability to produce the desired or intended
results. For teachers, this is the
belief the practices and structures they use and work in
contribute to student success. For
students, this is the belief they can use mathematics and are
prepared for college level work. The
conclusion of this study to improve perceived efficacy can be
specifically attributed to two areas:
1) Unclear expectations or vision of the mathematics program,
and 2) A need for more effective
strategies for reaching all learners in a classroom. Vision in
this context is defined as having a
plan to define clear goals and the methods to reach those goals.
I believe there are two main
areas which may develop this vision: 1) Defining the
institutions goals for secondary
mathematics,and2)Teacherpreparednessandsupport.
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 25
DeweyandDefiningVision
Literature (Harel, 2008, Hauk et al., 2010, Polya, 1954)
stresses in order to learn and use
mathematics one needs to provide both ways of thinking and
applying mathematics along with
knowing and practicing the procedures and formulas. The majority
of current mathematics
teachers were taught with traditional methods and in transition
to teaching in a more progressive
setting, it may be necessary for teachers and directors to
reflect on the questions Dewey (1938)
posedtoprogressiveeducators:
The problem for progressive education is: What is the place and
meaning of subjectmatter and of organization within experience? How
does subjectmatter function? Is there anything inherent in
experience, which tends towards progressive organization of its
contents? What results follow when the materials of experience are
not progressively organized? A philosophy
whichproceedsonthebasisofrejection,ofsheeropposition,willneglectthesequestions.
Dewey was trying to stress the importance of having a plan or
vision for student learning. The
setting schools possess a design principle of the teacher being
the primary designer of their
curriculum and assessments. This design principle allows for
teacher passion to infuse the
learning arena if teachers are excited about their lessons, the
students will also be excited. This
is a valid premise, however it does not preclude the necessary
standards or progression of
learning which needs to take place in order for students to
successfully gain mathematical
fluency.
In answering Deweys questions regarding subjectmatter functions,
organization and
progression, the CCSSM have been developed to guide educators to
what topics and practices a
student needs to understand and use to meet the goals of
secondary mathematics, but not how to
teach them. The setting schools have adopted the framework of
the CCSSM, but there are still
questions in how to incorporate them with the model of
projectbased learning. Teacher efficacy
is being affected by the pull of the school model, parent and
student desire (and their own) to
have the students perform well on gatekeeping exams, and their
own backgrounds in traditional
mathematicsinstruction.
Mathematicsteachersinthisresearchsettinghavebeengatheredfromvariousindustries
anddisciplinemajors.AsJimLewis,aprofessorofmathematicsattheUniversityof
NebraskaLincolnandresearcherwiththeMathematicsTeacherEducationPartnership(MTEP)
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 26
states,"Oneoftheideasisthatwhatyouneedtoknowinordertoteachwellisdifferentfrom
whatyouneedtoknowtobeayoungengineeroreconomist,"and,"Inmathematics,youare
oftentryingtosynthesizeknowledge.Asateacher,you'retryingtopullapartknowledgeand
understandwhypeoplehavedifficultylearning"(Sawchuk,2014).Thevariedbackgroundsof
teachersinthisresearchmaycontributetoalackofcommonunderstandingand/orhowtofoster
thedevelopmentofmathematicalpracticesforstudents.Teachersanddirectorswithinaschool
needtodefinewhatthosepracticesmeantothemanddevelopacommonlanguagetofacilitate
theirsuccessfulacquisitionbystudents.Asshownbymyexperienceswithinonesettingschool,
havingthedesireandtimetodefinethesepracticesdevelopedteacherefficacyifthereisaplan,
thereisawaytoknowifoneisbeing
effective.Theprocessweemployedwas
similartothedesignthinkingprocess
developedbyStanfordsd.school(see
Figure8)asuggestedmodelfortheprocess
canbefoundinAppendixE.Figure8:DesignThinkingModelSource:JoeyAquino/WordPress
However, the process would have benefited from more frequent
meetings and the use of
an instructional coach or mentor to guide the process
(BehrstockSherratt et al. 2014 Lemov,
2012). Additional areas for research and dialogue would be to
extend this conversation and
process to middle school and elementary teachers to align the
vision and practices of the K12
studentexperienceinmathematics.
ToTestorNotToTest
An additional area affecting teacher efficacy is standardized
testing. High school students
wishing to pursue a college degree are directly affected by the
outcomes of college entrance
exams. The setting schools are performing in line with state
averages. However, given the
advantages of smaller classes and more personalized instruction,
shouldnt the results reflect
those benefits? The CRESST center at UCLA stated the new SBAC
and PARCC tests reflect a
shift towards measuring deeper learning (Hermann & Linn,
2013), with similar revisions to the
college entrance exams also forthcoming. If these tests are
geared towards assessing deeper
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 27
learning and the CCSSM and those are the frameworks for the
mathematics program at the
setting and other progressive schools, will they be deemed as
important benchmarks for student
learning? There are conflicting messages being sent to the
educators in this study regarding the
importanceofstandardizedtests.
Progressive school educators and leaders need to decide if the
CCSSM and associated
gatekeeping exams are important to them as an institution. If
they value what gets measured,
then the teaching will follow with whatever modifications this
involves. This does not need to
translate to teaching to the test, however it does mean aligning
curriculum to the skills and
knowledge required to be successful. Teachers can still use
their individual passions to help
students discover the math, but alignment of vision and
practices is key across grade levels and
schools. If school leaders dont value what they measure, then
they have to be clear to their
stakeholders (teachers, students, parents and the community)
about that idea, and let the
stakeholders make an informed judgement regarding their decision
to be involved with the
institution. Having an aligned and public vision will foster
efficacy, and it does not have to trump
teacherasdesigner.Itsolelyprovidesaframeworkofunderstandingandpurposetothework.
ProfessionalDevelopment
New, and some experienced, teachers could benefit from
professional development to
assist them in their transition to the pedagogical aspects of
deeper learning and strategies for
differentiating instruction. Multiple conversations with
teachers, even those with masters in
mathematics, admitted to being unaware of the ways of thinking
(Harel, 2008) behind certain
mathematical knowledge they possess. As stated earlier, being
good at math or even using
math in the workplace is different from understanding what
strategies will help student learn. In
order to further develop a sense of efficacy, additional
content, discourse and anticipatory
knowledge (Hauk et al., 2010) is needed. The use of mentors was
deemed the most effective
method of support for new teachers from this research and others
(BehrstockSherratt et al, 2014,
Lemov, 2012). The use of instructional coaches as a professional
development strategy also
providesamentorthatisreadilyavailable.
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 28
InclusionIssues
The inclusive environment provides a layer of complexity for
educators. The majority of
educators in this setting expressed concern over reaching all
students. Researchers Powell et al.
(2013) found students with mathematics learning disabilities
(MD) need explicit instruction
which involves teacher demonstration of detailed stepbystep
instructions along with
independent practice. The methods needed to help MD students may
be in conflict with the
model of projectbased learning or they may need additional
strategies to reach competency. In
recent interviews with students regarding YouthTruth survey
results, those who find mathematics
easy expressed concerns regarding teachers focusing instruction
on the students who need
moresupporttherebyimpedingtheirabilitytomovedeeperand/orfasterthroughthematerial.
A strategy that may increase teacher and student efficacy is
increased dialogue and
cooperation with inclusion specialists or special educators.
Hobbs & Westling (2002) and
Monsen et al. (2014) found teacher efficacy improved when
special educators and teachers
worked together to form a support system for themselves and
students. Jointly reviewing case
studies helped them develop best practices and led to an
emphasis on cooperative learning and
teamdecisionmaking(Hobbs&Westling,2002,p.188).
ResearchSurroundingProjectBasedLearning
An area for additional research could focus on the complexities
of trying to teach
mathematics through the use of projects. Teachers may benefit
from specific teaching strategies
to improve the balance of instruction between thinking,
understanding and practicing
mathematics when attempting instruction through projects.
Questions surrounding time
allocation and the ability for students to effectively gain the
adopted standards through project
work could be key considerations. Also, research regarding the
definition and structure of
mathematics projects could help educators more effectively plan
and coordinate instruction.
Again, these future research considerations are in line with
defining the vision of a mathematics
program.
I believe the ultimate goal of mathematics education is to
create learning environments to
increase student efficacy. Student efficacy will hopefully
result in the successful attainment of
-
TEACHEREFFICACYINSECONDARYMATHEMATICS 29
the language of mathematics and its associated college and
career readiness. Efficacy appears to
be a cyclical event. Student efficacy can facilitate teacher
efficacy and vice versa. Teacher and
student efficacy can be facilitated by defining the goals of a
mathematics program and making
them transparent to stakeholders. Providing educators with
continued professional development
in their own transitions from traditional methodologies to a
more progressive or constructivist
approach to mathematics is key to improving their efficacy. If
students and teachers share
common vision, practices and language of mathematics across the
K12 spectrum, who knows
wherewecouldrankasanation.
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TEACHEREFFICACYINSECONDARYMATHEMATICS 30
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TEACHEREFFICACYINSECONDARYMATHEMATICS 34
AppendixA
TeacherSurveyQuestions
Thesequestionswereamixtureofsingleanswer,multiplechoice,ranking/scalingandopen
endedquestions:
1. Whatgradeleveldoyouteach?
2. Doyouteachacombinedclass(likePhysics/Math1)?
3. Howlongisatypicalclassperiod(inminutes)?
4. Isyourclassasemesteroryearlongcourse?
5. Doyoufeelyouhaveamanageableworkload?
6. Approximatelyhowmanystudentsdoyouteachinday?
7.
Howwouldyourateyourteachingstyle,frombeinghighlytraditionaltobeinghighly
progressive/constructivist?{scaledquestion}
8. Doyouuseanyparticulartextbookorprimarysourceinformation?
9.
Howmanyminutesperweekwillbeused/assignedtoreinforcemathematicalconcepts
(i.e.skillspractice)?
10. Whatpercentageofclasstimedoyouuseforopenendedquestions?
11.
TowhatextentdoesyourclassincorporatetheCommonCoreStateStandardsin
Mathematics(CCSSM)?
12.
Doyoufeel(ScaledquestionsfromDisagreecompletelytoAgreeCompletely)
a. freetoteachinwhatevermethodsyouchoose?
b.
thattheCCSSMprovideagoodframeworkforstudentstoobtainmathematical
proficiency?
c.
thatitispossibletoobtainadepthofunderstandingofmathematicalconcepts
throughprojects?
d.
itisimportanttoprovideclasstimetopracticemathematicalskillsandconcepts?
e.
thatallmathstudentsareadequatelychallengedinthecurrentschoolstructurefor
mathematics?
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TEACHEREFFICACYINSECONDARYMATHEMATICS 35
f.
thatallmathstudentsinyourclass(es)areadequatelychallengedand/or
supported?
g.
thatyourstudentswillbepreparedforcollegelevelmathematics?(junior&
seniorteachers)
h.
thatthemathteamatyourschoolsharesacommonvision/philosophyof
teaching/learningmathematics?
13. Whatisyourdefinitionofamathproject?
14.
Doeshavinganallinclusivemodelaffectyourteaching,andifso,how?
15.
Doeshavinganallinclusivemodelaffectstudentlearning,andifso,how?
16.
Whatdoyouthinkarethebestmethod(s)(instructionalorinstitutional)forstudentsto
understandandbeabletousemathematics?
17.
Whatdoyouthinkisthepurposeofsecondarymathematicsprograms?
18.
Arethereanyothercommentsorsuggestionsregardingthecelebrationsand/or
challengesintheteachingorstudentlearningofmathematicsthatyouwouldliketo
share?
19.
Wouldyouliketoshareyournameforpossiblefocusgroups,interviewsordiscussions?
(thisiscompletelyvoluntary)
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TEACHEREFFICACYINSECONDARYMATHEMATICS 36
AppendixB
InterviewQuestionsforDirectors
1. HowlonghaveyoubeenworkingfortheHTHorganization?
2. Howlonghasyourcurrentstructureformathematicsbeeninplace?
3.
Ifthestructurehaschangedunderyourtenure,whatwereyourreasonsformakingthe
change?
4.
Areyouplanningtomakeanyfuturechangestomathematicsinstruction?Ifso,whatand
why?
5.
Doyouhavepersonnel(coaches)thatareassignedspecificallytoassisteducatorsinmath
instruction?
6. Doyoufeelthat
i)
allmathstudentsareadequatelychallengedwiththecurrentschoolstructurefor
mathematics?
ii) itisimportanttopreparestudentforstandardizedtests?
iii)
junior/seniorstudentsarepreparedforcollegelevelmathematics?
iv)
mathinstructorsareadequatelytrainedtoteachinaprogressive/constructivist
manner?
v) mathconceptscanbeeffectivelylearnedthroughprojects?
vi) mathteachersfeelthatmathshouldbetaughtthroughprojects?
7.
Doyouhaveanyconcernsregardingthemathematicsprogramatyourschool?Ifso,whatare
they?
8.
Arethereanyothercommentsorsuggestionsregardingmathematicsinstructionor
preparednessthatyouwouldliketoshare?
{Additionalindividualfollowupquestionswereaskedbasedonresponsestotheabove
questions.}
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TEACHEREFFICACYINSECONDARYMATHEMATICS 37
AppendixC
CollegeAdvisorSurveyQuestions
1. HowlonghaveyoubeenworkingfortheHTHorganization?
2. Doyoufeel(Scaledquestions)
a.
thatallmathstudentsareadequatelychallengedwiththecurrentschoolstructure
formathematics?
b. thatstudentsarepreparedtoperformwellonstandardizedtest?
c.
thatjunior/seniorstudentsarepreparedforcollegelevelmathematics?
3.
Arethereanyothercommentsorsuggestionsregardingthemathematicsprogramsthat
youwouldliketoshare?
4.
Wouldyouliketoshareyourcontactinformationforpossiblefocusgroupsortoprovide
additionalcomments?(thisiscompletelyvoluntary)
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TEACHEREFFICACYINSECONDARYMATHEMATICS 38
AppendixD
MathematicalPractices
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TEACHEREFFICACYINSECONDARYMATHEMATICS 39
AppendixE
DesignThinking&DefiningPractices
Forthefullpresentation,selectthislink:FosteringEfficacy