teacher efficacy in secondary mathematics: fostering confidence and fluency

TEACHEREFFICACYINSECONDARYMATHEMATICS 1

TeacherEfficacyinSecondaryMathematics:FosteringConfidenceandFluency

KellyLynWilson

HighTechHighGraduateSchoolofEducation

AuthorNote

KellyLynWilsonisnowtheDirectorofInnovativeandEntrepreneurialProgramsatSevern

HighSchoolinAnnapolis,Maryland.Thisresearchwassupportedinpartbyafellowshipfrom

theWaltonFoundation.CorrespondenceconcerningthispapershouldbeaddressedtoKellyLyn

Wilson,SevernSchool,201WaterStreet,SevernaPark,MD,21146.

Contact:[email protected]


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.


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


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


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.


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


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)


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


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


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.


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


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:


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.


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).


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


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


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.


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.


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.


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


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.


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


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


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.


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)


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


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.


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


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.


References

2013STARTestResults.(2015).RetrievedFebruary2,2015,from

http://star.cde.ca.gov/star2013/Index.aspx

Aboutthestandards:Developmentprocess.CommonCoreStateStandardsInitiative.Web.

2014.http://www.corestandards.org/aboutthestandards/developmentprocess/

Bandura,A.(1994).Selfefficacy.InV.S.Ramachaudran(Ed.),Encyclopediaofhuman

behavior(Vol.4,pp.7181).NewYork:AcademicPress.(ReprintedinH.Friedman[Ed.],

Encyclopediaofmentalhealth.SanDiego:AcademicPress,1998).

BehrstockSherratt,E.,Bassett,K.,Olson,D.,&Jacques,C.(2014).FromGoodtoGreat

ExemplaryTeachersSharePerspectivesonIncreasingTeacherEffectivenessAcrossthe

CareerContinuum.RetrievedMay10,2015,from

http://www.gtlcenter.org/sites/default/files/Good_to_Great_Report.pdf

Bidwell,A.(2014).Commoncoresupportinfreefall.Web.20August2014.

http://www.usnews.com/news/articles/2014/08/20/commoncoresupportwaningmostno

wopposestandardsnationalsurveysshow

Commoncorestatestandardsadoptionmap.AcademicBenchmarks.Web.2014.

http://academicbenchmarks.com/commoncorestateadoptionmap/

Dewey,J.(1938).Experienceandeducation.NewYork:Macmillan.

EducatetoInnovate.(n.d.).RetrievedApril2,2015,from

https://www.whitehouse.gov/issues/education/k12/educateinnovate

FulfillingthePromiseoftheCommonCoreStateStandards.(2012).RetrievedMay2,2015,

fromhttp://www.ascd.org/ASCD/pdf/siteASCD/commoncore/CCSSSummitReport.pdf


Gurria,A.(2014).Pisa2012ResultsinFocus.RetrievedApril6,2015,from

http://www.oecd.org/pisa/keyfindings/pisa2012resultsoverview.pdf

Hanushek,E.,&Rivkin,S.(2012).TheDistributionofTeacherQualityandImplicationsfor

Policy.AnnualReviewofEconomics,131157.

Harel,G.(2008a).DNRPerspectiveonmathematicscurriculumandinstruction:Focuson

proving,PartI,TheInternationalJournalonMathematicsEducation,40,487500.

Harel,G.(2008).Whatismathematics?Apedagogicalanswertoaphilosophicalquestion.InB.

Gold&R.Simons(Eds.),Proofandotherdilemmas:Mathematicsandphilosophy(pp.

265290).Washington,DC:MathematicalAssociationofAmerica.

Hauk,S.,Jackson,B.&Noblet,K.(2010)Noteacherleftbehind:AssessmentofSecondary

mathematicsteacherspedagogicalcontentknowledge.Presentedat13thSIGMAA

conferenceonresearchinundergraduatemathematicseducation.

Herman,J.&Linn,R.(2013)Ontheroadtoassessingdeeperlearning:ThestatusofSmarter

BalancedandPARCCassessmentconsortia.NationalCenterforResearchonEvaluation,

Standards,andStudentTesting.UCLA.Jan(2013)

Hobbs,T.,&Westling,D.L.(2002).MentoringforInclusion:AModelClassforSpecialand

GeneralEducators.TeacherEducator,37(3),186201.

Hoy,A.,&Spero,R.(2005).Changesinteacherefficacyduringtheearlyyearsofteaching:A

comparisonoffourmeasures.TeachingandTeacherEducation,343356.

Jensen,E.(2006).Enrichingthebrain:Howtomaximizeeverylearner'spotential.San

Francisco:JosseyBass,AJohnWiley&SonsImprint.


Leslie,I.(2015,March11).Therevolutionthatcouldchangethewayyourchildistaught.

RetrievedMarch26,2015,from

http://www.theguardian.com/education/2015/mar/11/revolutionchangingwayyourchildt

aught

Lemov,D.(2012).Teachlikeachampion2.0:62techniquesthatputstudentsonthepathto

college.JosseyBass.(Seconded.).

MathForAmericaSanDiego.LunchwithPhilDaro.October16th,2014

McDonnell,L.&Weatherford,M.S.(2013)Organizedinterestsandthecommoncore. EducationalResearcher.201342:488Web.Nov13,2013

DOI:10.3102/0013189X13512676

Monsen,J.j.,Ewing,D.e.,&Kwoka,M.(2014).Teachers'attitudestowardsinclusion,

perceivedadequacyofsupportandclassroomlearningenvironment.Learning

EnvironmentsResearch,17(1),113126.doi:10.1007/s1098401391448

Neuroplasticity.(n.d.).RetrievedNovember6,2014,from

http://www.medicinenet.com/script/main/art.asp?articlekey=40362

Protheroe,N.(2008).Teacherefficacy:Whatisitanddoesitmatter?Principal,87(5),4245.

Polya,G.(1954).Inductionandanalogyinmathematics.Princeton,NJ:Princeton

UniversityPress.

Powell,S.,Fuchs,L.&Fuchs,D.(2013)Reachingthemountaintop:Addressingthecommon

corestandardsinmathematicsforstudentswithmathematicaldifficulties.Learning

DisabilitiesResearch&Practice,Vol.28,No.1.pp.3348.


Sawchuk,S.(2014).InLightofCommonCore,Ed.SchoolsLooktoTransformMathTeacher

Prep.Retrievedfrom

http://www.edweek.org/ew/articles/2014/11/12/12ccteacherprep.h34.html

Shaughnessy,M.(2004).AnInterviewwithAnitaWoolfolk:TheEducationalPsychologyof

TeacherEfficacy.EducationalPsychologyReview,153176.

Toughchoicesortoughtimes:ThereportoftheNewCommissionontheSkillsoftheAmerican

Workforce.(2007).ChoiceReviewsOnline,450406.RetrievedMay10,2015,from

http://www.helios.org/uploads/docs/ToughChoices_EXECSUM.pdf

WhatisDeeperLearning?(n.d.).RetrievedMarch10,2015,from

http://www.hewlett.org/programs/education/deeperlearning/whatdeeperlearning


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?


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)


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.}


AppendixC

CollegeAdvisorSurveyQuestions

1. HowlonghaveyoubeenworkingfortheHTHorganization?

2. Doyoufeel(Scaledquestions)

a. thatallmathstudentsareadequatelychallengedwiththecurrentschoolstructure

formathematics?

b. thatstudentsarepreparedtoperformwellonstandardizedtest?

c. thatjunior/seniorstudentsarepreparedforcollegelevelmathematics?

3. Arethereanyothercommentsorsuggestionsregardingthemathematicsprogramsthat

youwouldliketoshare?

4. Wouldyouliketoshareyourcontactinformationforpossiblefocusgroupsortoprovide

additionalcomments?(thisiscompletelyvoluntary)


AppendixD

MathematicalPractices


AppendixE

DesignThinking&DefiningPractices

Forthefullpresentation,selectthislink:FosteringEfficacy