Proposal - Bryan Harms - GSE 2013

7/28/2019 Proposal - Bryan Harms - GSE 2013

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Bryan Harms - Action Research Proposal 1

Table of Contents

Introduction ....................................................................................................................... 2

Understandings .................................................................................................................. 6

Problems, Exercises and Problem Based Learning ......................................................... 6

Support and Criticism of a Problem-based Learning Approach ..................................... 9

Relational Equity ........................................................................................................... 11

Setting ............................................................................................................................... 13

Location - High Tech Middle School, Chula Vista ....................................................... 13

Demographics ................................................................................................................ 14

Principles of Design, Three Key Integrations ............................................................... 16

My Classroom ............................................................................................................... 22

Methods ............................................................................................................................ 24Overview ....................................................................................................................... 24

Surveys .......................................................................................................................... 25

Interviews ...................................................................................................................... 25

Work samples ................................................................................................................ 25

Observations .................................................................................................................. 26

References ........................................................................................................................ 27


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Introduction

By removing the creative process and leaving only the results of that process, you

virtually guarantee that no one will have any real engagement with the subject.

(Lockhart, 2009)

I am an 8th grade mathematics teacher. I work at a school that allows me with to design my math

program in a way that meets my strengths as a teacher and the needs of my students. I have had a

certain amount of success teaching the standard procedures and concepts prescribed by the state

for students studying mathematics in 8th grade. Despite this success, I believe the approach I have

taken has misled many students about what mathematics is and what mathematicians do. If you

asked my students what it took to be good at math they might say something like this: Practice

is the most important thing, for every new topic you want to know every type of problem there is

and then you solve those kinds of problems as many times as you need to until you can do it right

about 80% of the time. In short, in my class the process of mathematics was: somebody shows

you how to it and you practice how to it that until you can do it on your own.

Maybe this approach is not a terribly offensive and on its surface has some appeal. Its appeal to

me as a teacher is pretty clear. I care about my kids; I want them to feel smart and successful. I

start out in the beginning of the year with all these exercises that I need to show kids how to

solve. I know they will be tested on them by the state, they will be judged on them by their

peers, parents and future teachers. So I show them how to do it, make them practice the steps

until they are proficient and repeat. There is a lot to cover and little time to waste.


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But implicit to this approach is the assumption that students should care about learning steps and

procedures to these things partially (perhaps sometimes wholly) because if they do not others

will see them as dumb or weak or lazy or not good at math. Also implicit is the idea that doing

something just because it is hard or requires discipline is reason enough to do it.

Anecdotally, from my own experiences as a student I realize this approach does not always

work. I was not a good student in my middle and early high school years. I remember little of

mathematics classes but I do have a vivid memory of lifting my head up from my desk one day

to ask a girl next to me why she was doing solving a problem the way she was and her

responding because that was what she was told to do. I asked why again and she responded

because that is how you do it. I put my head down again and that was that. There came a point

in my young life that the negative perceptions of others about my work ethic, academic ability

and intelligence became something that I saw as static. As a kid with why on my mind all the

time, just because was an exhausting answer.

But you should have seen me in front of a chess set. My elementary school principle and my

father introduced me to the game. My father would give me as many pieces as necessary to make

the game competitive. My principle would play with me at recess, often when I had to stay inside

because I hadnt completed some assignment or another. I could stare and strategize for hours. I

could ask why and what if. I could try something and it would work or it would not and I would

try something else. There were tactics that had to be understood and strategies that had to be

developed. There was mystery, consequence, collaboration and critical thought. Every problem

was new, every problem multi faceted. Early on, I did not solve these problems correctly in the


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sense that I did not win the games. But by the end of fifth grade I could beat my principle and by

the eighth grade I was winning citywide school championships. The wins were great but they

would not have come with out the problems, the need to understand why, and the opportunity to

create and discover.

The practice of procedures has a role in mathematics education, especially at the elementary and

secondary level, but what was missing from my class room was the point of practicing the

procedures, the opportunities for discovery, for solving new and novel problems, the public

debate of ideas, the proof and presentation thought.

The focus on procedural fluency in my classroom came at the expense of learning effective

problem solving strategies and habits of mathematicians. It came at expense of seeing math

exercises as the sum total of mathematics. We were drilling exercises but rarely solving

problems.

My first glimpse at another approach came when someone told me about Exeter problems that

you could get freely on the Internet. I was told that the problems were sometimes used as

challenge problems for kids who needed extra challenge but it wasnt until I started going

through the problem sets myself and creating a solution set that I really became intrigued by

them. They were not what they seemed to be, or what many people had described them to be.

The problems came in sets, and the sets were ordered. The problems fell into different categories

and had different goals. Sometimes the problems introduced new terminology. Sometimes the

problems were superficially simple to solve but revealed new ideas. Some made connections


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between arithmetic thinking and algebraic thinking. Some were abstract, some were concrete,

and some made connections between the two. None of the problems forecasted the skills needed

to solve them, but problem sets clearly focused on particular concepts and skills. The problems

built on themselves, adding little bits of complexity as they went along.

The problem sets covered multiple ideas in a single set, so students might be doing problems

about the implications of the distributive property and also deriving the rules for exponents at the

same time. The problem sets came without instructional texts explaining how to do them.

I was fascinated and began to research. I learned that everything but the initial problems sets

came from conversations about, and presentations of, the problems. Content was co-constructed

from the discussions and presentations of student work. I learned that the Exeter problems are

part of a larger learning approach called problem-based learning, an approach that centered on

constructing knowledge through solving, presenting and talking about problems. Problem-based

learning (PBL) is the approach to mathematics I wish to look at during my action research. In my

action research I will explore the following question: What effect does an approach to

teaching mathematics that explicitly emphasizes problems, equitable collaborations,

presentations of thought, and construction of knowledge; have on students ability to think

critically, the development of their problem solving skills and strategies, and on their

perceptions of themselves as mathematicians?


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Understandings

Problems, Exercises and Problem Based Learning

Because I will be researching an approach based on problems it is worthwhile to define what I

mean by problems and how a problem differs from an exercise. An exercise is designed

primarily to practice a technique that has already been introduced. It is parameterized so that by

changing the value of the parameters the technique can be practiced multiple times. The goal of

exercises is to develop fluency with techniques, and also to provide example of standard ways in

which the techniques are commonly used.

Paul Lockhart describes a good problem as something you dont know how to solve. (Lockhart

2009). That is a good place to start. A good problem is not just something you dont have the

answer for it is something your not quite sure how to solve. It requires strategies as well as

techniques. Where as exercises drill technique, the goal of a problem is often to discover

technique while practicing strategy.

Problem Based Learning was Developed 40 years ago for the health sciences curriculum to

address the intensive learning requirements of medical education. Medical faculty at McMaster

University in Canada introduced the process with the hopes that it would be a more effective and

humane approach to meeting the substantial requirements of medical education. The approach fit

their philosophy for structuring an instructional culture that promoted student centered learning,

multidisciplinary education and lifelong learning in professional practice (Boud and Feletti 1997,


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as cited by Savery, 2006, p. 9). There are many different definitions of problem-based learning

but for the purposes of my research I will focus on two.

Howard S. Barrows, an innovator and practitioner of PBL suggests that four components make

up problem-based learning:

Ill-structured problems are presented as unresolved so that students will generate notonly multiple thoughts about the cause of the problem, but multiple thoughts on how to

solve it.

A student-centered approach in which students determine what they need to learn. It is upto the learners to derive the key issues of the problems they face, define their knowledge

gaps, and pursue and acquire the missing knowledge.

Teachers act as facilitators and tutors, asking students the kinds of meta-cognitivequestions they want students to ask themselves. In subsequent sessions, guidance is faded.

Authenticity forms the basis of problem selection, embodied by alignment to professionalor real world practice. (Barrows 2002, as cited by Johannes and Barneveld, 2009,

p.46)

Although problem-based has its roots in health sciences at the university level, the approach has

been adopted at the secondary and elementary level as well and for disciplines besides the health

sciences. The second operational definition I will use comes from Dr. Carmel Schettino. Dr.

Schettino currently works in the mathematics department of Deerfield Academy and before that

was the chair of the mathematics department at Emma WIllard School and was also a

mathematics teacher at Exeter Academy. Dr. Schettino defines problem-based learning as:

An instructional approach of curriculum and pedagogy where student learning and

content material is constructed (and co-constructed) through the use, facilitation, and

experience of contextual problems in a decompartmentalized, threaded topic format in a


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discussion-based classroom setting where student voice, experience, and prior knowledge

are valued. (Schettino 2011)

ItisSchettinosfocusonstudentvoice,experienceandpriorknowledgehousedinanon-

hierarchicalenvironmentembracingarelationalpedagogythatspeakmoststronglyto

usingthisapproachtopromoterelationalequity.ThisisnotsurprisingsinceSchettino

focusesmuchofherresearchonissuesofequityinmathematicseducation.

Agoodexampleofwhataproblem-basedclassroomcanlooklikeisthemathematics

classroomsatExeterAcademy.Implementationsoftheclassroomwillvarybutinall

problem-basedclassrooms,problemsolvingiscentraltotheprocessoflearning

mathematics.Exeterfacultiesdefineproblemsolvingasinvestigating,conjecturing,

predicting,analyzing,andverifying,followedbyawell-reasonedpresentationofresults.

Thisproblemsolvingisdoneincollaborative,student-centeredclassrooms(PhillipsExeter

AcademyWebsite).

Theroleoftheteacherinaproblem-basedclassroomisdifferentthanroleofateacherina

classroomthatfocusesprimarilyondirectinstruction.Barrow(Barrows2002,ascitedby

JohannesandBarneveld,2009)referstotheteacherasafacilitatorandtutor,asking

questionswhichtheyhopestudentswillaskthemselves.Schettino(2011)highlightsthe

non-hierarchicalnatureoftheenvironment,whichhasimplicationsfortheteacheraswell

asthestudents.Theteacherisafacilitatorandacoach,encouragingandinspiring,

providingdirectionandpushingthinking.Thefacilitatorscaffoldsasnecessary,sensingthe

understandingoftheclass,probingforunderstanding,providingopportunitiesforstudent


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interaction.Thefacilitatoractivelyfades,relinquishingauthority,steppingbackasstudent

internalizetheprocessofpresentationanddiscussion.

Support and Criticism of a Problem-based Learning Approach

There have been many studies that look at the effectiveness of problem-based learning. In 2009,

Johannes Strobel and Angela Barneveld reviewed the majority of data from studies published in

the past 15 years relating to the effectiveness of problem-based learning. Their study used a

qualitative meta data synthesis of all the data to contrast the assumptions and findings of the

meta analytical research done on the effectiveness of PBL. Their findings indicated that PBL was

superior to traditional methods in the areas of long-term retention, skill development and student

and faculty satisfaction. (Strobel and Barneveld, 2009, p.53) Additionally, problem-based

learning is an effective approach in schools looking to develop the 21 st century skills of

collaboration, communication and technology literacy. Problem-based learning also incorporates

many of the Common Core State Standards such as problem solving and developing quantitative

reasoning skills(Schettino 2011).

The most recent criticism of problem-based learning comes in a study entitled Why Minimal

Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist,

Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. (Kirschner, Sweller, &

Clark 2006).

In one of their criticisms, Kirschner, Sweller, & Clark suggest that the cognitive loads associated

with the unguided exploration of new content, especially in novice learners, might be detrimental

to learning:


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Sweller and others (Mayer, 2001; Paas, Renkl, & Sweller, 2003, 2004; Sweller, 1999,

2004; Winn, 2003) noted that despite the alleged advantages of unguided environments

to help students to derive meaning from learning materials, cognitive load theory

suggests that the free exploration of a highly complex environment may generate a heavy

working memory load that is detrimental to learning. This suggestion is particularly

important in the case of novice learners, who lack proper schemas to integrate the new

information with their prior knowledge (Kirschner, Sweller, & Clark 2006).

This may be a valid criticism of wholly unguided environments. But, here it seems the authors

are conflating problem-based learning with other strategies with superficial similarities. Problem

based learning may share similar characteristics with some unguided learning strategies but it is

not unguided nor is it minimally guided. Problem-based learning provides extensive scaffolding

and guidance to facilitate student learning (Hmelo, Duncan and Chinn 2007). Additionally, my

own implementation of problem-based learning, heavily influenced by Schettino and others

emphasizes the importance of prior knowledge in both the design and exploration of new

problems, hopefully avoiding issues of students diving into content that is at a level that is too

complex for them to tackle.

Kirschner, Sweller, & Clark also suggest that a certain lack of clarity on the part of PBL

practitioners about the difference between learning a discipline and practicing a discipline might

explain why it maintains its appeal to educators.

All in all, a lack of clarity about the difference between learning a discipline and

research in the discipline...has led many educators to advocate a problem-based method

as the way to teach a discipline. (Kirschner, Sweller, & Clark 2006)


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It is important not to confuse being a mathematician with the content of the discipline itself. But

distilling the PBL approach as merely a way to study the practices and habits of mathematician is

a mischaracterization of its goals. Indeed, it focuses on a balance between practices and content

but neither is relegated to the other. If that is the case in some instances of PBL I will certainly

try to not make it the case in my own. In his book A Mathematician's Lament: How School

Cheats Us Out of Our Most Fascinating and Imaginative Art Form Paul Lockhart does a good

job of describing the importance of doing mathematics as a part of the study of mathematics:

Mathematics is the music of reason. To do mathematics is to engage in an act of

discovery and conjecture, intuition and inspiration; to be in a state of confusionnot

because it makes no sense to you, but because you gave it sense and you still don't

understand what your creation is up to; to have a break-through idea; to be frustrated as

an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.

(Lockhart 2009)

Relational Equity

Equitable collaborations are an integral part of the problem base classroom I wish to create. In

practice this will often involve how students interact with each other when presenting their ideas

and problem solving together. Jo Boaler has proposed the term relational equity as a way of

describing the types of equitable interactions that occur between students in a classroom, these

types of interactions include treating each other with respect and considering others viewpoints

fairly. This notion of equity suggested by Boaler moves the focus on equity away from

traditional measures of school achievement such as scores on standardized tests and instead

investigates the ways in which students interact with each other in the classroom (Boaler 2008).

Boaler reported that the relational equity identified in her study consisted of three strands


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Respect for others ideas - leading to positive intellectual relations Commitment to the learning of others Learned methods of communication and support

Equitable relations amongst students can be seen as an end in itself, but Boalers research

suggests that classrooms that actively promote relational equity also perform better in traditional

measures of school achievement. (Boaler 2008) Her findings suggest that an approach that

stresses promoting the types of interactions and collaboration necessary to create good citizens in

a 21st

century society can also improve academic performance.


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Setting

Location - High Tech Middle School, Chula Vista

High Tech Middle Chula Vista was established in 2011. We are in our second operational year.

The building is new and was also finished in 2011. Located on the edge of a large canyon, the

school sits at one of the southern most points of Southern California. At night the lights of

Tijuana reflect on our glass window.

Figure 1: Picture of the school and campus looking south.

Entering the school from the north, one is surrounded by the familiar markings of suburban

Southern California. There is a mall down the street and a housing development sits directly

across from the entrance. Upon exiting the school from the south side, one is immediately struck

by the vast undeveloped wilderness. Rattlesnakes and coyotes are still found in abundance. Wild

grasses dry to tumbleweed and mark the changing seasons. Hawks perch from the power lines

along Discovery Falls Drive; they face the south, towards the canyon, full from a meal of field

mice or preparing to hunt for another.


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The building is two stories and houses both a middle and elementary school. The two schools

share a common entrance. The middle school extends to the south of the entrance on both floors,

and the elementary school to the north. Our campus is shared with High Tech High Chula Vista.

The building was designed to be an educational learning space for its students, staff and the

community. Hallways serve as galleries of student work. The classroom walls facing the

hallways are made of glass. They provide a sense of openness and encourage work that is worth

seeing as well as doing. Much of the buildings light is provided naturally through the creative

use of day lighting. Our school is one of the most energy-efficient in the nation and 70% of the

energy used by our facility is self-generated through the use of rooftop solar panels.

Figure 2: Solar panels supply much of our energy needs.

Demographics

Our student body is admitted by lottery. This system allocates a number of spots for each zip

code based on its middle-school aged population compared to the corresponding population of

the community. In this way our student population closely mirrors the student population of the


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communities they come from. Forty-two percent of our students are eligible for free or reduced

lunch. The ethnicity of our student body is shown in the following table and chart.

Grade

Totalstudents

Asian

AfricanAmerican

Caucasian

Chinese

Filipino

Hispanic

AmericanIndian

Japanese

Korean

PacificIslander

Vietnamese

6 115 1 11 9 0 16 70 6 1 0 0 1

7111 0 10 11 0 15 64 3 3 2 3 0

8 113 1 10 11 3 16 64 3 1 1 3 0

339 2 31 31 3 47 198 12 5 3 6 1

Figure 3: Table of Ethnicities by Grade Level with Totals


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Figure 4: Graph of Ethnicity of Student Body

Principles of Design, Three Key Integrations

The three schools on our campus share more than proximity. We share a common set of guiding

principles that drive our work. The principles are shared by all of the High Tech schools, of

which there are now eleven. The ideas are not strictly static, but rather interpreted and

manifested in different ways across classrooms and schools. Despite this, they are clearly the

foundation of our shared vision and form the basis of our common language.


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The HTH principles of design are personalization, adult world connections, common intellectual

mission and teacher as designer. These principles are distilled from the earlier work of our

founder and CEO Larry Rosenstock. The ideas are infused into all aspects of our educational

approach at the middle school.

These principles of design highlight three key integrations, which are the signature of a High

Tech school:

1. Connecting the classroom and the community2.

Integrating students

3. Linking hands and minds

The design principles and the three integrations are implemented through a project-based

approach to learning. We learn by doing, by creating and by exhibiting our creations.

Connecting the Classroom and the Community

Our classrooms are not seen as separate from our community; our projects are always connecting

the classroom with the world around it in some way. Education is not something we provide in

isolation.

The connections take many forms but common implementations include:

o Consulting experts during the design phase of a project.o Inviting guest speakers and experts from the community to assist on projects.o Designing products that benefit or reflect the community.


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Some examples of how we have established this connection in the past year are:

o Inviting a scientist and participant in the show The Colony to assist students in buildingdevices to help them survive a hypothetical apocalypse scenario.

o Inviting radio producers from NPR to help students produce radio pieces about storiesfrom the border.

o Having student families and community members help create altars for a projectcelebrating Dia de los Muertos.

Figure 5:Bringing expert guests into the classroom help students connect what they do in

class to how it will be used in the outside world. Here, Amy West, scientist and contestant

on the television show "The Colony" poses with students during the Dystopia to Utopia

Project


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We also hold many public exhibitions of our students work, highlighting the importance of our

work outside of the classroom and the purpose of it being created for a wider audience than just

our academic community.

Figure 6: Exhibitions are one way of integrating the community into the classroom. This is

the poster from our schools first exhibition.

Integrating Students

Admission at HTCV is non-selective, and we do not track our students. We have a higher

percentage of special education students than the district as a whole and provide a fully inclusive

environment to all our students with special academic needs. Heterogeneous grouping is both a

point of pride and a challenge of our academic model. How to provide an appropriately

challenging curriculum to all students in a classroom with a wide spectrum of abilities, maturity

and previous skills is a question that is continually asked and constantly refined. Projects provide


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multiple entry points for students and varying degrees of challenge. Projects also give students

choices which cater to individual passions and strengths. Often the grouping of students at all

points of the academic spectrum leads to performance increases in all students. This is in part

due to the high standards expected of all and the personalization of the academic programs.

Figure 7: Admission to HTMCV in not selective and we do not track our students.

Linking Hands and Minds

Learning at HTMCV is applied. Our projects stress learning by doing in response to questions

asked. Here are some examples from my projects over the last year:

o How does the science air pressure effect the operation of a hovercraft?o How does a truss structure compare to an arch structure in the design and implementation

of a bridge?


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o How can you harness the power of the sun to cook food?o What is the difference between filtering water and purifying water?

The answer to these questions is both theoretical and applied. Students learn to successfully

explain theories and apply them to products. In response to the above questions, they must build

hovercrafts, water purification systems and solar cookers.

Figure 8: Learning is as much about doing as it reading.


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Figure 9: Concepts like scale and proportions can be integrated into projects.

My Classroom

I teach 8th graders: a team of 56 students broken into 2 classes of 28. I teach integrated math and

science. I share these 56 students with a humanities teacher, and as a team we spend most of the

day with them. Usually I spend about 2 hours with each group per day.

In 8th grade, we cover the physical sciences and applied mathematics including topics in algebra

and pre algebra. I am in my fifth year with the High Tech organization and my fourth year as a

teacher. My classroom is active. I am still, and will probably always be, refining the process of

integrating math and science through projects.

To accommodate the way I teach, the setup of my room is always changing; I have three

configurations that students know by heart. All of the desks are on wheels and the setup of the

room can be changed in less than four minutes. The different classroom setups are designed to

best utilize the space for the type of teaching or working we are doing. The two most common

setups are:


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1. Seminar setup - half the class works independently while the other half works in seminardiscussion groups.

2. Project groups - where students are setup in groups of four to collaborate more effectively

I am given a lot of freedom to decide how to go about designing a math program that meets my

strengths as a teacher and the needs of my students. I have tried many different approaches and

tweaks and with each year bring what I like and what has worked from years past to bear on the

current year.

In my first three years of teaching I used a program called Judo Math, developed by another

teacher in my organization. This year I have explored and experimented with problem based

learning along with computer aided procedural practice. I plan to expand and refine this model

next year.


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Methods

Overview

The action research question I am studying is: What effect does an approach to teaching

mathematics that explicitly emphasizes problems, equitable collaborations, presentations of

thought, and the construction of knowledge; have on students ability to think critically, the

development of their problem solving skills and strategies, and on their perceptions of

themselves as mathematicians? I will use the following methods to collect data:

surveys interviews work samples observations

The focus of my research will be my 8th grade mathematics students. I have a class of 56 eighth

graders. I will not focus on a single class because I often mix the classes up. I will also decide on

a smaller representative group consisting of four students. This group of four will be the focus of

my observational, work sample and interview data collection.

In order to reach a conclusion about the effects of problem based mathematics on my classroom I

will look at my students attitudes towards mathematics, their ability to identify and apply

strategies of problem solving, their opinions about what mathematics is and what math students

do, and their attitudes about their own mathematics abilities.


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Surveys

I will give a whole class survey three times, at the beginning middle and end of the research

period. The purpose of these surveys is to capture and compare at three points along the way

across my entire classroom. Topics to be surveyed include:

How they view mathematics (open ended, and scaled) How they view themselves as mathematicians How well math class meets their needs (pacing, attention, challenge)

Interviews

I will choose four students who are representative sample of my classroom and conduct three

interviews with each student throughout the research period. I will attempt to collect personal

reflections, which supply evidence about their attitudes toward mathematics and their attitude

about themselves as mathematicians. In addition I want to see how they are experiencing the

problem centered approach and what effect the different approaches we might try throughout the

year have on them.

Interviews will provide an opportunity to delve more deeply into questions that might not be

addressed in surveys or which lack detail in journals. The might also provide opportunities to

expand on ideas brought up in journals or during observations.

Work samples

All students will keep problem journals that will contain the evidence of their problem solving

work . Additionally, every other week they will choose one problem to journal about in more


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detail. I will look at the journals of my representative sample group for evidence of critical

thinking and the use of problem solving skills and strategies.

Observations

I will videotape my students during collaborative discussions and presentations of problem work.

I will study these videos and look for evidence of critical thinking, the use of problem solving

skills, and behavior that might suggest perceptions about mathematical dispositions and

confidence. Additionally, I will keep a journal during these discussions and record my own

observations of the same nature.


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References

Boaler, J. (2008). Promoting 'relational Equity' And High Mathematics Achievement Through

An Innovative Mixed-ability Approach.British Educational Research Journal, 34(2), 167-194.

Hmelo-Silver, C. E., Duncan, R., & Chinn, C. A. (2007). Scaffolding and Achievement in

Problem-Based and Inquiry Learning: A Response to Kirschner, Sweller, and Clark (2006).

Educational Psychologist, 42(2), 99-107.

Kirschner, P. A., Sweller, J. a., & Richard, E. (2006). Why Minimal Guidance During Instruction

Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based,

Experiential, and Inquiry-Based Teaching. Educational Psychologist, 41(2), 75 - 86.

Lockhart, P. (2009). A mathematician's lament. New York, NY: Bellevue Literary Press.

Mukhopadhyay, S. (2011). Participatory and Dialogue Democracy in U.S. Mathematics

Classroom. Democracy & Education, 18(3), 44-50.

Phillips Exeter Academy | Mathematics. (n.d.). Phillips Exeter Academy | Home. Retrieved May

1, 2013, from http://www.exeter.edu/academics/72_6532.as

Savery, J. (2006). Overview of Problem-based Learning: Definitions and

Distinctions.Interdisciplinary Journal of Problem-based Learning , 1(1), 9-20.


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Strobel, J. , & van Barneveld, A. (2009). When is PBL More Effective? A Meta-synthesis of

Meta-analyses Comparing PBL to Conventional Classrooms.Interdisciplinary Journal of

Problem-based Learning, 3(1).

Schettino, C. (n.d.). Problem-Based Learning | Carmel Schettino. Carmel Schettino |

Mathematics education research, math classroom tips and events. Retrieved May 1, 2013, from

http://www.carmelschettino.com/wp/research/problem-based-learning/

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Proposal - Bryan Harms - GSE 2013