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Deep experience of
Mathematics:
Impact on Teachers
Gail Burrill
Michigan State
University
(Park City
Mathematics
Institute)
Al Cuoco
Education
Development Center
(Focus on
Mathematics)
Jim Lewis
University of
Nebraska
(Math in the Middle)
Bill McCallum
Arizona Teacher
Institute
Facilitator: Glenn Stevens
Panelists
Deep Experience of Mathematics
• as exploration and problem-solving
• as an empirical science
• as a community activity
• as mathematicians experience it
Teachers experiencing mathematics
• emphasis on learning and doing mathematics
• strengthening mathematical habits of mind
• low threshold, high ceiling
• deeply personal engagement in mathematicalideas
Key Features
Beliefs about the
Nature of Mathematics
• Mathematics is natural
– The empirical nature of mathematics
– People do mathematics naturally
• Mathematics exists independent of us
– We can perform experiments, explore, and investigate
– We can test ideas and decide for ourselves
• Experience precedes formality
– Definitions and theorems are capstones
– Language is a tool for coming to terms with experience
• Mathematics is the study of structure
– Operations, order
– Shape
– Continuity
– Transformation
• Mathematics is the art of figuring things out
Claims / Questions
Experiences of “immersion” in mathematics have thefollowing impact on teachers:
! Teachers’ beliefs and attitudes about mathematics willchange.
• Mathematics as a part of nature / a language / a tool / an art.
• Mathematics is built around ideas with multiple threadsconnecting those ideas
• Mathematics is the art of figuring things out / sense-making /exploring / asking questions.
! Teachers’ beliefs and attitudes about themselves as teachers of mathematics will change.
! Mathematics as a professional / community / individualactivity.
! Mathematics as a lifelong learning experience.
! “Advanced mathematics” as an accessible learning experience
! Look for the big ideas and arrange classrooms accordingly
Claims / Questions
! Teachers’ beliefs and attitudes about their students aslearners of mathematics will change.
! All students can achieve at high levels in mathematics;
! Students can enjoy doing mathematics;
! Teachers gain deeper insight into how students think / reason /learn and solve problems;
! Teachers learn the “meaning” of student questions and developstrategies for mining student ideas;
Claims / Questions
A Few Questions
– What are the key parameters of a “deep mathematical experience”?
• mix of mathematics and pedagogy
• Beliefs about mathematics -- Body of knowledge vs. Habits of Mind
• “duration” of the experience?
– Who are these experiences for? (All teachers? Leaders?)
– Do/How do these experiences transfer to the classroom?
– What impact do mathematics-focused school communities have on teacherprofessionalism and on recruitment and retention?
– Experimentation with various forms of immersion
• Teacher Institutes
• Math circles and study groups
– What is the role of mathematicians?
– How do these strategies generalize to the other sciences?
Deep Experience of Math:
Impact on Teachers
Gail Burrill
Michigan State University
Institute for Advanced Study/Park City
Math Institute
Impact on Teachers
Michigan State University
Institute for Advanced Study/Park City
PD3
2008 PD3: Supported by NSF ESIE-0554309 and NSF Cooperative Agreement EHR-0324808
PCMI Programs• Three-week residential Summer Session/Core
Program & Cross-Program activities
• Publication series
Mathematics
Research
Graduate
Undergraduate
Undergraduate faculty
Mathematics Education
Secondary Teachers
International Seminar
Math Education Research
Cross-Program
ActivitiesActivities
• International Seminar/SSTPparticipants
• Research mathematicians conduct
discussion groups with SSTP
participants
• Pizza and Problem Solving
• Clay Institute Lectures
• SSTP Working Group takes
course for undergraduate math
faculty
PCMI: Secondary School
Teachers Program• Three Week Summer Session for Secondary
Teachers
• MSP PD3: PCMI and Districts Partner to DesignProfessional Development
• Professional Development Outreach Groups(PDO)
University of Minnesota
San Jose State University
Harvey Mudd College
St.Peters College, New Jersey
University of Washington
University of Utah
SSTP
55 to 60 Secondary Teachers from PD3, PDO and
at large - selected through an application
process
E-tables with PD3 sites
Groups of 5-6 per table, microphones, norms
Reflect on Practice
• Use artifacts of practice to ground discussion
• Work together to discuss and design problems
and lessons
• Consider research related to teaching and
learning mathematics
Working Groups
Investigating Geometry Learning from TeachingCases
Visualizing FunctionsReasoning from Data andChance
Implementing LessonStudy
Exploring DiscreteMathematics
Produce a resource for colleagues
Deepen knowledge of
mathematics
PCMI daily 2 hour course
using materials prepared
by EDC team
•Problem based approach using the Ross model
•Taught by former and current classroomteachers
•Work done collaboratively in groups consistingof 6th grade to calculus teachers
Deepen knowledge of
mathematics
PDO groups
New Jersey Shore Summer Program
PD3 academic year activities
PDO leaders
Courses at Texas State, Harvey Mudd, University of Washington
Other PD providers
PROM/SE MSP Summer Math Academy,
Honduras Summer Grad Class
Evidence Sources
Summer program exit surveys
Academic year interviews (TPC evaluation)
Site visits/baseline data about teachers’ beliefs and practices (PD3 evaluators)
Anecdotal information from applications,PDO groups, activities report
PCMI Math Forum list serve
Components of the course
Content
related to high schoolmath but not directlyin the curriculum,emphasizing connections to acentral math concept
Instruction
facilitated and managednot lecture
Context
working in groupsfacilitated by tableleaders (teachers andmathematicians)
Content examples
2008: From Algebra to Geometry investigates number
theory, algebraic geometry, and analytic geometry as a
springboard into the structure of different algebraic
systems and geometric curves. (Algebraic Geometry)
2007: Developing Mathematics: Probability Through
Algebra explores and makes connections among
questions about randomness, binomial expansion and
the probability that two positive integers, chosen at
random, have no common factor. (Statistical Mechanics)
2006: Some Applications of Geometric Thinking looks
at basic geometric habits of mind like continuous change
and things that don't change, and how these apply to a
wide variety of situations. (Topology)
mathforum.org/pcmi/
Impact on teachers’ math
changed ‘habits of mind.’
1) learning to think in a new way,
2) developing the habit of questioning
problems or concepts and asking how
to determine if a statement was true
and why,
3) seeing the elegance provided by a
deepened understanding of
mathematics, and
4) learning to stop and listen to others’
ways of thinking (TPC survey)
Impact on teachers’ math
Ratings consistently 3.8, 3.9 out of 4. 80% learnednew content. Teachers
• Made connections did not know exist
• Increased depth of knowledge
• Learned things “I did not know I did not know”
• Learned new content - fundamental theorem ofalgebra, geometry of complex numbers, Farreynumbers
• Learned that they “knew less mathematics thanthey thought they did”
• Had “forgotten how useful polynomials were”(Exit
survey; interviews)
Instruction: Reaching all
Organization of materials
Important stuff, neat stuff, tough stuff
Training and at least weekly reflection for tableleaders
Careful selection and regular rotation of groupsmatching both table leaders’ strengths andparticipants’ needs
Instructors key in supporting participants: listen,respond, highlight interesting strategies and struggles
Instruction
•Participants encouraged to take responsibilityfor their own learning
•Individuals’ thinking nurtured while, at the sametime, the daily work was collaborative
•Sufficient time for participants to explore themathematics before a class discussion
•While encouraging participants to work on theproblems on their own first and rarely providingstrategies to pursue, the instructors and thetable leaders simultaneously provided guidanceand helped participants deepen their thinkingand expand their ideas. (TPC interviews)
Impact on teachers’ thoughts
about instruction
• “How much learning can happen with so littleinstruction”
• “Learning by doing”
• “… both learning and doing”
• “Good to be reminded of what it means to bea student”
• Being challenged
• Learned to manage group work that isproductive
Exit surveys
Reflected on practice
• all right for students to work alone
• students working on own and in groups leads
to deeper learning than ‘telling’ them
• students who are quiet may know what is
going on.
• begin learning in concrete ways before
abstractions are introduced
• messing around with ideas and patterns
important in learning mathematics
• Need experiences that lead to deep
understanding (academic year interviews)
Approach to teaching
• Teaching for understanding
• More time for students to explore
mathematics
• Less time at the blackboard as a teacher and
more time walking around and working with
students
• Ask students to write more to explain their
thinking
• Used enriched problem sets with students
who struggle
• More care with language (academic year interviews)
Context for learning: a
community
1) The table design and rotating assignmentsencouraged participants to meet and workwith each other
2) The instructors and the table leaders referredparticipants to others working on the samequestions/problems (exit survey)
3) Residence math nights
4) PD3 groups work on problems togethersometimes led by mathematicians,sometimes by one of the group
Continuing the learning
community
• Contact with other participants- at the local level through PDOs
- at the national level to develop workshops or
other materials
• Returning participants felt connections withstaff and with other ‘veterans.’
• With instructors to develop a seminar
• With table leaders
• Math Forum list serveTPC survey/tracking list serve
Social Networks
Gathering Evidence:
Challenges
• Volunteer/selection nature of participants andknowledge of status quo
• Lack of neutral follow up during academicyear for at large and PDO participants
• Instruments
• Defining what to measure as success
• Measuring growth in content knowledgegiven nature of course
Math in the Middle
Institute Partnership
• A 25-month masters program that educates and supportsteams of outstanding middle level math teachers who willbecome intellectual leaders in their schools, districts, andESUs.
• A major initiative to provide evidence-based contributions toresearch on learning, teaching, and professionaldevelopment.
• A special focus on rural teachers, schools, and districts.
M2 Goal
Invest in high-quality teachers in order to improve K-12student achievement in mathematics and to significantlyreduce achievement gaps in the mathematical performanceof diverse student populations.
MM22 Partnerships Partnerships
People and OrganizationsPeople and Organizations
• All 14 ruralEducational ServiceUnits plus LPS
• 65 Local Districts
• 91 Schools
• 130 Teachers– 60 have earned
Masters Degree
– 95% retention rate
Typical Cohort 5th 6th 7th 8th 7-12 HS
32 teachers 7 7 5 7 3 3
The Challenge Our Institutes Face
• What Mathematics do Teachers “Need toKnow” and How Should They “Come toKnow” Mathematics?
– What does it mean to offer challenging courses andcurricula for math teachers?
– How do we help teachers translate the mathematicsthey come to know into classroom practice that leadsto improved student learning?
Math in the Middle Institute
Partnership
M2 courses focus on these objectives:
• enhancing mathematical knowledge
• enabling teachers to transfer mathematics
they have learned into their classrooms
• leadership development and
• action research
M2 Institute Courses
• Eight new mathematics and statistics courses designedfor middle level teachers (Grades 5 – 8) including:– Mathematics as a Second Language
– Functions, Algebra and Geometry for Middle Level Teachers
– Experimentation, Conjecture and Reasoning
– Number Theory and Cryptology for Middle Level Teachers
– Using Mathematics to Understand our World
• Special sections of three pedagogical courses:– Inquiry into Teaching and Learning
– Curriculum Inquiry
– Teacher as Scholarly Practitioner
• An integrated capstone course:– Masters Seminar/Integrating the Learning and Teaching of
Mathematics
Math in the Middle
Institute Design
Summer Fall SpringWk1 Wk2&3
Yr 1 M800T Teac801 & M802T M804T Teac800
Yr 2 M806T M805T & Stat892 Teac888 M807T
Yr 3 M808T Teac889/M809T
and the Masters Exam
- A 25-month, 36-hour graduate program.
M2 Courses
SUMMER
• Offer 1 and 2 week classes.
• Class meets from 8:00 a.m.
- 5:00 p.m.
• 35 teachers – 5 instructors
in class at one time.
• Substantial homework each
night.
• End-of-Course problem set
– Purpose – long term
retention of knowledge
gained.
ACADEMIC YEAR
• Two-day (8:00 – 5:00) on-
campus class session.
• Course completed as an on-
line, distance education
course using Blackboard
and Breeze.
– Major problem sets
– End-of-Course problem
set
– Substantial support
available for teachers
The Habits of Mind of a mathematical thinker
A person with the habits of mind of a mathematicalthinker can use their knowledge to make conjectures, toreason, and to solve problems.
Their use of mathematics is marked by flexibility ofthinking paired with the belief that precise definitions areimportant. They make connections between a problemthey are trying to solve and their mathematicalknowledge. When presented with a problem to solve,they will assess the problem, collect appropriateinformation, find pathways to the answer, and be able toexplain that answer clearly to others.
“Habits of Mind” definition cont.
While an effective mathematical toolbox certainlyincludes algorithms, a person with well developed habitsof mind knows both why algorithms work and underwhat circumstances an algorithm will be most effective.
Mathematical habits of mind are also marked by easeof calculation and estimation as well as persistence inpursuing solutions to problems. A person with welldeveloped habits of mind has a disposition to analyzesituations as well as the self-efficacy to believe that he orshe can make progress toward a solution.
This definition was built with help from Mark Driscoll’s book,Fostering Algebraic Thinking: A guide for teachers grades 6-10.
A sample
“Habits of Mind” problem
The Triangle Game: (Paul Sally, U. Chicago) Consideran equilateral triangle with points located at each vertexand at each midpoint of a side. The problem uses the setof numbers {1, 2, 3, 4, 5, 6}. Find a way to put one of thenumbers on each point so that the sum of the numbersalong any side is equal to the sum of the numbers alongeach of the two other sides. (Call this a Side Sum.)
– Is it possible to have two different Side Sums?
– What Side Sums are possible?
– How can you generalize this game?
(more) Habits of Mind Problems
• Math 802T: An (un)common solution: Find a positive
integer which if divided by 2 leaves a remainder of 1, divided
by 3 leaves a remainder of 2, divided by 4 leaves a remainder
of 3, divided by 5 leaves a remainder of 4, divided by 6 leaves
a remainder of 5, divided by 7 leaves a remainder of 6,
divided by 8 leaves a remainder of 7, and divided by 9 leaves
a remainder of 8. Is there more than one solution? (an infinite
number?) If so, find the smallest positive integer solution.
• Math 805T: There are 27 different three digit numbers that
can be made from the digits 1, 2, and 3 – 111, 121, 312, etc.
Use graph theory to determine how to place nine 1’s, nine 2’s,
and nine 3’s on a circle so that each of the 27 triples appears
exactly once when all sets of three consecutive digits around
the circle are read in a clockwise direction.
End-of-Course Problem Sets
• Math 800T: Superman flew from Metropolis to Gotham City at 300
km/hr. He flew back from Gotham City to Metropolis at 600 Km/hr
(traveling the same distance each way). What was his average
speed for the round trip?
• Math 800T: Argue that between any two rational numbers you can
find both a rational number and an irrational number.
• Math 804: A cube with edges of length 1 is inscribed in a sphere.
What is the radius of the sphere? For an extra credit, find the radius
of a sphere with an inscribed tetrahedron with edges of length 1.
• Math 804T: A circle passes through the vertices of an isosceles
triangle with two sides of length 3 and a base of length 2. What is
the area of the circle? Partial credit for a decimal approximation
(correct to two places); full credit for mathematical reasoning that
gives an exact answer.
• Two options for the Masters Degree
– MAT (Specialization in the teaching of middlelevel mathematics (Mathematics Department)
– MA (Teaching, Learning and Teacher Ed.)
• Masters exam in mathematics
– Take home exam (two math questions)
– Action Research Project Report (5-8 pages)
– An 8-10 page expository mathematics paper
– An oral presentation about the paper
M2 Masters Degrees
A Sample Masters Exam Question
A math class with “n” students sits in a circle to playmathematical chairs. The students choose anelimination number “d” and then count off in order, 1, 2,3, … . When the count gets to d, that student iseliminated from the game. The next student starts thecount over and the students count 1, 2, 3, … . Again,when the count gets to d, that student is eliminated.Continue in this manner until only one student is left.That student wins the game.
Where should you sit in order to win the game?
Hint: Solve the problem first for elimination number 2 or3 and then try to solve it for elimination number d.
Note: This is a version of The Ring of Josephus problem.
More sample Masters Exam Questions
• Find all instances where three consecutive
entries in a row in Pascal’s Triangle are in the
ratio 1 : 2 : 3.
• Suppose we roll some regular 6 sided dice.
– How many different outcomes are possible if we roll
• 8 identical dice?
• n identical dice?
• n identical dice and each of the numbers 1, 2, 3, 4, 5, and 6
show up at least once?
Sample Titles of Master’s Papers
• The Volume of a Platonic Solid
• Pythagorean Triples
• The Polygon Game
• Farey Sequences, Ford Circles, Pick’s Theorem
• Vigenere Cipher
• Heron, Brahmagupta, Pythagoras, and the Law
of Cosines
• Fourier Series
Side Sum Solutions for Hexagons
Side Sum 17: 3, 8, 6, 4, 7, 9, 1, 11, 5, 10, 2, 12
Side Sum 18: None
Side Sum 19: 6, 2, 11, 5, 3, 9, 7, 4, 8, 10, 1, 12
And 4, 10, 5, 8, 6, 2, 11, 1, 7, 9, 3, 12
And 5, 11, 3, 9, 7, 4, 8, 10, 1, 6, 12, 2
And 3, 9, 7, 11, 1, 10, 8, 6, 5, 2, 12, 4
Side Sum 20: 7, 11, 2, 8, 10, 4, 6, 9, 5, 3, 12, 1
And 9, 3, 8, 5, 7, 11, 2, 12, 6, 4, 10, 1
And 8, 2, 10, 4, 6, 9, 5, 3, 12, 7, 1, 11
And 10, 4, 6, 2, 12, 3, 5, 7, 8, 11, 1, 9
Side Sum 21: None
Side Sum 22: 10, 5, 7, 9, 6, 4, 12, 2, 8, 3, 11, 1
Patterns with Minimums & Maximums
Polygon Minimum
Side Sum
To find the
next
Minimum
Maximum
Side Sum
To find the
next
Maximum
Triangle 9 +3 12 +3
Square 12 +2 15 +4
Pentagon 14 +3 19 +3
Hexagon 17 +2 22 +4
Heptagon 19 +3 26 +3
Octagon 22 29
A Solution for an n-sided polygon, n odd
• General solution for an n-gon where
n = 2k + 1, n odd
• For a Heptagon Solution, n = 7; k = 3
To find the vertices begin with 1, move
clockwise by k each time, and reduce
mod n. The midpoints begin with 2n
between 1 and 1+k and move
counterclockwise, subtracting 1 each
time. For a heptagon, the
Side Sum = 5k + 4.
114
4
8
7
9
3106
11
2
12
513
M2 Research Questions
• What are the capacities of teachers to translatethe mathematical knowledge and habits of mindacquired through the professional developmentopportunities of M2 into measurable changes inteaching practices?
• To what extent do observable changes inmathematics teaching practice translate intomeasurable improvement in studentperformance?
Using the LMT to Measure Teacher
Knowledge
Cohort 1 Cohort 2
• Percentage Pre Post Pre Post
<40% 1 0 3 0
40s 1 1 6 0
50s 3 1 3 3
60s 5 5 2 7
70s 11 9 10 9
>=80 7 12 4 9
Total 28 28 28 28
Studying our teachers’ practice
• We have several qualitative research studies of teacher
practice being conducted by:
– Ruth Heaton (co-PI)
– Graduate Students
• Wendy Smith
• Yolanda Rolle
• David Hartman
• School Leadership is being studied in a joint project with
the RETA at Northwestern University
• A study of teaching mathematics in rural schools
Investigating student achievement
• A study of the impact of Math in the Middle
teachers on student achievement in LPS middle
schools is led by Walt Stroup.
• An “alternative assessment” seeks to learn
whether students taught by Math in the Middle
teachers are developing the desired “habits of
mind of a mathematical thinker.”
Teachers as
Mathematicians:
The case of
Focus on
Mathematics
Slides will be available at
http://focusonmath.org
http://www2.edc.org/cme/showcase
Getting the Language
right
.
FindingsScaffoldedLogically
ChallengesMotivatedInquiry
ResultsDrivenData
ClaimsBasedResearch
OutcomesSupportedEvidence
CBA
Pick one from each column
Focus on Mathematics
Our Approach
!Depth over breadth
Teachers experience sustained immersion in mathematics.
!Focus on mathematics
Everything we do revolves around mathematics.
!Capacity building
Teachers learn to drive professional development.
!Community building
Mathematicians, teachers, and educators work and learn together.
Focus on Mathematics
Our Programs
The programs are designed to…
! help teachers develop a profession-specific
knowledge of mathematics for teaching,
! engage teachers in rich and ongoing
mathematical experiences,
! and establish a lasting mathematical community
among mathematicians and teachers.
The Immersion Experience
• as a community activity
• as an empirical science
• as exploration
Teachers and mathematicians experiencing mathematics
• emphasis on learning
• strengthening mathematical habits of mind
• deeply personal engagement in mathematicalideas
Key Features
Focus on Mathematics
Examples of immersion:
• The summer program
• Study groups
To Think Deeply of
Simple Things
Arnold E. Ross
On the First Day
“The first weeks of the program, I could connect to
things I knew. Even if I was frustrated one day, the
next day I'd have an epiphany - there were lots of
ups and downs. Understanding math concepts
was not enough, you had to look at things in
different ways. It's not necessarily intuitive. I
learned a lot about my own patience. Every time I
felt frustrated, I realized something that I wouldn't
have realized without being frustrated.”
FoM Middle School Teacher
The Experience
“A lot of us didn't feel we were
prepared for the summer program . . .
Afterwards we felt we could do
anything.”
FoM Middle School Teacher
I have higher expectations for my students and am more
willing to give them time to struggle and provide time for
them to solve hard problems independently. More
generally, I have greater confidence in my students and in
myself.
Professionally, when I began teaching, I felt that teaching
and research were mutually exclusive. Happily, through
experience, PROMYS helped me to understand that
teachers can and should think deeply about higher
mathematics. Teaching and research are complements
rather than substitutes.
FoM High School Teacher
Focus on Mathematics
Study Groups
Evolving Roles of Participants and Mathematicians
! Mathematicians working with teachers as colleagues
! Sharing expertise
! Connecting to mathematics for teaching
! Increasing active involvement by teachers
! Teacher-led sessions
The Study Group at
Watertown High
The Study Group at
Wright Middle
School
The Study Group at
Lawrence High
Focus on Mathematics
3. The FoM Mathematical Community
What People are Saying
It is hard to capture with words theenthusiasm that FoM has created forworking together as “mathematicianswho are also teachers ofmathematics.”
Focus on Mathematics
What People are Saying
It is the best “professional development” that I have been involvedin throughout my 35-year teaching career. I guess the besttestament for the success of FoM comes from the continuedattendance of so many Lawrence High School teachers. Wecontinue to talk about the topics discussed at our study groups longafter the weekly session is over.
Focus on Mathematics
What People are Saying
To this end, it has transformed myteaching from one of a competitivenature amongst students to one ofinclusion. I adopted this viewpointafter my experience with thePROMYS program this pastsummer. I try to inject as muchenthusiasm into my teaching ofmathematics as I can, in conjunctionwith a lot of encouragement of mystudents. I think it's true that if you'retold often enough that you stink atsomething, you come to believe it.Much of my job as a teacher ofmathematics is trying to undo thedamage.
Focus on Mathematics
What People are Saying
FoM has changed the way I teach. My students spend a lotless time grinding through worksheets. I am also muchmore likely to answer a question from a student or peer witha question rather than an explanation.
Focus on Mathematics
What People are Saying
We talk about these study group problems when we are not in thestudy group. If we happen to have lunch together or we have thesame prep time we talk about them and the different things weare doing in class.
Focus on Mathematics
What People are Saying
We have terrific rapport among 2 high school teachers andmyself, both personal and intellectual. That we foundprograms we can work on together as equals is really apleasant surprise. It may sound arrogant, but I wouldn’thave thought it would work so well given that I’m the bigshot professor. Once you pick a problem that’s accessible,I can’t use my fancy tools and we’re all at the same level.
Focus on Mathematics
Effects on teachers: Evidence based findings
• Teachers’ mathematical knowledge provides
the confidence needed to adapt curricula.
• Teachers assume the role of program
decision maker versus ‘implementer’ of the
curriculum.
• Teachers collect, create effective problem
sets that work for the range of their students’
prior mathematical knowledge and
experience.
• Teachers design lessons in which students
engage in mathematical experiences.
Focus on Mathematics
Effects on teachers: Evidence based findings
• Teachers hold a strong belief that students can
learn effectively from one another.
• Teachers encourage students to explore
alternative approaches to solving problems, and
they use student presentations to advance the
learning of all students in the classroom.
• Teachers frequently spend considerable class
time observing, posing questions to, and working
with individual students and groups of students.
• Teachers plan extensively, but they continually
adapt their plans based on students’ work.
Focus on Mathematics
Effects on student achievement: Data driven challenges
• FoM’s immersion programs intervene at the teacher
level, so any measure of student achievement is
mediated by teachers.
• The immersion programs engage teachers over
time. At what point in time would you expect to see
changes in student achievement?
• Student exposure to teachers that have completed
an immersion experience is likely to be one year.
• There is an absence of adequate measures of
student achievement that align with the goals of the
immersion programs.
• There is no coherent research program on the influence
of immersion experiences for teachers on student learning.
Focus on Mathematics
But the effects are
visible:
Data supported claims
Focus on Mathematics
Directions for research:• The effects of immersion
• Immersion at the K–6 level
• The role of a mathematical community
• The effects on student achievement
• The effects on teacher-led professional development
ArizonaTeacherInitiative
WilliamMcCallum
Goals
Components
Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
The Arizona Teacher Initiative at the
Institute for Mathematics and Education
William McCallum1 Daniel Madden1 (PI)Rebecca McGraw1 Erin Turner 2 Roger Pfeuffer3
1Department of Mathematics, University of Arizona
2College of Education, University of Arizona
3Tucson Unified School District
MSP Learning Network Conference, 2008
ArizonaTeacherInitiative
WilliamMcCallum
Goals
Components
Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
Goals of ATI
Middle school teachers with a profound understanding ofmiddle school mathematics and with leadership skills
A sustainable, replicable Master’s program for producingmiddle school mathematics teacher leaders
University faculty able so support effective teacherpreparation and professional development
A distance-learning version of the Master’s program thatcan be implemented nationally
A national corps of high school teachers andmathematicians who can implement courses for theMaster’s program in their areas
ArizonaTeacherInitiative
WilliamMcCallum
Goals
Components
Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
Components of ATI
Master’s Degree in Middle School Mathematics Leadership
Certificate in Mathematics Mentoring
Postdoctoral Fellowship in Teacher Preparation
ArizonaTeacherInitiative
WilliamMcCallum
Goals
Components
Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
Master’s Degree
ParticipantsCohorts of 10–15 middle school teachers per year (mostlyelementary certified)
Program (part-time, 3 years)Content courses (16 units)
Number and Operations
Algebra
Geometry
Probability and Statistics
Leadership and mentoring (3–4 units)Mathematics Mentoring Methods
Mathematics Professional Development Models
Research (12 units)Research on Student Learning
Methods of Research
Thesis or practicum integrated into classroom teaching
ArizonaTeacherInitiative
WilliamMcCallum
Goals
Components
Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
Number and Operations
Yoga:See the development of number systems as based on asmall number of unifying mathematical lawsSee underlying abstract mathematical constructions inmiddle school mathematics materialsRead and understand new materials at levels above andbelow the middle school curriculum, and adapt newapproaches and ideas to the middle school curriculum
Content:The Natural NumbersThe IntegersThe Rational NumbersIrrational numbersReal Numbers
ArizonaTeacherInitiative
WilliamMcCallum
Goals
Components
Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
Algebra
Yoga:Read, contemplate, and interpret expressions and equationsDevelop algebraic intuition and foresightMake connection between algebraic representations andgraphical, numerical, and verbal representations
Content:algebraic expressions and equationsthe coordinate plane and graphinglinear functions and equationsexponential functions and equationsquadratic functions and equationslogarithmssystems of linear equations.
Sample activity
ArizonaTeacherInitiative
WilliamMcCallum
Goals
Components
Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
Sample activity from algebra course
Problem
The expression
0.6
�t1 + t2 + t3
3
�
is the contribution to a student’s final score from three testscores. What is a different way of writing this? Which wayshould a student use in order to
calculate the total test contribution to their final grade
calculate the effect of getting 10 more points on test 2
Responses
0.6
�t1 + t2 + t3
3
�, 0.2t1 + 0.2t2 + 0.2t3,
t15
+t25
+t35
, . . .
ArizonaTeacherInitiative
WilliamMcCallum
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Master’sDegreeNumberAlgebraResearch
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(1) 0.6
�t1 + t2 + t3
3
�(2) 0.2t1 + 0.2t2 + 0.2t3
Student A: I wrote (2) because I thought that the originalexpression said the average of the 3 tests was worth 60%, soeach test was worth 20%. But I’m not sure it is right.Student B: (1) and (2) are obviously the same!Student A: How you can see that just by looking at them?Student B: You just move the 3 over so it’s dividing the 0.6,which gives you 0.2, then distributed the 0.2.Instructor: How do you know you can move the 3 over? Whatrule says you can do that?Student B: Isn’t it because you only have division andmultiplication, so it’s the commutative law?Instructor: But division isn’t commutative.Student C: But you can write division as multiplication. Justwrite it as multiplication by 1/3.Student A: Oh yeah! [Discussion shifts to associative law.]
ArizonaTeacherInitiative
WilliamMcCallum
Goals
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Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
Research question
How do you measure this interaction?
ArizonaTeacherInitiative
WilliamMcCallum
Goals
Components
Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
Research on Student Learning
Course StructureOrganized so as to be responsive to participants interestsFocused on developing an investigative stance towardstudent thinkingIncluded individual and small group components
EvaluationFormative and summative componentsPublic presentation of knowledgeAssessment of oral and written communication
ArizonaTeacherInitiative
WilliamMcCallum
Goals
Components
Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
Certificate in Mathematics Teacher Mentoring
ParticipantsTwo secondary-certified mathematics teachers per year
Program (full-time, 1 year)Teaching/assisting in Master’s coursesUniversity mathematics course analysisApprenticeship in teacher mentoring program
ArizonaTeacherInitiative
WilliamMcCallum
Goals
Components
Master’sDegreeNumberAlgebraResearch
Certificateprogram
Postdocprogram
Postdoc in Mathematics Teacher Preparation
ParticipantsTwo post-doctoral fellows (Ph.Ds in mathematics)
Program (full-time, 3 years)Teaching/assisting in Master’s coursesTeaching departmental coursesLeading Certificate candidates’ course analyses