Deep experience of Mathematics: Impact on Teachers · Ratings consistently 3.8, 3.9 out of 4. 80% learned new content. Teachers • Made connections did not know exist • Increased

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.


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


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


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


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.”



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.



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.



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.



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.



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.


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.


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.


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.


But the effects are

visible:

Data supported claims


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


WilliamMcCallum

Goals

Components


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


WilliamMcCallum

Goals

Components


Certificateprogram

Postdocprogram

Components of ATI

Master’s Degree in Middle School Mathematics Leadership

Certificate in Mathematics Mentoring

Postdoctoral Fellowship in Teacher Preparation


WilliamMcCallum

Goals

Components


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


WilliamMcCallum

Goals

Components


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


WilliamMcCallum

Goals

Components


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


WilliamMcCallum

Goals

Components


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

, . . .


WilliamMcCallum

Goals

Components


Certificateprogram

Postdocprogram

(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.]


WilliamMcCallum

Goals

Components


Certificateprogram

Postdocprogram

Research question

How do you measure this interaction?


WilliamMcCallum

Goals

Components


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


WilliamMcCallum

Goals

Components


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


WilliamMcCallum

Goals

Components


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

Documents

Deep experience of Mathematics: Impact on Teachers · Ratings consistently 3.8, 3.9 out of 4. 80% learned new content. Teachers • Made connections did not know exist • Increased