InterMath Professional and Cognitive Development through Problem Solving and Technology http://www.intermath-uga.gatech.edu/

Evan Glazer Learning and Performance Support Laboratory University of Georgia [email protected]

http://www.arches.uga.edu/~eglazer/NECC2001.html

Overview

     My Background About InterMath Professional Development Cognitive Development Teachers’ Activities http://www.intermath-uga.gatech.edu/

Background

 Doctoral Candidate in IT at UGA http://www.intermath-uga.gatech.edu/

Baptism Bulldawg Style

Background

 Doctoral Candidate in IT at UGA  InterMath - problem solving with TEMIs.

http://www.intermath-uga.gatech.edu/

Under the direction of Jim Wilson and others http://www.intermath-uga.gatech.edu/

Background

 Doctoral Candidate in IT at UGA  InterMath - problem solving with TEMIs.

 Taught high school mathematics in Glenview, Illinois http://www.intermath-uga.gatech.edu/

You might be a “real” teacher if...

 You grade papers in the car, during commercials, in faculty meetings, in the bathroom, and (at the end of the six weeks) have been seen grading in church.  You have a disjointed neck from writing on boards without turning your back on the class.  You are written up in medical journals for the size and elasticity of your kidneys and bladder.

http://www.intermath-uga.gatech.edu/

You might be a “real” teacher if...

 You have been timed gulping down a full lunch in 2 minutes, 18 seconds. Master teachers can eat faster than that.  You know it is better to seek forgiveness than to ask permission.

 You buy Excedrin and Advil at Sam's.

 You know secretaries and custodians run the school http://www.intermath-uga.gatech.edu/

You might be a “real” teacher if...

 You hear the heartbeats of crisis; always have time to listen; know you teach students, not just a subject; and you are absolutely non-expendable.

http://www.intermath-uga.gatech.edu/

Background

 Doctoral Candidate in IT at UGA  InterMath - problem solving with TEMIs.

 Taught high school mathematics in Glenview, Illinois  Web-based development since 1993.

http://www.intermath-uga.gatech.edu/

Background

 Doctoral Candidate in IT at UGA  InterMath - problem solving with TEMIs.

 Taught high school mathematics in Glenview, Illinois  Web-based development since 1993  I have a passion for working with kids, teaching, learning, mathematics, and running http://www.intermath-uga.gatech.edu/

QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.

Mathematicians have the unique talent of determining how fast they are running, and how much faster they need to run by the end of the race in order to set a personal record.

http://www.intermath-uga.gatech.edu/

Unflattering portrait: How one child perceives mathematicians

Source: http://www.thetimes.co.uk/article/0,,2-61352,00.html

Background

 Doctoral Candidate in IT at UGA  InterMath - problem solving with TEMIs.

 Six years of teaching high school mathematics in Illinois  Web-based development since 1993  I have a passion for working with kids, learning, mathematics, and running  Research and writing interests include communities of practice and technology integration.

http://www.intermath-uga.gatech.edu/

Available September 1, 2001

Greenwood Publishing

About InterMath

    What is InterMath?

Who is involved in InterMath?

What is the rationale for InterMath?

What are the goals of InterMath?

About InterMath

Professional Development Cognitive Development http://www.intermath-uga.gatech.edu/

About InterMath

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What is InterMath?

•Who is involved?

•Rationale •Goals Professional Development Cognitive Development

What is InterMath?

    A five year, NSF-funded Internet based professional development project A community of middle school mathematics teachers and administrators An effort to strengthen teachers’ conceptual understanding A program that uses technology to explore problem solving situations http://www.intermath-uga.gatech.edu/

About InterMath

•

What is InterMath?

•Who is involved?

•Rationale •Goals Professional Development Cognitive Development

A Sad Excuse for Problem Solving

About InterMath

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What is InterMath?

•Who is involved?

•Rationale •Goals Professional Development Cognitive Development

A Consequence of Bad Problems

About InterMath

•What is InterMath?

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Who is involved?

•Rationale •Goals Professional Development Cognitive Development   

Who is Involved?

LPSL - UGA (Hannafin, Wilson)  Research scientists, SMEs, Web and curriculum developers CEISMC - Georgia Tech (Ohme)  Web designers and technical support Workshop Participants  Middle school teachers/administrators and instructors at UGA, Kennesaw State, Valdosta State, and state INTECH centers.

http://www.intermath-uga.gatech.edu/

Rationale for InterMath

 New approaches to teaching and learning

About InterMath

•What is InterMath?

•Who is involved?

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Rationale

•Goals Professional Development Cognitive Development

About InterMath

•What is InterMath?

•Who is involved?

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Rationale

•Goals Professional Development Cognitive Development

Learning over the Past 50 years

Teaching Math in 1950:

A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price. What is his profit?

Learning over the Past 50 years

About InterMath

•What is InterMath?

•Who is involved?

•

Rationale

•Goals Professional Development Cognitive Development

Teaching Math in 1960:

A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price, or $80. What is his profit?

Learning over the Past 50 years

Teaching Math in 1970: About InterMath

•What is InterMath?

•Who is involved?

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Rationale

•Goals Professional Development Cognitive Development A logger exchanges a set "L" of lumber for a set "M" of money. The cardinality of set "M" is 100. Each element is worth one dollar. Make 100 dots representing the elements of the set "M". The set "C", the cost of production contains 20 fewer points than set "M". Represent the set "C" as a subset of set "M" and answer the following question: What is the cardinality of the set "P" of profits?

Learning over the Past 50 years

Teaching Math in 1980:

A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20.

About InterMath

•What is InterMath?

•Who is involved?

•

Rationale

•Goals Professional Development Cognitive Development Your assignment: Underline the number 20.

Learning over the Past 50 years

Teaching Math in 1990:

By cutting down beautiful forest trees, a logger makes $20. What do you think of this way of making a living? Topic for class participation after answering the question:

About InterMath

•What is InterMath?

•Who is involved?

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Rationale

•Goals Professional Development Cognitive Development How did the forest birds and squirrels feel as the logger cut down the trees?

Learning over the Past 50 years

Teaching Math in 2000: About InterMath

•What is InterMath?

•Who is involved?

•

Rationale

•Goals Professional Development Cognitive Development A company outsources all of its loggers. The company claims that contracting workers will Ultimately save them on expenses for benefits and salaries when demand for their product is down. The union requests that you take a closer look at historical employment, salary, benefits, and contracting data located on the web.

Was outsourcing a good move? Use a cognitive tool like a spreadsheet or simulation to develop and support your reasoning.

Rationale for InterMath

 New approaches to teaching and learning  Insufficient content background in teacher training programs  Resilient and pervasive beliefs about teaching mathematics

About InterMath

•What is InterMath?

•Who is involved?

•

Rationale

•Goals Professional Development Cognitive Development http://www.intermath-uga.gatech.edu/

Is this what mathematics teaching is about?

About InterMath

•What is InterMath?

•Who is involved?

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Rationale

•Goals Professional Development Cognitive Development

Rationale for InterMath

About InterMath

•What is InterMath?

•Who is involved?

•

Rationale

•Goals Professional Development Cognitive Development     New approaches to teaching and learning Insufficient content background in teacher training programs Resilient and pervasive beliefs about teaching mathematics Escalating demands for teachers and little support http://www.intermath-uga.gatech.edu/

About InterMath

•What is InterMath?

•Who is involved?

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Rationale

•Goals Professional Development Cognitive Development

Teacher Support

Rationale for InterMath

About InterMath

•What is InterMath?

•Who is involved?

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Rationale

•Goals Professional Development Cognitive Development    New approaches to teaching and learning Insufficient content background in teacher training programs Resilient and pervasive beliefs about teaching mathematics  Escalating demands for teachers and little support  Difficulty attaining learning standards http://www.intermath-uga.gatech.edu/

About InterMath

•What is InterMath?

•Who is involved?

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Rationale

•Goals Professional Development Cognitive Development

Georgia Test Scores

About InterMath

•What is InterMath?

•Who is involved?

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Rationale

•Goals Professional Development Cognitive Development

The Truth

Compared to the Nation

About InterMath

•What is InterMath?

•Who is involved?

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Rationale

•Goals Professional Development Cognitive Development  “Georgia drops to 50th in nation in SAT scores

”

 “State Board of Education's goal of reaching a 1,000 mean score by 2001 is all but dead.”

-Savannah Morning News

http://www.intermath-uga.gatech.edu/

Goals for InterMath

About InterMath

•What is InterMath?

•Who is involved?

•Rationale •

Goals

Professional Development Cognitive Development     Promote innovative practices Revitalize teaching by modeling and applying innovative approaches Support reform of mathematics teaching and learning Establish a human and technological support infrastructure http://www.intermath-uga.gatech.edu/

Professional Development

    Participant Selection Workshop Procedures Our Use of Technology Additional Contributions About InterMath

Professional Development

Cognitive Development http://www.intermath-uga.gatech.edu/

Participant Selection

   Underserved schools will receive priority Schools where mathematical technologies are available Teams of middle-school teachers and administrators About InterMath

Professional Development

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Participant Selection

•Workshop Procedures •Use of Technology •Additional Contributions Cognitive Development http://www.intermath-uga.gatech.edu/

“If you can teach middle school, then you can teach anyone.”

About InterMath

Professional Development

•

Participant Selection

•Workshop Procedures •Use of Technology •Additional Contributions Cognitive Development

Workshop Procedures

About InterMath

Professional Development

•Participant Selection •

Workshop Procedures

•Use of Technology •Additional Contributions Cognitive Development    100 hours of training  45 hour workshop, 55 hour “follow along” development course Development of portfolio using a web page  Write-ups of investigations  Development of technology lessons Ongoing support system  Human (experts, community)  Electronic (listserv, bulletin board, email)  Documents (tutorials, guides)

About InterMath

Professional Development

•Participant Selection •Workshop Procedures •

Use of Technology

•Additional Contributions Cognitive Development

InterMath’s Use of Technology

…does not focus on finding answers.

InterMath’s Use of Technology

About InterMath

Professional Development

•Participant Selection •Workshop Procedures •

Use of Technology

•Additional Contributions Cognitive Development    Cognitive tool for building ideas, organizing thinking, solving problems, and exploring patterns Multiple technologies promote multiple representations Provides new learning opportunities http://www.intermath-uga.gatech.edu/

Additional Contributions

About InterMath

Professional Development

•Participant Selection •Workshop Procedures •Use of Technology •

Additional Contributions

Cognitive Development    Evolving database of investigations Teacher resources and lesson plans Discourse through electronic communities http://www.intermath-uga.gatech.edu/

Cognitive Development

    Rationale of Workshop Activities Workshop Content Investigations Teacher’s Activities About InterMath Professional Development

Cognitive Development

http://www.intermath-uga.gatech.edu/

Rational of Workshop Activities

About InterMath Professional Development

Cognitive Development

•

Rationale of Workshop Activities

•Workshop Content •Investigations •Teacher’s Activities   Intrinsically motivating  Appropriate level of challenge  Appeal to curiosity  Learner control  Realistic applications Ownership of learning  Open-ended exploration  Generative extensions of investigations

Workshop Content

About InterMath Professional Development

Cognitive Development

•Rationale of Workshop Activities •

Workshop Content

•Investigations •Teacher’s Activities    14 topics under four categories  Algebra  Geometry  Number Concepts  Data Analysis Adapted investigations from MTMS Support materials  Guides and tutorials  Database of solutions http://www.intermath-uga.gatech.edu/

About InterMath Professional Development

Cognitive Development

•Rationale of Workshop Activities •Workshop Content •

Investigations

•Teacher’s Activities

Investigation Criteria

 Multiple cases can be investigated using  technology.

Pre-Algebra students can rely on technology to investigate the situation.

 The investigation promotes generalizability or can be used as a springboard for further exploration.

 Multiple methods can be used to explore the situation.

 Multiple solutions are possible.

 The investigation is easy to start exploring.

 The investigation can be modified for use in a middle school classroom.

http://www.intermath-uga.gatech.edu/

Sample Investigation

About InterMath Professional Development

Cognitive Development

•Rationale of Workshop Activities •Workshop Content •

Investigations

•Teacher’s Activities Choose two numbers. Add them together and form a Fibonacci-like sequence, starting with your first two numbers, and ending with a total of ten numbers. For example, if your first two numbers are 3 and 5, then your third number is 8 (3+5), your fourth number is 13 (8+5), your fifth number is 21 (8+13), and so on. Determine a relationship using the seventh term and sum of the terms of your sequence. Is this true for

every

sequence of this nature? Explain.

http://www.intermath-uga.gatech.edu/

Sample Investigation

To make a pen for his new pony, Ted will use an existing fence as one side of the pen. If he has ninety-six meters of fencing, what are the dimensions of the largest rectangular pen he can make? About InterMath Professional Development

Cognitive Development

•Rationale of Workshop Activities •Workshop Content •

Investigations

•Teacher’s Activities (Source: Mathematics Teaching in the Middle School, Nov-Dec 1994).

http://www.intermath-uga.gatech.edu/

Sample Investigation

Quadrilateral ABCD is a rectangle with length to width in the ratio of 3 : 2. AN and BM are bisectors of angle A and angle B, respectively. What is the ratio of the area of the shaded region to the area of ABCD? Explain your solution. About InterMath Professional Development

Cognitive Development

•Rationale of Workshop Activities •Workshop Content •

Investigations

•Teacher’s Activities http://www.intermath-uga.gatech.edu/

Teacher Write-Up

Determining which method to use in solving this problem intrigued me. My students have been studying rotations, dilations, and reflections using a coordinate graph and triangles. This seemed a good opportunity for me to show them how these movements could be used to solve a problem….

About InterMath Professional Development

Cognitive Development

•Rationale of Workshop Activities •Workshop Content •Investigations •

Teacher’s Activities

http://www.intermath-uga.gatech.edu/

Creating Investigations

What are some issues?

 Support  Familiarity with software  Motivation  Prior knowledge  Use of technology  Structure and hints  Self and Peer-evaluation About InterMath Professional Development

Cognitive Development

•Rationale of Workshop Activities •Workshop Content •Investigations •

Teacher’s Activities

http://www.intermath-uga.gatech.edu/

Teacher Lessons

Your employer has put you in charge of the packaging for a new promotional item. If children send in 10 proofs of purchase along with $1.95 to cover shipping, they will receive a box of 125 cubic centimeters of SLIME! The packaging for the slime should be in the shape of a rectangular prism for easy mailing. What dimensions should you give this package? About InterMath Professional Development

Cognitive Development

•Rationale of Workshop Activities •Workshop Content •Investigations •

Teacher’s Activities

Problem created by Kim Seay, Dacula Middle School http://www.intermath-uga.gatech.edu/

A Reflection

About InterMath Professional Development

Cognitive Development

•Rationale of Workshop Activities •Workshop Content •Investigations •

Teacher’s Activities

... it's a different story to hear [from a teacher] "hey, I did this in my classroom, and the kids did awesome with it, and you should see what they did" than to hear somebody stand up at a lecture and they say this is how you should teach cause kids will do this and kids will do that. Um, a lot of stuff sounds good in theory, but you think, oh yeah, well how is that going to happen? You know, how could you do that? ... I think that since I have been able to do some of it, and been able to get the type of reactions from the kids, that makes me want to tell people, you know, and show people, here you can do this and it's really good.

About InterMath Professional Development

Cognitive Development

•Rationale of Workshop Activities •Workshop Content •Investigations •

Teacher’s Activities

Teacher Lessons

The bases on a baseball field are 90 feet apart and form right angles. The pitcher’s mound is 60 ft and 6 inches from home plate. If the third baseman is throwing the ball to the first baseman, should the pitcher duck his head if he is standing on the mound? Problem created by Peggy Reigle, Athens Academy

Next Steps

  What we have achieved (years 1-2)  Design and development of workshop investigations and website  Workshops in Athens/Atlanta  Teacher development of write-ups and lesson plans What we plan to achieve (years 3-5)  Implementation of workshops at various sites around Georgia  Ongoing support system of resources, tools, and people

Our Challenge Ahead

Our goal is to empower teachers with skills and strategies to use technology in a manner where mathematics is explored, learned, and applied in meaningful, useful, and resourceful ways. However, if we train teachers to use technology solely based on our vision without being aware of their learning and pedagogical needs, we may be in for an unpleasant and unfortunate surprise...

http://www.intermath-uga.gatech.edu/