Preparing Teachers for Integrated Mathematics

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Preparing Teachers for Integrated Mathematics

• Presented by
• John Olive
• Sandy Blount
• Chris Franklin
• Brad Findell
• Malcolm Adams

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PRISM Mathematics Curriculum Team Members

• Mathematics
• Malcolm Adams
• Sybilla Beckmann
• Dan Kannan
• Ted Shifrin
• NE GA PRISM
• Sandy Blount

• Mathematics Education
• John Olive
• Brad Findell
• Statistics
• Chris Franklin
• 6th -12th Teachers
• Bill Moore
• Darrel Presley

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Courses Under Review

• 14 Mathematics Courses
• 4 Math Education Courses
• 1 Statistics Course

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The Overall Goal

• We will identify opportunities
• in UGA Mathematics, Mathematics Education, and
  Statistics courses
• to highlight connections and ideas
• that help prepare teachers
• to teach algebra, geometry, and statistics in an
  integrated,
• standards-based approach.

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Worthwhile Mathematical Tasks
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The Goal for 2006-07

• The PRISM Mathematics Curriculum Team will
  develop units of instruction
• based on Math 1 and 2
• to be taught in
• Mathematics Education Courses beginning in Fall
  2008.

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Questions to Address

• What are the steps of data analysis?
• What are attributes of worthwhile mathematical
  tasks?
• How can mathematics be integrated at the high
  school level?
• How can a task be resized to fit a particular
  need?

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TASK 1

• Geometric Probability

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TASK 1 Geometric Probability

• Questions to Ponder
• Think of two different ways to calculate the area
  of the J as a fraction of the area of the 6X6
  grid.
• If you tossed the pair of dice twenty times, how
  many times would you expect to plot a point in
  the J? Would you be surprised if only one point
  landed in the J? Would you be surprised if 6
  points landed in the J?
• If you tossed the pair of dice 100 times, how
  many times would you expect to plot a point in
  the J? Would you be surprised if only 5 points
  landed in the J? Would you be surprised if 30
  points landed in the J?
• Which of the above results would be most
  surprising? Why?

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First Year Medical Student

• Today was the first day of the statistics
  module. You would have been speechless as to how
  little (if any!) statistics my classmates know.
  About 10 of the class said they had some
  statistics in high school/college with the other
  90 having NONE! When we began talking about the
  statistics I realized that of the 10 that had
  stats only about 20 (so 2 of the entire class)
  had a solid grasp on its methods.

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STATISTICAL THINKING versusMATHEMATICAL THINKING

• The Focus of Statistics on Variation in Data
• The Importance of Context in Statistics
• Guidelines for Assessment and Instruction in
  Statistics Education A Curriculum Framework for
  Pre-K-12 Statistics Education
• The GAISE Report (2007) The American Statistical
  Association http//www.amstat.org/education/gais
  e/

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THE FRAMEWORKUnderlying Principles

• PROBLEM SOLVING PROCESS
• Formulate Questions
•          clarify the problem at hand
•          formulate one (or more) questions that
  can be answered with data
•  
• Collect Data
•          design a plan to collect appropriate
  data
•          employ the plan to collect the data
•  
• Analyze Data
•          select appropriate graphical or
  numerical methods
•          use these methods to analyze the data
•  
• Interpret Results
• interpret the analysis taking into account
  the scope of inference based on the data
  collection design
•        relate the interpretation to the original
  question

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Across all Grade Levels

• Carrying the Statistical Problem Solving Process
  across all grade levels is crucial
• Its also important to carry across all grade
  levels the maturing of statistical concepts

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Mathematics and Statistics Working Together

• Statistics cannot be an add-on of convenience but
  must be integrated into the curriculum in a way
  that allows time for the development of key
  concepts at an appropriate grade level and for
  connecting and constant reinforcing of these
  concepts at higher grade levels.
• One of the benefits of teaching statistics and
  mathematics together is that statistics can
  enliven a class through real examples that
  motivate and illustrate virtually any topic in
  school mathematics.
• RESOURCES NCTM Navigation Data Analysis Books
  across all grade levels
• Quantitative Literacy (QL) Books and the
  Elementary QL Books Pearson Learning

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Conclusion

• A primary goal of emphasizing data analysis is to
  help students achieve the noble goal of being a
  sound statistically literate citizen. Instruction
  in data analysis needs to emphasize statistical
  concepts not simply tools, procedures, or
  terms. This must begin in the elementary grades
  allowing students to develop maturity in
  statistical concepts throughout their years of
  schooling.
• We cant expect our students to become
  statistically literate by taking one statistics
  course -- we spend 13 years helping our students
  mature into mathematical thinkers -- we must do
  the same for all our students to become
  statistical thinkers. The Georgia GPS has
  embraced the importance of statistical reasoning
  across the Pre-K-12 curriculum.

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TASK 1 Geometric Probability

• Questions to Ponder
• In the figure below would you expect different or
  similar results for landing a point in an orange
  square? Explain your answer.

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TASK 1 Geometric Probability

• Questions to Ponder
• What about the following arrangement of 6 orange
  squares?

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Integrated Mathematics

• Not just integration within a course, but
  integration within each unit
• New ways of thinking about mathematics
• New habits, questions, and perspectives
• Using geometry, number, and data analysis in
  service of algebra
• Using algebra, number, and data analysis in
  service of geometry
• Using algebra, geometry, and number in service of
  data analysis
• Graphs of functions provide opportunities for
  geometric reasoning
• A function does not have slope, but its graph
  does

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TASK 2

• The Olympics

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TASK 2 Olympics DataThe Mens 200m Dash

• The table on the handout lists the winning times
  (in seconds) for the mens 200 meter dash in the
  Olympics through 1996.
• Notice that parts b and c have been completed for
  you.
• Work in pairs or groups of three on the remaining
  parts

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TASK 2 Olympics Data Questions

• f) Which runners were ahead of their time? By
  how many seconds were they ahead? Explain your
  method.
• g) By how many years were they ahead? Explain
  your method.
• h) What is the slope of your model? What does
  it mean? Is it reasonable?
• i) What is the y-intercept of your model? What
  does it mean? Is it reasonable?
• j) What is the x-intercept of your model? What
  does it mean? Is it reasonable?

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TASK 2 Olympics Data Graph
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TASK 2 Olympics Data Extensions

• The independent variable year, year since 1900,
  or Olympic number.
• If data from 2000 and 2004 are included, how does
  the model change?
• Compare the slopes of regression lines for mens
  data with that of womens data.
• Are there any Olympic events for which the
  womens improvement is worse than mens?
• When a linear model seems an insufficient fit,
  what other models might be considered?
• Some Olympic data shows roughly the same
  improvement in the same two halves of the 20th
  century.
• Compare and contrast two kinds of Olympic events
  how fast vs. how far.

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Worthwhile Tasks

• Focus on important mathematics
• Big ideas like proportionality, function, rate
  of change
• Have multiple points of entry
• All students can get started
• Have multiple solution paths
• Provide opportunities for problem solving and
  reasoning
• Sometimes have multiple answers
• Depending upon interpretation, decisions about
  the context
• Have rich opportunities for extensions

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TASK 3

• The Greek Quiz

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Resizing a Task

• Grade Level
• Standards
• Class/Student Appropriate

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Some Grade 6-8 Standards (Skills)

• Data collection, frequency distributions
• Variation, outliers, mean, median, mode
• Theoretical probability, number of outcomes, find
  probability of simple independent events

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Some Math 1-2 Standards (Skills)

• Calculate and use simple permutations
• Use expected value to predict outcomes
• Compare summary statistics from different sample
  data distributions
• Mean absolute deviation
• Binomial coefficients, binomial theorem, algebra
  of polynomials

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In SummaryFour Main Points

• Data Analysis
• Integrated Mathematics
• Worthwhile Mathematical Tasks
• Resizing a Task

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Questions Answers

• Mathematics
• Malcolm Adams
• Math Education
• John Olive
• Brad Findell

• Statistics
• Chris Franklin
• NE GA PRISM
• Sandy Blount