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/gaise/ Christine Franklin

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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/gaise/Christine
Franklin Henry KranendonkNCSM Conference
March 19, 2007
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Outline of Presentation Overview of the GAISE
Report The Evolution of a Statistical
Concept - The Mean as Fair Share/Variation from
Fair Share - The Mean as the Balance
Point/Variation from the Mean - The Sampling
Distribution of the Mean/Variation in Sample
Means Summary
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Benchmarks in Statistical Education in the United
States (1980-2007)

• The Quantitative Literacy Project (ASA/NCTM Joint
  Committee, Early 1980s)
• Curriculum and Evaluation Standards for School
  Mathematics (NCTM, 1989)
• Principles and Standards for School Mathematics
  (NCTM, 2000)
• Mathematics and Statistics College Board
  Standards for College Success (2006)
• The GAISE Report (2005, 2007)

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GOALS of the GAISE Report

• Promote and develop statistical literacy
• Provide links with the NCTM Standards
• Discuss differences between Mathematics and
  Statistics
• Clarify the role of probability in statistics
• Illustrate concepts associated with the data
  analysis process
• Present the statistics curriculum for grades
  Pre-K-12 as a cohesive and coherent curriculum
  strand
• Provide developmental sequences of learning
  experiences

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Stakeholders

• Writers of state standards
• Writers of assessment items
• Curriculum directors
• Pre K-12 teachers
• Educators at teacher preparation programs 

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STATISTICAL THINKING versusMATHEMATICAL
THINKING
  The Focus of Statistics on Variation in
Data The Importance of Context in
Statistics  
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PROBABILITY Randomization
Sampling -- "select at random from a
population"   Experiments -- "assign at
random to a treatment"    
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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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Developmental Levels The GAISE Report proposes
three developmental levels for
evolving statistical concepts. Levels A, B,
and C
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The Framework ModelA Two-Dimensional
Model One dimension is the four components of
the statistical problem-solving process, along
with the nature of and the focus on
variability The second dimension is comprised
of three developmental levels (A, B, and C)
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THE FRAMEWORK MODEL
 
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THE FRAMEWORK MODEL
 
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Activity Based Learning The GAISE Report
promotes active learning of statistical content
and conceptsTwo Types of Learning
Activities Problem Solving Activities Concept
Activities
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The STN article illustrates a Problem Solving
Activity across the three developmental levels.
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The evolution of a statistical concept --
What is the mean? Quantifying variation
in data from the mean
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Level A ActivityThe Family Size Problem A
Conceptual Activity for Developing an
Understanding of the Mean as the Fair Share
value Developing a Measure of Variation from
Fair Share
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A QuestionHow large are families
today? Nine children were asked how many
people are in your family. Each child
represented her/his family size with a
collection snap cubes.
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Snap Cube Representation for Nine Family Sizes
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How might we examine the data on the family sizes
for these nine children?
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2 3 3 4 4 5
6 7 9
Ordered Snap Cube Numerical Representations of
Nine Family Sizes
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Notice that the family sizes vary. What if we
used all our family members and tried to make all
families the same size, in which case there is no
variability.How many people would be in each
family?
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How can we go about creating these new
families?We might start by separating all the
family members into one large group.
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• All 43 Family Members

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Step 1Have each child select a snap cube to
represent her/him-self.These cubes are
indicated in red.
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• Create Nine New Families/Step1

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Step 2Next have each child select one family
member from the remaining group.These new
family members are shown in red.
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• Create Nine New Families/Step2

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Continue this process untilthere are not enough
family members for each child to select from.
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• Create Nine New Families/Step4

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Discuss results The fair share valueNote
that this is developing the division algorithm
and eventually, the algorithm for finding the
mean.
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A New ProblemWhat if the fair share value for
nine children is 6? What are some different
snap cube representations that might produce a
fair share value of 6?
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Snap Cube Representation of Nine Families, Each
of Size 6
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Have Groups of Children Create New Snap Cube
RepresentationsFor example, following are two
different collections of data with a fair share
value of 6.
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• Two Examples with Fair Share Value of 6.
• Which group is closer to being fair?

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How might we measure how close a group of
numeric data is to being fair?
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• Which group is closer to being fair?
• The upper group in blue is closer to fair since
  it requires only one step to make it fair. The
  lower group requires two steps.

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How do we define a step? One step occurs
when a snap cube is removed from a stack higher
than the fair share value and placed on a stack
lower than the fair share value . A measure
of the degree of fairness in a snap cube
distribution is the number of steps required
to make it fair.Note -- Fewer steps indicates
closer to fair
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• Number of Steps to Make Fair 8
• Number of Steps to Make Fair 9

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Students completing Level A understand the
notion of fair share for a set of numeric
data the fair share value is also called the
mean value the algorithm for finding the
mean the notion of number of steps to make
fair as a measure of variability about the
mean the fair share/mean value provides a
basis for comparison between two groups of
numerical data with different sizes (thus cant
use total)
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Level B ActivityThe Family Size Problem
How large are families today? A Conceptual
Activity for Developing an Understanding of
the Mean as the Balance Point of a
Distribution Developing Measures of Variation
about the Mean
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Level B ActivityHow many people are in your
family?Nine children were asked this question.
The following dot plot is one possible result
for the nine children
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------------------ 2 3 4 5 6 7 8
9 10
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Have groups of students create different dot plot
representations of nine families with a mean of 6.
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------------------ 2 3 4 5 6 7 8
9 10
------------------ 2 3 4 5 6 7 8
9 10
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• In which group do the data (family sizes) vary
  (differ) more from the mean value of 6?

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1
2
4
2
1
0
1
2
3
------------------ 2 3 4 5 6 7 8
9 10
0
0
4
3
2
0
2
3
4
------------------ 2 3 4 5 6 7 8
9 10
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In Distribution 1, the Total Distance from the
Mean is 16. In Distribution 2, the Total
Distance from the Mean is 18.Consequently, the
data in Distribution 2 differ more from the mean
than the data in Distribution 1.
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1
2
4
2
1
0
1
2
3
------------------ 2 3 4 5 6 7 8
9 10
Note that the total distance for the values below
the mean of 6 is 8, the same as the total
distance for the values above the mean. For this
reason, the distribution will balance at 6 (the
mean)
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The SAD is defined to be The Sum of the
Absolute DeviationsNote the relationship
between SAD and Number of Steps to Fair from
Level A SAD 2xNumber of Steps
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• Number of Steps to Make Fair 8
• Number of Steps to Make Fair 9

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An Illustration where the SAD doesnt work!
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4
4
------------------ 2 3 4 5 6 7 8
9 10
1
1
1
1
1
1
1
1
------------------ 2 3 4 5 6 7 8
9 10
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The SAD is 8 for each distribution, but in the
first distribution the data vary more from the
mean. Why doesnt the SAD work?
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Adjusting the SAD for group sizes yields
the MAD Mean Absolute Deviation
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Measuring Variation about the Mean SAD Sum
of Absolute Deviations MAD Mean of Absolute
Deviations Variance Mean of Squared
Deviations Standard Deviation Square
Root of Variance
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Summary of Level B and Transitions to Level
C Mean as the balance point of a
distribution Mean as a central
point Various measures of variation about the
mean.
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The Mean at Level C At Level C, the notion of
the Sampling Distribution of the Sample
Mean is Developed. This development
connects probability and statistics and provides
the link between the descriptive statistics
students have learned at Levels A and B and
concepts of inferential statistics they will
learn at Level C.
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Eighty Circles/What is the Mean Diameter?
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Activity Students select samples of 10
circles they considered to be representative
of the 80 circles. The mean for each sample
is determined. Students select simple
random samples of 10 circles. The mean for each
sample is determined.
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How do the results from self-selection compare
with random sampling? Following are results
from two introductory-level statistics classes
(50 students).
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(No Transcript)
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Sampling Distributions provide the link to two
important concepts in statistical
inference. Margin of Error Statistical
Significance
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The STN article in your packet provides an
illustration of how a sampling distribution is
used to develop these statistical concepts.
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SUMMARY GOALS of GAISE Report

• Promote and develop statistical literacy
• Provide links with the NCTM Standards
• Discuss differences between Mathematics and
  Statistics
• Clarify the role of probability in statistics
• Illustrate concepts associated with the data
  analysis process
• Present the statistics curriculum for grades
  Pre-K-12 as a cohesive and coherent curriculum
  strand
• Provide developmental sequences of learning
  experiences