EXPECTED VALUE:
ORCHESTRATING UNDERSTANDING
Jim Short
[email protected]
Presentation at Palm Springs
11/6/15
Statistical Inference is Irrefutable!
Introductions
Take a minute to think about, and then be ready to
share with the others at your table:
 Name
 School
 District
 Something you really like about the Probability and
Statistics in the California CCS-Math
 One thing you hope to learn today
3
Workshop Goals


Deepen understanding of expected value – looking
at what it means, not the formula for computing it
Engage in hands-on classroom activities designed to
develop conceptual understanding of expected
value
 Special
thanks to Sherry Fraser and the other authors of
the Interactive Mathematics Program
4
Workshop Norms
1. Bring and assume best intentions.
2. Step up, step back.
3. Be respectful, and solutions oriented.
4. Turn off (or mute) electronic devices.
ATP Administrator Training Module 1 – MS/HS Math
Guidelines for Assessment and Instruction in
Statistics Education (GAISE) Report
Statistical problem solving is an investigative process that
involves four components:
I Formulate Questions
– clarify the problem at hand
– formulate one (or more) questions that can be answered
with data
II Collect Data
– design a plan to collect appropriate data
– employ the plan to collect the data
III Analyze Data
– select appropriate graphical and numerical methods
– use these methods to analyze the data
IV Interpret Results
– interpret the analysis
– relate the interpretation to the original question
Mathematical Modeling
•
•
•
What is mathematical modeling?
“Modeling is the process of choosing and using appropriate
mathematics and statistics to analyze empirical situations, to
understand them better, and to improve decisions.”
Process:
▫
▫
▫
▫
▫
▫
Identify variables and select those that are essential
Formulate a model to describe the relationships
Analyze and perform operations to draw conclusions
Interpret results in the light of the context
Validate the conclusions
Report on the conclusions and reasoning behind them
Importance of Probability and Statistics in K-12 Mathematics
Connecting Math Across Grade Levels
# OF
PEOPLE
Grades 3-5
High
School
3 3 3 3 4 4 4 4 4 4
6 7 8 9 0 1 2 3 4 5
LENGTH OF CUBIT
(CM)
Grades 6-8
|
|
|
|
|
|
36 37 38 39 40 41
|
|
|
|
42 43 44 45
Mean: 39.3 cm
Standard Deviation:
2.2 cm
Importance of Probability and Statistics in K-12 Mathematics
Access and Equity
• The study of statistics offers opportunities for
Culturally Responsive Instruction by allowing students
to collect and analyze real-world data relevant to their
lives
• The study of statistics requires teachers to attend to
issues of language through
–
–
–
–
Reading
Writing
Listening
Speaking
Importance of Probability and Statistics in K-12 Mathematics
Agreeing with Arthur Benjamin


Brief TED talk by Arthur
Benjamin:
Arthur Benjamin- Teach
statistics before
calculus! - Talk Video TED.com[via
torchbrowser.com].flv
Notice and Wonder
Statistical Reasoning Process

Questions


Collect Data



Analyze
Interpret


Is this a standard deck of
cards?
Pick one card at a time
with replacement and
record the results.
Calculate the probabilities
Use the probability to
draw your conclusion
Pick a Card!
X
P(X)
Black card
0.5
Interpretation
No big deal
Pick a Card!
X
P(X)
Interpretation
Black card
0.5
2nd Black
0.25 Still no big deal
No big deal
Pick a Card!
X
P(X)
Interpretation
Black card
0.5
2nd Black
0.25 Still no big deal
3rd Black
0.125 A little strange, but not unreasonable
No big deal
Pick a Card!
X
P(X)
Black card
0.5
No big deal
2nd Black
0.25
Still no big deal
3rd
Black
4th Black
Interpretation
A little strange, but not
0.125
unreasonable
Very strange, we wonder, but it’s
0.0625
possible
Pick a Card!
X
P(X)
Black card
0.5
No big deal
2nd Black
0.25
Still no big deal
3rd
Black
4th Black
5th Black
Interpretation
A little strange, but not
0.125
unreasonable
Very strange, we wonder, but it’s
0.0625
possible
0.03125 We want to check the deck!!
The 5% threshold in Statistics is not arbitrary!
Never Tell An Answer
Please remember the enormous responsibility
we all have as learners not to spoil anybody
else’s fun.
The quickest way to spoil someone else’s fun
is to tell them an answer before they have a
chance to discover it themselves.
Susan Pirie
Events With Different Values

Do “Rug Games”


Now do “Pointed Rugs”




What are we using to compute probabilities?
How has the previous problem been changed?
Do “Spinner Give and Take”
How are “Pointed Rugs” and “Spinner Give and Take” the
same? How are they different?
How could “Spinner Give and Take” be changed to make
it “fair”? What makes a game of chance “fair”?
Expected Value

“One-and-One”




Who can explain a “one-and-one” situation in basketball?
What is your intuition about the number of points Terry will
make for her team per one-and-one situation in the long run?
Working in groups of 3, at most 4, complete 50 simulations of
a “one-and-one” with Terry shooting, and use your data to
complete “A Sixty-Percent Solution”
Now create an area model to develop a theoretical analysis
of the situation. How many points per situation for Terry in
the long run?
From the Interactive Mathematics Program: Year 1, The Game of Pig. Copyright © 2009 by IMP, Inc.
Used by permission of the publisher, It's About Time, www.iat.com.
Conditional Probability


P(A|B) = PB(A) is the probability of A occurring given that B
has occurred.
Example:



What is the probability that you will cough at some point today?
What is the probability that you will cough at some point today if
you have a cold?
Roll a pair of dice, die G and die H


What is the probability that G = 2?
What is the probability that G = 2 given that G+H≤5?
Conditional Probability


Work in groups of 3 or 4, and roll a pair of dice (different
colors, G and H) 50 times, and record the values of G and G+H
 Use your results to calculate an experimental P 𝐺 = 2 and
P 𝐺 =2 𝐺+𝐻 ≤5
Now create an area model and complete the theoretical
analysis:





What is P
What is P
What is P
What is P
𝐺=2 ?
𝐺 + 𝐻 ≤ 5?
𝐺 =2 𝐺+𝐻 ≤5 ?
𝐺 = 2 𝐚𝐧𝐝 𝐺 + 𝐻 ≤ 5 ?
Hence the formula:𝑃 𝐴 𝐵 =
𝑃(𝐴∩𝐵)
𝑃(𝐵)
What Have We Done?



Begin with experiences
to build a conceptual
understanding
Build from there to the
formal mathematics
Allow for student
agency and authority
Evaluations



Thank you for attending this section
Please take a moment to provide feedback on the
session per the next two slides
Suggestions for improvement are welcomed!
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