EXPECTED VALUE:ORCHESTRATING UNDERSTANDING

Presentation at Palm Springs 11/6/15

Jim Shortjshort@vcoe.org

Statistical Inference is Irrefutable!

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

Introductions

3

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

Workshop Goals

4

ATP Administrator Training - Module 1 – MS/HS Math

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.

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 dataII 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 themImportance of Probability and Statistics in K-12 Mathematics

Connecting Math Across Grade Levels#

OF

PEO

PLE

36

37

38

39

40

41

42

43

44

45

LENGTH OF CUBIT (CM)

| | | | | | | | | | 36 37 38 39 40 41 42 43 44 45

Grades 3-5

Grades 6-8

High School

Mean: 39.3 cmStandard 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) Interpretation

Black card

0.5 No big deal

Pick a Card!

X P(X) Interpretation

Black card

0.5 No big deal

2nd Black 0.25 Still no big deal

Pick a Card!

X P(X) Interpretation

Black card

0.5 No big deal

2nd Black 0.25 Still no big deal

3rd Black 0.125

A little strange, but not unreasonable

Pick a Card!

X P(X) InterpretationBlack card

0.5 No big deal

2nd Black 0.25 Still no big deal

3rd Black 0.125A little strange, but not unreasonable

4th Black 0.0625Very strange, we wonder, but it’s possible

Pick a Card!

X P(X) InterpretationBlack card

0.5 No big deal

2nd Black 0.25 Still no big deal

3rd Black 0.125A little strange, but not unreasonable

4th Black 0.0625Very strange, we wonder, but it’s possible

5th Black0.0312

5We 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” What are we using to compute probabilities?

Now do “Pointed Rugs” 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

and Now create an area model and complete the

theoretical analysis: What is ? What is What is ? What is

Hence the formula:

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

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