Transcript "Is this die fair? An analysis of students` technology

Apply What You Know to a
Problem You Have Never
Seen: How AP Students Apply
Statistical Reasoning to Solve
a Task Using Empirical Data
Robin Rider, East Carolina University
[email protected]
and
Renea Baker, D. H. Conley High School
[email protected]
Background image is a screenshot of Probability Explorer (Stohl, 2002) http://www.probexplorer.com
1
Classical vs. Frequentist
• Classical - probability is a quantity determined a
priori trials being conducted and derived
theoretically from the sample space
VERSUS
• Frequentist - probability is a quantity
determined a posteriori trials being conducted
and derived experimentally from data,.
(Borovcnik, Bentz, & Kapadia, 1991)
2
Students’ Schoolopoly Assignment
Investigate whether the die sent to you by the
company is, in fact, fair. That is, are all six
outcomes equally likely to occur? You will need
to create a poster to present to the School
Board. The following three questions should be
answered on your poster:
•
•
•
Would you recommend that dice be purchased from the
company you investigated?
What evidence do you have that the die you tested is “fair”
or “unfair”?
Use your experimental results to estimate the theoretical
probability of each outcome, 1-6, of the die you tested.
Stohl & Tarr, 2002
3
Simulation vs. Physical Dice
• Each pair of students investigated two
situations
• Computer Simulated Dice
– Probability Explorer (Stohl, 2002)
– Assigned company could produce weighted or
unweighted dice
• Physical Dice
– Assigned dice could be weighted or
unweighted
4
Schoolopoly Digital Dice Companies
Pairs of students were assigned one company to
investigate.
Actual Probabilities (Unknown to Students)
Companies
P(1)
P(2)
P(3)
P(4)
P(5)
P(6)
Luckytown Dice Co.
0.1500
0.1500
0.1500
0.1500
0.1500
0.2500
Dice R’ Us
0.1250
0.1875
0.1875
0.1875
0.1875
0.1250
High Rollers, Inc.
0.1333
0.2000
0.1333
0.2000
0.1333
0.2000
Dice, Dice Baby!
0.1111
0.1667
0.2222
0.2222
0.1667
0.1111
Pips and Dots
0.1667
0.1667
0.1667
0.1667
0.1667
0.1667
Slice N’ Dice
0.1600
0.2000
0.2000
0.2000
0.0400
0.2000
5
Schoolopoly Physical Dice
• Dodgy Dice
– Available from Highland Games
– http://www.halfpast.demon.co.uk/
– Either weighted toward a particular number or
fair
– Weights unknown
6
Research Questions
• How are students reasoning about fairness,
sample sizes, and the probability of each
outcome on the die?
• What do students consider to be evidence
to support their claim about the fairness of
the die?
• What are the difference in reasoning with
physical dice vs computer simulated dice?
7
Sources of Data
•
•
•
•
•
Video of computer actions and conversation
Video of students at desk
Transcripts
Students’ final report to the school board
Students’ poster and video of presentation to
class
8
Sample Data Collection
9
Sample Poster Artifacts
10
Sample Poster
11
Sample Poster
12
Preliminary Results
• All students used an
appropriate statistical test
(Chi Square) for the data
collected
• Students who collected
multiple samples were
attentive to variation
between trials and used
variation to support their
claims of fairness
• All students had some
incorrect application of the
CLT
– Collected ONE sample and
used the fact that their
sample size was greater
than or equal to 30 to justify
that it was large enough to
make conclusions about
probability
– After collecting a sample
students immediately
applied a hypothesis testing
procedure
13
Preliminary Results
• With both the physical
dice and the simulation
students made comments
as to a “race” between
outcomes, cheering
outcomes which were
“behind”
– “Two’s are catching up”,
“Three’s are taking a strong
lead”, “No clear cut winner”
– Seems to indicate a desire
for the dice to be fair
– This “racing” phenomena
has been noted in middle
school students also
• Used similar reasoning
for both physical and
computer simulated dice
– Students who tested the
physical dice first tended to
use the same sample size
for the computer
simulation, students who
did the simulation first
used much larger samples
for the simulation than for
the physical dice
• Lee, Rider, & Tarr (under
review)
14
Not surprising
• Students were very
procedural in their
approach to the problem
– Immediately applied a HT
to the data
– Collected few, relatively
small samples based on
lack of understanding of
the CLT
• Students were unfamiliar
with the process of
having to collect their own
data to make conclusions
and had difficulty in how
to approach the problem
– Students particularly had
difficulty in determining
how to estimate the
theoretical probability
Surprising
• Because of the small
samples (even though it
met the “rule of thumb” for
Chi Square) a surprising
number of pairs made Type
II errors (not rejecting a
false null hypothesis)
– Limit power of the Chi
Square Test for Goodness of
Fit
• Only one group of students
took multiple (3) samples
and averaged the
proportions for each
outcome to get a better
estimate of the theoretical
probability
15
Where Do We Go from Here?
• Curricular focus on
procedures in AP
Statistics
– Develop more
opportunities to
integrate empirical
data collection
activities to reason
about theoretical
probabilities
– Better understanding
of CLT and Law of
Large numbers
• Replication of study
after integrating more
empirical data
collection activities
– Will students have a
deeper conceptual and
practical
understanding of CLT
and Law of Large
Numbers?
16
References
• Borovcnik, M., Bentz, H.-J., & Kapadia, R. (1991). A
probabilistic perspective. In R. Kapadia & M.
Borovcnik (Eds.), Chance encounters: Probability in
education (pp. 27-71). Boston: Kluwer Academic
Publishers.
• Lee, H. S., Rider, R. L., & Tarr, J. E. (2005). Making
connections between empirical data and theoretical
probability: Students’ generation and analysis of
data in a technological environment. Manuscript
currently under review.
• Stohl, H. & Tarr, J. E. (2002). Developing notions of
inference with probability simulation tools. Journal of
Mathematical Behavior 21(3), 319-337.
17