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Teaching Uncertainty to
High School Students
Roberta Harnett
MAR 550
Current curriculum

Only 2% of college-bound H.S. students
had statistics courses (1988)
– 160 statistics courses in 13 departments at one
university
Biology
 Physics
 Math

– NCTM Principles and Standards for School
Mathematics (http://www.nctm.org/standards)
– Uncertainty is part of NYS math standards for
all grades
Nature of Science

Science is a search for the ”right” answer
– Authoritative, objective, and factual
– Uncertainty in science is counterintuitive, and
often not expressed explicitly in problems
– The “true” value of something can be
measured, deviations from this are errors
caused by students

Point reasoning vs. set reasoning
– More students in point than set reasoning
category
Misconceptions
 Marble
task:
– Two bags have black and white counters
 Bag
J: 3 black and 1 white
 Bag K: 6 black and 2 white
Which bag gives the better chance of picking a
black counter?
A) Same chance
B) Bag J
C) Bag K
D) Don't know
Why?________________________
Answer
 Correct
answer:
A (¾ vs 6/8 = ¾ black counters)
 50%
chose C because there were
more blacks in bag K (39%)
 Ratio concept in probability
 Little improvement with age
Randomness
 Students
asked to identify which
distribution of snowflakes and which
sequences of 0's and 1's were random
 Students expected patterns in
randomness
 Sequence of coin tosses
– Can the teacher guess which is random,
and which is designed by the student?
Kahneman and Tversky
 Representativeness
– Even small samples should reflect
distribution or the process which
produced the random event you’re
looking at
– Neglect of sample size
 Chance
of getting 7 out of 10 heads is same
as chance of getting 70 out of 100 heads
– Sequence of children born BGGBGB vs.
BBBBGB vs. BBBGGG
Representativeness
 Assume
that the chance of having a
boy or girl baby is the same. Over the
course of a year, in which type of
hospital would you expect there to be
more days on which at least 60% of
the babies born were boys?
A) In a large hospital
B) In a small hospital
C) It makes no difference
Representativeness
 Assume
that the chance of having a
boy or girl baby is the same. Over the
course of a year, in which type of
hospital would you expect there to be
more days on which at least 60% of
the babies born were boys?
A) In a large hospital
B) In a small hospital
C) It makes no difference
Judgemental Heuristics
 Availability
– People judge probability of event based
on how well they remember instances of
that event
– Our ideas of probability are often biased
because we don't remember frequencies
of events that happen to us the same
way we remember events that happen to
other people
Conditionals

Urn problem
– P(W1|W2) vs P(W2|W1)

Students understand conditionals when
they can use a causal relationship
– How can conditioning be done based on event
that happens after the event it conditions?

Misconceptions can be corrected by
simulations of the problems
Outcome-oriented
 Each
trial of an experiment is a
seperate, individual phenomenon
 Students think that they should
predict for certain what will happen,
instead of what is likely to happen
 Maintain original predictions even
when evidence contradicts them
Understanding means
 Students
believe samples should be
representative, regardless of sample
size
 No difference between sample and
population mean
 Students don't understand how to
weight means by sample size
Addressing Problems
 NCTM
standards to address problems
in math
 NCLB has caused changes to be made
in curriculum in all subjects
 Science and Technology standards
 Students must be confronted with
their misconceptions
– Simulations
Constructivism
 Students
must construct their own
ideas
 Construct knowledge to fit what they
already know or believe about the
world
 Difficulty replacing old ideas
– Inquiry based learning
– 5E lesson style
 Engage,
explore, explain, elaborate, evaluate
Constructivion vs. Acception


Construction leads
to understanding
details of a problem
Can use concept in
new situation


Accepting facts
(without
constructing
knowledge)
focuses on
superficial details
Can only solve
problems which
are presented the
same way
Cognitive factors
 Field-dependant
 Reflective
 Sensory
vs. field-independent
vs. impulsive
modality
VARK
 Traditional
teaching methods apply
mostly to A/R learners
 Research has shown that teaching to a
particular sensory modality doesn’t
help much
 Center for the Study of Learning and
Teaching Styles at St. John's
University
Teaching probability

Students must be forced to confront their
misconceptions directly
– Write down predictions, then compare with
results
– Students who do not explicitly make
predictions beforehand may actually rely on
misconceptions even more

Teachers need to understand probability
– Teachers who don’t feel confident about a
subject they are teaching are less likely to
correct students when they’re wrong
– Need to confront nonnormative beliefs about
probability in students and themselves
Including uncertainty in science

Environmental Science Interactive with
Ramas eLab
– Online course for AP or college level students

Simulation studies
– Antibiotic resistant TB, beak size in Darwin’s
finches

Interdisciplinary subjects
– Climate change

Online resources for teachers
– www.cdc.gov/excite
In class demonstrations
 Fisher
and Richards (2004)
– Percentage of boys and girls in a
population
– Can be done with simulated data
– Students demonstrate understanding
beyond what is explained, after
discussion
– Altered problem
 Age-guessing
Summary
 Students
are not being taught much
about probability before college
 Students hold many misconceptions
about probability
 Misconceptions can be corrected if
students are forced to confront them
with data
– Simulation programs
– Hands-on activities
References
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Fisher, L.A. and D. Richards. 2004. Random Walks as Motivational
Material in Introductory Statistics and Probability Courses. The
American Statistician 58, 4, 310-316.
Gelman, A. and M.E. Glickman. 2000. Some class participation
demonstrations for introductory probability and statistics. Journal
of Educational and Behavioral Statistics 25, 1, 84-100.
Hall, B. 2006. Teaching and learning uncertainty in science: the
case of climate change. Planet, 17, 48-49.
Sandoval, W.A. and K. Morrison. 2003. High School Students’
Ideas about Theory and Theory Change after a Biological Inquiry
Unit. Journal of Research in Science Teaching, 40, 4, 369-392.
Stroup, D.F., R.A. Goodman, R. Cordell, R. Scheaffer. 2004.
Teaching Statistical Principles Using Epidemiology: Measuring the
Health of Populations. The American Statistician, 58, 1, 77-84.
Wilson, Patricia S. Ed. Research Ideas for the Classroom: High
School Mathematics.MacMillan Publishing Company, New York,
1993.
http://usny.nysed.gov/teachers/nyslearningstandards.html
http://www.nctm.org/