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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
 Physics
 Math
– NCTM Principles and Standards for School
Mathematics (
– 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
 Marble
– 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
 Correct
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
 Students
asked to identify which
distribution of snowflakes and which
sequences of 0's and 1's were random
 Students expected patterns in
 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.
 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
 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
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
 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
 No difference between sample and
population mean
 Students don't understand how to
weight means by sample size
Addressing Problems
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
 Students
must construct their own
 Construct knowledge to fit what they
already know or believe about the
 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
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
 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
Teaching probability
Students must be forced to confront their
misconceptions directly
– Write down predictions, then compare with
– 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
Interdisciplinary subjects
– Climate change
Online resources for teachers
In class demonstrations
 Fisher
and Richards (2004)
– Percentage of boys and girls in a
– Can be done with simulated data
– Students demonstrate understanding
beyond what is explained, after
– Altered problem
 Age-guessing
 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
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,