Teaching Uncertainty to Teaching Uncertainty to High School StudentsHigh School Students

Roberta HarnettRoberta Harnett

MAR 550MAR 550

Current curriculumCurrent curriculum Only 2% of college-bound H.S. students had Only 2% of college-bound H.S. students had

statistics courses (1988)statistics courses (1988)– 160 statistics courses in 13 departments at one 160 statistics courses in 13 departments at one

universityuniversity BiologyBiology PhysicsPhysics MathMath

– NCTM Principles and Standards for School NCTM Principles and Standards for School Mathematics (Mathematics (http://www.nctm.org/standardshttp://www.nctm.org/standards))

– Uncertainty is part of NYS math standards for Uncertainty is part of NYS math standards for all gradesall grades

Nature of ScienceNature of Science

Science is a search for the ”right” answerScience is a search for the ”right” answer– Authoritative, objective, and factualAuthoritative, objective, and factual– Uncertainty in science is counterintuitive, and Uncertainty in science is counterintuitive, and

often not expressed explicitly in problemsoften not expressed explicitly in problems– The “true” value of something can be The “true” value of something can be

measured, deviations from this are errors measured, deviations from this are errors caused by studentscaused by students

Point reasoning vs. set reasoningPoint reasoning vs. set reasoning– More students in point than set reasoning More students in point than set reasoning

categorycategory

MisconceptionsMisconceptions

Marble task:Marble task:– Two bags have black and white countersTwo bags have black and white counters

Bag J: 3 black and 1 whiteBag J: 3 black and 1 whiteBag K: 6 black and 2 whiteBag K: 6 black and 2 white

Which bag gives the better chance of picking a Which bag gives the better chance of picking a black counter?black counter?

A) Same chanceA) Same chance B) Bag JB) Bag J

C) Bag KC) Bag K D) Don't knowD) Don't know

Why?________________________Why?________________________

AnswerAnswer

Correct answer: Correct answer: A (¾ vs A (¾ vs 66//88 = ¾ black counters) = ¾ black counters)

50% chose C because there were 50% chose C because there were more blacks in bag K (39%)more blacks in bag K (39%)

Ratio concept in probabilityRatio concept in probability Little improvement with ageLittle improvement with age

RandomnessRandomness

Students asked to identify which Students asked to identify which distribution of snowflakes and which distribution of snowflakes and which sequences of 0's and 1's were randomsequences of 0's and 1's were random

Students expected patterns in Students expected patterns in randomnessrandomness

Sequence of coin tossesSequence of coin tosses– Can the teacher guess which is random, Can the teacher guess which is random,

and which is designed by the student?and which is designed by the student?

Kahneman and TverskyKahneman and Tversky

RepresentativenessRepresentativeness– Even small samples should reflect Even small samples should reflect

distributiondistribution or the or the processprocess which which produced the random event you’re produced the random event you’re looking atlooking at

– Neglect of sample sizeNeglect of sample sizeChance of getting 7 out of 10 heads is same Chance of getting 7 out of 10 heads is same

as chance of getting 70 out of 100 headsas chance of getting 70 out of 100 heads

– Sequence of children born BGGBGB vs. Sequence of children born BGGBGB vs. BBBBGB vs. BBBGGGBBBBGB vs. BBBGGG

RepresentativenessRepresentativeness

Assume that the chance of having a Assume that the chance of having a boy or girl baby is the same. Over the boy or girl baby is the same. Over the course of a year, in which type of course of a year, in which type of hospital would you expect there to be hospital would you expect there to be more days on which at least 60% of more days on which at least 60% of the babies born were boys?the babies born were boys?A) In a large hospitalA) In a large hospital

B) In a small hospitalB) In a small hospital

C) It makes no differenceC) It makes no difference

RepresentativenessRepresentativeness

Assume that the chance of having a Assume that the chance of having a boy or girl baby is the same. Over the boy or girl baby is the same. Over the course of a year, in which type of course of a year, in which type of hospital would you expect there to be hospital would you expect there to be more days on which at least 60% of more days on which at least 60% of the babies born were boys?the babies born were boys?A) In a large hospitalA) In a large hospital

B) In a small hospitalB) In a small hospital

C) It makes no differenceC) It makes no difference

Judgemental HeuristicsJudgemental Heuristics

AvailabilityAvailability– People judge probability of event based People judge probability of event based

on how well they remember instances of on how well they remember instances of that eventthat event

– Our ideas of probability are often biased Our ideas of probability are often biased because we don't remember frequencies because we don't remember frequencies of events that happen to us the same of events that happen to us the same way we remember events that happen to way we remember events that happen to other peopleother people

ConditionalsConditionals

Urn problemUrn problem– P(WP(W11|W|W22) vs P(W) vs P(W22|W|W11) )

Students understand conditionals when Students understand conditionals when they can use a causal relationshipthey can use a causal relationship– How can conditioning be done based on event How can conditioning be done based on event

that happens after the event it conditions?that happens after the event it conditions? Misconceptions can be corrected by Misconceptions can be corrected by

simulations of the problemssimulations of the problems

Outcome-orientedOutcome-oriented

Each trial of an experiment is a Each trial of an experiment is a seperate, individual phenomenonseperate, individual phenomenon

Students think that they should Students think that they should predict for certain what will happen, predict for certain what will happen, instead of what is likely to happeninstead of what is likely to happen

Maintain original predictions even Maintain original predictions even when evidence contradicts themwhen evidence contradicts them

Understanding meansUnderstanding means

Students believe samples should be Students believe samples should be representative, regardless of sample representative, regardless of sample sizesize

No difference between sample and No difference between sample and population meanpopulation mean

Students don't understand how to Students don't understand how to weight means by sample sizeweight means by sample size

Addressing ProblemsAddressing Problems

NCTM standards to address problems NCTM standards to address problems in mathin math

NCLB has caused changes to be made NCLB has caused changes to be made in curriculum in all subjectsin curriculum in all subjects

Science and Technology standardsScience and Technology standards Students must be confronted with Students must be confronted with

their misconceptionstheir misconceptions– SimulationsSimulations

ConstructivismConstructivism

Students must construct their own Students must construct their own ideas ideas

Construct knowledge to fit what they Construct knowledge to fit what they already know or believe about the already know or believe about the world world

Difficulty replacing old ideasDifficulty replacing old ideas– Inquiry based learningInquiry based learning– 5E lesson style5E lesson style

Engage, explore, explain, elaborate, evaluate Engage, explore, explain, elaborate, evaluate

Constructivion vs. AcceptionConstructivion vs. Acception

Construction leads Construction leads to understanding to understanding details of a problemdetails of a problem

Can use concept in Can use concept in new situationnew situation

Accepting facts Accepting facts (without (without constructing constructing knowledge) knowledge) focuses on focuses on superficial detailssuperficial details

Can only solve Can only solve problems which problems which are presented the are presented the same waysame way

Cognitive factorsCognitive factors

Field-dependant vs. field-independentField-dependant vs. field-independent

Reflective vs. impulsiveReflective vs. impulsive

Sensory modalitySensory modality

VARKVARK

Traditional teaching methods apply Traditional teaching methods apply mostly to A/R learnersmostly to A/R learners

Research has shown that teaching to Research has shown that teaching to a particular sensory modality doesn’t a particular sensory modality doesn’t help muchhelp much

Center for the Study of Learning and Center for the Study of Learning and Teaching Styles at St. John's Teaching Styles at St. John's University University

Teaching probabilityTeaching probability Students must be forced to confront their Students must be forced to confront their

misconceptions directly misconceptions directly – Write down predictions, then compare with Write down predictions, then compare with

resultsresults– Students who do not explicitly make Students who do not explicitly make

predictions beforehand may actually rely on predictions beforehand may actually rely on misconceptions even moremisconceptions even more

Teachers need to understand probabilityTeachers need to understand probability– Teachers who don’t feel confident about a Teachers who don’t feel confident about a

subject they are teaching are less likely to subject they are teaching are less likely to correct students when they’re wrong correct students when they’re wrong

– Need to confront nonnormative beliefs about Need to confront nonnormative beliefs about probability in students and themselvesprobability in students and themselves

Including uncertainty in scienceIncluding uncertainty in science

Environmental Science Interactive with Environmental Science Interactive with Ramas eLabRamas eLab– Online course for AP or college level studentsOnline course for AP or college level students

Simulation studiesSimulation studies– Antibiotic resistant TB, beak size in Darwin’s Antibiotic resistant TB, beak size in Darwin’s

finchesfinches Interdisciplinary subjectsInterdisciplinary subjects

– Climate changeClimate change Online resources for teachersOnline resources for teachers

– www.cdc.gov/excitewww.cdc.gov/excite

In class demonstrationsIn class demonstrations

Fisher and Richards (2004)Fisher and Richards (2004)– Percentage of boys and girls in a Percentage of boys and girls in a

populationpopulation– Can be done with simulated dataCan be done with simulated data– Students demonstrate understanding Students demonstrate understanding

beyond what is explained, after beyond what is explained, after discussiondiscussion

– Altered problemAltered problem Age-guessing Age-guessing

SummarySummary

Students are not being taught much Students are not being taught much about probability before collegeabout probability before college

Students hold many misconceptions Students hold many misconceptions about probabilityabout probability

Misconceptions can be corrected if Misconceptions can be corrected if students are forced to confront them students are forced to confront them with data with data – Simulation programsSimulation programs– Hands-on activitiesHands-on activities

ReferencesReferences Fisher, L.A. and D. Richards. 2004. Random Walks as Motivational Fisher, L.A. and D. Richards. 2004. Random Walks as Motivational

Material in Introductory Statistics and Probability Courses. Material in Introductory Statistics and Probability Courses. The The American StatisticianAmerican Statistician 58, 4, 310-316. 58, 4, 310-316.

Gelman, A. and M.E. Glickman. 2000. Some class participation Gelman, A. and M.E. Glickman. 2000. Some class participation demonstrations for introductory probability and statistics. demonstrations for introductory probability and statistics. Journal Journal of Educational and Behavioral Statisticsof Educational and Behavioral Statistics 25, 1, 84-100. 25, 1, 84-100.

Hall, B. 2006. Teaching and learning uncertainty in science: the Hall, B. 2006. Teaching and learning uncertainty in science: the case of climate change. case of climate change. PlanetPlanet, 17, 48-49. , 17, 48-49.

Sandoval, W.A. and K. Morrison. 2003. High School Students’ Sandoval, W.A. and K. Morrison. 2003. High School Students’ Ideas about Theory and Theory Change after a Biological Inquiry Ideas about Theory and Theory Change after a Biological Inquiry Unit. Unit. Journal of Research in Science TeachingJournal of Research in Science Teaching, 40, 4, 369-392., 40, 4, 369-392.

Stroup, D.F., R.A. Goodman, R. Cordell, R. Scheaffer. 2004. Stroup, D.F., R.A. Goodman, R. Cordell, R. Scheaffer. 2004. Teaching Statistical Principles Using Epidemiology: Measuring the Teaching Statistical Principles Using Epidemiology: Measuring the Health of Populations. Health of Populations. The American StatisticianThe American Statistician, 58, 1, 77-84. , 58, 1, 77-84.

Wilson, Patricia S. Ed. Research Ideas for the Classroom: High Wilson, Patricia S. Ed. Research Ideas for the Classroom: High School Mathematics.MacMillan Publishing Company, New York, School Mathematics.MacMillan Publishing Company, New York, 1993.1993.

http://usny.nysed.gov/teachers/nyslearningstandards.htmlhttp://usny.nysed.gov/teachers/nyslearningstandards.html http://www.nctm.org/http://www.nctm.org/