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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 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/