Teaching High School Statistics and use of Technology
SUGGESTIONS FOR BEST PRACTICES FOR STATISTICS EDUCATION AND USING STATISTICAL SOFTWARE AS AN EDUCATIONAL TOOL IN THE CLASSROOM1 Simon KING High School Statistics Teacher – Cary Academy, Cary, NC Texas A & M Department of Statistics STAT685
BACKGROUNDo Most of high school statistics classes incorporate the graphing calculator with occasional use of statistical software (school resources permitting) to give students experience of seeing statistical output.o High school statistics curriculum follow the College Board AP Statistics Curriculum (College Board, 2001)o “Students are expected to bring a graphing calculator with statistical capabilities to the exam, and to be familiar with this use.” (College Board, 2005).o Designed Advanced Statistics and Analytics course that did not follow the College Board AP Statistics curriculum and used JMP® as the main statistical and educational tool. No graphing calculator. College Board. (2001). www.collegeboard.com. Retrieved from www.collegeboard.com: www.collegeboard.com 2 College Board. (2005). Calculators on the AP Statistics Exam. Retrieved from apcentral.collegeboard.com: http://apcentral.collegeboard.com/apc/members/exam/exam_information/23032.html
SECTION 1 - SUGGESTIONS FOR BEST PRACTICES IN THE CLASSROOM 1.1 Most of the students taught in statistics class will not be statisticians. Use the course to explore student concerns and interests 1.2 Give students experience of research and reading of journal papers and articles. Explore “statistical thinking” 3Discussed in STAT641
SECTION 1 - SUGGESTIONS FOR BEST PRACTICES IN THE CLASSROOM 1.3 Teacher as the statistical consultant o As an end-point for the course, for final student experiment/paper the teacher is used as a statistical consultant and the student applies what they learned throughout the year 1.4 Use psychology to teach statistics o Psychology is an example, but rather teaching statistics as an applied subject, present context first 1.5 Promote „statistical literacy‟ o Beyond course content, it is crucial students develop and retain this skill. 1.6 The statistics teacher „living and breathing‟ statistics o If we want our students to enjoy and be curious about the discipline, then we need to role model the behavior we want to see in them 1.7 Statistics should not be taught like a mathematics course o “In mathematics, context obscures structure. In data analysis, context provides meaning” (Cobb & Moore, November 1997) 4Cobb, G. W., & Moore, D. S. (2000). Statistics and mathematics: Tension and cooperation. American Mathematical Monthly, August-September, 615-630.
SECTION 1 - SUGGESTIONS FOR BEST PRACTICES IN THE CLASSROOM1.8 Teach without the textbook o General overreliance on textbooks in US education o facilitates creativity by the teacher o Teacher is in full control of learning o Not recommended for new to statistics education teachers1.9 Student feedback o Role-model being a reflective learner o Collect anonymous feedback on the course from the students o Share the feedback and make reasonable changes to the course1.10 Incorporate a multicultural curriculum o Through context and interpretation1.11 Collect datasets o Plan curriculum first then consider which datasets will best support the learning objectives1.12 Use of statistical applets o Must be purposeful application with reflection on learning o Better applets generally use real data or actual context1.13 Play games to collect student data in-class 5 o Fun, but must be linked to learning objectives
SECTION 2 – USING ANALYTICAL SOFTWARE AS AN EDUCATIONAL LEARNING TOOLThis section presents examples of how statisticalsoftware (in this case, JMP®) can be used as aneducational tool.This section also discusses the challenges ofadopting such software that the teacher has toconsider including how to assess students and the„the black box‟ issue of statistical software. 6
2.1 SUPPORTING THE USE OF STATISTICAL SOFTWARE 7 Example of video tutorial for use of JMP®
2.2 VISUALS Explore history of visualsCharles Joseph Minard, 1869 8Discussed in STAT604 Florence Nightingale, 1857
Explore poor visualso Poor media use of visuals o poor visual created in JMP® o Students spend time learning best practices 9Discussed in STAT604 and STAT641
Kinesthetic learning – manipulating visuals o Students explore good and bad influential points by excluding points and refitting a regression line 10 o While the data is not „real‟, it is a context thatDiscussed in students can relate to.STAT608
2.3 EXPLORE TEST ASSUMPTIONS o JMP® script confidence intervals (sample size of two) from a normal „population‟ 11Discussed in STAT641 and STAT642
2.4 CREATE AND EXPLORE VISUALS NOT IN A TRADITIONAL HIGH SCHOOL CURRICULUMo bubble plot of year, median house price and median house income (data: U.S. Census Bureau, 2009) 12 U.S. Census Bureau. (2009). Income. Retrieved 10 15, 2011, from www.census.gov: http://www.census.gov/hhes/www/income/income.html
2.5 TEACHING CONTENT BEYOND THE HIGH SCHOOL CURRICULUM o Exploring normality with histogram, qq plot and Shapiro-Wilk test 13
2.6 Exploring large datasets with multiple variables Bivariate Plot of TOTAL SAT score versus Percent Taking by STATE with added indicator of US region (Data: College Board, 2001) 14 College Board. (2001). www.collegeboard.com. Retrieved 10 15, 2011, from www.collegeboard.com: www.collegeboard.com
15o Correlation matrix of body measurements (Data: SAS) SAS . (n.d.). JMP-SE 8.01 - Body Measurements.jmp.
2.7 NATURAL VARIABILITYo Twice done exercise. First time when students are exploring „Natural Variation‟. Second time with chi-square goodness-of-fit test 16o This supports student understanding of natural variation and that it is measurable.
Typical JMP® output from „dodgy dice‟ exerciseo When exploring natural variability, they compare the distributions to the expected distributions and „draw a line in the sand‟; if the expected and observed distributions are too far apart, they will reject the dice/coins as „dodgy‟.o When applying chi-square goodness-of-fit, they enter into JMP® 17 what the expected values should be in order to measure natural variability.
2.8 EXPLORE DISTRIBUTIONSo Rather than explore individual binomial probabilities, explore and visualize entire distributions to examine concepts in more depth. 18
Typical student JMP® output for Zener Cards exerciseThis question provokes student reflection and class discussionon the following conceptual topics either already covered or tobe covered in class:• Probability density functions• Cumulative probability• Sum of expected outcomes• Null Hypothesis setting on a value for alpha• Natural variability 19
2.10 “ASSOCIATION IS NOT CAUSATION” (ALIAGA, ET AL., 2010) Bubble plot of „Storks Deliver Babies‟ by countryThe data set “Storks deliver babies” ” (Matthews, 2000), will show anassociation between Storks (pairs) and Birth Rate (1000‟s/yr). By adding thevariable (country) Area (km2) and visualizing the data through a bubble plotwhere the country area is the size of the bubble, the presence of a covariateis evident. 20• Matthews, R. (2000). Storks Deliver Babies. Teaching Statistics, Vo. 22, No. 2, pages 36 - 38.• Aliaga, M., Cobb, G., Cuff, C., Garfield, J., Gould, R., Lock, R., et al. (2010). GAISE: Guidelines for Assessment and Instruction in Statistics Education: College Report. American Statistical Association.
3D Scatter plot “Storks Deliver Babies” with a fourth variable (Humans (millions)) added.o In JMP® a 3D scatterplot can be rotated, etc. It is less effective 21 represented as a 2D image.
2.9 „THE BALANCING ACT‟ – USING STATISTICAL SOFTWARE PURPOSEFULLY AND HOW TO ASSESS STUDENTSo GAISE report “We caution against using technology merely for the sake of using technology” (Aliaga, et al., 2010)o “Rather than let the output be the result, . . . , it is important to discuss the output and results with students and require them to provide explanations and justifications for the conclusions they draw from the output and to be able to communicate their conclusions effectively” (Chance, Ben-Zvi, Garfield, & Medina, 2007)o “Conceptual understanding takes precedence over procedural skill” (Burrill & Elliott, 2000)o In a traditional statistics course, all too often procedure blurs concept; some students can use formulae to get correct answers, but cannot tell you why they are doing what they are doing.o Align a course and assessment to conceptual understanding and interpretation, supported through “statistical literacy” and “statistical thinking”.• Aliaga, M., Cobb, G., Cuff, C., Garfield, J., Gould, R., Lock, R., et al. (2010). GAISE: Guidelines for Assessment and Instruction in Statistics Education: College Report. American Statistical Association. 22• Chance, B., Ben-Zvi, D., Garfield, J., & Medina, E. (2007). The Role of technology in Imporving Student Learning of Statistics. Technology Innovations in Statistics Education, 1(1).• Burrill, G. F., & Elliott, P. C. (2000). Thinking and Reasoning with Data and Chance. National Council of teachers of Mathematics.
2.9 „THE BALANCING ACT‟ - ASSESSMENT 23 Exercise – Weight Loss Programs
Typical output for Exercise – Weight Loss ProgramsThe following reflective questions are then asked:• What type of distribution are the four „bell curves‟ on the right of the output and what is their relationship with the t-statistic and „degrees of freedom‟?• Given that the Null Hypothesis for each test is initially true, what does the p-value tell us (hint: think natural variability)?• As the mean weight loss increases over the four weight loss programs, how and why does this effect: 24 • The t-test statistic? • The p-value? • The Null Hypothesis?
Assessment examples: z-score• What does a z-score measure? Include a sketch to help explain.• For the z-score formula, what is the purpose of the numerator and denominator?• If a z-score of 1 equals a p-value of 0.84 and a z-score of 2 equals a p-value of 0.975, then does a z-score of 1.5 equal (0.84+0.975)/2? Give your answer and explain your reasoning (a sketch would be useful)• A student calculates a p-value for a corresponding z-score of 2.8 for a normal distribution to be 0.997. Does this result seem reasonable? Justify your reason.• It was found that the mean IQ of the population is 100 with a standard deviation of 15 (Neisser, 1997). Discuss how you would calculate the percentage of the population with an IQ between 69 and 130. The visual below is to help you if required. 25 Neisser, U. (1997). Rising Scores on Intelligence Tests. American Scientist, 85 (440-7).
Assessment examples: binomial distributionA study conducted in in Europe and North America indicated that the ratio of births of male to female is1.06 males/female. (Grech, Savona-Ventura, & Vassallo-Agius, 2002). This results in the probability ofa giving birth to a boy as approximately 51.5%. Presuming this article is accurate, if we distribute theexpected probability of number of boys born out of 10 births, we get the following bar graph: Figure 1 – distribution of expected probabilities of number of boys out of 10 birthsa. What type of probability is being used to model this distribution and why?b. What assumption do we have to make to be able to use this type of probability and why is this assumption important?c. Do you expect this distribution to be symmetric? Justify your decision.d. Show how you would calculate one expected probability outcome from the example (but do not actually calculate it). 26 Grech, V., Savona-Ventura, C., & Vassallo-Agius, P. (2002). Unexplained differences in sex ratios at birth in Europe and North America. BMJ (Clinical research ed.), 324 (7344): 1010–1.
FUTURE WORKWrite a paper to address the following:“Students are expected to bring a graphing calculator with statistical capabilitiesto the exam, and to be familiar with this use.” (College Board, 2005).While high school statistics education will be permanently indebted to the CollegeBoard for the introduction of the AP Statistics curriculum and examination, Ibelieve the above policy regarding graphing calculators slows the development ofK-12 statistics education as teachers and school systems have no pressing needto explore adoption of statistical software.While the use of the most generic graphing calculators has not really changedthat much in statistics class since 1993, statistical software has evolved andcontinues to evolve at a fast pace.By removing the above policy, it could be argued that teachers and schoolsystems would be more motivated to seek out statistical software and thusfacilitate more innovation in statistics education.College Board. (2005). Calculators on the AP Statistics Exam. Retrieved 10 15, 2011, from apcentral.collegeboard.com: 27http://apcentral.collegeboard.com/apc/members/exam/exam_information/23032.html