teaching high school statistics and use of technology

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SUGGESTIONS FOR BEST PRACTICES FOR STATISTICS EDUCATION AND USING STATISTICAL SOFTWARE AS AN EDUCATIONAL TOOL IN THE CLASSROOM

Simon KINGHigh School Statistics Teacher – Cary Academy, Cary, NCTexas A & M Department of StatisticsSTAT685

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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

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

http://www.collegeboard.com/

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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”

Discussed in STAT641

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1.3 Teacher as the statistical consultanto 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 statisticso 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)

Cobb, G. W., & Moore, D. S. (2000). Statistics and mathematics: Tension and cooperation. American Mathematical Monthly, August-September, 615-630.

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1.8 Teach without the textbooko General overreliance on textbooks in US educationo facilitates creativity by the teachero Teacher is in full control of learning o Not recommended for new to statistics education teachers

1.9 Student feedbacko Role-model being a reflective learnero Collect anonymous feedback on the course from the studentso Share the feedback and make reasonable changes to the course

1.10 Incorporate a multicultural curriculumo Through context and interpretation

1.11 Collect datasetso 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 learningo Better applets generally use real data or actual context

1.13 Play games to collect student data in-classo Fun, but must be linked to learning objectives

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SECTION 2 – USING ANALYTICAL SOFTWARE AS AN EDUCATIONAL LEARNING TOOL

This section presents examples of how statistical software (in this case, JMP®) can be used as an educational tool.

This section also discusses the challenges of adopting such software that the teacher has to consider including how to assess students and the ‘the black box’ issue of statistical software.

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2.1 SUPPORTING THE USE OF STATISTICAL SOFTWARE

Example of video tutorial for use of JMP®

http://castatistics.wikispaces.com/Getting+Started+with+JMP


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2.2 VISUALSExplore history of visuals

Florence Nightingale, 1857

Charles Joseph Minard, 1869

Discussed in STAT604


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Explore poor visuals

o Poor media use of visuals

o poor visual created in JMP®o Students spend time learning best

practices

Discussed in STAT604 and STAT641

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Kinesthetic learning – manipulating visuals

o Students explore good and bad influential points by excluding points and refitting a regression line

o While the data is not ‘real’, it is a context that students can relate to.Discussed in

STAT608

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2.3 EXPLORE TEST ASSUMPTIONS

Population Characteristics

Demo Characteristics

Run Simulation

Population Data

JMP Data Table

Distribution of Sample Data Sample Summary Table

Sample Confidence Interval

Confidence Intervals for Population Mean

Percentage of Confidence IntervalsContaining the Population Mean

Discussed in STAT641 and STAT642

o JMP® script confidence intervals (sample size of two) from a normal ‘population’

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2.4 CREATE AND EXPLORE VISUALS NOT IN A TRADITIONAL HIGH SCHOOL CURRICULUM

o bubble plot of year, median house price and median house income (data: U.S. Census Bureau, 2009)

U.S. Census Bureau. (2009). Income. Retrieved 10 15, 2011, from www.census.gov: http://www.census.gov/hhes/www/income/income.html

http://www.census.gov/

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2.5 TEACHING CONTENT BEYOND THE HIGH SCHOOL CURRICULUM

Yearso Exploring normality with histogram, qq plot and Shapiro-Wilk

test

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2.6 Exploring large datasets with multiple variables

Bivariate Fit of Percent taking By TOTAL SATBivariate Plot of TOTAL SAT score versus Percent Taking by STATE with added indicator of US region (Data: College Board, 2001)College Board. (2001). www.collegeboard.com. Retrieved 10 15, 2011, from www.collegeboard.com: www.collegeboard.com

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Scatterplot Matrixo Correlation matrix of body measurements (Data:

SAS)SAS . (n.d.). JMP-SE 8.01 - Body Measurements.jmp.

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2.7 NATURAL VARIABILITY

o Twice done exercise. First time when students are exploring ‘Natural Variation’. Second time with chi-square goodness-of-fit test

o This supports student understanding of natural variation and that it is measurable.

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Typical JMP® output from ‘dodgy dice’ exercise

o 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® what the expected values should be in order to measure natural variability.

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2.8 EXPLORE DISTRIBUTIONSo Rather than explore individual binomial

probabilities, explore and visualize entire distributions to examine concepts in more depth.

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0123456789101112131415161718192021222324252627282930Total

Level 00000000000000000000000000000001

Count0.0012379400390.0092845502950.0336564948180.0785318212420.1325224483460.1722791828500.1794574821350.1538206989730.1105586273870.0675636056250.0354708929530.0161231331610.0063820735430.0022091793030.0006706437170.0001788383250.0000419152320.0000086296070.0000015581230.0000002460190.0000000338280.0000000040270.0000000004120.0000000000360.0000000000030.0000000000000.0000000000000.0000000000000.0000000000000.0000000000000.0000000000001.000000000000

Prob

N Missing 031 Levels

FrequenciesNumber of correct guesses

Typical student JMP® output for Zener Cards exercise

This question provokes student reflection and class discussion on the following conceptual topics either already covered or to be covered in class:• Probability density functions • Cumulative probability• Sum of expected outcomes• Null Hypothesis setting on a value for alpha• Natural variability

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2.10 “ASSOCIATION IS NOT CAUSATION” (ALIAGA, ET AL., 2010)

The data set “Storks deliver babies” ” (Matthews, 2000), will show an association between Storks (pairs) and Birth Rate (1000’s/yr). By adding the variable (country) Area (km2) and visualizing the data through a bubble plot where the country area is the size of the bubble, the presence of a covariate is evident. • 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.

Bubble plot of ‘Storks Deliver Babies’ by country

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Scatterplot 3D3D 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 represented as a 2D image.

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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.

• 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.

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2.9 ‘THE BALANCING ACT’ - ASSESSMENT

Exercise – Weight Loss Programs

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-10 -5 0 5 10 15

MeanStd DevStd Err MeanUpper 95% MeanLower 95% MeanN

0.3463814.91990840.8982483

2.183505-1.490743

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MomentsHypothesized ValueActual EstimateDFStd Dev

00.34638

294.91991

Test StatisticProb > |t|Prob > tProb < t

0.3856

t Test

0.70260.35130.6487

-3 -2 -1 0 1 2 3

Test MeanProgram A

-10 -5 0 5 10 15


1.41904054.647558

0.84852413.1544672-0.316386

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01.41904

294.64756


1.6724

t Test

0.10520.05260.9474

-3 -2 -1 0 1 2 3

Test MeanProgram B

-10 -5 0 5 10 15


2.42933114.2568999

0.77724.01888360.8397785

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02.42933

294.2569


3.1257

t Test

0.0040 *0.0020 *0.9980

-3 -2 -1 0 1 2 3

Test MeanProgram C

-10 -5 0 5 10 15


5.04072764.50800030.82304456.72404263.3574126

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05.04073

294.508


6.1245

t Test

<.0001*<.0001*1.0000

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Test MeanProgram D

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:• The t-test statistic?• The p-value?• The Null Hypothesis?

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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.

Neisser, U. (1997). Rising Scores on Intelligence Tests. American Scientist, 85 (440-7).

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Assessment examples: binomial distributionA study conducted in in Europe and North America indicated that the ratio of births of

male to female is 1.06 males/female. (Grech, Savona-Ventura, & Vassallo-Agius, 2002). This results in the probability of a giving birth to a boy as approximately 51.5%. Presuming this article is accurate, if we distribute the expected 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 births

a. 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).

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.

Number of boys born

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FUTURE WORKWrite a paper to address the following:

“Students are expected to bring a graphing calculator with statistical capabilities to the exam, and to be familiar with this use.” (College Board, 2005).

While high school statistics education will be permanently indebted to the College Board for the introduction of the AP Statistics curriculum and examination, I believe the above policy regarding graphing calculators slows the development of K-12 statistics education as teachers and school systems have no pressing need to explore adoption of statistical software. While the use of the most generic graphing calculators has not really changed that much in statistics class since 1993, statistical software has evolved and continues to evolve at a fast pace.

By removing the above policy, it could be argued that teachers and school systems would be more motivated to seek out statistical software and thus facilitate more innovation in statistics education.

College Board. (2005). Calculators on the AP Statistics Exam. Retrieved 10 15, 2011, from apcentral.collegeboard.com: http://apcentral.collegeboard.com/apc/members/exam/exam_information/23032.html