Suggestions for Best Practices for Statistics Education and Using Statistical                  Software as an Educational ...
research projects, some are fulfilling a general quantitative reasoning requirement)” (Aliaga, et al., 2010,p. 10). A high...
Psychology is one example, but in actuality many disciplines (or a blend) could be used to teach statistics(e.g. business,...
Figure 1 – The ‘continuous loop of experimental design’1.5     Promote ‘Statistical Literacy’A statistics teacher needs to...
Figure 2 – Telephone poll results concerning Terri Schiavo case - CNN/USA TODAY/GALLUP POLL(CNN)Students also need to be e...
simply ‘right or wrong’ like in a traditional mathematics course. Students often also struggle initially tounderstand the ...
Part of a teacher’s responsibility is to demonstrate to their students how to be a reflective learner. Themost effective w...
explore a concept. The best way to approach this challenge is for the teacher to spend a little time eachweek looking for,...
of least-squares regression. It is accompanied by some very good reflective questions. The only issue isthe lack of real d...
Figure 7 – Applet ‘Simulating the probability of a head with a fair coin’(Webster West)1.13    Play games to collect stude...
Figure 8 – Concentration Memory GameSECTION 2 – Using analytical software as an educational learning toolA high school sta...
2.1     Supporting the use of statistical softwareCareful consideration needs to be taken when deciding what analytical so...
Figure 8 – example of video tutorial for use of JMP®2.2     Visuals“Technology has also expanded the range of graphical an...
Figure 10 - JMP® SE 8.01 – help (SAS®)Kinesthetic learning about is usually about manipulating physical objects. For many ...
2.3 Explore Test AssumptionsStatistical software provides an opportunity to explore ideas in more depth. For example, when...
2.4 Create and explore visuals not in a traditional high school statistics class curriculumUsing statistical software, stu...
Figure 13 – Pareto Plot for auto sales, July 2009 (Motorsales, 2009)2.5     Teaching content beyond the high school statis...
Figure 14 – exploring normality2.6     Exploring large datasets with multiple variablesWhile multivariate analysis is prob...
However, adding the category of region (in the US) to the scatterplot (Figure 16) paints a differentpicture:Figure 16 – Bi...
Figure 17 – correlation matrix of body measurements (data: (SAS ))2.7     Natural VariabilityVariability and its measureme...
Figure 18 – Exercise – ‘Dodgy Dice’When the students attempt this exercise at the start of the year they explorethe distri...
2.8     Explore DistributionsThe binomial distribution is mostly taught at high school level as a probability application ...
Figure 21 – typical student JMP® output for Zener Cards exercise2.10    “Association is not causation” (Aliaga, et al., 20...
Figure 22 – Bubble plot of “storks deliver babies”
Figure 23 – interactive 3D scatter plot “Storks Deliver Babies”2.9     ‘The Balancing Act’ – using statistical software pu...
It can be argued that while there is no real gain from students calculating the standard deviation, thereis great benefit ...
Figure 24 – Exercise – Weight Loss ProgramsThe followingconcept and application questions that are then asked:       What ...
Figure 25 – typical student output for ‘Exercise – Weight Loss Programs’
Teachers can still have students calculate in classwork and homework, however, if the purpose is toreinforce a concept, bu...
The College Board® expecting all students to sit the Advanced Placement Statistics examination        with a graphing calc...
ReferencesAliaga, M., Cobb, G., Cuff, C., Garfield, J., Gould, R., Lock, R., et al. (2010). GAISE: Guidelines for        A...
Lienhard, J. H. (1998 - 2002). Nightingales Graph. Retrieved 11 1, 2011, from Engines of our Ingenuity:       http://www.u...
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Suggestions for best practices for statistics education and using statistical software as an educational tool in the classroom v2 10242011

  1. 1. Suggestions for Best Practices for Statistics Education and Using Statistical Software as an Educational Tool in the ClassroomSimon KINGAbstractThe high school course that should be affected the most from the advent of technology in the classroomis probably statistics. However, statistics classes mostly use graphing calculators with perhaps theoccasional use of statistical software to give students experience of statistical output. While thiscircumstance is often due to school resources and their availability, many schools with computerresources are not often utilizing statistical software in their statistics classes.The use of technology and statistics in and away from the classroom is accelerating at a fast pace, and itwould be prudent to keep that pace in the classroom.This paper recommends best practices for statistics education in the classroom and demonstrates howto use statistical software as an educational tool in the classroom, including ideas for assessment.BackgroundLike many other high school mathematics teachers, my career in teaching statistics happened byaccident; a hole in class scheduling that needed plugging and no-one else in the mathematicsdepartment desiring to teach it. What followed in my first year of teaching statistics was a steep learningcurve where I was working hard to stay ahead of my students and my closest friends were the statisticstextbook and my graphing calculator. Once you survive the first year, you reflect and start planning for anew year, asking the question, “Now what?”What becomes obvious in the first year of teaching statistics is the potential of a statistics course; itallows a teacher to be creative in designing lessons, limited to the time to plan and the resources theschool has available (and often the limited access to them). Probably more than any other high schoolcourse, the teaching and learning of statistics in the classroom is highly influenced by the use oftechnology.Section 1 will recommend best practices in the classroom. In section 2, this paper will specifically outlinehow to best use statistical software in the classroom.SECTION 1 – Suggestions for best practices in the classroom1.1 Most of the students taught in a statistics class will not be statisticiansAt a college level, “Today’s introductory statistics course is actually a family of courses taught acrossmany disciplines and departments. The students enrolled in these courses have different backgrounds(e.g. mathematics and, psychology) and goals (e.g., some hope to do their own statistics analyses in
  2. 2. research projects, some are fulfilling a general quantitative reasoning requirement)” (Aliaga, et al., 2010,p. 10). A high school statistics course gives students opportunities to explore interests through data.While a statistics teacher need not react and adapt to every social trend, “Data analysis may be a way tohelp bridge the gap between what students learn in school and their everyday experiences and concernsand thus enrich their school experiences.” (Barnes, 2009, p. 614)1.2 Give students the experience of research and reading of journal papers and articlesIt is important that students in a statistics class understand that statistics is an applied discipline. Whenhaving students read papers and articles, it is important to initially provide ones that are well writtenand accessible. A paper that is difficult to read deflects the student away from the learning experiencethey should be getting. This experience also helps students understand what a scientific paper looks likebefore they attempt their own.It is also important for students to read articles that do not have statistical significance to demonstratethat these papers still contribute to our understanding of a given topic. Many statistics students willconduct experiments that will indicate no statistical inference, but on reflection, they should understandthat this is not a ‘failure’ and can often provoke more discussion (e.g. type II error, experiment designissues, hypothesis validity, etc.).When the students read papers, they can also learn about ways to test a particular hypothesis. They canadapt or replicate a test they have researched (see section 1.4). This supports the students in thedevelopment of “statistical thinking”.1.3 The teacher as a statistical consultant The GAISE: College Report says that students should know, “When to call for help from a statistician.” (Aliaga, et al., 2010, p. 13),thus a good finishing point at the end of a high school statistics course is to have students research, design and conduct an experiment with statistical inference and write a paper of their findings. The teacher plays the role of the statistical consultant to advise and guide the process. The following items are crucial in the process: Before designing and conducting the experiment, the students must have researched journal articles (or similar) on the topic. The experimental design needs to be approved by the ‘consultant’. The ‘consultant’ will support the students in appropriate use of statistical inference. The scientific method is a ‘continuous loop’ (figure 1). “We ultimately need to engage students in all phases of the investigative cycle of statistics, including data gathering, data analysis, and inference” (Groth & N.Powell, 2004, p. 106)1.4 Use psychology context to teach statistics
  3. 3. Psychology is one example, but in actuality many disciplines (or a blend) could be used to teach statistics(e.g. business, biology, engineering, etc.). What makes psychology effective is the accessibility of thesubject matter to high school students. Simple exampleswhich students can study and test includeresearching and testing the effect of the ‘Mozart Effect’ or chewing gum on concentration, or brandpreference (i.e. “The Pepsi Challenge”).What the teacher has an opportunity to do is to present the topic of research first and then find thestatistical inferential solution. Wiberg (2009) wrote, “Instead of teaching statistical methods and thenapplying them to psychological problems and research questions, the order was swapped. The studentswere presented with a psychology research article containing statistics which they would encounterduring the course” (section 4.2)A good example of this is to have students study the ‘Matching Hypothesis’ which states, “The matchinghypothesis proposes that we don’t seek the most physically attractive person but that we are attractedto individuals who match us in terms of physical attraction.” (Eysenck, 2009, p. 4). The students readthree journal articles on the topic, which demonstrate the development of the understanding of thehypothesis over time and how to test for it. These articles help the students understand goodexperimental designs to test for the matching hypothesis and just as importantly how some designshave failed. Usually, the student experimental design solution involves a ranking test which is theintended statistical learning component.This promotes the understanding of “statistical thinking”, or students’ understanding of the why and thehow of conducting of statistical experiments. Additionally, they gain experience of the experimentalprocess.
  4. 4. Figure 1 – The ‘continuous loop of experimental design’1.5 Promote ‘Statistical Literacy’A statistics teacher needs to consider what information they want their students to retain in the future -weeks, months and even years after they have completed the statistics course. Most content, if not usedor reviewed on a regular basis is not retained after completing a course (although reviewing of notesshould recall them), but it is reasonable to suppose that the skill of statistical thinking (ability to thinklogically and analytically) stays with the student for much longer. The development of a statistics coursecurriculum should have ‘statistical literacy’ as one overarching aim that can be supported through thecourse objectives (i.e. content). One technique to promote ‘statistical literacy’ is to have students look athow the media uses statistics. What is key to this process is having the students appropriately reflect onwhat is presented. For example the following prompts might be used to explore the graphic (Figure 2)(CNN) What conclusions can you come to about the topic by exploring the media? How effective is the visual in conveying the important conclusions about the topic? Has the data been used appropriately? Explain.
  5. 5. Figure 2 – Telephone poll results concerning Terri Schiavo case - CNN/USA TODAY/GALLUP POLL(CNN)Students also need to be encouraged to explore thehow the data is used objectively versus subjectively.When students design and conduct their own experiments, they need to approach their analyses anddiscussion objectively. They also need to be aware that, in the end “data beat anecdotes”(Aliaga, et al.,2010, p. 11).1.6 The statistics teacher ‘living and breathing’ statistics Outside of the classroom, it is crucial that a high school statistics teacher furthers his interest in thesubject. There are many ways of doing this: Read “The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century” (Salsburg, 2002) Become a member of the American Statistical Association Take further courses in statistics (e.g. enroll part-time in M.S. Statistics program)Statistics teachers enjoy working with students who have curiosity about the world of statistics and lovelearning; the teacher can help the students develop these characteristics by role modeling the behaviorthey want to see in them.1.7 Statistics should not be taught like a mathematics course“In mathematics, context obscures structure. In data analysis, context provides meaning” (Cobb &Moore, 2000). Many high school statistics teachers come into statistics teaching with a traditionalmathematics background and it can take time to understand that context is fundamental in statisticseducation. The teacher also needs to be aware that many students taking a statistics course for the firsttime struggle with the transition from a traditional mathematics course to a statistics course. They oftenget frustrated with the expectations that they will be writing and that solutions to problems are not just
  6. 6. simply ‘right or wrong’ like in a traditional mathematics course. Students often also struggle initially tounderstand the difference between descriptive statistics and inferential statistics.It might be prudent for a statistics teacher to spend time at the start of the year giving tasks that directlyaddress these issues. One example of how to do this would be through the exploration of the USelection polltracker (figure 3) (BBC, 2008) Figure 3 – US Election Polltracker(BBC, 2008)The 2008 US general election provides context. As an interactive visual, the students can explore thecontext of changes in the time graph. The concept of statistical inference can also be explored as marginof error and sample size is indicated (the students can also explore the idea of a statistical ‘dead heat’).1.8 Teaching without the textbookUS education in general is tied to the heavy use of textbooks (and the other resources that now comewith them) for learning. Overusing a textbook can often lead a teacher away from using creativity todesign lessons and units. Moving away from teaching with a textbook challenges a teacher toindividualize their approach and pedagogy and ultimately gives them full control of the student learning.Teaching without a textbook is not recommended for teachers new to statistics education as it takestime master course content and collect resources.1.9 Student Feedback
  7. 7. Part of a teacher’s responsibility is to demonstrate to their students how to be a reflective learner. Themost effective way to do this is by obtaining student feedback periodically and making reasonablechanges to the course on sections completed (for future classes) and sections not yet covered. Studentbuy-in to this feedback is essential. There are a few steps that make this an effective process: The student feedback should be anonymous The student feedback responses are discussed in class Some changes to the course, however small should be agreed upon with the students as a result of the feedback Students must understand they need to be constructive and respectful when giving feedback The statistics teacher is not defensive about constructive criticism1.10 Incorporating a multicultural curriculumA statistics course presents a teacher with a great opportunity to shape the whole student. By usingappropriate datasets, a statistics course can be enhanced as a multicultural curriculum. For example,exploring a sample of thirty countries from the world prison population data (data: Walmsley, 2007)gives a right-skewed distribution (figure 4) Figure 4 – JMP® output of a sample of 30 world prison population data(Data: Walmsley, 2007)The outlier in this sample is USA with 714 people in prison per 100,000 of national population. Topromote discussion about multi-cultural issues, the students can be asked, “What are the reasons thatUSA has 714 people in prison per 100,000 of the national population? Research is encouraged.”Students need to be encouraged to be respectful and justified in their responses, but the demographicsof the 714 is usually discussed (i.e. race, gender, age, etc.), as well as societal and cultural norms.1.11 Collecting DatasetsThe applications of statistical concepts can often very much depend on the quality of the dataset used.Whenever possible, real data should be used to give a better context. When a teacher plans acurriculum there are always difficulties when a teacher needs a certain type of dataset to help students
  8. 8. explore a concept. The best way to approach this challenge is for the teacher to spend a little time eachweek looking for, finding and storing datasets. The teacher should also be familiar with thecharacteristics of datasets they have collected. So, rather than have to find a new dataset to help withthe application of a new concept, they will instead use one from their dataset library. Datasets can alsobe ‘recycled’. That is, datasets can re-used in a course to explore different concepts and theirapplication. This has the added advantage of the student becoming familiar with the dataset and itscharacteristics. The ‘GAISE: College Report’ recommends that a statistics teacher, “search for good, rawdata to use from web data repositories, textbooks, software packages, and surveys” (Aliaga, et al., 2010,p. 16).1.12 Statistical AppletsWeb-based applets serve a purpose of helping students explore concepts kinesthetically and visually.However, the statistics teacher needs to be careful in their use. “What these tools often gain invisualization and interactivity, they may lose in portability. And while they can be freely and easily foundon the Web, they are not often accompanied by detailed documentation and activities for student use.The time required for the instructor to learn a particular applet/application, determine how to bestfocus on the statistical concepts desired, and develop detailed instructions and feedback for thestudents may not be as worthwhile as initially believed.” (Chance, Ben-Zvi, Garfiel, & Medina, 2007, p.7). The ‘Regression by Eye simulation’ (Rice Virtual Lab in Statistics) (Figure 5) is accompanied withinstructions and a short exercise. The simulation is very good both visually and interactively, but thechallenge for the teacher is to link this into learning. Another issue is that it does not use ‘real’ data andso can be abstract for many students. Figure 5 – ‘Regression by Eye Simulation’ Applet (Rice Virtual Lab in Statistics)The applet ‘Understanding the Least-Squares Regression Line with a Visual Model: Measuring Error in aLinear Model’ (NCTM)(Figure 6) is a wonderful interactive visual to support the student understanding
  9. 9. of least-squares regression. It is accompanied by some very good reflective questions. The only issue isthe lack of real data so some students may struggle with its abstract nature.Figure 6 – Applet ‘Understanding the Least-Squares Regression Line with a Visual Model: MeasuringError in a Linear Model’(NCTM)The applet ‘Simulating the probability of a head with a fair coin’(Webster West) has some instructionsfor use. What makes this a very useful educational tool is that it uses a real application as a simulation(i.e. coin flips) and in that regard a teacher could easily add a few prompts to support the studentdiscovery of The Law of Large Numbers.
  10. 10. Figure 7 – Applet ‘Simulating the probability of a head with a fair coin’(Webster West)1.13 Play games to collect student data in-classUsing flash applets and games there are many ways to quickly and efficiently collect and analyze data in-class. For example, one experiment to conduct in class would be to test for the improvement ofconcentration and performance on a task by chewing gum. To collect data for this experiment, allcontrol and treatment groups would play a concentration game (MathIsFun.com, 2011) and an analysisconducted (bias and confounding would be part of the discussion). Provided it is a timed task, theteacher has many appropriate flash games to choose from. What becomes evident on analysis, is howwell the students know game data; in particular variability and the effect of outlying values. The phrase‘know your data’ comes to mind. However much students enjoy collecting data in this way, the learningpoint cannot be lost, however. “In order for fun to avoid being (or perceived as) frivolous or unrelated tocourse objectives, it is important to present it with structure and intention.” (Lesser & Pearl, 2008).There is nothing simpler than generating data through playing an online game in class for use of analysis(for example, males versus females in a comparison of two means). The GAISE: College Reportrecommends “Us*ing+ class-generated data to formulate statistical questions” (Aliaga, et al., 2010, p.16). While using ‘real data’ is good, having students generate and analyze their own data is one stepbetter.
  11. 11. Figure 8 – Concentration Memory GameSECTION 2 – Using analytical software as an educational learning toolA high school statistics course is usually centered round delivering the College Board AdvancedPlacement Statistics curriculum (College Board) and the use of a graphing calculator. According to theCollege Board, “Students are expected to bring a graphing calculator with statistical capabilities to theexam, and to be familiar with this use.” (College Board, 2005). This creates a dilemma for the statisticsteacher who is fortunate enough to have access to technology, specifically statistical software. While thestatistical software presents a wonderful opportunity to enhance the learning experience of thestudents, using the software more extensively at the expense of the time students might spend with agraphing calculator would potentially put the students at a disadvantage in the College Board APStatistics examination. Therefore, when statistical software (e.g. JMP®, Minitab®, R) in the classroom isused, it is mostly to give the students the experience of seeing and interpreting statistical output. Thisapproach is reflected in most high school statistics textbooks, which heavily incorporate graphingcalculators and usually focus on the use of analytical software at the end of the chapter or assupplemental material. This is a reasonable approach to take considering the College Board policy ongraphing calculator use in the AP Statistics examination and the challenge of limited resources in manypublic schools in USA.The GAISE College Report (Aliaga, et al., 2010, p. 20) indicates, “Regardless of thetools used, it is important to view the use of technology not just as a way to compute numbers but as away to explore conceptual ideas and enhance student learning as well.”This considered, this section will present examples and ideas for how statistical analytical software canbe used as an educational learning tool beyond solely an analytics tool. This section will specifically giveexamples using JMP®, however these examples and ideas can be adapted for other analytical software.
  12. 12. 2.1 Supporting the use of statistical softwareCareful consideration needs to be taken when deciding what analytical software to adopt for theclassroom. Having students use software that requires programming can deflect student attention awayfrom the aims of the course. The use of graphical calculators and statistical software moved the teachingof statistics towards analysis and interpretation and away from pen and paper number crunching. Itcould be argued that excessive time spent on programming again distracts from the real purpose of thecourse.It is important that if we are assessing student analytical and reasoning skills and conceptualunderstanding, they should not be penalized because they cannot recall how to program a statisticaltest or what buttons to press.Using new software can be challenging. Most software we have learned to use (e.g. web browser, wordprocessor, etc.) is through experimentation and the occasional use of the ‘help option’. For studentsusing statistical software we need to follow the same path with perhaps one little addition. For studentsto learn new software the teacher needs to consider the following: Give students time and opportunity to play and explore particularly with graphing Provide step by step instructions for a given statistical process Provide short video tutorials (example – figure 8). These should be no more than five minutes long and tabbed if possible
  13. 13. Figure 8 – example of video tutorial for use of JMP®2.2 Visuals“Technology has also expanded the range of graphical and visualization techniques to provide powerfulnew ways to assist students in exploring and analyzing data and thinking about statistical ideas, allowingthem to focus on interpretation of results and understanding concepts rather than on computationalmechanics.” (Chance, Ben-Zvi, Garfield, & Medina, 2007, p. 4). Just as technology has changed how weteach statistics, visuals have evolved such that we can explore data far more effectively. An effectiveclassroom experience is spending time learning about the history and evolution of visuals (e.g. FlorenceNightingale(Lienhard, 1998 - 2002)), the representation of multiple random variables in one visual (e.g.bubble plots, Charles Joseph Minard(Wikipedia, 2011)) and the more recent phenomena of interactivevisuals. In addition, having the students focus on what constitutes a good (or bad) visual is time wellspent. For example, Figure 9 below is discussed by studentsand they have to come up with a bettervisual recommendation. Figure 9 – example of poor visual - JMP® output of student test scoresJMP® is a useful product in that it makes one pay particular attention to the classification of the randomvariables (Figure 10 - JMP® SE 8.01 – help (SAS®)). If a random variable is wrongly classified, the output isusually different from what was expected.
  14. 14. Figure 10 - JMP® SE 8.01 – help (SAS®)Kinesthetic learning about is usually about manipulating physical objects. For many people it is anengaging way to learn. “I hear and I forget, I see and remember, I do and I understand” (Confucius, 551B.C.). Pressing buttons on a graphing calculator or computer is not kinesthetic learningA good example of kinesthetic learning is exploring bivariate data in JMP®. Teachers and students havean opportunity to exclude points and re-plot the regression line in the same graph. This is an excellentway to explore influential points (figure 10). Figure 10 – JMP® Fit Y by X Analysis. Excluding points and re-fitting the regression line.
  15. 15. 2.3 Explore Test AssumptionsStatistical software provides an opportunity to explore ideas in more depth. For example, whendiscussing the Central Limit Theorem (CLT), a student is usually informed that the sample size should beat least thirty if the population is non-normal. This, as most statisticians know is a simplification and inreality one needs to know as much as possible about the distribution of the population. Using JMP® anda provided script, students can explore different ‘population’ distributions and make discoveries aboutwhat sample size is appropriate. This can lead to a better understanding of what conditions are neededfor the CLT. For example, it can be shown that a sample size of two from a normal distribution (UCLAStatistics, 2008)provides confidence intervals that capture the mean 98% of the time (Figure 11). Figure 11 – JMP® script - confidence intervals (sample size of two) from a normal ‘population’The students also explore non-normal populations to see if a sample of thirty holds. For the learningprocess, students reflect on why they got the results they did (percentage of confidence intervalscontaining the population mean) and using real data provides better context for them to be able tounderstand their results.
  16. 16. 2.4 Create and explore visuals not in a traditional high school statistics class curriculumUsing statistical software, students have the opportunity to explore visuals not in a traditional highschool curriculum. Many teachers have students look at and explore interesting visuals online, but rarelydo students create anything other than traditional visuals. Again, analytical software presents newopportunities for students to create visuals from bubble plots (figure 12) to Pareto plots (figure 13).Using a Pareto plot, students can explore the ‘80-20 rule’.Figure 12 – bubble plot of year, median house price and median house income(U.S. Census Bureau, 2009)
  17. 17. Figure 13 – Pareto Plot for auto sales, July 2009 (Motorsales, 2009)2.5 Teaching content beyond the high school statistics curriculumResources such as analytical software open the availability of statistical inference beyond the scope of aregular high school statistics course. Examples of content that might be added include bootstrapping,Fisher’s exact test and ANOVA. Perhaps the best rule of thumb regarding what content to add is that theconcepts of any new content added needs to be accessible to student. For example, Fisher’s exact testneeds a good grasp of probability. If the course has a high degree on probability content, then perhapsthis is an option. Another potential option is a more in-depth exploration of normality. Adding a Shaprio-Wilk goodness-of-fit test add an extra dimension of analysis for students (Figure 14)
  18. 18. Figure 14 – exploring normality2.6 Exploring large datasets with multiple variablesWhile multivariate analysis is probably beyond the scope of a first year statistics course, statisticalsoftware provides opportunities for students to explore large datasets with multiple variables. Thismirrors the work of a statistician, who often has to explore covariates in bivariate analysis. Visualizationof multiple variables often paints a very different picture than if we limited ourselves to one or twovariables. For example, when we create a scatter plot of average SAT score versus percentage ofstudents taking by State (Figure 15) we see what looks like a negative association.Figure 15 – Bivariate Plot of TOTAL SAT score versus Percent Taking by STATE (College Board, 2001)
  19. 19. However, adding the category of region (in the US) to the scatterplot (Figure 16) paints a differentpicture:Figure 16 – Bivariate Plot of TOTAL SAT score versus Percent Taking by STATE with added indicator of USregion (College Board, 2001)With a little more investigation by students it becomes clear that there are two populations in onescatterplot; one population (top left oval) is of the states where college bound students take only theSAT. The second population (bottom right oval) arethe regions where the students take the ACT as theprimary test and elect to take the SAT.A correlation matrix (figure 17) explores multiple relationships between different body measurementsand mass. The matrix provides a useful exploration of which body measurements most correlate to massin addition to body measurements that correlate to each other.
  20. 20. Figure 17 – correlation matrix of body measurements (data: (SAS ))2.7 Natural VariabilityVariability and its measurement are fundamental to a high school statistics course. Students need tospend time discovering and identifying natural variability. Key to statistical inference is themeasurement of variability. “Variability is natural, predictable and quantifiable” (Aliaga, et al., 2010, p.11). Technology can be used to explore this concept, with particular attention paid to visualization. Anexample of how students can explore the concept of natural variability is by giving them simulated dataof rolls of dice and coins and they have to determine which of the dice and coins are biased (Figure 18 –‘Dodgy Dice’ exercise).
  21. 21. Figure 18 – Exercise – ‘Dodgy Dice’When the students attempt this exercise at the start of the year they explorethe distributions of theevents (Figure 19). What they usually do is compare these distributions to what they think the expecteddistributions of the events would look like. The further away the simulated distribution is from their ideaof the expected distribution, the more likely it is that they will say the coin or dice are dodgy. What theyhave done is draw ‘a line in the sand’ - one side is natural variation and the other side a biased coinand/or dice. The students frequently disagree where to draw the line which is a helpful conversation forwhen the arbitrary nature of alpha=0.05 is introduced. The second time they attempt this exercise isapplying chi-square goodness-of-fit test when they can accurately measure variability. Figure 19 – typical student JMP® output from ‘dodgy dice’ exercise
  22. 22. 2.8 Explore DistributionsThe binomial distribution is mostly taught at high school level as a probability application because of thetype of questions expected in the multiple-choice section of the College Board AP Statistics examination(College Board). However, using statistical software it can be explored as one of the family ofdistributions. Crucially, to have students explore the binomial distribution visually rather than calculateindividual probabilities opens opportunities for students to explore statistical concepts. The Zener Cardsexercise (Figure 20) demonstrates this idea. Figure 20 – Exercise Zener CardsThe students will obtain output as shown in Figure 21. Part c. of the exercise creates a good discussion;because of people’s skepticism of psychic abilities, they consider their friend not to have psychic abilitiesunless there is there is good evidence otherwise (i.e., null and alternate hypotheses). They need todecide what the probability outcome needs to be less than to consider their friend to be psychic (i.e.,alpha value). This is an excellent discussion to review before introducing hypothesis testing.
  23. 23. Figure 21 – typical student JMP® output for Zener Cards exercise2.10 “Association is not causation” (Aliaga, et al., 2010, p. 11)Statistical software is an excellent tool for exploring association versus causation. An excellent classexercise can be taken from the paper and dataset, “Storks deliver babies” (Matthews, 2000). Amultivariate data set here is crucial, to not only explore the association between “pairs of storks” and“birth rate per year (1000’s/yr)” but to visually explore covariates of these variables to a third variable orfourth variable. The bubble plot (Figure 22) shows the covariate “Size by Area km²”, while the 3D scatterplot (figure 23) shows the three previously mentioned variables with the addition of the covariate“Humans (millions)”. While 2D representations of 3D scatter plots are limiting as visuals, JMP® allowsusers to interact with the 3D scatterplot by rotating its axes, etc.
  24. 24. Figure 22 – Bubble plot of “storks deliver babies”
  25. 25. Figure 23 – interactive 3D scatter plot “Storks Deliver Babies”2.9 ‘The Balancing Act’ – using statistical software purposefully and to assess studentsWhen you look at all high school and first year college statistics courses, the variety in pedagogy andavailable resources are vast. This paper concentrates on the use of statistical software as the centrallearning tool in the classroom.The GAISE College report (Aliaga, et al., 2010, p. 20) states, “We cautionagainst using technology merely for the sake of using technology”. Concern remains that using statisticalsoftware becomes ‘too easy’ and students just end up pressing buttons and ‘chasing a p-value’ withoutfully understanding where this value comes from. “Menu driven is commonly easier for most users as itallows the user to navigate using the mouse and to hunt and peck a bit more, which has bothadvantages (students don’t feel lost) and disadvantages (often trial and error strategy rather than realthought when choosing a command).” (Chance, Ben-Zvi, Garfield, & Medina, 2007, p. 5). This highlightsthe potential greatest loss of using statistical software;poorly used resources can result in a loss ofconceptual understanding and the mechanics of statistics.
  26. 26. It can be argued that while there is no real gain from students calculating the standard deviation, thereis great benefit in them seeing the formula and being asked to explore its purpose.Sccording to theGAISE: College Report, “ helps students understand the role of standard deviation as ameasure of spread and to see the impact of individual values on ”(Aliaga, et al., 2010, p. 18).While examples of the pedagogical use of statistical software have been presented in this paper,educators are hesitant to explore the use of statistical software because traditional tests that have anumber of procedural skill questions don’t fit with what is happening in the classroom when usingstatistical software. The aims of the class therefore need to be revisited. I would propose developingnew aims for the statistics course and to align assessment with those aims: conceptual understanding,interpretation, and statistical thinking.“Conceptual understanding takes precedence over procedural skill.” (Burrill & Elliott, 2000, p. 315) In atraditional statistics course, all too often procedure blurs concept; some students can use formulae getcorrect answers, but cannot tell you why they are doing what they are doing.Using statistical software, concepts and output can be combined to great effect. “Rather than let theoutput be the result, . . . it is important to discuss the output and results with students and require themto provide explanations and justifications for the conclusions they draw from the output and to be ableto communicate their conclusions effectively.” (Chance, Ben-Zvi, Garfield, & Medina, 2007, p. 16)In this exercise (Figure 24), the students are asked not only to produce output (see Figure 25), butanswer conceptual questions based on the t-distribution and hypothesis testing.
  27. 27. Figure 24 – Exercise – Weight Loss ProgramsThe followingconcept and application questions that are then asked: What types of distributions 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 affect: o The t-test statistic? o The p-value? o The Null Hypothesis?
  28. 28. Figure 25 – typical student output for ‘Exercise – Weight Loss Programs’
  29. 29. Teachers can still have students calculate in classwork and homework, however, if the purpose is toreinforce a concept, but it is important to have students explore why they are following a certainprocedure. For example, to help students think about z-scores we can use the following prompts: 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)Classwork, homework and assessments need to be aligned. Tests can be given where the students donot use a calculator, but rather answer only conceptual questions and statistical output is usedconceptually and interpretively.Nothing should stop the statistics teacher from showing students statistical tables. Linking a normaldistribution table and the formula for the Gaussian function and calculating z-scores and p-values inWolfram Alpha (Wolfram Alpha LLC, 2011) in figure 26 has merit in the understanding ideas, history andconcepts of the normal distribution. However, as the scientific calculator took away the need forlogarithm tables so probably should technology remove the need for normal distribution tables.Student projects, experiments and papers serve an important purpose (see 1.2, 1.3 & 1.4) of developingstatistical thinking. If at the end of the year, students are expected to research, design experiments,collect data and write-up their results with the teacher supporting as a statistical consultant, thengradual learning steps need to be added throughout the year with more support and structure with theintention of gradually removing the stabilizer wheels. Most students have little previous statisticalexperience at the start of a high school statistics course.SUMMARYSection 1 gives ideas for best practice for statistics teachers, but it is not an exclusive list. There aremany other suggestions for best practice that have merit, such as supporting ‘active learning’ in theclassroom and giving students frequent feedback on how they are doing.Section 2 demonstrates that statistical software can be an integrated learning tool beyond just ananalytic tool in the classroom that will help high school statistics education evolve and keep pace withthe practice of statistics in business and research. However, there are many barriers to teachersadopting such software as a central pedagogical tool. These barriers include the following: limited access to statistical software in many schools textbooks having graphing calculators as the central statistical tool the learning curve required by a teacher and the students in order to use statistical software ‘re-visioning’ of class objectives and assessment
  30. 30. The College Board® expecting all students to sit the Advanced Placement Statistics examination with a graphing calculator (College Board, 2005).Integrating statistical software potentially changes the focus of a statistics course and studentassessment, resulting in a shift in conversation in the classroom. Conversations in class are morefrequently centered about statistical concepts and their application, with less time spent on process andmechanics. Figure 26 – p-value calculation in WolframAlpha(Wolfram Alpha LLC, 2011)
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