Quito, Ecuador 2012 Navigating Statistics and Data Analysis in the Secondary Mathematics Curriculum Roxy Peck

A Bit of History…  In many countries, statistics has been a recommended part of the secondary school mathematics curriculum for a LONG time.  For example, in the United States Curriculum and Evaluation Standards (NCTM, 1989), Standard 10 Principles & Standards for School Mathematics (NCTM, 2000), Data Analysis and Probability Standard Quito, Ecuador 2012

A Bit of History…continued BUT…  Impact? Not so much.  Implementation has been elusive.  Where statistics has been incorporated, it has often been done in an ad hoc way.  All things considered, disappointing and little impact. Quito, Ecuador 2012

A Bit of History…continued So what has changed? In the United States Common Core State Standards in Mathematics. Statistics and probability standards give statistics and probability a more prominent role and place much more emphasis on conceptual understanding. Expected to have major impact on K-12 curriculum beginning in Will also impact AP Statistics course and university level introductory statistics course. Quito, Ecuador 2012

Secondary Statistics Standards Include: (Subset of the Common Core Standards) Summarize, represent, and interpret data on a single count or measurement variable S-ID.1. Represent data with plots on the real number line (dot plots, histograms, and box plots). S-ID.2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. S-ID.3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Quito, Ecuador 2012

The Standards – There’s More! Summarize, represent, and interpret data on two categorical and quantitative variables S-ID.5. Summarize categorical data for two categories in two- way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. Quito, Ecuador 2012

The Standards – But Wait! There’s Even More! Make inferences and justify conclusions from sample surveys, experiments, and observational studies S-IC.3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. S-IC.4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. S-IC.5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. S-IC.6. Evaluate reports based on data. Quito, Ecuador 2012

And not just in the U.S.  Similar content is showing up in the secondary curriculum of many other countries, including  U.K.  Australia  New Zealand  And others! Quito, Ecuador 2012

The Challenge…  Integration of this content presents a challenge for many secondary mathematics teachers.  How is statistical reasoning different from mathematical reasoning?  How can statistical reasoning be developed? Quito, Ecuador 2012

Consider the map of counties shown below. The number in each county is last month’s incidence rate for a disease in cases per 100,000 population. (Dick Schaeffer, 2005)

Quito, Ecuador 2012 Statistical Thinking Versus Mathematical Thinking  Mathematical Thinking Explain patterns Often a deterministic way of thinking  Statistical Thinking Search for patterns in the presence of variability Acknowledge role of chance variation Often involves “Ruling Out Chance” as an explanation

Quito, Ecuador 2012 The BIG Idea…  Ruling Out Chance!  The plan for the rest of today’s session: Three classroom activities that illustrate the connections between probability and statistical inference help students develop statistical thinking

Quito, Ecuador 2012 Ruling Out Chance… Activity 1: The Cookie Game (My favorite classroom activity!)

Quito, Ecuador 2012 Discussion Points  Cookie Game illustrates the thought process that underlies almost all of statistical inference Could this have happened by chance when… Competing claims about a population, one of which is initially assumed to be true (the null hypothesis) Sample from the population A decision based on whether the observed outcome would have been likely or unlikely to occur BY CHANCE when the null hypothesis is true

Quito, Ecuador 2012 Discussion Points  Convincing evidence vs. proof  Relationship between probability assessment and choice of significance level

Quito, Ecuador 2012 Ruling Out Chance…  Activity 2: Inappropriate Dress  A CareerBuilders ( press release dated June 17, 2008 claims more than one third of employers have sent an employee home for inappropriate attire. Suppose that in a random sample of 40 employers, 15 report that they have sent an employee home to change clothes. Do the data provide convincing evidence that the CareerBuilders claim is correct?

Quito, Ecuador 2012 Sample proportion = =.38 Is chance variation from sample to sample (sampling variability) a plausible explanation for why the sample proportion is greater than 1/3?

Quito, Ecuador 2012  To be convinced, we must see a sample proportion not just greater than 1/3, but one that is enough greater than 1/3 that it is not likely to have occurred just by chance due to sampling variability.  Ruling Out Chance: What kind of sample proportions would not be convincing? What kind of sample proportions would we expect to see just due to chance when the population proportion is 1/3 ?

Quito, Ecuador 2012 The dotplot

Quito, Ecuador 2012 Discussion Points  Can’t rule out chance…  What does this conclusion mean? Have we shown that the proportion is NOT greater than 1/3?

Quito, Ecuador 2012 Ruling Out Chance…  Activity 3: Duct Tape to Remove Warts  Some people seem to believe that you can fix anything with duct tape. Even so, many were skeptical when researchers announced that duct tape may be a more effective and less painful alternative to liquid nitrogen, which doctors routinely use to freeze warts. The article “What a Fix-It: Duct Tape Can Remove Warts” (San Luis Obispo Tribune, October 15, 2002) described a study conducted at Madigan Army Medical Center. Patients with warts were randomly assigned to either the duct tape treatment or the more traditional freezing treatment. Those in the duct tape group wore duct tape over the wart for 6 days, then removed the tape, soaked the area in water, and used an emery board to scrape the area. This process was repeated for a maximum of 2 months or until the wart was gone.  Data on handout

Quito, Ecuador 2012  Duct tape was more successful (84% successes) than liquid nitrogen (60% successes), but is this convincing evidence that the duct tape treatment is superior? Could this have happened by chance just due to the random assignment?  Ruling Out Chance: Suppose that there is no difference between the treatments and that these 36 people would have had successful removal no matter which treatment was applied. If this is the case, the difference between the 15 successes for the liquid nitrogen group and the 21 successes for the duct tape group is just due to the “luck of the draw” when the random assignment to groups was done.

Quito, Ecuador 2012 The Frequency Distribution Based on 100 trials Number Of Successes Count Percent

Quito, Ecuador 2012 Discussion Points  Observed result “unlikely” to have occurred by chance just due to the random assignment of subjects to experimental groups (reference to Cookie Game activity for what constitutes “unlikely”)  “Ruled out” chance. What does this conclusion mean? Can we be certain?

Quito, Ecuador 2012 Later, in AP Statistics or University Level Statistics…  Can move from activities like these to the more formal approaches, linking to the activities Sampling distributions become our way of deciding whether we can rule out chance in situations where our intuition isn’t adequate (as it was in the cookie game) or when we get tired of simulating (in the case of proportions). Can then move from situations involving proportions where simulation is more straight forward to situations involving means that don’t lend themselves to simulation. Theoretical results rather than simulation provide the information needed to decide if we can “rule out chance”.

Quito, Ecuador 2012 Linking to Traditional Methods  Hypothesis testing logic and types of conclusions that can be drawn—students remember the cookie game activity and the logic involved  Convincing evidence and significance levels (the cookie game activity makes the customary choices for significance levels seem intuitively reasonable—What does it take to rule out chance?)  P-Values as a basis for drawing a conclusion (the activities motivate why it is reasonable to base conclusions about a population based on the P-Value as a measure of whether it is likely or unlikely that we would observe a sample result as extreme as what was observed just by chance if the null hypothesis is true)

Quito, Ecuador 2012 Thanks for coming! Feel free to contact me with any questions.