�Some Hows and Whys for Integrating Statistical Thinking into Mathematics Courses

Beth Chance

Cal Poly – San Luis Obispo

bchance@calpoly.edu

www.rossmanchance.com/chance/

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Feb, 2023

�Outline

• What are we doing right?
• Who is “we”?
• You
• NCTM/ASA/CAUSE: www.causeweb.org
• Statistics educators

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�Outline

• What are “we” doing right?
• What is missing?
• What are the next steps?
• Principles and Examples from own course, new grant
• Example HS modules (DRK12 grant with EDC)

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Where I think “we” are at

• Better understanding of many of the distinctions between statistical and mathematical thinking
• Appreciation for how data-based examples can build student interest in mathematical topics
• Appreciation for how technology can help with analysis, visualization, and deepening student understanding
• Technology tools getting more and more accessible (cost, learning curve, capabilities)
• Introducing data topics at younger and younger grades
• Increasing opportunities for students to build their communication and collaboration skills, even in math courses

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Examples: Statistical vs. Mathematical thinking

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Examples: Statistical vs. Mathematical thinking

• Does writing in cursive improve your SAT score?
• In 2005-6, students who wrote in cursive on the SAT score statistically significantly higher than students who wrote in printed block letters.
• One essay was written, and the same essay was given to each in a group of graders.  Half the graders were randomly assigned to read the essay in cursive. The other half read the essay in printed block letters.  The average score for the cursive essay was statistically significantly higher than the average score for the printed essay.

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Recommendations

• Common Core State Standards
• NCTM Guidelines
• Mathematical practices (e.g., does your answer make sense, modeling, apply knowledge to new scenarios)
• GAISE Guidelines

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Recommendations

• Common Core State Standards
• NCTM Guidelines
• Mathematical practices
• GAISE Guidelines (College)
• Teach statistical thinking
• Focus on conceptual understanding
• Use real data
• Foster active learning
• Use technology to explore concepts and analyze data
• Use assessments to improve and evaluate student learning

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Need to continue to increase awareness of these recommendations among teachers, teacher preparation programs

Moral #1:

What’s missing?

• Data Science
• Where do we start?
• What is data?
• Where do data come from?
• Do these data answer this research question?
• Who benefits from these data? Who is harmed?
• How do we prepare data for analysis?
• How do we validate our analysis?
• How do we explain our results to someone else?
• What are the next steps?

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�Example (HW exercise)

• The Current Population Survey (CPS) is “one of the oldest, largest, and most well-recognized surveys in the United States.  It is immensely important, providing information on many of the things that define us as individuals and as a society – our work, our earnings, and our education.”  (Optional: video overview)
• CPS webpage https://www.census.gov/programs-surveys/cps.html. Follow the links for Technical Documentation and then Methodology.
• Current Population Survey Datasets. Follow the link for Annual Social and Economic Supplements. Download a csv file.
• Data Dictionary. Find the description of the A_HRSPAY variable. (How did you find it? What did you search on?)  What does this variable measure?  Who is measured for this variable?

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Example (G6 – Summarizing distributions)

What is the average hourly pay?

• Look at mean or median?

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• For those who earned an hourly wage

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• And non-zero hourly wage?

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Use the Data Dictionary!

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Example

• CDC’s Vital Statistics Online Data Portal allows you to download birth records for all births in the U.S. in a particular year.
• January 2022 birth weights (n = 305,536 births)
• Make a graph!
• “Normal distribution”

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Focus on the principles, not the shiny tools�

Moral #2:

What’s missing?

• GAISE Guidelines (2015)
• Teach statistical thinking
• Teach statistics as an investigative process of problem-solving and decision making
• Focus on conceptual understanding
• Integrate real data with a context and purpose
• Foster active learning
• Use technology to explore concepts and analyze data
• Use assessments to improve and evaluate student learning

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�What is the “statistical investigative process”?

• Are many variations, but key idea is helping students see the entire process from beginning to end to replication/revision

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• GAISEIIPreK-12

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Example (G6: Summarize distribution)

• Researchers wanted to investigate whether pamphlets containing information for cancer patients are written at a level that the cancer patients can comprehend.

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• They asked the statistician to compare the means
• Median reading level = 9
• Median pamphlet readability level = 9

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�Example – Informal Inference (G7)

• Can dogs detect cancer from breath samples?
• Marine was repeatedly given sets of 5 bags, one from a patient with colorectal cancer
• Marine was correct in 30 out of 33 attempts
• Could Marine have been simply guessing?

applet

“I don’t think he was guessing because the chances of him getting 30/33 by guessing is like 0.00001. “

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Focus on the statistical investigation process (Research question, data collection, analysis, “looking forward and behind”)��

Moral #3:

What’s missing?

• GAISE Guidelines (2016)
• Teach statistical thinking
• Focus on conceptual understanding
• Integrate real data with a context and purpose
• Foster active learning
• Use technology to explore concepts and analyze data
• Use assessments to improve and evaluate student learning

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Example (G6: Summarize distribution)

• What’s happening here?

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• 2020 Men’s Olympic Rowing Team?

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Focus on the why, not just the how���

Moral #4:

Example (G8: bivariate data)

• 1971 Draft Lottery
• CBS News Report:

https://www.youtube.com/watch?v=-p5X1FjyD_g

• 366 blue capsules, each corresponding to a unique (1952) birthdate, were successively drawn from a container. The first date drawn, September 14, was assigned rank 1. This process was repeated until all of the capsules were chosen.

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• Was this a fair, random process?

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Example

• Draft Lottery

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How do I teach this in my math classes?

• Data on stopping distances of cars at various speeds

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Example: Stopping distances

• In driving class, you may be taught the “three second rule”: leaving three seconds of space between your vehicle and the vehicle driving in front of you. The three second rule is said to apply in “good, daylight conditions.” But is there scientific basis for this recommendation? Can this recommendation be supported by data?
• Suppose you want to gather data on stopping distances (in feet) for cars driving at different speeds (miles per hour). Provide instructions to a researcher for obtaining these stopping distance measurements.

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Should we find the mean distance at a specific speed?

Will some estimates be better than others?

Or use a model?

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Example: Stopping distances

• Physics: stopping distance ∝ speed².

applet

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Example: Stopping distances

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Example: Kentucky Derby speeds

• After data cleaning (e.g., change in track length!)

Quadratic Shifted log

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Which model is better?

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Example: Data vs. Model (S-ID)

• Remember the U.S. birthweights in 2022?
• Removing missing values, premature births…

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Example: Margin of Error (S-IC)

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Don’t leave out the math

Moral #5:

How do I teach this in all classes?

• Stopping distances vs. speed
• Mini-experiment: How many drops of water can you fit on the head of a penny? What if you put soap in the water first?
• What is the average length of words in the Gettysburg address? What about more recent State of the Union speeches?
• Who is Robert Galbraith?
• How/Why have immigration patterns changed over time?

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Look for opportunities to combine discussion across courses����

Moral #6:

What’s next?

• GAISE Guidelines (2016)
• Teach statistical thinking.
• Teach statistics as an investigative process of problem-solving and decision making
• Give students experience with multivariable thinking.
• Teach statistical thinking.
• Focus on conceptual understanding
• Integrate real data with a context and purpose
• Foster active learning
• Use technology to explore concepts and analyze data
• Use assessments to improve and evaluate student learning

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�What is multivariable thinking?

• Recognize that variation in a response comes from many, many sources and be able to identify possible sources
• Studies can be designed to more effectively control for the many variables that may influence the response
• Analyze data in the presence of confounding and bias
• Draw appropriate conclusions from a data analysis and convey the limitations of these conclusions

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�Why multivariable thinking?

• Why?
• The world is multivariable; data-centric decision making is on the rise
• Making good decisions informed by data requires multivariable understanding
• Should not wait for only the students taking advanced statistics courses

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�Example: Berkeley admissions

• Two-way table

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• Proportion of females accepted 113/449 = 0.252
• Proportion of males accepted 533/1198 = 0.445

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Example: Berkeley admissions

• Program A

Program F

Program A

applet

�Example: Prediction

Height from hand span

Height from foot length

r = 0.577

r = 0.686

, + 3.2 in

What are the units?

�Example: Prediction

Height from foot length, sex

Height from foot length

s = 3.1 in

r = 0.686

, + 3.2 in

Example: Stopping distances

• Why don’t the observations all fall on the curve?

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• Why not use the same driver, car, course?

data = “model” + unexplained variation

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It’s all about the variability��

Moral #7:

What’s next?

• Give students opportunities to explore, invent, have opinions

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�Example

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• Check out the World Income Inequality database, produce a graph for the USA over time and comment on what it reveals.
• Work though Introductions to Databases and/or Introduction to Querying at the Databases for Many Majors website.

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�Example student response

• The article discusses other concrete factors (which we didn’t explore in the homework) that might potentially affect the difference in median hourly wages by gender and race, such as workforce experience, industry/occupation, and standardized tests scores. Even controlling for these factors, the article points out there are still gaps in median hourly wages by gender and race, which could be attributed in part to racial or gender discrimination. However, there are what are referred to as “unmeasured differences” that could be due to factors other than �discrimination, for example women or minority groups may be discouraged from entering high-paying STEM fields. I though this reference to “unmeasured differences” was cool, since the �researchers are essentially talking about potential confounding variables, but using a more accessible name for it!

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What’s next?

• Use technology tools that are appropriate for the student audience, build coding and algorithmic thinking skills over time

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�Technology for good

• Focus on technology for learning
• Ease of use, free, platform-independent
• Able to handle large (new) datasets
• Encourages student exploration and discovery
• Data visualization
• Multiple variables
• Interactive and dynamic
• Provides meaningful feedback to/back-and-forth with the student
• Doesn’t behave like a “black box”
• Can be used to teach as well as do statistics
• Conceptual understanding
• Communication tool

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�Example - FEV

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Multiple Variables applet

• www.rossmanchance.com/applets/index2021.html

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What’s missing?

• GAISE Guidelines (2016)
• Teach statistical thinking
• Focus on conceptual understanding
• Integrate real data with a context and purpose
• Foster active learning
• Use technology to explore concepts and analyze data
• Use assessments to improve and evaluate student learning

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�Examples

• Castaneda v. Partida – underrepresentation of Mexican-Americans on grand juries in Hidalgo County
• Hazelwood School District vs. United States – underrepresentation of African Americans among high school teachers
• What is the comparison group?
• Title IX lawsuit against Brown University
• U.S. v. Burton
• Kristin Gilbert
• Flint, MI water crisis
• Removal of “unusual observations”
• Why Binomial Distributions Do Not Work as Proof of Employment Discrimination

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�Rouncefield (1995)

(p. 3) “Using this dataset, students can ask real questions about real-life situations. These in turn raise ethical and moral questions, which motivate students’ learning, making the subject matter more relevant and interesting. Just as the teacher of history or literature would not avoid moral issues in her lessons, the statistics teacher likewise should not avoid them.”

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The statistics of poverty and inequality. Journal of Statistics Education, 3(2).

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Engage, Empower, Enrage�

Moral #8:

�Final Example

• Strengthening Data Literacy Across the Curriculum
• Funded by NSF DRK-12 (Award no. 1813956)
• In collaboration with researchers at Education Development Center (EDC) and The Concord Consortium
• Develop modules for use in non-AP Statistics courses in racially and ethnically diverse urban high schools
• Raw data from U.S. Census and American Community Survey (IPUMS)
• Common Online Data Analysis Platform (CODAP)
• Team investigation

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SDLC – Data Portal

• New “plugin”: Microdata portal
• Students can select from dozens of attributes, different states, different years
• Can compare sample to sample variation
• Can compare with actual ACS survey
• Definitely opportunities for data cleaning

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�Module 1 - Income Inequality�

• How is income measured (e.g., vs. wages)
• What do we mean by “income inequality”?
• What does the income distribution look like in 2017?
• Random sampling
• Data cleaning
• Mean vs. Median
• Percentiles (e.g., top 25% vs. bottom 25%), Boxplots
• How has the distribution changed over time?
• Adjusting for inflation
• Mean/Median ratio
• Males vs. Females (79 cents to dollar)
• Males vs. Females after adjusting for education

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$38,000

$30,000

30/38 = .79

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�Module 2 - Immigration

• Myths (TeachingTolerance.org)
• Estimating proportion of population that are immigrants
• Margin of error
• Changes over time
• Where have immigrants come from over time
• Immigration Timeline
• Where have immigrants settled
• Conditional proportions
• Are immigrants as likely to be employed as U.S. born
• Employment rate vs. Labor force participation
• Are immigrants as likely to be employed as U.S. born after adjusting for education

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Module 2 – Immigration cont.

• Row vs. Column percentages

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Final Data Investigation

Part A: Choose one question from the following to investigate

1. What types of occupations are immigrants most likely to hold compared to U.S.‐born individuals?

2. How does the typical wage of immigrants compare to the typical wage of U.S.‐born individuals?

3. Are immigrants in 2017 less likely than immigrants in 1980 to speak English well?

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Part B: Choose a third attribute to extend your analysis. In particular, you will explore how, if at all,

your findings from Part A change when you adjust (or control) for this third attribute.

• Sex

• Race/ethnicity

• Education

• Age

• Marital status

• U.S. region

• Birthplace (Option for Question 3 only)

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Feedback

• Student Elmer: https://www.youtube.com/watch?v=aYpuJA6_1TY
• Teacher Sara: https://www.youtube.com/watch?v=0BMAc1L_-Cw

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SDLC Student quote: Useful, fun, and interesting

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“It helped me put the pieces together when people say you’ll never use this in the real world. This was very helpful to understanding and it made me have a new appreciation for math and especially statistics in general. It just helped me be way more engaged throughout the year, and this was probably the most fun I’ve had in math in a very long time and the most interesting thing I’ve done in years.”

Strengthening Data Literacy across the Curriculum

SDLC Student quote: Real-world relevance

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“It was more akin to the real world and it had more to do with stuff that you see on the news or that you hear. Everyone needs to know about income and what you make and what you expect to make… I thought it was just really interesting compared to the regular put numbers on paper and see what comes out.”

Strengthening Data Literacy across the Curriculum

SDLC Student quote: Focus on current social issues

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“I enjoyed the lessons that discussed current issues such as income inequality between men and women and people with different levels of education. It connected the lesson with the real world and helped me understand both statistics and society in more in-depth ways.”

Strengthening Data Literacy across the Curriculum

�Teaching with Data - Lessons learned

• Need careful consideration of how to have these conversations in class (e.g., “myths”)
• Make sure students feel empowered for change
• Need to balance data cleaning/messiness of real data with more productive struggle, open exploration
• Importance of scaffolding experiences across the grades and across disciplines

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Teaching with Data - Morals

1. Need to continue to increase awareness of these recommendations among teachers, teacher preparation programs
2. Focus on the principles, not the shiny tools
3. Focus on the statistical investigation process (Research question, data collection, analysis, “looking forward and behind”)
4. Focus on the why, not just the how
5. Don’t leave out the math
6. Look for opportunities to combine discussion across courses
7. It’s all about the variability
8. Engage, Empower, Enrage

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Teaching with Data

Can bring together the mathematical habits of mind we want students to develop in an engaging way that will develop essential skills for our data-based society

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– if we work together

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New grant

• “Integrating the statistical investigation process, data visualization, and simulation into high school statistics” (NSF DRK-12, 6/22-5/25)
• Algebra and Geometry courses
• Year 1 cohort: ~8 teachers, class testing first modules in May
• Co-developing lessons, instructor supports (e.g., office hours, videos)
• If interested in potentially participating, email PIs
• Soma Roy (soroy@calpoly.edu)
• Nathan Tintle (nathan.tintle@isi-stats.com)

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�Resources

• Establishing Classroom Ground Rules Washington University in St. Louis, 2009
• Facilitating Challenging Conversations in the Classroom. 
• Four Ways to Facilitate Teaching About Immigration ProQuest, 2017
• Making the Most of “Hot Moments” in the Classroom University of Michigan
• Speak Up At School: How to Respond to Everyday Prejudice, Bias and Stereotypes Teaching Tolerance, 2018
• Race Talk: Engaging Young People in Conversations about Race and Racism Anti-Defamation League, n.d.
• Teaching Race: Pedagogy and Practice Vanderbilt University, 2019

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�References

• Gutstein, E. (2003). Teaching and learning mathematics for social justice in an urban, Latino school. Journal for Research in Mathematics Education, 37–73.
• Lesser, L. M. (2007). Critical values and transforming data: Teaching statistics with social justice. Journal of Statistics Education, 15(1), 1–21.
• Linnenbrink-Garcia et al. (2010). Measuring situational interest in academic domains. Educational and Psychological Measurement, 70, 647–6
• Manyika, J., Chui, M., Brown, B., Bughin, J., Dobbs, R., Roxburgh, C., & Hung Byers, A. (2011). Big data: The next frontier for innovation, competition, and productivity (pp. 1–143). McKinsey Global Institute.
• Priceonomics. (2017, September 28). The data science diversity gap. Forbes. Increasing demands for data fluency, including in K-12 curriculum (e.g., McKinsey Report, 2016)
• Voss, R., & Rickards, T. (2016). Using Social Justice Pedagogies to Improve Student Numeracy in Secondary School Education. Journal of Education and Practice, 7(15), 40–47.
• Sproesser, Engel, & Kuntze (2016). Fostering self-concept and interest for statistics through specific learning environments. Statistics Education Research Journal, 15(1), 28–54.

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Example Resources

• Guidelines for Assessment and Instruction in Statistics Education (GAISE) http://www.amstat.org/education/gaise/gaiseprek-12_full.pdf
• EngageNY (www.engageny.org/resource/grade-6-mathematics-module-6-topic-lesson-1)
• Illustrative Mathematics (www.illustrativemathematics.org)
• NCTM Reasoning and Sense Making series
• Focus on Statistics: Investigations for the Integration of Statistics into Grades 9–12 Mathematics Classrooms (https://www.statisticsteacher.org/files/2021/12/Focus_Final.pdf)
• High School Mathematics Lessons to Explore, Understand, and Respond to Social Injustice
• Digging into data: Illustrating a Data Investigation Process, Statistics Teacher (https://www.statisticsteacher.org/2022/03/23/diggingdata/)
• https://skewthescript.org/

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Example (Tech) Resources

• RossmanChance applet collection: https://www.rossmanchance.com/applets/index2021.html
• GapMinder: https://www.gapminder.org/
• CODAP: https://codap.concord.org/
• Desmos: https://www.desmos.com/
• Tuva Labs: https://tuvalabs.com/
• Statistics in Schools: https://www.census.gov/schools
• Census at School: https://ww2.amstat.org/censusatschool/
• What’s going on in this graph?: https://www.nytimes.com/column/whats-going-on-in-this-graph

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�Thank you!

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• Questions or Comments?

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• bchance@calpoly.edu

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