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SESSION 1A · Saturday, August 1 · 10:00–11:30 am

Listening to Learners

Using Student Reasoning and Authentic Data

to Teach Correlation and Regression

Jen McNally · Laura Callis

DISCUS-IS · Homework, Attendance, & Final Exam Grade

NSF DUE 2314358

NSF DUE 2235355

INSTITUTE FOR INTRODUCTORY STATISTICS INSTRUCTORS�SPONSORED BY EAPOST: EXPANDING THE ART & PRACTICE OF STATISTICAL THINKING

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Quantitative Data: �The Relationship Between Final Exams, Attendance, Homework

Dr. Laura Callis, Curry College

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Activity Context

In my class, we cover these topics in 3 class sessions
The videos aren’t a lesson – it’s a clinical interview activity we do with students to understand their thinking in our research project
I’ve adapted it as an activity for you all to explore the applets, but you could break it into several days for students, with more scaffolds.
The handout will need editing by you for your students
I’ll demo the applet, you try it alongside me, and then we’ll look at student thinking

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A COMMON QUESTION

“What do you think I’ll get on the final exam?”

Why grades? We needed contexts that were super-obvious to students – they all understood grades!

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Step 1: Ask a research question��How well can we predict final exam grade from homework grade & attendance?

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Step 2: Design a Study & Collect Data

46 students in my classes:

Final Exam Grade
Percent of Classes Missed (out of 100%)
Overall Homework grade

Students are allowed to re-do homework assignments up to 3 times to earn a higher grade.
Assignments are approximately weekly, available at the beginning of the semester

🗨️How do you think

these 3 variables

are related?

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Step 3: Explore The Data

What do you notice? Wonder?

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TEACHING: �WHAT DO STUDENTS THINK?��Note: These Are Interviews, Not Teaching Moments��Let’s develop cognitive empathy with students

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STUDENT THINKING�INSTITUTE_1A_CLIP1.MP4 �INSTITUTE_1A_CLIP3.MP4�INSTITUTE_1A_CLIP2.MP4 ��How are these students approaching the scatterplot? �What do you wonder about how the third student is thinking about the regression equation?

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Step 3: Explore The Data

Data: https://tinyurl.com/Correlation2NSF

Applet:

https://www.rossmanchance.com/applets/2021/regshuffle/regshuffle.htm

Or Google “Rossman Chance Applets”

Click “Analyzing Two Quantitative Variables”

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Step 3: Explore The Data

Remember to click “Use Data”
Choose FinalExam as response variable
Click “Show Descriptive Statistics”
You may want to have a separate tab open with the same set up, but with a different Explanatory variable.
🗨️What do you think? Is there a relationship between these variables? What kind of relationship?

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ASSOCIATION

Positive Association	Observational units that have above-average values for the explanatory variable tend to have above-average values for the response variable. Observational units that have below-average values for the explanatory variable tend to have below-average values for the response variable.
Negative Association	Observational units that have above-average values for the explanatory variable tend to have below-average values for the response variable. Observational units that have below-average values for the explanatory variable tend to have above-average values for the response variable.
Association Strength	The degree to which the above tend to be true

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Using the definition of positive and negative association, what type of association do you think there is between these variables?

There seems to be a stronger association between homework and final exam grade, though not all points follow the pattern. It is a positive association.

There is a slight negative association with PercentMissedClass and FinalExam, but this seems to be really driven by how far some points are from their means.

These kinds of graphs can be used to explain the process of finding the correlation coefficient – how far from the mean for x for each data point? How far from the mean for y for each data point? Standardized (make into z scores), so we don’t need to worry about units. (In other words, the mean lines become our new x and y axes [and get stretched or condensed]). Then, multiply the corresponding differences for each data value together and add them up. Notice that if most of the points are in Quadrants 1 and 3, there’s lots of positive products 🡪 big positive number. If there’s lots of points in Quadrants 2 and 4, lots of negative products 🡪 negative number of large magnitude. But, if the points are balanced in all four quadrants, then the positive and negative products will cancel each other 🡪 sum is close to zero. But, it can be influenced by points that are extremely far away from the means, ie 1-2 points can over inflate the correlation coefficient.

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Correlation Coefficient

Do the correlation coefficients match your predictions and your informal examination of the data?
The correlation coefficient is impacted by values very far away from the mean values for x and y.

See the impact of deleting a point like the student who missed over 50% of class.
Now delete a student who missed 15% of the class. Does r change as much?

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Line of Best Fit

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Line of Best Fit

Move the line by using the green squares to make a better prediction line.
Compare your line with others. How are you deciding whose line is the best predictor?

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Use Your Line to Predict

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Sum Of Absolute Errors

To figure out which line is the best prediction line, it would make sense to add up all of these residuals or errors – but, we should make them all positive so that the negatives and positives cancel out.

1. Click on “Show residuals.” What are all of those blue lines about?

2. Use the Sum of Absolute Errors (SAE) to compare your prediction lines with others. Who has the best line? How are you deciding?

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Sum Of Square Errors

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Did you do better than using �the mean Final Exam score?��

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COMPUTER’S ANSWER

Click “Show Regression Line”

Click “Show Squared Residuals”

Did the computer beat you at creating a better prediction line? How can you tell? Compare your SSEs.

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A comparable measure

The Sum of Square Errors depends on units and it depends on the number of data points

Add a data point that doesn’t sit directly on the line to see how the Sum of Square Errors changes

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Explained variation

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Compare Predictors

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Step 5: Make inferences

How to Simulate:

Put each of the students’ final exam grades on cards and shuffle them
Put each of the students’ total homework grades on a card and shuffle them.
Randomly match 1 card from the final exam grade pile to 1 card from the homework grade pile until all the cards are in pairs.

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SIMULATE

Click “Show Shuffle Options” on the Right

Use the drop-down under “Choose Statistic” to Choose “Correlation”

Under “Select Display” choose “Plot”

Shuffle once. Compare your scatterplot with a neighbor. Is that what you expected? Is it similar or different from our actual data?

Notice that the correlation coefficient from this shuffle gets plotted on the graph. Is this what you expected? How does it compare to the actual data? What about your groupmates?

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MANY REPETITIONS

Increase the number of Shuffles to 1000.

How often did you get a correlation coefficient as larger as the one we got in our actual data?

What does that tell you about whether these results could be due just to chance?

In a separate tab, conduct a similar analysis for the other predictor variable.

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Step 6: Draw Conclusions��Explore Multivariable Applet: https://www.rossmanchance.com/applets/2021/multreg/multreg.html

Which variable, Homework grade or Percent Missed Classes, seems to be a better predictor of Final Exam Grade? How are you deciding?

What if we use both explanatory variables? How does that change our predictions?

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STUDENT THINKING INVESTIGATIONS

Sign up at inclusivestatistics.com

Once you gain access, look through some of the videos of student thinking

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11:25–11:30

Before you leave this session —

write on your index card:

One thing you learned about what students actually do with correlation and regression.

One question you're leaving with about how to teach it.

Keep this card — you'll use it in action planning at 2:00 pm

Session 2A begins at 11:30