Applying Japanese Lesson Study Principles to an Upper-level Undergraduate Statistics Course

Figures & data

Table 1. Short timeline of the study lesson plan

Time	Steps in Study Lesson
Day One	Discuss the general problem: How would an M&M manufacturer decide whether the colors of M&Ms are being produced in the correct proportions?
	Discuss potential sample results: How much deviation is too much?
	Pass out M&M samples and form groups of two. Each group must devise a test statistic to measure the deviance of their sample from what they would expect if the process is working correctly.
	Groups of two combine to form groups of four, and each group of four selects and presents one of their test statistics (with rationale) to the class. The rationale should be based on deliberations about what defines a good test statistic.
	Pose the next problem: Based on their chosen test statistic, would they conclude that their original sample of M&Ms contains convincing evidence that the manufacturing process is malfunctioning?
	Begin to investigate empirical sampling distributions and p-values with hand calculations from simulated samples generated by S-Plus under the null hypothesis.
At-home before Day Two	Generate an empirical sampling distribution for the group test statistic and also for the chi-square goodness-of-fit statistic.
	Find empirical p‐values for group's original data and 10 prototype samples designed to illustrate the performance of the test statistic under specific cases.
Day Two	Discuss results from in-class and take-home assignments. Think about criteria for good test statistics.
	Groups work on the Fumble Problem (Larsen and Marx 2001, p. 253) — students investigate how to extend the chi-square goodness-of-fit test to discrete probability distributions.
	Guide groups to think about issues such as estimating model parameters, adjusting degrees of freedom, and avoiding small cells.
At-home before Day Three	Conduct simulations to see the value in adjusting degrees of freedom in the chi-square distribution when parameters are estimated.
	Examine how the simulation extends goodness-of-fit tests to continuous probability distributions.
Day Three (not officially part of the study lesson)	Discuss results from Days One and Two, and the take-home assignment for Day Three.
	Develop the chi-square test of independence for two-way tables.

Table 2. Four-column lesson plan for first part of Day One

Learning Activities and Key Questions	Student Activity and Expected Responses	Teacher's Response and Things to Remember	Goals and Evaluation
Day One Introduce general problem: How would an M&M manufacturer decide whether the colors of M&Ms are being produced in the correct proportions?		A candy manufacturer is told to make 13% brown, 14% yellow, 13% red, 24% blue, 20% orange, and 16% green candies, but he believes the manufacturing process is malfunctioning.
Brainstorm plan	Suggest ways to evaluate the claim	Get students to suggest collecting data seeing how the data matches the claimed values evaluating the statistical significance of the discrepancy May discuss power / sample size ‖ e.g. Is one sample good enough?	How easily do students consider sampling variability?
Discuss potential sample results: How much deviance is too much?	Expect students to be okay with a little deviance from null, but unsure of where to draw the line.	Present possible ways multinomial sample of size 40 could turn out — ask if each one provides significant evidence of malfunctioning.	Do they understand that some variability from what is expected is natural?
Introduce data	Each student receives a bag, work in pairs to get the tally for the first 40 M&Ms	Blindly take 20 candies from the big bag of M&Ms. If asked, tell them to ignore broken ones. If asked, tell them the bag were randomly purchased from a local store.
Examine sample	Students tally the colors	Look at your sample results. Do they support the manufacturer's claim?	Do students think beyond the sample?
Develop “custom” test statistics: While we expect some discrepancy, how can you decide if your sample is “too different” from expected? How can you measure how “deviant” your sample is? Can you express this as one number?	Students brainstorm ways to measure the deviation. Groups of 2 for 5 minutes.	What are some properties of your measurement technique? Do you expect the results to be large, small? Positive, negative? Pass out Handout #1. Students who want to use z-scores need to combine them in some way to come up with one number.	Which ideas from course do students latch onto? Do their custom statistics separate samples which agree with the null from those which agree with alternative?
Combine with another group: Decide which of two test statistics is preferable.	Prepare to defend choice to class. Groups of 4 (2 groups of 2) for 5 minutes	Encourage groups to be able to defend choice based on desirable properties.	What are seen as good properties of a test statistic?
Share with class	“Defend” their test statistic (and its properties) to the rest of the class	The formula you have come up with is a “test statistic”.

Table 3. Examples of prototype samples

Sample	Blue	Orange	Green	Yellow	Red	Brown
A	10	8	7	5	5	5
B	10	8	7	5	9	1
C	14	4	7	5	5	5
D	13	11	10	2	2	2
E	26	22	20	4	4	4
F	40	0	0	0	0	0

Larsen, R. J., and Marx, M. L. (2001), An Introduction to Mathematical Statistics and Its Applications, 3rd Ed., Upper Saddle River, NJ: Prentice-Hall.

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