Date Modeling: From Design Experiments to Scale

Implementation Rich Lehrer- Vanderbilt University

Anthony Petrosino- University of Texas

The Waterbury Summit: Quantitative ReasoningAugust 8, 2013

Quantifying Chance

“If you don’t understand statistics, you don’t know what is going on” (Thompson, 2010, Wired).

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School Instruction

• Variability is given little attention in USA K-12 Instruction

• When it is…it is fairly simple:– Brief exposure– Some statistics (mean and standard

deviation)– No focus on data modeling

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Trouble with Current Practice

• Much of current science instruction fails to push beyond simple comparison of outcomes

• If the numbers are different, the treatments are presumed to differ

• Since tools for understanding ideas about sampling, distribution, or variability are not immediately accessible to students, teachers stick closely to “investigations”

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Data Modeling

• Participation in contexts that allow students to– Develop questions– Consider qualities of measures and attributes

relevant to a question– Structure data– Make inferences about their questions

a single measurement

True measure

measurement error

Contextualizing Chance:Measure of Attribute by Multiple

Measurers

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Context of the Investigation

• A series of tasks and tools aimed at helping students consider error as distributed and as potentially arising from multiple sources.

• Introduced distribution as a means of displaying and structuring variation among student observations of the “same” event

• GOAL: to view variability as distribution and not simply as a collection of differences among measurements.

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Attribution and Mechanisms

• How might the process of measuring produce sources of variability?

• Students measured a number of things calling attention to variability and structure including:– The schools flagpole– A pencil– Model Rockets at the apex of their flight

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Random-Systematic Error

• Multiple sources of random and systematic error were identified and investigated

• The goal was to determine whether the differences between the resulting distributions in height of rockets was consistent with random variation or could be attributed to a design feature (shape of nose cone)

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Contributions to Measurement Variability?

• Different procedures for measuring

• Precision of different measuring tools

• Trial by trial variability

Original Flag Pole Data

Center estimates true height

“Clumpiness” indicates precision

Quantifying Precision of Measure

Revisit Rounded vs. Pointed Activity

Experiment

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Comparing Distributions

Rounded vs Pointed

Pointed- BLUERounded- RED

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Student Reasoning 1) Pencil and paper

1) 1997 NAEP Items

2) Clinical Interviews1) Reasoning about and critiquing experiment2) Influences of process on distribution

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Results

On NAEP items…– Far better than National Sample of 4th Graders on

“Data and Measurement”– Same Level as National Sample of 8th Graders

From clinical interviews- we learned that students generated distributions of measures consistent with process of measurement. Their distributions varied with precision of the instrumentation yet retained the symmetry expected when considering error of measure.

NAEP Item (#MO61905)

NAEP Item (#MO61905) Results

NAEP Item (#M04900)

NAEP Item (#M04900) Results

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Summary

• Lessons learned about variability were used to reason about experiment– Trial to trial variation – Measurement variation – Effect variation

Sum of distances from each measurement to the median

Inventing Statistics of Signal and Noise

Measure Review: Developing an Aesthetic of Measure

• What about the distribution is attended to by the invented measure (the statistic)?

• What is the relation between the symbolic structure of the invented measure and the characteristics of the distribution?

• What is the meaning of different values of the statistic?

• What happens to the statistic when changes to the sample distribution are imagined?

“What I did was I circled the median and mode, Our numbers that closeto it. … All the numbers inside here (pointing to outlined region in the graph). See how small they are? If we agree on them, they will get bigger. But if we disagree on them, they will get smaller (pointing again to the interior of the outlined region in the graph).”

Precision: Clumps and Regions

Shakira: First you get the median and mode. Okay. For the median and mode here 49 and 48. So? those two numbers you keep automatically. Okay. What are the numbers that closest to 49??

Amanda: 47.

Lena: And 48.

Mikela: 47…, 48…, 50…, 51…

Shakira: No. They are close to 49 up here. 50 is one right after 49, and 51 is going after 50. So those two are closest to the 49. They agree the most.

Renee: Okay. How can you put 47 and 49? But you can’t put 37 and 39??

Shakira: Huh??

Renee: In the two little circle things you have up there, in the little groups how can you put 47 and 49 but not 37 and 39??

…

Shakira: Okay. Okay. 37 isn’t the one closest numbers to 39.

Renee: But 47is??

Negotiating Meaning

S: So um my my method was to first find the range of your data and um to find out how much um how ho:::w much the graph or the people agree is uh the range is the key. Um. If the range would be like one like stack= ((moving his right hand up and down)) Teacher: That S: =it would be the range would be zero and like zero is the best? Sort of like grading of it? And then.<< >> and then infinity?((laughing))Infinity is like the worst.

Evan: Why can’t they have two way outliers (on less variable of two data sets).(3.0)John (Author): It doesn't matter.T transforms data to follow Evan’s suggestion((Alex looking at distributions, decides that one is still clearly less variable than the other))Ethan: OH. Not now. Not by his way.John suggests repair by trimming outliers, then computing the range statistic. Others disagree. When to stop “chopping?”

Modeling Variability

Model-based Inference

1. Empirical sampling distribution of a measure (statistic) of the model

1. Compare sample statistic to sampling distribution

Data Display: Case ->Aggregate

Data Display as a Scientific Practice• Grade 3 Investigations of Organism Growth• Context: Social Studies, Role of Silkworms

– Students investigations of• Conditions for Hatching Eggs• Growth

– Structure-function (mouthparts)– Indicators of growth (measure length)– Challenge: For a day of growth, create a display of the data that

shows something that you noticed about the measurements—some pattern or trend—so that someone else looking at your display can see what you notice. (n = 241 larvae)

– Display review. Class compares displays re show and hide.

Knowledge-Practice

• Shape of Data as bridge between organism and population levels of thinking, a key to reasoning about evolution.

• Shape of data may provoke new forms of inquiry.

Case Value

DaD 2

DaD 4

Case Value (DaD2)

Invented Display: Groups of Similar Values (DaD 3)

Scale (DaD 4)

They all want the food

Display-based Reasoning What about the tail?

Sampling Natural Systems

• Samples and Material Means• Sample to Sample Variability (6th grade)

“It’s like sampling, like the armspan. So we were taking samples of Mr. R’s armspan. Other people would put in their measurement and make a bigger graph. Put it together, so it would be more exact. Like at the Silent Street Pond, taking samples, it will get better.”

Height of a single person

?????

Naturally occurring variability

Natural Variation:Heights of People