9. Prediction
Preamble: The remaining 5 activities represent my attempt to practice drawing students and colleagues into areas of mathematical thinking that range from quite settled for me to still quite unsettled. By Activity 13 we will get to the farthest reaches of my explorations of ways that "changes in work, technology, commerce, and social life [are changing] our needs and capacities for mathematical thinking."
In previous activities, we have seen prediction from two angles: in a
deterministic sense of population growth (on the two islands) and in a
statistical sense of relating the height of a son to the height of their father. The deterministic angle could have been addressed with algebra and calculus, but we skipped that in a (spreadsheet) simulation that proceeds in discrete steps and allows for response to feedback. (Analogy: calculus allows the calculation of, say, the trajectories of missiles, but not the adjustment of the trajectory in response to feedback about whether, say, the missile passed in front of or behind the moving target.) Critical thinking, in the sense of in the sense of understanding ideas and practices better when we examine them in relation to alternatives, was warranted in all three cases. The two angles and critical thinking also come into this Activity.
Preparation
Choose
one of the following sets of items and come prepared to convey the essence of the items, especially in relation to how they illustrate critical thinking in the sense of understanding ideas and practices better when we examine them in relation to alternatives.
1. Models for deterministic prediction
Rainey, "Predicting Irma's Path," indicates how well a huge array of variables integrated into weather models can predict what is to come.
"What is Calculus Used For?,"
https://www.youtube.com/watch?v=_Idra8rVS1I presents three types of modeling and highlights the value of modeling for when experiments cannot be done.
"The Calculus of Catching Fly Balls,"
https://www.youtube.com/watch?v=-J1qryj6kdg alerts us that using heuristic rules not solving fancy models using calculus may be what is going on in the real world.
2. How associations between variables measured at an aggregate level may not match associations between variables measured at the individual level
"Ecological fallacy,"
https://www.youtube.com/watch?v=8Mpi10MRhDU
Krimmel and Rader, "Opposition to Federal Spending"
3. Why was Galton concerned about regression to the mean?
"Regression to the mean,"
https://www.youtube.com/watch?v=1tSqSMOyNFE
Taylor, "Why was Galton so concerned..."
4. Regression equations are not causation, but are they any more than tightness of packing of associated variables?
- Regression equations are unfortunately named because they are used almost always without reference to regression to the mean (which was itself unfortunately named; see #3). A better name would be "equations of best prediction."
Download this spreadsheet:
http://www.faculty.umb.edu/peter_taylor/CorrelationScatter.xlsx
Recreate for yourself the explanation given in the paragraph that begins "When statisticians today use the term regression and deviation" at the bottom of the first column of page 16 of Taylor, "Why was Galton so concerned..."
Goals
- Pursue critical thinking in the sense of understanding ideas and practices better when we examine them in relation to alternatives
- Contrast deterministic with statistical prediction and, for both angles, identify alternatives to standard approaches.
Activities
1. Warm-up (10 minutes)
a. Use chat to inform instructor of which item you prepared.
b. Consider the pollination of a field of flowers by a bee or other insect. Suppose someone tells you they (i.e., the person, not the bees) have used calculus and computer models to find the optimal path the pollinator takes. What heuristic (rule-of-thumb) rule could you invent for the pollinator to take? (Hint: Imagine being a pollinator using a lot of energy to fly from flower to flower, which makes you hungry and frustrated when the flower doesn't have much pollen to replenish your energy.)
c. Recall that, for the heights data, the association that had most social significance was one that did not admit to the obvious causal idea of a heredity connection, namely, mother's and father's heights.
2. Jig-saw discussion of items. Meet in breakout groups (links below) with others who prepared the same set to compare--and thus refine--your understandings (15 minutes). Then meet in breakout groups with others who prepared different sets of readings to: a) convey the essence of the set you read, especially with respect to the goals of the activity (i.e., tension with alternatives and deterministic vs. statistical prediction); and b) learn from the others (30 minutes).
3. Compose your summary of the significance of the items you read (as refined by your jigsaw discussions) and post to the blog, http://wp.me/p98Z1M-w. (15 mins)
4. Plus-delta feedback on activity (5 mins)
References
Krimmel, K. and K. Rader (2017). "Opposition to Federal Spending Is Driven by Racial Resentment," Harvard Business Review, https://hbr.org/2017/09/research-opposition-to-federal-spending-is-driven-by-racial-resentment (viewed 10 Sep 17)
Rainey, J. (2017), "Predicting Irma's Path Is Giving Supercomputers a Challenge," https://www.nbcnews.com/storyline/hurricane-irma/predicting-irma-s-path-giving-supercomputers-challenge-n798961 (viewed 8 Sep 17)
Taylor, P.J. (2008) "Why was Galton so concerned about 'regression to the mean'?—A contribution to interpreting and
changing science and society" DataCritica, 2(2): 3-22. http://www.faculty.umb.edu/peter_taylor/taylor07dGalton.pdf
