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SESSION |
CONTENT |
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1.- 00.02.14 |
Discussed students' commitment to frequent interviews and to maintaining a daily journal whose entries are about their thoughts & difficulties with ideas discussed in class. Explained what we expected in journals and that they are to be turned in daily. Instruction: Slide presentation/discussion on ideas of chance & prob.; discussion focused on having students reflect on and unpack meaning of statements reporting statistical facts (ex. risky rides, vitamin use, Gustav's bad luck, car colors & temperatures) Asked: What does this mean to you? & How do you suppose someone determined these facts? Stressed the big idea: unreasonable to make probabilistic statements about any one particular event/outcome, probability has to do with what we expect to happen in the long run, thus probabilistic statements only make sense when they are about a process that can be repeated over and over. Touched on Rashad & Betty example and idea of conceiving situations as probabilistic Homework: Journal; Uncertainty module: p. 5 questions, read pp. 6-8, answer questions 1-3. |
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2.- 00.02.15 |
Homework submitted by all but 3 students; Pat reminds them of importance of their doing the assigned reading and homework. Discussed "probabilistic situations" and practiced re-conceiving situations probabilistically ( PowerPoint examples: prob. that next U.S. Attorney General is a woman; prob. that it will snow tomorrow; prob. that Titans go to SuperBowl XXXV, etc.) Determining Probabilities: - empirical prob. inferred from data - theoretical prob. inferred from situation and knowledge of process that will be repeated Coin Toss Activity: toss 3 pennies, what is prob. that you get (exactly) 2 heads? Step 1. re-phrase question/situation in probabilistic terms step 2. students determine empirically by repeating experiment, Pat displays data in a table. Homework: Journal entry; Uncertainty module: read pp. 11-14, prepare to discuss a-j in activity representing situations as drawing balls from a bag (pp. 14-15); Exercises 2.2, 2.3, 2.5, 2.10 (pp.15-16)
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3.- 00.02.16 |
Collected journals and homework. Reviewed homework: representing situations as drawing balls from a bag (p. 15, a) predicting sex of a child; b) deciding on chances of successful space shuttle launch; f) predicting how many accidents can be expected each day of the week; g) shooting an arrow at a balloon and hitting it, by a beginner;). Discussed kinds of assumptions needed in order to simulate situations using balls in a bag. ActivStats activity with randomness tool: students in 3's at computers go through section 12.1 activities, students asked to "explain what you observe about what happens in the long run" (simulation is of a cursor landing on one of two colored regions of a bar, idea is to show that despite unpredictability of where cursor lands on each trial a pattern of where it lands emerges in the long run, students should track emergence of pattern and use it to estimate relative proportion of two colored areas). Repeated same activity with multiple randomness tool as a way to gather data more efficiently. Homework: complete ActivStats sections 12.1-12.2, use Random Outcomes with Data Desk to do 12.8 in Active Practice of Stats. |
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4.- 00.02.17 |
Discussion of questions raised in journals/interviews: a) What does it mean that "the probability of a specific outcome on a specific event" is 0 or 1?, b) What is the difference between theoretical and empirical probability? Revisit randomness tool demo (ActivStats 12-2): discuss how simulation works, what is happening, what is meant by "long term behavior" when we repeat a process over and over. Pat points out several things: emerging graph shows history of cumulative results (cum. relative frequency), individual samples vary from one to the next as much early as late BUT cumulative results begin to vary less and less with increasing number of trials, horizontal axis is on logarithmic scale - long discussion ensues on what that means, stu's have great difficulty understanding why equal lengths on scale do not represent equal magnitudes. Ran ActivStats 12-2 Data Desk simulation: "generate random sample" and pointed out decrease in variability of sample proportions with increase in sample size (as shown by the updated pie chart). |
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5.- 00.02.18 |
We discussed simulations and whether we try to simulate all characteristic of probabilistic situations, or just certain essential ones. Luis brought up the idea of simulating repeatedly throwing a six sided dice by repeatedly drawing a ball (w/replacement) from a bag containing six different colored balls, asked if we could use the former to simulate the latter, why? Then we discussed ProbSim and the basic metaphor of drawing elements from a mixer, and what each part in the ProbSim window denotes. Before having them start the Prob Sim activity, we jointly set up Prob Sim to simulate drawing 2 balls from a bag containing 2 red balls and 3 blue balls. We looked at the fraction of the time that we drew a red ball followed by a blue ball (w/replacement) as a way to build a reasonable expectation of the probability of that event. Stu's then engaged in simulating tossing 3 coins 15 times, and the Vanderbilt fans situation. Homework: Use Prob Sim to simulate Vanderbilt fans situations; read pp.25-28 of Uncertainty module and PLAN to use Prob Sim on Exercises 3.2.1, 3.2.2. |
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6.- 00.02.22 |
Students at computers in 3s and work on implementing Prob Sim simulation they were to have planned for 2/18 homework. Some discussion of results of Ex. 3.2.1 simulation, and consequences of sampling with or w/o replacement in Ex. 3.2.2. Pat introduces Rodney King problem and in-class discussion takes place. Students experience difficulty conceiving the situation probabilistically (at first they don’t attend to, or seem to see importance of assuming, population composition), and seem confused about the issues Pat is trying to raise. Students run Prob Sim simulation of Rodney King situation. Students see results of simulation, Pat asks about fraction of the time that all white patrols can occur, but many stu’s seem to continue thinking of a single time that it can occur. Stu’s can’t decide on cutoff level as a lower or upper bound. Homework: develop answer to cutoff level issue (question 2 of Rodney King scenario), and design simulation to investigate the HIV scenario. |
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7.- 00.02.24 |
Pat leads discussion on Rodney King homework questions: "what can it mean that chance is a reasonable explanation for all pursuing officers being white?" Pat stresses that the focus is on the process and not on the result (because the result occurred), and that this is a long-run scenario. Discussion of HIV homework problem. Pat rephrases question and all students agree that equivalent question is: "what fraction of those people who test positive actually have HIV?" Pat then proceeds to use area diagram to illustrate how the population is split up and points out regions relevant for answering the question. Homework: U.S. demographics situation and questions. |
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8.- 00.02.25 |
Luis leads class. Reviews HIV situation and has students again re-phrase situation in probabilistic terms. Stresses the representation of sub-populations on area diagram again, in particular focuses students’ attention on those regions depicting portion of population that tested positive. Constructs the ratio from the diagram by connecting numerator and denominator to corresponding regions on diagram. Luis then introduces 2 other representations, contingency table and tree diagram , as ways other ways to show the same information as the area diagram. Discussion of U.S. demographics homework. Students are perplexed by the categories Hispanic and Non-hispanic, it seems odd to them to classify Americans in this way. Individual students are first asked to re-phrase one of the questions in probabilistic terms and to then say what information in the table they would use to answer the question. Problematic questions were those where one has to think about drawing from a sub-population of U.S. and then think about a fraction of that sub-population. Rho, Omega, and Zeta had some difficulty with these questions. Luis wraps up session by briefing students on what we are trying to have them think about when they think about or encounter probability questions. He stresses that although it may at times seem to them that we are playing a game of getting them to just say what we want to hear, that instead we are trying to help them come to think of situations in a particular way. Part of being able to think this way is to be able to articulate clearly this way of thinking. |
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9.- 00.02.28 |
Pat revisits what it means for an event to be statistically unusual. Students talk about it not happening very often. Pat then re-frames Rodney King scenario from 3 perspectives: prosecutor, juror, judge. Students seem tired of this discussion and appear to be tuning out. Pat revisits U.S. demographic data table but from a sampling perspective. Uses PowerPoint demo to make the link between long-term behavior and sample space; emphasizes need to imagine long runs as containing multitude of outcomes and their relationship to one another. Students again show uneasiness with the Hispanic & Non-Hispanic categories in demographic data chart; journals suggest that this was a real stumbling block for some. |
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10.- 00.02.29 |
Students take Quickie Quiz. Pat then discusses questions with them (contingency table showing Vanderbilt student population composition). Many found Questions 3a & 4a most difficult (items involving conditional statements). Not much indication, in the discussion, of students having had serious difficulty with the questions, but their test papers show otherwise (as though they were unaware that they had misunderstood). Pat introduces probability notation, in the context of the notation X denotes the event "choose a person at random and get a student from the South", then P[X] is to be interpreted as "the fraction of the time that we expect event X to occur in a long series of trials" Homework: read pp. 30-35 of Uncertainty module, do activities in boxes, &Ex. 1, 3, 5, 6. |
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11.- 00.03.01 |
Luis leads class. Puts discussion of last homework on hold indefinitely. Assigns students the Marilyn vos Savant scenario as a mini-project. Gives students good part of class time to work on the problem in pairs without much interference or guidance. Students seem quite engaged with the problem. Towards end of class Luis hands out instructions on what they are expected to submit for mini-project. Homework: submit written report of solution and present it in class. |
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12.- 00.03.02 |
Class period reduced by 1/2 hours due to school assembly. Three students present their project reports, only one (Rho) had a clear understanding of problem and of how to proceed. |
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13.- 00.03.03 |
Continuation of project reports presentations and discussions. It appears that most students have no clear understanding of the question they need to answer. Luis begins discussion of crime statistics contingency table. Intended to have students to conceive of population as data ( a collection of crime reports each of which records the value of the same three attributes: race of attacker, race of victim, seriousness of the crime). Think of drawing a random sample from this population and then partition the sample into sub-parts and restrict attention to these subparts in conditional probability questions. |
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14.- 00.03.06 |
Luis re-does discussion of crime statistics. In question 3 Luis points out how Alpha restricted his attention to the sub-population of white attacker/white victim, and how he found fraction of this subpart involving Fatal crimes. Uses this to introduce idea of conditional probability. Students then go through ActivStats section 14-1 demo on birth weight & smoking contingency tables and take quiz in that section. For homework, they are to not only answer the probability questions in the quiz but to interpret the notations P[A and B], P[A|B], etc. in probabilistic terms, and to explain what/how they used the information in contingency tables to answer the questions. Also for homework, answer questions about Engineer/Scientists & Education statistics contingency table. |
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15.- 00.03.07 |
Luis goes over the ActivStats section 14-1 quiz on birth weight & smoking contingency table, asking students to elaborate the difference between P[A and B] and P[A|B]. He tries to point out the definition of conditional probability in symbolic form by having students elaborate the structure of their calculation when they are determining P[A|B]. Points out how relationship P[A|B] = P[A and B]/P[B] "falls out" of the calculation. Has students go back over Quickie Quiz and Demographic contingency tables and practice expressing the questions, and the calculations they used to answer them, formally. Some students reveal their difficulties in formalizing the questions, in particular they experience confusion distinguishing P[A and B] from P[A|B]. Short discussion ensues about the distinction. |
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16.- 00.03.09 |
Pat gives presentation of drug use/test accuracy problem using GSP dynamic diagram; he tries to help students build imagery of percentages in the problem by having them make binary comparisons between various shaded regions in area diagrams. Pat presents the next series of assignment problems – packet of 8 probability problem situations. In groups of 2-3, students are to work 5 of the 8 problems (with #1 & #2 mandatory for all) over the next several class periods. Students begin working on #1 (determine what drug test accuracy would give no more than 1% error rate). This problem seems immediately problematic for most (or all); they appear not to understand what question they must try to answer. |
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17.- 00.03.10 |
Pat makes rounds as students continue working probability problems assigned yesterday. Group 1: Delta, Theta, & Zeta as well as Group2: Alpha, Omega, & Rho continue to really struggle with problem #1 (even just trying to understand clearly what question they must answer). Group 3: Beta & Gamma appear to have done #1 & #2, they seem to focus most of their energies on working #4 (highway stats & racial profiling). |
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18.- 00.03.13 |
Students present their solutions to problems 1 and 2. In problem 2 (Prisoner’s Dilemma) students overwhelmingly think that prisoner’s chances of being increase from 1/3 to 1/2 given that he knows one person who will not be released. |
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19.- 00.03.14 |
Pat recaps problem 1: scaffolds students through construction of expression for test + error rate, first conceptually then formally in terms of variables. Gives demo showing contour plot of error rate function. Emphasizes two things: 1) this is not showing them what they are expected to do in the problem, rather he is using computer as an electronic chalkboard; 2) the computer calculates values for z (error rate) as point on x-y plane is selected with cursor (an thus as values for x and y change). Pat tries to help them imagine the resulting path of the distance z varying along a direction perpendicular to board (i.e., the "fairy dust trace" metaphor) as a way to help them anticipate visualizing the surface. He then shows the 3-d graph, relates it to contour graph and points out how changes in z on the surface relate to changes x and y, and has them interpret this in terms of the test + error rate and its relation to accuracy rate and user rate. Pat starts recap of problem 2 (prisoner’s dilemma). He begins by having students simulate the problem scenario: each pair of students (one plays Micheal, the other the Warden) is given a paper bag containing 3 labels "Micheal", "Omega", and "Quincy". It quickly becomes clear that students do not understand the processes they must simulate. Students barely have time to think of what to do before bell rings. |
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20.- 00.03.15 |
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21.- 00.03.16 |
Homework: first stab at the Cab Problem to be submitted next class |
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22.- 00.03.17 |
(no class for us — joint project meeting, no videotape) Students work the Cab Problem a second time in class (they seem to have collaborated) |
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23.- 00.03.27 |
First day back after spring break, students mourning death of schoolmates killed in road accident over the weekend. Many are totally unresponsive (class cancelled for next two days). After short bereavement counseling by Cooperating Teacher (and Pat), Pat discusses relation between each probability notation (P[A|B], P[A and B], P[ A or B]) and its statement, its meaning, its pictorial representation(s), and its formula. Homework: read Uncertainty module section on Mathematical Expectation |
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24.- 00.03.30 |
(no videotape) Pat distributes handout summarizing 00.03.27 discussion on the relation between a probability notation’s symbol, its statement, its meaning, its pictorial representation, and its formula (P[A|B], P[A and B], P[ A or B]). Pat introduces idea of expected value in the context of playing a game involving payoffs. Pay $1 for each play of the following game: flip two coins, if both land heads you receive$.50, if both land tails you receive $2, if a head and a tail you receive $1. Expected value of the game is defined as "the amount per play you win or lose over the long run". Pat asks students "Over the long run, how much would you expect to net per play?". After a short interchange in which it is evident that students have difficulty with the question, Pat suggests that they simulate the situation (by actually performing the experiment 15 times each) as a way to investigate the issue. Pat writes results of each person’s simulation (i.e., number of games in which they each got 0 heads, 1 head, 2 heads, and net payoff for each of the 3 outcomes) in a table on board. Calculates the average payoff per play as (sum of net payoffs)/(number of plays) = 9/105. Some students (Gamma, I think) surprised at this result because they thought each outcome was equally likely, turns out they hadn’t unpacked the outcome "one head and one tail" and realized that it is sum of two probabilities each 1/4. Pat brings this out by constructing a tree diagram and listing all outcomes and probabilities. Discussion then turns to how to think about the game’s expected value without having to consider the number of plays (since this info not provided). Pat asks students "how to turn the given info (net payoff per outcome and probability of each outcome) into average amount we net per play?" Some students (Rho, Gamma) suggest calculating average of (gain per play) [they calculate (1+ 0 + 0 + -.5)/4], whereas Pat was thinking of (average gain) per play. He points out how their calculation answers a different question. Class winds down with Pat’s derivation of the expression for (average gain) per play in terms of number of games played. He points out how cancellation of this variable in expression implies that expectation is independent of number of games played. After pointing this out, Pat further points out how expected value can be expressed in terms of probabilities and payoffs: (1/4)(-.5)+(1/2)(0)+(1/4)(1) = P[lose $.50](-$.50)+P[gain $0]($0)+P[gain $1]($1). He ends by stressing how understanding this rests on their making the connection between P[A] and its meaning "the fraction of the time we expect outcome A to occur over the long run". Homework: read Uncertainty module section on Mathematical Expectation and work the section examples. |
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25. — 00.03.31 |
(No videotape) No students had read the assigned Uncertainty module chapter on expected value. Luis recaps idea of expected value discussed in yesterday’s class, and relates "definition" given in module to expression derived yesterday; he stresses how module definition is actually a verbal description of the calculation given in derived formula and tries to get students to make the correspondence between the two. Luis also points out that in derived general formula the last index of x should be changed from n to k ; he asks why it is important to do so but students not receptive. He thus explains what quantity each symbol in the formula refers to, trying to make sure students especially understand what xs stand for (by appealing to yesterday’s coin flipping game)– they are the amounts you can win on any one play, he’s careful to point out that xI is not the amount you win on ith play of the game. He also stresses that formula is useful for calculating expected values, whereas they should think of the definition when trying to understand meaning of expected value. It became clear that students had not read the 3-page summary of expected value lesson handed out yesterday. Luis tells them that they absolutely need to read it and understand it. In this discussion, Beta comments that the text is "really confusing", Luis interprets her to mean that for understanding, it needs to be read carefully and not in a cursory manner. It’s as though she’s thinking "if I can’t make sense of what it says by glancing at it, then it’s confusing so I’ll stop trying" – this is exemplary of a chronic problem with our students; they’ve bought into the "fast food" culture of school mathematics: the tacit understanding that if you can’t make sense of, or solve, a problem in a few minutes, then give up trying because it must be unduly hard and we’re not expected to have to think too hard in school. Luis distributes first handout (Expected Value Defined) and class discusses point made on the handout: expected value doesn’t predict what value will actually be in any number of repetitions, rather it is an average. He makes the analogy with average speed of a car traveling from point A to point B at different speeds on different segments of trip as the hypothetical constant speed at which car would have to travel in order to cover same distance in same time. Luis hands out 3 expected value problem situations, and asks students to start working problem 1 in groups. He orients students to what they will have to think about:
Students spend rest of class trying to work problem 1. Their first (and perhaps biggest) hurdle is to conceive of what constitutes a game, an outcome or its value. Gamma asks almost right away "but it’s the expected value of what, what is a value here?" Alpha thinks it might be an average number of games, I ask them to read the problem information carefully (there’s a lot of "time off task") and to think about what constitutes an outcome. As class time runs out, and students appear not be progressing much, Cooperating Teacher jumps in and tells them to consider using tree diagrams. From then on he practically takes students through all steps of the problem (which he outlines on blackboard). Given the time constraint and that it was Friday, this may have been a good move. On the other hand, it definitely robbed students of the opportunity to struggle through the issues that are important for conceiving a situation in terms of expected value. As bell rings Luis tells students to try the other problems for Monday, he tells them to try a simulation for clown/card problem (i.e., actually do the clown’s card game many times in order to estimate probabilities of outcomes). This may have fallen on deaf ears, don’t expect much on Monday. |