Mar 19

AP Statistics the Genghis Khan way 1

AP Statistics the Genghis Khan way 1

Recently I enjoyed reading Jack Weatherford's "Genghis Khan and the Making of the Modern World" (2004). I was reading the book with a specific question in mind: what were the main reasons of the success of the Mongols? Here you can see the list of their innovations, some of which were in fact adapted from the nations they subjugated. But what was the main driving force behind those innovations? The conclusion I came to is that Genghis Khan was a genial psychologist. He used what he knew about individual and social psychology to constantly improve the government of his empire.

I am no Genghis Khan but I try to base my teaching methods on my knowledge of student psychology.

Problems and suggested solutions

Steven Krantz in his book (How to teach mathematics : Second edition, 1998, don't remember the page) says something like this: If you want your students to do something, arrange your classes so that they do it in the class.

Problem 1. Students mechanically write down what the teacher says and writes.

Solution. I don't allow my students to write while I am explaining the material. When I explain, their task is to listen and try to understand. I invite them to ask questions and prompt me to write more explanations and comments. After they all say "We understand", I clean the board and then they write down whatever they understood and remembered.

Problem 2. Students are not used to analyze what they read or write.

Solution. After students finish their writing, I ask them to exchange notebooks and check each other's writings. It's easier for them to do this while everything is fresh in their memory. I bought and distributed red pens. When they see that something is missing or wrong, they have to write in red. Errors or omissions must stand out. Thus, right there in the class students repeat the material twice.

Problem 3. Students don't study at home.

Solution. I let my students know in advance what the next quiz will be about. Even with this knowledge, most of them don't prepare at home. Before the quiz I give them about half an hour to repeat and discuss the material (this is at least the third repetition). We start the quiz when they say they are ready.

Problem 4. Students don't understand that active repetition (writing without looking at one's notes) is much more productive than passive repetition (just reading the notes).

Solution. Each time before discussion sessions I distribute scratch paper and urge students to write, not just read or talk. About half of them follow my recommendation. Their desire to keep their notebooks neat is not their last consideration. The solution to Problem 1 also hinges upon active repetition.

Problem 5. If students work and are evaluated individually, usually there is no or little interaction between them.

Solution. My class is divided in teams (currently I have teams of two to six people). I randomly select one person from each team to write the quiz. That person's grade is the team's grade. This forces better students to coach others and weaker students to seek help.

Problem 6. Some students don't want to work in teams. They are usually either good students, who don't want to suffer because of weak team members, or weak students, who don't want their low grades to harm other team members.

Solution. The good students usually argue that it's not fair if their grade becomes lower because of somebody else's fault. My answer to them is that the meaning of fairness depends on the definition. In my grading scheme, 30 points out of 100 is allocated for team work and the rest for individual achievements. Therefore I never allow good students to work individually. I want them to be my teaching assistants and help other students. While doing so, I tell them that I may reward good students with a bonus in the end of the semester. In some cases I allow weak students to write quizzes individually but only if the team so requests. The request of the weak student doesn't matter. The weak student still has to participate in team discussions.

Problem 7. There is no accumulation of theoretical knowledge (flat learning curve).

Solution. a) Most students come from high school with little experience in algebra. I raise the level gradually and emphasize understanding. Students never see multiple choice questions in my classes. They also know that right answers without explanations will be discarded.

b) Normally, during my explanations I fill out the board. The amount of the information the students have to remember is substantial and increases over time. If you know a better way to develop one's internal vision, let me know.

c) I don't believe in learning the theory by doing applied exercises. After explaining the theory I formulate it as a series of theoretical exercises. I give the theory in large, logically consistent blocks for students to see the system. Half of exam questions are theoretical (students have to provide proofs and derivations) and the other half - applied.

d) The right motivation can be of two types: theoretical or applied, and I never substitute one for another.

Problem 8. In low-level courses you need to conduct frequent evaluations to keep your students in working shape. Multiply that by the number of students, and you get a serious teaching overload.

Solution. Once at a teaching conference in Prague my colleague from New York boasted that he grades 160 papers per week. Evaluating one paper per team saves you from that hell.


In the beginning of the academic year I had 47 students. In the second semester 12 students dropped the course entirely or enrolled in Stats classes taught by other teachers. Based on current grades, I expect 15 more students to fail. Thus, after the first year I'll have about 20 students in my course (if they don't fail other courses). These students will master statistics at the level of my book.

Aug 16

The pearls of AP Statistics 21

Turn a boring piece of theory into a creative exercise

Here is Figure 6.5 from Agresti and Franklin:

Figure 6.5I say: why make students memorize this? Why give it as another axiom when they can deduce this themselves? I do it in two steps, to separate the logic from arithmetics. First, I ask two questions: 1) If you have a symmetric bell-shaped distribution and you know the area of the left tail, what will be the area of the right tail symmetric to it? 2) With the same distribution and tails, how much will be the area between the tails if you know the area of one tail? I didn't see a single student who wouldn't be able to answer these questions. Then I tell the students to look up the tail area in the statistical table in Appendix A and find the areas in Figure 6.5. Initially this can be done for the standard normal (μ=0, σ=1) and then generalized for other normals.

Certainly, this takes more time than just giving a ready recipe, but who said the shortest way is the best? By the way, this exercise satisfies the requirements for exercises used for team competitions.

Wolfram Alpha is a good free resource about anything related to Math (if you have access to the Internet). In particular, if you want your students to visualize what happens when σ changes, tell them to enter the command

Plot[Evaluate@Table[PDF[NormalDistribution[0, \[Sigma]], x], {\[Sigma], {.75, 1, 2}}], {x, -6, 6}, Filling -> Axis]

Similarly, to see the effect of changing μ, they can enter

Plot[Evaluate@Table[PDF[NormalDistribution[\[Mu], 1.5], x], {\[Mu], {-1, 1, 2}}], {x, -6, 6}, Filling -> Axis]
Feb 16

Teaching methodology dilemma: Is lecturing good or bad?

Teaching methodology dilemma: Is lecturing good or bad?

teach or notI have come across a nice paper by G.Gibbs Twenty terrible reasons for lecturing, SCED Occasional Paper No. 8, Birmingham. 1981, available here. The author discusses one by one the following statements:

1.1 "Lectures should last an hour. If I can stay awake for an hour, so can they".

1.2 "Its the only way to make sure the ground is covered".

1.3 "Lectures are the best way to get facts across".

1.4 "Lectures are the best way to get students to think".

1.5 "Lectures are inspirational: they improve students' attitudes towards the subject, and students like them".

1.6 "Lecturers make sure that students have a proper set of notes".

1.7 "Students are incapable of, or unwilling to, work alone, so its good for them to have full timetables".

1.8 "The criticisms one can make of lecturing only apply to bad lecturing".

1.9 "The value of lectures can only be judged in the context of other teaching and learning activities which make up the course".

The main conclusion is

"I believe both institutions and validating bodies ought to be asking serious questions about courses which appear to be based primarily on lecturing as the dominant teaching method"

and it is supported with a deep analysis and references to many researches done at US universities.

Jan 16

Active learning - away from boredom of lectures

Active learning is the key to success.

Active learning

Figure 1. Excel file - click to view video

The beginning of an introductory course in Statistics has many simple definitions. The combination simplicity+multitude makes it boring if the teacher follows the usual lecture format. Instead, I suggest my students to read the book, collect a sample on a simulated random variable and describe that sample. The ensuing team work and class discussion make the course much livelier. The Excel file used in the video can be downloaded from here. The video explains how to enable macros embedded in the file.

The Excel file simulates seven different variables, among them deterministic and random, categorical and numerical, discrete and continuous; there is also a random vector. When you press the "Observation" button, the file produces new observations on all seven variables. The students have to collect observations on assigned variables and provide descriptive statistics. They work on assignments in teams of up to six members. The seven variables in the file are enough to engage up to 42 students.