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Nov 18

## Little tricks for AP Statistics

This year I am teaching AP Statistics. If the things continue the way they are, about half of the class will fail. Here is my diagnosis and how I am handling the problem.

On the surface, the students lack algebra training but I think the problem is deeper: many of them have underdeveloped cognitive abilities. Their perception is slow, memory is limited, analytical abilities are rudimentary and they are not used to work at home. Limited resources require  careful allocation.

### Terminology

Short and intuitive names are better than two-word professional names.

Instead of "sample space" or "probability space" say "universe". The universe is the widest possible event, and nothing exists outside it.

Instead of "elementary event" say "atom". Simplest possible events are called atoms. This corresponds to the theoretical notion of an atom in measure theory (an atom is a measurable set which has positive measure and contains no set of smaller positive measure).

Then the formulation of classical probability becomes short. Let $n$ denote the number of atoms in the universe and let $n_A$ be the number of atoms in event $A.$ If all atoms are equally likely (have equal probabilities), then $P(A)=n_A/n.$

The clumsy "mutually exclusive events" are better replaced by more visual "disjoint sets". Likewise, instead of "collectively exhaustive events" say "events that cover the universe".

The combination "mutually exclusive" and "collectively exhaustive" events is beyond comprehension for many. I say: if events are disjoint and cover the universe, we call them tiles. To support this definition, play onscreen one of jigsaw puzzles (Video 1) and produce the picture from Figure 1.

Video 1. Tiles (disjoint events that cover the universe)

Figure 1. Tiles (disjoint events that cover the universe)

### The philosophy of team work

We are in the same boat. I mean the big boat. Not the class. Not the university. It's the whole country. We depend on each other. Failure of one may jeopardize the well-being of everybody else.

You work in teams. You help each other to learn. My lectures and your presentations are just the beginning of the journey of knowledge into your heads. I cannot control how it settles there. Be my teaching assistants, share your big and little discoveries with your classmates.

I don't just preach about you helping each other. I force you to work in teams. 30% of the final grade is allocated to team work. Team work means joint responsibility. You work on assignments together. I randomly select a team member for reporting. His or her grade is what each team member gets.

This kind of team work is incompatible with the Western obsession with grades privacy. If I say my grade is nobody's business, by extension I consider the level of my knowledge a private issue. This will prevent me from asking for help and admitting my errors. The situation when students hide their errors and weaknesses from others also goes against the ethics of many workplaces. In my class all grades are public knowledge.

In some situations, keeping the grade private is technically impossible. Conducting a competition without announcing the points won is impossible. If I catch a student cheating, I announce the failing grade immediately, as a warning to others.

To those of you who think team-based learning is unfair to better students I repeat: 30% of the final grade is given for team work, not for personal achievements. The other 70% is where you can shine personally.

### Breaking the wall of silence

Team work serves several purposes.

Firstly, joint responsibility helps breaking communication barriers. See in Video 2 my students working in teams on classroom assignments. The situation when a weaker student is too proud to ask for help and a stronger student doesn't want to offend by offering help is not acceptable. One can ask for help or offer help without losing respect for each other.

Video 2. Teams working on assignments

Secondly, it turns on resources that are otherwise idle. Explaining something to somebody is the best way to improve your own understanding. The better students master a kind of leadership that is especially valuable in a modern society. For the weaker students, feeling responsible for a team improves motivation.

Thirdly, I save time by having to grade less student papers.

On exams and quizzes I mercilessly punish the students for Yes/No answers without explanations. There are no half-points for half-understanding. This, in combination with the team work and open grades policy allows me to achieve my main objective: students are eager to talk to me about their problems.

### Set operations and probability

After studying the basics of set operations and probabilities we had a midterm exam. It revealed that about one-third of students didn't understand this material and some of that misunderstanding came from high school. During the review session I wanted to see if they were ready for a frank discussion and told them: "Those who don't understand probabilities, please raise your hands", and about one-third raised their hands. I invited two of them to work at the board.

Video 3. Translating verbal statements to sets, with accompanying probabilities

Many teachers think that the Venn diagrams explain everything about sets because they are visual. No, for some students they are not visual enough. That's why I prepared a simple teaching aid (see Video 3) and explained the task to the two students as follows:

I am shooting at the target. The target is a square with two circles on it, one red and the other blue. The target is the universe (the bullet cannot hit points outside it). The probability of a set is its area. I am going to tell you one statement after another. You write that statement in the first column of the table. In the second column write the mathematical expression for the set. In the third column write the probability of that set, together with any accompanying formulas that you can come up with. The formulas should reflect the relationships between relevant areas.

Table 1. Set operations and probabilities

 Statement Set Probability 1. The bullet hit the universe $S$$S$ $P(S)=1$$P(S)=1$ 2. The bullet didn't hit the universe $\emptyset$$\emptyset$ $P(\emptyset )=0$$P(\emptyset )=0$ 3. The bullet hit the red circle $A$$A$ $P(A)$$P(A)$ 4. The bullet didn't hit the red circle $\bar{A}=S\backslash A$$\bar{A}=S\backslash A$ $P(\bar{A})=P(S)-P(A)=1-P(A)$$P(\bar{A})=P(S)-P(A)=1-P(A)$ 5. The bullet hit both the red and blue circles $A\cap B$$A\cap B$ $P(A\cap B)$$P(A\cap B)$ (in general, this is not equal to $P(A)P(B)$$P(A)P(B)$) 6. The bullet hit $A$$A$ or $B$$B$ (or both) $A\cup B$$A\cup B$ $P(A\cup B)=P(A)+P(B)-P(A\cap B)$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$ (additivity rule) 7. The bullet hit $A$$A$ but not $B$$B$ $A\backslash B$$A\backslash B$ $P(A\backslash B)=P(A)-P(A\cap B)$$P(A\backslash B)=P(A)-P(A\cap B)$ 8. The bullet hit $B$$B$ but not $A$$A$ $B\backslash A$$B\backslash A$ $P(B\backslash A)=P(B)-P(A\cap B)$$P(B\backslash A)=P(B)-P(A\cap B)$ 9. The bullet hit either $A$$A$ or $B$$B$ (but not both) $(A\backslash B)\cup(B\backslash A)$$(A\backslash B)\cup(B\backslash A)$ $P\left( (A\backslash B)\cup (B\backslash A)\right)$$P\left( (A\backslash B)\cup (B\backslash A)\right)$ $=P(A)+P(B)-2P(A\cap B)$$=P(A)+P(B)-2P(A\cap B)$

During the process, I was illustrating everything on my teaching aid. This exercise allows the students to relate verbal statements to sets and further to their areas. The main point is that people need to see the logic, and that logic should be repeated several times through similar exercises.