Today I had a chance to observe the Posa ( pronounced posha) method. A mathematics colleague that I met at the Polya conference delivered the BSME class, using the Posa method. There is a Hungarian way of teaching the gifted, and that is the Posa method, but my colleague Dr. Joseph Lewis (this is a pseudonym) is involved in a research project where he is teaching using this method to research subjects in a high school.
The Posa method follows the interactive approaches to mathematics teaching we believe is the best way to teach mathematics. My colleague kept on saying “do you understand” and I did not completely understand even though I have had number theory. I wonder if the American students in the class, taking it for a grade, understood? I should have understood, but for the first time in my life I did not feel upset because I did not understand. I want to understand, but I would have never said I did not understand. (What does that say about me?)I already went to the internet to find proofs that I remember doing years ago, but remember, I am not sitting in this class for a grade. I am sitting in the class to better understand the POSA method. This is how I understand the POSA method.
Keep in mind, I fully respect the problem solving approach to teaching mathematics and I think this is the way we should be teaching all mathematics classes.
1. Problems are given to students to work on before the formal lecture. There is no textbook. There are carefully selected problems for the purpose of doing the problem solving approach and in this case, the problems are what I call fun problems and proofs. Fun problems are the ones where you can via trial and error work out the answer. Usually these problems have the feedforward and feed-backward. You know if the answer it correct without the telling you. Sometimes the process of working on the problem is the full mathematical experience.
2. Step 2, the students arrive to class with some version of the completed work. Some students have completed the work, and know what the teacher is talking about and confirms the teacher’s work or engaged explanation of the work.
3. Step 3. The teaching of the lesson is a series of questions, with responses to lead students through the process. Here is an example of the questions I documented:
a. “Who has a nice example?”
b. “What would you do now?”
c. “What do you think?”
d. “What if it is nonlinear but bounded?”
e. “How many of them can be counted?”
f. “Who has the construction so that the product that does not equal 72?”
g. “Who can extend the set , so that we have enough elements and the product is divisible by 72?”
h. “If you understand, what is the next construction?”
i. “What if you don’t have any two’s?”
j. “Can you extend the this subset so that the product is not…?”
k. “If you don’t like 3’s then what?”
m. “How do you this is the maximum?”
n. “Should I give you a hint?”
o. “Is it clear?”
(I am going to compare these questions to the type of questions asked by the high school teachers. )
4. Step 4, the inclass interplay is with the students who understand the problems completed the class before.
5. Step 5, introduce a hands on problem that requires manipulating some sort of manipulative. This can be done individually or in small groups. This provides some concrete -non abstract examples of the work discussed abstractly.
6. Students engage in some small in class writing and then the homework for the next time is assigned.
What a wonderful day I had with BSME. There is a lot of support for our American students to learn about the problem solving approach to teaching mathematics. I expect to return to the classroom to see more next week.