| Lesson Observations In NYC Public Schools : A Tool For Punishing Teachers | ||||||||||||||||||||||||||||||||||||||||||
Adonai came in my classroom on 9/30/04 : the class was created two days earlier, weeks into the semester, because, as always they were grappling with programming chaos : much of each semester is always wasted as they try to balance classes (some classes overcrowed others nearly empty), and make sure that students' program cards are not in conflict with rosters. Students, unhappy for being transferred out were very disruptive. I thought Adonai was coming to help after I complained about the situation. Her presence did not deter disruptive students from going wild. I had no choice but start the lesson. She observed it and gave me a U-rating -- ugly, within and without. Mathematics vs Mathematics Education " In the world of modern mathematics, Dr. Peter D. Lax, ranks among the giants. As a teenage refugee from Nazis, he worked on the Manhattan Project at Los Alamos, where met the likes of Hans Bethe, Richard Feynman and Edward Teller. As a young mathematician, he was a protégé of John von Neumann, a father of modern computing. (…) This month, the Norwegian Academy of Science and letters announced that Dr. Lax , who is 78, would receive its third Abel Prize, created to compensate for the absence of mathematics category among the Nobel Prizes. Q : Do you believe that high school and college math are poorly taught ? A : By and large, that's correct. I would like to see the schools of education teach much more math than methods of teaching and educational psychology. In mathematics, nothing takes the place of real knowledge of the subject and enthusiam for it. (New York Times 3/29/05) Dr. Wiltshire is a typical math education "doctor" : long on psycho-educational-whatever, but short on the real thing. He observed two lessons of mine : the main reason he gave for rating them unsatisfactory betrayed his ignorance.Throughout the observation report , Dr. Wiltshire strives to find faults, usually by innuendos. It was safe for him as long as he confined himself to peripheral issues: "homework sheet was three days late" ( why not, if you have enough exercises for two , three sessions); "students talking"" (were they disruptive ? I would not allow that) ; "no aim was written on the board" (self-evident : practice problems handouts were distributed for a test review) ; "teacher was working on his laptop" (why not ? he was taking attendance using Excel spreadsheet, the official attendance bubble sheet being scandalously inaccurate), etc. The moment "Doctor" Wiltshire ventured into what really mattered -- math concepts and how they were presented -- he betrayed his ignorance : a strong indicator of the limitations of math education, as opposed to academic math. He wrote : You then asked for volunteers to come to the board and write the different ways in which a triangle can be congruent. Several students volunteered to go to the board and did what you had asked them to do. The information that the students wrote on the board was as follows:1. SAS Secondly, "No student wrote that a right triangle could also be proven congruent using the HY.leg postulate" because they are darn right, and you don't know what you are talking about, Dr. Wiltshire, when you consider that as an omission, when you consider that as a professional mistake deserving a U-rating. Your ignorance is getting costly, sir. Here is why your statement is nonsensical : "HY.leg postulate", where HY presumably stands for Hypotenuse, is a theorem , not a postulate : if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and the corresponding leg of another right triangle, then the triangles are congruent . Therefore, my students did not list it as a postulate. What is a postulate ? It is a general statement accepted without proof -- like an axiom, which has a broader meaning --, and the basis for proving theorems. In other words, if a statement is provable, then it is not a postulate. To bridge your deficiency an example of postulate is in order : : There is one and only one straight line that goes through any two points . (Euclid) I will now use this postulate to prove the following theorem : "Two lines intersect in at most one point. " Proof : Suppose two lines intersect in two points A and B.Then, both lines go through A and B, contradicting the postulate : There is one and only one straight line that goes through any two points. Triangle congruence postulates are : 1. SAS : If two sides and the included angle of one triangle are congruent to the corresponding sides and angle in another triangle, then the triangles are congruent. 2. SSS: If the three sides of one triangle are congruent to the corresponding three sides of another triangle, then the triangles are congruent. 3. ASA: If two angles and the included side of one triangle are congruent to the corresponding angles and sides in another triangle, then the triangles are congruent. AAS is NOT a triangle congruence postulate, contrary to what Wiltshire thought -- he did not object to AAS being listed on the board as postulate. I reminded the student who wrote it that AAS is derived from the ASA postulate as shown below. The proof of the Hy.leg (HL) theorem is straightforward ( it was done in class ):
Likewise, the proof of the AAS theorem is straightforward ( it was done in class ) :
I hope Dr. Wiltshire will carefully read this material and learn from it, and consider seriously taking more math courses : he has too many education credits to our misfortune. Not all doctorate degrees are equal. Doctorate in education impresses only the uninformed -- parents and the public at large. Facts are stubborn : “Consistently, for decades, those college students who have majored in education have been among the least qualified of all college students, and the professors who taught them have been among the least respected by their colleagues elsewhere in the college or university. The word ‘contempt' appears repeatedly in discussions of the way most academic students and professors view their counterparts in the field of education. At Columbia Teachers College, 120 th Street is said to be “the widest street in the world" because it separates that institution from the rest of Columbia University. (…) Education schools and education departments have been called ‘intellectual slums' of the university (…) Hard data on education student qualifications have consistently shown their mental test scores to be at or near the bottom among all categories of students. This was as true of studies done in the 1920's and 1930's as of studies in the 1980's. Whether measured by the Scholastic Aptitude Tests, ACT tests, vocabulary tests, reading comprehension tests, or Graduate Record Examinations, students majoring in education have consistently scored below the national average.” ( Thomas Sowell, Inside American Education The Free Press 1993 p.24).
|