Friday article summary - what is "good" teaching?

When Good Teaching Leads to Bad Results: The Disasters of "Well Taught" Mathematics Courses
Schoenfield, A. Educational Psychologist. (1988)

Original article is available here.

Judging from how often I've come across people referring to this article, everybody but me must have read it long ago. Probably during teacher training? Teacher education in Sweden is ridiculous and at least mine did not include any research articles of any kind. So I read this article today for my own sake, just to figure out what it is everyone else is talking about.


Quick summary

Background: When this article was published in the late 80s, there had already been substantial research done investigating the relationship between mastery of procedures and understanding of concepts and problems. For example, Wertheimer had shown how students proficient in the four arithmetic operations solved the silly long way questions such as (274+274+274)/3, and similar results were found in other areas of mathematics as well. Schoenfield and others had identified some beliefs that students hold about mathematics learning:

• Belief 1: The processes of formal mathematics (e.g. 'proof') have little or nothing to do with discovery or invention.  Corollary: Students fail to use information from formal mathematics when they are in 'problem-solving mode.'
• Belief 2: Students who understand the subject matter can solve assigned mathematics problems in five minutes or less.  Corollary: Students stop working on a problem after just a few minutes because, if they haven't solve it, they didn't understand the material (and therefore will not solve it).
• Belief 3: Only geniuses are capable of discovering, creating, or really understanding mathematics.  Corollary: Mathematics is studied passively, with students accepting what is passed down 'from above' without the expectation that they can make sense of it for themselves.
• Belief 4: One succeeds in school by performing the tasks, to the letter, as described by the teacher.  Corollary: Learning is an incidental by-product to 'getting the work done'"

Method: "When good teaching leads to bad results" was a case study of a class in a well-to-do suburban upstate New York school.  Students in this school consistently scored very high on the New York Regents examinations, and the teacher of this class had a good reputation among students and colleagues. The class was very well run, with students spending a large percentage of time on-task and carefully following teacher directions.  Schoenfield observed this class regularly during the year, and filmed the unit on geometric constructions which took place just before the Regents exams.

Results: Schoenfield mainly reports about the unit on geometric constructions, in which he observed how the teacher consistently emphasized accuracy and memory rather than understanding of why the methods of construction were successful. At no point were the students asked to, or encouraged, to apply what they had learned earlier in the year about proofs to understand and create their own constructions. The teacher was teaching for maximum performance on the Regents exams, in which questions likewise required no understanding or creativity in constructing.

Conclusion: Even very good teachers with strong students teach in such a way that the above mentioned beliefs are reinforced, and that a main reason for this is the nature of high-stakes assessments.

Some thoughts: obviously, high stake exams matter to how students and teachers approach learning and teaching. Because 80% of my students' marks are determined by their performance on the final IB exam, over which I have no influence, it is very tempting to follow the syllabus and exam format very closely. This is a double-edged sword: it keeps standards high, but narrowly procedure oriented. I'd like to include more conceptual questions on our in-class exams, but it just wouldn't fly with the students or, I think, school leaders.
Within these restrictions, however, there is a massive amount of freedom, only limited by the "contract".
Schoenfield also points out that many teachers and students use the textbook as an authority on what to learn and how to learn it. With courage and effort the textbook can be reduced to a helpful preparation for exams.

High stake exams are necessary - this I firmly believe from watching the inflation of marks and deflation of mastery and understanding among Swedish schools competing with each other. But the exams need to include questions that check conceptual understanding, in order to encourage teachers and textbooks teach accordingly. In the mean while, it is possible to teach that way anyway, and trust that the skills necessary for high marks on the exam will follow naturally.