On Classroom Contracts

There's a long-time trend among some educators to hang their hats on classroom "contracts" a lot of the time. In particular: syllabi and behavior contracts on the first day of class -- that students really are expected to sign and be held to. I've always thought that this is enormously stupid, and my operating assumption is that such educators have never worked in industry or dealt with an actual contract.

Among my problems is that it mis-communicates the condition of being a student in a particular classroom, with a particular instructor. The registration itself compels the student to be held to the standards stipulated by the instructor (hopefully documented in the syllabus), and for the instructor to enforce those standards. Supposedly "signing a contract" doesn't change that in any way, and effectively constitutes fraud on the part of the instructor.

What if a student refuses to sign? Are they then not held to those standards? Are the expelled from the course? Assuming this never happens, is the real lesson then one of training students to mindlessly sign anything put in front of them by an authority figure? *

Here's a quote I ran into today from the website of CIO Magazine, from someone who actually is experienced with these issues in the business setting: 

... the purpose of contracts isn’t to define relationships -- it’s to define what happens when there’s no trust and something goes seriously wrong.

When most educators roll out "contracts" they phrase it in terms of, "a contract is a guarantee that can never be broken", but the truth is they're exactly the opposite. In my game-engineering days, we once worked on a project where work started, progressed, and actually completed while the contract was still being negotiated between our executive and the outside company. That is: the contract was signed after the job was already done. The job itself commenced based on mutual trust and good faith on both sides. The contract was only a foundation for any legal actions that would occur afterward if one party truly screwed the other one over. That's what contracts are really for; evidence in a court case when a dispute does occur. Contracts only matter when they are broken.


* This is an enormously corrosive habit in a civil society. Many times I've fantasized about a reverse lesson where I give a contract on the first day with some clause like "give me your firstborn child", point it out after signing, and then rip them all up in front of students. However, this would rather obviously be too much of a mind-fuck on day one, when the primary challenge is one of building trust at that time.

Good Teaching, Bad Results

A provocative article that I just discovered: Schoenfeld, Alan H., "When Good Teaching Leads to Bad Results: The Disasters of 'Well-Taught' Mathematics Courses" (Educational Psychologist, 1988). From the abstract:

This article describes a case study in mathematics instruction, focusing on the development of mathematical understandings that took place in a 10-grade geometry class. Two pictures of the instruction and its results emerged from the study. On the one hand, almost everything that took place in the classroom went as intended—both in terms of the curriculum and in terms of the quality of the instruction. The class was well managed and well taught, and the students did well on standard performance measures. Seen from this perspective, the class was quite successful. Yet from another perspective, the class was an important and illustrative failure. There were significant ways in which, from the mathematician's point of view, having taken the course may have done the students as much harm as good. Despite gaining proficiency at certain kinds of procedures, the students gained at best a fragmented sense of the subject matter and understood few if any of the connections that tie together the procedures that they had studied. More importantly, the students developed perspectives regarding the nature of mathematics that were not only inaccurate, but were likely to impede their acquisition and use of other mathematical knowledge. The implications of these findings for reseach on teaching and learning are discussed.

In particular, Schoenfeld's observations are largely predicated on "teaching to the test" of standardized finals (esp.: Regents testing in New York State), with students memorizing standardized procedures for each particular problem (including a rote repertoire of geometry proofs and constructions), and generally not being able to think through any problems outside those narrowly-formulated items.

Little did he dare imagine how much more corrosive standardized testing would be 30 years later! A colleague and I were just discussing this issue (narrow and fragile problem-solving knowledge of students) just yesterday.

Hat tip to Daniel Hast on StackExchange ME for the link.

More Reading Fractions as Decimals

Last December, we speculated that many students who are weak in understanding fractions may read them incorrectly as decimals (for example: thinking that 1/2 = 1.2).

For the spring term, I added a question on my first-day diagnostics regarding this topic. Specifically: "Graph the fraction on a number line: 2/3." Four multiple-choice options were given in graphical form: (a) between 0 and 1 [the correct answer], (b) b/w 1 and 2 [at 3/2], (c) b/w 2 and 3 [at 2.3], (d) b/w 3 and 4 [at. 3.2].

Results:

• Remedial intermediate algebra class (N = 26): (a) 62%, (b) 8%, (c) 23%, (d) 8%.
• Credit college algebra class (N = 21): (a) 86%, (b) 5%, (c) 10%, (d) 0%.

Conclusions: In both cases, item (c), the result of thinking that 2/3 = 2.3, was indeed the most commonly selected incorrect response. While most students in both classes selected the correct answer, approximately one-quarter of the intermediate algebra class instead picked the location of 2.3. Students registered for the college algebra class clearly had stronger incoming knowledge of fractions.