Monday, June 19, 2017

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.

Monday, June 5, 2017

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].

  • 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.

Monday, May 22, 2017

Eugene Stern: How Value Added Models are Like Turds

Eugene Stern critiques the Value Added Model for teacher assessment thusly:
So, just to take another example, if I decided to rate teachers by the size of the turds that come out of their ass, I could wave around a lovely bell-shaped distribution of teacher ratings, sit back, and wait for the Times article about how statistically insightful this is.
Read more at MathBabe. 

Monday, April 10, 2017

Mercator Projection All the Way Down

Map facts: The Mercator projection is technically infinitely tall, and more warped as it goes down, so it must always be cropped somewhere. Below is a cropping somewhat lower than normal, so you can see: (1) Antarctica, (2) buildings at the Amundsen–Scott South Pole Station, and finally (3) individual snowflakes.

Hat tip: Geoawesomeness.

Monday, April 3, 2017

No, we probably don’t live in a computer simulation

A lovely rant by Sabine Hossenfelder:
All this talk about how we might be living in a computer simulation pisses me off not because I’m afraid people will actually believe it. No, I think most people are much smarter than many self-declared intellectuals like to admit. Most readers will instead correctly conclude that today’s intelligencia is full of shit. And I can’t even blame them for it. 

At Backreaction

Monday, March 27, 2017

How To Ruin Your Favorite Sitcoms With Simple Math

Math does not exist to make things better. It exists to empower you to tear things apart.

I support this message. 

Monday, March 20, 2017