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

Monday, March 6, 2017

It Can Never Lie To You

An very nice interview with Sylvia Serfaty, Paris-based mathematician, and winner of the Poincaré Prize:
“First you start from a vision that something should be true,” Serfaty said. “I think we have software, so to speak, in our brain that allows us to judge that moral quality, that truthful quality to a statement.”

At Wired.

Monday, February 13, 2017

Francis Su: Math as Justice

“Every being cries out silently to be read differently.” 

To Live Your Best Life, Do Mathematics

Monday, February 6, 2017

Milliken on CUNY Connected and Remediation

As a follow-up to last week's post, CUNY Chancellor James Milliken has this week unveiled out a new strategic plan called "CUNY Connected". Among the promises are increased graduation rates. In the subsection on remediation reform (again), he writes:
Each fall, approximately 20,000 students—over half of all CUNY freshmen– are assigned to developmental education in at least one subject, usually mathematics. In associate degree programs, 74 percent of freshmen were assigned to developmental education in math in fall 2015, 23 percent in reading, and 33 percent in writing. But CUNY’s one-size-fits-all approach to preparing students has not worked. In fall 2015, just 38% of the 14,215 students in remedial algebra successfully completed it.

Implementing these reforms, the number of students placed in remediation will decline by at least 15 percent. The number of students determined to be proficient after one year of remediation will increase by at least 5 percentage points in year one and will increase as we move to scale.

Under the reforms, 20,000 students per semester will receive tutoring and supplemental instruction and 4,000 will be enrolled in courses with faculty who have been newly trained. Another one thousand students will enroll in immersion programs or new developmental workshops. All students will have access to instructional software.

CUNY will bring to scale two developmental options of proven efficacy: 1) co-requisite courses—credit-bearing courses with additional mandatory supports in the form of workshops or tutoring, and 2) alternatives to math proficiency other than algebra for students pursuing majors or courses of study that do not require algebra. College algebra is necessary for many but not all majors.

We will also end the practice of requiring all students to pass common tests in algebra, writing and reading to exit developmental education. Grades, it has been found, are a better predictor of proficiency and success. CUNY will continue the use of standardized common final exams that count for 35 percent of the final course grade.

These are dictates that were communicated internally at CUNY within the last year. It's interesting that higher passing rates can be dictated in advance by fiat. To be clear: Most CUNY graduates will not need to be algebra proficient, most will not take a course which uses algebra skills, and those who do will not need to succeed on any particular assessment or test to be declared proficient. Another point of clarification: While "college algebra" is mentioned in this section, college algebra is not actually a remedial course (most students already have never taken college algebra at CUNY); the remedial/general education expectation which is being removed is at the level of elementary algebra, around 8th-9th grade level skills as identified in the U.S. Common Core and other curricula.

Monday, January 30, 2017

Milliken on NY1

CUNY Chancellor James Milliken gave an interview Friday night on NY1. Among the things he said:
The urban 3-year community college graduation rate in this country is 16%. CUNY 17½. We're committed to doubling that.
Consider what institutional mechanisms have radically changed highly politicized statistics like that in the past.Full video here (quote at 3:55):

NY1 Online: CUNY's Future

Monday, January 9, 2017

Operations Before Numbers

Most elementary algebra books start on page one with a description of different sets of numbers that will be in use (naturals, integers, rationals, and reals). Then soon after they discuss the different operations to be performed on those numbers, the conventional order-of-operations, etc. This seems satisfying: you get the objects under discussion first, and then modifiers to be performed on those objects (nouns, then prepositions).

But the problem that's irked me for some time is this: the sets of numbers are themselves defined in terms of the operations. Most obvious is the fact that rationals are quotients of integers: a/b (b nonzero); so this presumes knowledge of division beforehand. Integers, too, are really differences of natural numbers (though usually expressed as something like "signed whole numbers"); they are fundamentally a result of subtraction. So in my courses I resolve this by coming out of the box on day one with a review of the different arithmetic operations, names of results, and their proper ordering; then on day two we can discuss the different sets of numbers thus generated.

Now, in other mathematical contexts  -- where you are only discussing one field at a time -- it is conventional to discuss the elements of a set first, and then the operations that we might apply on them second. That makes sense. But at the start of an elementary algebra course we tend to be cheating a bit by trying to consolidate a presentation of at least 4 different sets all at once. It would be fairly rigorous to present naturals and their operations (add, subtract, multiply, divide, etc.), and then integers (and their addition, subtraction, multiplication, etc.), then rationals and their operations (etc.), and then finally a separate discussion of real numbers and their operations (etc.). But that would take an inordinate amount of time, and the operations are so very similar that it would seem repetitive and wasteful to most of our students (outside of difference in closures, etc.).

So if the elementary algebra class wants to cheat in this fashion and present the whole menagerie of number categories in one lecture, I would argue that we need to abstract out the operations first, and then have those available to describe the differences in our sets of numbers second.

Thoughts? Are you still satisfied with describing numbers before operations?

Monday, January 2, 2017

The Nelson-Tao Case

A case that I read in the past, and have searched fruitlessly for months (or years) to cite-reference -- which I just found via a link on Stack Exchange (hat tip to Noah Snyder). Partly so I have a record for my own purposes, here's an overview:

In 2011 Edward Nelson, a professor at Princeton, was about to publish a book demonstrating a proof that basic arithmetic theory (the Peano Postulates) was essentially inconsistent. This started a discussion on the blog of John Baez, in which the eminent mathematician (and superb mathematical writer) Terry Tao spent some time trying to explain what was wrong with Nelson's proof. After about three cycles of back-and-forth, the end result was this:
You are quite right, and my original response was wrong. Thank you for spotting my error.

I withdraw my claim.

Posted by: Edward Nelson on October 1, 2011 1:39 PM

This is one of the best examples of what I personally call "the brutal honesty of mathematics". Read the whole exchange here on John Baez' site.