Quiz on Finding Intercepts at Automatic-Algebra

We added a new quiz to the Automatic-Algebra site recently: a speed drill in finding intercepts for linear equations written in standard form. This supports speed-graphing lines written in this same defining format, or as usually presented for systems of linear equations (and, of course, is a commonly assessed skill on basic algebra exams). Please check it out and send any feedback that you might have!


Bill Gates Tries Again

Announced last week: Bill Gates will pouring another $1.7 billion into various education initiatives in the next few years. He has previously spent over $5 billion on various initiatives which he admits hasn't shown much in the way of results. This time:
He said most of the new money — about 60 percent — will be used to develop new curriculums and “networks of schools” that work together to identify local problems and solutions, using data to drive “continuous improvement.” He said that over the next several years, about 30 such networks would be supported, though he didn’t  describe exactly what they are. The first grants will go to high-needs schools and districts in six to eight states, which went unnamed.
Sounds a heck of a lot like Achieving the Dream (the network for community colleges).

More at Washington Post.


NY Times on Coding Boot Camp Closures

Recently several large coding boot camp institutes closed their doors, suggest that we may have a bursting bubble in that sector. Among them are (1) Dev Bootcamp, bought by Kaplan, with 6 schools, and (2) The Iron Yard, backed by Apollo Education (Phoenix University), at 15 campuses.

This article asserts that the average course lasts 14 weeks and costs $11,400. Some courses last 26 weeks and cost $26,000. The sector is apparently transitioning such that about half of the registrants are individuals paying on their own, and half are companies paying for employees to up-skill.

Among the difficulties are that the boot camp model only works with intense, face-to-face interactions, and therefore has difficulty scaling to modern profitability levels (contrast this with the MOOC model which seeks to cheaply automate learning for hundreds of thousands, but has failed catastrophically at trying to create success for low-skilled and remedial students). While the Flatiron School in New York has an online offering, it costs $1,500 per month, and personal instructors online throughout the day (the article includes a story of the vice president making a phone call to one panicked student).
“Online boot camp is an oxymoron,” said Mr. Craig of University Ventures. “No one has figured out how to do that yet.”

New York Times.


Thesis Defense Horror Stories

A rich assortment of STEM thesis defense horror stories of the format, "Famous professor X disproved this guy's thesis at his defense in thirty seconds". An example:
A case I know of first-hand: A doctoral student in engineering developed some powerful pattern matching theorems based on various transformations including one that was introduced in a major conference paper. At the student's defense one of the examiners pointed out an example showing that the transformation doesn't have one of the key claimed properties. The student sat silent for a minute and then simply said that his thesis is wrong. The examiners were shocked and assured him that the situation couldn't be that bad. It turned out that it was that bad and the student did not complete his doctorate. Fortunately his advisor helped him land a good job in which he has established a successful career.

Computational Complexity


The Difference Between Humantities and Mathematics

Consider the charts of Polish national high-school exit exams below.

From Imgur. Discussion on Reddit.


Matt Might on How to Get Tenure

Matt Might, faculty at U. Alabama (and several other institutions), writes on how to get tenure, or more generally, how to approach an academic career with a sense of inspiration. Key takeaway:
Life is too precious and too fleeting to waste my time on bullshit like tenure. I didn’t become a professor to get tenure. I became a professor to make the world better through science. From this day forward, I will spend my time on problems and solutions that will matter. I will make a difference.
Matt Might.


Katherine Johnson on Double-Checking

A very short clip of an interview with Katherine Johnson -- subject of the book and movie Hidden Figures -- who had her 99th birthday last week. Here she talks about John Glenn demanding, before his first spaceflight, that she double-check by hand the trajectory calculations from the digital computer in use at the time.

I feel like this might be an excellent starting point for a classroom discussion on, "Why would anyone ever want to double-check a computer calculation? How could it be wrong? Was John Glenn totally insane?" Granted that I become more and more convinced that the topic of sanity/double-checking may be the most fundamental sense-making theme that runs through all the classes we teach at a community college.


Thursday Thought: Beginnings

Schiller in a letter to Goethe, February 5th, 1796:
I always think it better, whenever possible, not to begin at the beginning, as it is always the most difficult part.
Found in the Introduction to Numbers by K. Lamotke (Springer-Verlag, 1991).


Grade Inflation and Corporatized College

Ed Burmila has an incisive piece on how "college run as a business" inevitably promotes grade inflation and the other ills we deal with daily:
When grades are inflated, everyone appears to win. Students are happy for obvious reasons. Administrators are happy that students are staying enrolled. Teachers are happy their already hefty workloads aren’t being increased further. Yet the collective outcome — contra econ 101 — is suboptimal.

As the number of bachelor’s degrees awarded rises, employers can demand additional credentials (which universities are only too happy to offer at eye-watering prices). The only ones who benefit are those who can afford to distinguish themselves from the pack of three-point-something GPAs by buying costly elite credentials.



California State Ends Remediation

At the start of the month, the chancellor of California State pushed the button on the "nuclear option", ending all requirements for remedial-level skills in math and English in their college system. Much like CUNY, they are promising a doubling in graduation rates (from 20% to 40%).
Cal State plans to drop placement exams in math and English as well as the noncredit remedial courses that more than 25,000 freshmen have been required to take each fall — a radical move away from the way public universities traditionally support students who come to college less prepared than their peers.

In an executive order issued late Wednesday, Chancellor Timothy P. White directed the nation’s largest public university system to revamp its approach to remedial education and assess new freshmen for college readiness and course placement by using high school grades, ACT and SAT scores, previous classroom performance and other measures that administrators say provide a more accurate and comprehensive understanding of students’ knowledge.

Cal State will no longer make those students who may need extra help take the standardized entry-level mathematics (ELM) exam and the English placement test (EPT).

LA Times.


NY State backs off from Common Core

NY State is backing away from implementing the Common Core Learning Standards. Of course, the direction they're going is to make standards easier. Keep in mind: The great majority of NY state high school graduates cannot pass 6th-grade reading or arithmetic tests; and within the last decade, NY had the 3rd-lowest GED math success rates, above only Mississippi and Washington, DC.

As reported by NYSUT on the newly-named "Next Generation Learning Standards":
Committees went standard-by-standard, grade-by-grade. Some learning standards were thrown out; others were merged; some standards were moved to different grade levels... an Early Learning Standards Task Force with 30 educators and parents issued a set of pre-K–2 recommendations, such as incorporating "play" as an instructional strategy... To provide more time for students to develop deep levels of understanding of math content, the review committees suggested major grade movements in statistics and probability at the middle level and in algebra at the high school level. Some changes are semantic. For example, the new standard calls for "exploring" a concept without the expectation of mastering the concept at that grade level.



The empty promise of school choice

Charter school outcomes are, on average, worse the public school outcomes. Some people may be surprised by this -- but likely only if they are in the flock of the American free-market ideology. Fredrik deBoer, who researches academic assessment at CUNY, writes on how little sense this makes on his blog The ANOVA:
There are no specific pedagogical mechanisms that are tied to the doctrine of choice. There’s no secret book titled “Actually Good Pedagogy” that only charter schools get to buy. There is no experimental approach to learning long division that is only employed in private schools. Your average child moving from one type of school to another might find that their social world has changed, that their commute is easier, or that they like the cafeteria food better. But there is no consistent differences in how charter school teachers teach, no fundamental difference in approach between public and private. When people ask me why the results from some atypical charter success story fail to scale up in places like Detroit or Chicago or Nashville, I often say “because ‘charter school’ is not a meaningful experimental condition.”... What the designation “charter school” shares broadly is only the ability to fire teachers with impunity, that’s all. And that could only possibly work to improve outcomes, again, in a world where teachers control student outcomes in a straightforward and uncomplicated way. They don’t.



Traditional practice best for struggling 1st-graders

The researchers, led by Paul L. Morgan at Pennsylvania State University, analyzed U.S. Department of Education data from about 14,000 students across the United States who entered kindergarten in 1998. They first looked at how the students performed on math tests in kindergarten. The data included teacher surveys, allowing the researchers to track the methods that the kids’ subsequent first-grade teachers said they used. And finally, they had the students’ first-grade math scores.

The researchers found that the higher the number of struggling students, who scored in the bottom 15 percent in kindergarten, in a first-grade teacher’s classroom, the more likely the teachers were to use manipulatives (hands-on materials), calculators, music and movement (See Table 3 on page 12 in the study). The fewer the struggling students, the more likely that teachers stuck with traditional methods, such as showing the whole class how to solve something one way from the chalkboard and then having students practice the method using worksheets.

Yet, at the end of first grade, the researchers found that struggling students who were given traditional instruction posted significantly higher math score gains than the struggling students who had been taught by the progressive methods. Gains are measured by how much students math scores rose between kindergarten and the end of first grade. (See Table 5 on page 15 in the study.)

“Routine practice is the strongest educational practice that teachers can use in their classroom to promote achievement gains,” Morgan said.

Education By The Numbers


Laptops in class considered harmful

Together, these led to a number of important insights into computer use in the classroom. First, participants spent almost 40 minutes out of every 100-minute class period using the internet for nonacademic purposes, including social media, checking email, shopping, reading the news, chatting, watching videos, and playing games. This nonacademic use was negatively associated with final exam scores, such that students with higher use tended to score lower on the exam. Social media sites were the most-frequently visited sites during class, and importantly these sites, along with online video sites, proved to be the most disruptive with respect to academic outcomes.

Scientific American


Scientific American: Math is now a key weapon against disease

Maths has become a vital weapon in the scientific armoury we have to tackle some of the most pressing questions in medical, biological and ecological science in the 21st century. By describing biological systems mathematically and then using the resulting models, we can gain insights that are impossible to access though experiments and verbal reasoning alone. Mathematical biology is incredibly important if we want to change biology from a descriptive into a predictive science – giving us power, for example, to avert pandemics or to alter the effects of debilitating diseases. 

Scientific American


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].

  • 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.


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. 


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.


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


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. 



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.


Francis Su: Math as Justice

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

To Live Your Best Life, Do Mathematics


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.


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


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?


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.