Monday, August 31, 2015

Two unsung heroes of higher-level thinking

Every year when teaching my "autism and reasoning" class, I have another chance to delve into cognitive science. And each time, I'm reminded of how much is involved even in apparently "meaningless" tasks like memorizing and reproducing complex shapes like (excuse the low fidelity) this one:


Unless you have a photographic memory, incorporating this figure--or at least as much as possible of it--into long term memory involves a rather high-level skill: coming up with some sort of organizational structure. Perhaps it's a house with a weather vane on top lying on its side with its base to the left and a dormer window on the top, marked with an X, an incomplete copy of itself on the left, and a button-like porthole to the right, and flanked by crosses along its straightest edges. It's much less fruitful to simply memorize it as a bunch of specific lines at specific angles.

This is similar to another skill often dismissed as meaningless: speed. As I noted earlier in connection with math tests:
many people assume that speed tests (especially multiple choice speed tests) measure only rote knowledge. But they’re also a great way to measure conceptual understanding. Performance speed reflects, not just rote recall, but also efficiency, and efficiency, in turn, is a function of reasoning, strategizing, and number sense.
When it comes to our computers, we place high value on speed and memory capacity; perhaps for the same reason, we increasingly dismiss these same things in humans. But they correlate, not just with those skills that are being supplanted by computers, but with higher level skills still matter for the foreseeable future.

Saturday, August 29, 2015

More revolutionary ideas for classroom change

One of the book reviews I most looked forward to reading in last week's New York Times book review was Lisa Miller's review of “Most Likely to Succeed: Preparing Our Kids for the Innovation Era,” by Harvard’s Innovation Lab's Expert-in-Residence Tony Wagner and venture capitalist Ted Dintersmith. In Miller's words, the book:

argues that the only way to ensure any kind of future security for our children is to totally upend the education system and rethink what school is for.
First come the shocking revelations about what K12 education is like in America:
Public education in America is based on antiquated late-19th-century priorities, on the need “to educate large numbers of immigrants and refugees from farms for basic citizenship and for jobs in a growing industrial economy.” Most of the stuff children are forced to know, and on which our culture’s sense of achievement is based, is unnecessary in the age of Google. But tests and test-makers still run the show, and kids are required to “jump through hoops” and drill and drill to assimilate reams of facts (“content”) instead of learning the skills that will keep them employed and employable for years to come…
Gosh, I can't imagine where I've ever heard that before. Thank goodness Wagner and Dintersmith are getting the word out.

Equally astounding are Wagner and Dintersmith's notions of the revolutionary ways that things might change:
After the revolution Wagner and Dintersmith imagine, college will no longer be a scandalously expensive universal requirement but an option for only the most academically minded. They propose an overhaul of the SAT scoring system in which adolescents would be sorted into categories of collegiate preparedness: “In Good Shape,” “It Won’t Be Easy” and “Think Different.” Those in the last two categories might be satisfied, and indeed better served, in free or low-cost apprenticeships or by taking vocational courses.
A time when college wasn't expensive; a time when only the most academically minded attended it; a three-level grading scale (good/fair/poor; A/B/C; "meets expectations"/"needs assistance"/"struggles")--when have we ever seen these things before?

The authors also suggest "an interdisciplinary approach; hands-on, project-based learning; student-directed curriculums." How disruptive! How revolutionary!

It's interesting that all these upending ideas have already been tried out--either in previous centuries, right now, or all along. The only thing that might possibly qualify as novel is Wagner and Dintersmith's idea that all students should be molded into entrepreneurs. But that just adds one more layer of implausibility and impracticality to an idea that's already long been popular: the idea that all students should be molded into leaders.

Far more revolutionary than all of this is a proposal by Alfred North Whitehead:
When you are criticizing the philosophy of an epoch, do not chiefly direct your attention to those intellectual positions which its exponents feel it necessary explicitly to defend. There will be some fundamental assumptions which adherents of all the variant systems within the epoch unconsciously presuppose. Such assumptions appear so obvious that people do not know what they are assuming because no other way of putting things has ever occurred to them. 
(Science and the Modern World, 1926).
For an introduction to this epoch's underlying assumptions, Wagner & Dintersmith is a great place to start.

Thursday, August 27, 2015

Math problems of the week: 5th grade Common Core-inspired test questions vs. Singapore Math

I. A sample end-of-5th grade test question from Pennsylvania's revised PSSA tests.

Note: The state doesn't publicize what percentage correct is necessary for a score of "proficient" or "advanced," but rumor has it that it's significantly below the 80% benchmark set by the Singapore Math placement tests.




II. From the Singapore Math 5A placement test, for which 80% correct says you're ready for the second half of the fifth grade curriculum:


III. Extra Credit

Compare the problems in terms of (1) how well defined the questions are and (2) how obvious the first steps are.

First hint: How well defined is "1/3 of a bowl of rice"?

Second hint: In which problem is the first arithmetic operation immediately obvious, and in which one do you need first to devise an overall strategy?

Tuesday, August 25, 2015

21st century humanities majors

In my post below, I discussed what I think are the most job-relevant communication and collaborative skills. Boiled down a bit differently, they are:

(1) attending to and understanding directions
(2) being competent enough to fulfill them
(3) getting your own points across clearly.

It is these skills--rather than the more social aspects of communication and working together--that are in increasingly short supply.

And all three of them relate, in part, to something over which there's been a lot of handwringing recently: the value of a liberal arts education. What with the decline in the numbers of humanities majors, and, in particular, of English and literature majors, along with all the forces out there "disrupting" traditional education, more and more people are wondering what, if anything is being lost.

Here's my answer. Fewer humanities majors, and fewer traditional (reading and analytical writing-based) humanities classes, means fewer students reading significant quantities of challenging prose, writing significant numbers of analytical essays, and getting significant amounts of feedback on their reading and writing skills. What's being lost, in other words, are

(1) careful reading skills, including the ability to sustain attention while reading
(2) writing skills, including the ability to make clear points and coherent arguments

There relate directly to the workplace skills deficiencies I note above.

In addition, as far as literature majors in particular are concerned, there is some reason to think that reading, discussing, and analyzing literature fine-tunes empathy and ethical reasoning. Reading--especially in nonfiction-intensive courses like history--also substantially enhances general knowledge. Empathy, ethical reasoning, and general knowledge, in turn, probably enhance performance in a whole variety of vocations, including 21st century jobs.

Ironically, some people think that college needs to be rethought in light of how much today's jobs are changing. We never know which specific skills are going to be important in the future, so we should focus on more general ones like flexibility, creativity, grit, and team work. But I'm guessing that most jobs still require the ability to follow complex directions and get your points across clearly: indeed, these are some of the most general, generalizable skills there are.

If these skills are so important, why are so many students defecting from the departments that, traditionally, have fostered them the most? Perhaps students are ill-informed about what the humanities can offer them, assuming that the more "pre-professional" majors--business, communications, interactive media?--provide more relevant vocational training. Or perhaps the humanities departments themselves are at fault: perhaps, as I've suggested earlier, they are no longer focused on informational content, complex characters, ethical subtleties, or, most importantly, on developing students' reading, writing, and analytical skills.

Sunday, August 23, 2015

The importance of communicating well and working well with others

In an earlier post on HR departments and hiring, I considered whether it's ever appropriate to discriminate against people on the basis of personality:

Should corporations never discriminate against people on the basis of personality? Ironically, the personality type that thrives best in interviews--particularly the informal ones that are so popular today--is the type that is potentially the most toxic of all: the narcissistic, manipulating backstabber who charms his superiors, undermines his equals, and takes credit for the work of his underlings, advancing through the corporation and undermining morale and productivity. 
Given just how toxic the psychopathic employee can be for the workplace, one can appreciate how important it is to consider personality and social behavior. The problem, though, is that people often conflate two different types of socially problematic personalities:

(1) jerks
(2) people who mean well--or at least don't mean ill--but are socially awkward.

Personality does matter, but the aspect that matters most is decency, not charm.

In reading the comments on my earlier post, I've realized that two other general factors that seem important in making hiring decisions can be bifurcated in similar ways.

Anonymous/bj writes:
the ability to communicate and work with others is being shown to be an important part of successful performance of many jobs, including the ones that include significant technical skills.
As with personality, so, too, with the ability to communicate and the ability to work with others. Each has a more social aspect, and a more job-relevant aspect. For the former:

(1) conversational skills; being fun to talk to
(2) content-based communication skills: understanding directions, getting your points across clearly.

For the latter:

(1) getting along with people socially; behaving such that they enjoy your company
(2) understanding what your role is in a collaboration and being competent and conscientious enough to fulfill it.

The more job-relevant aspects of communication and working with others aren't trivial. A programmer on a software development team I was part of--someone who appeared to have gotten hired in part because he was a drinking buddy of one of the team leaders--set us back about a year (and many paychecks worth of funds) because it turned out he wasn't able to follow our directions or communicate what was confusing to him. I'm sure he was fun to hang out with after hours at bars. A lawyer friend of my regularly laments how her firm hires "team players" who don't pull their weight because, for all their Ivy League training, they're lacking in reading and writing skills.

Maybe I'm insufficiently "21st Century" in my thinking, but I'd take a decent, competent coworker who understands directions and can get his or her point across clearly, however socially awkwardly, over someone who's engaging to talk to and fun to be around but, like so many people these days, is a sloppy, inaccurate reader; an inarticulate writer; an inattentive listener; a poor follower of complex directions; and a lazy, responsibility-deflecting coworker--even if s/he isn't also a manipulative, backstabbing psychopath.

Friday, August 21, 2015

Math problem of the week: a 5th grade Common Core-inspired test question

A sample 5th grade test question from Pennsylvania's revised PSSA tests, which resulted in a huge drop in test scores test-wide.




[The correct answer is starred, and the incorrect answers have italicized annotations]

Extra Credit:

Compare this question's mathematical demands with its linguistic demands. Identity the dangling modifier, the ambiguity of "at the end," and the ambiguity of what "from the end of" and "combined" apply to.

Monday, August 17, 2015

Feedback loops: it's not just students who need them

Many jobs have built-in feed-back loops. You produce lousy work, and you get negative critiques and poor sales. You interact poorly with customers, and you get indignant reactions and diminishing patronage. Online reviews of all sorts of professionals are proliferating around the internet--from doctors to realtors to carpet cleaners to college professors.

So jobs that lack feed-back loops really stand out now. One of them, I recently realized, is psychiatric evaluation. Conducting and writing up evaluations, for some psychotherapists, is their main occupation. In the absence of follow-up treatments or longitudinal tracking, these people receive no information about the accuracy of their prognostications. Most insulated from feedback are those evaluators who contract primarily, not with private customers (who might vote with their feet), but with public agencies (which tend toward inertia and cronyism). Having this realization helped take the sting out of J's most recent psych eval. But more on that in my next post.

Certain key pockets of the education sector also lack feedback loops. Take admissions departments. However carefully and comprehensively they review applicants, how often do they receive feedback, years later, about how particular acceptances or rejections panned out? How often do they hear about accepted applicants who failed out, were kicked out, or who tormented their classmates and professors? Whenever a problematic student comes along who lacks basic reading skills, basic writing skills and or a basic work ethic, or makes extravagant excuses, unreasonable demands, and nasty accusations (the kinds of things that often accompany being severely lazy or under-skilled), or who cheats their way through assignments (ditto), I wish there were official channels for reporting back to the admissions committee. In the worst cases, when I find myself and others spending dozens of hours on a single problematic student, I wish the admissions department were required to dip into its budget and reimburse us for at least some of our extra time.

On the flip side, how often do admissions departments hear about rejected applicants who thrived elsewhere and made great contributions later on in life?

Then there are K12 classroom teachers. While the occasional K12 teacher gets reviewed on sites like ratemyteachers.com, most aren't, and even if they were, there's little opportunity to for students/parents to vote with their feet. And while testing and other assessments take up huge chunks of class time, how often do they function as meaningful, teacher-directed feedback? Standardized tests scores provide only a couple of quantitative data points (e.g., general scores for reading, writing, and math), even these measurements are only as reliable as the (often problematic) tests themselves are. As for the potentially more granular and meaningful class-specific assessments, many schools and teachers seem to treat these exclusively as feedback for students and parents, and not as feedback that might motivate adjustments to teaching practices. On one occasion at our local school, for example, a near 90% failure rate on an in-class science test resulted, not in the science teacher considering that maybe she should reteach the lesson in a different way, but in her sending out an blast email to parents complaining about how badly their children did and how they clearly hadn't studied hard enough.

This last example is giving me some feedback: I'm realizing now that I need to reformulate what I'm suggesting here. It's not enough to have a feedback loop; there also needs to be some built-in motivation to actually pay attention to that feedback and make appropriate adjustments.

Saturday, August 15, 2015

Conceptual structures vs. learning conceptual structures = not homomorphic


It's become an obsession. Everywhere I look now, I see cases where people dwell on basic concepts that are relatively easy to understand in isolation and challenging only within complex situations.  Time and again I wish that instructors would move on more quickly and save time for the more complicated applications that lie ahead.

Typically, this concept-dwelling happens at the beginning of a course or unit and takes up the first couple of classes. For example, in an MIT OpenCourseware course on basic chemistry that I'm having J sit through in preparation for Chem 101 in college, the instructor begins with an overly long discussion of what's known about the basic structure of atoms. While she asks the students whether they've ever had plum pudding, I think about the challenging problems that lie ahead and the time that could have been reserved for working through some of these in class.

The problem, I think, is a combination of the difficulty that many instructors have remembering what it's like to be the student, and the default assumption that the conceptual structure of a given subject maps neatly onto to the process of learning that subject. What's foundational to a subject, in other words, may not be the same as what's foundational to learning that subject. Now I don't remember much at all about chemistry. But I'd venture to guess that, while learning the structure of atoms is part of what's foundational to learning chemistry, but there are a number of other key concepts that are also foundational to learning chemistry, and more time-consuming to master (including, for example, some basic algebra).

Another fallacy that afflicts introductory classes is a sort of ontogeny recapitulates phylogeny assumption. This is the idea that the history of the field's development over time maps onto the student's personal trajectory of understanding it. But while the history of the various notions of elementary particles is interesting in and of itself (a great topic for a history of science class), it probably doesn't replicate the student's own developing understanding of elementary particles, and going over that history may or may not actually facilitate that understanding.

Of course, all of these are empirical questions that could be addressed by applying the foundational concepts of yet another field--in new and complex ways.

Thursday, August 13, 2015

Math problems of the week: 4th grade Common Core-inspired test questions




Extra Credit: Discuss the ratio of concepts to procedures.

Tuesday, August 11, 2015

Personnel departments and personality discrimination

As questions about J's future employability nag with growing urgency, I've been chatting with various professionals who specialize in the employability of individuals on the autism spectrum. One person I talked to stressed one of Temple Grandin's longstanding themes: if you're applying to a job at a large corporation, it's absolutely essential to bypass the HR department. Ironically, the very entity whose raison d'être is, in part, to ensure compliance with laws against job discrimination tends itself to discriminate against certain classes of individuals--namely, those who don't interview well. The poorest of these performers, naturally, are smack on the autism spectrum.

In corporate settings (as opposed to academia), HR serves, for some reason, as the initial screener rather than the final arbiter. This means that an autistic individual who, say, is a highly qualified software developer may never have a chance to be considered by the department for which he would actually be working--and which might actually want to hire him.

With more and more firms using informal interviews that seek "cultural fit," things are harder than ever--even for those who are only marginally socially awkward. While this may make for a more "collegial" workplace with lots of camaraderie and after-hours socializing, it's ultimately bad news, not just for neurodiversity, but for workplace productivity and creativity. Just like K12 chools, HR departments are systematically bypassing real talent; over-emphasizing non-cognitive "21st century" skills at the expense of timeless skills like reading, writing, and quantitative reasoning; and confusing social savvy with the ability to collaborate professionally.

Should corporations never discriminate against people on the basis of personality? Ironically, the personality type that thrives best in interviews--particularly the informal ones that are so popular today--is the type that is potentially the most toxic of all: the narcissistic, manipulating backstabber who charms his superiors, undermines his equals, and takes credit for the work of his underlings, advancing through the corporation and undermining morale and productivity.

So, actually, social skills do matter very much in this century (and a few others). But not in the ways that most HR departments think they do.

Sunday, August 9, 2015

Plus ça change, plus c'est la même choCCSS

In a recent Op-Ed in the New York Times, mathematician Jordan Ellenberg argues that the Common Core Standards change little: they're just the latest instantiation of standardized school-accountability testing; that even states that have officially abandoned the Common Core still use tests that, de facto, are Common Core-aligned; and that the supposed shifts in math priorities wrought by the Common Core reflect trends that have long been in place--e.g., such concepts as "number sentences" and "making tens."

But these eminently reasonable observations don't justify Ellenberg's reassuring tone. The main effect of the Common Core, I've argued, is to further entrench dominant trends--most of which happen to be high problematic. While supporters of these trends are forever citing the Common Core State Standards as justification, they would endure even if the CCSS were to suddenly vanish. Constructionism would continue to extend its tentacles into more and more public, private, and parochial schools; math would continue to water itself down; group activities would continue to grow; and grit and other non-cognitive, so-called "21st century skills" would continue to dominate K12 learning objectives--well into the 21st century.

The latest reports I've heard from the trenches--from the latest batch of classroom teachers I've had as students--have confirmed this. Many of them submit lesson plans as their final projects, and these lesson plans invariably list several CCSS goals at the top--right above the "Essential Question" and the "Learning Objectives" sections. Sometimes it's not entirely clear to me why a particular lesson maps, say, to CCSS.ELA-LITERACY.W.5.2.C ("Link ideas within and across categories of information using words, phrases, and clauses") rather than CCSS.ELA-LITERACY.W.5.2.D ("Use precise language and domain-specific vocabulary to inform about or explain the topic"). This year I probed a bit, asking my students about the challenge of mapping lessons to goals and ensuring that each of the many goals for whatever grade level they're teaching is reflected in at least one of their lessons.

"Usually we create the lesson first, and then look up the goals and pick the most relevant one," one student explained.

"How do you make sure you eventually cover all of them?"

"No one really checks, but usually we put in several standards per lesson, so eventually we probably cover them."

Retrofitting lessons to standards might seem odd--until you realize what's actually retrofitting what. And see the standards for what they are in practice: one gigantic retrofit to what's already, ever more firmly, in place.

Friday, August 7, 2015

Math problems of the week: Common Core-inspired 4th grade test questions

More problems from the Smarter Balanced Assessments, a Common Core-inspired, standardized test consortium now consisting of about 12 states: the next three problems on the sample 4th grade practice test.




Wednesday, August 5, 2015

Stop belaboring the concepts: the limits of "conceptual understanding"

In an earlier post, I discussed two mathematical concepts that are easy to grasp in isolation and that therefore shouldn't be belabored ad nauseam: the different multiplicative groupings that produce highly divisible numbers like 24; and the fact that subtraction represents not just removal but measurement differences. The more I think about this, the more similar mathematical concepts I come up with: concepts that are easy to grasp via concrete examples, but often excessively belabored by teachers, delaying the more challenging abstractions and applications of these concepts to more mathematically complex situations.

Such concepts include:

1. The number line
2. fractions
3. negative numbers
4. place value
5. the axioms of arithmetic
6. sets and subsets
7. functions; domain and range
8. slope

What's challenging and interesting about negative numbers, for example, isn't that they represent numbers less than 0 or numbers on a particular side of 0 on the number line, or that they have such concrete instantiations as distances below sea level or temperatures below freezing. What's challenging about negative numbers is grasping that a negative times a negative is a positive and correctly distributing and multiplying out the negative numbers in a complex expression. A class that spends two weeks on ways on which negative numbers correspond to distances relative to sea level is wasting precious time and making students think that negative numbers are boring.

What's challenging and interesting about place value isn't the concept of groups of 10, 100, 1000, etc., or how 123 is 1 hundred plus 2 tens plus 3 ones, but the use of place value by the standard algorithms.

And what's challenging and interesting about sets and functions and slope aren't the concrete examples that teachers, rightly, use to introduce them, but their more mathematically abstract instantiations: for example, the connection between if A then B and A ⊆ B, or the slope of a slope in a non-linear function.

In general, when students struggle to do problems involving these various concepts, the answer is to spend more time, not on the concepts themselves, but on worked examples and practice problems. The best way to get better at math problems, in other words, isn't to spend hours depicting and discussing what a fraction is, what a function is, or multiple ways to multiply numbers, but to do lots of math problems that involve these concepts in mathematically challenging and interesting ways.

Monday, August 3, 2015

Leadership vs. Advanced Placement

How nice to see the Philadelphia Inquirer finally running an article on the best science and STEM school in city, as opposed to the science leadership school that gets all the local buzz and national attention. And how nice to see this school (yes, J's former high school) finally get some monetary recognition from the Philadelphia School Partnership:
Carver High School for Engineering and Science, which is expanding to serve 120 seventh and eighth graders in September, has been awarded $200,000 from the Philadelphia School Partnership, officials announced Thursday.

That's on top of a $147,000 grant that PSP, a deep-pocketed nonprofit, already awarded to Carver to fund planning for its middle school.

The newest award will support more planning as the school develops at 16th and West Norris Streets, principal Ted Domers said.

"There's a void of meaningful STEM [science, technology, engineering, and math] education in the city and across the country," Domers said. "We think this is an opportunity to be doing something that no one else is doing."
Carver's middle-school students will take engineering and computer science classes from the moment they walk in the door. They'll study algebra as eighth graders. Eventually, that will mean more advanced classes for them as high schoolers.

"The only reason our kids can't accelerate quicker is because we can't expose them quickly enough," Domers said.

Going forward, there's no reason a sophomore Carver student won't be able to take a class like Advanced Placement computer science, Domers said.
Mr. Domers, who, I'm pretty sure, is the best principal in the entire Philadelphia school district, is absolutely right that no other Philadelphia public school is doing this. With a largely low-SES population, with much less selective admissions than its leadership school counterpart, Carver High School for Engineering and Science has been achieving much higher scores on math, science, and computer science AP tests.

Perhaps next year Arne Duncan will invite Mr. Domers, rather than the science leadership leaders, to the "Principals at ED" program that, in the words of the earlier Inquirer article, "brings groups of highly innovative and successful principals from across the country to the Education Department to learn more about federal programs and to share experiences from their jobs as school leaders."

Saturday, August 1, 2015

Don't dwell on the concepts; use them to build things

About a year ago, Elizabeth Green published a piece in the New York Times magazine entitled Why Do Americans Stink at Math? Just a couple of months ago, the Notices of the American Math Society published some reactions to her piece. And I, in turn, have some reactions to what's in some of these reactions--in particle to Bill Jacob's discussion of how teachers should promote conceptual understanding:

Imagine a third-grade class being asked, "How many legs on there on three spiders?" Children who draw three spiders may first count the legs, but the context can elicit many strategies. Three groups of eight legs can e viewed as six groups of four when four legs are drawn on each side of a spider and viewed as a unit. A row of three spiders could be viewed as having two rows of twelve legs (top and bottom), or the legs could be counted as twelve pairs. A skilled teacher can pull from various groupings of the legs a spatial understanding of why the equivalence of 8 × 3 = 4 × 6 = 2 × 12 = 12 × 2 arises, beyond merely having the same value.
Jacobs also writes about the virtues of emphasizing how subtraction can be thought of as "difference on a number line" so that later students will understand why slope can be expressed as (y1 - y0) / (x1 - x0):
Learners who only understand subtraction as removal and not as difference in a measurement context will miss the meaning of (y1 - y0) and (x1 - x0) in the slope expression.
Both of these concepts--the different multiplicative groupings that produce highly divisible numbers like 24; and the fact that subtraction represents not just removal but measurement differences--strike me as relatively easy to grasp in isolation, even for third graders. In other words, they seem like the kinds of concepts that, like function and slope, are relatively easy to understand in and of themselves, especially when presented concretely, as Jacobs is advocating.

What's challenging about representing slope as (y1 - y0) / (x1 - x0), I'm pretty sure, isn't the concept of subtraction as measurement difference, but the symbolic abstraction involved in the algebraic expression.

And what's challenging about different multiplicative equivalences like 8 × 3 = 4 × 6 = 2 × 12 = 12 × 2 is remembering that you can use them to as tools to quickly simplify complex expressions.

These and other concepts are like some of the simple building blocks of computer programming--like div and mod and lists and arrays--or, for that matter, the simple building blocks of engineering--like levers and pulleys and valves and gears. All these are simple concepts in and of themselves, but very powerful tools for solving complex problems--especially when understood abstractly. The challenge is to figure out when and how to use them.