Tuesday, April 30, 2013

The best creativity requires the "left brain," too

In an article in a recent Wall Street Journal article entitled For Innovation, Dodge the Prefrontal Police, Alison Gopnik discusses what it takes to have creative ideas--for example, about how to use a Kleenex tissue.

A recent experiment shows that subjects were able to think of more ideas when their prefrontal cortex, specifically their left prefrontal cortex, was disrupted:

The researchers got volunteers to think up either ordinary or unusual uses for everyday objects like Kleenex. While the participants were doing this task, the scientists either disrupted their left prefrontal cortex with tDCS or used a sham control procedure. In the control, the researchers placed the electrodes in just the same way but surreptitiously turned off the juice before the task started.

Both groups were equally good at thinking up ordinary uses for the objects. But the volunteers who got zapped generated significantly more unusual uses than the unzapped control-group thinkers, and they produced those unusual uses much faster.
This sounds like yet another argument for how the left brain is an obstacle to creativity--and yet another argument for the importance of playing around and doing open-ended activities rather than analyzing things:
Portable frontal lobe zappers are still (thankfully) infeasible. But we can modify our own brain functions by thinking differently—improvising, freestyling, daydreaming or some types of meditation. I like hanging out with 3-year-olds. Preschool brains haven't yet fully developed the prefrontal system, and young kids' free-spirited thinking can be contagious.
But, as Gopnik points out:
There's a catch, though. It isn't quite right to say that losing control makes you more creative. Centuries before neuroscience, the philosopher John Locke distinguished two human faculties, wit and judgment. Wit allows you to think up wild new ideas, but judgment tells you which ideas are actually worth keeping. Other neuroscience studies have found that the prefrontal system re-engages when you have to decide whether an unlikely answer is actually the right one.
I first noticed this division of labor between inspiration and analysis while, of all things, working out problems for a highly abstract logic course. I'd spend hours struggling to come up with some sort of proof. Eventually, I'd take a break. And it was typically at some point during this break that an inspiration would come. I'd return to the problem with a new awareness of how to proceed. But as soon as I put pencil to paper, I'd see that, while the inspiration had pointed me in the right direction, I still needed to proceed step by logical step. Left-brain redux.

My experience with pre-analytical inspiration in this particular course suggests an addendum to this blog post: to wit, it's not just that creativity requires the "left brain", too, but also that even "left brain" fields like logic, for all the initial analytical legwork they require, also involve some kind of pre-analytical creativity.

Sunday, April 28, 2013

Letter from Huck: I Sneak in the Back Window and Teach How to Attend to Precision

Out in Left Field proudly presents the twelth in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names have been changed to protect privacy.  


Dedicated readers of my letters may recall my reaction to the Common Core’s “Standards for Mathematical Practice” (SMPs). I continue to see these SMPs posted in various math classrooms where I sub, though I try to ignore them as much as I can as I continue to drift down the ideological divide, otherwise known as math teaching.

The SMPs came to my attention again recently while subbing at a middle school. For me the day started as sub assignments usually do: by reading the teacher’s directions to the sub. That day, the teacher wanted me to administer a “district assessment” for each of her classes and then have them start on their homework.

I glanced at the assessments and saw some mention of “Common Core” on the front of the answer sheet. Neither the answer sheets nor the instructions that I had to read aloud gave the reason for these tests, other than that the students would be evaluated as to how they solved and analyzed problems. I expected that students would ask me if their tests figured into their grades, so I called the front office before first period class and asked.

“I’m not sure,” said the person in the front office. Pause. “Well, let me think.” Pause. “No. I’m pretty sure they don’t figure into their grade.”

“Thanks,” I said. “I will tell the class what you told me.” Accountability did not seem to faze her and she said “OK.” (I since found out that, yes, it does affect their grades in class, and their placements next year, and is a performance task like what they will see with the Common Core when it goes into effect--next year.)

The test had four problems which students had to answer by showing their work directly on the exam and “explaining their answers.” Before my first period class started the test, one boy raised his hand. “Our teacher always tells us to ‘Attend to precision,’” he said, pointing to that particular SMP which was posted on the wall behind me. “Could you say it please?” The class looked at me expectantly.

"Attend to precision.”

"Thank you," he said.

“You know, I don’t even know what that means.”

“It’s one of the things she has on the wall,” the student said.

“Yes, I know. But it’s just so vague. Here’s what it means to me. It means use the right math vocabulary, and show your work. So for this test, if you just write a number down for your answer, it won’t be enough. You have to write what you did so someone else can follow it.” This seemed to satisfy the class, which is to say that probably not one person focused on what I said.

The questions had to do with discounts and percentages, which they seemed comfortable with. One question asked them to say whether an item was marked down 25% four times in a row, explain whether or not the final price after the four discounts would be $0, and provide the reason for their answer. From what I could see when I glanced at the tests while collecting them, most students didn’t get the answer right, but they had no problem putting down an explanation.

The situation was quite different with the eighth grade algebra classes, however. Their test only had two questions. One involved an L shaped figure with dimensions like x+15 on one side; i.e., no simple numbers were used. The question asked for the students to write an expression for the area of the figure. This involved splitting the figure into two rectangles, figuring out what various missing dimensions were, and then writing the area as the sum of the area of two rectangles—something that would amount to an algebraic expression.

After a few minutes of quiet, there was suddenly a flurry of indignant questions: "Do they want us to calculate the actual area? Like a number?” “What does 'expression' mean? Is that like an equation?" I was struck by the difference between the seventh graders who simply wrote down their process, and the eighth graders who were confused by what was required of them. One student put it into perspective for me. "What do they mean 'explain your reasoning'? I just do it." I took this to mean that all year they’ve learned how to express ideas algebraically, with showing their work being sufficient explanation. Given that that's what they thought they were doing to begin with, requiring them to “explain their reasoning” made no sense.

Of course, the students would not know that there are people who view those who can't "explain their reasoning" (however correctly they solve a complex problem) to be doing "rote work" and lacking "understanding." But it seems to me that if we really want students to do such explaining, then we should tell them how. Simply telling students to "explain your reasoning and attend to precision" is not likely to accomplish much. Knowing how to explain something precisely doesn't come automatically with understanding. And students are not likely to pick it up by themselves working in groups and the like.

As it was, I had a class full of eighth graders at a near-riot level. I stood in front of the class and said, "OK, I'm going to show you something." They were still talking. "Please listen," I said and then added "Actually, please don't listen because I'm probably going to get fired and/or shot for doing this." The class immediately got quiet.

I drew a square on the board and labeled two of the sides with an "x". I said "This is a square with sides of x units. How do we express the area?" The class said "X squared".

“Can you explain why?”

Someone said “Area is length times width.”

"That's all I'm going to tell you,” I said and erased the figure.

I'm still waiting for someone to knock on my front door and put the handcuffs on me.

Friday, April 26, 2013

Math problems of the week: 6th grade Connected Math vs. Singapore Math

A continuation of last week's problems-- the next pages in the chapters on circles sections of 6th grade Connected Math and Singapore Math:

I. From Connected Math "Going Around in Circles" section [click to enlarge]:



II. From Singapore Math Primary Mathematics 6 Workbook "Circles" chapter [click to enlarge]:



III. Extra Credit

Discuss 21st Century Geometry as an empirical science, complete with hypotheses about, and tests conducted upon, circular objects.

Wednesday, April 24, 2013

How far does a Constructivist education get you? Getting by via google, cut & paste, guess & check, and b.s.-ing the rest

Catherine Johnson recently put up on kitchentablemath one of the most concerning blog posts I've seen in a while. It's mostly a testimonial from palisadesk, who has spent some time field-testing recent K12 assessments, and who has noted that many of them are contrived to favor students who've undergone Constructivist curricula like Reform Math, balanced literacy, and Lucy Calkins' style writer's workshop (complete with journaling, personal reflections and peer editing), as opposed to students who've had traditional math, phonics, and grammar-focused, teacher-directed writing instruction.

As palisadesk reports:

Unfortunately, it is true that using Lucy Calkins' methods can raise test scores, due to the design of the current generation of "authentic assessments" (aka holistic assessment, standards-based assessment, performance assessment). I know several schools (including my own) where test scores rose substantially when they STOPPED doing systematic synthetic phonics and moved to a workshop model instead.
How can this be? As Catherine Johnson notes, "somehow I had assumed that, basics being basic, absence of basics would make any test hard to pass." But palisadesk observes that:
It's really not all that unbelievable, if you consider how the testing has changed. Schools used to use norm-referenced measures (like the IOWA, the
CTBS, Metropolitan Achievement Test, etc.) which also have definite limitations, but different ones.

Once they replaced those (as many states have done) with "constructed-response" item tests, variously known as 
performance assessments, holistic assessments, standards-based assessments 
and so on, a more fuzzy teaching approach also yielded benefits. These open-response items are usually scored on a rubric basis, based on anchor 
papers or exemplars, according to certain criteria for reasoning, conventions of print, organization, and so forth. These are variously weighted, but specifics like sentence structure, spelling, grammar,
 paragraph structure etc. generally carry less weight than such things as
"providing two details from the selection to support your argument."

The open responses often mimic journal writing -- it is personal in tone, calls for the student to express an opinion, and many elements of what we would call good writing (or correct reading) count for little or even nothing.

Some of this may have to do with how instructional time is allocated:
[A] school where I worked implemented good early reading instruction with a strong decoding base (and not minimizing good literature, either),
but saw its scores on the tests go down almost 25%. I think the reason
for that is that teaching children to write all this rubbish for the "holistic assessments" is very time consuming, and if you spent your instructional time teaching the basic skills -- which aren't of much value on these tests -- your kids will do poorly.

So much for language arts. And math?
The same is true in math. A local very exclusive private school which is famous for its high academic achievement recently switched from traditional math to Everyday Math and saw its test scores soar on these assessments (probably not on norm-referenced measures, but they aren't saying).
Several commenters provide additional insights. Glen writes:
I've seen this issue in multiple domains, where people seemed to do just fine without much foundation underneath their skills. I'm still often surprised by how many professional programmers are former English majors who can't divide 1/4 by 2/3 to save their lives but seem to do fine at their programming jobs. They learn how to use the systems they work with and do similar tasks over and over, occasionally googling for some online code samples, which they modify for their purposes. It mostly works, and I find myself having to stretch uncomfortably, and often unsuccessfully, to come up with examples of why their lack of what I consider foundational skills matters in the "real world."
Citing foreign language, Glen says:
I've had several interesting experiences with people who had just learned by communicating ("the modern, natural way") and seemed like near natives in proficiency but whose apparent nativeness collapsed surprisingly when something forced them out of their well-practiced comfort zones. But they really did quite well in daily life.
This reminds me of what David Sedaris has written about learning Japanese through Pimsleur. :
Instead of being provided with building-blocks which would allow you to construct a sentence of your own, you're left using the hundreds or thousands of sentences you have memorized. That means waiting for a particular situation to arise in order to comment on it; either that or becoming one of those weird non-sequitur people, the kind who, when asked a question about paint color, answer, "There is a bank in front of the train station,"or, "Mrs. Yamada Ito has been playing tennis for fifteen years."
Or, in the case of Sedaris and his cab driver:
I tell him that I have three children, a big boy and two little girls. If Pimsleur included “I am a middle-aged homosexual and thus make do with a niece I never see and a very small godson,” I’d say that. In the meantime, I work with what I have.
Back to Glen:
I wish there were an easy and persuasive demonstration of the value of building up from foundational knowledge and skills, but people who go right to the task at hand rather than building up to it can be surprisingly successful. Tests that evaluate their "real world" abilities can find them quite competent, and tests that challenge their foundational skills can easily be derided as "artificial."
To this, Magister Green adds:
Unfortunately what this approach has to lead to, ultimately, is a complete ossification of society. Lacking a command of the basics must lead to a lack of ability to deal with novel situations, situations in which one's pre-fabricated responses are inadequate. As such I fail to see how innovation can continue, beyond those ever-so-few geniuses who do understand the basics and thus can see beyond the immediate into the potential.

Teaching the basics is hard, and testing for them is even harder since you often can't test for the basic concept itself but rather are testing for evidence that understanding of the basic concept exists.

So long as nothing in our world ever changes or otherwise requires anyone to work beyond their pre-determined zone of competence, testing and teaching of this sort will do just fine. The depressing thing is that your average medieval peasant had a more thorough and basic education than our kids will receive under this new system.
SteveH and Allison address math scores in particular. SteveH notes:
If you give enough partial credit in math, you can get an 'A' without ever getting one problem correct.
And Allison writes:
Our state (MN) tests have constructed responses too, and again, Everyday Math did better on some scores of some of those tests because now students are given credit for "explaining their thinking". in words. The ELL kids got hammered.

but you can only fake reality for so long. Sooner or later, someone wants you to actually know something, not solve-by-google.
SATVerbalTutor brings up the standardized college admissions tests:
Even on the SAT, it's perfectly possible to score very well on the essay without having anywhere near mastered the basic conventions of correct writing. (If you're curious, go on College Confidential and read some of the "12" essays that people have posted -- they're utterly horrifying).

...
Even the French AP has been revamped to focus more on holistic "real world" skills; the grammar-based format of the old test was apparently just too hard. The result is that kids can bullshit their way through, and even ones who have no real understanding of the language can pass.
However, as SATVerbalTutor notes, this strategy doesn't seem to work on the Critical Reading section of the SAT:
You see a lot of kids who can score super-high in Math and Writing and then just fall down on CR. They can memorize formulas and apply them to a certain point, but when so many elements are in play at once, they're in way over their heads. Very often these kids will complain that they get down to two answers "but always guess the wrong one" when in fact they have no real understanding of what the passage was actually saying (as evidenced by the fact that one of the answers is exactly the *opposite* of the point of the passage).
Which is perhaps why, when the SAT was recentered in 1995, it was especially the Verbal Reasoning is (now the "Critical Reading") test whose scores were boosted, for most scores by around 60-70 points out of a possible 800. Perhaps another recentering is close at hand.

Even with the recentering, it's crucial to have SATs and the ACTs around as normed tests that nearly all college-bound students end up taking. And, given that there must be a strong temptation for each testing company to change with the Constructivist, "constructive response" times, it's also crucial for the two companies to be competing with one another. Unless both go Constructivist simultaneously, the brightest students can flock to whichever one holds out.

Returning to math, we now have a world-famous scientist arguing in the Wall Street Journal that math skills aren't that important for science (except, he concedes, for "most of physics and chemistry, as well as a few specialties in molecular biology"):
When something new is encountered [in science], the follow-up steps usually require mathematical and statistical methods to move the analysis forward. If that step proves too technically difficult for the person who made the discovery, a mathematician or statistician can be added as a collaborator.
..
If your level of mathematical competence is low, plan to raise it, but meanwhile, know that you can do outstanding scientific work with what you have.
...
For every scientist, there exists a discipline for which his or her level of mathematical competence is enough to achieve excellence.
(A critical response to Wilson's article in Slate notes that some of Wilson's peers have found a decisive lack of excellence in his own recent research on group selection, in particular a possible error in the underlying math.)

To some extent, you can get by in this 21st century world with a Constructivist education. Solve by google, read via key-word searches, answer questions via cut and paste (a high-tech version of the sort of regurgitation that Constructivists claim to abhor), "program" software via cut and paste, do math by guess & check and plug & chug, and, when you need to produce decent writing or do real math, hire a ghost writer or a mathematician to do it for you.

But what if you need to critically read? or critically think? Or understand the complex legalese of the tax code, the fine print, and life and death jury instructions?  Or come up with a new, complex thought all by yourself? And what if your society as a whole needs more than a few elite scribes and problem solvers and idea generators to maintain its place in the modern age?

Monday, April 22, 2013

Simplifying history

Today's high school texts consist mostly of short, simple sentences. Here, for example, are a few from McDougal Littell's World History:

For hundreds of years, peasants had depended on oxen to pull their plows. Oxen lived in the poorest straw and stubble, so they were easy to keep. Horses needed better food, but a team of horses could plow three times as much land in a day as a team of oxen. (p. 387)
A single sentence would get this point across in significantly fewer words:
For hundreds of years, peasants had depended on oxen, which could survive on mere straw and stubble, rather than on horses, which, though they plow three times as much land in a day, required better food.
Though shorter, however, the revision requires more sustained attention. If you get distracted in the middle of the sentence, your comprehension suffers more than if you get distracted in the middle of the choppy original.

Writers of earlier textbooks assumed readers with longer attention spans. Consider, for example, the opening paragraph in Outines of European History (James Harvey Ronbinson, 1907):
If a peasant who had lived on a manor in the time of the Crusades had been permitted to return to earth and travel about Europe at the opening of the eighteenth century, he would have found much to remind him of the conditions under which, seven centuries earlier, he had extracted a scanty living from the soil.
It's not just long, but also complex. The sentence starts with an if-clause that contains two embedded clauses: a lengthy relative clause ("who had lived on a manor in the time of the Crusades") and an even longer infinitival clause ("to return to earth and travel about Europe at the opening of the eighteenth century"). All this is followed by a main clause in the conditional mode of the present perfect tense, which contains a lengthy infinitival clause ("to remind him of...") which, in turn, contains a relative clause ("under which...") which, in turn, is interrupted by an appositive ("seven centuries earlier").

Modern history texts do occasionally contain somewhat long sentences, but they aren't as long, or, more importantly, as complex, as the kind of sentence one finds regularly in Outlines of History. Here, from the same section as the first excerpt from McDougal Littell's World History, is one of the longest sentences to be found there:
As traders moved from fair to fair, they needed large amounts of cash or credit and ways to exchange many types of currencies. (p. 389)
Here there are no multiple embeddings to keep track of and no need to hold interrupted phrases in working memory. Most of the complexity comes from simple co-ordination--the "or" and "and" conjunctions that conjoin the what follows the verb "need". This kind of structure doesn't involve the kind of embedding that burdens working memory.

Many of the complex sentences of older textbooks can be broken up into smaller sentences. For example, a modern textbook editor might chop up the Outlines of History sentence as follows:
Imagine a peasant who had lived on a manor in the time of the Crusades. Suppose he been permitted to return to earth seven centuries later, at the opening of the 18th century, and to travel around Europe. Much of what he would have found at this time would remind him of the conditions under which he, long ago, had extracted a scanty living from the soil.
Of course, once again, the resulting simplicity costs significantly more words.

But not all complex sentences can be broken up with extra words as the only downside. Consider the second sentence in this excerpt from a later passage in Outlines of History:
The chief effects of the Napoleonic occupation of Germany were three in number. First, the consolidation of territory that followed the cessation of the left bank of the Rhine to France, as explained previously, had done away with the ecclesiastical states, the territories of the knights, and most of the free towns. Only thirty-eight German states, including four free towns, were left when the Congress of Vienna took up the question of forming a confederation to replace the defunct Holy Roman Empire. Second,... (Outlines of History, p. 249)
Here's a stab at breaking up that second sentence:
The chief effects of the Napoleonic occupation of Germany were three in number. First, the consolidation of territory that followed the cessation of the left bank of the Rhine to France had done away with three things. First, it did away with the ecclesiastical states. Second, it did away with the territories of the knights. Third, it did away with most of the free towns. (All this was explained previously). Only thirty-eight German states, including four free towns, were left when the Congress of Vienna took up the question of forming a confederation to replace the defunct Holy Roman Empire. Second, ...
Simplifying the second sentence by breaking it up has complicated the paragraph as a whole. For one thing, now we have three items embedded within the first of three items. Secondly, since each of the three new sentences has its own separate focus, the focus of the paragraph as a whole flits around all over the place. Thirdly, the increased verbiage between the first "First" and the final "Second" makes it harder to remember, by the time we get to that "Second," that we're in the middle of a list of "chief effects of the Napoleonic occupation of Germany." A modern day student with attention difficulties is not going to fare any better with this modern update than with the original.

Verbal complexity isn't simply a matter of mercurial trends in textbook stylistics that modern editors can edit away without losing content. As I noted earlier:
Some thoughts are too complex to be captured in sentences that avoid ... attention-demanding complexity. It’s alarming to think that sentences that express such thoughts are no longer accessible to many readers. Even more alarming is the possibility that people are too distracted to even think them on their own.

Saturday, April 20, 2013

What could be more important than grit? Listen and learn.

The basic problem with all this focus on "grit" and "character" in K12 education--for which Paul Tough's recent book How Children Succeed is both the latest inspiration, and the latest incarnation of one of our culture's most persistent memes-- is that it simply adds to the litany of things that are diluting academics. The more we dilute actual academics, the more we limit students' opportunities to develop such learning-enhancement traits as curiosity, skepticism, and, yes, grit (a.k.a. perseverence)--traits that, in fact, develop alongside academic development.

You develop grit by doing a hard math problem whose solution requires lengthy puzzling out; by answering comprehension questions that make you work through challenging sentences and paragraphs; by revising an essay in response to requests for clarity, economy, and coherence. It's true that students today, more than ever before, need grit. But the reason for this has less to do with the vicissitudes of the world outside of school than with what is no longer happening at school. And the solution isn't to send teachers to grit workshops, cover classrooms with grit slogans, and to interrupt academics with grit rallies.

I've recently realized, however, that there is one element of "character" that absolutely must be well established before academic instruction can take off. In fact, I realize this approximately once a week: whenever I'm teaching in our after-school math enrichment program. I realize it specifically whenever I'm trying to explain a concept that requires students to attend to my words for more than a few seconds in a row. For the more restless of these students--which tend to be those who most need the explanation in the first place--those few seconds are a few seconds too long.

To teach someone anything, you need, at minimum, a window of joint attention with that person. As I know from raising an autistic son, when this window of joint attention is rare and fleeting, so, too, are opportunities for direct instruction. Joint attention with neurotypical kids is typically much more frequent and extended, but they, too, are potentially distracted away before you finish what you're saying.

The key word here is potentially. By the time neurotypical kids are in the early grades of grade school, they can potentially be quite attentive. How attentive depends both on the immediate environment--how distracting it is--and on longitudinal circumstances--whether the kids are acquiring habits that foster attention. Are they sitting facing the teacher, or facing one another in "pods"? Is the class full of distractions like group activities, laptops, and a couple of highly disruptive students? Have the students come to view the teacher as the "sage on the stage," or merely as the "guide on the side"? Have they been held accountable for paying attention to what their teachers say and for not disrupting others? In other words, has the environment in the early grades been conducive to the development and maintenance of teacher-focused attention?

For too many of my students, the answer would appear to be "no." But I suspect that many people would take a quick look and blame either the kids themselves--surely they need attention-enhancing mediation--or their surely disruptive home lives. Only if you looked at the classrooms that these children (mostly 5th graders) have attended daily for the last several years--where, in fact, students sit in pods and student-centered activities and distractions predominate--might you get an inkling of what's really going on here.

One educational paradigm that has caught on is KIPP. Its SLANT (Sit up, Listen, Ask and Answer Questions, Nod, and Track the Speaker) is all about extending the window of attention so as to make it possible for students to learn from their teachers.

The sad fact is that today's student-centered educational paradigm, which sees itself as the best model for developing all those "essential, non-academic" skills, and which is forever vilifying KIPP's SLANT, is, in fact, stunting the growth of the one trait that matters the most for learning. And making it very difficult for those of us who seek to remediate the academic consequences of this non-clinical, but no less debilitating, deficit.

Thursday, April 18, 2013

Math problems of the week: 6th grade Connected Math vs. Singapore Math

The first pages in the chapters on circles sections of 6th grade Connected Math and Singapore Math:

I. From Connected Math "Going Around in Circles" section [click to enlarge]:



II. From Singapore Math Primary Mathematics 6 Workbook "Circles" chapter [click to enlarge]:


III. Extra Credit

Which is more mathematically instructive: constructing and measuring circles, or going around in them?

Tuesday, April 16, 2013

Autism and Abstract Thinking III: "Hard to define but easy to give an example"

In my Autism, Language and Reasoning class I'm constantly underlining the distinction between words whose meanings are hard to grasp and words whose meanings are simply hard to teach. For kids with autism, both of these challenges arise more often than they do for neurotypical kids.

On the one hand, AS kids have trouble grasping a whole host of everyday meanings that presuppose a neurotypical level of social awareness: "friend," "respect," "jealousy," and "fairness," to name just a few. On the other hand, AS kids have trouble learning a much broader range of words whose meanings are simply difficult to teach explicitly: abstract nouns ("shape"), as well as many more words that aren't nouns ("jump," "around," "because," "and"). These meanings aren't inherently difficult for AS kids to grasp; the problem is that they're difficult to teach.

This teaching obstacle is something most of us don't appreciate--or even need to think about. Neurotypical kids will pick up the meanings of "jump," "and," etc., without us having to explain them directly: they simply tune in to conversations and pick up cues from social context. The more tuned-out of AS children, on the other hand, will learn these words only if someone explicitly teaches them.

Some words, of course, are much easier to teach than others. Pointing to a dog and saying "dog" is one thing, but how do you point to the meaning of "jump" or "and" or "because"? Many people, including the experts, have observed that the vocabularies of children with autism are dominated by concrete nouns, with many fewer verbs, function words, or nouns with abstract meanings. Not appreciating how dependent AS kids are on direct instruction, and how difficult it is to teach most word meanings directly, all too many people, experts included, conclude that part of autism is difficulty with abstract concepts.

The distinction between conceptual issues and instructional issues is itself a rather subtle, abstract concept. So I was delighted to see J grasping one facet of this distinction just the other day, when I, just for fun, asked him to define the words "left" and "right." He took a stab at it, then observed that "Some words are hard to define but easy to give an example."

I asked him for another example of this. He looked off toward the kitchen door, and then got up, opened the door, and said "open." I suggested that we look up the dictionary definition of "open," and we spent the next 10 minutes chuckling over the tautological or circular definitions you get when you chase down the definitions of the key defining terms of each next definition. He particularly enjoyed looking up such basic but elusive words as "is" and "I"--both words he specifically picked, and also words that once totally eluded him--until such time as I managed, marshaling all my linguistic knowledge, to find a way to teach them explicitly. But that's a whole nother story.

Sunday, April 14, 2013

Alternative entry points in High School English

My son recently came home with this 10th grade English assignment:

Greetings, you have a collage assignment for a character or title chosen by yours truly :).
Here is what your collage must include:
1) Character's name
2) Words, phrases, or adjectives that describe your character
3) Drawn pictures
4) Printed pics
5) Pics from magazines, newspapers, old postcards
6) Material or a partial background using something other than the background of the paper
7) Colored pencils
8) Crayons
9) Markers
10) Embellishment--I'm thinking 3D (i.e. feathers, buttons, earrings, a miniature book)
I though it would be a nightmare getting this done; one of those high-ratio-of effort-to-learning assignments. But he was surprisingly independent and did most of it, fairly quickly, on his own (with some help from SparksNotes), despite the fact that we didn't have stick glue and didn't read step 6 until he'd already covered the poster board as per steps 1-5.

Had J instead been given the kind of antiquated literary analysis assignment that I received back in my day (the book in this case being James Baldwin's Go Tell it On the Mountain), the requisite effort would been a thousand times greater with not necessarily that much more learning in the process (though, of course, any positive number is greater than zero).

So in some ways this assignment exemplifies the alternative entry point ideal: it makes literary appreciation "accessible" to those with very weak verbal skills.

But it's also a classic case of accommodation instead of remediation. Assigning my son collages rather than essays will give him a shot at a decent grade in Honors English; it will do nothing to improve his ability to write a coherent essay--whether in pencil, marker, or crayon.

Friday, April 12, 2013

Math problems of the week: 6th grade Everyday Math vs. Singapore Math

I. 6th grade Everday Math, from the "Rates and Ratios" chapter of the 6th grade Everyday Mathematics Student Math Journal, Volume 2:

A. The first problem involving distance, rate, and time (p. 290) [click to enlarge]:


B. The one other problem from this section involving distance, rate, and time (p. 249):




II. 6th grade Singapore Math, from the "Speed" chapter of the Primary Mathematics 6A Workbook.

A. The first problem set (p. 74) [click to enlarge]:


B. Another problem from about 17 additional problems involving distance, rate and time--all word problems (p. 79)


III. Extra Credit

What sorts of higher-level thinking do Singapore students miss out on by not being asked to create number sentences in the first problem set?

Wednesday, April 10, 2013

Do the authors of the Common Core really want us to replace step-by-step math with words, pictures, and applications?

Education Week, perhaps the most widely-read K12 education publication (circulation: 50,000+ paid copies per week; registered online users: approx. 725,000), has been consistently interpreting the Common Core as favoring Reform Math, discovery learning, and "real-life" applications--all staples of the dominant education paradigm known as Constructivism.

Here's the third of three articles in last week's education week that read Constructivist goals in the Common Core Math standards (and, in this particular case, also in the National Board Certification process): an article by Algebra 2 and AP Calculus teacher Alison Crowley. Describing her realization that "'teaching math' and 'covering textbook sections' were not synomymous," she writes [all boldfaces are mine]:

Before I started implementing, or had even heard about, the Common Core State Standards, I had already begun shifting my instructional practices to include more hands-on activities and group work, and less book work. Project-based learning began trending in my math teacher circles, and pursuing National Board certification forced me to rethink my instructional practices. Were my students actually learning the material for mastery or were they just good at following directions and memorizing steps?
One might ask the same of teachers. Here again we have the unquestioned contrast between "mere memorization"=book work vs. actual learning=hands-on group work + project-based learning.

Crowley asks us to consider the following situation:
If a teacher is explaining how to solve a system of equations using the substitution method, she might list on the board a set of steps for students to follow. Step 1: Solve one of the equations for one of the variables; Step 2: Substitute the value or equation found in Step 1 into the other equation. If you peeked inside her classroom on this particular day, you would likely see all of the students copying notes, and then probably completing a worksheet with problems similar to the example. From an observer's perspective, you might think the lesson was going very well.
But do the students really have a solid understanding of the mathematics they are using? And more importantly, do they understand why they're using it? Do they have a graphical understanding of what it means to solve a system of equations? Can they explain their methodology to another student? Can they apply it to real-world situations? Is their knowledge transferable so that they will be able to draw upon it when they are solving more difficult systems of equations in future math classes?
Again there's the assumption that putting things into words and applying them to the real world is the best test of understanding--even initial understanding. But why wouldn't initial guidance through solving systems of equations build a foundation for solving more difficult systems later on?

Crowley, naturally, guesses it doesn't--a mere guess being apparently enough of a foundation for an article in Education Week. Drawing on the untested assertions of Ann Shannon, a mathematics consultant with the Bill and Melinda Gates Foundation (itself a notorious promoter of Constructivist practices), Crowley writes:
My guess is that the answer to most of these questions is "no." What Ann Shannon would say is that in this particular situation, the students have been GPS-ed from problem to solution. Just like when I drive in a new city using my global positioning system, I can follow the directions and get to where I need to go. But I can't replicate the journey on my own. I don't have a real understanding of the layout of the city. If a road were blocked due to a parade, for example, I would be in trouble because I have no real understanding of the city's geography.
A cute metaphor, and I'm sure it's only a matter of time before "GPS-ing" will be buzzing all over the edworld. But cute and catching don't mean true. The GPS gives specific directions that aren't supposed to generalize, and they don't. "Drive 5.8 miles, then turn left on Cherry Street" is not analogous to "Step 1: Solve one of the equations for one of the variables; Step 2: Substitute the value or equation found in Step 1 into the other equation." If Cherry Street is blocked, the GPS's directions are useless; if the first equation has 5y instead of 2x, the teacher's directions still work. GPS-ing is not the same as step-by-step instruction.

But, when you have a great new buzzword, who cares? Especially you find justification for it in the Common Core math standards:
So, how can we keep from GPS-ing our students, so that they understand the mathematics behind a series of steps? How can teachers help them grasp the why, instead of just the how?
The good news is that the common-core standards provide an open playing field that encourages teachers to move away from the step-by-step model.
Consider the following standard for solving systems of equations:
A.R.EI.11 Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equations f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
It should be noted that the Common Core standards also include A.REI.5 Solve systems of equations using the elimination method (sometimes called linear combinations) and A.REI.5 Solve a system of equations by substitution (solving for one variable in the first equation and substitution it into the second equation).Furthermore, these two standards are the first two under the "Systems of Equations" section; there are five more intervening standards before the standard Crowley cites. But, even though the her very own source suggests that it should come later on in the curriculum, Crowley prefers the graphical to the symbolic, and the real-world application to the step by step procedure:
Remember the earlier example about the teacher showing the students how to solve a system of equations using a set of steps? The first sentence of the new standard, "Explain why the x-coordinates where the graphs intersect are the solutions," really pushes the teacher to introduce and explain a new concept in a way that goes beyond one-dimensional instruction. What is an x-intercept and what does it look like on a graph? How is that related to the algebraic equation? Perhaps rather than starting a lesson with the steps for solving the equations, the teacher might first have students consider graphs of related equations, or better yet, a real-world example of a system of equations and what the values of the x-intercepts mean in that situation. This standard also challenges the teacher to present multiple types of equations from the beginning of the lesson so that the students can apply the concept of an x-intercept to many types of functions.
Multiple types of equations all at once as an introduction to systems of equations? This sounds like a pedagogical nightmare for all concerned.
For many teachers, myself included, this is a fairly significant change in instructional practice. Although I have taught lessons on solving systems of equations using real-world applications and emphasizing graphical connections, I have not yet truly focused my instruction on the "why" behind the mathematics or given students opportunities to create their own understanding.
But the "why" isn't inherently graphical; nor does it necessarily emerge from real-world applications any more perspicuously than from algebraic applications. As for students creating their own understanding, this is one of those things that looks better in theory than in real-world application.
So how do math teachers make that shift away from GPS-ing students a reality? It won't be easy, and we can't do it alone. We need opportunities to collaborate, plan, and reflect with colleagues, both in our buildings and nationwide. We need quality resources and relevant, engaging professional development.
The more I read, the more it strikes me that the big winners under the Common Core aren't the children, but the professional developers (and the developers of all those "quality resources").
We need time to learn from teachers who are already successfully implementing the common core, like Kansas educator Marsha Ratzel, who recently shared insights in her piece "The Talking Cure: Mathematical Discourse," on what happened when she gave her students time and space to have conversations about math. We need administrators and parents to support us and play an active role in helping us transform our classrooms into places where students are truly engaged in what they are learning.
True engagement with x = having converations about x. Now we just need to solve for x.
In my daily classroom instruction, I am still sometimes guilty of GPS-ing students. But I am hopeful that as I learn how to fully implement the common standards, I will become less and less dependent on steps and crossing standards off a poster. After all, my students really deserve to navigate themselves.
Do the authors of the Common Core also agree? It would seem so. As I noted above, Education Week may be the most widely read publication on K12 education. If its articles are misinterpreting the Common Core, surely the CC authors would have noticed and spoken up by now.

Monday, April 8, 2013

Do the Common Core authors really want Reform Math for "high needs" students?

It's increasingly clear that the Common Core is being interpreted, not just as further justification for Reform math and discovery learning, but also as a one-size-fits-all set of goals that rules out providing those who need it with below grade-level work.

We see both of these assumptions at work in an article in the past week's Education Week entitled
Teachers Break Down Math Standards for At-Risk Pupils [bold-face is mine throughout]:

Many accomplished teachers are enthusiastic about the common-core math standards' emphasis on mathematical reasoning and strategic expertise over rote computation, but some say the transition to the new framework poses daunting challenges for students who are already behind in the subject.
...
More than half the respondents in a recent survey of K-12 teachers who are registered users of edweek.org said they feel unprepared to teach the common standards to high-needs students.
Here we should keep in mind that, since "social promotion" trumps academic preparedness, and since poor instruction is rampant, many of these "high needs" students aren't special needs students, but simply students who aren't academically ready for their current grade level.
Despite often lacking support and clear guidance, however, teachers aren't necessarily ready to throw in the towel. Some math educators are taking steps to refine their practices and adopt creative methods to help struggling students make the shift to the new instructional paradigm.
In other words, no one is telling teachers what to do when their students are too far behind the Common Core Standards for their assigned grade level to be able to reach these goals. And so teachers are figuring things out on their own. Are they providing these students with math at their actual grade levels? Or are they shoehorning them into the Common Core Complex?
One approach teachers commonly cite, for example, is to maintain the common core's emphasis on abstract reasoning and conceptual understanding while, at least at first, using word problems that require less-advanced math skills.
"It's OK if you need to start more basic," said Mr. Arcos, explaining that initially he used two-digit addition without regrouping his 5th graders, many of whom were at a 2nd or 3rd grade level in math.
The key is to "avoid focusing on the algorithm or any tricks," he said, so that the students have to work through the problems strategically. He noted that students at his school have daily problem-solving classes in this vein, as well as computation-skills practice two mornings a week.

Similarly, Todd Rackowitz, a math teacher at Independence High School in Charlotte, N.C., noted that, in integrating the common standards into an Algebra 1 course for students who are behind grade level, he "focuses on problems that don't involve complex computation at first." Even using basic math, students can begin to "make connections between the key elements of algebra, like slope and parallel lines and rate of change," he said.
So we have the usual dichotomies: algorithms = tricks = bad; "strategic" thinking = "making connections" = good. And the reduction of "key" algebraic concepts to concrete, arithmetic ones ("plug-and-chug" algebra). The idea here seems to be that what "high needs" students need isn't remediation, but Reform Math concentrate. Here's more:
"You have to help kids understand how to justify solutions, through discussion, interaction, and close guidance," said Mr. Arcos, adding that his school has adjusted scheduling to allow for more small-group and one-on-one instruction in math.
...
To build students' problem-solving and abstract–reasoning skills, he has also found it helpful to have students work out solutions and understanding through "group discussion and discovery." To spark engagement with problems, Justin Minkel, a 2nd and 3rd grade teacher at Jones Elementary School in Springdale, Ark., noted that he has his students "do a lot of writing in math." That practice, he said, helps students see the conceptual underpinnings of the problems they are working on and, with his assistance, see how words and phrases can relate to mathematical notations.

Yup, explaining answers, writing about math, and working in groups--it's all there. The only promising measure here is one-on-one instruction. But if it's not at the student's grade-readiness level, how far does it get him?
Mr. Minkel, whose school has a high percentage of low-income students, said he also makes an effort to give his students problems that have "practical applicability" to the real world. He noted that he has had success, for example, in having his students use what they were learning in math in an economics unit that involved determining the costs of materials for a building project against a budget.

Such activities can help students "make sense of problems"—the first of the common core's Standards for Mathematical Practice—and begin thinking about the ways math relates to their own lives, Mr. Minkel said.

...
Real-life applications, interdisciplinary projects--this is apparently what the Common Core authors want, especially when it comes to "high needs" students. Because if it isn't what they want, surely they'd have said so publicly--especially after high-profile articles like this one.
"It's harder to teach this way than just teaching algorithms and steps," said Mr. Minkel. "It forces you to go deeper... Sometimes, we realize that we don't understand things as well as we thought."
The only question is: when will you realize this?

Saturday, April 6, 2013

Learning algebra vs. "learning algebra"

In my weekly math comparisons, I've recently been exploring what today's Reform algebra texts leave out. Here's a partial list:

-Factoring polynomials of degree greater than two.
-Rational expressions with more than two terms in the numerator and denominator.
-Multiplication of more than two polynomials or of polynomials that contain more than two terms.
-Polynomial long division and other multi-step methods for simplifying multi-termed rational expressions.
-Systems of equations that involve more than two variables.
-Problems that require manipulation of polynomials in order to simplify them or convert them to special (often abstract) patterns--e.g., by adding/subtracting the same term to different parts of the expression/equation, or by rearranging terms and grouping them together in a structural hierarchy (see the second problem set in last week's Problems of the Week).
-Word problems requiring students to define multiple variables and set up multiple equations.

Not only is much of Reform algebra limited to one or two terms, one or two variables, powers of one or two, and problems involving only one or two steps; many of the problems only require students to plug in numbers or push buttons on their graphing calculators. Essentially, Reform algebra doesn't get you much beyond Reform arithmetic.

Luckily, some students are still using more traditional algebra texts. But this means there's great variation in what different students are learning in the name of "algebra"--the focus of an article in this week's Education Week. It describes a study by the National Center of Education Statistics of high school algebra and geometry courses that analyzed transcript data from 17,800 students and content data from "120 Algebra I, Geometry, and integrated math textbooks used at the 550 public schools those students attended." Its conclusion:

Students taking Algebra 1 and Geometry classes are getting considerably less substance than their course titles would suggest.  
... 
The study found that, on average, two-thirds of topics covered in Algebra 1 and Geometry courses focused on core content topics in each of those subjects, while the other third covered topics in other math areas. Researchers also gauged the rigor of classes based on the topics and questions covered in each book. A course categorized by researchers as beginner-level algebra had more than 60 percent of its material on elementary and middle school math topics such as basic arithmetic and pre-algebra problems such as basic equations. By contrast, a rigorous Algebra 1 courses included more than 60 percent of material on advanced topics such as functions and advanced number theory, as well as other higher-level math subjects such as geometry, trigonometry, and precalculus.  
“We found that there is very little truth-in-labeling for high school Algebra 1 and Geometry courses,” said Sean P. “Jack” Buckley, the NCES commissioner, in a statement on the study.  
...  
For example, a student taking a rigorous Algebra 1 course covered 11 topics in advanced number theory, compared with only six for students in courses with the same name that researchers classified as beginner- and intermediate-level classes. A student in an Algebra 1 class ranked by the study as beginner-level had no exposure to advanced functions, and more than a quarter of the class was devoted to basic arithmetic and pre-algebra. A student in a rigorous Geometry class likewise covered significantly more topics in coordinate and vector geometry, and significantly fewer topics in basic arithmetic and pre-geometry, than a student in a beginner-level Geometry class.
The article turns to what this means, not for the worthiness of today's Reform books, but for the racial/ethnic achievement gap. It observes that, while differences in how many math credits appear on the transcripts of white vs. black graduates have narrowed, differences in what different ethnic groups are actually learning is another story:
There were no significant differences in the proportion of students of different racial groups who took rigorous Algebra 1 courses—roughly a third of each group—though Hispanic and Asian and Pacific Islander students were more likely than other groups to take beginner-level algebra courses. However, the NCES found that more white students in honors Geometry classes, 37 percent, covered rigorous topics, compared with 21 percent of black and 17 percent of Hispanic students in similarly titled classes.
Perhaps our math texts should be less concerned with multiculuralism, and more with multiplication, unlike what this Integrated Math textbook index (reposted from last week's Problems of the Week) suggests:
 
 

Thursday, April 4, 2013

Math problems of the week: writing functions in 1930s math vs. the Interactive Math Program

The first assignments in which students are asked to write functions to model hypothetical situations:

I. From A Second Course in Algebra (Virgil S. Mallory, 1937), Chapter IV, the "Functional Relations" section [click to enlarge]:


II. From the Interactive Math Program Year 4 (Dan Fendel, Diane Resek, Lynne Alper and Sherry Fraser, 2000) "World of Functions" chapter [click to enlarge]:



(Interactive Math Program is a four year high school math curriculum).

III. Extra Credit

Compare how much of the symbolic modeling is spoon-fed to the students in the Interactive Math Program vs. the traditional math problem sets. Then discuss what this means for preparing students for 21st century jobs.

Tuesday, April 2, 2013

Education news round up: The NY Times vs. the Atlantic

Here's some of the latest on education from New York Times Op-Ed columnist Tom Friedman. First, from three weeks ago:

Institutions of higher learning must move, as the historian Walter Russell Mead puts it, from a model of “time served” to a model of “stuff learned.” Because increasingly the world does not care what you know. Everything is on Google. The world only cares, and will only pay for, what you can do with what you know.
Therefore, we have to get beyond the current system of information and delivery — the professorial “sage on the stage” and students taking notes, followed by a superficial assessment, to one in which students are asked and empowered to master more basic material online at their own pace, and the classroom becomes a place where the application of that knowledge can be honed through lab experiments and discussions with the professor.
[All this in an article showcasing MOOCs--massive open online courses--which in fact are the apotheosis of sage on the stage instruction and of superficial learning writ large. Friedman,perhaps a humanities major, seems to have forgotten all about the problem sets, problem sessions, labs, and discussion sections that supplement lectures in the "traditional" model of tertiary education.]

This past weekend, Friedman's hero was "Harvard education specialist" Tony Wagner, whose views I blogged about earlier and about whom several astute commenters weighed as well.
“We teach and test things most students have no interest in and will never need, and facts that they can Google and will forget as soon as the test is over,” said Wagner. “Because of this, the longer kids are in school, the less motivated they become. Gallup’s recent survey showed student engagement going from 80 percent in fifth grade to 40 percent in high school. More than a century ago, we ‘reinvented’ the one-room schoolhouse and created factory schools for the industrial economy. Reimagining schools for the 21st-century must be our highest priority. We need to focus more on teaching the skill and will to learn and to make a difference and bring the three most powerful ingredients of intrinsic motivation into the classroom: play, passion and purpose.”  
... 
“Teachers,” he said, “need to coach students to performance excellence, and principals must be instructional leaders who create the culture of collaboration required to innovate. But what gets tested is what gets taught, and so we need ‘Accountability 2.0.’ All students should have digital portfolios to show evidence of mastery of skills like critical thinking and communication, which they build up right through K-12 and postsecondary. Selective use of high-quality tests, like the College and Work Readiness Assessment, is important. Finally, teachers should be judged on evidence of improvement in students’ work through the year — instead of a score on a bubble test in May. We need lab schools where students earn a high school diploma by completing a series of skill-based ‘merit badges’ in things like entrepreneurship. And schools of education where all new teachers have ‘residencies’ with master teachers and performance standards — not content standards — must become the new normal throughout the system.”
Yeah, yeah, we've heard it all before. Boring facts; looking it up on Google, portfolios and play. Forget specific content and content standards.

The New York Times, whose education columnists are mostly armchair intellectuals who mouth the dominant memes,can be trusted to publish only the politically ascendant side of the education debate, whether it's Friedman on "21st century skills," David Brooks on grit and character education, or Susan Engel on cooperative groups and student-centered discovery learning--or all three on how we need to stop focusing, in one way or another, on core academic subjects.

Thank goodness for the online Atlantic. It has consistently published stuff outside the dominant paradigm--not from armchair intellectuals, but from those with some familiarity with what goes on in classrooms. Within the last few months, it has published pieces on the virtues of grammar instruction, the downsides of Writer's Workshop, the problematic emphasis on social skills over academics, and the problems with Reform Math and the Common Core math standards. Most recently, there's a new piece by Barry Garelick, the author of the last two pieces, on the virtues of grouping students by ability--a practice that, even though it's been making a quiet comeback, has been derided by the education establishment for decades.