Saturday, March 31, 2012

Cutting curbs for Aspies--and for students in general

Finally an Edweek Commentary I can relate to. In her piece on the challenges her Asperger's son faces at school, Anita Charles, the director of teacher education at Bates College, writes:

I have seen him get perfect answers in math, only to lose many points for not "showing" the work adequately. (In elementary school, the start of the math troubles, he was sent back from the math enrichment classroom because he kept showing up without a pencil.) I have seen him freak out over tests based exclusively on corrected homework—that somehow didn't manage to stay with his notebook, thereby earning him that toxic zero. Never mind the points lost for the disastrous notebook itself. (And this for a kid who was solving square roots in his head at the age of 5. Nowadays? "I can't do math," he tells me. What I tell him is that he's perfectly capable of doing "math"—he just doesn't "do school" as well.)
I have watched my son get a D on a science lab because he couldn't express enough details, had too many crossed-out errors in his lab report, and didn't list the full names of his lab partners.
...
He can only take yet another low mark for "group participation," "eye contact," "presentation," "neatness," "organization," and try not to fall apart at the seams.
...
We have asked our "Aspies" to try to "figure out" the way school functions for too long. What would it take for the rest of us to try to "figure out" the way kids with Asperger's function? What would it take to put a few ramps and railings in place, to find alternative assessments, to speak gently to these gentle souls and ask them what the world feels like to them? It's not that difficult, honestly. And the gifts we uncover might astound us.
Amen.

As director of teacher education at Bates College, Charles is well-positioned to make a difference. I'm hoping that she's encouraging Bates' teachers of future teachers to question whether group participation and eye contact should figure in anyone's grades; whether it's fair to impose heavy organizational demands on anyone (not just the Aspies) who happens to be at the lower end of normal in their Executive Function development; whether neatness should ever be a requirement when students no longer learn penmanship; whether trivial nitpicks should ever add up to a D; and whether students should ever get points off for not showing their work when their answers are consistently correct.

Much as I sympathize with autism families in particular, these requirements aren't just frustrating the Aspies. Indeed, just as many have benefited from other accommodations intended originally for the disabled (everything from curb cuts to speech recognition software), many might benefit, in particular, from classroom accommodations for Aspies.

Thursday, March 29, 2012

Math problems of the week: 2nd grade Investigations vs. Singapore Math

Two approaches to subtraction practice:

I. A 2nd grade Investigations sheet, from mid-way through the 2nd grade curriculum, Putting Together and Taking Apart, p. 135 [click to enlarge]:


II. A  subtraction exercise from early on in the 2nd grade Singapore Math Curriculum, Primary Mathematics 2A, p. 35 [click to enlarge]:

III. Extra Credit:
1. Map the assignment to the teaching protocol:
a. Telling you which strategy to use
b. Providing feedback about whether your answer is correct

2. Which assignment is more Constructivist?

Tuesday, March 27, 2012

Here we go again!

Here we go again. This time it's the New York Times' David Brooks rhapsodizing about the New American Academy in Crown Heights, which he nicknames the Relationship School:

When you visit The New American Academy, an elementary school serving poor minority kids in Crown Heights, Brooklyn, you see big open rooms with 60 students and four teachers. The students are generally in three clumps in different areas working on different activities. The teachers, especially the master teacher who is floating between the clumps, are on the move, hovering over one student, then the next. It is less like a factory for learning and more like a postindustrial workshop, or even an extended family compound.
...
[Founder Shimon Waronker] has a grand theory to transform American education, which he developed with others at the Harvard School of Education. The American education model, he says, was actually copied from the 18th-century Prussian model designed to create docile subjects and factory workers. He wants schools to operate more like the networked collaborative world of today.
Sound familiar? This the same school in which, according to the earlier (January, 2011) Times article:
While waiting for her teacher to come by, one little girl arranged the pennies she had been given to practice subtraction into a smiley face. Another shook her pennies in a plastic bag. A high-pitched argument broke out over someone’s missing quarter.
“We don’t know what we are supposed to be doing, but we are learning about math,” Thea Burnett, 6, said.
...
When [teacher Jennifer McSorley] leaned forward out of her chair to write a word on an easel, a 6-year-old boy moved it, and she fell when she tried to sit back down.
...Then another boy ran off to hide under an easel. Someone grabbed someone else’s pennies. The noise snowballed.
...
In the first two months of school, a student pulled a chunk of an adult’s hair out, and an ambulance crew was called twice to calm a child. Eight weeks into the year, the only student work visible on the blue-painted walls was a poster with finger-painted hand prints and the words “Hands Are Not for Hitting.”
...
By January, three children who were violent had been moved to more-structured environments; seven other first graders moved away or withdrew, reducing the class size to 50.
Apparently, things have improved in the last 15 months:
[The lack of structure] was a problem at first, but Waronker says the academy has learned to get better control over students, and, on the day I visited, the school was well disciplined through the use of a bunch of subtle tricks.
For example, even though students move from one open area to the next, they line up single file, walk through an imaginary doorway, and greet the teacher before entering her domain.
Lining up in front of imaginary doorways, apparently, doesn't yield the same depths of docility as lining up in front actual ones does.

Brooks notes that:
The New American Academy has two big advantages as a reform model. First, instead of running against the education establishment, it grows out of it and is being embraced by the teachers’ unions and the education schools. If it works, it can spread faster.
I'm sure that's true, though given who Waronker's (shocking!) backers are, I'd replace "if" with "whether or not."
Second, it does a tremendous job of nurturing relationships. Since people learn from people they love, education is fundamentally about the relationship between a teacher and student. By insisting on constant informal contact and by preserving that contact year after year, The New American Academy has the potential to create richer, mentorlike or even familylike relationships for students who are not rich in those things.
Maybe I'm just a cold-hearted left-brainer, but I'd say that people learn from those who know the material in question and how to teach it--whether or not there's love involved. If there is, I'd take love of teaching and learning and love of the material over interpersonal love.

Brooks concludes with the usual disclaimer (italics mine):
It’s too soon to say if it will work, especially if it’s tried without Waronker and the crème-de-la-crème teachers he has recruited, but The New American Academy is a great experiment, one of many now bubbling across the world of education.
If it's too soon to say if it will work, and if what's really making the difference is the crème-de-la-crème teachers, why give it so much attention? Instead, why not look further afield and ask (1) whether this model is truly original, and (2) whether, minus the the crème-de-la-crème teachers, and Beatles' bromides  aside, it has ever worked anywhere.

Sunday, March 25, 2012

Letter from Huck Finn: I Learn the Ropes and Give Things Some Serious Thought

Out in Left Field proudly presents the fifth 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.


I Learn the Ropes and Give Things Some Serious Thought

During my student teaching, I tended to stay away from discussions of educational issues. Judging by the reaction to my last letter, discussion of educational issues can result in disagreements expressed at great volume. I stayed away from politics and religion as well. I did venture a bit into football, though, which is not without its risks. The principal was a Notre Dame and Michigan State fan, and I went to University of Michigan. She didn’t like Ohio State, though, so we got along just fine.

While I don’t consider student teaching similar to football, I was often reminded of my experience in the Michigan Marching Band. I was in the Band when it was directed by the legendary perfectionist William D. Revelli. The Band’s high step marching was physically demanding. I had enough wind to enable me to march but I couldn’t play a note on my clarinet all season. I kept that my dirty little secret, but when I rejoined the band the next year, I was delighted to find that I had the physical endurance to march and play at the same time, and incorporate all the nuances and instruction hollered at us during rehearsals.

I hoped that what happened in the band would happen in teaching: that I would be able to simultaneously teach and automatically do with ease what I now had to be told to do. My supervising teacher, Tina, often interrupted my teaching. She would whisper things like: “Circulate; see what they’re writing—you’re chained to the desk.” “You take too long to answer the question; go for the big idea.” And in the discovery-based algebra class: “You’re telling them too much. You have to ask them questions; if you do it all for them, they’ll never learn.” This last one was perplexing: was it an issue of proper questioning—an important skill to learn—or was it an application of the ed school adage that if students aren’t struggling, they aren’t learning?

I began teaching first and third period pre-algebra classes my fourth week. Third period was an honors class. Though I liked teaching both, I found the honors class more motivated and responsive than the first period class. I admit to feeling guilty in saying that the honors class was more fun to teach, though I suspect I'm not alone. Some teachers, given a choice, might not want to teach struggling students, but then fall prey to being called elitist. Neither teachers nor schools want to be accused of that. So now in the name of equity for all, schools have mixed ability/performance classes. I’m happy to say my school grouped students by ability/performance. I don't believe honors and gifted classes are elitist. I think all students need attention.

That said, first period was a bear to teach. I was always nervous before it began; fifteen minutes before the bell I would have to run for the bathroom. Once the bell rang, and I swung the door open, I would relax. As students entered the classroom, the following exchange would continually take place:

Student: “Will we need our books today?”

Me: “Yes, get your books and warm-up folders, please.”

After answering this same question about five times, I would eventually tell them to ask their neighbor, thus attempting to make good use of the groups of four desks clustered together.

Because of the non-responsiveness of the first period class, Tina and I decided to have them take out a sheet of paper first thing, to write down specific items (vocabulary, formulas), and then have them do a problem. “With this class you’ve got to get them doing something,” Tina told me. “Otherwise they’ll fall asleep on you. And you’ve got an evaluation coming up.”

She was referring to the evaluation by the local university person who ultimately would give me my grade. He was a kind man who had taught high school math for over 30 years. I did well the first time, but that was the honors class. He now wanted to see me with the first period class.

The day he came I taught a lesson on inequalities. I had the class write down the symbols for less than, less than or equal to, greater than, greater than or equal to, which they already knew. I then added the terms: “Not as much as”, “No more than/at most”, “More than”, “No less than/at least”. I gave examples and then showed them a picture of a sign that you see at amusement parks that read “To go on this ride you must be at least 48 inches tall.” I asked “If H is the height of a child, how would we write in symbols that the height, H, meets the requirements?”

Dead silence.

“For example,” I said, as if nothing was wrong, “Would we write it like this?” I wrote H < 48. The class said “No-o-o-o.”

A boy in front named Francisco said softly “H is greater than or equal to 48.”

“Could you say that louder?” I asked. Francisco shook his head. Someone else gave the right answer.

“Yes,” I said and turned to Francisco. “You see, you were right! You have to have the courage of your convictions. In math, people often disagree. And sometimes they express these disagreements at great volume.” I stopped my pep talk there, thankfully. I didn’t need anyone falling asleep during my evaluation.

I continued the lesson, students stayed awake, and I received a good evaluation which made Tina happy. Later during 4th period which was her “prep” she talked about things to come. “I think pretty soon we’ll have you start teaching some algebra classes. Eventually, you’ll be in charge of all three classes—doing the planning, quizzes, and tests. And the discipline—it can’t just be me.”

I excused myself and ran for the bathroom. The idea of being fully in charge was overwhelming. If I found two classes to be demanding how would I handle three? Relax, I told myself. I would never get that far. Tina would be watching every move. Sooner or later I'd get into a discussion with her about math education and say things at great volume. I'd be politely booted out and advised to do something normal for my age, like be a Walmart greeter.

Friday, March 23, 2012

Math problems of the week: 3rd grade Investigations vs. Singapore Math

Addition / Subtraction Games:

I. From the 3rd grade (TERC)  Investigations Math Curriculum [click to enlarge]:


II. From the 3rd Grade Singapore Math Curriculum (Primary Mathematics 3A) [click to enlarge]:

III. Extra Credit

Which game does your child prefer?

Wednesday, March 21, 2012

21st century schools: moving beyond knowledge and skills

In a recent piece  in the New York Times Education supplement, former Harvard professor Larry Summers argues that:

Education will be more about how to process and use information and less about imparting it. This is a consequence of both the proliferation of knowledge — and how much of it any student can truly absorb — and changes in technology. Before the printing press, scholars might have had to memorize “The Canterbury Tales” to have continuing access to them. This seems a bit ludicrous to us today. But in a world where the entire Library of Congress will soon be accessible on a mobile device with search procedures that are vastly better than any card catalog, factual mastery will become less and less important.
Summer neglects to mention a couple of other skills, besides the ability to recall facts, commonly cited as obsolete thanks to 21st century technology:

- Calculation and graphing skills (via the increasingly ubiquitous 21st century calculators)
- Penmanship (via the increasingly ubiquitous and mobile 21st century keyboards)
- Spelling and grammar (via the 21st century's increasingly sophisticated spell-checkers and grammar checker)
- Accurate note taking (via the 21st century Echo Pen)

To these I’d like to add a few more that, to my knowledge, have yet to be appreciated as obsolete:

- Reading (sounding out words, aka “decoding”)
- Writing (even via a keyboard)

After all, a person’s ability decode a language’s written form, or to encode his or her words in written form, will soon be obviated by soon-to-be-ubiquitous and mobile speech-to-text and text-to-speech apps.

And:

- Foreign language instruction (it’s only a matter of time before Google Translate-like apps combine with text-to-speech and speech-to-text functionality, facilitating even spontaneous face-to-face interactions with non-English speakers)

It’s ironic that educators haven’t yet focused on the growing obsolescence of these last three skills, given how tedious and time consuming they are to master. Think of all the hours and hours of class time that could be freed up for higher level activities like interdisciplinary group research projects if, in particular, we didn’t have to teach kids how to read and write!

Monday, March 19, 2012

Making people squirm--or inspiring them

For all my qualified support for art in academics, making things relevant, and child-centered classrooms, there are two trendy educational practices in which I see absolutely no virtue: making students share personal feelings, and making them work in groups. Not only are these activities particularly ill-suited to children on the autistic spectrum; they also detract from all students’ educational experiences and appear to irritate a much broader range of kids than teachers may realize.

Particularly irritating are the practices of a teacher-blogger (thanks to FedUpMom for the link) who appears to think it’s a good thing when personal reflections assignments make her middle school students squirm.

As for group work, several recent articles explain just how misguided the typical group activities assigned by today’s classroom teachers are. A study recently reported on by the Wall Street Journal discusses how certain people tend to clam up in groups and become temporarily less intelligent:

If we think others in a group are smarter, we may become dumber, temporarily losing both our problem-solving ability and what the researchers call our "expression of IQ."
The clamming-up phenomenon seems to be more common in women and in people with higher IQs, according to the report, published in January in the journal Philosophical Transactions of the Royal Society B.
And in a lengthy article in the January 30th New Yorker, Jonah Lehrer, recapping some of what Susan Cain says in a January 13th New York Times opinion piece, discusses evidence that brainstorming only stimulates novelty (rather than stifling it) if people work apart and later pool their ideas, and if they are specifically instructed not to withhold criticism (the standard protocol), but to engage in debate and critique one another’s points. Moreover, the best brainstorming happens not in contrived meetings, but in relatively brief, incidental exchanges when coworkers (or workers in related fields) bump into one another in hallways. Overall, people are most productive and creative, Lehrer notes, when they spend most of their time working independently.

The most successful workplace collaborations thus stand in stark contrast to the regular, lengthy, teacher-imposed, teacher-contrived, heterogeneous groupings that predominate (and dumb down, stifle, and intimidate) in so many of today’s classrooms--at the same time that educators keep claiming that these activities are preparing students to be successful real-world collaborators.

Saturday, March 17, 2012

Heterogeneous groups, revisited

From a recent New York Times article on teacher evalutions:

Steve Ball, executive principal at the East Literature Magnet School in Nashville, arrived at an English class unannounced one day this month and spent 60 minutes taking copious notes as he watched the teacher introduce and explain the concept of irony. “It was a good lesson,” Mr. Ball said.
But under Tennessee’s new teacher-evaluation system, which is similar to systems being adopted around the country, Mr. Ball said he had to give the teacher a one — the lowest rating on a five-point scale — in one of 12 categories: breaking students into groups.
Presumably the teacher's fault lay in spending an entire class teaching the students as a whole instead of dividing them into groups. And if you know anything about current trends in education, you can guess what sorts of groups Tennessee's new teacher evaluation system expects. Only mixed-ability groups, I'm willing to bet, would have earned the teacher full points.

When it comes to heterogenous-ability-based grouping, there are all sorts of problems. The more prepared kids are bored, the less-prepared kids are intimidated; the former resent being expected to teach the latter instead of moving ahead at their own rates. These problems are particularly bad if by "grouping" one means making students work together in a group, as opposed to grouping them into different classrooms.

When it comes to classroom-level grouping, there's an alternative sort of heterogeity that (infrequent though it is) is much more effective--in part because it introduces into classrooms a novel, healthy diversity that not only avoids the problems of mixed-ability groups, but overcomes a common segregation. What I have in mind is heterogenous-age-based grouping. A school that does what it should and assigns students to particular subject levels according to readiness (i.e., according to their cognitive and academic readiness with respect to a given subject/task) will end up with heterogeneous-age classrooms.

I've seen such classrooms work quite well: the older kids provide more mature social and emotional role models for the younger kids, with the result that everyone behaves better. This also helps break down the age-segregation that pervades so much of childhood, giving all children opportunities to observe and engage with intellectual peers who are at all different levels of social and emotional maturity.

As far people in general are concerned, there are other sorts of heterogeneity we're missing out on--ones involving varieties of diversity whose benefits we rarely discuss. There's view point diversity, rarer and rare in today's sorted society; and there's neurodiversity. For all we love to rhapsodize about acceptance and full inclusion, the tolerance of our increasingly right-brained, socially-oriented society for socially awkward Aspies, rivaling our tolerance for those who don't share our political views, may be at an all-time low.

Thursday, March 15, 2012

Math problems of the week: 5th grade Trailblazers vs. Singapore Math

I. From 5th grade Math Trailblazers Student Guide, Unit 9, lesson 6: Calculator Strategies: Addition and Subtraction of Fractions, final problem set (p. 321) [click to enlarge]:



II. From the 5th grade Singapore Math, Primary Mathematics 5B, Fractions chapter, about half-way through (pp. 58-59)  [click to enlarge]:



III. Extra Credit:

What are Singapore Math students missing out on by never receiving direct instruction in "calculator strategies"?

When it comes to calculator strategies, which is better: direct instruction, or child-centered discovery?

Tuesday, March 13, 2012

Relevance, revisited

In the course of various recent discussions I've had, I've started to suspect that when some people speak of the importance of relevance, what they actually seem to have in mind is accessibility. A colleague of mine observed, for example, that The Great Gatsby isn't sufficiently relevant to many of today's students because they know little about the Jazz Age, the concept of nouveau riche, etc. And I agree. But the problem, as I see it, isn't that the book is irrelevant to what students know from personal experience. The problem is that students lack the background knowledge they would need to make sense of the book.

There are entire genres of fiction--namely, Sci Fi and Fantasy--that are largely irrelevant to the circumstances of students' personal lives, but that nonetheless are often quite acessible, in part because the books themselves must provide the necessary background knowledge. The Harry Potter series manages to be both highly irrelevant and highly accessible (the more so with its easy-to-relate-to child protagonists). In addition, of course, to being highly popular. Indeed, the popularity of the Harry Potter books stems, in part, from their combination of total irrelevance and total accessibility.

One place where irrelevance and inaccessibility converge is in books whose main characters a child cannot relate to. For growing numbers of children, this is yet another problem with The Great Gatsby. (It baffles me that, in our current Age of Relevance, this book has remained such a staple of high school literature).

In my earlier post on relevance, I wrote that "rather than making things relevant by keeping them close to home, why not make things relevant by taking children there?" As long as there's enough information about what "there" is and how to get there, even the most exotic book can be accessible--especially if there's a main character to whom the child can relate enough to imagine him/her as her proxy.

Another relevant sense of relevance is relevance to what you already know--a much broader category than relevance to your personal life. At kitchentablemath, Catherine Johnson cites Larry Squire on a new study about memory consolidation:

We learn and remember better when new material can be related to what we already know. Professional athletes can remember details of particular plays that occurred in a long match. Experienced poker players can reconstruct the card distribution and betting sequence that occurred in previous hands. This is possible because these individuals have a rich background of relevant experience and therefore can organize new material into meaningful and orderly patterns.
But this is an argument not for making everything relevant to students personal lives, but for expanding what they know: for teaching them a rich body of interconnected facts in various core subjects that they revisit year after year.

As a study reported on in yesterday's New York Times suggests, a core knowledge-focused curriculum (as in the curriculum designed by E.D. Hirsch Jr.’s Core Knowledge Foundation) has a second benefit. It leads to substantial gains in reading comprehension. Comprehension depends on accessibility; teaching content goes hand in hand with teaching reading.

Sunday, March 11, 2012

Letter from Huck Finn: Thinking about Inverting, Multiplying, Understanding, and Hanging

Out in Left Field proudly presents the fourth in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names (with the exception of Miss Katharine's) have been changed to protect privacy.





Thinking about Inverting, Multiplying, Understanding, and Hanging

Miss Katharine gave me an earful after my last letter. She was bent out of shape over my statement that I am of “average intelligence”. I tried to tell her that enough people believe I'm an idiot or of otherwise low intelligence because of my opinions on education to allow me to get away with saying that. She mumbled something I couldn't hear about who the real idiots are and then told me to "clarify matters". So here goes:

I may be above average in intelligence, but I am neither a math genius, nor am I brilliant. I make this distinction because I do not believe that traditional math serves only those people who are brilliant and/or are pre-destined to study mathematics. I believe it provided many people with sufficient preparation to take calculus in high school or college.

Some think that the old ways of teaching math were just rote memorization, with no understanding. To be blunt, I happen to think that’s a pile of crap. To be more refined, I happen to believe that procedural fluency leads to understanding; once you’re able to do certain procedures, it’s easier to understand why they work. Obviously, there is a lot of back and forth on this. Tina, my supervising teacher, believes that students shouldn’t just be told how to do procedures; they should understand the “why.” She is, however, a firm believer in having students do many problems to gain procedural fluency.

“Do you think you’d like to try teaching a class?” Tina asked me one day. I said I was ready, and we decided on a day that I would do it, which turned out to be the day that fractional division came up in the pre-algebra class. “That one’s a doozy,” she said. “Are you sure you want to take that on?”

“It’s my favorite topic,” I said.

She was intrigued with this and told me about people not being able to explain how the “invert and multiply” rule works. She is correct. In discussions, arguments, and brawls over “understanding”, the invert and multiply rule for dividing fractions is the poster child. For those who view understanding as paramount, the fact that many people are unable to explain why the rule works is considered as another piece of evidence that traditional methods have failed.

“The students already learned how to do this, in fifth grade,” she said. “But for pre-algebra I really like to make sure they understand why it works.” And I agree. Explaining the derivation of the invert and multiply rule is an appropriate topic for a pre-algebra class.

I suggested that since we had just finished the chapter on solving one step equations, we could build on that knowledge. Since a/b divided by c/d equals some number x, then we know that x times (c/d) = a/b. And since they know that c/d times d/c equals 1, we can then isolate x by multiplying each side by d/c, resulting in x = a/b x d/c.

She thought a moment and said she likes to use the “complex fraction” method. So if the problem is 7/8 divided by 3/32 , it’s written as a complex fraction: (7/8) / (3/32).

To simplify, we want to get rid of the 3/32 in the denominator. The easiest way is to multiply by the reciprocal, 32/3 which makes the denominator equal to 1. But if we multiply the denominator by 32/3 we have to do the same for the numerator. This sequence of steps ends up looking like: (7/8 x 32/3) / (3/32 x 32/2) = (7/8 x 32/3)/1 = 7/8 x 32/3.

And there it is; invert and multiply in all its glory. To ensure they really see it, she has students use this method in their homework problems rather than simply inverting and multiplying. This appealed to me because of my belief that procedure leads to understanding, and that the repetition of doing it “the long way” at the very least would make them grateful for just inverting and multiplying. Every time they did it the short way, they would remember with thankfulness not having to go through the rigmarole—which just happened to be the derivation.

Tina taught the lesson to the first period class, and then had me teach the same lesson for the third period class. Unlike the discovery-based algebra class, the pre-algebra classes were taught in a traditional manner. Nevertheless, when my time came to teach, I was a bit nervous, a bit rushed, and probably stood in front of the whiteboard too many times, blocking my own writing.

To make sure they connected with what they already knew, I started off with a question: “How many of you remember how to divide a fraction by a fraction?” Many hands went up. I called on Emilio. I waited. He said nothing.

“Well, Emilio, how do you do it?”

“Oh, I thought you were just asking if we remembered how to do it so I raised my hand.” He will either go into math or law, I remember thinking.

“Well, as long as we’re here, why don’t you tell us?”

“You flip the second fraction upside-down and you multiply,” he said.

“Correct,” I said. “But now we’re going to find out why it works.” I went through the explanation, asking questions as I did so to make sure they were following. I gave them some fraction division problems to do, instructing them to do it as I had done on the board.

In the end, the students knew what problems are solved by fractional division as well as the procedure. But I’d also say, that in the days that followed, once they were again allowed to use the short way (invert and multiply), the derivation did not matter much and probably only a few could reproduce it if asked. That doesn’t bother me.  I'd rather they know how to solve problems than be able to reproduce an explanation they don’t fully understand for a procedure they cannot perform.  A teacher friend of mine told me not to say that too loud or they’ll hang me. Well then, I guess when judgment day comes in the education world you’ll find old Huck hanging from a tree

Friday, March 9, 2012

Math problems of the week: 1900's algebra vs. Core-Plus Math

Introducing simultaneous equations.

I. From Wentworth's New School Algebra (published in 1898), "Simultaneous Simple Equations" chapter, pp. 174-175, p. 177, p. 178 [click to enlarge]:





II. From Core-Plus Mathematics Project, Course 1, "Linear Equations and Inequalities" chapter, pp. 227-228 [click to enlarge]:




III. Extra Credit:

Core Plus Course 1 (9th grade) contains about a dozen simultaneous equations problems, none involving more than two variables, and all with one of the variables isolated on the left side of the two equations as above. Thus, one variable is already "solved for," and the other one, appearing in two expressions that are equal to one another, can be solved for in a few easy steps.

Wentworth's New School Algebra contains many hundreds of simultaneous equations, many involving three variables, some with the variables in the denominators. Solving them involves multiple algebraic manipulations.

Relate this contrast to the amount of explanation given by the Core-Plus textbook (the entirety of which you see here) to the amount of explanation given by Wentworth above for just one of several algebraic methods.

Wednesday, March 7, 2012

Art in academics, revisited

In the last couple of years, more and more evidence has surfaced that the best way to learn things is by actively attempting to recall the material (a.k.a. "retrieval practice") rather than by passively rereading texts or studying notes and concept maps. Standard recall exercises include the much-maligned use of flashcards, the much maligned use of practice tests, and the much maligned taking of actual (closed-book) tests--the last of which turns out to be a learning experience unto itself. Other, less maligned options are closed-book, oral or written summaries or retellings. Or, in when it comes to foreign language learning, oral and written production, which are more effective than passive listening and reading--including, as my collaborator and I have found, for language-impaired autistic children learning English.

Yet another medium for retrieval practice occured to me a couple of months ago during my day in New York City with my daughter. It was a sunny, mild Boxing Day afternoon, with many people out and about. We walked northwestwards through Central Park and stopped for some time to watch a charcoal portraitist flesh out the lips and cheeks of a smiling and dimpled young woman. We enjoyed watching how specific lines and shadings would capture her likeness on page, and especially when the those lines took unexpected paths, highlighting micro-features we hadn't picked up on.

My daughter had her brand new sketch kit in hand, and was anxious to put her own charcoal pencils to paper. An hour later she had her chance. We'd continued northwestwards through the park to the Natural History Museum, and, after shuffling through the tremendously long lines, found ourselves sitting before dioramas of East Asian ruminants. And as I watched her sketching first the Asian deer, and then the water buffalo, I got out my own notebook and tried sketching them myself. Unlike her, I'm a terrible artist, and I struggled to render the correct proportions and angles of legs, flanks, horns, and muzzles.

As I attended to these anatomical details as never before, I realized for the first time that there is a place for art in academic subjects. Not collages, not dioramas, not abstract express-yourself art, but representational sketches. And not in math, English, or Spanish, but in subjects whose content is highly visual, like biology, chemistry, geology, geography, or engineering. Here, drawing the organelles in the cell, or the organs in the body, or the shape of a water molecule, or the tectonic plates of the Pacific, or a map of Africa, or the components of an internal combustion engine, are great ways to master material--especially if you close the book (or simply look away from the diorama and down at your sketchpad) and try drawing from memory.

Suddenly I remembered how much I learned about the foliage of Chestnut trees from the drawings I made in a 6th grade biology class in France. For all our "creative" and "colorful" science posters, and Lorax-inspired skits about endangered species, and dance performances about photosynthesis, I've never seen art deployed in this way here in U.S. classrooms. If only our education world cared as much about the content students remember as it does about how "creative" they are, what an effective exercise in visual retrieval practice this could be.

Monday, March 5, 2012

More on good uses of Constructivist ideals: art, relevance, heterogeneous groups, and child-centered ideas

In a series of ealier posts, culminating here, I argued that various problematic trends in K12 instruction still have potential virtues: in particular, trends pertaining to creativity, personal connections, verbalizing answers, self-esteem, multiple solutions, showing your work, group discussions, and hands-on, discovery learning. To this list I now add art in academics, relevance, heterogeneous grouping, and child-centered ideas. Once again, I use these terms with utmost caution, and intend them only in very specific ways that are far, far removed from the ways in which the edworld has bastardized them. I'll  begin with child-centered classrooms in which children "take ownership".

Many aspects of child-centered learning are misguided. Children aren't little scientists, mathematicians, and authors; they depend on direct, structured instruction of a body of basic skills and content; they don't know enough to know what they need to know. Given a choice, many would opt to avoid hard math problems, to produce a poster rather than an essay, or to watch the movie version of Charlotte's Web rather than reading the book.

But there are two things that even the younger children can be surprisingly wise about: what motivates them, and what helps them behave well. So why not involve children in the formulation of class rules and incentives? In fact, I've seen this work quite well, and, for all my criticisms of child-centered learning and children "taking ownership," I want to make sure I don't forget about this one highly promising way in which children can help direct their classrooms.

Saturday, March 3, 2012

Autism Diaries XXXIII: J as editor, provocateur, and philosopher

The practical jokester:

Take your sister's shoes out of the closet, pull the laces as tight as possible, and tie two really tight knots at the base of the laces, just above the top lace hole.

(Somehow this seems like a standard practical joke, but I've never seen it written up anywhere).

Keep your face straight afterwards and deny all wrong doing.

The editor:

I've never edited Wikipedia before, but noticed recently that some of the articles I'd assigned to my class had some embarrassing typos in them. No big deal, I thought; I'll just click [edit page] and edit them away. Instead a notice popped up saying that my i.p. address had been banned from editing Wikipedia.

Apparently J. has returned to his old editing habits--which ranged from such inoccuous edits as adding "some beachhouses have ceiling fans" to Wikipedia's "beachhouse" page, to more draconian ones like deleting Wikipedia's "delete key" page.

The provocateur:

He helped clean up the trash around his school on Martin Luther King Day, but couldn't help asking me beforehand what would happen if he yelled "Whites Only" during the event.

The introspector:

"Sometimes when I want to do something bad part of my brain says 'no' because it knows I will get in trouble."

Whereupon I praise him for the good voices in his head.

The metaphysician:

"If electricity prefers metal, why do people get a shock if they touch a wire?"

Unsure of the answer, I instead introduce the term anthropomorphize and ask "Does electricity have feelings?"

"No."

"Do your cells have feelings?"

"No."

"Do your brain cells have feelings?"

Without missing a beat, he gets my drift. "One brain cell does not have feelings. Many brain cells together have feelings."

Ah, he's grasped emergent properties. Time to teach him that term as well.

Thursday, March 1, 2012

Math problems of the week: introductory algebra in CPM vs. Singapore Math

A continuation of last week's problem of the week:

I. The 4th and 5th pages in College Preparatory Mathematics (CPM)'s Foundations for Algebra, Year 2 [click to enlarge]:



II. The 4th and 5th pages in 6th grade Singapore Math, Primary Mathematics 6A [click to enlarge]:



III. Extra Credit:
When I was an algebra student, we not only lacked CPM's social skills training activities and Tool Kit, but we also didn't take notes. What did we miss out on? How well would we function in today's 21st century jobs?