Saturday, August 30, 2014

Charles Dickens on discovery learning

‘NOW, what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir!’
When people turn to Dickens for a critique of educational practices, what they cite is almost always this opening passage from Hard Times. And, like all of Dickens' parodies, its a great one, gaining momentum as it continues:
The scene was a plain, bare, monotonous vault of a school-room, and the speaker’s square forefinger emphasized his observations by underscoring every sentence with a line on the schoolmaster’s sleeve. The emphasis was helped by the speaker’s square wall of a forehead, which had his eyebrows for its base, while his eyes found commodious cellarage in two dark caves, overshadowed by the wall. The emphasis was helped by the speaker’s mouth, which was wide, thin, and hard set. The emphasis was helped by the speaker’s voice, which was inflexible, dry, and dictatorial. The emphasis was helped by the speaker’s hair, which bristled on the skirts of his bald head, a plantation of firs to keep the wind from its shining surface, all covered with knobs, like the crust of a plum pie, as if the head had scarcely warehouse-room for the hard facts stored inside. The speaker’s obstinate carriage, square coat, square legs, square shoulders, — nay, his very neckcloth, trained to take him by the throat with an unaccommodating grasp, like a stubborn fact, as it was, — all helped the emphasis.

‘In this life, we want nothing but Facts, sir; nothing but Facts!’
The speaker, and the schoolmaster, and the third grown person present, all backed a little, and swept with their eyes the inclined plane of little vessels then and there arranged in order, ready to have imperial gallons of facts poured into them until they were full to the brim.
The fact that there probably isn't a single classroom left here in America that involves bare walls and fact-filled teachers determined to pour gallons of those facts into passive, vessel-like students doesn't stop people from using these excerpts to evoke strawmen images of traditional classrooms.

Given actual practices today, a more relevant Dickensian excerpt is this one from Martin Chuzzlewit. It describes the educational practices of a Mr Pecksniff, who runs a studio for aspiring architects:
Mr Pecksniff's professional engagements, indeed, were almost, if not entirely, confined to the reception of pupils... His genius lay in ensnaring parents and guardians, and pocketing premiums. A young gentleman's premium being paid, and the young gentleman come to Mr Pecksniff's house, Mr Pecksniff borrowed his case of mathematical instruments (if silver-mounted or otherwise valuable); entreated him, from that moment, to consider himself one of the family; complimented him highly on his parents or guardians, as the case might be; and turned him loose in a spacious room on the two-pair front; where, in the company of certain drawing-boards, parallel rulers, very stiff-legged compasses, and two, or perhaps three, other young gentlemen, he improved himself, for three or five years, according to his articles, in making elevations of Salisbury Cathedral from every possible point of sight; and in constructing in the air a vast quantity of Castles, Houses of Parliament, and other Public Buildings. Perhaps in no place in the world were so many gorgeous edifices of this class erected as under Mr Pecksniff's auspices; and if but one-twentieth part of the churches which were built in that front room, with one or other of the Miss Pecksniffs at the altar in the act of marrying the architect, could only be made available by the parliamentary commissioners, no more churches would be wanted for at least five centuries. [Emphasis added.]
Yes, with his mockery of student-centered, discovery-based learning, Dickens was way ahead of his time. Of course, part of this satire involves not just this particular pedagogy, but also the easy financial rewards that accrue from it to the Power(s) that Be, in this case, Mr. Pecksniff. But I'm not sure whether this makes the passage any less relevant today--despite the fact that, somehow, it doesn't seem to be cited quite as often as the "facts" passage from Hard Times.

Thursday, August 28, 2014

Math problems of the week: Common Core-inspired algebra problems

From Free Common Core Math Standards practice, one of the first sites that come up when you google "Common Core math problems."

How clear are these problems  in their presentation, their terminology, and their solutions? You can submit answers and get your score on the original site.

Tuesday, August 26, 2014

More right-brained science: discovery as dreaming and serendipity

Archimedes in his bathtub; Newton under an apple tree. Our right-brained culture prefers see these brief moments of serendipitous inspiration as the engines of mathematical and scientific discovery. It's romantic to think so; and so much more preferable than to harp on the importance of long, hard years of knowledge acquisition and failed or inconclusive experiments.

Most recently perpetuating this myth is Rivka Galchen in her Bookends essay in last week's New York Times Book Review:

I wonder if we can really teach someone to be a biologist. I mean, sure, we can say, This is what a cell is, and here’s this thing called RNA, and here’s this thing called DNA, and here’s this technique called agarose gel electrophoresis that will separate your DNA and RNA fragments by size — but will teaching really produce the next Charles Darwin or Rachel Carson or Francis Crick? A real scientist follows her own visionary gleam. Penicillin was discovered when Alexander Fleming returned to his messy lab after a long vacation and made sense of a moldy petri dish most people would have thrown out as contaminated. The structure of the benzene ring came to the chemist Friedrich August Kekule after a daydream about a snake biting its own tail. You can’t teach that kind of dreaming. [Emphasis added]
Science as gleams and dreams. It takes only a milligram of skepticism to identify two highly questionable claims here, and only a moment of Internetting to see whether it's justified.

As this excerpt from Wikipedia explains, returning to the messy lab was only the first step in "discovering" penicillin, and the "sense" that Fleming made of his mold was rather preliminary:
Fleming recounted that the date of his discovery of penicillin was on the morning of Friday, September 28, 1928. It was a fortuitous accident: in his laboratory in the basement of St. Mary's Hospital in London (now part of Imperial College), Fleming noticed a Petri dish containing Staphylococcus plate culture he mistakenly left open, was contaminated by blue-green mould, which formed a visible growth. There was a halo of inhibited bacterial growth around the mould. Fleming concluded the mould released a substance that repressed the growth and caused lysing of the bacteria. He grew a pure culture and discovered it was a Penicillium mould, now known to be Penicillium notatum. Charles Thom, an American specialist working at the U.S. Department of Agriculture, was the acknowledged expert, and Fleming referred the matter to him. Fleming coined the term "penicillin" to describe the filtrate of a broth culture of the Penicillium mould. Even in these early stages, penicillin was found to be most effective against Gram-positive bacteria, and ineffective against Gram-negative organisms and fungi. He expressed initial optimism that penicillin would be a useful disinfectant, being highly potent with minimal toxicity compared to antiseptics of the day, and noted its laboratory value in the isolation of Bacillus influenzae (now Haemophilus influenzae). After further experiments, Fleming was convinced penicillin could not last long enough in the human body to kill pathogenic bacteria, and stopped studying it after 1931. He restarted clinical trials in 1934, and continued to try to get someone to purify it until 1940. [Emphasis added.]
And as this second Wikipedia article excerpt makes clear, the supposed daydream was only a small step in the process--and a highly questionable one at that, with a possible origin in parody. When Kekulé spoke of how he came up with his theory
He said that he had discovered the ring shape of the benzene molecule after having a reverie or day-dream of a snake seizing its own tail (this is a common symbol in many ancient cultures known as the Ouroboros or Endless knot). This vision, he said, came to him after years of studying the nature of carbon-carbon bonds. This was 7 years after he had solved the problem of how carbon atoms could bond to up to four other atoms at the same time. It is curious that a similar, humorous depiction of benzene had appeared in 1886 in the Berichte der Durstigen Chemischen Gesellschaft (Journal of the Thirsty Chemical Society), a parody of the Berichte der Deutschen Chemischen Gesellschaft, only the parody had monkeys seizing each other in a circle, rather than snakes as in Kekulé's anecdote. Some historians have suggested that the parody was a lampoon of the snake anecdote, possibly already well known through oral transmission even if it had not yet appeared in print. (Some others have speculated that Kekulé's story in 1890 was a re-parody of the monkey spoof, and was a mere invention rather than a recollection of an event in his life. [Emphasis added.]
And as the New York Times itself reported over 25 years ago:
[A]t least one historian now believes that Kekule never dreamed the snake dream, and that, in any case, the benzene ring had already been described by other chemists at the time Kekule claimed to have discovered it. That is the conclusion of Dr. John H. Wotiz, a professor of chemistry at Southern Illinois University who has made an exhaustive study of the documents and lore Kekule left to his scientific heirs. [Emphasis added.]
Of course, it takes less than a milligram of skepticism to realize that we shouldn't treat fiction writers like Galchen as our primary sources for the history of science. But when a highly articulate writer repeats, in such a highly visible forum as the New York Times, ideas that are already popular misperceptions, particularly ones that resonate so fully with today's education trend-setters, those misperceptions become all the more convincing, and the misguided practices they foster, all the more entrenched.

Sunday, August 24, 2014

Conversations on the Rifle Range 7: Winds and Currents, Formative Assessments, and the Eternal Gratitude of Dudes

Barry Garelick, who wrote various letters under the name Huck Finn and which were published here is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number seven:

All my classes were getting ready to take their first quiz later in the week. My second period class was the second-year Algebra 1 class. We were working on systems of linear equations covering the various ways of solving two equations with two unknowns.

I was preparing for my second period class by looking over the upcoming quiz and identifying the questions that most students would likely get wrong. As I reached the disturbing conclusion that this would be almost all the questions, Sally, the District person who talked to the math teachers about Common Core the day before school began, stuck her head in the door and asked if I had done any of the activities she had talked about that day. These were discovery-oriented projects that lead students to explore certain topics (specifically: probability, repeating decimals, and solving systems of equations) while allowing teachers to do formative assessments. Which means evaluating students by observing them “communicate and defend their thinking”.

What I wanted to say: “I’ve got other things to worry about than doing something that’s only going to tell me what I already know: that the majority of students shouldn’t have been placed in an algebra class in the first place. I’ve got quizzes coming up in all three of my classes that many students are likely to fail. And as far as providing students the forum to communicate and defend their thinking I prefer they communicate using math language that they are actually taught.”

What I did say: “Not yet.”

I explained that I didn’t know if I’d be able to do any of the activities except the systems of equations for my second year algebra 1 class. “We’re not covering probability, and the activity on repeating decimals is a problem.”

“Why is that a problem?” she asked.

“I have students who can’t divide.”

She nodded in a way that vaguely suggested sympathy. “Yes, that would be a problem,” she said and then vaguely brightened. “Why don’t you give it a try? Maybe have them use calculators or something.”

“Yeah, I might be able to do something like that,” I said with vague optimism, and she left. A few minutes later, my second period students started filing in.

The second period class was a mix of maturities; all had made it through the first half of Algebra 1 the previous year. Only a third of the class had sufficient mastery and maturity to handle the second half. There were about six students on the JV football team and all but one was struggling. Lately one of the boys, named Gray, was making an extra effort. With the upcoming quiz and football game, he was suddenly more cooperative. This was the first quiz and thus would constitute his provisional grade for the class. An average grade of D or below would mean he couldn’t play in the game—until it came back up to at least a C. His grandparents were coming from out of town to see him play, he told me.

My first meeting with Gray had been an auspicious one. I accidentally kicked his backpack as I circulated around the room. “Hey! Watch where you’re going!” he said.

“You need to learn some politeness, young man,” I said.

“Well you kicked my backpack.”

“It was an accident and I’m sorry; and you need to learn some manners.”

He wore the chip on his shoulder for a few more classes but I made a point of working with him to gain his trust. We were now working on word problems and he was struggling with wind and current problems. I asked the class if anyone could help set up the equations for the following problem: “An airplane flying against the wind can travel 3000 miles in 6 hours. Flying with the wind, it travels the same distance in 5 hours. Find the speed of the airplane in still air and the speed of the wind.”

“Who can set up the equations for this problem” I asked. Gray raised his hand. “I’ve got this,” he said. “When it’s going against the wind its 6r - w = 3000 and with the wind it’s 5r + w = 3000.”

I wrote the equations on the board and asked the class: “Do we agree with how we represent the wind speed?”

Gray suddenly shouted: “Dude! It’s right. You subtract the wind speed in the first case, and add it for the second equation. I know it’s right!”

“Dude! You’re wrong,” I said.

It isn’t my habit to be so blunt but it was the first time I had been called “Dude” and wanted to take advantage of it. I worked the problem at the board showing how (r+w) and (r-w) are the speeds of the plane and went over the distributive property. But working with Gray later, I saw that his understanding was hit or miss.

Quiz day came the next time we met. The strugglers in the class continued to struggle. Gray made a valiant effort but was not grasping key concepts. At one point he raised his hand and asked me, “How do you do this problem?” It was a boat and current problem.

“How do you represent the speed when it’s going with the current?” I asked. He wrote “r + c”.

“OK,” I said. “So now, how do you represent distance?” He made the same mistake and did not distribute the multiplication of time across “r + c”. I did the unthinkable. I wrote down one of the equations.

My justification was weak but it boiled down to this: based on the other work I saw on the quiz, I could tell he was not going to get a passing score. I did the same for others who were in similar danger. I’m fairly certain I’m not the only teacher in history to do this.

The next day was my off day. As was my custom, I came in to prepare for the next days’ lessons. Gray came in before the first period.

“What’d I get on the quiz?” he asked.

“You got a 45 percent.” He looked sullen.

“Is there anything you can do? I mean I know you can’t raise my score; I’m not asking for that. But I have to play in the game on Friday.”

“I’ll tell you what,” I said. “I won’t enter the grades until the weekend. So you’ll be OK for this week.” He looked at me as if I had found a cure for cancer.

“Dude!” he said.

“But you know as well as I do that this is temporary, don’t you?”

“Dude!” he said and left.

The other football dudes who failed the quiz were in a similar state of temporary eternal gratitude.

Friday, August 22, 2014

Math problems of the week: Common Core word problems from New York State

An ongoing series: for all that Common Core advocates claim about what is and isn't stipulated in the Common Core goals, what ultimately matters is how actual people actually implement them in actual classrooms.

Here, via the Associated Press and EngageNY, New York's Common Core curriculum, are some sample Common Core inspired word problems:

Grade 2 addition:
Solve using your place value chart and number disks, composing a 10 when necessary: 53 + 19

Grade 2 subtraction:
Craig checked out 28 books at the library. He read and returned some books. He still has 19 books checked out. How many books did Craig return? Draw a tape diagram or number bond to solve.

Grade 4 multiplication:
Represent the following expressions with disks, regrouping as necessary, writing a matching expression, and recording the partial products vertically: 3 x 24

Grade 4 word problem:
Cindy says she found a shortcut for doing multiplication problems. When she multiplies 3 × 24, she says, "3 × 4 is 12 ones, or 1 ten and 2 ones. Then there's just 2 tens left in 24, so add it up and you get 3 tens and 2 ones." Do you think Cindy's shortcut works? Explain your thinking in words and justify your response using a model or partial products.


Extra Credit:
Some Common Core-inspired curriculum writers believe they have found a strategy to reach Common Core math goals. Their strategy involves requiring students to solve problems using number disks, number bonds, tape diagrams, matching expressions, vertical recordings of partial products, and explanations of their thoughts about other people's strategies. Do you think their strategy works? Explain your thinking in words, and justify your response using a model or diagram.

Wednesday, August 20, 2014

Will damning studies reform the reformers?

Catherine Johnson recently posted on Kitchentablemath some excerpts of the first major study of the longitudinal effects of Reform Math. Published in the August 2014 issue of Economics of Education Review, this study examined the effects of the province-wide imposition of Reform Math in schools throughout Quebec in the early 2000's. Its main points, which should be circulated as far and as wide as possible, include the following:

1.  Before the reforms began:

the performance of students in the province of Quebec was comparable to that of students from the top performing countries in international assessments.
2. The reform program:
relied on a socio-constructivist teaching approach focused on problem-based and self-directed learning. This approach mainly moved teaching away from the traditional/academic approaches of memorization, repetitions and activity books, to a much more comprehensive approach focused on learning in a contextual setting in which children are expected to find answers for themselves.
More specifically, the teaching approach promoted by the Quebec reform is comparable to the reform-oriented teaching approach in the United States... supported by leading organizations such as the National Council of Teachers of Mathematics, the National Research Council, and the American Association for the Advancement of Science.
[The] approach was designed to enable students to "find answers to questions arising out of everyday experience, to develop a personal and social value system, and to adopt responsible and increasingly autonomous behaviors."
In the classroom, students were expected to be more actively involved in their own learning and take responsibility for it. Critical to this aspect was the need to relate their learning activities to their prior knowledge and transfer their newly acquired knowledge to new situations in their daily lives. "Instead of passively listening to teachers, students will take in active, hands-on learning. They will spend more time working on projects, doing research and solving problems based on their areas of interest and their concerns. They will more often take part in workshops or team learning to develop a broad range of competencies." (MELS, 1999).
3. Within Quebec province, reform was universal and uniform:
Whether private or public, English speaking or French speaking, all schools across the province were mandated to follow the reform according to the implementation schedule. This implies that all children in Quebec were treated according to same timeline, and that parents were not able to self-select their children into or out of the reform, except by moving out of the province which they did not.
4. Summary results:
We find strong evidence of negative effects of the reform on the development of students’ mathematical abilities. More specifically, using the changes-in-changes estimator, we show that the impact of the reform increases with exposure, and that it impacts negatively students at all points on the skills distribution.
So here's my question: how will American Reform Math advocates respond if/when presented with this article? Will they:

a. attribute the results in Quebec to "poor implementation"?
b. attribute the results in Quebec to cultural differences between Quebecois students and U.S. students?
c. say "That's interesting but there are plenty of studies that support Reform Math," and then quickly forget about this one?
d. transfer their newly acquired knowledge to new situations in their daily lives and reconsider their support for Reform Math?

Monday, August 18, 2014

Autism Diaries: ethics

There are ethicists who specialize in ethical conundrums. And there are ethicists who specialize in the ethical treatment of people with disabilities. But are there any ethicists who specialize in how ethical conundrums are handled by those with disabilities--in particular, those on the autistic spectrum?

After all, AS individuals are thought to lack the kinds of empathy and perspective taking skills that inform the ethical views of neurotypicals.

Take J, for example. Much of what guides his behavior isn't an internal moral compass, but the threat of punishment. Thou shalt not kill; thou shalt not steal; thou shalt not torment animals; thou shalt not bother people or waste things--it's all because thou would get punished, if not imprisoned, if not executed.

And yet, J sometimes shows glimmers of ethical awareness. Recently, for example, we were talking about zoos, and I explained to him that many people don't like zoos because of how they coop up wild animals. This idea clearly troubled him, because he immediately started rationalizing about why zoos still might serve a purpose:

"But people like to look at the animals."

"But they can watch animals on nature videos and see them in their natural habitats."

A while later:

"But maybe it's like waterfalls. You don't want to just watch a waterfall in a video. You also want to go to the waterfall."

It occurred to me, then, to present him with one of those "trolley car" dilemmas:

"What happens if a trolley is accelerating out of control, and about to hit a group of people, but you could flick a switch so that it only hits one person. Then would you flick a switch and cause that one person to die?"

"Maybe I would do that. But maybe not if it was someone I know."

"What about if you're standing on a bridge above the trolley, and there's a man on the bridge next to you, and you could stop the trolley from hitting the group of people by throwing the man over the bridge in front of the trolley. Would you do that?"

"I don't think I would do that because it would seem like murder."

Fairly typical arguments, I thought--from the standpoint of neurotypical ethics.

But then there was this exchange:

"Is that right," he asked "it's OK to do experiments on animals but not on people?"

"But what about new medicines?" I asked. "How can we know if they will work on people unless they are tested on people?"

His response was disconcertingly swift:

"Maybe people should do experiments on criminals in jail."

And I'm still not sure how to answer this in terms he will understand.

Saturday, August 16, 2014

Conversations on the Rifle Range 6: Grant’s Tomb and the Benefits of Boredom

Barry Garelick, who wrote various letters under the name Huck Finn and which were published here is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number six:

The block schedule alternated even and odd period classes every other day. My classes were all even period which meant I taught every other day. Between my fourth and sixth period classes came the lunch break during which, in nervous anticipation of my sixth period class, I would walk from one end of the high school complex to the other.

On the day I was teaching about powers and roots I thought about the upcoming lesson during my walk. An ongoing difficulty in teaching this topic is explaining what happens when you square a square root, cube a cube root and so on. How do I get students to make the connection that (Ö2)² equals 2? I recalled how I tried to get my daughter to understand this when she was taking algebra. She stared blankly at the problem. “When you square something what are you doing?” “Multiplying a number by itself,” she said.

“Right, so we can write it like this, right?” I wrote down Ö2 ·Ö2.

She still stared blankly.

“Come on, it’s like asking who’s buried in Grant’s Tomb,” I said.

Her response: “Who’s Grant?”

Despite this setback, I still decided to go into teaching after I retired.

What I learned from teaching my daughter years ago is that students need to know the connection between a square root and the square of a number. But as I also learned from teaching my fourth period class earlier in the day, the definition of “square root” should not be the one in the Holt Algebra textbook: "A number multiplied by itself to form a product is the square root of that product." The difficulty is not only the convoluted wording, but the book introduces and defines perfect squares after the definition of square roots. Definitions should be introduced in an order that makes it possible for students to use “prior knowledge”. Not that they hadn’t learned about these things before, but still. Matt, the ocarina player, relied on a purely procedural approach: “Oh, when you square a square root, the radical sign disappears.”

While I have no problem with procedural understanding, I thought I could do better with sixth period. I would lead with what perfect squares are, and then give my own definition of square roots: "A square root of a given number is another number whose square is the given number." When I returned from my lunchtime walk, I started writing down my order of attack.

As was her custom, Elisa came into the classroom ten minutes before it started and asked for a piece of computer paper. I gave her a piece which she took to her seat and quietly drew one of her many pictures of dogs and wolves.

I jotted down a few more notes until the class started to file in. Patrick sauntered up to me and complained about the homework assignment sheet that I handed out at the beginning of the semester. “My mother says it’s hard to figure out,” he told me. “Do you know what assignment is due today?” I asked. He pointed to it. “Good; not so hard to figure out. Now show me your homework so I can check it in.”

“I didn’t do it,” he said and snickered.

Patrick sat next to Elisa and liked spending his time making snide remarks about my attempts to teach the class. Although Elisa giggled at his remarks she told me once after class that Patrick was a wise ass. She was probably one out of five or six students who paid attention in class. When I did my teaching, I taught to them.

The class was boisterous as usual but after homework check-in and other rituals, they were sufficiently less noisy so that I could begin the lesson. I had defined perfect squares, and square roots, gave some examples and then asked: “Who can tell me what this is equal to?” I wrote (Ö2)².

A few guesses, all of them wrong.

“Let me try this,” I said and wrote  (Ö16)². “What is the square root of 16?” I asked.

Someone said “Four”, and I wrote (4)² and asked Patrick what that equals.

“Sixteen”, he said and whispered something to Elisa who giggled. “Correct,” I said and wrote (Ö16)² = (4)² = 16 on the board. I did this a few more times with other numbers: (Ö9)²,(Ö4)², (Ö36)², (Ö25)², hoping that they would see the emerging pattern. When I started hearing “Ohhh, I get it” I put up the original problem again:. (Ö2)². 

Elisa gave the answer: “Two,” she said.

“Absolutely right,” I said. I went into cubes, cube roots, and higher powers but the amount of paper wads being thrown (an ongoing problem in that class) was increasing. Roots and powers were no match for their unrest so I set them to work on their homework. I allowed them to get in groups of their choosing. Some worked on their homework, many did not. Elisa was one of the students who worked on her homework; she sat by herself. She summoned me over to her desk. She was stuck on finding the cube root of eight.

“You know how exponents work, right?” I asked. She said “I think so.”

“OK, if I write 33, what is that?”

“It’s three times three times three.”

“Good. Which is what?”

She thought a minute. “Twenty seven!”

“Right! So now, to find the cube root of 8 we work backwards: we want to know what number cubed equals 8.”

“What do you mean ‘work backwards’ ?” she asked. I explained how we're reversing what we do with exponents and suddenly the light went on. I asked her to try some numbers to see if we could find the cube root of eight. "Obviously, it's not 1, so try the next one up." She multiplied two by itself three times and got eight. We tried another: The fourth root of eighty-one. This took a little trial and error but she got it. “Oh. Three! Yeah, I get it now,” she said.

After a moment she asked: "Did mathematicians invent this stuff because they're really bored?"

I burst out laughing, but gave her a serious answer. I said these things were invented to solve certain types of problems and sometimes out of curiosity.

“I think they were just bored," she said.

“And I think you probably like math more than you realize,” I said, knowing I had nothing whatsoever to base this on other than a wishful hunch.

“I think you’re crazy,” she said, and I moved on to others on the rifle range.

Thursday, August 14, 2014

Math problems of the week: Common Core inspired 4th grade math

A continuation of last week's problem of the week. For all that Common Core advocates claim about what is and isn't stipulated in the Common Core goals, what ultimately matters is how actual people actually implement them in actual classrooms.

From a 4th grade problem set included in a list of "common core math word problems" on the LakeShore Central School District website:

Barry Garelick's recent article, "A Common-Sense Approach to the Common Core Math Standards, elaborates the "explain how the algorithm works" standard and discusses how it might best be implemented.

Tuesday, August 12, 2014

Why do journalists stink at math education news?

I spent the last three weeks out of the country--in what was my first big break from my computer in a very long time.  One result: I found myself watching helplessly (from my iPhone) as my blog fired off prewritten posts automatically while an article that I would have loved to have blogged about immediately sent shockwaves through the blogosphere.  That article, of course, was Elizabeth Green's New York Times Magazine piece Why Do Americans Stink at Math?

Many people have critiqued this article, most recently Barry Garelick. I'd like simply to contribute a list of errata, by which I mean egregious journalist errors of the sort that define all too much education journalism.

1. On Brazilian street vendors

Green writes:

Studies of children in Brazil, who helped support their families by roaming the streets selling roasted peanuts and coconuts, showed that the children routinely solved complex problems in their heads to calculate a bill or make change. When cognitive scientists presented the children with the very same problem, however, this time with pen and paper, they stumbled. A 12-year-old boy who accurately computed the price of four coconuts at 35 cruzeiros each was later given the problem on paper. Incorrectly using the multiplication method he was taught in school, he came up with the wrong answer. Similarly, when Scribner gave her dairy workers tests using the language of math class, their scores averaged around 64 percent. The cognitive-science research suggested a startling cause of Americans’ innumeracy: school.
No: the cognitive science research suggests that contextual skills of the Brazilian street vendor sort only get you so far, and that decontextualized learning, of the traditional classroom sort, are essential for abstract math. (See, for example, Anderson et al's Situated Learning and Education.)

2. On Japanese vs. U.S. classrooms:

In discussing Reform Math, Green writes that "..countries like Japan have implemented a similar approach with great success." Key elements of Reform Math, of course, are child-centered discovery and group work

But claims like this were debunked years ago by Alan Siegel's analysis of Stigler et al's highly influential TIMSS Videotape Classroom Study. Reviewing the raw data, Siegel reviews and analyzes the excerpts that were the basis for this study, explaining what they actually do show, as opposed to what they were claimed to show:

  • The excerpts do not support the suggestion that in Japan, “[The] problem . . . comes first  [and] . . . the student has . . . to invent his or her own solutions.”
  • The evidence does suggest that in Japan, “Students rarely work in small groups to solve problems until they have worked first by themselves.”
  • Similarly, the evidence gives little weight to the notion that “Japanese teachers, in certain respects, come closer to implementing the spirit of current ideas advanced by U.S. reformers than do U.S. teachers.” 
  • The evidence does confirm that, “In other respects, Japanese lessons do not follow such reform guidelines.  They include more lecturing and demonstration than even the more traditional U.S. lessons . . ..”
Green also writes:
By 1995, when American researchers videotaped eighth-grade classrooms in the United States and Japan, Japanese schools had overwhelmingly traded the old “I, We, You” script for “You, Y’all, We.
Where "you y'all we" entails:
a single “problem of the day,” designed to let students struggle toward it — first on their own (You), then in peer groups (Y’all) and finally as a whole class (We).
But what the TIMMS video tapes of Japanese classrooms actually show, per Siegel's paper, are not you y'all we, but I you (optional y'all) I/we. The teachers go over what the class did yesterday spends a fair amount of time setting up a new problem (I); they then have students work on their own (you), with peer groups just one of several options for weaker students and not required optional (y'all); and back and forth with whole class, carefully directed by teacher every step of the way ("We," with a heavy dose of "I").

As Siegel puts it:
The excerpts show Japanese classes featuring a finely timed series of mini-lessons that alternate between grappling-motivated instruction on how to apply solution methods, and well chosen challenge exercises designed to instill a deep understanding of the solution methods just reviewed. No other interpretation is possible.
3. On American math classes

Green characterizes "Most American math classes" as  "focusing only on procedures." She adds that American math classes don't "look much different than they did before the reforms" and that "Textbooks, too, barely changed, despite publishers’ claims to the contrary."

One has only to walk into a math class at a model elementary or middle school, or look at an Investigations or Everyday Math (etc.) textbook, to see how wrong these claims are.

Indeed, a classroom lesson the Green describes as an exemplary role model is, unfortunately, all too common:
One day, a student made a “conjecture” that reflected a common misconception among children. The fraction 5 / 6, the student argued, goes on the same place on the number line as 5 / 12. For the rest of the class period, the student listened as a lineup of peers detailed all the reasons the two numbers couldn’t possibly be equivalent, even though they had the same numerator.
While her eager claims about how effective this "innovative" classroom is are supported by the vaguest and most anecdotal of "data"--if you can even call it that:
Over the years, observers who have studied Lampert’s classroom have found that students learn an unusual amount of math. Rather than forgetting algorithms, they retain and even understand them. One boy who began fifth grade declaring math to be his worst subject ended it able to solve multiplication, long division and fraction problems, not to mention simple multivariable equations. It’s hard to look at Lampert’s results without concluding that with the help of a great teacher, even Americans can become the so-called math people we don’t think we are.
4. On the cure
To cure our innumeracy, we will have to accept that the traditional approach we take to teaching math — the one that can be mind-numbing, but also comfortingly familiar — does not work. We will have to come to see math not as a list of rules to be memorized but as a way of looking at the world that really makes sense.
A total non sequitur, followed by the obligatory strawman caricature of traditional math.

A better way to conclude the article would have been with a quote hidden deeply within it--one by Magdalene Lampert, whose "innovative" classroom was described above:
“In the hands of unprepared teachers,” Lampert says, “alternative algorithms are worse than just teaching them standard algorithms."

Sunday, August 10, 2014

The miraculous 9 percent--and the autism "miracle cures" they inspire

In an article in last week's New York Times Magazine entitled The Kids who Beat Autism, Ruth Padawer reports on research suggesting that a significant subset of kids diagnosed with autism eventually no longer meet the diagnostic criteria.

To those who buy into the various autism miracle cure stories--with cures ranging from psychotherapeutic (Bettelheim's fortress rescue, Greenspan's Floor Time) to behaviorist (Applied Behavioral Analysis; "rapid prompting") to culinary (gluten-free diets) to chemical (chelation) to auditory (auditory integration therapy) to tactile (hugging therapy) to mammalian (riding on horses; swimming with dolphins)--this may be no surprise.

But I once asked a clinician at our local autism center whether they'd ever seen a child lose his or her diagnosis, and she replied that, out of the 98 children they'd seen thus far, only two had possibly outgrown the diagnosis, and, in one of the cases, it wasn't clear whether the child had truly met the criteria in the first place.

The article cites two studies cited, one of which was a retrospective study that examined the early medical files of 34 kids who don't currently meet the criteria for autism to verify that they did, in fact, once do so. But retrospective studies aren't random, and rely on past records that can no longer be independently verified. More compelling is the second study, a prospective one

that tracked 85 children from their autism diagnosis (at age 2) for nearly two decades and found that about 9 percent of them no longer met the criteria for the disorder.
This is the extent of the article's interesting revelations. What causes this recovery--or exactly which children will recover--remains unknown. What is known is that the recovered kids, along with those who stay autistic but make the most overall progress, tend to have higher IQs and to engage in more socially imitative behaviors to begin with. But this is neither news nor surprising.

I've long suspected that, to the extent that there were kids who fully recovered from autism, these were kids who would probably have recovered regardless of what specific therapy they underwent. Just as autism involves a neuro-developmental program that unfolds in the course of brain development, so, too, may be the case with autism recovery. In certain kids, it may be preprogrammed, at least to some extent, in their brain development.

Of course, rigorous therapies may also play key role. But it's hard to know just how much we should credit the particular therapy that a recovered child happened to be undergoing while he or she was simultaneously undergoing recovery.

Indeed, the existence of this mysterious 9 percent may explain the persistence of autism miracle cures. Every once in a while, someone in this group happens to have parents who happen upon one of the latest untested therapies, and, as a result, whatever this therapy happens to be--whether it's oxytocin inhalers or eye saccade training or barefoot heel walking or interactive labyrinths--will inspire screaming headlines, best-selling memoirs, and tons of airtime on daytime talk shows.

Friday, August 8, 2014

Math problems of the week: Common Core-inspired Math problems

For all that Common Core advocates claim about what is and isn't stipulated in the Common Core goals, what ultimately matters is how actual people actually implement them in actual classrooms.

Following the money, one place to look at what's on the horizon is the educational software arm of the Educational Industrial Complex. Here's what awaits us from one of the most popular educational software companies, Kidspiration. From one of its "sample lessons":

One has to wonder: how on earth did seven to eleven-year-olds solve daunting problems like this one before the age of computers in classrooms???

Wednesday, August 6, 2014

Conversations on the Rifle Range 5: Division by Zero, a Burning Question, and the Ocarina Player

Barry Garelick, who wrote various letters under the name Huck Finn and which were published here is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number five:

Discipline was getting to be a big problem, particularly for my sixth period class. So I confided in my across-the-hall neighbor Mrs. Rodriguez that I was having some problems with discipline. She had taught for many years and had a no-nonsense attitude. She advised me to threaten to hold the class in for a minute after the bell if they continued with whatever bad behavior was going on. “Tell them that if they don’t understand what you’re saying you can arrange to have the principal come in and explain your policy to them. That ought to do it,” she said.

I had occasion to try this in my fourth period class—first year Algebra 1. This class was generally well-behaved except for a few students, but certain students could be very rude. While writing something on the board, I heard someone pop their gum. There is a “no gum” policy in the school rules, and my teacher, in her classroom rules, has added to that by placing a ban on “gum popping” as well.

Without turning fully around I said “There is no gum popping in this class. In fact there is no gum in this class, so I advise you to either spit out the gum, or just stop the gum popping.” No one spit out their gum, so I went on with my lesson.

I heard the pop again. "OK," I said. "First of all there's to be no gum in class. You have elected not to spit it out. So if I hear you pop the gum again, the whole class will stay in one minute after the bell. And if you need the principal to come in and explain this rule I can arrange that. Is that sufficiently clear?"

A boy named Kenny raised his hand. He had a tendency to speak in a very precise fashion. “I have a question,” he said.


“How do you pronounce your name?"

Not quite what I was expecting, but I answered his question and, hearing no further questions or gum popping, decided to continue on with the lesson.

Kenny was a good student—one of the few students on the Junior Varsity football team who was doing well in the fourth period class. I gauged what I could or couldn’t teach in fourth period by what Kenny could understand. If he had difficulty, the rest of the class would, and I needed a different approach. Of my two first year Algebra 1 classes, the fourth period class was the only class where I could go into more detail and take mathematical “side trips.”. And if the fourth period class didn’t understand something, I knew I had to slow it down and teach it differently in sixth.

The day of the gum popping incident I showed that division by zero is impossible, among other topics related to multiplication and division. When I first learned about division by zero in my Algebra 1 class, fifty years ago, I found it remarkable—not because it is impossible, but because I wondered how I had gone so long without ever noticing this. My teacher, Mr. Dombey, then showed us that there was no number by which you could divide a non-zero number and get zero. He wrote 1/x on the board and asked us what the answer was if x was 1/2, then 1/4, then 1/10, then 1/100, and so on. Then he went the opposite direction; he made x larger and larger, showing the resulting quotient growing increasingly smaller—approaching zero. As x approaches zero, he explained, the quotient gets infinitely large. And as x gets infinitely large, the quotient approaches zero. The entire discussion took about ten minutes. There was no “Let’s explore and get into groups and written an essay about it.” And despite the lack of collaboration, group work and student-centered activities, the discussion had opened my eyes to aspects of math I hadn’t been aware of. One year later I decided I would major in math.

I used the same approach that Mr. Dombey had for my fourth period class. I showed why division by zero is impossible (something that critics of traditional math say is not done in traditional math) and then I thought I’d see how far I got with additional discussion. As Mr. Dombey had done, I asked the class what is 1 divided by 1/2, then 1/4, 1/10, 1/1000. Kenny and perhaps two other students supplied the answers which told me that the class’ overall facility with fractional division was weak. But for the most part, the class got the point. I knew I would not get that far with my sixth period class. I announced there would be a quiz next week. The class suddenly got nervous, especially the football players. They needed to maintain a C average in all classes or they could be suspended from the team, so the first quiz of the semester posed an immediate threat.

“Get started on your homework,” I said and started circulating around, answering questions. A boy named Matt who sat up front stared listlessly at his paper.

“Are you going to do your work?”

“Yeah, I guess,” he said. He was a nice boy but didn’t like to do his work in class. He told me on the first day that this was the second time he was taking the course, adding that he wasn’t very bright.

“Is that an ocarina?” I asked, pointing to a flute/whistle-like instrument on his desk.

“You know what an ocarina is?” he asked, his amazement no less than mine had been when I was learning about division by zero. “No one knows what an ocarina is!” he said.

“I’ve seen them before.”

“Can I play it for you?”

“I want you to do at least four problems in this set and show me your work and then I’ll decide,” I said.

A few minutes later he called me over and showed me that he had done some problems. I let him play. He played Greensleeves. I savored the moment, knowing that my sixth period class coming up later was not going to offer anything this nice.

It’s hard to know if anyone’s curiosity was stirred that day in that class by anything other than the question of how to pronounce my name. But then again, Mr. Dombey had no idea that his little discussion stirred so much curiosity on my part either.

Monday, August 4, 2014

Don't just assist; instruct, III

I've been working on my part of a forthcoming book on linguistic technologies for people with language impairments, and have just finished writing up a section on assistive technologies. There a lot of new devices in the wings that raise various new concerns, and I thought I'd share some of what I've written here:

Currently, the most typical assistive communication devices for classroom communication involve buttons designating certain words and phrases which the user pushes to communicate certain basic thoughts—typically basic desires:

More ideal would be to go beyond preprogrammed buttons without making the device so open-ended as to offer little or no assistance--as would be the case if you replaced an array of word buttons with an array of letters, as in a standard alphanumeric keyboard. Furthermore, to be truly assistive the device would have to somehow know what the child wants to say before he or she starts saying it.

There is, in fact, one type of software that actually approaches this capability. Known as word cuing software, these are programs that one can plug into a word processing program, either on a desktop or a tablet, and that anticipate the one’s likely next word from the words and letters that one has already typed in. These programs go far beyond the typical autocomplete, which waits until the word is almost complete and determined, and, accordingly, only gives you one choice.

Here’s are two screenshots from one popular program--Read&Write Gold:

In these programs, the word choices in the pop-up window are selected based on what is syntactically and morphologically appropriate given what the user has typed so far, and also what is likely given general statistics about word occurrence as well as the individual statistics that it collects about particular users and their word selection habits.

Some of these programs have additional features: they watch for common spelling errors, providing a pop-up list of alternative options, or they offer examples of how a given word might be used. Some allow users to select particular topics--topics as specific as Polish architecture--and then invokes a dictionary for that topic, and presents word choices (nouns, verbs, adjectives, etc.) specific to this topic. And, to help students who have no idea what to write about on a particular topic, there are a word banks (with words sized by frequency) to help you get started:

For language-impaired students, word prediction software is a powerful tool. In particular, it can help students who struggle with grammar produce grammatical sentences. For example, a student who types “I want” will see the word “to” as one of his choices, which steers him away from errors like “I want go.” A student who types a plural subject like “my friends” will only see verbs in the plural form (for example, “are” rather than “is”). A student who types the article “an” will only see noun choices that begin with vowels—as we see in the first image from ReadWrite Gold. Or, as we see in the second image, a student who types in “an example” will see, as his or her first choice, the preposition that most often follows the word example “example” (“of”).

These programs are powerful, but they are also problematic. In general, the more efficacious the assistive technology, the more it potentially reduces the urgency of teaching the skills that are being assisted. It is essential that assistive technology be only treated as such—namely, as assistive—and not used as a reason to adjust teaching priorities and eschew necessary remediation.

When it comes to autistic children in particular, another concern is the extent to which technology deprives them of the face-to-face social interactions on which they may be especially dependent for their social development. Too often, whether the students are autistic or not, one finds classrooms and other settings in which students are mostly looking at and interacting with screens rather than with one another.

The proliferation of technology in the classroom heightens both of these concerns. So do the unprecedented pressures that today’s teachers are under. Most language impaired students are included in regular classes with same-aged peers, and their teachers are increasingly under pressure to teach to the new Common Core State Standards and tests. These standards set high expectations for reading and writing and take a one-size-fits all approach to students at a given grade level. In light of this, fewer and fewer teachers, even special education teachers, feel that they have time to remediate basic skills--the more so when assistive devices make remediation seem increasingly less urgent.

Remediation and accommodation shouldn't be at odds ideally, they go hand in hand. The ultimate goal, after all, is to optimize the learning environment such that students reach their potential, and, ultimately, are liberated from assistive technology to the largest extent possible.

Saturday, August 2, 2014

Processing information vs. Cutting and Pasting , II

If I could do college over again, I’d major in computer science. I realized this during senior year when I took a class in artificial intelligence and learned LISP, the first programming language that really resonated with me (how I reveled in recursion), and then (in my final semester) a computation theory class in which I wrote rules for symbolic languages via formal grammars, and “virtual” programs (i.e., on actual paper rather than as software) via Turing Machines:

The funny thing was that I took these classes for enjoyment, while the computer science majors (pretty much everyone else in the room) took them because they had to. Most of them preferred other aspects of computer science that I didn’t find so inspiring: data bases, systems, and other real-world applications.

But if I were doing college today, I almost certainly wouldn’t major in computer science. That’s because real-world applications have edged out theory, and even basic programming techniques like recursion are, or so I’ve heard, becoming dying arts. I first wrote about this phenomenon six (!) year ago. But I had some new revelations about bad things have gotten more recently when I started looking for someone to help me port my GrammarTrainer software into a tablet-based platform. Surely the place to find someone was a computer science or digital media program, where students learn coding systems at the frontiers of today’s computational technologies.

But time after time these young people let me down. It turned out that they knew how to write code for iPads, and they knew how to use all sorts of high level programming packages, but they couldn't figure out how to get these packages to interface with my not-so-high-level code. Even though I set things up so that they didn’t have to actually deal with this code directly, but simply take as inputs to their code a rather small set of possible outputs from mine, they seemed to lack the necessary flexibility in programming and rigor in logic to write code that, however high level it needed to be (i.e., XCode), simply wasn’t that complex from a logical standpoint. Basic Boolean formations and basic decoding techniques seemed to elude even a bright computer science major from a top engineering school. What are kids learning these days, I kept wondering.

Sometimes one finds help in unexpected places. The person who has turned out to be up to the GrammarTrainer conversion tasks—very much so—is someone from my generation; someone who got in-depth training in actual programming, including all that fun theoretical stuff. Many of you know her as FedUpMom. She’s great! She can do anything!

Kids today may be learning about fancy graphical interfaces and how to plug one off-the-shelf program into another. But, in the end, no amount of cutting and pasting from one high-level package to another can match the flexibility of a Turing Machine. As I learned almost 30 years ago, no set of logical operations is too elaborate for those primitive Turing Machines—or for those students who spend time playing around with them.