Monday, September 1, 2014

A lament for geometry proofs

One of many casualties of Reform Math--along with simultaneous equations involving more than two variables, rational expressions involving polynomial denominators with more than two terms and of degree > 3--are traditional geometry proofs like this one

Proofs are gone for many reasons. Some people find them soul-killing. Here, for example, is what Paul Lockhart's (in)famous Lament has to say about this particular proof:
In other words, the angles on both sides are the same. Well, duh! The configuration of two crossed lines is symmetrical for crissake. And as if this wasn’t bad enough, this patently obvious statement about lines and angles must then be “proved.”
Instead of a witty and enjoyable argument written by an actual human being, and conducted in one of the world’s many natural languages, we get this sullen, soulless, bureaucratic formletter of a proof. And what a mountain being made of a molehill! Do we really want to suggest that a straightforward observation like this requires such an extensive preamble? Be honest: did you actually even read it? Of course not. Who would want to?
(Everyone I know loved to construct these proofs--precisely because they forced you to abandon intuition and work things out logically, building up from Euclid's postulates, in the wonderfully exotic, unnatural language of math.)

Others, I suspect, find these proofs too difficult to teach--especially in the era of Reform Math, with incoming geometry students having spent so little time with problems that demand any kind of rigorous, multi-step logic in the unnatural language of math.

Finally, there are the Common Core Math Standards. The only mention of geometry proofs anywhere in the math standards in the Introduction to the High School Geometry section:
During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms.
Within this introduction, the only other mention of proof is in a subsection entitled Connections to Equations, where the type of proof under discussion appears to be algebraic rather than geometric:
Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof.
Nowhere in any of the specific goals for geometry do we find the word "proof."

And yet, surely the Common Core authors, with their obsession with higher-level thinking and deep understanding, would use the powers vested in them by Bill Gates, The National Governor's Association, and Education Industrial Complex, to revive this dying art?

Or perhaps we can come up with a proof as to why this isn't happening:

Geometry proofs don't lend themselves to standardized tests.
Proofs can't be captured in multiple choice format; only expensive experts could grade them.

The Common Core authors don't like goals that aren't easily measured by standardized tests.
Most of them are affiliated with the educational testing industry (see here).

Or, as Lockhart would put it, in terms that are so much less bureaucratic, and in one of the world's many natural language: "Well, duh!"

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?