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.