Thursday, January 31, 2013

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

More on skirting actual calculation (and higher-level problem solving) in Reform Math:
statistics in TERC/Investigations vs. Singapore Math

I. The closest the TERC/Investigations Math Student Activity Workbook gets to the topic of averages [click to enlarge]:

II. The third problem set involving averages in the Singapore Math Primary Mathematics 5B Workbook [click to enlarge]:
III. Extra Credit:

Comparing the relative difficulty of the two problem sets, speculate on why 5th grade Investigations avoids even simple problems involving averages.

Tuesday, January 29, 2013

"Number sense" with minimal number work

In my recent post on vocabulary, I wondered whether what holds for vocabulary words also holds for math facts:

not just that knowing them fosters higher test scores and broader intelligence, but also that they are best learned not as lists of "math facts," but in the course of doing actual, systematically structured math.  
In an earlier post on the abysmal mastery of math facts seen in children whose only exposure to math is Reform Math, I estimate that Reform Math children are getting less than one tenth the embedded practice with math facts as their traditional and overseas peers are.

It's interesting to see how often Reform Math problem sets skirt the issue of actual calculation in their attempts to nurture number sense. Here below, for example, we have, from 5th grade TERC / Investigations, (1) a problem set that substitutes multiplication for estimation (where Singapore Math would have students do both):

(2) one of many problem sets in a chapter on decimals that simply ask students to put specific decimals in order of size (while ordering of fractions may require actual calculation, ordering of decimals does not).

(3) a multiplication/division "skill check" that almost looks like it's asking for calculations, but carefully STOPS you before you begin:

I'm reminded of those foreign language software programs that promise fluency in a language without making you practice grammar rules. As with language, people would love to think there's a silver bullet for arithmetic mastery and algebra preparedness that avoids the productive drill and practice that so many elemenary school teachers assume is as tedious for their students as it is for them. But what I've seen even with basic number sense suggests that a fair amount of productive drill ("embedded learning" in the best sense of the term) is, for most students, absolutely necessary.

Sunday, January 27, 2013

Yet another false dichotomy: facts, facts, facts, vs. higher-level thinking:

I was astounded to read in this week's Education Week an article on a study that points to the benefits of memorizing math facts:

Students who performed well on the math section of the PSAT showed more activity in brain areas linked to memory of math facts. Those with lower math PSAT scores had less brain activity in those areas and more in areas associated with processing number quantities.

The findings suggest that the high-achieving students knew the answers by memory, while lower-performing students were calculating even low-level problems.
Less astounding, but no less compelling, is an article in the winter issue of the City Journal by E.D. Hirsch, Jr, on the benefits of factual knowledge and vocabulary size (thanks to one of my readers for alerting me to this article). Vocabulary size, writes Hirsch, predicts intellectual aptitude and future earnings:
Vocabulary size is a convenient proxy for a whole range of educational attainments and abilities—not just skill in reading, writing, listening, and speaking but also general knowledge of science, history, and the arts.
Vocabulary doesn’t just help children do well on verbal exams. Studies have solidly established the correlation between vocabulary and real-world ability.
There’s no better index to accumulated knowledge and general competence than the size of a person’s vocabulary. Simply put: knowing more words makes you smarter. And between 1962 and the present, a big segment of the American population began knowing fewer words, getting less smart, and becoming demonstrably less able to earn a high income.
Why is vocabulary so important? As Hirsch explains:
The space where we solve our problems is called “working memory.” For everyone, even geniuses, it’s a small space that can hold only a few items in suspension for only a few seconds. If one doesn’t make the right connections within that space, one has to start over again. Hence, one method for coping and problem solving is to reduce the number of items that one has to make sense of at any moment. The psychologist George A. Miller called that process “chunking.”

...As long as you hold in your long-term memory a lot of associations with that name [a particular vocabulary item], you don’t need to dredge them up and try to cram them into your working memory. The name serves as a brief proxy for whatever aspects will turn out to be needed to cope with your problem. The more readily available such proxies are for you, the better you will be at dealing with various problems. Extend this example to whole spheres of knowledge and experience, and you’ll realize that a large vocabulary is a powerful coping device that enhances one’s general cognitive ability.
But our nation's vocabularyis nowhere near what it once was. While scores on vocabulary tests rose steadily for 50 years up until 1967, ever since then, thanks to the thorough penetration in our schools of a so-called "progressive" educational philosophy that proclaims "the unimportance of factual knowledge and book learning," we've had a “historically unprecedented" decline in verbal SAT scores. And, since the 1980s, an persistent nadir in those scores.

Some have tried to blame this on the widening pool of SAT test-takers, which include more and more students from disadvantaged backgrounds. But the same decline has occured on the Iowa Test of Educational Development, a test, Hirsch  notes, that is "given to all Iowa high school students, who were 98 percent white and mostly middle-class in attitude."

Correlated with the anti-book learning philosophy of so-called "progressivism," and with the decline in our nation's verbal scores, is the "dumbing-down of American schoolbooks." Cornell sociologist Donald Hayes, Hirsch writes, has found that:
publishers, under the influence of progressive educational theories, had begun to use simplified language and smaller vocabularies. Hayes demonstrated that the dilution of knowledge and vocabulary, rather than poverty, explained most of the test-score drop.
The remedy involves a thorough reversal of current trends. As Hirsch notes:
The fastest way to gain a large vocabulary through schooling is to follow a systematic curriculum that presents new words in familiar contexts, thereby enabling the student to make correct meaning-guesses unconsciously.
Vocabulary tests, one of the few remaining relics from the past, simply don't cut it:
Spending large amounts of school time on individual word study is an inefficient and insufficient route to a bigger vocabulary. There are just too many words to be learned by 12th grade—between 25,000 and 60,000.
A large vocabulary results not from memorizing word lists but from acquiring knowledge about the social and natural worlds.
Again, we need a complete reversal of current trends. We need to move away from theme-based materials and assignments and generic, de-contextualized "skills," "processes," and "higher-level thinking." We must return to systematic, fact-rich, content-based instruction.

Returning to the first article, I suspect that some of what E.D. Hirsch says about vocabulary applies to math facts as well: not just that knowing them fosters higher test scores and broader intelligence, but also that they are best learned not as lists of "math facts," but in the course of doing actual, systematically structured math. Which, like systematic, fact-rich, content-based instruction, would be another complete reversal of current trends.

Wednesday, January 23, 2013

Autism Diaries XLIII: the fan of fans

J especially enjoyed this year's New Year's parties, making himself the life of the party by interviewing everyone present about their ceiling fans. If he dicovered someone whose fans he hadn't yet immemorialized via digital camera, he'd try to set up a visit. If the person happened to live a house or two down from the party, he'd request an immediate film date. So during a lull at one party J and I, along with Jim, headed over to Jim's house to get some ceiling fan footage. Parties for J, in otherwords, represent present and future opportunities to augment an already vast video library of fans spinning at slow, medium, fast, and reverse.

But what if it turns out that one of your prospective hosts has high ceilings, no chains on their fans, and has misplaced the remote controls?

A week into the new year I came home to a very polite answering machine message from a lady at Home Depot for Mr. ___. Hmm, what is this "spare part" my husband has "inquired about," I wondered.

It didn't take too long to figure out what was really going on, especially later on that day when my friend K forwarded me a message she'd received from J.

(In case you're wondering why J has K's email address, it's because one day I accidently left my gmail open and unattended long enough for J to set up automatic forwarding to his account of all my incoming messages. I discovered this only after he accidentally replied to one of those messages: one from Daddy about taking him to the Eastern State Penitentiary on Halloween. His response when I called him on it: "Don't worry--I only looked at some of your messages.")

Here's J's message to K, with appropriate redactions:

I contacted Home Depot about getting a new remote for your dining room ceiling fan, and he said it was $16 including shipping.

---------- Forwarded message ----------
From: <___"">
Date: 9 January 2013 10:17
Subject: Replacement remote ceiling fan / Reference # ______

Mr. ___,

I was following up with you concerning your inquiry for a replacement remote. I checked with our parts department and the replacement remote is available for $16.00 including shipping. Please contact me directly at _____ if you would like to order the remote. Thanks in advance.


Resolution Expediter- Proprietary Brands

The Home Depot – Store Support Center

Customers FIRST!
J was delighted when I alluded to his message to K. "How did you find out?!" he shrieked, knowing full well the answer. "How did you find out which remote is compatible?" I asked back. It turned out he'd done a fair amount of research, including some preliminary back and forth with K.

"When do you think they will get the new remote?" he asked me.

"What makes you think they care about getting a new remote? They've been happy without one for many years."

"But it's only 16 dollars."

"Then why don't you buy it for them as a present?"

And so he did. He gave me some bills from his bedroom stash and had me call up the number in the email message. Every afternoon after I placed the order, he was ready to pounce on the mail the moment it went through the slot. When the package arrived, he tore it open, wrapped it up with Christmas paper, and wrote up a New Year's card: "I meant to give this present to you on New Year's Day but it arrived late."

He's currently giving K some time to set up the remote, and then he'll be back on her case for a film date asap.

Monday, January 21, 2013

Don't just assist (and/or medicate); instruct, II

Every time I look through a new special education catalog or teach another round of my autism class, I encounter descriptions and discussions of the latest round of assistive technologies and medications. And my reaction is always the same. However helpful they may prove to be overall, might some of these technological and pharmaceutical innovations--as well as other sorts of acommodations--end up detracting from, rather than enhancing, the education of special needs children?

My personal area of expertise is autism, and as I noted earlier in reference to Assistive and Augmentative Communication devices (AACs):

In autism, the most commonly used AACs [are used] for language--devices like the DynaVox, which gives users a menu of common vocabulary and phrases to select in order to communicate basic needs. The devices can revolutionize a child's basic functioning and psychological well-being, improving substantially his or her classroom behavior and teachability.

But no one should see AACs as a panacea for language instruction. However much they assist children in deploying their current language skills, it's far from clear that they actually teach them new ones.
Furthermore, the Dynavox and the like may inflate teachers' impressions of student achievement. For example:
When a child pushes "JUICE" on his or her DynaVox, many may see this as the child's intended shorthand for a full-fledged grammatical sentence--"I want juice"--or question--"Can I have juice?"--when he or she hasn't actually acquired general subject-verb-object order or the syntax of question inversion. Clicking on preset key words and phrases rather than constructing one's own phrases and sentences from scratch may mean that one has learned simple associations between stimuli and sounds, but not the linguistic skills prerequisite for intentional linguistic communication and thinking in full-fledged propositions.
Worse yet:
The ease of AACs, and how much they help teachers manage a classroom full of AS students, may sometimes breed complacency about the students' continuing needs for direct instruction in language. Managing AS kids is one thing, but, when it comes to meeting their legally-mandated educational needs, it is only the first step.
Other strategies for other types of academic challenges are similarly concerning. Consider, for example, what an October, 2012 New York Times article says about what's happening to low-income children who struggle in school:
When Dr. Michael Anderson hears about his low-income patients struggling in elementary school, he usually gives them a taste of some powerful medicine: Adderall.

The pills boost focus and impulse control in children with attention deficit hyperactivity disorder. Although A.D.H.D is the diagnosis Dr. Anderson makes, he calls the disorder “made up” and “an excuse” to prescribe the pills to treat what he considers the children’s true ill — poor academic performance in inadequate schools.

“I don’t have a whole lot of choice,” said Dr. Anderson, a pediatrician for many poor families in Cherokee County, north of Atlanta. “We’ve decided as a society that it’s too expensive to modify the kid’s environment. So we have to modify the kid.”

Dr. Anderson is one of the more outspoken proponents of an idea that is gaining interest among some physicians. They are prescribing stimulants to struggling students in schools starved of extra money — not to treat A.D.H.D., necessarily, but to boost their academic performance.
A somewhat different problem emerges in a profile in last week's Philadelphia Inquirer of a school for children with dyslexia and other "language-based differences." This private, college-prep school, AIM Academy, uses an "arts-based approach that includes costumes, games, activities, and classrooms decorated as medieval castles and prehistoric caves":
With a password and secret hand signal, all who enter Tom Waitzman's classroom gain passage to the Middle Ages.

Once inside, Max Lentz, dressed in a white cloak, no longer is a fourth grader at the AIM Academy in Whitemarsh Township, Montgomery County, but "William, duke of Normandy." Sitting nearby are classmates "Charlemagne, king of the Franks," and "Eric the Red."

"Why are the Vikings leaving Scandinavia?" asked Waitzman, whose classroom incarnation is "Merlin the wise."

"Overpopulation," Max, 10, said. "Good word," Waitzman told him.
Again we see key words (is "overpopulation" really the answer here?). But what about complete thoughts?
"I'm not just going to give a quiz or a work sheet or have them remember a lecture," Waitzman said. "It doesn't work with students with language-based differences."

What works at the 217-student academy is the philosophy "live it and learn it," said Patricia Roberts, the executive director, CEO, and cofounder of the K-12 school.
The technique is partly designed to keep students curious and engaged at a time when frustration with their reading skills has the potential to overwhelm.

Sophomore Insaf Sydnor of University City knows that feeling well. "You see the word the, I see go," Sydnor, 15, said of the years before she enrolled in AIM. "You see bat, I see hit."

The experience left her feeling lost and frustrated.
And if the school's arts and experience-based classrooms never instruct this 15-year-old how to read "bat," vs. "hit," she will continue to feel lost and frustrated, not only in college, but also in life.

Saturday, January 19, 2013

Language and culture: what matters most?

In stressing real-world relevance, 21st century skills, and personal connections, today's most outspoken education experts have forgotten the notion of learning for learning's sake. Explicitly or implicitly, many of them dismiss the possibility that students might enjoy a subject simply because it's interesting, and not because it contains obvious, practical connections to their personal lives and future vocations.

The latest of example of this is seen in an article in this past week's Education Week. Entitled Language Education We Can Use By, it assumes that the only purpose of foreign language instruction is to enable students to communicate proficiently and understand "internationally":

Here's how authors David Young and J.B. Buxton open their piece:

As the global nature of work and life in the 21st century becomes clearer by the day, calls for a greater focus on international education and language learning are growing louder. Leaders from the education, business, and national security communities are agreed: International understanding and second-language proficiency are critical to individual and national interests—and our K-12 system must do more to promote them.
Echoing claims by advocates of other types of "reform" that most classrooms are still defined by desks in rows, teacher in the front, rote drills, and pen and paper activities, Young and Buxton claim that foreign language classrooms are stuck in the past in other ways as well--ways that, in particular, fail to foster oral communication and international understanding:
For too many years, we have maintained a language-learning strategy that simply does not work. In programs using outdated pedagogies focused on grammar and translation and coupled with low expectations, students take foreign languages with goals that seemingly include everything except actually learning to speak the language. If graduates of our high schools regularly reflected that, after four years of mathematics, they couldn't solve for an unknown variable, we would be outraged. But we share a laugh when someone says, "I took four years of a language, but I can't really speak it."
Of course, many high school graduates can't solve for an unknown variable after four years of mathematics, precisely because classrooms have changed (something that the "we" of Education Week is decidedly not outraged about). Such detrimental changes extend to foreign language classrooms, where, in fact, there has long been a decreasing emphasis on grammar in favor of oral communication, and where translation exercises have mostly vanished to make room for "culture." And, perhaps most worryingly, where multi-year training in one language has been supplanted, at many schools, by "world languages" programs in which, echoing Reform Math's "an inch deep and a mile wide," students gain "awareness" of multiple languages by spending a mere year or semester on each.

For Young and Buxton, there are two goals of foreign language instruction. One is true oral proficiency, which they rightly argue emerges only from intense immersion environments--of the sort that schools, unless they are dedicated, multi-year bilingual education programs, simply can't provide.

The other goal of foreign language instruction, the one that applies to non-bilingual schools, should be "cultural awareness and sensitivity, global knowledge, and exposure to the target language."
It is time to convert existing courses to a classroom experience that provides a combination of introductory language exposure, cultural studies, and deep, experiential learning about the countries that speak the target language. These middle and high school language courses would have the following three components:
• Specific, real-life language instruction narrowed to focus on survival travel skills and with the goal of teaching a subset of the current language curriculum to greater depth and understanding—with relevance and utility as guiding principles;
• A cultural-studies framework that teaches students how to understand a country's cultural identity and to compare and contrast countries; and
• Global knowledge through the study, comparison, and contrasting of countries that speak the target language.
To be clear, students will not leave these classes with advanced language proficiency. What they will obtain, however, are the language skills needed to travel in countries that speak the language, an understanding of other countries and cultures, and an awareness of the global issues that impact both those countries and our own.
"Cultural awareness" sounds suspiciously like the "science appreciation" that is increasingly eclipsing science learning. "Deep, experiential learning" about countries and cultures sounds better than mere "awareness," but what makes the authors think that classrooms can provide this any more than they can provide linguistic immersion? After all, what holds for linguistic proficiency also holds deep cultural understanding: the best way to acquire it is to spend time immersed in the culture. Short of this, the next best strategies are:

-Watching films and TV shows produced by and for members of the culture in question (with English language captions).

-Reading popular novels and magazines (ideally not in translation; students can start with texts written for children).

Popular entertainment, especially shows and texts that include realistic portrayals of the culture, provide highly revealing windows into such key aspects of culture as popular taste, popular humor, daily life, family life, social dynamics, and popular pastimes. These are things that classrooms and "comparative culture" lessons simply cannot capture, and they are more "deep," meaningful, and experienced-based than the sort of formulaic and reductionist food- tradition- and etiquette-based instruction that often masquerades as meaningful multi-culturalism.

Yes, teachers could assign novels in translation and spend class time showing movies, but a much more effective role for foreign language teachers is to focus on foreign language instruction. Young and Buxton are right that "advanced language proficiency" will elude most students. But the most advanced language skills are productive skills: speaking and writing. Comprehension (reading and listening) is another matter. A class that emphasizes such "outdated" things as grammar, translation, vocabulary building, and auditory training (instead of the kinds of social studies topics that Young and Buxton favor) can vastly improve comprehension skills. And as their oral and reading comprehension skills improve, students can gain more and more from those cultural artifacts (shows and texts) that, to those who aren't immersed in it, reveal the most about the culture as a whole. For those who eventually do have an opportunity for cultural immersion, a classroom focused on foreign language instruction (as opposed to "cultural awareness") will give them a leg up in maximizing this immersion experience.

Culture, of course, is inherently interesting. But so is language. Young and Buxton don't seem to realize that one of the great pleasures of learning a foreign language comes from learning for learning's sake. Especially for the more linguistically minded students, and there are more of us out there than you might think, there's something deeply interesting in realizing that English structures are fundamentally different from those of other languages, and in learning the details of how a particular language deviates from English in how it constructs words, phrases, sentences, and thoughts. For this, too, an emphasis on grammar and translation, however "traditional," is absolutely necessary.

As another article that came out last week (this one in the Economist) reveals:
A series of announcements over the past few months from sources as varied as mighty Microsoft and string-and-sealing-wax private inventors suggest that workable, if not yet perfect, simultaneous-translation devices are now close at hand.
That is, the "universal translator" that has allowed science fiction humans to communicate face-to-face with aliens will soon be available to real humans. When that happens, many of the practical arguments for learning a foreign language will lose their force. The learning for learning's sake argument, on the other hand, will endure as long as humans, and languages, exist.

Thursday, January 17, 2013

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

I. The final problem set from the "Measuring Polygons" unit of the 5th grade TERC/Investigations Student Activity Workbook (Unit 5, p. 63) [click to enlarge]:

II. The final exercise from the "Perimeter, Area and Surface Area" unit of the 5th grade Singapore Math Primary Mathematics 5A Workbook (Unit 5, p. 122) [click to enlarge]:

III. Extra Credit:

What do you think is meant by "size" in the Investigations problem above? Does doubling the sides of a swimming pool actually double its "size"?

Tuesday, January 15, 2013

Conceptual understanding under Everyday Math

Picking up where I left off in my last post on Everyday Math students, Ah, but surely their "conceptual understanding" is deeper.

Yes, Everyday Math has eradicated most of those "boring," "tedious" calculation drills of traditional math, substituting the rote-calculating pre-21st Century mind with 21st Century technology like calculators, supposedly freeing the 21st Century mind for 21st Century pursuits like "conceptual understanding" and "higher-level thinking."

So what is the state of conceptual understanding among those whose entire experience with math consists of Everyday Math?  The students in our 3rd-5th grade after school program, selected for good behavior, interest in learning, and parental interest, and who otherwise show no signs of learning disabilities, include a number of students who probably now meet the diagnostic criteria for dyscalculia. Not only are they still counting on their fingers and struggling with the standard algorithms, but also (even many of the 5th grade cohort):

- Lack a basic number sense: for example, are stumped, totally stumped, when asked to give a number between 4000 and 5000.

- Lack a basic understanding of fractions: for example, that 1/1 = 2/2 = 3/3 = 4/4 = 1, or that 1/2 is smaller than 3/4

- Lack a basic understanding of decimals, and how to convert simple fractions to decimals and vice versa--e.g. 1/4 to .25 or .5 to 1/2

Number sense, and conceptual understanding more generally, are supposed to be the pride and glory of Reform Math. They are supposed, somehow, to emerge from finding multiple solutions to a small number of basic problems, inventing your own methods, using lengthy procedures that "break things down" like the partial-sums method and the partial quotients algorithm, and explaining your answers in words, numbers and pictures.

But there's growing anecdotal evidence that average students who are educated exclusively through Reform Math are more deficient in number sense than their traditionally educated peers.

For those educational researchers who are both intellectually honest and ethically minded (as opposed to those who are not), these anecdotes cry out for well-designed studies exploring Wu's hypothesis that basic skills versus conceptual understanding is a bogus dichotomy.

Sunday, January 13, 2013

Two requests for your input

Barry Garelick, co-founder of the U.S. Coalition for World Class Math, is looking for anonymous feedback from teachers, specifically about Reform Math and the Common Core Standards. As he writes:

Teachers are unwitting victims and it has become the norm for fuzzies to blame the elementary teachers for not understanding the math they have to teach. That may be true for some teachers, but it's too convenient an excuse, like blaming the students. Everyday Math, TERC [Investigations] and the foolishness of how Common Core Standards are going to be implemented will have a bad effect on students, and some teachers are forced to toe the line to implement things like Everyday Math, and Connected Mathematics Project with "fidelity" or face being fired--and these are teachers who know math.
Barry wishes to protect your identity, and asks that you send your responses/stories through me, at katharine dot p dot beals at gmail dot com.

Meanwhile an anonymous blogger at the new blog Wasting Time in School is seeking:
anecdotes of ridiculous time-wasting activities/assignments/projects. All contributions will be kept anonymous unless someone specifically requests otherwise.

Friday, January 11, 2013

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

I. The most challenging division caculations in the 5th grade Investigations Student Activity Book (from the second-to-last chapter in the book) [click to enlarge]:

II. The most challenging division calcuations in the 5th grade Singapore Math Primary Mathematics Workbook (from the second chapter in the book) [click to enlarge]:

III. Extra Credit:

Which is a better preparation for 21st century algebra: the "division starter" method, or traditional long division?

Wednesday, January 9, 2013

More educational malpractice: the sad legacy of Everyday Math, II

Here's a more personal follow-up to the topic of educational malpractice.  A couple of months ago, I wrote about working with a group of French African immigrant children whose school-based math education consists of Everyday Math. I noted how they weren't able to subtract 91 from 1000 because they didn't know how to borrow (regroup) across multiple digits.

As I noted then,

You can’t blame the mathematical deficiencies of these 4th and 5th graders on their parents: both the private school and the after school program select for parents who care about education. You can’t blame it on the kids: my kids, who clearly wanted to learn, had been admitted [to our after school program] in part based on their behavior.
You also can't blame it on language problems; these kids are fluent in English. In fact, there's really only one thing outside the Everyday Math curriculum that one can possibly point a finger to, and that is that these immigrant parents (many of them don't speak English) don't realize what many native-born parents already know: namely, that they can't count on the schools to fully educate their children.

So these kids are a case study in what happens when you leave math instruction entirely up to Everyday Math practitioners. And the answers to this question are slowly coming in.

For several of the 5th graders I work with, it turns out that not only do they not know how to borrow across multiple digits; they also don't know their basic addition and subtraction "facts." In other words, they don't automatically know that, say 5 plus 7 is 12, or that 15 - 8 is 7; instead they count on their fingers.

This got me thinking about addition and subtraction "facts." Back in my day, there was no issue of kids  learning these facts as such. Yes, we memorized our multiplication tables. But we never set about deliberately memorizing that 5 plus 7 is 12. Why? Because the frequency of the much-maligned "rote" calculations we did ensured that we, in today's lingo, constructed this knowledge on our own.

Back in my day, a typical third grade arithmetic sheet looked something like this:

And a typical fourth grade arithmetic sheet looked something like this:

But in Reform Math programs like Everyday Math, such pages filled with calculations are only occasional, and each problem involves a much shorter series of calculations. Here's a set from 4th grade Everyday Math:

Each multi-digit addition problem amounts to a series of simple addition problems. For example, adding two two-digit numbers involves adding at least two pairs of numbers; three if one is regrouping. Adding three three-digit numbers can involve 8 iterations of simple addition. Some of the problems in the second traditional math sheet involve as many as 17 iterations of simple addition.

In the traditional 4th grade math scenario, we may have had 25 problems per day like those in the first two sheets above, 5 days a week. With Everyday Math, you might get, at best, 25 problems like those in the second two sheets above per week.

Putting it all together, the resulting difference in the amount of practice with basic addition "facts" is quite large.  5 (days) times 25 (problems) times (say, as an average of iterations of simple addition) 10 for the traditional math curriculum versus 1 (day) times 25 (problems) times (average iterations) 3 for the Everyday Math curriculum. Assuming I'm not screwing up my arithmetic, that's 1,250 vs. 75 basic addition calculations per week. No wonder so many of those who are educated exclusively through Everyday Math don't know their "addition facts" by grade 5!

Ah, but surely their "conceptual understanding" is deeper. Note the calls for "ballpark estimate" at the bottom of each Everyday Math problem, where traditional math simply has you calculate. Stay tuned: in my next post on this topic, I'll discuss the state of conceptual understanding in my Everyday Math mal-educated 5th graders.

Monday, January 7, 2013

Educational malpractice for the sake of Reform Math

...Thus ends, a few days ago now, my Favorite Comments of '12. And it seems fitting to follow these, and belatedly usher in the year 2013, with a digest of one the most compelling email messages I've received in a very long time. It comes from James Milgram, an emeritus Professor of Mathematics at Stanford University.

Professor Milgram is known in the education world for his comprehensive critique of a study done by Jo Boaler, an education professor at Stanford, and Megan Staples, then an education professor at Purdue. Boaler and Staples' paper, preprinted in 2005 and published in 2008, is entitled Transforming Students’ Lives through an Equitable Mathematics Approach: The Case of Railside School. Focusing on three California schools, it compares cohorts of students who used either a traditional algebra curriculum, or the Reform Math algebra curriculum The College Preparatory Mathematics (CPM). According to Boaler and Staple's paper, the Reform Math cohort achieved substantially greater mathematical success than the traditional math cohorts.

In early 2005 a high ranking official from the U.S. Department of Education asked Professor Milgram to evaluate Boaler and Staples' study. The reason for her request? She was concerned that, if Boaler and Staples' conclusions were correct, the U.S. department of education would be obliged, in Milgram's words, "to begin to reconsider much if not all of what they were doing in mathematics education." This would entail an even stronger push by the U.S. educational establishment to implement the Constructivist Reform Math curricula throughout K12 education.

Milgram's evaluation of Boaler and Staples' study resulted in a paper, co-authored with mathematician Wayne Bishop and statistician Paul Clopton, entitled A close examination of Jo Boaler's Railside Report. The paper was accepted for publication in peer-reviewed journal Education Next, but statements made to Milgram by some of his math education colleagues caused him to become concerned that the paper's publication would, in Milgram's words, make it "impossible for me to work with the community of math educators in this country"--involved as he then was in a number of other math education-related projects. Milgram instead posted the paper to his Stanford website.

This past October a bullet-point response to Milgram's paper, entitled "When Academic Disagreement Becomes Harassment and Persecution," appeared on Boaler's Stanford website. A month ago, Milgram posted his response and alerted me to it. I have his permission to share parts of it here.

Entitled Private Data - The Real Story: A Huge Problem with Education Research, this second paper reviews Milgram et al's earlier critiques and adds several compelling updates. Together, the two papers make a series of highly significant points, all of them backed up with transparent references to data of the sort that Boaler and Staple's own paper completely lacks.

Indeed, among Milgram et al's points is precisely this lack of transparency. Boaler and Staples refuse to divulge their data, in particular data regarding which schools they studied, claiming that agreements with the schools and FERPA (Family Educational Rights and Privacy Act) rules disallow this. But FERPA only involves protecting the school records of individual students; not those of whole schools. More importantly, refusals to divulge such data violate the federal Freedom of Information Act. Boaler's refusal also violates the policies of Stanford University, specifically its stated "commitment to openness in research" and its prohibitions of secrecy, "including limitations on publishability of results."

Second, Milgram et al's examination of the actual data, once they were able to track it down via California's education records, shows that it was distorted in multiple ways.

1. Boaler and Staple's chosen cohorts aren't comparable:

It appears, from state data, that the cohort at Railside [the pseudonym of the Reform Math school] was comprised of students in the top half of the class in mathematics. For Greendale, it appears that the students were grouped between the 35th and 70th percentiles, and that the students at Hilltop were grouped between the 40th and 80th percentiles. [Excerpted from Milgram; boldface mine]
2. Boaler and Staple's testing instruments are flawed:
Our analysis shows that they contain numerous mathematical errors, even more serious imprecisions, and also that the two most important post-tests were at least 3 years below their expected grade levels.  [Excerpted from Milgram; boldface mine]
3. The data comparing test scores on California's standardized tests (STAR) comes from a comparison of test scores from students not involved in Boaler and Staple's study:
The students in the cohorts Boaler was studying should have been in 11th grade, not ninth in 2003! So [this] is not data for the population studied in [Boaler and Staple's paper]. This 2003 ninth grade algebra data is the only time where the Railside students clearly outperformed the students at the other two schools during this period. There is a possibility that they picked the unique data that might strengthen their assertions, rather than make use of the data relevant to their treatment groups.   [Excerpted from Milgram; boldface mine]
4. The most relevant actual data yields the opposite conclusion about the Reform Math cohort's mathematical success relative that of the traditional math cohorts:
o The most telling data we find is that the mathematics remediation rate for the cohort of Railside students that Boaler was following who entered the California State University system was 61%
o This was much higher than the state average of 37%
o Greendale's remediation rate was 35% o and Hilltop's was 29%.
5. School officials at "Railside" report that the results of the reform math curriculum are even worse than Milgram et al had originally indicated:
A high official in the district where Railside is located called and updated me on the situation there in May, 2010. One of that person's remarks is especially relevant. It was stated that as bad as [Milgram et al's original paper] indicated the situation was at Railside, the school district's internal data actually showed it was even worse. Consequently, they had to step in and change the math curriculum at Railside to a more traditional approach.

Changing the curriculum seems to have had some effect. This year (2012) there was a very large (27 point) increase in Railside's API score and an even larger (28 point) increase for socioeconomically disadvantaged students, where the target had been 7 points in each case.
6. Boaler’s responses to Milgram et al provide no substantiated refutations of any of their key points

In response to comments on an article on Boaler's critique of Milgram, Boaler states:
"I see in some of the comments people criticizing me for not addressing the detailed criticisms from Milgram/Bishop. I am more than happy to this. [...] I will write my detailed response today and post it to my site."
However, as Milgram notes in his December paper:
As I write this, nearly two months have passed since Boaler's rebuttal was promised, but it has not appeared. Nor is it likely to. The basic reason is that there is every reason to believe [Milgram et al's paper] is not only accurate but, in fact, understates the situation at "Railside" from 2000 - 2005.
In a nutshell: under the mantle of purported FERPA protection, we have hidden and distorted data supporting a continued revolution in K12 math education--a revolution that actual data show to be resulting, among other things, in substantially increased mathematics remediation rates among college students. Ever lower mathematical preparedness; ever greater college debt. Just what our country needs.

Nor is Boaler's Reform Math-supporting "research" unique in its lack of transparency, in its lack of independent verification, and in its unwarranted impact on K12 math practices. As Milgram notes,
This seems to be a very common occurrence within education circles.

For example, the results of a number of papers with enormous effects on curriculum and teaching, such as [Diane Briars and Lauren Resnick's paper "Standards, assessments -- and what else? The essential elements of Standards-based school improvement"] and [J. Riordan and P. Noyce's paper, "The impact of two standards-based mathematics curricula on student achievement in Massachusetts"] have never been independently verified.

Yet, [Briars and Resnick's paper] was the only independent research that demonstrated significant positive results for the Everyday Math program for a number of years. During this period district curriculum developers relied on [Briars and Resnick's paper] to justify choosing the program, and, today, EM is used by almost 20% of our students. Likewise [Riordan and Noyce's paper] was the only research accepted by [the U.S. Department of Education's] What Works Clearinghouse in their initial reports that showed positive effects for the elementary school program ``Investigations in Number, Data, and Space,'' which today is used by almost 10% of our students.
As Milgram notes:
Between one quarter and 30% of our elementary school students is a huge data set. Consequently, if these programs were capable of significantly improving our K-12 student outcomes, we would surely have seen evidence by now.
And to pretend that such evidence exists when it doesn't is nothing short of educational malpractice.

Friday, January 4, 2013

Favorite Comments of '12: Anonymous, Auntie Ann, and Niels Henrik Abel

On Processing Sandy

Anonymous said...

I've been in museums when school field trips were in progress and I haven't been impressed that the kids are getting much value from the experience. Most simply wander around aimlessly. Some have lists/questions and some have adult guides (chaperons, teachers or docents) but, even a science museums with interactive features, most kids seem to push/pull without any real thought. The exception has been homeschooling families/groups (I've asked the adults what kind of group they were), with advance preparation and specific objectives.

I think that classroom use of a good art DVD series, with appropriate classwork would be a better use of time, particularly for kids who are academically behind. For kids in cities, by 7th grade, they can and should be encouraged to visit local cultural/government/historical site on their own time - writing a report/answering questions on same could be extra credit.
Auntie Ann said...
This also ties in to the studies that show that effective teachers are effective in large part because they have their students spending more time actually on task than less-effective teachers. A fun classroom tends to be distracting, more-chaotic (let's make up a rap!--a recent option at our kids school,) and disjointed. Constantly churning the classroom to go to different stations, or to assemble and reassemble in small groups, or to have the day chopped up into short classroom bursts interspersed with short enrichment classes (not knocking art or science or gym, just the way the day is scheduled into 45 minute or shorter sessions,) leaves less time for tedium to build; less time for skills to build too.

Which then ties into the studies that show having ineffective teachers two years in a row is disastrous for kids, with literally a lifetime of consequences.

No drill-and-kill, because it's boring.

No long periods of concentration, because (come on!) the kids can't handle that; not when they're used to flashing lights, loud noises, and fast visuals. Asking them to actually focus, and working to develop their attention spans and concentration is just boring!

No chalk and talk, because some studies of high school science students show that they learn science better with hands-on experiments; therefore, it must also be true of all other skills, from basic phonics to art history. And besides, lectures are boring!

No corrections on homework and no revisions required, because the kids' tender feelings can't handle criticism and they have a panic attack at the sight of red pen marks on their papers. Besides, revisions are so boring!

Everything about school these days seems to be exactly structured to *not* teach or to *not* allow kids to actually learn anything.
Niels Henrik Abel said...
Regarding the teacher-as-therapist: How is it that they can get away with essentially practicing psychotherapy without a license? Seminars do not a therapist make. Besides, there's entirely too much emphasis on "feeling," and nowhere near enough on "thinking" - note how often people preface their opinions with "I feel..." rather than "I think..."

Thursday, January 3, 2013

Favorite Comments of '12: Catherine Johnson

On Boring high school history texts

Catherine Johnsonsaid...
The difference is AMAZING.


Second passage is vastly superior.

btw, reading these two side-by-side, I'm wondering whether excessive summarization automatically leads to "disordered topic" progression....(or whatever it should be called).

In the older passage, the grammatical subjects of the sentences inside a paragraph are often the same.

In the new passage, a paragraph can have almost as many different subjects as it does main clauses.
Catherine Johnsonsaid...
Here's my current rewrite: Which paragraph is better and why?

The principle:

Make the TOPIC of the paragraph the GRAMMATICAL SUBJECT of most (not necessarily all) of the MAIN CLAUSES in that paragraph.

The more I think about it, the more I think that compression makes cohesion much more challenging & difficult to achieve.
Catherine Johnsonsaid...
I need to look at both passages more closely, but the first couple of paragraphs in the older text use sentences that are pretty simple:

Frederick I was the first ruler to call his lands the Holy Roman Empire.

However this region was actually a patchwork of feudal territories.

His forceful personality and military skills enabled him to dominate the German princes, yet whenever he left the country, disorder returned.

Following Otto’s example, Frederick repeatedly invaded the rich cities of Italy.

His brutal tactics spurred Italian merchants to unite against him.

He also angered the Pope, who joined the merchants in an alliance called the Lombard League.

I'm pretty sure that the difference in readability stems from the difference in cohesion.
Catherine Johnsonsaid...
1. Frederick I
2. this region (from end of preceding sentence)
3. His forceful personality and military skills
4. Frederick
5. His brutal tactics
6. He

5 GRAMMATICAL SUBJECTS are "Frederick" or something belonging to Frederick (e.g.: "his forceful personality")
1 GRAMMATICAL SUBJECT follows directly from the end of the preceding sentence:
Frederick I was the first ruler to call HIS LANDS the Holy Roman Empire.
However THIS REGION was actually a patchwork of feudal territories.

I'm not aware of a scale for measuring the cohesiveness of a passage, but if one exists it would show that the older text has high cohesion while the MacDougal Littell text has low cohesion.

Favorite Comments of '12: kcab

On Constructivizing the Common Core, II

kcab said...

My middle-schooler is getting hit by a similar type of assignment. He's supposed to write a book for his math class, and part of it is supposed to involve a particular concept that has been covered in class already.

I don't really like the assignment in general, but I'm also concerned because this is a class in which he's subject-accelerated and is 2 grades behind and as much as 4 years younger than the majority of his classmates. For math itself this difference doesn't present a problem, but his writing is on grade level and he'll probably be dinged for that. I'm concerned that this spreading out of language arts to all subject areas will end up restricting the ability of students to work at the appropriate level in other subjects.

Favorite Comments of '12: Deirdre Mundy

On Left-brain meditation, II

Deirdre Mundysaid...
This is why Catholics, for instance, use the Rosary as a form of meditation. Set words, images from the Bible to reflect on, and a way to focus the mind on God, rather than emptying it (because, as an old priest once said "when you empty your thoughts and open the door, you never know WHAT will come in.)

Wednesday, January 2, 2013

Favorite Comments of '12: AmyP and Anonymous

On More Tough memes: grading students for “character”

AmyP said...

It's established practice to give quite a number of deportment grades for things like respect for adults, respect for fellow students, organization, neatness, etc.

I don't see why you'd need a separate grade for "grit," self-control, and motivation, as those are the characteristic that are already on view when a child shows up at school with their homework done and ready to work all day. Of course, it's hard to draw the line between where the kid's grit ends and mom and dad's grit begins.

Basically, these new insights about "grit" just mean giving the same kids two sets of bad grades for the same problems. For good or ill, it is certainly not the case that current academic grading is just evaluating raw intelligence--it's much more evaluating compliance, diligence and the ability to deal with frustration and boredom.

Actually, now that I think of it, Brooks has it backwards. Maybe we need to introduce some grades just for raw intelligence.    
Anonymous said...
You've made a good distinction. Grading on character should be for behaviors that are within the child's control, at least in principle: respect, cooperation, at least an acceptable level of perseverance, etc. Motivation is to some extent inborn, and to some extent home-generated. Let's leave that one out. Flexibility, the same. While it's true that the characterristics we grade children on in the "deportment" category, which has been with us forever, are somewhaet compliance-oriented, that's not a reason to reject the idea. Compliance IS important in a group setting, and being able to comply (whether one chooses to comply or not, in later life) is important to school success. "Grit" is just another word for persistence; it can be modeled, it can be rewarded, and it can be encouraged, but I see no reason to award a grade for "grit."

Favorite Comments of '12: Magister Green

On The unsung virtues of multiple choice tests


As someone who A) likes multiple-choice tests and B) quizzes his students every day, I've come to appreciate the power of a well-put together m/c test. Especially with distractors...I always try to put one distractor in each question that could be correct but is subtly wrong. You can really force students to reflect on why they are picking one answer over another; it makes for great post-quiz discussion and has been a potent tool for review and practice in class. I'm a fan.

Favorite Comments of '12: C T, Anonymous and David Foster

On "Embodied learning"

C Tsaid...

And perhaps someday, we could even figure out a way to get children out of their seats...perhaps even out of the classroom, outside where they could release pent-up energy...and view something called "nature"...We could call this experiential learning aid "recess".
Ah, but I am indulging in dreams today. ;)
Anonymous said...
After lengthy conversations with our school principal about the lack of adequate education for gifted students in math, she finally admitted that the math program is designed to be "egalitarian" and meet the math needs of the majority of kids who will go into liberal arts fields and will not need advanced problem solving abilities. She said that for the STEM oriented kids the school has purchased SmartLab, which is an elaborate series of computers with advanced graphics and movie making software and robotics equipment that cost around 300K, for kids to play with. Some bright shinny techie things instead of math education.
David Fostersaid...
Apparently a lot of schools have been replacing actual lab science with computer simulations of lab addition to the point that this approach doesn't work as well in terms of getting kids out of their seats and making the learning truly hands-on, it also kind of misses the whole point of science, which is that *you can always, at least in principle, see for yourself* don't have to trust an "authority."

And a black-box computer model is every bit as much an authority today as Aristotle's books were during the Middle Ages.

Tuesday, January 1, 2013

Favorite Comments of '12: Anonymous

On Whose intuition should we trust?

Anonymous said...

 Our private school has recently gone off the constructivist deep end, driven by two administrators who have no children of their own and were educated in the 60's. Neither has been in the classroom for 25 years or more, but they love the educational theory! The classroom teachers, most of whom are young parents, are much more pragmatic in their approach. Interestingly, many of them have their own kids in more traditional schools.

Favorite Comments of '12: Rivka, 1crosbycat, and Auntie Ann

On Constructivizing the Common Core

Rivka said...

I have a second grader. It never occurred to me that I should be encouraging her to read And Then There Were None, which is full of grisly murders, just because the words themselves aren't difficult.

I recently read a different handwringing article about "American high school students reading on a 5th grade level." That one was based on Accelerated Reading metrics, which rely almost entirely on syllabification. Here are some ratings I pulled up:

William Faulkner, As I Lay Dying: 5.4
Ernest Hemingway, The Sun Also Rises: 4.4
Ernest Hemingway, For Whom the Bell Tolls: 5.8
John Steinbeck, The Grapes of Wrath: 4.9
John Steinbeck, East of Eden: 5.3
Toni Morrison, Song of Solomon: 5.0

Kind of hard to believe that the Nobel Prize kept being awarded to fourth and fifth grade books, isn't it? Why don't they just award one to Beverly Cleary and be done with it? ...OR, maybe there's more to reading than the difficulty of the individual words? Just perhaps?
1crosbycat said...
I noticed this sort of thing with the Olympics, where some panel of people came up with scores for every possible movement, and deductions for every possible mistake, it seems in order to take human judgment out of scoring. There are qualities to gymnastics and high diving that cannot be quantified. It is funny that librarians want to be replaced by computers in determining book grade level, and scary that most public school administrators read EdWeek without utilizing any of those critical thinking skills they value so much in our kids.

Auntie Ann said...
I saw this a while back. It's a blog post referenceing an analysis of the Flesch-Kincaid reading levels of popular adult novels.

Smith's results surprised him. Despite deliberately choosing a mix of commercial and literary writers, he found that many of the results fell into the same range. The average in four separate categories was as follows:

The amount of passive voice the writers used ranged from 2.3% to 13.43%.

The number of characters per word ranged from 3.72 to 4.58.

The readability ranged from 72.34% to 91.84%, with an average of 83.1%.

Finally, on the Flesch-Kincaid Grade Level scale, the range was 2.68 to 6.3, with an average grade level of 4.4.

 In other words, he found that the bestselling writers were aiming their prose, prose that is read by a majority of adult readers in the country, at a fourth grade level.

Favorite Comments of '12: cranberry and AmyP

On This American Life on Tough

cranberry said...

social skills, impulse control, resilience, optimism, and grit

Is there any proof that these are more malleable than cognitive skills? I don't think they are. I think temperament is pretty hard wired from the start. When I look at my kids, their relative levels of energy and optimism seem stable from infancy on.

We don't set out to teach non-cognitive skills in school, but that doesn't mean they can be taught.

How would you teach grit? Isn't it possible the things one thinks teach grit are really a sorting mechanism?
AmyP said...

Is the message, "We have trouble teaching A. Let's teach B!" without first figuring out if they have any better shot at B?
cranberry said...

I fear that this book will lead to a resurgence of emphasis on school-based attempts to teach "character." It will be the new-old thing; there are any number of packaged programs schools can purchase. It won't have anything to do with Paul Tough's book, but I'm certain there will be chapters on grit added to the materials.

I've seen different schools define, and teach, character in very different ways. In our children's public school, character was called citizenship. It wasn't defined, but there was a mysterious rating which would appear on the report card for "citizenship." Three categories: poor, good, excellent.

One year a friend's son received a poor rating. No one would tell her why he received that rating. It was a mystery, wrapped in an enigma. How could he improve his citizenship, when the school refused to tell the family the basis for his grade? His mother didn't even know who had given him the grade.

I would define the functioning definition of citizenship in the public school as "serve on student council, or have a parent on the school committee."