‘Discovery Math’ is Weird but a Good Idea Nonetheless

This was brought to my attention by Jerry Becker
From thestar.com, Saturday, September 3, 2016. SEE
No, teaching math the “old-fashioned way” won’t work: Paul Wells
In response to the latest EQAO report, many parents insist that “discovery math” is the cause of low test scores in Ontario.
By Paul Wells  (National Affairs)
According to the latest EQAO report, half of Ontario Grade 6 students don’t meet the curriculum standard in math. That’s a problem. But it’s not the only one.

What worries me is that only 13 per cent of students who didn’t meet the provincial standard when they were in Grade 3 manage to catch up so they meet the standard for Grade 6. That’s the lowest number on that indicator in five years.

If you fall behind in math you stay behind. That’s why it’s important to get it right, not just at some vague moment in the future, but for kids who are in Ontario schools right now.

Fortunately, every parent in Ontario is sure they know how to teach math. Many parents want to get rid of “discovery math,” broadly defined as “doing it weird.” If only that loopy Liberal government would teach math the way we learned it when we were kids, the theory goes, there’d be no problem.

Sure, great, except for one thing. Very few parents I’ve met can perform more than the most rudimentary arithmetic for themselves. If you all learned math so well, why do you inch toward Junior’s algebra homework with a cross and a bulb of garlic?

Discovery math, to the extent it means anything, is an attempt to apply in a formal setting the insights about numbers that good mathematicians use routinely. People who are comfortable with numbers use all sorts of strategies to work with them. Confidently, through a kind of learned intuition.

So subtracting 272 from 836 is an altogether different proposition from subtracting 998 from

1,002. In the first case, you’re likelier to write it all out, solve the ones column first, carry 100 to the 10s column so you’re subtracting seven from 13, and so on. In the second case, I’d count up four from the lower number to the higher. It’s a really big drag on a kid to make her do the second problems the same way as the first. And parents who read “add to subtract” on a homework sheet, chuckle and roll their eyes, are committing malpractice.

This summer I made my stepson spend some time on Khan Academy, an educational website, to brush up his math before he enters Grade 8. He was briefly baffled by questions that asked, say,

6 1/4 – 3 3/4. One way to do it is to convert both sides to improper fractions. But it’s easier if you simply recognize that 6 1/4 is the same as 5 5/4. You can do the differences in your head in about two seconds.

The question is, how do you produce the kind of students who will make that insightful leap? All I know for sure is that you don’t do it by teaching a bunch of rules students will learn by rote – the beloved “old-fashioned way.” That may work for basic math facts. I did make our son practice his basic addition, subtraction and times tables one summer until he knew them from memory. I wish schools would take more time to nail those basic facts down. Since our school wouldn’t take the time, I did.

But very quickly, math becomes so complex you can’t have a rule for everything. Khan Academy teaches and tests 111 different skills at the fifth-grade level alone. You’d go crazy learning a rule for each skill. You must be able to intuit a useful method for each situation.

Modern curricula recognize, and try to teach, that flexibility. I refuse to say that’s a mistake. There is even empirical evidence it’s not. A March report from PISA, the international testing organization, found that in countries where students say they rely heavily on memorization, they scored starkly lower on complex advanced math questions than students who memorize less. “To perform at the very top,” the report concludes, students must learn to do math “in a more reflective, ambitious and creative way.”

What’s to be done about those declining EQAO scores? First, Ontario should support teachers by sharing best teaching practices more widely. In some countries, like Japan, teachers spend far more time mentoring younger newcomers to the profession, and sharing techniques among colleagues. Ontario schools should follow suit.

Second, support students by giving them more practice time. The only way to learn how numbers work together is by tackling incrementally more difficult questions, lots of them, over time. Kids need to practice insight just as their parents practiced times tables. If they do, they may just grow up knowing how to do math, not just how to complain about math teachers.

Paul Wells is a national affairs writer. His column appears Wednesday, Friday and Saturday.
It never appears to occur to either journalists or educational conservatives (or political ones) or to those deeply invested in undermining public education in the interest of turning it into a for-profit investment that curricula come and go due to fluctuations in standardized test scores, but the one sacred cow that is NEVER seriously interrogated is the testing process or its concomitant methods. Give me control of the tests and how they are scored and I ABSOLUTELY GUARANTEE that I can make results fluctuate to suit any political agenda and outcomes one might wish to see.
Mathematics itself has changed almost not at all when it comes to the content of K-12 curricula in most countries (and certainly in the US and Canada). Blaming decreasing test scores entirely on a teaching approach to math that is politically unpopular misses almost entirely that if assessments are skewed away from the kinds of thinking that teachers are trying to help students develop, it’s a slam dunk that scores wlll go down. And when assessments are developed to reflect more conceptual understanding (and scores go up), the conservatives and nay-sayers scream that the tests are “fuzzy.”
Once this sort of politicization of education is allowed to dominate the conversation, as it clearly is doing in this article and in many of the accompanying comments, there’s no chance for thoughtfu educators to pursue anything but lock-step, computation-dominated “math” teaching. Only that’s not math, and my Smart Phone does all of that vastly quicker, more accurately and more easily than nearly every human who has ever lived or ever will. If you want
 kids to be adept at replicating donkey arithmetic, so be it, but no one I teach will be encouraged to limit herself in that way.
Published in: on September 15, 2016 at 10:20 pm  Comments (1)  
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Math and Sex

You may not think that math and sex don’t mix, but I show here that they really do:


Published in: on September 1, 2016 at 12:38 pm  Leave a Comment  
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Different DC middle schools gave their students totally different PARCC math tests

Digest that: some DC middle schools gave a general math PARCC test to their students. Others administered an Algebra 1 PARCC test. Others gave a PARCC geometry test.

And not even Superintendent Hanseul Kang seems to know which schools administered what test.

This all comes from Valerie Jablow’s blog.

But all schools will be held ‘accountable’ to the same standard.



Remedial College Courses and Real Problems

From a recent discussion on the Concerned4DCPS list about a recent NYT article on the numbers of students taking remedial courses at the college level. I have taken the opportunity to revise and extend my remarks. If you want to read these in chronological order, start at the bottom.


(From me:)

Judge in NY State Throws Out ‘Value-Added Model’ Ratings

I am pleased that in an important, precedent-setting case, a judge in New York State has ruled that using Value-Added measurements to judge the effectiveness of teachers is ‘arbitrary’ and ‘capricious’.

The case involved teacher Sheri Lederman, and was argued by her husband.

“New York Supreme Court Judge Roger McDonough said in his decision that he could not rule beyond the individual case of fourth-grade teacher Sheri G. Lederman because regulations around the evaluation system have been changed, but he said she had proved that the controversial method that King developed and administered in New York had provided her with an unfair evaluation. It is thought to be the first time a judge has made such a decision in a teacher evaluation case.”

In case you were unaware of it, VAM is a statistical black box used to predict how a hypothetical student is supposed to score on a Big Standardized Test one year based on the scores of every other student that year and in previous years. Any deviation (up or down) of that score is attributed to the teacher.

Gary Rubinstein and I have looked into how stable those VAM scores are in New York City, where we had actual scores to work with (leaked by the NYTimes and other newspapers). We found that they were inconsistent and unstable in the extreme! When you graph one year’s score versus next year’s score, we found that there was essentially no correlation at all, meaning that a teacher who is assigned the exact same grade level, in the same school, with very similar  students, can score high one year, low the next, and middling the third, or any combination of those. Very, very few teachers got scores that were consistent from year to year. Even teachers who taught two or more grade levels of the same subject (say, 7th and 8th grade math) had no consistency from one subject to the next. See my blog  (not all on NY City) herehere, here,  here, herehere, here, here,  herehere, and here. See Gary R’s six part series on his blog here, here, here, here, here, and here. As well as a less technical explanation here.

Mercedes Schneider has done similar research on teachers’ VAM scores in Louisiana and came up with the same sorts of results that Rubinstein and I did.

Which led all three of us to conclude that the entire VAM machinery was invalid.

And which is why the case of Ms. Lederman is so important. Similar cases have been filed in numerous states, but this is apparently the first one where a judgement has been reached.

(Also read this. and this.)

Advanced Math Among American Students

An article in the Atlantic discusses the growing phenomenon of American students studying and succeeding in a wide variety of advanced mathematics courses and competitions. This includes organizations like MathCounts (which I coached at the JHS level for many years) as well as special summer math programs like MathPath, as well as math circles and AP calculus and statistics courses.

However, as the author notes:

National achievement data reflect this access gap in math instruction [between US poor kids and US rich kids – gfb]  all too clearly. The ratio of rich math whizzes to poor ones is 3 to 1 in South Korea and 3.7 to 1 in Canada, to take two representative developed countries. In the U.S., it is 8 to 1. And while the proportion of American students scoring at advanced levels in math is rising, those gains are almost entirely limited to the children of the highly educated, and largely exclude the children of the poor. By the end of high school, the percentage of low-income advanced-math learners rounds to zero.

Published in: on February 10, 2016 at 4:56 pm  Comments (2)  
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A Concise Primer on Privatization from Marion Brady

This is a concise primer, written by Marion Brady, on how the 1/100 of 1% have been privatizing our schools and getting away with it. -GFB

Advice column for pundits and politicians


Privatizing public schools: A primer for pundits and politicians


When, about thirty years ago, corporate interests began their highly organized, well-funded effort to privatize public education, you wouldn’t have read or heard about it. They didn’t want to trigger the debate that such a radical change in an important institution warranted.

If, like most pundits and politicians, you’ve supported that campaign, it’s likely you’ve been snookered. Here’s a quick overview of the snookering process.


The pitch


Talking Points: (a) Standardized testing proves America’s schools are poor. (b) Other countries are eating our lunch. (c) Teachers deserve most of the blame. (d) The lazy ones need to be forced out by performance evaluations. (e) The dumb ones need scripts to read or “canned standards” telling them exactly what to teach. (f) The experienced ones are too set in their ways to change and should be replaced by fresh Five-Week-Wonders from Teach for America. (Bonus: Replacing experienced teachers saves a ton of money.) (g) Public (“government”) schools are a step down the slippery slope to socialism.




Education establishment resistance to privatization is inevitable, so (a) avoid it as long as possible by blurring the lines between “public” and “private.” (b) Push school choice, vouchers, tax write-offs, tax credits, school-business partnerships, profit-driven charter chains. (c) When resistance comes, crank up fear with the, “They’re eating our lunch!” message. (d) Contribute generously to all potential resisters—academic publications, professional organizations, unions, and school support groups such as PTA. (e) Create fake “think tanks,” give them impressive names, and have them do “research” supporting privatization. (f) Encourage investment in teacher-replacer technology—internet access, I-pads, virtual schooling, MOOCS, etc. (e) Pressure state legislators to make life easier for profit-seeking charter chains by taking approval decisions away from local boards and giving them to easier-to-lobby state-level bureaucrats. (g) Elect the “right” people at all levels of government. (When they’re campaigning, have them keep their privatizing agenda quiet.)




If you’ll read the fine-print disclaimers on high-stakes standardized tests, you’ll see how grossly they’re being misused, but they’re the key to privatization. The general public, easily impressed by numbers and mathematical razzle-dazzle, believes competition is the key to quality, so want quality quantified even though it can’t be done. Machine-scored tests don’t measure quality. They rank.

It’s hard to rank unlike things so it’s necessary to standardize. That’s what the Common Core State Standards (CCSS) do. To get the job done quickly, Bill Gates picked up the tab, got the CCSS “legitimized” by getting important politicians to sign off on them, then handed them to teachers as a done deal.

The Standards make testing and ranking a cinch. They also make making billions a cinch. Manufacturers can use the same questions for every state that has adopted the Standards or facsimiles thereof.

If challenged, test fans often quote the late Dr. W. Edward Deming, the world-famous quality guru who showed Japanese companies how to build better stuff than anybody else. In his book, The New Economics, Deming wrote, “If you can’t measure it, you can’t manage it.”

Here’s the whole sentence as he wrote it: “It is wrong to suppose that if you can’t measure it, you can’t manage it—a costly myth.”


Operating the weapon


What’s turned standardized testing into a privatizing juggernaut are pass-fail “cut scores” set by politicians. Saying kids need to be challenged, they set the cut score high enough to fail many (sometimes most) kids. When the scores are published, they point to the high failure rate to “prove” public schools can’t do the job and should be closed or privatized. Clever, huh?

The privatizing machinery is in place. Left alone, it’ll gradually privatize most, but not all, public schools. Those that serve the poorest, the sickest, the handicapped, the most troubled, the most expensive to educate—those will stay in what’s left of the public schools.


Weapon malfunction


Look at standardized tests from kids’ perspective. Test items (a) measure recall of secondhand, standardized, delivered information, or (b) require a skill to be demonstrated, or (c) reward an ability to second-guess whoever wrote the test item. Because kids didn’t ask for the information, because the skill they’re being asked to demonstrate rarely has immediate practical use, and because they don’t give a tinker’s dam what the test-item writer thinks, they have zero emotional investment in what’s being tested.

As every real teacher knows, no emotional involvement means no real learning. Period. What makes standardized testslook like they work is learner emotion, but it’s emotion that doesn’t have anything to do with learning. The ovals get penciled in to avoid trouble, to please somebody, to get a grade, or to jump through a bureaucratic hoop to be eligible to jump through another bureaucratic hoop. When the pencil is laid down, what’s tested, having no perceived value, automatically erases from memory.


Before you write…


If you want to avoid cranking out the usual amateurish drivel about standardized testing that appears in the op-eds, editorials, and syndicated columns of the mainstream media, ask yourself a few questions about the testing craze: (a) Should life-altering decisions hinge on the scores of commercially produced tests not open to public inspection? (b) How wise is it to only teach what machines can measure? (c) How fair is it to base any part of teacher pay on scores from tests that can’t evaluate complex thought? (d) Are tests that have no “success in life” predictive power worth the damage they’re doing?

Here’s a longer list of problems you should think about before you write.



America’s schools have always struggled—an inevitable consequence, first, of a decision in 1893 to narrow and standardize the high school curriculum and emphasize college prep; second, from a powerful strain of individualism in our national character that eats away support for public institutions; third, from a really sorry system of institutional organization. Politicians, not educators, make education policy, basing it on the simplistic conventional wisdom that educating means “delivering information.”

In fact, educating is the most complex and difficult of all professions. Done right, teaching is an attempt to help the young align their beliefs, values, and assumptions more closely with what’s true and real, escape the bonds of ethnocentrism, explore the wonders and potential of humanness, and become skilled at using thought processes that make it possible to realize those aims.

Historically, out of the institution’s dysfunctional organizational design came schools with lots of problems, but with one redeeming virtue. They were “loose.” Teachers had enough autonomy to do their thing. So they did, and the kids that some of them coached brought America far more than its share of patents, scholarly papers, scientific advances, international awards, and honors.

Notwithstanding their serious problems, America’s public schools were once the envy of the world. Now, educators around that world shake their heads in disbelief (or maybe cheer?) as we spend billions of dollars to standardize what once made America great—un-standardized thought.

A salvage operation is still (barely) possible, but not if politicians, prodded by pundits, continue to do what they’ve thus far steadfastly refused to do—listen to people who’ve actually worked with real students in real classrooms, and did so long enough and thoughtfully enough to know something about teaching.


Note: I invite response, especially from those in positions of influence or authority who disagree with me.

Marion Brady mbrady2222@gmail.com

View this on Basecamp

Even the Chancellor Calls the Results ‘Sobering’

The Washington City Paper has an article on the PARCC results with way more graphs and charts than I do, and they quote even Chancellor Kaya Henderson as saying the results were ‘sobering’.

Please remind me why she still has a job?

She and several other speakers said that the PARCC results were more ‘honest’ than the old DC-CAS results, probably because the new ‘passing’ scores are lower than the old ones. I guess that means that it’s more ‘honest’ to say that students are doing worse than we were previously led to believe, under the current regime of all-testing-all-the-time and turn-half-the-students-over-to-unregulated charters?


PARCC Results Released in DC

I just got back from watching the public release of the results of the PARCC test that students in Washington DC took about 7 months ago.

(Let that sink in: it took the testing company, and their consultants, and the city’s consultants, over HALF A YEAR to massage the data into a releasable form. So much for having these tests be able to be used to ‘inform instruction’ or help teachers figure out what kind of help their students need. It’s now the last day of November, and the students have been in school since August. What kind of help is that to teachers or parents? And tho I haven’t looked at the released school scores or samples of what the teachers will see, I’m not optimistic. If the past is any guide, the scores themselves will be essentially useless as well…)

(It won’t take so long next time, we were assured…)

I got to see Mayor Bowser, Councilman Grosso, Chancellor Kaya Henderson, [powerless] Superintendent Hanseul Kang, and Deputy Mayor for Education Jenny Niles, and charter honcho Scott Pearson perform and answer some mostly-lame questions from some members of the media.

What we saw were that advanced students in DC (largely white ones) do exceedingly well on this PARCC battery of tests, and that others (blacks; hispanics; SPEDs; students on free or reduced lunch; ELLs; or Students At Risk) do much worse. Which of course is  the very same result we’ve seen on the NAEP for a couple of decades.

In fact, of all the cities and states measured on the NAEP, Washington DC has the very widest gaps in test scores between the Upper Caucasia Haves and the Have-Nots everywhere else, and those gaps are if anything getting wider.

It was interesting to hear Henderson’s defenses of the results, which still showed very low percentages of most students “passing” the PARCC. She said, among other things, that

(1) since students at the lower grades generally scored higher than those at the upper grades, that show’s we are on the right path [seems to me it shows the exact opposite; the longer that students have been exposed to “Reform”, the worse they do… and

(2) It takes a long time, you can’t just expect to turn a switch and have everything be wonderful overnight, we need lots of wrap-around services and a longer school day and school year and support for teachers.

Regarding the latter excuse: isn’t that exactly what teachers were condemned for saying under Chancellor Rhee, whose understudy was none other than Kaya Henderson? Didn’t Rhee imply that the only reason that poor students did poorly in school was that their greedy, lazy teachers, empowered by their evil union, refused to teach them anything? And that anybody who said that it’s a lot harder to teach impoverished students of color with chaotic families (if any) than it is to teach middle-class children with educated parents – why those people were just making excuses for poverty?


Mental Math, Traditional Math, and Best Methods

James Tanton is one of the most insightful teachers of teachers of math I’ve come across in a long time.

In this video and this essay, he explains how some parents feel mystified by some of the newer style of math problems that children are bringing to their parents for help. Clearly, some of the problems (like the 32-12 one which fills up an entire page in a child’s workbook) are being done in a tortured, time-wasting method. However, if you take a more difficult problem, such as 103 – 87, you can do this completely in your head by adding small increments (MENTALLY) to the 87 until you get to 103.

This is a reasonable thing to do, much like people count out (or used to count out) change into a customer’s hand.

In this case, from 87 to 90 (which is a much ‘nicer’ number than 87) we need to go up by 3.

Then from 90 to 100 (which is getting closer to our goal), we go up by 10. So we’ve gone up by 13 all together.

Then from 100 to 103 is three more, so we’ve gone up by 16, which means that 103 – 87 is 16.

I would be crazy to write all those steps out! – which is unfortunately what the poor child was asked to do.

But doing it quickly in your head makes a lot of sense.

As Tanton says, let’s get rid of laborious, tortured, time-wasting algorithms and examples, but let’s keep the thinking and understanding.

Click on the picture for a larger view; it’s a slightly-modified still from his video.

james tanton on thinking

Published in: on October 25, 2015 at 11:31 am  Comments (1)  
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