Part Two: Cheating in DCPS

DC Education Reform Ten Years After, 

Part 2: Test Cheats

Richard P Phelps

Ten years ago, I worked as the Director of Assessments for the District of Columbia Public Schools (DCPS). For temporal context, I arrived after the first of the infamous test cheating scandals and left just before the incident that spawned a second. Indeed, I filled a new position created to both manage test security and design an expanded testing program. I departed shortly after Vincent Gray, who opposed an expanded testing program, defeated Adrian Fenty in the September 2010 DC mayoral primary. My tenure coincided with Michelle Rhee’s last nine months as Chancellor. 

The recurring test cheating scandals of the Rhee-Henderson years may seem extraordinary but, in fairness, DCPS was more likely than the average US school district to be caught because it received a much higher degree of scrutiny. Given how tests are typically administered in this country, the incidence of cheating is likely far greater than news accounts suggest, for several reasons: 

·      in most cases, those who administer tests—schoolteachers and administrators—have an interest in their results;

·      test security protocols are numerous and complicated yet, nonetheless, the responsibility of non-expert ordinary school personnel, guaranteeing their inconsistent application across schools and over time; 

·      after-the-fact statistical analyses are not legal proof—the odds of a certain amount of wrong-to-right erasures in a single classroom on a paper-and-pencil test being coincidental may be a thousand to one, but one-in-a-thousand is still legally plausible; and

·      after-the-fact investigations based on interviews are time-consuming, scattershot, and uneven. 

Still, there were measures that the Rhee-Henderson administrations could have adopted to substantially reduce the incidence of cheating, but they chose none that might have been effective. Rather, they dug in their heels, insisted that only a few schools had issues, which they thoroughly resolved, and repeatedly denied any systematic problem.  

Cheating scandals

From 2007 to 2009 rumors percolated of an extraordinary level of wrong-to-right erasures on the test answer sheets at many DCPS schools. “Erasure analysis” is one among several “red flag” indicators that testing contractors calculate to monitor cheating. The testing companies take no responsibility for investigating suspected test cheating, however; that is the customer’s, the local or state education agency. 

In her autobiographical account of her time as DCPS Chancellor, Michelle Johnson (nee Rhee), wrote (p. 197)

“For the first time in the history of DCPS, we brought in an outside expert to examine and audit our system. Caveon Test Security – the leading expert in the field at the time – assessed our tests, results, and security measures. Their investigators interviewed teachers, principals, and administrators.

“Caveon found no evidence of systematic cheating. None.”

Caveon, however, had not looked for “systematic” cheating. All they did was interview a few people at several schools where the statistical anomalies were more extraordinary than at others. As none of those individuals would admit to knowingly cheating, Caveon branded all their excuses as “plausible” explanations. That’s it; that is all that Caveon did. But, Caveon’s statement that they found no evidence of “widespread” cheating—despite not having looked for it—would be frequently invoked by DCPS leaders over the next several years.[1]

Incidentally, prior to the revelation of its infamous decades-long, systematic test cheating, the Atlanta Public Schools had similarly retained Caveon Test Security and was, likewise, granted a clean bill of health. Only later did the Georgia state attorney general swoop in and reveal the truth. 

In its defense, Caveon would note that several cheating prevention measures it had recommended to DCPS were never adopted.[2] None of the cheating prevention measures that I recommended were adopted, either.

The single most effective means for reducing in-classroom cheating would have been to rotate teachers on test days so that no teacher administered a test to his or her own students. It would not have been that difficult to randomly assign teachers to different classrooms on test days.

The single most effective means for reducing school administratorcheating would have been to rotate test administrators on test days so that none managed the test materials for their own schools. The visiting test administrators would have been responsible for keeping test materials away from the school until test day, distributing sealed test booklets to the rotated teachers on test day, and for collecting re-sealed test booklets at the end of testing and immediately removing them from the school. 

Instead of implementing these, or a number of other feasible and effective test security measures, DCPS leaders increased the number of test proctors, assigning each of a few dozen or so central office staff a school to monitor. Those proctors could not reasonably manage the volume of oversight required. A single DC test administration could encompass a hundred schools and a thousand classrooms.

Investigations

So, what effort, if any, did DCPS make to counter test cheating? They hired me, but then rejected all my suggestions for increasing security. Also, they established a telephone tip line. Anyone who suspected cheating could report it, even anonymously, and, allegedly, their tip would be investigated. 

Some forms of cheating are best investigated through interviews. Probably the most frequent forms of cheating at DCPS—teachers helping students during test administrations and school administrators looking at test forms prior to administration—leave no statistical residue. Eyewitness testimony is the only type of legal evidence available in such cases, but it is not just inconsistent, it may be socially destructive. 

I remember two investigations best: one occurred in a relatively well-to-do neighborhood with well-educated parents active in school affairs; the other in one of the city’s poorest neighborhoods. Superficially, the cases were similar—an individual teacher was accused of helping his or her own students with answers during test administrations. Making a case against either elementary school teacher required sworn testimony from eyewitnesses, that is, students—eight-to-ten-year olds. 

My investigations, then, consisted of calling children into the principal’s office one-by-one to be questioned about their teacher’s behavior. We couldn’t hide the reason we were asking the questions. And, even though each student agreed not to tell others what had occurred in their visit to the principal’s office, we knew we had only one shot at an uncorrupted jury pool. 

Though the accusations against the two teachers were similar and the cases against them equally strong, the outcomes could not have been more different. In the high-poverty neighborhood, the students seemed suspicious and said little; none would implicate the teacher, whom they all seemed to like. 

In the more prosperous neighborhood, students were more outgoing, freely divulging what they had witnessed. The students had discussed the alleged coaching with their parents who, in turn, urged them to tell investigators what they knew. During his turn in the principal’s office, the accused teacher denied any wrongdoing. I wrote up each interview, then requested that each student read and sign. 

Thankfully, that accused teacher made a deal and left the school system a few weeks later. Had he not, we would have required the presence in court of the eight-to-ten-year olds to testify under oath against their former teacher, who taught multi-grade classes. Had that prosecution not succeeded, the eyewitness students could have been routinely assigned to his classroom the following school year.

My conclusion? Only in certain schools is the successful prosecution of a cheating teacher through eyewitness testimony even possible. But, even where possible, it consumes inordinate amounts of time and, otherwise, comes at a high price, turning young innocents against authority figures they naturally trusted. 

Cheating blueprints

Arguably the most widespread and persistent testing malfeasance in DCPS received little attention from the press. Moreover, it was directly propagated by District leaders, who published test blueprints on the web. Put simply, test “blueprints” are lists of the curricular standards (e.g., “student shall correctly add two-digit numbers”) and the number of test items included in an upcoming test related to each standard. DC had been advance publishing its blueprints for years.

I argued that the way DC did it was unethical. The head of the Division of Data & Accountability, Erin McGoldrick, however, defended the practice, claimed it was common, and cited its existence in the state of California as precedent. The next time she and I met for a conference call with one of DCPS’s test providers, Discover Education, I asked their sales agent how many of their hundreds of other customers advance-published blueprints. His answer: none.

In the state of California, the location of McGoldrick’s only prior professional experience, blueprints were, indeed, published in advance of test administrations. But their tests were longer than DC’s and all standards were tested. Publication of California’s blueprints served more to remind the populace what the standards were in advance of each test administration. Occasionally, a standard considered to be of unusual importance might be assigned a greater number of test items than the average, and the California blueprints signaled that emphasis. 

In Washington, DC, the tests used in judging teacher performance were shorter, covering only some of each year’s standards. So, DC’s blueprints showed everyone well in advance of the test dates exactly which standards would be tested and which would not. For each teacher, this posed an ethical dilemma: should they “narrow the curriculum” by teaching only that content they knew would be tested? Or, should they do the right thing and teach all the standards, as they were legally and ethically bound to, even though it meant spending less time on the to-be-tested content? It’s quite a conundrum when one risks punishment for behaving ethically.

Monthly meetings convened to discuss issues with the districtwide testing program, the DC Comprehensive Assessment System (DC-CAS)—administered to comply with the federal No Child Left Behind (NCLB) Act. All public schools, both DCPS and charters, administered those tests. At one of these regular meetings, two representatives from the Office of the State Superintendent of Education (OSSE) announced plans to repair the broken blueprint process.[3]

The State Office employees argued thoughtfully and reasonably that it was professionally unethical to advance publish DC test blueprints. Moreover, they had surveyed other US jurisdictions in an effort to find others that followed DC’s practice and found none. I was the highest-ranking DCPS employee at the meeting and I expressed my support, congratulating them for doing the right thing. I assumed that their decision was final.

I mentioned the decision to McGoldrick, who expressed surprise and speculation that it might have not been made at the highest level in the organizational hierarchy. Wasting no time, she met with other DCPS senior managers and the proposed change was forthwith shelved. In that, and other ways, the DCPS tail wagged the OSSE dog. 

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It may be too easy to finger ethical deficits for the recalcitrant attitude toward test security of the Rhee-Henderson era ed reformers. The columnist Peter Greene insists that knowledge deficits among self-appointed education reformers also matter: 

“… the reformistan bubble … has been built from Day One without any actual educators inside it. Instead, the bubble is populated by rich people, people who want rich people’s money, people who think they have great ideas about education, and even people who sincerely want to make education better. The bubble does not include people who can turn to an Arne Duncan or a Betsy DeVos or a Bill Gates and say, ‘Based on my years of experience in a classroom, I’d have to say that idea is ridiculous bullshit.’”

“There are a tiny handful of people within the bubble who will occasionally act as bullshit detectors, but they are not enough. The ed reform movement has gathered power and money and set up a parallel education system even as it has managed to capture leadership roles within public education, but the ed reform movement still lacks what it has always lacked–actual teachers and experienced educators who know what the hell they’re talking about.”

In my twenties, I worked for several years in the research department of a state education agency. My primary political lesson from that experience, consistently reinforced subsequently, is that most education bureaucrats tell the public that the system they manage works just fine, no matter what the reality. They can get away with this because they control most of the evidence and can suppress it or spin it to their advantage.

In this proclivity, the DCPS central office leaders of the Rhee-Henderson era proved themselves to be no different than the traditional public-school educators they so casually demonized. 

US school systems are structured to be opaque and, it seems, both educators and testing contractors like it that way. For their part, and contrary to their rhetoric, Rhee, Henderson, and McGoldrick passed on many opportunities to make their system more transparent and accountable.

Education policy will not improve until control of the evidence is ceded to genuinely independent third parties, hired neither by the public education establishment nor by the education reform club.

The author gratefully acknowledges the fact-checking assistance of Erich Martel and Mary Levy.

Access this testimonial in .pdf format

Citation:  Phelps, R. P. (2020, September). Looking Back on DC Education Reform 10 Years After, Part 2: Test Cheats. Nonpartisan Education Review / Testimonials. https://nonpartisaneducation.org/Review/Testimonials/v16n3.htm

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[1] A perusal of Caveon’s website clarifies that their mission is to help their clients–state and local education departments–not get caught. Sometimes this means not cheating in the first place; other times it might mean something else. One might argue that, ironically, Caveon could be helping its clients to cheat in more sophisticated ways and cover their tracks better.

[2] Among them: test booklets should be sealed until the students open them and resealed by the students immediately after; and students should be assigned seats on test day and a seating chart submitted to test coordinators (necessary for verifying cluster patterns in student responses that would suggest answer copying).

[3] Yes, for those new to the area, the District of Columbia has an Office of the “State” Superintendent of Education (OSSE). Its domain of relationships includes not just the regular public schools (i.e., DCPS), but also other public schools (i.e., charters) and private schools. Practically, it primarily serves as a conduit for funneling money from a menagerie of federal education-related grant and aid programs

More on the “false positive” COVID-19 testing problem

I used my cell phone last night to go into the problem of faulty testing for COVID-19, based on a NYT article. As a result, I couldn’t make any nice tables. Let me remedy that and also look at a few more assumptions.

This table summarizes the testing results on a theoretical group of a million Americans tested, assuming that 5% of the population actually has coronavirus antibodies, and that the tests being given have a false negative rate of 10% and a false positive rate of 3%. Reminder: a ‘false negative’ result means that you are told that you don’t have any coronavirus antibodies but you actually do have them, and a ‘false positive’ result means that you are told that you DO have those antibodies, but you really do NOT. I have tried to highlight the numbers of people who get incorrect results in the color red.

Table A

Group	Total	Error rate	Test says they are Positive	Test says they are Negative
Actually Positive	50,000	10%	45,000	5,000
Actually Negative	950,000	3%	28,500	921,500
Totals	1,000,000		73,500	926,500
Percent we assume are actually positive	5%	Accuracy Rating	61.2%	99.5%

As you can see, using those assumptions, if you get a lab test result that says you are positive, that will only be correct in about 61% of the time. Which means that you need to take another test, or perhaps two more tests, to see whether they agree.

The next table assumes again a true 5% positive result for the population and a false negative rate of 10%, but a false positive rate of 14%.

Table B

Assume 5% really exposed, 14% false positive rate, 10% false negative
Group	Total	Error rate	Test says they are Positive	Test says they are Negative
Actually Positive	50,000	10%	45,000	5,000
Actually Negative	950,000	14%	133,000	817,000
Totals	1,000,000		178,000	822,000
Percent we assume are actually positive	5%	Accuracy Rating	25.3%	99.4%

Note that in this scenario, if you get a test result that says you are positive, that is only going to be correct one-quarter of the time (25.3%)! That is useless!

Now, let’s assume a lower percentage of the population actually has the COVID-19 antibodies, say, two percent. Here are the results if we assume a 3% false positive rate:

Table C

Assume 2% really exposed, 3% false positive rate, 10% false negative
Group	Total	Error rate	Test says they are Positive	Test says they are Negative
Actually Positive	20,000	10%	18,000	2,000
Actually Negative	980,000	3%	29,400	950,600
Totals	1,000,000		47,400	952,600
Percent we assume are actually positive	2%	Accuracy Rating	38.0%	99.8%

Notice that in this scenario, if you get a ‘positive’ result, it is likely to be correct only a little better than one-third of the time (38.0%).

And now let’s assume 2% actual exposure, 14% false positive, 10% false negative:

Table D

Assume 2% really exposed, 14% false positive rate, 10% false negative
Group	Total	Error rate	Test says they are Positive	Test says they are Negative
Actually Positive	20,000	10%	45,000	2,000
Actually Negative	980,000	14%	137,200	842,800
Totals	1,000,000		182,200	844,800
Percent we assume are actually positive	2%	Accuracy Rating	24.7%	99.8%

Once again, the chances of a ‘positive’ test result being accurate is only about one in four (24.7%), which means that this level of accuracy is not going to be useful to the public at large.

Final set of assumptions: 3% actual positive rate, and excellent tests with only 3% false positive and false negative rates:

Table E

Assume 3% really exposed, 3% false positive rate, 3% false negative
Group	Total	Error rate	Test says they are Positive	Test says they are Negative
Actually Positive	30,000	3%	45,000	900
Actually Negative	970,000	3%	29,100	940,900
Totals	1,000,000		74,100	941,800
Percent we assume are actually positive	3%	Accuracy Rating	60.7%	99.9%

Once again, if you test positive in this scenario, that result is only going to be correct about 3/5 of the time (60.7%).

All is not lost, however. Suppose we re-test all the people who tested positive in this last group (that’s a bit over seventy-four thousand people, in Table E). Here are the results:

Table F

Assume 60.7% really exposed, 3% false positive rate, 3% false negative
Group	Total	Error rate	Test says they are Positive	Test says they are Negative
Actually Positive	45,000	3%	43,650	1,350
Actually Negative	29,100	3%	873	28,227
Totals	74,100		44,523	29,577
Percent we assume are actually positive	60.7%	Accuracy Rating	98.0%	95.4%

Notice that 98% accuracy rating for positive results! Much better!

What about our earlier scenario, in table B, with a 5% overall exposure rating, 14% false positives, and 10% false negatives — what if we re-test all the folks who tested positive? Here are the results:

Table G

Assume 25.3% really exposed, 14% false positive rate, 10% false negative
Group	Total	Error rate	Test says they are Positive	Test says they are Negative
Actually Positive	45,000	14%	38,700	6,300
Actually Negative	133,000	10%	13,300	119,700
Totals	178,000		52,000	126,000
Percent we assume are really positive	25.3%	Accuracy Rating	74.4%	95.0%

This is still not very good: the re-test is going to be accurate only about three-quarters of the time (74.4%) that it says you really have been exposed, and would only clear you 95% of the time. So we would need to run yet another test on those who again tested positive in Table G. If we do it, the results are here:

Table H

Assume 74.4% really exposed, 14% false positive rate, 10% false negative
Group	Total	Error rate	Test says they are Positive	Test says they are Negative
Actually Positive	38,700	14%	33,282	5,418
Actually Negative	13,300	10%	1,330	11,970
Totals	52,000		34,612	17,388
Percent we assume are really positive	74.4%	Accuracy Rating	96.2%	68.8%

This result is much better, but note that this requires THREE TESTS on each of these supposedly positive people to see if they are in fact positive. It also means that if they get a ‘negative’ result, that’s likely to be correct only about 2/3 of the time (68.8%).

So, no wonder that a lot of the testing results we are seeing are difficult to interpret! This is why science requires repeated measurements to separate the truth from fiction! And it also explains some of the snafus committed by our current federal leadership in insisting on not using tests offered from abroad.

 

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EDIT at 10:30 pm on 4/25/2020: I found a few minor mistakes and corrected them, and tried to format things more clearly.

The Math Teacher’s Job is Neither to Teach the Lesson, Nor to Help Individual Students Who are Struggling!

….but rather, to prepare a lesson from which ALL the students can learn!

… according to the way that Japanese math teachers are taught their craft, as described below. You will find that these methods, which include Lesson Study, are pretty much the exact opposite of American “Direct Instruction” or “Teaching Like A Champion.”  Given that nobody claims that Japanese students lag behind American ones in math or science, perhaps we in the US could profit from examining how other nations’ teachers do it. Note also that this description is of mathematics lessons in elementary school, not middle or high school.

Please read the following description and leave comments on what you think.

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From Tom McDougal. Lesson Study Alliance, Chicago [and brought to my attention by Jerry Becker. – GFB]

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It’s not the teacher’s job to teach the students!

By Tom McDougal

What?? You might be thinking. What else could the teacher’s job be but to teach?

The teacher’s job is to ensure that students learn, all of them, we hope, though we know we will usually fall short.

In Japan, most (elementary) math lessons are designed as  “teaching through problem solving” lessons (TtP). A teaching through problem solving lesson typically includes the following parts:

 

1.  introduce the problem

2.  explicitly pose the task for students

3.  students work on the task (5-10 minutes)

4.  share student ideas

5.  compare and discuss the ideas for the purpose of learning new mathematics

6.  summarize major points from the lesson

7.  student reflections

(There is sometimes overlap, and a back-and-forth between some of these, e.g. #4 & #5 may be combined.)

While students are working on the task (#3), the teacher walks around the room, monitoring their progress. Japanese educators have a term for this, kikkan shido, or  “providing] guidance between the desks.” They recognize that there are different ways to do kikkan shido, and it is often a subject of discussion in Lesson Study. During planning, for example, a team will usually discuss how – or whether  – the teacher should respond to a student who exhibits a particular misconception; during the post-lesson discussion, there may be argument about whether the kikkan shido was effective. And, it is considered a skill that new teachers need to develop.

Teachers who are inexperienced with TtP lessons often make an unfortunate error while doing kikkan shido: they see a student who is struggling, or who has done something wrong, and they stop and help that student. After several minutes the teacher moves on, encounters another student who is having trouble, helps that student, and so on. Then, suddenly, time is up, and the lesson ends.

There are at least four important drawbacks to this type of kikkan shido. First, as my description suggests, it uses up a lot of time. The teacher may never get around to all of the students, and other students who need help may never get it. Second, by addressing misconceptions privately rather than publicly, the teacher deprives other students of the opportunity to analyze those misconceptions and learn why they are incorrect. Any experienced teacher knows that certain misconceptions are very common, so when one student makes an error that stems from a common misconception, that offers an opportunity to “inoculate” other students against making the same error sometime later.

The third problem with tutoring students individually is that it conflicts with the whole premise of teaching through problem solving. You expect that some, or even all, of the students will have difficulty with the task; that’s why it’s called “problem solving” and not “practice.” Teaching through problem solving involves an expectation that students will have difficulty, but that the comparison and discussion phase will address their difficulties and that, by the end of the lesson, all (or almost all) of the students will have learned what they need to know.

And fourth, we want to help students learn to give viable arguments and to critique the reasoning of others, the third Standard for Mathematical Practice in the Common Core State Standards. To accomplish this, we need for students to share and discuss different, perhaps conflicting solutions. Students need to do the critiquing, not the teacher.

 

Of course, some errors are simply the result of sloppiness, or otherwise unrelated to the main learning goals of the lesson. So when the teacher sees an error while conducting kikkan shido, he or she has to decide: should this be addressed privately or publicly? What should I say to this student? Do I expect that, by the end of the lesson, this student will understand what he or she has done wrong? This is a tricky decision, and an important part of lesson planning is anticipating different student responses, correct and incorrect, and deciding ahead of time how to handle them.

Caring teachers naturally feel drawn to help struggling students: they feel like it is their duty to help those students right now. To counteract that impulse, I say, bluntly:

It is not the teacher’s job to teach the students. It’s the teacher’s job to create a lesson that teaches the students.

 

‘Discovery Math’ is Weird but a Good Idea Nonetheless

This was brought to my attention by Jerry Becker

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From thestar.com, Saturday, September 3, 2016. SEE

https://www.thestar.com/news/canada/2016/09/03/no-teaching-math-the-old-fashioned-way-wont-work-wells.html

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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.

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Paul Wells is a national affairs writer. His column appears Wednesday, Friday and Saturday.

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ALSO THIS RESPONSE TO THE EARLIER POSTING, FROM Michael Paul Goldenberg:

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.

Is Math Necessary?

This is worth reading. It’s a fact that we do NOT have a shortage of trained STEM grads, and it’s also true that very, very few people will ever use any concepts from advanced math in their work or in their day to day lives.

 

(As a former math teacher, I rejoice when I find a way to use relatively advanced math, eg algebra 2 or above, in the real world – which shows you that it doesn’t happen every day, even for someone who’s actively looking for it.)

 

So why do we require every single HS grad to master whatever the current Algebra 2 curriculum consists of?

via Mike Simpson  (remove)

Algebra II Drives Dropout Rates and Is Mostly Useless. Here’s a Plan for Getting Rid of It.

In his new book The Math Myth: And Other STEM Delusions, political scientist Andrew Hacker proposes replacing algebra II and calculus in the high schoo …

SLATE.COM

On the War in Vietnam, and its Memorial

Vietnam Veterans Memorial

This is a starkly geometrical monument to the tens of thousands of American fighting men and women who died in the Southeast Asian theater of war, from 1957 to 1975.

At first, it just looks like a long, black, polished granite V or chevron half-buried in the ground. When you get closer, you begin to see that each of the stone panels has names that were carefully carved into the stone, five names per line, in no immediately obvious order.

But the names are in fact carefully catalogued in chronological order by date of death (or initial injury/wound/disappearance). The order starts in panel 1E, just to the right of the center bend, and continues to the right as you face the wall, up until the single line of names on panel 70E, for May 25, 1968, and then continuing with another single line of names on panel 70W, also for 5/25/68, and continuing from left to right up to panel 1W.

If you want to look up a person who died in that war, you can look their name up alphabetically online or in a printed directory that is the size of a large old-fashioned telephone book. There you can find their branch of service, rank, casualty date, state of origin, and a code giving the panel and line number on which his name is engraved.

Math connection:

1. How many names are carved on the wall?
2. What was the midpoint of the war, as far as US combat deaths are concerned?

The second question is by far the easiest to answer: it was May 25, 1968, because the panels on the left of the vertex are virtually identical to the ones to the right of the vertex in terms of numbers of names.

The first question is a bit harder.

It was suggested by an elementary-school child that one could simply think of each half of the V as a triangle; if we were to use our imagination and flip, slide, or rotate one side above the other one, we would get a rectangle. Then, count the number of lines of names in the tallest panel, multiply by the number of panels, and then by five (because there appear to be five names per line, no matter how long or short the names) and we should have a pretty good estimate.

Here’s the idea:

Using that rule, the tallest panel that is all names (no extra inscriptions) has 137 lines of names. The shortest panel has a single line of names. There are 70 panels to the east, and 70 to the west. Re-arrange them to make a rectangle, and we have 70 panels with about 138 lines of names, each with 5 names per line. Multiply all that and we get 48,300 names.Unfortunately, this number is much too low: more than 55,000 American service personnel died in Vietnam.

What went wrong?

Let’s look at the wall again, more carefully.

Each side is in fact NOT a triangle. They are quadrilaterals. The bottom edge makes a bend at about panel 20, a little bit to the right of the head of the man in blue looking at the wall. It’s easy to miss.

If you spend some time counting and recording the number of lines on a sample of panels, you will notice that the number of lines does not go up uniformly.

I carefully counted and recorded the number of lines of names for about a few dozen panels, and found that my data, when graphed, looked more like this:

(By the way, I didn’t count every single line in all of those panels, except for the ones to the right of panel 30. I found patterns that allowed me to figure out how many names there were in panels 3 through 17. How did I do it?)

But how do we calculate the total number of names now?

My original triangle was easy enough, but the new, green triangle was pretty difficult to measure the area of, so I decided to cut the figure up differently, as shown here:

We have a light blue trapezoid (or close enough) on the left, which will be matched with another trapezoid rotated from the west side, producing a rectangle whose base is 20 panels long and that is about 137+128 lines tall; so it has 20 times 265 lines of names, or 5300 lines of names.

The darker blue triangle will be matched with its twin, also rotated from the west side, to produce a 50 by 128 rectangle with 6400 lines of names.

That’s a total of 11,700 lines of names.

Five names per line give 58,500 American military deaths from the Vietnam War.

Which is quite close to the official figure of 58,272 names.

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Obviously there was more than one country involved in this war. If you want to see memorials to the Vietnamese, Cambodians, Laotians, South Koreans, Australians, and other nationalities, both fighters and civilians, who were killed in that same conflict, you will need to look elsewhere, overseas.

Estimates of the number of Vietnamese, Laotian, and Cambodian killed during and right after the war vary widely and are not very accurate. It is safe to say that several million Southeast Asians were killed during the war and its aftermath.

If one were to build a memorial to all those Southeast Asian casualties with the names spaced in the same way as in this one, how large would that memorial need to be?

For scale: If we begin counting at 1960 and end at 1980, that’s about 20 years, or 240 months. If three million Southeast Asian combatants and civilians were killed during that period, then that means that about 12,500 Vietnamese, Cambodians and Laotians were killed by bombs, bayonet, napalm, bullets, knives, drowning, and so on, each and every single month. In four months, that’s about 50,000 dead, which is pretty close to the total number of American combat deaths for the entire war.

Two hundred forty months divided by 4 months gives 60, which means that one would need approximately sixty memorials the size of this one to honor the memories of the millions of Southeast Asian civilians and fighters killed during and after the Vietnam War.

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Post script:

While touring this memorial on Saturday, November 9, 2013 (i.e. two days before Veterans Day / Armistice Day) with four other math teachers, I was shown a couple of features I hadn’t noticed before:

1. Every ten rows, in every second panel, there is a dot; this makes it much easier to count the total number rows in each panel.
2. A few lines of names actually do have six names. It does not happen very often, but is definitely enough to throw the count off by a bit.

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Some grim calculations in honor of today being Veterans Day (actually, the 11^th day of the 11^th month of 1918 was the day that an armistice (a cease-fire) was proclaimed to stop the fighting on the Franco-German front during World War One.

58,000	Rough estimate of US war dead in Vietnam theater
1,218,000	square feet at 3 fet by 7 ft per coffin
43,560	one acre (in square feet)
28	acres neded to bury all American Vietnam War dead
2,000,000	rough number of Vietnamese, Cambodian and Laotian war dead
42,000,000	square feet at 3 fet by 7 ft per coffin
43,560	one acre (in square feet)
964	acres neded to bury all Southeast Asian Vietnam War dead
1.51	number of square miles needed

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And here is what I just wrote on the time line of a colleague, who had the bad fortune to get drafted during Vietnam and served over there:

“As someone who lived through the Vietnam War era dreading the day when I would be ordered by my draft to report for induction in a war that I knew for a fact was completely unjustified, I am sorry that you and your buddies were drafted and had to fight and risk or lose [your] lives over there and fight against one side in a civil war. (Some wars are justified, many wars are not.) I am glad that my lottery # ended up being a little too high to get called up. I did my best to try to stop the war… I suspect that our efforts, along with the [much more important] actions of many soldiers etc who saw no point in the war, may have saved a lot of lives in getting the US out of there. In any case, I’m glad you made it back safely, Ben.”

A quick look at some of the Common Core math standards, grades 7 & 8

I was prepared to be appalled by the Common Core math standards, but I’m not.

The CC math standards — at first glance, anyway — actually look quite a lot better than the old middle-school math standards we used to have in DC, which had interminable lists of many minute details kids were supposed to know — and which lists repeated themselves over and over again in grades 4, 5, 6, 7 and 8.

Mile wide, centimeter deep it was.

So far, I don’t see any sign of that ridiculous nonsense in the new CC standards.

However, I can just see teachers requiring their students to copy and recite turgid prose like this, which is a direct quote from page 56 of the PDF. It means something to me, but to how many other adults?

“Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

For example, collect  data from students in your class on whether or not they have a curfew on  school nights and whether or not they have assigned chores at home. Is  there evidence that those who have a curfew also tend to have chores?

A Page For Folks Who Might Find that they Like Math

A web page with very fun explorations that show what real mathematicians do:

http://mathlesstraveled.com/2012/02/09/17×17-4-coloring-with-no-monochromatic-rectangles/

(Hint: long division, balancing checkbooks, and adding fractions aren’t such a big part. I suspect that if we taught some of this stuff in class, kids would see that, in fact, math has some fun aspects — it’s not all drudgery. There is a lot of “gee-whiz” stuff that is accessible to the average layperson or young’un.

More Problems With DCPS Curriculum and DC-CAS

Upon taking a closer look at the DCPS standards and the DC-CAS, I submit that they should probably both be ignored by any teacher who actually wants to do right by students. If you are doing a good job teaching the things that students should actually know, it won’t make much difference on their DC-CAS scores. Conversely, if you teach to the DC-CAS, you are short-changing your students.

Case in point: Standards in Geometry and Algebra 1 ostensibly covered on the 10th grade DC-CAS. Recall that all 10th graders at this point in DCPS have supposedly finished and passed Algebra 1, and are enrolled in at least Geometry by 10th grade.

I have prepared a little chart giving the standards (or learning objectives) for Geometry: the ones listed in the DCPS list of learning standards, and the number of questions that I found on the page of released DC-CAS questions that supposedly address that standard. There is almost no correlation at all. In fact, if you threw a dart at the topics and chose them randomly, you would do a better job than the test-writing company did.

Why Does EVERYBODY need to learn Advanced Algebra?

An interesting article by Dennis Redovich:

http://www.ednews.org/articles/294-what-is-the-rationale-for-requiring-higher-mathematics-proficiency-for-all-k-12-students.htm