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MathInSalinaKansasPart2 23 Jun 2006 - 13:28 CatherineJohnson
re: MathInSalinaKansas Wow. I spoke yesterday to a mathematics professor at a university here in New York state. When I asked him what level of mathematical knowledge entering freshmen bring to their course work, he said, "We can't assume that a student knows anything we would want him to know." Specifically, his students can't do algebra. They can't set up a two-variable word problem and solve it. These are college freshmen. Posted on May 07, 2005 @ 11:21
MathsInEngland 26 Jul 2005 - 17:10 CatherineJohnson
I have no idea how I got to this link, so can't give credit....
Maths in England sounds even worse than here, if that's possible, which I suppose it isn't:
Lost count of gloomy reports about the state of maths in schools and universities? For more than a decade mathematicians have been moaning and the government has responded with inquiries, changes in the curriculum, numeracy hours in primary schools, golden hellos for maths teachers and a plethora of other initiatives in England.
Golden hellos, you say. Sounds good to me. Think I'll knock off here and go learn some more Russian Math. Which is an especially good idea given the paragraphs that follow:
Where will the next generation of UK mathematicians come from, asks the group, drawn from university maths departments around the country, learned societies and the government's curriculum watchdog. At the moment the answer seems to be "from Russia and Hungary". In many university maths departments nine out of 10 of appointments go to candidates from abroad, while the shortage of maths teachers in schools has got so bad that the Department for Education and Skills has stopped collecting the figures.
Oh, boy. This next part jibes unpleasantly with Loveless's report on the importance of ability tracking for the most talented students:
There is also agreement on the need - outlined by Adrian Smith's report Making Mathematics Count - to boost the numbers of pupils taking A-level maths, the pool from which science graduates (and future maths teachers) will come. Maths has gone from the largest A-level entry to third place as numbers have dropped by nearly half from 80,000 in 1989 to 49,000 in 2002. A curriculum for the most able 25% of pupils is needed to encourage them to progress to A-level, says the report, which also suggests awarding more university admissions points for a maths A-level than other subjects. Dr Gardiner wants a national debate. He argues that in the last 15 years or so, "much of our mathematics teaching, and most of our assessment at all levels, have become fragmented - with multistep tasks being routinely reduced to (and assessed as) a collection of unrelated 'one-step routines'". The upshot, he says, is that maths undergraduates cannot solve the kind of problems that 13-year-olds used to be expected to do. He adds: "Students in general are no longer required to combine simple techniques in the most basic ways - so they no longer understand that the power of elementary mathematics lies in the integration of simple techniques into larger wholes.
This is an interesting assessment of the problem, in terms of Saxon Math versus Singapore Math. From the get-go--and I mean from the 1st or 2nd grade--the Singapore curriculum (the old one, at any rate) asks children to do multi-step problems. That strikes me as the right way to go, but of course I can't base such judgments on anything more than what I think I see in Christopher & me as we learn math. Nevertheless, the one aspect of Saxon Math that makes me feel chronically nervous is the one-stepness of the word problems. Christopher and I are now working through Saxon 8/7, which is in theory a 7th grade book, and the word problems are either one-step, or they're two-step problems that we're told upfront are two steps. That can't be right. otoh, I had a fun moment the other day when Christopher, who is, after all, still only 10 years old, solved a problem (probably in the Primary Mathematics 3A Workbook) and then tossed off the comment, 'It's a two-parter,' like some guy in a bar casually mentioning he just wrestled a bear. He thought he was hot stuff, doing a two-parter. I loved it. Macho in a 10 year old boy--especially macho about a story problem--is awfully sweet. (OK, maybe that's a mother's perspective.) Still, if he gets manly I-wrestled-a-bear-feelings from doing maths, I say that's a good thing.
Challenging Word Problems Grade 3 book, and I know there are problems in there he's not going to be able to do.
maths in England
maths in England, part 2
more maths in England, part 2
top students in England, US, & Singapore
why do kids like math?
another brilliant person who liked getting right answers (scroll down)
Catherine's cousin talks about Everyday Math
Call for national debate on maths teaching GUARDIAN
Where will the next generation of UK mathematicians come from? (GOVT REPORT: pdf file)
TwoMathEdBlogs 27 Jul 2005 - 19:26 CatherineJohnson
Stephanie just sent me a link to a fascinating list of prerequisites for college math, which includes a terrific Comments thread, at Tall Dark and Mysterious, a blog written by "Twentysomething curmudgeon seeking employment teaching college math in BC." And btw, these are not prerequisites for a serious college math course:
A year ago, I would have posted that list under a heading more along the lines of “Things Students Should Know By Grade Nine”, but alas, experience as extinguished such optimism on my part.
This is long, but it's so valuable I'm quoting the entire list, which I'll probably 'archive' over on....the 'math lessons' page? Another Content Question for the folks at Information Architecture, Inc. (Definitely read the Comments section as well):
Based on my experiences, students graduating from high school should, in order to succeed in even the most basic college math classes: 1.Be able to add, subtract, multiply, and divide fractions. Moreover, they should understand that the horizontal bar in a fraction denotes division. (Seem obvious? I thought so, too, until I had a student tell me that she couldn’t give me a decimal approximation of (3/5)^8, because “my calculator doesn’t have a fraction button”.) 2.Have the times tables (single digit numbers) memorized. At minimum, they should understand what the basic operations mean. For instance, know that “times” means “groups of”, which will enable them to multiply, for instance, any number by 1 or 0 without a calculator, and without putting much thought into the matter. This would also enable those students who have not memorized their times tables to figure out what 3 times 8 was if they didn’t know it by heart. 3.Understand how to solve a linear (or reduces-to-linear) equation in a single variable. Recognize that the goal is to isolate the unknown quantity, and that doing so requires “undoing” the equation by reversing the order of operations. Know that that the equals sign means that both sides of the equation are the same, and that one can’t change the value of one side without changing the value of the other. (Aside: shortcuts such as “cross-multiplication” should be stricken from the high school algebra curriculum entirely - or at least until students understand where they come from. If I had a dollar for every student I ever tutored who was familiar with that phantom operation, and if I had to pay ten bucks for every student who actually got that cross-multiplication was just shorthand for multiplying both sides of an equation by the two denominators - I’d still be in the black.) 4.Be able to set up an equation, or set of equations, from a few sentences of text. (For instance, students should be able to translate simple geometric statements about perimeter and area into equations. ) Students should understand that (all together now!) an equation is a relationship among quantities, and that the goal in solving a word problem is to find the numerical value for one or more unknown quantities; and that the method for doing so involves analyzing how the given quantities are related. In order to measure whether students understand this, students must be presented, in a test setting, with word problems that differ more than superficially from the ones presented in class or in the textbook; requiring them only to parrot solutions to questions they have encountered exactly before, measures only their memorization skills. 5.Be able to interpret graphs, and to make transitions between algebraic and geometric presentations of data. For instance, students should know what an x- [y-]intercept means both geometrically (”the place where the graph crosses the x- [y-]axis”) and algebraically (”the value of x (y) when y [x] is set to zero in the function”). 6.Understand basic logic, such as the meaning of the “if…then” syllogism. They should know that if given a definition or rule of the form “if A, then B”, they need to check that the conditions of A are satisfied before they apply B. (Sound like a no-brainer? It should be. This is one of those things I completely took for granted when I started teaching at the college level. My illusions were shattered when I found that a simple statement such as “if A and B are disjoint sets, then the number of elements in (A union B) equals the number of elements in A plus the number of elements in B” caused confusion of epic proportions among a majority of my students. Many wouldn’t even check if A and B were disjoint before finding the cardinality of their union; others seemed to understand that they needed to see if A and B were disjoint, and they needed to find their cardinality - but they didn’t know how those things fit together. (They’d see that A and B were not disjoint, claim as much, and then apply the formula anyway.) It is a testament to the ridiculous extent to which mathematics is divorced from reality in students’ minds that three year olds can understand the implications of “If it’s raining, then you need an umbrella”, but that students graduating from high school are bewildered when the most elementary of mathematical concepts are juxtaposed in such a manner.) 7.More generally: students should know the basics of what it means to justify something mathematically. They should know that it is not enough to plug in a few values for x; you need to show that an identity, for instance, is true for all x. Conversely, they should understand that a single counterexample suffices to show that a claim is false. (Despite the affinity on the part of the high school text I am working for true/false questions, the students I am working with do not understand this.) Among the educational devices to be expunged from the classroom: textbooks that suggest that eyeballing the output of a graphing calculator is a legitimate method of showing, for instance, that a function has three zeroes or two asymptotes or what have you.
Another blog by a college calculus professor: Learning Curves
MikePiscalOnPublicSchools 01 Aug 2005 - 21:13 CatherineJohnson
Go read Mike Piscal right now. You might want to scroll down and begin with his first post, which ends with this:
There are four special interests that have blocked, clogged, and undermined reform for decades. It is all about money, control, and power. It is diseased value system that leaves our kids uneducated, exposed to violence and drugs, and with too few or zero opportunities to pursue the American Dream. Who are the four? Emphatically, I name names: the teacher’s unions, the University Schools of Education, the bureaucracies, and (unbelievably) the PTA’s. In my blogs, I will name the leaders of these entities and expose their lies, their self-interest, and their unwillingness to change the status quo.I'm looking forward to hearing what he has to say about the PTA.
Here's Thomas Toch, of Brookings:
The PTA has particularly strong ties to teacher unions. Charlotte Frass, chief Washington lobbyist for the American Federation of Teachers, said, "We often lobby together." Ties are even close to the nation's other leading teachers union, the National Education Association. One of the PTA's three Washington lobbyists is married to an N.E.A. lobbyist, and from the founding of the PTA's Washington legislative office in 1978 through 1993, its lobbyists were housed in rent-reduced offices in the N.E.A.'s headquarters a few block from the White House. Like the unions, the PTA pushes relentlessly for more federal education financing. Earlier this year more than 200 PTA political activists descended on Capitol Hill, urging members of Congress to back the Clinton administration's proposals for $25 billion in federally subsidized school-construction bonds and $5 billion in grants to reduce public school class sizes. The organization rejects the belief of many would-be school reformers today that public schools would work harder to improve if they had to compete for students and financing. "There are always winners and losers in a marketplace," Maribeth Oakes, the PTA's legislative director, said, "and we shouldn't have an education system where there are losers." The group has backed charter school laws only if they require that the hybrid public schools report to traditional school boards. Critics contend that strips the schools of the very independence that is the basis of the charter concept.
And here's Chester Finn:
[the PTA has] been politicized, ideologized, bureaucratized and, at least in the PTA's case, has become part of the public-education establishment, more interested in propping up institutional claims and employee interests than advancing the interests of parents and kids. 'All T and no P' is how I've come to describe the National PTA and its state affiliates. ... I can't name a single policy issue of consequence at the state or national level where the PTA's testimony doesn't mirror that of the NEA and/or AFT.(thanks to Illinois Loop)
BasicCollegeMathematics 02 Aug 2005 - 01:14 CatherineJohnson
A round-about path from Vlorbik to Tall, Dark, and Mysterious to Basic College Mathematics at mathnotes.com. Scroll down.
HighSchoolAlgebraTexts 02 Aug 2005 - 15:49 CatherineJohnson
Temple and I are writing an op-ed about American high schools, and I just came cross a treasure trove of PowerPoint slides filled with Horror Statistics, so naturally I had to stop dead in my tracks and get one posted on ktm......
This is a case where PowerPoint has a distinct advantage when it comes to conveying the Bad News. The whole entire key to conveying bad news on PowerPoint is:
source: PowerPoint presentation on U.S. high schools at U.S. Department of Education
NobodyKnowsPhysicsEither 11 Aug 2005 - 21:11 CatherineJohnson
So today I'm sick. I'm sick thanks to the 5 hours I spent sitting on United Airlines Flight 682 breathing recirculated air. Plenty of time to breathe and re-breathe ambient viral particles and really make them stick. Speaking of 5-hour stints in the penalty box, I gather from the Chicago Tribune coverage of this event that not only does no one carrying a United States passport know Thing One about math, we're clueless on the subject of physics, too:
A wingtip-to-wingtip brush between two United Airlines planes waiting to take off from Chicago O'Hare International Airport on Monday frayed the nerves and patience of many of the 223 passengers aboard the two aircraft, but it injured no one.OK, I don't know any physics, either....but I do know that two objects the size of an Airbus 320 cannot be said to brush. No, indeed. Two objects the size of an Airbus 320, when they come into contact, can only be said to collide. Especially when we are describing this event from the point of view of the tiny human passengers sitting inside. (Of course, if we are describing the event from the point of view of official airline spokespeople, which apparently we are, that's different.)
Ah-hah. Just as I suspected.
An object with small inertial mass changes its motion more readily, and an object with large inertial mass does so less readily.Thank you, Wikipedia. In the case of our wingtip-to-wingtip brush, my plane, the one sitting there in the penalty box minding its own business, was actually pushed sideways across the tarmac, causing most of us to believe, for a split-second, that we were crashing, just like the plane in Toronto! or even just like the plane that fell into Rockaway! My point being: if you're going to hit a stationary Airbus hard enough to move it sideways, we're not talking wingtip-to-wingtip brush. We're talking large inertial mass. We're also talking scaring the bejesus out of the folks inside.
While we're on the subject of Media Inaccuracies, let me add that there were also no ensuing pleasantries from the flight crew, as this passage seems to suggest:
Passengers on Flight 682 had the benefit of soft drinks and a film that was in progress, as that flight had already been delayed for more than an hour.OK, yes, we 'had' the 'benefit' of soft drinks and a film that was in progress, MONSTER IN LAW, starring Jennifer Lopez and Jane Fonda. But all that was in the past. The film was ending when the other plane hit, the soft drinks were long gone, and the stewardess had refused to hand out pretzels until we were 'in the air.' I hadn't eaten since 9 AM that morning; it was now 3:00. Finally, around 5, a stewardess brought around some cups of juice. And that was it. There was so little interest displayed in our well-being or wishes—all of us wanted off the plane—that the guy next to me said they were probably going to give us a goodbye kick when we finally walked out the door. The skies are not so friendly at United these days, I think.
StatisticsAndMiscarriageOfJustice 18 Aug 2005 - 01:13 CatherineJohnson
Here is the heart of John Kay's column about the counterintuitive nature of Bayesian statistics:
Last month, the General Medical Council struck off Professor Sir Roy Meadow, the paediatrician, from the medical register. He had given misleading evidence in the criminal prosecution of Sally Clarke, whose two infants died in their cots. When Mrs Clarke was charged with their murder, Sir Roy told the jury that the chances of two successive cot deaths in the one family was “one in 73m”. But although the disciplinary committee heard evidence from distinguished statisticians, it does not appear that they understood the application of probability theory to such cases any better than Sir Roy. The committee found that he had underestimated the incidence of cot deaths, and that he had not taken account of genetic and environmental factors that mean a household that experiences one cot death is more likely than average to suffer another. But even if you recognise these effects, his key conclusion remains valid. It is unlikely that such an accident would have happened at all. It is very unlikely indeed that such an accident could have happened twice in the same family. Of course it is unlikely. The events that give rise to criminal cases are always unlikely, otherwise the courts would be unable to deal with the backlog. If Osama bin Laden is ever brought to justice, the question will not be “is it likely that two aircraft hit the World Trade Center on September 11?” – to which the answer is no – but “given that two aircraft did hit the World Trade Center on September 11, is it likely that bin Laden was responsible?” Confusion of these two separate issues has become known as “the prosecutor’s fallacy”. A cot death in a family increases the probability that there will be another, but a murder in a family may well increase the probability of another murder by even more: wicked parents may continue to be wicked. Sir Roy might have been right to conclude that two cot deaths were more suspicious than one. But the Court of Appeal, releasing Mrs Clarke, was certainly right to have concluded that this statistical evidence could never, on its own, establish guilt beyond reasonable doubt. You should not trust doctors, or lawyers, with probabilities; and be very hesitant about trusting yourself. Adversarial legal proceedings are a bad forum for unravelling technical issues. And we cannot expunge collective responsibility for mistakes by excoriating selected individuals. The business and financial system, more than Bernie Ebbers and Henry Blodget, was to blame for the dotcom boom and bust. Failures in legal processes, rather than over-confident professors, led to the unjust conviction of women such as Sally Clarke.
I'm going to add this story to my collection of Cautionary Tales illustrating Why People Should Learn Math. Until today it hadn't occurred to me people should learn math so they don't get their license to practice medicine yanked when they bumble a statistics and probability question in court. Let that be a lesson to us.
low birth weight paradox (& Monty Hall)
Monty Hall, part 2
false positives, part 2
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
CollegeReadiness2005 18 Aug 2005 - 16:56 CatherineJohnson
Many Going to College Are Not Ready, Report Says
poll on education (probably available only to subscribers):
The most starkly polarizing subject is math, with 37 percent citing it as their least favorite subject and 23 percent citing it as their favorite. The respondents' disdain for math almost triples that of science, which came in at 13 percent. English was respondents' second-favorite subject: Twenty-one percent liked it best and 18 percent liked it least.
AlanGreenspanOnRisingInequality 21 Aug 2005 - 02:25 CatherineJohnson
I'm going to start posting this email from NYC Math Forum at NYC HOLD once a month:
In the matter of preaching to the choir, C-Span has a video of Alan Greenspan's testimony to the House Joint Economic Committee. There is a fascinating exchange between Greenspan and Senator Reed about the divergence in income between skilled/supervisory workers and unskilled workers. They agree this is a very serious problem. At one point, Reed asks what short term policies can be implemented to "enhance the incomes of most of the workers of America. I transcribed about two minutes of testimony which you can hear for yourselves, starting around minute 34:00 of the video clip. Greenspan: Well, Senator, I don't think there are short term policies, other than the ones we typically use to assuage those who fall into unemployment or policies in the tax area in which we endeavor to redistribute income. The basic problem, as we have discussed previously, as best I can judge, goes back to the education system. We do not seem to be pushing through our schools our student body at a sufficiently quick rate to create a sufficient supply of skilled workers to meet the ever-rising demand for skilled workers which means that wage rates are accelerating. But the very people who have not been able to move up into the education categories where they become skilled overload the lesser skills market and cause wages to be moving up well below average. The consequence, of course, is an increased concentration of income. And, as I have often said, this is not the type of thing which a capitalist democratic society can really accept without addressing. And as far as I am concerned, the cause is very largely education. It is not the children because at the 4th grade they are above the world average. Whatever it is we do between the 4th grade and the 12th grade is obviously not as good as what our competitors abroad do because our children fall below, well below, the median in the world, which suggests that we have to do something to prevent that from happening and I suspect, were we able to do that, we will indeed move children through high school, into college, and beyond in adequate numbers. As indeed we did in the early post WW II period, such that we do not get the divergeance in income which is so pronounced in the data we currently looked at.
Rising inequality has been Topic A for months now (make it years) with the WALL STREET JOURNAL & the NEW YORK TIMES both running major several-part series on the subject. Rising inequality alond with declining social mobility. Well, what is the reason for rising inequality and declining social mobility? Is it just that the rich get richer? (Which seems to be the thesis of everything I read, but don't go by me.) I'm with Alan Greenspan. It's basic supply and demand. If you don't have enough highly educated people to fill jobs requiring highly educated people, those wages go up. If you have too many highly uneducated people to fill jobs where advanced education isn't a requirement, those wages go down. Now I'm going to indulge in some psychologizing, which generally speaking I don't approve of. I think the reason journalists don't bring up this possibility is that journalists, being highly educated, and NOT being highly educated when it comes to math & economics (I speak from experience), just naturally tend to assume that of course the wage gap between them and the custodial staff is widening; what journalists do is lots more valuable. (I'm only dinging journalists here because I'm talking about journalism. I'll hazard a guess that just about every highly educated person other than Alan Greenspan thinks the same thing.)
Alan Greenspan on rising inequality
rising inequality, part 2
rising inequality, part 3
median income families UCSC students
another statistics question
channeling the Wall Street Journal
Financial Times on US college costs
Economist on US higher ed
The Economist on rising inequality in universities
MerckVioxxCase 25 Aug 2005 - 22:57 CatherineJohnson
Massive judgment against Merck last Thursday:
Jurors who voted against Merck said much of the science sailed right over their heads. "Whenever Merck was up there, it was like wah, wah, wah," said juror John Ostrom, imitating the sounds Charlie Brown's teacher makes in the television cartoon. "We didn't know what the heck they were talking about."
Yeah, that explains the $253 million verdict.
Then there's this, from the Chicago Tribune:
Stringer, 38, of Chicago, suffers from a spinal condition that causes nerve pain, particularly in her hands. She lives with pain all of the time and gets little relief from other drugs. She is saving her last three precious Vioxx tablets for the really bad days. "Some people say other drugs work just as well, but they don't understand pain," Stringer said in a telephone interview last week from her home on Chicago's Northwest Side. "Mentally, you want something else to work, so if there is some psychosomatic thing, then it should work. But other drugs don't." Stringer is among thousands of patients who swear by Vioxx despite its recently publicized health risks--including one 2000 study that found its use increased the risk of heart attack by a factor of five when compared to someone taking Naproxen, an older painkiller. More than 20 million people took Vioxx for relief of arthritis and other types of pain. An estimated 75 million Americans, or one in four, are living with chronic pain, according to The National Pain Foundation. Stringer and others cannot obtain more Vioxx now because Merck pulled the arthritis drug off the market in September after a study linked its steady use with increased risk of heart attack and stroke.
EducationReporting 27 Aug 2005 - 18:39 CatherineJohnson
Alright, I HAVE to GO PACK..... BUT FIRST! A terrific post at eduwonk by guest blogger Charles Pyle, of the Virginia Department of Education, on NCLB:
Does the public understand the law? I know I have spoken with quite a few reporters over the last few days who are still trying to grasp its finer points.... Education reporting these days is all about data. I have been amazed during the week since Virginia released its preliminary AYP ratings at the number of newsrooms at newspapers in medium and smaller cities in the state without Excel! These reporters can't open online or attached spreadsheets or download data.
SecondCareerTeacher 29 Aug 2005 - 17:27 CatherineJohnson
I just found this Comment:
I am a 2nd career math teacher in my third year in the classroom. I finished the work for my Masters of Education in the spring. During the discussion with my advisor about my final portfolio he told me he expected multiple samples of evaluations which were "non-threatening and non-judgmental"! This is what is being taught to our new teachers today. I am 51-years-old and know better than to believe this nonsense. However, my resistance nearly kept me out of the classroom. My student teaching "mentor" told me more than once that I better shape up or that I would never teach a day. We ended the term with me repeating verbatim his 1st period Algebra 2 "lecture" to the other 4 classes. Unbelievably, he limited the amount of time he spoke to 5 minutes per period. The balance of the hour was spent on "collaborative" projects. I could go on, but my point has been made. As parents we can not give up the good fight. It is our duty to our children and to our communities.
I hope this commenter will tell us more. I'm struck by the fact that his or her professor was doing no lecturing at all, contrary to what most of us had assumed. (I know I had thought constructivism was being taught via direct instruction....) I have questions. What caused your 'mentor' to warn you about shaping up? And why did you have to repeat his algebra lecture? (Or do I misunderstand? Did you repeat his algebra lecture to algebra students? As part of student teaching?) Can I ask what your first career is? Educational background? Did you work in a math-related field??? I'm intensely curious! I agree with you about a duty to our communities. It's more than just our own kids. It's the country. This is why I'm never quite able to tell myself just to stop talking about it and focus on Christopher. I don't like what I see happening to some of the children I know (though certainly not all). I should probably add that in my own school district the problems aren't exactly about constructivism, discovery learning, & overall fuzziness. I guess my feeling about my own school is that we have the same American failings everyone else has. I think I would say there are three glaring weaknesses here in Irvington: curriculum, which just got much worse with the adoption of TRAILBLAZERS, of course; lack of transparency; and lack of formative assessment. Kids fall through the cracks, and by the time a parent figures it out a lot of time has been lost. I've been told, on the QT, that it is unofficial policy here to tell parents their children are doing better than they are, the better to keep parents at bay. I suspect there's a fair amount of truth to this view. What I know for a fact is that Christopher failed 2 out of 3 unit tests in 4th grade math and we never heard 'boo' from the school. No one mentioned it; no one brought it up. I was working so hard on my book that I missed the Unit 5 failure; if I'd missed the Unit 6 test, too, Christopher would have been in serious trouble.
KleinOnMultiplicationTablesInLA 02 Mar 2006 - 22:14 CatherineJohnson
via NYC HOLD, A major math mess by David Klein (registration required):
In March 2000, math specialists in Los Angeles Unified School District estimated that 60 percent of L.A.'s eighth-graders did not know the multiplication tables.
At Cal State Northridge (CSUN), where I am a math professor, many students enroll with mathematical skills below the fifth-grade level. Some of them do not know the multiplication tables and rely on calculators instead. Through spring semester 2002, the CSUN math department controlled the remedial math program. It was well-run by one of my colleagues, with a passage rate of 81 percent. The program was regarded as a model by other institutions. Given the weak math skills of entering students, it is hard to imagine a higher honest success rate. Here, unfortunately, is where racial politics enters the picture. The 81 percent passage rate - however impressive in context - was not high enough for the Pan African Studies and Chicana/o Studies departments at CSUN. Both departments wrote open letters denouncing the math department. Pan African Studies wrote on behalf "of black and brown student clientele regarding the structure of the program, the ambivalence and/or elitist attitudes of some of its instructors and the high failure rates in the developmental math courses." In criticizing the failure rate, Chicana/o Studies argued "that the math department has developed a culture that rejects students who are not math majors," and wrote, "the reaction of the math department is surprising since we believed that the university had progressed in the past 30 and some years." [snip] Besides citing the failure rate of 19 percent, the math department's critics gave no other evidence to support charges of racism, elitism or other accusations. Many of the remedial math instructors were themselves Latino, and all worked tirelessly to help the students, including tutoring outside of class. Not only did the math department have a paper trail to prove the effectiveness of its program, it also had extremely high student evaluations to match. Nevertheless, attempts by the math department to defend itself from charges of racial insensitivity, etc., were ignored by the CSUN administration. Control of the program was taken away from the math department - and now no one complains about passage rates. That's because the problem of remedial math education was solved largely by defining it out of existence. In academic circles, any suggestion of racial insensitivity or "whiteness" typically settles an argument in favor of the accuser, with no further questions asked. Unfortunately, not only is mathematics education susceptible to race-identity politics, it is also undermined by corporations and the federal government. Corporate foundations and federal bureaucrats have awarded multimillion-dollar grants for the development of math programs that include multicultural platitudes but which undermine arithmetic and algebra competence. Meanwhile, other CSUN policies also drive the cycle of remediation. Much to my chagrin, students on my campus are allowed to use calculators during the arithmetic final exam for future elementary-school teachers. Math professors who teach the arithmetic course for future elementary-school teachers, such as myself, are required to allow all students to use their calculators on the exam that tests their understanding of how and why arithmetic "works." The inescapable fact is that California expects more competence in arithmetic from its elementary-school students than CSUN expects from its future teachers. Since 1998, schoolchildren have not been allowed to use calculators on the state's annual standardized tests, and with good reason. Through its own policies, CSUN drives the cycle of remedial math by sending teachers into the field who sometimes lack proficiency in basic arithmetic. Many of the CSUN- trained elementary school teachers are highly qualified, excellent teachers, but others squeak through with teaching credentials in spite of not knowing arithmetic. Ethnic studies departments, corporate foundations and at least one Cal State University campus have found common cause in supporting educational programs that ultimately deprive California's future elementary school teachers of basic arithmetic skills. These misguided agendas should be confronted directly by the public and by its elected representatives.
SchmidtCoherentCurriculum 26 Sep 2005 - 20:03 CatherineJohnson
It had been awhile since I'd last read William Schmidt's American Educator article, Coherent Curriculum. I'd forgotten this section:
Some people might ask, “What difference does it make if we can’t do fancy math problems?” It does make a difference. A typical item on the TIMSS 12th-grade math test shows a rectangular wrapped present, provides its height, width, and length, as well as the amount of ribbon needed to tie a bow, and asks how much total ribbon would be needed to wrap the present and include a bow. Students simply need to trace logically around the package, adding the separate lengths so as to go around in two directions and then add the length needed for the bow. Only one-third of U.S. graduating seniors can do this problem, however. This is serious.
Lately I've been seeing the claim that our seniors blow off the TIMSS test, while Asian kids spend weeks in grueling preparation. Color me not impressed. A 17 year old should be able to do this problem in his sleep.
ItCouldBeALotWorse 10 Oct 2005 - 15:05 CarolynJohnston
The story of progressive math education in Israel since the 1960s, as Ron Aharoni tells it, is both horrible and incredible.
As an academic subject, mathematics education is very young, and all of us have had the misfortune of being its guinea pigs. Arguably, Israel has paid a higher price than anywhere else in the world for this experimentation. The American "new math" reform of the 1960s was brought to Israel in a most strange and extreme form, called "structuralism" by its authors. It was imposed on practically all Israeli schools for a full quarter of a century. Its story may reflect on the politics of education, not only in Israel. ... In this case, new math "structuralism" meant that no concept or operation was taught directly or through its meaning. For every concept there was a "representation", or substitute, whose study was supposed to lead to an understanding of the original concept. The four operations were taught using Cuisenaire rods. A face-like picture into which three numbers are put, two in the places of the eyes and one at the mouth, was supposed to teach children when to add and when to subtract. If the two numbers were at the eyes, and the missing number was at the mouth, it was an addition problem. If one number was at an eye and another at the mouth, it was a subtraction problem. (The children were made to recite: "Eye and eye is plus; mouth and eye is minus.") Division was taught as the reverse operation of multiplication, using so-called "multiplication rectangles." Most extreme was the teaching of the decimal system. Instead of the principle of the collection of tens, strange creatures were invented called bodytails, which had bodies representing the tens and tails representing the units. And those are just a few of many such devices. In international mathematics assessments, Israel dropped from first in the world in 1964 to 29th place in 1999 -- behind everyone except the developing nations.Excerpted from Aharoni, Ron, What I learned in Elementary School. American Educator, Fall 2005.
Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)
HotUnderTheCollar 13 Nov 2005 - 00:11 CarolynJohnston
AnotherCMPStory 14 Nov 2005 - 14:22 CarolynJohnston
This was posted on the PenfieldInTheNewYorkTimes thread today by CharlesH. In the spectrum of mistakes made in the Connected Math series, this is at the other end from MySpecialNumber: projects that are so difficult and time-consuming that, in the end, the child learns nothing from having done them. Here's Charles:
Giving people a choice is the democratic thing to do and is also a good political strategy. It should satisfy everyone. It's an inoffensive offensive. But I doubt that zealous educationists in a position of power will go along. Being responsive to reasonable popular wishes is not their thing. I also suspect that many parents are not conversant with the math issues and won't know what to do with choice. Just yesterday I talked to a parent of a sixth grader I am tutoring in math who had no clue of fuzzy math. (I tutor disadvantaged kids after hours in addition to my regular classes.) I was helping the kid do homework. Part of the homework required the kid to cut a sheet of paper into strips to make various fractions. The parent was aghast and thought it was a time-waster. I had to explain the purpose of the exercise. It was all news to her. The school the kid is in uses the fuzzy series Connected Math. The homework assignment was quite demanding and way beyond the kids abilities. She had neither a conceptual understanding of the task nor the requisite tools (computational skills, procedural knowledge, math facts) to accomplish the task had she had a conceptual understanding of the problem. This is a key problem with fuzzy math. It is quite pretentious on the one hand, and refuses to teach the necessary skills on the other. The result: the kid was hopelessly drowning and getting straight F's. Now what was the task? It was a real-world problem. A class was holding a fundraiser to raise $300.00 in ten days. The progress was shown in the form of thermometers showing progress in two-day increments. The thermometers were all 8 1/2 inches long and showed the money raised so far on the various days in red. The fraction strips were to be used to determine the amount of money raised so far on the various days and then to plot the progress in a coordinate plane. The kid was to make the strips and mark fractions from 1/2 to 1/12 on the various strips. Making fractions strips of 1/2, 1/4 and 1/8 is of course easy. It's not so easy to come up with 1/3, 1/5, 1/12. You could do time-consuming trial-and-error folding. Or you could divide 8 1/2 by the INVERSE of the various fractions. This adds another layer of complexity far beyond the capabilities of the mathematically crippled kid. Dividing by the fractions tells you how many fractions should be on the strip, e.g. 8 1/2 divided by 1/5 gives you 42.5 Dividing by the inverse gives you the length of the fraction (1.7). Now convert 1.7 to 1/16's so you can use the inch ruler!!! Now try doing this with the remaining fractions up to 1/12's as required by the CMP task used as a homework assignment! Even if you can get the numbers they don't work well with an inch ruler. You could approximate. Suppose you (meaning the kid) could accomplish all that. Then what? She was supposed to find the strip with the right fractions and measure the red thermometer bar (money raised). Then she had to add the fractions and know how to calculate the decimal from the red bar to total thermometer length ratio to finally determine the amount raised to begin plotting. All this without instructions in CMP and without computational skills and procedural knowledge. No wonder the kid is drowning. What a tragedy.
TheEndOfCivilization 30 Nov 2005 - 23:20 CarolynJohnston
Instructivist links to a commentary posted in EdWeek by some "learning specialists" who have noted an alarming trend toward students learning too much stuff. The article is full of hysterically funny snippets. Here's one:
Standardized testing holds teachers and students "accountable" for mastering only skills and knowledge that are established and uncontroversial, asks them to address only questions with a single correct answer. They are not encouraged or allowed to explore the ambiguous, the uncertain, the mysterious; the wonders of the world. History offers sobering examples of what can happen when standardized achievements are elevated over open-ended abilities.What the heck are they talking about? Is there even one example of a society that bit the dust because its people were too high-achieving?
RudbeckiaHirtaAtJoannes 01 Dec 2005 - 00:00 CatherineJohnson
It's taken me awhile to put Rudbeckia Hirta together with her blog, Learning Curves. Or rather, I should say, it's taken me awhile to keep them together....since I used to know that RH writes Learning Curves. Then I forgot. In any case, I've got it now. Learning Curves is fantastic today. There's a terrific math horror story (I keep a collection), an intriguing homework story, a calculator lament, a freshman haiku, and an excellent proposal for ed research that RH may not have the patience for, but I hope someone some day will. Here's the Math Horror Story:
A few years ago I was at JoAnn [Fabrics] (the one on route 35, just south of Red Bank), and there was a woman at the cutting table. She was holding a roll of home-dec fabric and a pattern. The clerk asked her how much fabric she wanted cut. The woman said she didn't know. She was making covers for her dining room chairs, the pattern said that each chair needed 5/8 of a yard of fabric, and she had eight chairs. The clerk didn't know either. They were not wondering whether you could get by with less than five yards of fabric if you arranged the pattern pieces cleverly. No, they had NO IDEA how much fabric she needed.
This is the kind of thing I have to have dust-ups with my husband about. A few months ago, I was obsessing over Bad Fraction Knowledge In American Students, when Ed said, 'Nobody uses fractions.' I'm sure he says these things on purpose. He says he doesn't, but I think he does. Anyway, I pointed out, logically, that I use fractions all the time. When I cook, for instance. Say I want to modify a recipe. I will use fractions. So Ed says, 'Nobody modifies recipes.' Again, typical. One of the Themes of our marriage is the outlier-ness of me. Yes, the implication is, you modify recipes. But you're different. Nobody else does the crazed, obsessive, over-the-top, recipe-modifying, fraction-using stuff you do. Which is hilarious, seeing as how I'm the least outlier-ish person I know. I am practically a walking cliche, I'm so mainstream. UPDATE 9-23-2006: Except when it comes to TV sci fi. So Ed says, 'Nobody modifies recipes,' and I say, 'There's a whole big bestselling book on how to make your own mixes. You know, like my pancake mix. The women who wrote it give lectures and presentations all over the country. They appear on morning talk shows. They use fractions!' And he stopped arguing about fractions and conceded the point! It was great!
This is what gets to me. Ed has an excuse. He's sitting at his kitchen table on a weekend morning, his wife is obsessing over fractions, and he's long since lost all interest in whether American students can or cannot add, subtract, multiply, and divide fractions. He already knows they can't; it's not a burning mystery in his life. He comes up with things like Nobody uses fractions just to liven things up. Plus he hasn't figured out the difference between over the top and outlier. I am over the top. I am not an outlier. Case in point, modifying recipes. Modifying recipes is mainstream behavior, which occurs not infrequently on television cooking shows. At least, I think it does. So Ed has an excuse when he comes up with things like, Nobody uses fractions. But what's everyone else's excuse? Why do we keep hearing that people don't use fractions, or don't use long division, or don't use quadratic equations in everyday life? OK, yes, it seems to be the case that people don't use quadratic equations in everyday life. But fractions? Long division? People don't use this stuff in everyday life? Maybe Ed is right. Maybe I live in a parallel universe where people are ceaselessly modifying recipes or purchasing fabric at JoAnn's or altering knitting patterns or buying paint or laying carpet or what have you, all the while using fractions & long division to do it. Because obviously, this kind of thing doesn't happen in the real world.
InnumeracyInApHistory 03 Mar 2006 - 19:41 CatherineJohnson
Via joannejacobs, a link to Innumeracy and the Persistence of Memory at A Shrewdness of Apes:
All of my grades are based on percentages. I'm not one of these teachers who wants to convert someone's scores in my head, so I just weight grades differently. But all grades are based on 100 possible points. I can tell at a glance how a student is doing this way. But this habit often makes it interesting when students are trying to figure out their grades on quizzes. I usually have a rather simple number of questions in terms of being able to calculate grades easily: 5, 10, 12, 20, 25, or 33 items. As I watched several of my AP students struggle with figuring out their grades, I had to suppress a groan of frustration. It was a 20 item quiz-- therefore each question would be worth 5 points, right? Young Frederick wanted to pull out his calculator to figure out what his score would be if he missed 7. "No calculator. You can do this," I urged. He couldn't begin to figure out how to determine his grade without a calculator. He is 16 years old and taking pre-calculus and other college-track classes (I never took a course beyond algebra 2, much to my chagrin). He doesn't immediately know that 7x5=35, and then subtract 35 from 100, nor can he figure out that 13x5=65. As a matter of fact, he stumbled over the 100-35 part and insisted the answer was 75. It is obvious that his only problem is NOT that he didn't do his reading for my AP US history class carefully enough last night. His problem begins with a basic innumeracy. Of course, many would say that he is a victim of a larger educational trend which I pray to God is finally being placed on the pyre of idiotic educational theories: that rote memorization is bad, bad, baddety bad bad. Frederick has to THINK about what 6x9 is, and he doesn't get that 6x9 is the same as 9x6 is the same as 3x2x9 is the same as (3 cubed) x2, and so on-- that's a related but different problem we could talk about all day. I think it's a crime that Frederick has to waste valuable thinking time on matters such as 6x5, much less 100-35. Frederick has much more complex things to think about, but by the time he gets there, his poor little thinker is all worn out on information he should have committed to recall 7 or 8 years ago. The greatest civilizations of the ages depended upon rote memorization. The Torah was preserved through the power of memory for hundreds of years. The Iliad and the Odyssey were memorized and sung for generations. But somewhere along the line in the last forty or so years, memorizing was a skill that became shameful and vilified by someone among the educational cognoscenti. In the words of some of my students, I would like to find this dude and kick him in the shins. I still remember huge chunks of poetry and music that I had to memorize over twenty years ago.
I mentioned earlier that my mom said I should have Christopher memorize poetry as a way to increase his writing skills. We're probably past that point now (i.e. we're past the point at which I can tell him to memorize something and make it stick), which is too bad. I'm going to buy the book I searched out for this purpose anyway.
why Johnnie can't multiply The comments thread is interesting. Some commenters say Frederick's problem is constructivist math; at least one other says his problem is rote learning:
I totally agree with you. And as to the first poster, even a Business/Consumer Math class requires the basic skill of muliplication...In my opinion, the constructivist approach to math is to blame for what you're (and the rest of us) are seeing. + + + I teach a class in high school called "Numeracy" for students who are still below a 7th grade (and most below 5th grade) level. They have "learned" all of the algorithms that have been pushed into them by rote work and filling out worksheets, but they have no idea which algorithm to use when, or why. I would bet that your student knows how to do the multiplication problem, or could easily do the subtraction problem if you lined up the numbers nicely like on a worksheet. The deficit is in his ability to think mathematically. The real key to numeracy is being able to see into the mathematical nautre of a problem, and select the appropriate response - either something that has been learned and memorized, or a creative response synthesized from understanding the way numbers work. I completely agree that memorization of facts and algorithms is critical - but in math, this only has value once a concept is really understood. Anyone who thinks that the majority of students can really learn to divide fractions (in a lasting way, in a way that will be applicable to rational functions in algebra, to solving word problems, etc.) by "invert and multiply" has never really tried to teach.
I've never learned to find a percent on a calculator (apart from doing the obvious division problem).* Are these kids using a special 'percent' key or function? Is that part of the reason this student needs a calculator to find out what his grade will be if he misses 7 out of 20? Now that I've worked my way through nearly 3 superb K-8 math texts, I'm amused by my own experience with percents. Sometime during my years in school, I learned to set up percent problems as proportions. I was aware that there was a faster way of doing it, and that this way involved DIVIDING. But I absolutely could not remember, for sure, what it was. So I spent my entire adult life setting up percent proportions and solving them via cross-multiplication. I did this a lot. I was constantly finding percents of this or that; I can't even imagine how Richard Cohen has managed to get through an entire adult life not being able to calculate a percent. Otoh, it's entirely possible that the reason I was constantly finding percents was that I wanted to know what percent something was — like 'what percent of this book have I finished reading?' for example. In other words, it's entirely possible that the reason I was constantly finding percents was that I had a bit of a fixation on statistics, and percent was basically the only statistic — along with simple averages & medians — I knew how to figure out for myself. So, uh, I guess I can imagine how a normal human being could get through an entire life not knowing how to figure percent. Maybe. The point is: I had immensely fragmented knowledge, though my fragmented knowledge was not without meaning. Not only could I not remember how one would find percent the simpler, faster, more straightforward way, but I didn't recognize the fact that the simple, faster, more straightforward way was the exact same thing I was already doing setting up a formal, written-down proportion, but just skipping the first couple of steps. If any of you Math Brains out there are wondering what fragmented knowledge looks like, that's it.
Liberty Common School I'm wondering whether Carolyn, Doug, Greta, Chris (& any other ktm readers & contributors in CO) are familiar with this school? Skimming through school policy on math (pdf file) the school sounds wonderful. Here's a school newsletter. (pdf file) Haven't looked at it yet.
*Actually, I think one of my books may have taught a lesson on doing percents on the calculator, but I forgot.
-- CatherineJohnson - 02 Mar 2006
ThoughtsAboutUnderachievement 27 Apr 2006 - 03:26 CarolynJohnston
In response to my post the other day about the effect of Kafkaesque classes on teenagers, Catherine wrote: I think (don't know!) that punitive grading and crazy teachers may be worst for high achiever types.....like Christopher.....who are going to narrow down their interests prematurely... .. and this observation sparked, in turn, some thoughts about underachievement as a protective mechanism. I was a horrendous underachiever. I did horribly in classes everyone suspected (but couldn't prove) I could have aced easily. I was underachieving to the point where it deeply offended my teachers. I remember, one time, getting sent in fury from my Spanish class to the principal because I'd cracked open a book under my desk the instant class began; I hadn't even spared her the few seconds it would have taken me to get bored. I turned around in my senior year of high school and started getting As. I couldn't tell you exactly why, except that my math teacher had finally granted me an "F" in my last quarter of algebra 2 in my junior year, and it shocked me. I was also, I think, bored with myself and wanting to sink my teeth into something. Although it's hard to examine the motives of a teenager even when the teenager is your former self, I think what was going on was that I was hoarding the secret of my potential. If noone knows what you can do -- not even yourself -- then the sky is the limit. Why work hard, when the reward might be that you run up against your limitations? Better to keep silent and have people suspect you are stupid, than to open your mouth and remove all doubt. You might suspect I had low self-esteem, and you might be right (I don't any more, though). Catherine and I have also talked about how she never took any writing courses in high school, and I always completely avoided choir and other venues where I would have to sing. Writing is Catherine's greatest natural talent (she believes), and singing is mine (I believe). We each think, in retrospect, that we were protecting our best-loved activities from being "Kafkafied". Anyway, for what it's worth -- if there's an underachiever in your life -- think for a moment about the advantages underachievement confers the underachiever, and whether the benefits of achievement outweigh them. I don't know if there's anything you can offer to motivate someone who is really afraid they are going to fail if they try -- I also don't know whether Kafkaesque classes can turn a competitive person into a 'protective underachiever'. But it's worth some thought.
-- CarolynJohnston - 25 Apr 2006
TheSongOfTheAmazonBird 11 May 2006 - 22:32 CatherineJohnson
Still on my Quest, I've come across this reader review of Daybook of Critical Reading and Writing by Fran Claggett:
My Opinion..., April 14, 2002
Reviewer: A reader
The reason I gave this book two stars, is because we use this book in our class all of the time. Most of the stories and poems in here are hard to understand and complicated.
I know that you are supposed to use your mind, and there is no right or wrong answer, but you can not use your mind if you dont know what is going on. I keep getting zero's on my daybook assignments, because all I can put in the margins or the pages to write what you think, is that I can't write anything because it was hard to understand, so I get zero's for not understanding, and that to me isnt fair! So, I think that if to this book you tell your opinion, I think that if your opinion is that you didnt understand it, than that should still be counted as "no right or wrong answer".
But besides that, it is a good idea.
the song of the Amazon bird
the song of the Amazon bird, part 2
the song of the Amazon bird, part 3
-- CatherineJohnson - 11 May 2006