Forbes has an interesting guest editorial on its online page. Mr. Joseph Tartakovsky has decided to write an op-ed on the uselessness of studying math. I suppose that Forbes decided to run this in the interest of equal time, given that they've linked to an essay by Diane Ravitch. Tartakovsky is a law student who has no use whatsoever for math, stating "Why teach math in the age of the calculator? The device is available everywhere, from cellphones to fashionable watches."
Like any good lawyer, he backs up his thesis with facts:
"Once a visitor to the Indian prodigy Srinivasa Ramanujan (1887-1920) noted that his cab number, 1729, seemed "rather a dull one." "No," replied Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways." He did that in his head. So what? Give me two minutes and my calculator watch, and I'll do the same without exerting any little gray cells. "
Look Tartakovsky, I hate to be the one to rain on your parade, what with publication in Forbes and all, but I think that even with two calculators and a laptop you wouldn't be able to prove Ramanujan's thesis.
He goes on and completes his proof of the uselessness of math by showing that both Benjamin Franklin and Winston Churchill were bad at math but went on to illustrious careers in spite of it.
Other than the fact that Tartakovsky has no use for math, I don't know what point he is trying to make. I only hope that as our recession worsens, he doesn't write an essay scolding big business for hiring engineers from China and India when so many people in the U.S. need jobs.
Saturday, February 7, 2009
Friday, February 6, 2009
Democrats for Education Reform on the stimulus
But we are very worried that Congress and the Obama administration haven't learned from the recent banking bailouts (bonuses for failing corporate executives?) that massive infusions of cash must be accompanied by significant reform if this is going to be anything along the lines of change we can believe in.
Nowhere is this issue more important than in the bailout which is poised to play out for public education (a bailout that we strongly support, so long as significant changes are made in the ways this money is spent to improve student learning and ensure equity.)
February 3, 2009
talking points:
DFER is encouraged that the House bill specifically includes funds for:
- Teacher Incentive Program ($200 million) – to boost teacher pay, change pay schedules, and improve the quality of teaching in high-poverty, high-minority schools
- Improving State Data Systems ($250 million)
The amounts for each of these functions are relatively (and in some cases, ridiculously) small. They approach rounding error in the context of the $145 billion education package.
- Charter Schools Facilities ($30 million)
We urge the Senate to fund each of these three functions at a level above that which can be found in the House bill.
DFER strongly prefers the more specific language in the House bill on the use of state funds to:
- revamp state assessments; and,
We urge the Senate to adopt the House language on incentives and innovation.
- getting qualified, experienced teachers to high-poverty, high-minority schools.
Talking Points on American Recovery and Reinvestment Plan February 3, 2009
hmmm
The talking points don't seem to jibe with the post introducing the talking points.
Or am I missing something?
In any event, I personally am against pouring more money into the education sector, which now consumes 1/2 trillion dollars a year every year, year in and year out.
Absent profound and sweeping changes in institutional structure and culture, money won't fix what ails our schools. The curriculum and teaching practices in wealthy suburban schools are evidence enough of that.
another helicopter mom
[Michael] Steele once told me a story that speaks volumes about his determination to succeed: Steele attended college at Johns Hopkins University. In his freshman year, Steele said, his grades were so bad that the dean of students told him he would not be invited to return. His mother forced him to go back to the dean and insist he was prepared to do anything to continue his studies. The dean said "no," but Steele's mother made him confront the dean again. This went on for weeks until the dean, in desperation, agreed to readmit Steele with the proviso that any bad grades would end in his dismissal, mother or no mother. Steele went on to graduate near the top of his class at Johns Hopkins and was elected student body president.
Democrats: Beware of Michael Steel
by Bob Beckel
February 06, 2009
realclearpolitics
She sounds like a real pistol, that one.
I like that in a mother.
Wednesday, February 4, 2009
gizmo idolatry
excerpt from:
Gizmo Idolatry by Bruce Leff & Thomas E. Finucane
JAMA 2008; 299(15): 1830-1832
Offhand, I would say the Partnership for 21st Century Skills has an advanced case of gizmo idolatry.
Gizmo Idolatry by Bruce Leff & Thomas E. Finucane
JAMA 2008; 299(15): 1830-1832
It seems that "gizmo idolatry" now exists in the practice of medicine...In this article, gizmo is used to refer to a mechanical device or procedure for which the clinical benefit in a specific clinical context is not clearly established, and gizmo idolatry refers to the general implicit conviction that a more technological approach is intrinsically better than one that is less technological unless, or perhaps even if, there is strong evidence to the contrary. The credulous acceptance and rapid diffusion of frontal lobotomies in the 1930s and 1940s led to great harm, and to a Nobel Prize for Egas Moniz in 1949 “for his discovery of the therapeutic value of leucotomy in certain psychoses.”1 ....
[snip]
Seven overlapping categories of incentives may encourage clinicians and patients to favor the use of gizmos.
Common Sense Appeal
Many gizmos make so much sense, in the absence of evidence or even the presence of evidence to the contrary, that their value or utility is persuasive prima facie.
Human Love of Bells and Whistles
Increasing the technological complexity of treatment appears to increase the significance of an illness and the appeal of an intervention.
Exploits vs. Uneventful Diligence
In The Theory of the Leisure Class, Veblen10 noted that in early societies, the upper leisure class performed high-prestige hunting and military exploits, while the lower classes performed menial work, such as agriculture, child-rearing, and cooking, which was arguably more important to survival of the society. Vestiges of this construct persist in medicine where surgical exploits are valued more highly than uneventful diligence or watchful waiting of primary care. Recovery
from backache can be transformed into a patient’s exploit if magnetic resonance imaging is obtained, and even more so if this leads to surgery. [ed.: this one is likely to be overblown for patients, at least. See: Personality: What Makes You the Way You Are by Daniel Nettle]
Gizmo Utilization as Proof of Competence
The cutting edge or first on the block use of a gizmo can bestow on the physician a mantle of expertise, competence, and preeminence. [SMART Boards!]
Gizmo as Source of Quantifiable, Objective Information
Gizmos are used to provide objective, quantifiable information, often used to rule out a diagnosis. For patients who are frail and face the risk of surgery, technological preoperative testing shows that the patient received a thorough evaluation, although this testing does not produce better outcomes than a thorough clinical examination, assessment of functional capacity, and basic laboratory testing.11 [Data-driven instruction!]
Proof Against Negligence
The risk of malpractice litigation depends heavily on a physician’s communication skills,12 but a totemic belief persists that gizmo deployment reduces the risk of litigation.....the death of a very thin, demented, bedfast nursing home resident with pressure sores may be less likely to lead to litigation if that patient died with dietary supplementation being infused through a percutaneous endoscopic gastrostomy as he or she lay on a specialized bed. [irrelevant to schools]
Channeling Money
Business models, having more to do with money than health care, are created around gizmos. A prostate irradiation therapy known as intensity-modulated radiation therapy is being marketed with great potential to enhance practice revenue for urologists, who reportedly are reimbursed at $47 000 per patient treated.15 Thousands of physicians have purchased, are using, and bill profitably for a handheld device that checks patients for nerve disease.16 Although evidence
of benefit to the patient is uncertain, profits to practitioners and corporate vendors for successful gizmos such as these can be substantial.17 [vendors!]
Offhand, I would say the Partnership for 21st Century Skills has an advanced case of gizmo idolatry.
returns to education: twin study
I had never heard of this study --
Now that Goldin & Katz's book has been published, I would like to see Education Week simply assume that education has an "impact" on productivity.
This paper uses a new survey to contrast the wages of genetically identical twins with different schooling levels. Multiple measurements of schooling levels were also collected to assess the effect of reporting error on the estimated economic returns to schooling. The data indicate that omitted ability variables do not bias the estimated return to schooling upward, but that measurement error does bias it downward. Adjustment for measurement error indicates that an additional year of schooling increases wages by 16%, a higher estimate of the economic returns to schooling than has been previously found.From the Princeton Weekly Bulletin:
Estimates of the Economic Returns to Schooling from a New Sample of Twins
Alan Krueger & Orley Ashenfelter
One of Ashenfelter’s best-known areas of study is the relationship between education and income. Research into the impact of education had been complicated by the fact that many other factors, such as social class or innate intelligence, could contribute to someone’s earning power. Ashenfelter and Princeton colleague Alan Krueger found a way to bypass those factors through an unusual field study.
Beginning in 1991, the researchers traveled to the Twins Day Festival in Twinsburg, Ohio — which attracts thousands of twins each year, young and old — and interviewed more than 600 attendees over three years to collect data on their schooling and earnings. They were assisted by graduate students and, in one year, a pair of identical Princeton undergraduates.
Ashenfelter and Krueger, who also collaborated with Rouse on the study, found that each year of additional schooling equaled an additional 12 to 16 percent in earnings. Because the study compared genetically identical subjects, excluding other factors that were difficult to measure, it was widely considered the strongest analysis to date of the impact of education on earnings.
“What drove it was the fact that we realized there was a cheap and easy way to collect data,” said Ashenfelter, “and it was hysterically funny to go to a festival like that.”
Krueger said, “The twin study was great fun. What I took away from the experience is that Orley never treats research as finished. He thinks long and hard and deeply about economic problems. He also makes research fun. Our paper made a splash because we collected new data … and used simple as well as sophisticated econometric methods to answer a longstanding question in economics.”
Doing What Comes Naturally
by Eric Quiñones
Education Week:
[T]the twin who put in more school time tended almost always to be the higher-wage earner later in life. That was also true, subsequent studies by Mr. Krueger and others revealed, even when the difference in schooling time amounted to just a few months.
“If you look at the evidence at the individual level, it’s overwhelming,” said Barry P. Bosworth, a senior fellow at the Brookings Institution, a Washington think tank. “I’m far more inclined to say that it can be aggregated up to gains for society as a whole. But they don’t let economists run controlled laboratory experiments, so it’s hard to know for sure.”
Mr. Krueger agreed. “I think education matters in the long run,” he added. “In the short run, it’s the business cycle that matters.”
Researchers Gain Insight Into Education's Impact on Nations' Productivity
by Debra Viadero
Education Week April 23, 2008
Now that Goldin & Katz's book has been published, I would like to see Education Week simply assume that education has an "impact" on productivity.
Utah's NASA Math initiative
Thought you might like to know about what Oak Norton has been working on in Utah: (fyi-anything in bold is my emphasis; in red, my addition)
Here's the section on teacher competency testing (starting at line 149):
Have at it with the placement tests here:
6th grade Primary Math 6b placement test.
8th grade New Elementary Math 2 placement test.
What has been called the NASA Math Initiative has now been finalized and assigned a number (SB159). Please read this bill and check out the many features we have been able to pull together in the bill. It's going to be a bit of an uphill struggle still to get it passed but there are some strong supporters of the bill.One of the best features of this bill is that schools that do not get grant money can still adopt the plan and implement it without being required to also follow the Utah standards. They will still take the UPASS test as required by state law, but they won't be viewed on the same playing field as schools on the Utah core. The standards being adopted in the bill are the 2001 Singapore Math Standards!!! Please go read the full text which includes many other features including tests for teachers to ensure they are competent to teach the math they teach.(starting at line144) (b) A school district or charter school may deliver instruction to students who do not participate in the Math Education Initiative in accordance with the math curriculum standards adopted under this section in lieu of other math curriculum standards prescribed by the board.
Link to the bill:
Here's the section on teacher competency testing (starting at line 149):
53A-13-407. Math competency tests for Singapore math teachers.
The board shall make rules in accordance with Title 63G, Chapter 3, Utah Administrative Rulemaking Act, that require a teacher who teaches Singapore math in a school district or charter school that participates in the Math Education Initiative to demonstrate math competency by passing a test as follows:
(1) a teacher who teaches Singapore math in grades kindergarten through three shall pass a grade level six Singapore math test; and
(2) a teacher who teaches Singapore math in grades four through eight shall pass a grade level eight Singapore math test.
Have at it with the placement tests here:
6th grade Primary Math 6b placement test.
8th grade New Elementary Math 2 placement test.
Tuesday, February 3, 2009
IQs up, verbal intelligence down
Andrew Gelman on Science, Nature, & Nature Neuroscience:
Politically speaking, the article, Technology and Informal Education: What Is Taught, What Is Learned, serves my purposes (my purposes being: I'd like my district to stop buying SMART Boards and start assigning lots of good books for homework). But even on a skim-through, I found problems.
Greenfield's most interesting claim, which I believe until I learn otherwise, concerns the famous rise in IQ scores:
In short, it's not "intelligence" that's gone up, it's scores on the visual scales and basic vocabulary.
Verbal intelligence -- that would be the SAT-V kind of intelligence that predicts college success (and, yes, SATs do predict college success) -- is down.
Unfortunately, Greenfield attributes the decline in verbal intelligence to technology, kids, and parents, not schools. Visual media have replaced books at home; therefore kids can't read challenging texts or think because "reading for pleasure" is the key to critical thinking.
So what should the schools do now that kids can't read?
They should stop using books & switch to PowerPoint:
+++++
Number one: I'm aware of no evidence that "reading for pleasure" produces high verbal intelligence scores (though I assume it helps), and I am aware of informal evidence that reading for school is essential.
Number two: you can't teach verbal disciplines by means of pictures. Period. Ed was working on K12 education back when California teachers were attempting to deliver "sheltered instruction" to ELL students. In theory, sheltered instruction involves a number of strategies, but in practice it meant trying to teach language-based disciplines using pictures instead of words. It didn't work.
PLUS, when it comes to PowerPoint, I am as one with Colonel H.R. McMaster (now Brigadier General McMaster):
Kids aren't doing any reading outside school so they should do less reading inside school, too?
Gelman is right.
As Seth points out, the authors write that many of the mistakes appear in "such prominent journals as Science, Nature, and Nature Neuroscience." My impression is that these hypercompetitive journals have a pretty random reviewing process, at least for articles outside of their core competence of laboratory biology. Publication in such journals is taken much more of a seal of approval than it should be, I think. The authors of this article are doing a useful service by pointing this out.Coming across this observation was a case of synchronicity, because I had just yesterday put the finishing touches on a citizen's op ed re: technology vs critical thinking, which I was inspired to write after skimming an article in SCIENCE.
Suspiciously high correlations in brain imaging studies
Politically speaking, the article, Technology and Informal Education: What Is Taught, What Is Learned, serves my purposes (my purposes being: I'd like my district to stop buying SMART Boards and start assigning lots of good books for homework). But even on a skim-through, I found problems.
Greenfield's most interesting claim, which I believe until I learn otherwise, concerns the famous rise in IQ scores:
In the midst of much press about the decreasing use of the print medium and failing schools, a countervailing trend may come as a surprise: the continuing global rise in IQ performance over more than 100 years. This rise, known as the Flynn effect, is concentrated in nonverbal IQ performance (mainly tested through visual tests) but has also occurred, albeit to a lesser extent, in verbal IQ (1-5).... Increasing levels of formal education and urbanization were particularly important [as causes of the rise in IQ] in the United States and Europe in the first half of the 20th century (9, 10). More recently, technological change may have taken the dominant role. The changing balance of media technologies has led to losses as well as gains. For example, as verbal IQ has risen, verbal SATs have fallen. Paradoxically, omnipresent television may be responsible for the spread of the basic vocabulary (11) that drives verbal IQ scores, while simultaneously the decline in recreational reading may have led to the loss of the more abstract vocabulary driving verbal SAT scores (6, 12, 13).
The changing balance of media technologies has led to losses as well as gains. For example, as verbal IQ has risen, verbal SATs have fallen. Paradoxically, omnipresent television may be responsible for the spread of the basic vocabulary (11) that drives verbal IQ scores, while simultaneously the decline in recreational reading may have led to the loss of the more abstract vocabulary driving verbal SAT scores (6, 12, 13).
Technology and Informal Education: What Is Taught, What Is Learned
by Patricia Greenfield
Science 2 January 2009:
Vol. 323. no. 5910, pp. 69 - 71
In short, it's not "intelligence" that's gone up, it's scores on the visual scales and basic vocabulary.
Verbal intelligence -- that would be the SAT-V kind of intelligence that predicts college success (and, yes, SATs do predict college success) -- is down.
Unfortunately, Greenfield attributes the decline in verbal intelligence to technology, kids, and parents, not schools. Visual media have replaced books at home; therefore kids can't read challenging texts or think because "reading for pleasure" is the key to critical thinking.
So what should the schools do now that kids can't read?
They should stop using books & switch to PowerPoint:
Talk about the bad becomes normal.Schools should make more effort to test students using visual media, she said, by asking them to prepare PowerPoint presentations, for example.
"As students spend more time with visual media and less time with print, evaluation methods that include visual media will give a better picture of what they actually know," said Greenfield, who has been using films in her classes since the 1970s.
"By using more visual media, students will process information better," she said.
+++++
Number one: I'm aware of no evidence that "reading for pleasure" produces high verbal intelligence scores (though I assume it helps), and I am aware of informal evidence that reading for school is essential.
Number two: you can't teach verbal disciplines by means of pictures. Period. Ed was working on K12 education back when California teachers were attempting to deliver "sheltered instruction" to ELL students. In theory, sheltered instruction involves a number of strategies, but in practice it meant trying to teach language-based disciplines using pictures instead of words. It didn't work.
PLUS, when it comes to PowerPoint, I am as one with Colonel H.R. McMaster (now Brigadier General McMaster):
McMaster is a humanist, with a doctorate in history, who is allergic to the military’s culture of PowerPoint presentations where the jargon and diagrams do the thinking for you. He once told me that if an idea couldn’t be put in paragraph form, it didn’t deserve consideration.And number three: wtf?
Kids aren't doing any reading outside school so they should do less reading inside school, too?
Gelman is right.
voodoo correlations in social neuroscience
I've always been skeptical of big behavioral claims based on brain scan data.
Turns out I was right.
Here's Andrew Gelman (and here, too).
Brain imaging studies under fire (naturenews)
interview: Have the Results of Some Brain Scanning Experiments Been Overstated? (Scientific American)
Turns out I was right.
Here's Andrew Gelman (and here, too).
Brain imaging studies under fire (naturenews)
interview: Have the Results of Some Brain Scanning Experiments Been Overstated? (Scientific American)
LEHRER: What is a "voodoo correlation"?Of course, now I'm wondering whether there is anything I think I know about brain & behavior that is not based on non-independent analysis.
VUL: We use that term as a humorous way to describe mysteriously high correlations produced by complicated statistical methods (which usually were never clearly described in the scientific papers we examined)—and which turn out unfortunately to yield some very misleading results. The specific issue we focus on, which is responsible for a great many mysterious correlations, is something we call “non-independent” testing and measurement of correlations. Basically, this involves inadvertently cherry-picking data and it results in inflated estimates of correlations.
To go into a bit more detail:
An fMRI scan produces lots of data: a 3-D picture of the head, which is divided into many little regions, called voxels. In a high-resolution fMRI scan, there will be hundreds of thousands of these voxels in the 3-D picture.
When researchers want to determine which parts of the brain are correlated with a certain aspect of behavior, they must somehow choose a subset of these thousands of voxels. One tempting strategy is to choose voxels that show a high correlation with this behavior. So far this strategy is fine.
The problem arises when researchers then go on to provide their readers with a quantative measure of the correlation magnitude measured just within the voxels they have pre-selected for having a high correlation. This two-step procedure is circular: it chooses voxels that have a high correlation, and then estimates a high average correlation. This practice inflates the correlation measurement because it selects those voxels that have benefited from chance, as well as any real underlying correlation, pushing up the numbers.
One can see closely analogous phenomena in many areas of life. Suppose we pick out the investment analysts whose stock picks for April 2005 did best for that month. These people will probably tend to have talent going for them, but they will also have had unusual luck (and some finance experts, such as Nassim Taleb, actually say the luck will probably be the bigger element). But even assuming they are more talented than average—as we suspect they would be—if we ask them to predict again, for some later month, we will invariably find that as a group, they cannot duplicate the performance they showed in April. The reason is that next time, luck will help some of them and hurt some of them—whereas in April, they all had luck on their side or they wouldn’t have gotten into the top group. So their average performance in April is an overestimate of their true ability—the performance they can be expected to duplicate on the average month.
It is exactly the same with fMRI data and voxels. If researchers select only highly correlated voxels, they select voxels that "got lucky," as well as having some underlying correlation. So if you take the correlations you used to pick out the voxels as a measure of the true correlation for these voxels, you will get a very misleading overestimate.
This, then, is what we think is at the root of the voodoo correlations: the analysis inadvertently capitalized on chance, resulting in inflated measurements of correlation. The tricky part, which I can’t go into here, was that investigators were actually trying to take account of the fact they were checking so many different brain areas—but their precautions made the problem that I am describing worse, not better!
Monday, February 2, 2009
Backwards Identities and Identities in Context
I ran into an understanding problem with my seventh grade son a while back. (I teach him algebra at home because it would really screw up his
schedule if he took the 8th grade class at school. The school was flexible about this and it was better than the online course the school
offered.) He had to take an equation and change it into the slope-intercept form of a line:
y = mx + b
(where 'm' is the slope and 'b' is the y-intercept)
and then graph it. When he rearranged the equation, he came up with:
y = 3 - x/2
He couldn't figure out what 'm' was. We've been over this in the past, but not so much in context. He couldn't "see" the equation properly. I
know that I instantly see things in equations that he doesn't see, but I found (once again) that I have to reinforce the message. I think a big
problem is that the identities are so simple. What could be simpler than:
a/1 = a
What about what I call backwards identities, like:
a = a/1
My son know this because I've drilled it into his head. Everything is a fraction or a rational expression; if you don't see a dividing line, you
can put one in there. This was big back when I taught him about dividing fractions. What if you have something like this:
5 / (3/4)
The invert and multiply works fine if you know that 5 = 5/1.
Another simple identity is: a*1 = a
What about: a = 1*a ?
for something like x/2, you can look at it as (1*x)/2
Although this :
(a/b) * (c/d) = (a*c)/*(b*d)
is not a basic identity, many don't see it in reverse.
I could rewrite
(1*x)/2
as
(1*x)/(2*1) = (1/2)*(x/1) = 1/2 * x
By the time his textbook gets to the slope-intercept problems, they assume that these simple(?) things are not an issue. But, I've mentioned before that it took me until about my junior year (trig class) in high school before I really mastered all of these things.
(Side note: I hate text math even though I'm forced to do that in my programs. Math is 2D. It's graphics, not text, and I can "see" so much more when I write equations on a piece of paper. Dividing lines are horizontal, not slashes, and there is no place for a '^' in math. Kids need to do
math with a pencil and paper. OK. I got that out of my system, almost.)
So, lately, I've been trying to teach my son how I "see" equations and all of my tricks or understandings I had to figure out myself. I've talked
about these things a little bit in the past. I'll see if I can hit the highlights.
When you look at an equation:
1. Look for the terms. These are the big "chunks" of the equation that are separated only by a '+', '-' , or '=' sign. It doesn't matter how complicated the terms are, circle them and ALWAYS include the preceeding '-' sign with the term.
All equations look like this:
... (term) + (term) = (term) + (term) ...
Since the sign belongs with the term, there will always be a '+' between terms. I've told students to look at it this way:
25 - 3x is really 25 + (-3x)
and
15 - x is really 15 + (-1*x)
This helps when you get to something like:
15 - 3(x-2) or 15 - (x - 2)
Many students just don't know what to do with the '-' sign. If you think of it as:
15 + (-3)(x-2) or 15 + (-1)(x-2)
Then there should be less mistakes.
2. You can change the position of any term on one side of the "=" sign just by moving it. (a+b = b+a) The key is that you have to keep the sign
with the term. (If you don't see a sign, it's a '+'.)
Some students have a hard time doing this:
3 - 10x = -10x + 3
They just don't know what to do with the '-' sign. I automatically write equations so that they don't start with a minus sign, but I should probably not do that.
3. You can change the position of a term from one side of an equal sign to the other by changing the sign of the term. Even if you see
complicated integral signs floating around, this still works. I'm not sure why, but I've always thought of it as swinging (?) a term from one
side of an equation to the other and changing the sign. I've noticed that because my son can be sloppy, he makes simple mistakes when he formally subracts or adds terms from both sides. My "swinging" method reduces writing and his sloppy mistakes.
This brings up another topic. I've been really strict with my son about clearly writing each algebraic change on a separate line. Doesn't
everyone remember this from 7th and 8th grade? I've told my son that this prevents silly mistakes and makes it easier for the teacher to read. He
doesn't like to rewrite equations, so he makes the next change to the same equation. If he makes a silly mistake and the teacher can't see it,
then it will be marked completely wrong. If it's clear, he might get partial credit.
The next level is to "see" the terms.
4. Find and circle all of the factors in each term. these are the things that are simply multiplied and divided together. If you don't see the
dividing line, divide the term by 1. (a = a/1) If you see multiple dividing lines, make sure you know what is the overall numerator and
denominator. All terms are fractions or rational expressions. Note that it's harder to "see" the factors of a rational expression in text using
'/' as the dividing line.
(Side Note 2: Order of operations is really all about math as text. The first time it was formally taught to me was when I started to program.
You should never see anything like 4/8/2 in math. Even with programs, nobody should ever write something like that even if you know exactly how the compiler handles order of operations. It should be written as (4/8)/2. But that's really ugly. Paper and pencil math is beautiful.)
Factors are easy to see if you're talking about somehting like 3x/2. The numerator has (3)(x) and the denominator has (2).
But what about [3x^2(3x-5)^1/3] / [(x-5)(y+1)]
Note that I can't "see" this term as well as if it was in a nice 2D, graphic form.
The exponent always belongs to the factor just to the left. My son has had trouble with this. It's (3) * (x^2), not (3x)^2.
You have to see the factors instantly: (3), (x^2), (3x-5)^1/3 for the numerator, and (x-5), (y+1) for the denominator.
5. Any factor in the numerator can change position with any other factor in the numerator. Likewise for the denominator. It's surprising that
when terms get more complicated, students freeze up. They don't see it as a*b = b*a.
I think that order of operations affects this. It's driven into kids heads. They are given problems where they have to evaluate expressions in one particular order. What they think of as "understanding" is really a rote process. Order of operations is mostly for computer language compilers because they have to have some rote process to correctly evaluate expressions from left to right.
When I look at a complicated term (or equation), I never think of order of operations. I think about what I can do with it. I can rearrange factors however I want in the numerator or denominator as long as I know the rules, and the rules aren't defined by order of operations.
Here is something else to "see".
6. You can move any factor to or from the numerator or denominator by changing the sign of the exponent. If you don't see an exponent for a
factor, then it's 1. These sorts of steps come from what I call backwards identities or rules. Students have to know that the rules work both
ways.
My son knows very well the distributive rule, but he still has a hard time factoring:
(3x - 12x^2) = 3x(1 - 4x)
He doesn't feel comfortable with what to do when the 3x is factored out. He has to realize that 3x = (3x)*(1), a backward identity.
Most learn something like this:
x^(-1) = 1/x
but can't see that this means that you can bring a factor up from the denominator into the numerator by changing the sign of the exponent; the
backwards form of this rule.
One might talk about order of operations when you have two operations next to each other, but to apply them to a term or an equation is wrong.
This is a little bit of how I see equations. I hope it helps.
schedule if he took the 8th grade class at school. The school was flexible about this and it was better than the online course the school
offered.) He had to take an equation and change it into the slope-intercept form of a line:
y = mx + b
(where 'm' is the slope and 'b' is the y-intercept)
and then graph it. When he rearranged the equation, he came up with:
y = 3 - x/2
He couldn't figure out what 'm' was. We've been over this in the past, but not so much in context. He couldn't "see" the equation properly. I
know that I instantly see things in equations that he doesn't see, but I found (once again) that I have to reinforce the message. I think a big
problem is that the identities are so simple. What could be simpler than:
a/1 = a
What about what I call backwards identities, like:
a = a/1
My son know this because I've drilled it into his head. Everything is a fraction or a rational expression; if you don't see a dividing line, you
can put one in there. This was big back when I taught him about dividing fractions. What if you have something like this:
5 / (3/4)
The invert and multiply works fine if you know that 5 = 5/1.
Another simple identity is: a*1 = a
What about: a = 1*a ?
for something like x/2, you can look at it as (1*x)/2
Although this :
(a/b) * (c/d) = (a*c)/*(b*d)
is not a basic identity, many don't see it in reverse.
I could rewrite
(1*x)/2
as
(1*x)/(2*1) = (1/2)*(x/1) = 1/2 * x
By the time his textbook gets to the slope-intercept problems, they assume that these simple(?) things are not an issue. But, I've mentioned before that it took me until about my junior year (trig class) in high school before I really mastered all of these things.
(Side note: I hate text math even though I'm forced to do that in my programs. Math is 2D. It's graphics, not text, and I can "see" so much more when I write equations on a piece of paper. Dividing lines are horizontal, not slashes, and there is no place for a '^' in math. Kids need to do
math with a pencil and paper. OK. I got that out of my system, almost.)
So, lately, I've been trying to teach my son how I "see" equations and all of my tricks or understandings I had to figure out myself. I've talked
about these things a little bit in the past. I'll see if I can hit the highlights.
When you look at an equation:
1. Look for the terms. These are the big "chunks" of the equation that are separated only by a '+', '-' , or '=' sign. It doesn't matter how complicated the terms are, circle them and ALWAYS include the preceeding '-' sign with the term.
All equations look like this:
... (term) + (term) = (term) + (term) ...
Since the sign belongs with the term, there will always be a '+' between terms. I've told students to look at it this way:
25 - 3x is really 25 + (-3x)
and
15 - x is really 15 + (-1*x)
This helps when you get to something like:
15 - 3(x-2) or 15 - (x - 2)
Many students just don't know what to do with the '-' sign. If you think of it as:
15 + (-3)(x-2) or 15 + (-1)(x-2)
Then there should be less mistakes.
2. You can change the position of any term on one side of the "=" sign just by moving it. (a+b = b+a) The key is that you have to keep the sign
with the term. (If you don't see a sign, it's a '+'.)
Some students have a hard time doing this:
3 - 10x = -10x + 3
They just don't know what to do with the '-' sign. I automatically write equations so that they don't start with a minus sign, but I should probably not do that.
3. You can change the position of a term from one side of an equal sign to the other by changing the sign of the term. Even if you see
complicated integral signs floating around, this still works. I'm not sure why, but I've always thought of it as swinging (?) a term from one
side of an equation to the other and changing the sign. I've noticed that because my son can be sloppy, he makes simple mistakes when he formally subracts or adds terms from both sides. My "swinging" method reduces writing and his sloppy mistakes.
This brings up another topic. I've been really strict with my son about clearly writing each algebraic change on a separate line. Doesn't
everyone remember this from 7th and 8th grade? I've told my son that this prevents silly mistakes and makes it easier for the teacher to read. He
doesn't like to rewrite equations, so he makes the next change to the same equation. If he makes a silly mistake and the teacher can't see it,
then it will be marked completely wrong. If it's clear, he might get partial credit.
The next level is to "see" the terms.
4. Find and circle all of the factors in each term. these are the things that are simply multiplied and divided together. If you don't see the
dividing line, divide the term by 1. (a = a/1) If you see multiple dividing lines, make sure you know what is the overall numerator and
denominator. All terms are fractions or rational expressions. Note that it's harder to "see" the factors of a rational expression in text using
'/' as the dividing line.
(Side Note 2: Order of operations is really all about math as text. The first time it was formally taught to me was when I started to program.
You should never see anything like 4/8/2 in math. Even with programs, nobody should ever write something like that even if you know exactly how the compiler handles order of operations. It should be written as (4/8)/2. But that's really ugly. Paper and pencil math is beautiful.)
Factors are easy to see if you're talking about somehting like 3x/2. The numerator has (3)(x) and the denominator has (2).
But what about [3x^2(3x-5)^1/3] / [(x-5)(y+1)]
Note that I can't "see" this term as well as if it was in a nice 2D, graphic form.
The exponent always belongs to the factor just to the left. My son has had trouble with this. It's (3) * (x^2), not (3x)^2.
You have to see the factors instantly: (3), (x^2), (3x-5)^1/3 for the numerator, and (x-5), (y+1) for the denominator.
5. Any factor in the numerator can change position with any other factor in the numerator. Likewise for the denominator. It's surprising that
when terms get more complicated, students freeze up. They don't see it as a*b = b*a.
I think that order of operations affects this. It's driven into kids heads. They are given problems where they have to evaluate expressions in one particular order. What they think of as "understanding" is really a rote process. Order of operations is mostly for computer language compilers because they have to have some rote process to correctly evaluate expressions from left to right.
When I look at a complicated term (or equation), I never think of order of operations. I think about what I can do with it. I can rearrange factors however I want in the numerator or denominator as long as I know the rules, and the rules aren't defined by order of operations.
Here is something else to "see".
6. You can move any factor to or from the numerator or denominator by changing the sign of the exponent. If you don't see an exponent for a
factor, then it's 1. These sorts of steps come from what I call backwards identities or rules. Students have to know that the rules work both
ways.
My son knows very well the distributive rule, but he still has a hard time factoring:
(3x - 12x^2) = 3x(1 - 4x)
He doesn't feel comfortable with what to do when the 3x is factored out. He has to realize that 3x = (3x)*(1), a backward identity.
Most learn something like this:
x^(-1) = 1/x
but can't see that this means that you can bring a factor up from the denominator into the numerator by changing the sign of the exponent; the
backwards form of this rule.
One might talk about order of operations when you have two operations next to each other, but to apply them to a term or an equation is wrong.
This is a little bit of how I see equations. I hope it helps.
Sunday, February 1, 2009
The "Global Achievement Gap" muddle
An editorial by Sandra Stotsky, member of the National Mathematics Advisory Panel, reveals misrepresentation in the author's "study" of MIT graduates. He supposedly found that only a few MIT students mentioned anything "more than arithmetic, statistics and probability" as useful to their work.
It's a great read!
It's a great read!