Won't Get Khanned Again: How Can Education Help Democracy Trump Capitalism?

The other evening, I saw an article about Salman Khan's latest plans to expand his "education" empire into the world of brick and mortar schooling (how unrevolutionary of him, I must say), and it set me to thinking. One sentence caught my attention in particular:

"They played a 'paranoia' version of the game Risk to understand the theory of probabilities using Monopoly money, where kids trade securities based on the outcome of the game."

There's nothing obviously new about this: there have been teachers offering stock market simulation "games" in various grade bands for decades. So what's the big deal? Maybe nothing, maybe something significant. Here we have Sal Khan, former Wall St. hedge fund analyst (not a job for which I hold a great deal of respect, for some reason, particularly in connection with the education deform movement), giving summer campers, some of whom by his own words, "couldn't see the board" (which I assume means that they were quite young), the opportunity to find out how our capitalist system works. Not that the word "capitalist" or its variants ever gets mentioned of course. But it's certainly Sal's prerogative to indoctrinate   proselytize   rationalize  um, expose students to The Market.

What popped into my head was, "How do educators (in mathematics or any other subject) who are concerned not only about social justice and the obvious inequities (and iniquities) of the current American system give students the opportunity to critically examine the assumptions about what it means to be human that are inherent in our so-called "free-market" capitalist system and how that system impacts our alleged belief in "core democratic values."

For those of you not playing along at home, let me remind you that the reason we've fought wars over the last 60 years in places like Korea, Vietnam, Panama, Nicaragua, Somalia, Kosovo, Iraq, Afghanistan, and Libya (to name just some of the low-lights; for a more thorough list of our military interventions, both foreign and domestic, over the last 120 years or so, go here), is to spread our "democratic core values" and bring freedom to oppressed people throughout the world (or so I've heard it said). Yet, some people lately have been making a lot of noise that includes the notion that we don't really have democracy or anything vaguely like it here at home. And some folks are linking capitalism and the activities of Wall Street, banking, multinational corporations, globalization, and much else that I suspect Mr. Khan finds perfectly fine, to the absence of meaningful democracy, social justice and equity in these United States.

So while I can't stop Sal Khan from expanding his influence into the hearts and minds of our children, particularly since Bill Gates isn't backing my on-line screeds financially or otherwise, nor has the O'Sullivan Foundation offered me $5 million in grant money to spread my vision to physical classrooms, I guess I'm still free (until the Internet comes under complete control of the government and its corporate and oligarchical masters) to try to get others interested in offering kids a different viewpoint. In particular, I invite people to offer ideas (information on and/or links to already-existing projects, speculations on projects that might be, or inklings of possibilities) on non-didactic education (how's that for an oxymoron?) on the human and humane implications and costs of unchecked capitalism.

My initial thought last week was that I wanted a game that allowed students to explore various economic, social, and political arrangements and systems in ways that made it likely (perhaps unavoidable) that players would need to think hard about what it costs people to live and work in our system, not just Americans, of course, but people all over the world, and not just monetarily, but in terms of physical, intellectual, emotional, ethical, spiritual, and other aspects of what might be called human health and well-being. Naturally, the environment in which we live, the planet we inhabit, would likely need to be considered carefully as well.

Lest I appear more completely ignorant than I actually am (which is, of course, quite ignorant of a host of things), I should mention that I'm aware that Bucky Fuller was up to something at least in part like what I'm raising above with his World Game. I'm most certainly not in his league, but I'm thrilled to have discovered in reading about his game ideas that during the 1960s, Fuller several times proposed them as the core curriculum for Southern Illinois University. I'd like to see his college curricular idea and raise him a K-16 and beyond curriculum, one that begin as close to the start of formal education as possible and  finds ways to lead students into conversations about the what happens to us when we operate in a capitalist mindset. I think there are fundamental ethical questions and assumptions that kids of school age care about and are quite capable of discussing intelligently.

I do put in that oxymoron about non-didactic education advisedly. There's no question that it's possible to readily create lessons the entire point of which is to propagandize an anti-capitalist moral. That's not what I'm interested in, however. I think that if such a curriculum is to be useful, it needs to be much more open-ended than what being the flip side of Mr. Khan's little market game seems grounded in. But maybe I'm chasing a chimera here. What do you think?

Huh? The ETS Discourages Thinking Again

Every day, the ETS/College Board kindly  e-mails to subscribers an "SAT Problem of the Day": about one in every four is a math problem. Twisted soul that I am, I frequently start my day by answering the problem of the day, and while I'm generally able to nail better than 98% of them, I particularly look forward to the math problems, and even more so to ones that relatively few people who've attempted them have gotten right (yes, I'll admit to still having a sad little place in my soul that is stroked by such silliness).

This morning's poser was definitely challenging for a majority of respondents. As of this writing, with nearly 80,000 folks attempting the problem, only 40% had answered it correctly. Before we continue, here it is for your smoking enjoyment:

The stopping distance of a car is the number of feet that the car travels after the driver starts applying the brakes. The stopping distance of a certain car is directly proportional to the square of the speed of the car, in miles per hour, at the time the brakes are first applied. If the car’s stopping distance for an initial speed of  miles per hour is  feet, what is its stopping distance for an initial speed of  miles per hour?

(A)  feet   (B) 51 feet  (C) 60 feet  (D) 68 feet  (E) 85 feet

Okey-doke. I'll give you some time to think about it, preferably before you look further down the screen for the official answer and explanation. And it's that explanation I want to address. No peeking now! After all, your self-esteem is riding on how well you do here (not to mention your entire future). 

All done? Great. Now before we see the official explanation/answer, let's take this opportunity to do what the folks in Princeton and Berkeley (East and West Coast headquarters of the ETS) apparently don't want us to do: think!

First of all, were this problem to appear in an actual test (despite my enormous collection of past actual SAT tests, I don't recall having seen this one before), my best guess is that it would be one of the last ten problems or so in a section of mathematics problems, based on the 40% correct response rate (I've seen math problems where fewer than 10% of students attempting it got the right answer; as I have no way of knowing what percent of the folks who answered this one today are actually high school students, this might actually be a bit harder for the intended audience than the numbers indicate). It seems fair to suggest that this isn't something that is ridiculously difficult, but that many high school students would nonetheless get wrong for various reasons.

One reason, of course, is literacy: if students read this carelessly or are unable to comprehend the point, they may overlook or simply fail to understand the significance of the words "directly proportional to the square of the speed of the car." If so, one obvious trap answer, a distractor, if you will, that will draw a disproportionately high number of incorrect replies, is (A). It's relatively easy to miss/ignore the word "square" and figure that since 40 mph is twice the speed 20 mph, the stopping distance will be twice that of the slower car and hence 2 x 17 ft = 34 ft. 

But if you're on your toes, even if you're not a math whiz you should be suspicious of this answer. It's just too damned easy to come up with. A late-appearing SAT or ACT math problem isn't going to be quite so simple.  And in fact, any simple equation you set up here that ONLY involves the given numbers and an unknown will be wrong. That "square" thing isn't put in for show, and it's going to come into play somehow or other. 

Sometimes it's possible on problems at this level of difficulty to do more reasoning and eliminate additional choices after avoiding the "big trap" answer. I've not thought this one through further other than to solve it, but perhaps interested readers can offer ideas as to why the other three wrong answers were chosen by the test-makers as possible replies. For me, the problem itself spoke to such a simple solution that I just cut to the chase and answered it. And then, I eagerly awaited learning what the experts had to say as to why my answer was (of course) correct. ;^)  So with no further ado, here is the official explanation from the ETS/College Board:

 

Explanation

The stopping distance is directly proportional to the square of the initial speed of the car. If  represents the initial speed of the car, in miles per hour, and  represents the stopping distance, you have that the stopping distance is a function of  and that , where  is a constant. Since the car’s stopping distance is  feet for an initial speed of  miles per hour, you know that . Therefore, , and the car's stopping distance for an initial speed of  miles per hour is  feet.

Say what? Gosh, but that seems like a hell of a lot of work to me, and quite frankly, 0.0425 never came into play in my calculations.  Having done (way too) many SAT/ACT math problems, when I see something that involves ratios (usually geometry problems, but the underlying issue here is the same), I think about dimensions: am I looking at a linear/linear ratio (if it's a challenging problem, probably not)? If not, is it linear/square? Linear/cubic? Some other ratio where the dimensions change? In this case we're looking at a proportion in which we have a linear ratio (speed to speed) that is being set equal to another linear ratio (braking distance to braking distance), but we're told that the second varies as the SQUARE of the first. So we've got (20/40)^2 which simplifies to (1/2)^2 or 1/4. Thus, the other side of this proportion needs a number on the bottom (conveniently, our unknown) that is 4 times the number on top (17) - of course, you might set things up another way. As 4 x 17 = 68, that's the right answer. Doable, I might add, in your head if you have a bit of facility with mental arithmetic. On the other hand, I challenge you to do the number-crunching in the official explanation in your head. Not impossible, of course, but not exactly as simple as squaring (1/2) and then multiplying 17 by 4 (one way is to take (15 + 2) x 4 = 60  + 8 to get a total of 68.

My point is, of course, that the best place to go for explanations on SAT/ACT math problems is not the test-makers. Considering that these are timed tests, they will NEVER do a "process of elimination" approach, which is necessary when you don't know how to proceed (and when you have to deal with certain question structures, both in math and elsewhere, where it's much easier to eliminate wrong answers than arrive at the right one). They'll never talk about the errors they presume you'll make or the whole concept of distractors. Nope, that would be a bit too helpful. And it would undermine the nice illusion that want us all to harbor that these tests are simply logical extensions of the sorts of things we teach/learn in school.

 But of course, a word to the wise guy should suffice: these aren't school tests, never have been, never will be. They're part of a game that's been set up to reward those most adept at cutting through the baloney. It's not absolutely necessary to be able to do that in order to do well, but it certainly helps, particularly on a timed test, where the longer you take to do any given problem, the less you can spend on others, some of which may take more than the average allotted time per problem (about 60 seconds on math). The better you are at cutting to the chase, the more time you buy for the ones that you really need a little more time to figure out, and the more likely you are to actually be able to see and think about all the problems. That's an edge every student needs, but few actually get. And if everyone follows the "official" explanations, few ever will.

Duncan, USDOE Stonewall on His Ties to SALF

Arne Duncan and Carol Spizzirri

Regular readers of this blog recall that in June of this year I sent an open letter to US Secretary of Education Arne Duncan in an attempt to get his input on open questions surrounding his involvement with the Save-A-Life Foundation and its founder/president, Carol Spizzirri. What ensued was a four-month runaround from various functionaries working for Mr. Duncan which concluded with a classic non-answer answer that advised me to seek the information I requested about Mr. Duncan's views from . . . wait for it! . . . officials in the Chicago Public Schools. How they were supposed to be able to read Mr. Duncan's mind was not specified. But I didn't need to be a mind reader to recognize that I wasn't going to get a meaningful reply from Arne Duncan. Was I disappointed? You bet. Was I surprised that someone who appears to have skeletons in his closet he'd prefer the American public would either forget or never hear rattling in the first place was  not forthcoming about them? Not in the least.

Yesterday, Peter Heimlich, who has  been vigorously pursuing this story, posted his findings on his blog, The Sidebar:

Haunted by his ties to tainted nonprofit, Education Secretary Arne Duncan ignores questions about $174,000 "phantom" program he arranged for the group

If you've not been following this story here, Peter's piece is probably the best summary of the entire scandal, and even if you have, you would do well to read his expert take on Duncan and SALF.

A Partial Bridge Over Troubled Mathematical Waters: Mumford and Garfunkel Try To Fix US Math Education

William R. Robinson

 William R. Robinson, my former mentor at University of Florida's Department of English, used to say that the further someone was from getting it right, the more useful it is to find something in what they have to say that is “heuristic” (by which he meant ‘thought-provoking’) and that the closer someone is to ‘getting it right,’ the more significant are the ways in which they ‘get it wrong.’

That useful binary construct came to mind again last week as I read NEW YORK TIMES opinion piece by Sol Garfunkel and David Mumford, "How To Fix Our Math Education."

Sol Garfunkel

David Mumford

Much of what Garfunkel and Mumford have to say is praiseworthy. Certainly, their main point - that a 'one-size-fits-all' approach to mathematics curricula is a bad idea - is mostly well-made and sensible. It is also true that many students would benefit by a more practical, applied approach to teaching and learning mathematics (though I would suggest that all students should have the chance to see the beauty of mathematics in itself, to see it as a living, growing, very human body of knowledge that not only has practical power, but also profound and stunning beauty. Neither the traditional American approach to mathematics for the vast majority of its students nor a strictly applied or modeling approach addresses the aesthetics of mathematics).

Nonetheless, my fundamental complaints regarding Garfunkel and Mumford's piece do not revolve around pure mathematics and mathematical aesthetics. Rather, my shock and disappointed came from  their failure to debunk some of the assumptions of the current educational deform movement that they raise and then pass over without any examination at the beginning of their proposed remedies.

First, the authors state unquestioningly that

widespread alarm in the United States about the state of our math education. . . can be traced to the poor performance of American students on various international tests, and it is now embodied in George W. Bush’s No Child Left Behind law, which requires public school students to pass standardized math tests by the year 2014 and punishes their schools or their teachers if they do not.

So much nonsense left unexamined in one paragraph. First, how widespread is this "alarm," exactly, and who accepts that the sky is falling when it comes to US mathematics education? While it is true that we are not doing justice when it comes to educating most of our students meaningfully about mathematics, this is not a recent problem and likely can be claimed about any era of American education one cares to examine closely. Readers of Gerald Bracey's work are familiar with the fact that our "Sputnik moments" in mathematics, science, and other areas are not limited to now or the late 1950s. Education punditry going back at least as far as 18th century America has found fault with public schooling, though the claimed consequences of the alleged failures have varied from the decay of the moral fabric of the nation to the current received wisdom that our economy and future ability to "compete in the global marketplace" are at risk because public schools are so poor and our youngsters so uneducated, ignorant, and intellectually lazy.

These notions have been debunked repeatedly by less credulous scholars, from Bracey to Richard Rothstein to Diane Ravitch; many others have or are starting to look more critically at such claims, including the ludicrous notion that economic success or failure depends directly and primarily on the quality of public education. Many have noted, too, that none of our successes are ever attributed by the critics to something that public education may have done right. Teachers, schools, colleges of education, and professors make increasingly-convenient scapegoats for sins that are more obviously attributable to people and institutions far removed from the world of public education and teacher preparation.

But even if there were truth to the causal linking of K-16 education to our economy (and even if one accepts the very arguable notion that the main or sole purpose of public education is to provide career preparation to and the winnowing of young people for the cost-free benefit of businesses - like many other historical facts the punditry and deform machine conveniently ignore, the fact that it used to be the responsibility of companies to train workers for their jobs and to figure out who was best fit for what kinds of work has be neatly pushed under the rug), the notion that comparative performance on international tests of mathematics and science gives sufficient and meaningful information about how well or poorly the schools in a country are doing is simply ridiculous. This is another bit of received wisdom that has repeatedly been debunked by a host of experts and scholars, going back at least as far as the criticism the mathematician Banesh Hoffman leveled in THE TYRANNY OF TESTING, his 1962 critique of standardized tests. For Mumford and Garfunkel to take at face value that these international tests are telling us anything of importance seems like another important opportunity missed.

Finally in the quoted paragraph, we have an accurate description of what NCLB is about: punishing public schools, teachers, administrators, students, their parents, and the communities that are their homes. But not a word of criticism from the authors to this shocking and cynical policy. Nothing about the deeply-flawed mathematics through which the inevitable branding of all schools as "failing," sooner or later, is the consequence, if not the outright goal of all of its authors and supporters. This oversight by two mathematicians is difficult to understand.

As stated earlier, Garfunkel and Mumford do a commendable job of outlining reasons to rethink and reorganize what mathematics should be available to students and how mathematics can be made more relevant and appealing to many young Americans. But then they offer near the end this disastrous assertion:

It is true that our students’ proficiency, measured by traditional standards, has fallen behind that of other countries’ students, but we believe that the best way for the United States to compete globally is to strive for universal quantitative literacy: teaching topics that make sense to all students and can be used by them throughout their lives.

The authors make two egregious errors here: first, they accept that we've fallen behind based on traditional standards. But in fact, many experts have shown that when apples are compared to apples, American students more than hold their own on such tests. The problem has been that the education deform movement has made much of results where we have a broad sample of American students, including those from the neediest, most impoverished urban and rural schools being compared with a far less representative sample of students and schools from many of the countries against which we are being judged and, ostensibly, found wanting. It simply makes no sense to draw conclusions about our schools, teachers, or students based on such invalid comparisons.

Second, Garfunkel and Mumford, sadly continue to accept the notion that we should make educational policy on a national level (a questionable notion in itself that I will not go into here) based on the idea that we educate students to compete (and, of course, conquer) in a global competition. This sort of social Darwinist, survival of the fittest nonsense is a huge key to how the education deformers get their agenda accepted by so many politicians and uncritical members of the public. Not only does it go against many of the basic principles of child-rearing and education, but it also undermines the collegiality among educators that is crucial to how many of the countries (Japan and Finland immediately come to mind) that are held up as "beating" us construct their schools.

Finally, let me make clear again that while I support the authors' suggestions about bringing applied mathematics and modeling much more deeply into what is commonly available to students, I don't think they suffice to give every student a fair chance at numeracy or at a variety of pursuits (not merely jobs) that can emerge from mathematics. I believe that they have properly tried to get more leverage for aspects of mathematics that have been mostly pushed aside over the last century or more in our schools, but they've left out some things of vital importance for enriching students lives. And of course, they've either overlooked or willfully ignored some key assumptions and myths that, left unchallenged, guarantee that the very changes Garfunkel and Mumford advocate will never be seriously considered by the real educational policy makers, let alone actually implemented.

Wrong Again, Jonathan: Mr. Alter Doesn't Get Public Education (and neither does Obama)

Once upon a time, back in the Ronald Wilson Reagan era, I used to look forward to reading Jonathan Alter's column in NEWSWEEK. He seemed to be one of the guys who got it. I particularly remember what he wrote about Gary Hart during his rise and fall in the 1984 campaign.

Sad to say, Mr. Alter seems to have completely jumped the shark. Every post he makes on education is so blindly wrong, so clearly ignorant of what's going on with the current education deform movement in which Mr. Duncan and, at least passively, Mr. Obama, are willing partners, that as an educator who has worked most of his career with students, parents, teachers, and administrators in districts, schools, and communities devastated by extreme poverty, I cannot help but be appalled.

Looking at his latest education commentary ("Obama Shows Spunk Pushing Brave Education Plan"), there is simply no way that Mr. Alter could be truly in touch with what's going on in places like Chicago, New Orleans, Detroit, Los Angeles, Atlanta, Philadelphia, Boston, Miami, or New York City, given how shallow and erroneous is his analysis of education and what Obama's administration is up to. The recent waiver offer by Mr. Duncan is a classic case of figuring out long after everyone else has done so that a horrid piece of legislation like NCLB is about to crash and potentially bring loads of hard-working people down with it. This impending disaster was completely predictable for anyone who had a grasp of the nonsensical mathematics informing "annual yearly progress" and the absurd notion of 100% proficiency in math and literacy AT ANY POINT IN TIME. But I believe the authors of NCLB for the most part knew exactly what they were doing: creating a way to destroy public education and turn even the most outstanding public schools into "failing schools" through its impossible demands.

When Obama put Duncan in charge, we saw another strategic move, cynically called "Race To The Top," that was nothing more than openly bribing states to comply with anti-union, anti-teacher, anti-student, pro-testing, pro-charter, pro-privatization policies in exchange for the federal dollars, or otherwise be denied the funding to which ALL kids should have equal access, regardless of where they live. It doesn't get any more undemocratic, any less fair, any more patently wrong.

In light of NCLB and RttT, many people have correctly counseled states to refuse to take the waivers being "boldly" offered by Duncan/Obama. In fact, the absolutely sanest strategy for every state, district, administrator, school, teacher, parent, and student would be to say, "No!" to high-stakes testing abuse and insanity, to refuse to allow, administer, or participate in these heinous tests. Please look at the Bartleby Project and join those of us who will not bend to the forces of billionaires and politicians who care nothing about learning, kids, education, or democracy, but only about lining their pockets and those of their friends, colleagues, and masters. The folks pulling Mr. Obama and Mr. Duncan's strings (and one merely need look at how they operated in Chicago with public schools there to know that what's happened at the federal level since they took power was all too predictable) have to be rubbing their hands with glee at the "spunk" being shown in the US Dept. of Education. If this is the best Mr. Obama and his minions can come up with, we need a different nominee in 2012.

The Three Most Important Words in Education: Assessment, Assessment, Assessment.

Today (August 10, 2011), Alfie Kohn posted a piece entitled, "Teaching Strategies That Work! (Just Don't Ask 'Work to Do What?')"

As I read it (with the usual enjoyment and anger Alfie Kohn's posts elicit from me), I found myself thinking about this paragraph in particular: "Thus, 'evidence' may demonstrate beyond a doubt that a certain teaching strategy is effective, but it isn't until you remember to press for the working definition of effectiveness -- which can take quite a bit of pressing when the answer isn't clearly specified -- that you realize the teaching strategy (and all the impressive sounding data that support it) are worthless because there's no evidence that it improves learning. Just test scores." 

In countless arguments I've had on-line with people about education and assessment in general, and mathematics education and testing in particular, invariably my antagonists (and I use that word advisedly) would reject any curricular materials, pedagogical strategy, tool, task, theory, activity, etc., by stating, "Where is your gold standard research that shows that X is effective?" And as night follows day, when pressed, they would make clear that inside that "gold standard" was what for them comprised a platinum standard: only 'objective' (and hence machine-scored, multiple choice tests if given on a wide-scale, or, if the assessment was small and local (e.g., in one classroom), only tests that were scored with no partial credit, no discretionary judgment or rubrics for the scorer, but rather those that had single answers that were either 100% right or 100% wrong, so that the results couldn't be shaped by the scorer (who might, of course, be inclined to be subjective or, even worse, fuzzy!) 

Now, I think we all want reliable and valid tests, but I find it intriguing that these folks were SO suspicious of any test that allowed for a "human" factor in the scoring (let alone one that had human factors in the tasks themselves, of course!), and so absolutely convinced that given practicality, costs, and the fuzziness factor they so abhorred, only those nationally-normed, multiple-choice standardized tests would count. They were the true measurement of anything one might wish to measure in education. 

Alfie Kohn raises the opposite question: what good are your 'results' if all they are is improving test scores, not learning? And these ideologues I finally gave up arguing with about 15 months or so ago only want to talk about just that: test scores, and a very particular type of test at that. With no wiggle room at all. As an advocate for increased intelligent use of meaningful formative assessment (see the work of Paul Black, Dylan Wiliam, et al.), I find myself realizing with increasing dismay that everything I value about education is precisely what is dismissed by the folks I'm trying to either convince or, yes, defeat in the court of not only reason, public opinion, and school policy, but in the halls of government and the meeting rooms of the monied and powerful. If I and others who agree with my viewpoint are not able to get people to see that improving scores on lousy tests is an utter waste of time by ANY reasonable criteria one might choose to use, then US public education is doomed.

 And we cannot afford to let that happen, to allow control to be ceded to self-interested greedy profiteers and people with various political, social, or especially religious agendas that are at odds with our democratic core values. Assessment, what it means, and how we do it isn't the ONLY issue we need to struggle with, but it tends to be the one that touches upon where the rubber meets the road for a lot of folks.
Naturally, I agree with people like Alfie Kohn, Marian Brady, and others who want to focus clearly on WHAT we're teaching and why we are teaching it. There's no getting away from content and the curriculum that frames it. But I think assessment remains key because it is still the one thing that people are guaranteed to pay attention to: kids, parents, teachers, administrators, politicians, media, and the general public. If we can't win the assessment fight, we're in very deep trouble.

Trust Us, We're The ETS (And we have a nice bridge for sale in Brooklyn, too!)

This was today's (July 24, 2011) SAT question of the day from our good friends at the College Board and Educational Testing Service:

Read the following SAT test question and then click on a button to select your answer. 

If a, b, and c are numbers such that a / b = 3 and b / c = 7, then (a+b) / (b+c) is equal to which of the following?

A. 7 / 2

B. 7 / 8

C. 3 / 7

D. 1 / 7

E. 21

It's an old problem. I remember it from the '80s (some of the problems that appear in the Problem of the Day go back as far as the '70s). Nothing particularly wrong with it, nothing particularly great about it. But here's what bothers me, though it doesn't surprise me based on experience. Here's the explanation they offer:

Explanation

From a / b = 3 is implied that (a+b) / b = 4 (1)

And from b / c = 7 is implied that b / (b+c) = 7 / 8 (2)

If we multiply (1) and (2) together, we have that (a+b) / (b+c) = 4 times (7 / 8) = 7 / 2 ( b is canceled out).

Hmm. Well, maybe I don't know high school kids all that well after tutoring SAT math for 30-odd years, but I don't think very many of them would get that explanation or think to do the problem that way, assuming they tried to solve it. And that's one of my gripes with many of the ETS solutions to their own problems: they're generally not the quickest or most intuitive way to get at the answer.

My approach was to solve for a and c in terms of b, the variable that's common to both of the equations. It follows that a = 3b. Substituting for a in the expression (a + b) yields 3b + b = 4b. The solving for b in the second statement gives c = b/7 and substituting for c in the expression (b + c) yields b + b/7 or 8b/7. So the entire expression (a + b) / (b + c) in terms of b is 4b/ (8b/7). Indeed, the b's simplify to 1 (it's implied in the original problem that none of the variables are zero), and the rest simplifies to 4 * 7/8 or 7/2.

Maybe that's not "better," that what ETS offers, but I think it's more clear as to what's going on and why the approach works. The ETS explanation feels a bit like magic to me, looking at it from a student's perspective. That kind of thing in textbooks always frustrates me, and I know it loses a host of kids at the starting line. While this sort of answer from them may not be their most egregious sin, it is one example of why I warn students to not assume that when ETS explains something in math (or anything else) that it's always going to be coin of the realm. It's not wrong, but how helpful is it, really, for most kids?