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This article suggests that attempts over the last twenty years to improve mathematics education in North America have been wrong-headed non-starters, and that people don't need as much math as the education establishment says they do. I'm going to address the second point first.
I can certainly see the case for not needing the full gamut of mathematics available to the end of high school; that's why they're called electives. But the author (who, sadly enough, is a professor emeritus of mathematics) has made a fundamental error in logic: he has taken the mathematically-illiterate society we have, where being unable to calculate a tip is seen as normal and acceptable, and claimed that people don't need math to get along in that world. He's right. The reason, however, is not that math is useless; it's that we're comfortable as a society with the failings we've built into the system to accommodate people who can't do math.
The better question is: what harm does rampant mathematical illiteracy do to our society, that could be remedied with better mathematical literacy? This is a much harder question, because it requires both a diagnosis of a problem that currently exists and a proposed solution.
From my somewhat less-exalted position, I see several ills in society that could be improved with better mathematical literacy.
The first is our societal inability to understand and use statistics. This begins with journalists and trickles down to everyone who reads what they write. When experts in various branches of science or finance or politics begin to talk about the effects of their discoveries, most people can't parse the numbers. Even those who can figure out what those numbers mean in context can't always break them down to decide for themselves if the numbers seem reasonable, which means they're unable to use their own knowledge to verify or deny the claims with any accuracy. Does this mean they don't try? No, it doesn't - it means when they do try to parse those claims, they get it wrong, and most people DON'T NOTICE. The result is an inability to separate a well-spun lie with statistics from a truth that could improve our lives.
One lie perpetuated this way is the myth about stranger danger, leading to us bubble-wrapping our kids far later than any society in the past has ever done. Politicians use these lies all the time to drum up support for this or that cause - I suspect the recent health care debate in the U.S. is part and parcel of this problem, and I'm quite certain the Tea Party movement is based in large part on the participants' inability to work the math.
The second, closely related problem is the general scientific illiteracy in society. Again, it leaves us unable to interpret scientific data with accuracy, which then leads to devaluing it. Thus NASA is defunded while the war in Iraq continues to grind on.
The third is our societal inability to deal with finances. When money is almost completely computerized, the inability to manipulate numbers and understand what's happening to our money is a significant problem for many people. This is evidenced by the willingness to overspend using credit.
Our society would be better off if every young person got to the end of grade ten with a thorough understanding of all the math concepts presented up to that point. Which brings me to my second point: that the good professor is overstating the change in mathematics education and therefore has understated its impact on understanding.
Sea changes in education take decades. We've been engaged in this particular change since the mid-eighties, so a little more than two decades. I'm in a position to know much more than he is: many teachers haven't caught on yet. They think they're doing it right when they use manipulatives, but they haven't got a firm grasp on the entire span of changes to teaching that needs to happen for the most effective math learning to take place. It's still very common for teachers to stop using manipulatives almost entirely in the junior grades. It's still very common for teachers to be unable to diagnose the fault in understanding that a child is experiencing well enough to scaffold their learning and fix the problem. When teachers are stressed, they fall back to teaching the way they were taught.
At the moment, most standardized tests in America are asking for factual knowledge of discrete facts. Teachers need to teach to this because their jobs are on the line if their kids aren't doing well. So they teach using methods that will increase discrete knowledge and skills. The research into mathematics education (and all other branches of education, for that matter) says that this is backwards; we need to be teaching to the big ideas and then engaging our students to break down those ideas to get to the little bits. This is not yet a consistent thing in North American classrooms, and part of the reason is that the test scores are being misused to blame teachers instead of supporting them. (And we're back to the first point: this is a classic example of the misuse of mathematics to make a political point, and a huge section of society has swallowed it whole, unable to break it down because they don't understand the statistics.)
Mathematics education has changed, but not enough. Not every teacher is using it. Not every teacher has the supports in place - notably the classroom supplies - to use it. Many, perhaps even most teachers still rely on textbooks that short-circuit real problem-solving in favour of giving an example and asking students to do it that way. We can't see yet what a society full of mathematically-literate people looks like; at least, not in North America.
My vision is to see how a mathematically literate generation of students would change the face of society. How would our civilization be different if more than half of our kids were able to read a series of political stats and call out the person who deliberately misrepresented them as a liar? How would our society improve if every young person determined that living off credit was a deal with the devil and they weren't going to do it? How would our society be better if a huge scientific discovery in the area of astrophysics generated a demand for more funding to pursue it further? How would our society be better if math were again seen as art, integral to and interconnected with other arts?
As in any discipline, if you ask the wrong questions, you get the wrong answers. The author of the article asked the wrong questions, because he lacks a vision of the way things could be.
I can certainly see the case for not needing the full gamut of mathematics available to the end of high school; that's why they're called electives. But the author (who, sadly enough, is a professor emeritus of mathematics) has made a fundamental error in logic: he has taken the mathematically-illiterate society we have, where being unable to calculate a tip is seen as normal and acceptable, and claimed that people don't need math to get along in that world. He's right. The reason, however, is not that math is useless; it's that we're comfortable as a society with the failings we've built into the system to accommodate people who can't do math.
The better question is: what harm does rampant mathematical illiteracy do to our society, that could be remedied with better mathematical literacy? This is a much harder question, because it requires both a diagnosis of a problem that currently exists and a proposed solution.
From my somewhat less-exalted position, I see several ills in society that could be improved with better mathematical literacy.
The first is our societal inability to understand and use statistics. This begins with journalists and trickles down to everyone who reads what they write. When experts in various branches of science or finance or politics begin to talk about the effects of their discoveries, most people can't parse the numbers. Even those who can figure out what those numbers mean in context can't always break them down to decide for themselves if the numbers seem reasonable, which means they're unable to use their own knowledge to verify or deny the claims with any accuracy. Does this mean they don't try? No, it doesn't - it means when they do try to parse those claims, they get it wrong, and most people DON'T NOTICE. The result is an inability to separate a well-spun lie with statistics from a truth that could improve our lives.
One lie perpetuated this way is the myth about stranger danger, leading to us bubble-wrapping our kids far later than any society in the past has ever done. Politicians use these lies all the time to drum up support for this or that cause - I suspect the recent health care debate in the U.S. is part and parcel of this problem, and I'm quite certain the Tea Party movement is based in large part on the participants' inability to work the math.
The second, closely related problem is the general scientific illiteracy in society. Again, it leaves us unable to interpret scientific data with accuracy, which then leads to devaluing it. Thus NASA is defunded while the war in Iraq continues to grind on.
The third is our societal inability to deal with finances. When money is almost completely computerized, the inability to manipulate numbers and understand what's happening to our money is a significant problem for many people. This is evidenced by the willingness to overspend using credit.
Our society would be better off if every young person got to the end of grade ten with a thorough understanding of all the math concepts presented up to that point. Which brings me to my second point: that the good professor is overstating the change in mathematics education and therefore has understated its impact on understanding.
Sea changes in education take decades. We've been engaged in this particular change since the mid-eighties, so a little more than two decades. I'm in a position to know much more than he is: many teachers haven't caught on yet. They think they're doing it right when they use manipulatives, but they haven't got a firm grasp on the entire span of changes to teaching that needs to happen for the most effective math learning to take place. It's still very common for teachers to stop using manipulatives almost entirely in the junior grades. It's still very common for teachers to be unable to diagnose the fault in understanding that a child is experiencing well enough to scaffold their learning and fix the problem. When teachers are stressed, they fall back to teaching the way they were taught.
At the moment, most standardized tests in America are asking for factual knowledge of discrete facts. Teachers need to teach to this because their jobs are on the line if their kids aren't doing well. So they teach using methods that will increase discrete knowledge and skills. The research into mathematics education (and all other branches of education, for that matter) says that this is backwards; we need to be teaching to the big ideas and then engaging our students to break down those ideas to get to the little bits. This is not yet a consistent thing in North American classrooms, and part of the reason is that the test scores are being misused to blame teachers instead of supporting them. (And we're back to the first point: this is a classic example of the misuse of mathematics to make a political point, and a huge section of society has swallowed it whole, unable to break it down because they don't understand the statistics.)
Mathematics education has changed, but not enough. Not every teacher is using it. Not every teacher has the supports in place - notably the classroom supplies - to use it. Many, perhaps even most teachers still rely on textbooks that short-circuit real problem-solving in favour of giving an example and asking students to do it that way. We can't see yet what a society full of mathematically-literate people looks like; at least, not in North America.
My vision is to see how a mathematically literate generation of students would change the face of society. How would our civilization be different if more than half of our kids were able to read a series of political stats and call out the person who deliberately misrepresented them as a liar? How would our society improve if every young person determined that living off credit was a deal with the devil and they weren't going to do it? How would our society be better if a huge scientific discovery in the area of astrophysics generated a demand for more funding to pursue it further? How would our society be better if math were again seen as art, integral to and interconnected with other arts?
As in any discipline, if you ask the wrong questions, you get the wrong answers. The author of the article asked the wrong questions, because he lacks a vision of the way things could be.