An apocryphal joke has a medical student failing physics and questioning why he should ever have to solve useless mechanics problems that he will never again see in his life. The physics professor reassures the hapless student.
"These problems are terribly important: They save lives."
"How?" cries the student.
"They keep thousands of idiots like you out of medical school."
There are many reasons why we teach various parts of Physics and Mathematics but not all of them are obvious. Physics centers around finding a simple set of universal laws that govern the universe at the most basic level. The skill set that physics is trying to teach medical students is the ability to disassemble a complicated problem into smaller, more easily solvable component parts, use some of those laws to understand the parts and then reassemble the pieces into a whole. The human body is one of the most complicated machines we have ever studied and, if a doctor cannot understand the workings of a simple mechanics problem, then he really will kill people.
Andrew Hacker recently argued with some force, in the New York Times, that we should not be torturing the minds of high-school students with algebra. The primary burden of the piece is to catalog the abysmal performance of US students and to evoke sympathies of the vast majority of people who never use algebra after high school. While recognizing the need for an intellectual elite that can do algebra, Dr. Hacker goes on to argue that something that is so useless should not be holding back students who might be able to make remarkable contributions elsewhere in our culture. He advocates that students should be taught things that are more real, like how the CPI is constructed and the meaning of statistics. "This need not involve dumbing down. Researching the reliability of numbers can be as demanding as geometry." So presumably another set of differently-abled people will be held back and many more useless (and annoyingly difficult) things like Shakespeare, French and Astronomy can safely be dropped from the curriculum. I sympathize with the critical need for everyone to know what statistics and margins of error mean and for them to be able to compute their mortgage payments, but I also believe it is crucial that high-school students also learn very basic algebra.
One of the less obvious goals in algebra is to get people to think more abstractly. Very elementary mathematics is all about "real things" and initially employs realia to help us add, subtract and multiply. From this experience we learn the language and some of the basic rules of mathematics. We abstract and generalize the experience and learn that, when we manipulate one side of an equals sign then the equality is only true if we do the same thing to the other side. Algebra makes a major intellectual leap: It names and labels things that we do not immediately know and that sometimes lie outside our direct experience. There are certainly other studies that involve abstractions like love, empathy and ethics, but in algebra we learn to handle abstractions that are not part of visceral human experience. We learn not only to be comfortable with such external unknowns but how to master them.
In algebra we develop essential life skills. We learn dispassionate analysis of external realities: how to simplify the things that we know and reduce the things that we do not; to see that some problems are unsolvable as presented; to identify exactly what data is needed to solve a problem entirely; to recognize extraneous data that is irrelevant to our problem; to identify data that conflicts with what we already know about a problem. By learning algebra we all become far better thinkers and even the majority who never use algebra again will still have enriched their life experience and expertise by grappling with difficult abstraction.
A limited understanding of one's passions and of the real things that can be manipulated by hand were sufficient to the needs of peasantry in medieval times. Today's society requires us to think in abstractions, to understand why an invisible, odorless gas that we breathe out every moment of our lives might be killing us all through climate change. We need to manipulate these abstractions to reasonably determine whether something is a fad or whether we must change our life-style. Does vaccination cause an unacceptable risk of autism? Does your body mass index affect your long-term health? What further data do we need to make an informed decision? What are the unknowns we should try to corral and eliminate before we make a critical decision or before we vote?
Algebra was developed by the Arab cultures as Western Europe was emerging from the Dark Ages. Algebra is not just the language of mathematical elites, it is one of the cornerstones by which we have emerged from a peasant society, ruled by the small elites sometimes capable of abstract thought, to become a complex, vibrant democracy. Algebra has helped us to rise beyond the simple understanding of immediate, tangible experiences and frame questions and look for the essential data that will give us deeper understanding. Only authoritarian and reactionary politicians benefit from a population for whom abstractions have no meaning. Such a population will be satisfied by sound bites and flag waving and will be placated by bread and circuses while their economy is subverted and their democracy implodes. Like mechanics problems in physics, the study of algebra, and the skills it develops, are not just critical to our long-term health individually but to our survival as a society.
~Oscar
Why is that, Dr. Warner? Answer that question, and this article will become much more interesting.
http://motls.blogspot.com/2012/08/hackers-fight-against-algebra-in-new.html?m=1
Even if algebra were "useless to the majority", it's still very important for the system because it actually distinguishes people who can think accurately and reliably enough from those who can't. That brings good grades to some, bad grades to others, and some people become dropout because of algebra.
Those bad grades and abandoned schools may be bad for those who are rated in this unflattering way but they're good and I would say vital things for the society because they help to allocate human resources more properly. In some cases, such allocation really saves human lives. It also increases the GDP. Grades for most other subjects tend to be jokes. They're either grade-inflated "predominant" A's or they reflect how much the teacher likes a student personally, if I avoid mentioning more intimate relationships.
And bad grades for algebra really mean that algebra should be taught more, not less, while other subjects where people already can get A's are being overtaught and too much time is being wasted with those things that most people learn rather quickly, anyway. (And many other subjects boil down to mere memorization which is much more acceptable an activity than abstract thinking for many folks - which is another yet related story.)
I don't think there is any higher reason for a student to learn algebra or geometry other than that it is useful. Try telling a struggling algebra student about "dispassionate analysis of external realities." Algebra doesn't make us better people, it makes us more effective people. Yes, I agree that physics teaches problem solving (but for a doctor it is also useful foor understanding things like pressure or the optics of the eye, not to mention how all the modern medical tools work), but if we need to project into the future to make things useful for students, we probably are not doing a very good job of teaching.
Algebra is , simply put, useful. What needs to change is HOW and WHEN it is taught. Math teachers should be bringing the theory into a context students can relate to - this is a teaching issue, not an algebra issue. In this context, the Times article is right on.
This person's work might be in the right direction:
http://www.edweek.org/tsb/articles/2011/04/04/02meyer.h04.html
Dr. Hacker's piece does contain a condemnation of the reality of how algebra is taught. None of us disagree with the data, only with the conclusion he wants to make. That is what needs a careful and thoughtful response.
And that DOES represent a problem. But it's not a problem that has anything in particular to do with how algebra is taught.
It was real!
Then another day I thought of pi, and again, something clicked and I realized that it was a relationship number that would never change.
Even if we make all things in whole number amounts, say my 3 inch ditch, the relationships always end in fractions. Things that are not whole, nice numbers, at least at this time I think that is so.
You can use pizza ads to find that the area (the amount of pizza you get) works out with the
A=pi rsquared every time. Use the calculator, square roots are no fun the other way.
Will you get more pizza from a large pizza (put in the size they give you) or two medium? Again use their numbers. A large isn't always a large in another store.
This is baby simple stuff and I am sure you are all way beyond this, but I hope you laughed a bit at it.
I think the problem its not so much algebra but the preparation of the kids before they encounter algebra. Do they have their multiplication tables at the level of an autonomic response? Are they confident of their ability to add, subtract, multiply and divide. Do they have fairly good control of how to do these things with fractions? If a decent level of competency is not attained in these basic techniques they everything is simply going to come apart in algebra, geometry and trigonometry.
I cannot play the violin. I know in great detail how the violin works. I have listened to a very large number of performances. I know the steps required to reach a certain level of competency and yet I cannot even make a handful pleasant-sounding notes. This is because I have never practiced even the most basic technique. I suspect the problem is the same in math: kids know the steps but do not get, or do, the essential exercises that will raise their technical ability to where it must be to get them to the next level.
What sometimes amazes me is that kids (and parents) understand this simple fact when it comes to sport and yet they do not seem to think it applies in academics, particularly math.
Another thing that schools are doing in California where algebra is taught in 8th grade is passing students into geometry as freshmen, even if they failed algebra. This gives students a chance to be successful and see other applications of math. (those that fail geometry return to algebra, with perhaps a broadened perspective).
More practice is not the answer (we need to know how to practice and why we practice - that is more obvious in sports). Proper instruction, relevance of material, and remediation (without playing the blame game of preparedness) is what is needed to improve success in math.
I guess you can reasonably deduce which side of the debate I'm on.
She loves numbers, especially magic 0 which can make a 1 into a 10. So they put her in the Girl Scout Cookie book at 6 boxes of cookies, and she added 2 magic 0's. They caught the 600 box order before it went out.
There are many types of abstractions in the world, but why so much focus on algebra? I teach teachers at a university and was walking to the parking lot with a mathematician I didn't know, and he asked why we are so preoccupied with algebra in K-12 education. We both agreed that understanding statistics and probability has far broader everyday uses, such as judging intelligently the claims of opposing politicians. Basic statistics and probability are far more important for an educated citizenry, but for some reason, it gets neglected while students get two years of algebra. How odd.
Math is less about memorizing formulas like the Pythagorean Theorem or Margin of Error. It's about understanding how those relationships work, and how to get there. And without a foundation in algebra, it’s difficult to gain a true understanding of more “useful” disciplines like statistics and probability.
Rid the world of 24/7 entertainment media and we're halfway back to being a respectable race of beings.
For the most part, nothing higher is needed than Algebra I, which is good for figuring out how much money you're going to have to pay back by the time you're done paying off your car note. Those going to college are re-taught geometry and Algebra II anyway (usually in a far more concise and practical way than we learned it in high school), and those who aren't going certainly don't need to be tortured with it. There are very few reasons to need to know the surface area of a right cylinder.
Don't underestimate the value of basic geometry: estimation skills are very useful.
I did not really like geometry when I took it. I had to review it 2 years ago when my 13 year old daughter asked for help on questions she didn't understand during a summer independent study course on it. But I was able to figure out and show her how to all but one problem - and I haven't done geometry in 45 years.
The real issue with math and the STEM subjects is their cumulative nature. They build in depth and breath as you learn more and holes in your understanding propagate and grow, creating obstacles to further learning. As a student, I was notorious for taking advanced science courses without prerequisites, but I really had to work to fill in the material that I had skipped.
If students have not taken an adequate background for STEM material, they have foreclosed those options in the future.
How could we divide polynomials with binomials when I was still trying to understand where in real life they all existed. I learned to manipulate the numbers, eventually it just worked.
Without high school Algebra, I lacked the basic to see where this was all going. I finally ended College Algebra with a sound C. I could do better today.