In the humanities, you are taught how to understand what people are trying to do with language, so I can't help reading your comment from that perspective.
You say people rarely fail algebra and succeed at other subjects. To put that another way, let's assume that there are four sets. Set A consists of a group of people who are successful at one or more subjects, and Set B consists of people who are successful at no subjects. Set C consists of people who are good at math, and Set D consists of everyone else. Your statement implies that, for the most part, A = C and B = D.
Thus you seem to be arguing that excellence at math is the sine qua non of an educated person -- it is a defining characteristic.
What might you be trying to do with this line of reasoning? It seems fair to assume that you yourself are skilled at math, so really you are making a claim about yourself and other people who are similar to you. Your model of an educated person is based on yourself. In society, educated people are people whose judgments should be trusted and listened to, so we can conclude that the purpose of your argument is to try to get those benefits for yourself and people like you. Conversely, we can also conclude that you are trying to have those benefits removed from people who are different from you.
"Thus you seem to be arguing that excellence at math is the sine qua non of an educated person -- it is a defining characteristic."
No[1]. I am saying that a person who is on a path to being educated won't let something as simple as algebra stand in their way.
The context is that some people are failing in pretty major ways in life, and the question is whether that's due to algebra requirements or not. I say that's ridiculous -- nobody that's on a path toward education will give up because of something as simple as algebra... there are other problems going on there that we are not seeing. And we should be grateful that algebra does show us that the student has fallen off the path to education, and we should be trying to identify that situation even earlier.
I am willing to be proven wrong empirically, if you can make the case that some significant fraction of people are well on their way to an education, but then it gets derailed because of an algebra requirement.
[1] I am inclined to believe that algebra (or something very similar) is fundamental to an education, particularly those parts of an education that involve reasoning, but I don't state that as a fact or even my opinion.
"Conversely, we can also conclude that you are trying to have those benefits removed from people who are different from you."
No, you can't. There are a long string of false assumptions and logical flaws in your post. You should reflect on your own reading and writing.
Here’s what I said in a discussion about it elsewhere.
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These things are all about (1) experience piling on experience, through deliberate practice, and (2) having the right mindset to approach learning and problem solving. Every “I’m just bad at math” student I tried to tutor in high school algebra was entirely capable of understanding the concepts, when forced to focus by 1-on-1 help and when provided with some slight bits of guidance. Most of their inability to do the work was psychological blocks, reinforced over time by a destructive self-vision and -message. I.e. repeat enough times “some people just can’t do math, and I’m one of them”, and, little surprise, the prophecy fulfills itself.
In my firm opinion, the best to approach complex abstract systems is by building a thorough and cohesive mental model of how they work. To add a new concept or abstraction into the existing model requires testing the new part against all the existing parts in many combinations to see how they fit together, to think about how they might be used with each-other, to find out where the mismatches are or where prior understandings were faulty... But the learner must start with a thorough and cohesive model before it’s possible to fit anything new into it. So this is a process which takes years, and can’t be magic-wanded later.
Unfortunately, it is extremely hard to convince a classroom of students to adopt the right mindset and then focus serious amounts of attention for long periods of time on learning, if they haven’t figured it out already. And it’s easy for external factors – stress, sleep deprivation, distractions – to throw things off. So instead, many teachers try to teach atomized and easier-to-test chunks of “plug-and-chug” type material. This is in the long term entirely destructive, because without something to attach it to, students forget what they learned last time, and must start from scratch on each new formula or method. Instead of understanding the meanings of what they’re doing, or having any clear motivation (other than “if you fail the test you’ll spend your life serving fast food”), students are left with only a mechanical process to copy. I think of the relationship between this and real mathematics as something like the relationship between candyland and chess, or between filling in a color-by-numbers picture and learning to paint.
The ones who for whatever reason are slightly better at copying the mechanical processes (in whatever subject) are praised, feel bits of accomplishment, and are willing to devote a bit more time. Some of them even start to piece together more comprehensive models on their own, or play with the parts a bit for fun, and are promoted through this work into the ranks of the mathematically literate. The ones who start out really stuck, who have difficulties performing the mechanical processes, don’t see the point, and don’t receive any further help, just start to adopt a defeatist view of the whole thing. They aren’t willing to spend time on something so unrewarding and painful. They start to doubt their basic competence. Avoidance becomes a more and more attractive alternative to effort.
The idea that there are a sizable proportion of people just innately incapable of abstract thought is complete nonsense. Some people have obvious severe mental handicaps (down syndrome or similar), but everyone else is plenty capable of learning high school algebra.
"No, you can't. There are a long string of false assumptions and logical flaws in your post."
Stated as fact, but you give no reasons why anyone should believe it.
I thought this part was quite striking: "algebra does show us that the student has fallen off the path to education."
Isn't this essentially my point? For you, algebra is the defining characteristic. If you don't know algebra, you don't know anything. The claim is not that math is an important part of an education, but a much stronger one: math is the foundation of all other types of knowledge. That's how you're able to conclude that failing at algebra means failing at every other subject.
To repeat my earlier point, this is not persuasive. Everyone wants to believe that they're particular skillset is vital and important, because everyone wants to feel vital and important. It seems important to think against the grain of our own biases instead of pandering to them.
"Stated as fact, but you give no reasons why anyone should believe it."
I said something fairly simple and you were the one that made the surprising and radical claims (not to mention personal attacks). It's up to you to justify them, and you'll get no help from me.
"For you, algebra is the defining characteristic. If you don't know algebra, you don't know anything."
You keep repeating this, but I have not said it.
A dead canary shows us that the mine is no longer safe. But a live canary is not a defining characteristic of a safe mine.
Do you not see the irony in that your own argument above used set theory to prove your point about algebra?
You say people rarely fail algebra and succeed at other subjects. To put that another way, let's assume that there are four sets. Set A consists of a group of people who are successful at one or more subjects, and Set B consists of people who are successful at no subjects. Set C consists of people who are good at math, and Set D consists of everyone else. Your statement implies that, for the most part, A = C and B = D.
I think you're reading the argument exactly backwards. The perception is that algebra (and advanced mathematics) is much harder than other subjects and the perception is that people can be good at a lot of things and still be bad at math. His argument is that in reality everything is taught poorly but it's easier to fudge the results when grading is necessarily subjective.
I am incredibly surprised that what I said was so controversial. Objective measures highlight (and therefore prevent or contain) fraud in all matters, and education is no different.
Yes, the argument that objective subjects are better judges of someone's education sounds plausible at first glance, but on closer inspection, I think the opposite is actually true. Suppose we have 10 completed math tests that have been graded, and all 10 are 100% correct. We should expect that the answers will be identical (or at least mathematically identical.) On the other hand, 10 essays that have all been graded as 100% would all be different.
One effect is that it is much easier to cheat at math than cheat at writing an essay. A good teacher can more easily identify if a student turned in work that they could never have written themselves. This is because each piece of writing is unique to the student. In math, a correct answer is a correct answer, so you can't distinguish cheating, which is why math tests aren't just about writing down the objective answers, but showing your work. Graders look at how you got to an answer, what techniques you used, whether you got to the answer in a straight-forward way or not. These factors are going to be much more varied between any two tests, and probably give you a better sense of who has mastered the material even when the final answers are identical.
To me, what this shows is that knowledge is highly personalized. Even when it is objective, and has clear right and wrong answers, the precise way you actually put knowledge to use is very much unique to each person. The fact is that understanding is a subjective experience -- everyone knows that there is a feeling associated with discovering the truth, figuring out a problem and so on.
"One effect is that it is much easier to cheat at math than cheat at writing an essay."
Interesting point, but the context is people failing algebra, and I don't immediately see how the ability to cheat contributes to that problem. Can you elaborate?
"Graders look at how you got to an answer, what techniques you used..."
And although free-form answers do make it a little "softer" of a subject than it may seem at first glance, there are good objective standards to go by. The fact is, a good multiple-choice test can tell you a lot about how well a student understands algebra; it's hard to devise such a test for subjects like writing.
The point about cheating is this: what makes cheating possible? It's because there is a gap between answers written on a piece of paper, and what is inside someone's head. This is true of all kinds of tests, but it is more true of answers that are identical for every student. If the names on the tests were somehow mixed, this would be undetectable by graders, where it would be immediately apparent for a written exam.
The significance of that has nothing to do with preventing cheating. It just means that the gap between right answers and real knowledge is greater when the answers are all identical. A concrete example: in my college physics classes, I studied with a friend who solved problems by memorizing which type of problem required which formulas. I figured out which formula to use by visualizing the problem, which is a much more efficient way of doing it and leads to a better general understanding of physics.
These important differences are undetectable just by looking at whether we both got the right answers on a test. You can account for this indirectly, by limiting the amount of time, so that my study partner would never be able to finish the whole test by using his method. But the information he gets about the incorrectness of his answers does nothing to help him fix his inefficient method. He did very well on the homework, the issue was only revealed at the midterm.
Physics education would be improved if taught people how to visualize problems, which means making it more qualitative and less quantitative.
The issue is teachers cheating by grading more easily than they should, not students cheating. You are completely missing the point.
Graders look at how you got to an answer, what techniques you used, whether you got to the answer in a straight-forward way or not. These factors are going to be much more varied between any two tests, and probably give you a better sense of who has mastered the material even when the final answers are identical.
For the most part, they don't. I taught calculus for several years, and different students picked different techniques. I observed no correlation between particular choices and student quality. Good students answered more and harder problems, bad students answered only a few easy ones.
The only reason we demanded they show work is to make sure they solved the problem rather than copying the answer of the guy sitting next to them.
Exactly. I have a liberal arts degree and studied math basically only through first year calculus. The strange thing is, compared to most people around me, I am relatively skilled at math. I can do algebra, some calculus perhaps at least pulling some estimates of integrals and derivatives, but my real love is in humanities.
The reason why algebra is important has nothing to do with the GP's idea that it is the most objective of studies. Algebra is useful. That's it. People who know basics of algebra can use it constantly. I could see replacing geometry with a deductive logic class since that's all one studies with HS geometry anyway. I could see teaching less plane trig and more spherical trig too (I tried to teach myself spherical trig in order to better follow some writings of Ptolomy and others, and failed). But these aren't going to happen. But if you don't know algebra these doors are all closed.
What I think the GP is getting at is that it may not be the case that these students are really that stellar at the other subjects that are failing algebra. It's just easier to get passed along. A teacher can more easily allowing gibberish to pass for an analysis of The Great Gatsby than let pass someone's completely wrong attempt at solving a linear equation. It happens. Everyone knows it; it's not a controversial thing.
"The reason why algebra is important has nothing to do with the GP's idea that it is the most objective of studies."
I said neither that algebra is important, nor that it is the most objective.
What I'm saying is very non-controversial: failing at algebra is not just a problem but also a symptom of a general educational failure in an individual.
Even if algebra served no other practical or intellectual purpose at all, the fact that it reveals educational problems is valuable in and of itself. And that quality is due to its objective nature.
This is true, but when you look at geometric proofs all you are doing in HS geometry is deductive logic using an abstraction which is a general approximation of the real world. That's why I could see getting rid of HS geometry and just having a deductive logic class too, perhaps with a unit of Euclidean Geometry included in it.
Algebra and calculus are different though. They are tools for finding unknown information and thinking about changing values.
No, I think you take this too far. No claim is being made proficiency at mathematics is the sine qua non of an educated person. The issue at principii is teaching basic algebra. An educated person, in general, should not have trouble with basic algebra. We're not discussing PDEs, Cantor's diagonal argument, topology, numerical analysis, or any of that.
From a philosophical perspective, I think this is a strawman.
You say people rarely fail algebra and succeed at other subjects. To put that another way, let's assume that there are four sets. Set A consists of a group of people who are successful at one or more subjects, and Set B consists of people who are successful at no subjects. Set C consists of people who are good at math, and Set D consists of everyone else. Your statement implies that, for the most part, A = C and B = D.
Thus you seem to be arguing that excellence at math is the sine qua non of an educated person -- it is a defining characteristic.
What might you be trying to do with this line of reasoning? It seems fair to assume that you yourself are skilled at math, so really you are making a claim about yourself and other people who are similar to you. Your model of an educated person is based on yourself. In society, educated people are people whose judgments should be trusted and listened to, so we can conclude that the purpose of your argument is to try to get those benefits for yourself and people like you. Conversely, we can also conclude that you are trying to have those benefits removed from people who are different from you.