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"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 p

"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.

jacobolus 14y ago
Here’s what I said in a discussion about it elsewhere. * * * 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.
mrxd 14y ago
"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.
- jeffdavis 14y ago
  "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.
- chris_wot 14y ago
  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.

jacobolus 14y ago

Here’s what I said in a discussion about it elsewhere. * * * 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.

mrxd 14y ago

"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.

jeffdavis 14y ago
"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.
chris_wot 14y ago
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.