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Mathematics = thinking clearly

On the Azimuth blog I appended this to a comment I made:

My New Year’s resolution is to have another go to sell the idea that “The subject matter of Mathematics is how to think clearly about problems (mostly excluding human interaction issues like culture)”. Teachers and students are hopelessly confused by an education system that treats mathematics as a collection of facts (about Platonic entities) which is sometimes useful in the real world. My definition will give Mathematics its rightful place in the core of a modern education. I’m not going to make any progress until I can find a real Mathematician to endorse the idea.

I was then delighted to get an endorsement from John Baez himself. This turned my project from a fantasy to something that might actually happen. So I’m going to use this forum thread to do some elaboration and some planning for how we can maximize our chances of taking over the world. I welcome any forumites to join the discussion, whether you agree or disagree. John has indicated that if the discussion is reasonably coherent he will use it as the basis of a blog post. That will significantly increase the number of people reached and contributing positive and negative views. Hopefully soon after that we’ll have a plan and we can start in earnest.

“you find sometimes that a Thing which seemed very Thingish inside you is quite different when it gets out into the open and has other people looking at it.” A.A.Milne

Comments

  • 1.

    The objective isn’t really to (re)define what mathematics is, though that is the proposed strategy. Rather it is to get the message out that: (a) Mathematics does not have limited applicability, but is the methodology for thinking clearly about the full range of problems that don’t involve complex motivation [and the better you are at it, the more likely you are to think clearly about any problem you come to]; (b) Teachers and students need to know that Mathematics is not about facts, nor even about specific methods, but about understanding and clear thinking, and it is the core of a modern education.

    Comment Source:The objective isn’t really to (re)define what mathematics is, though that is the proposed strategy. Rather it is to get the message out that: (a) Mathematics does not have limited applicability, but is the methodology for thinking clearly about the full range of problems that don’t involve complex motivation [and the better you are at it, the more likely you are to think clearly about any problem you come to]; (b) Teachers and students need to know that Mathematics is not about facts, nor even about specific methods, but about understanding and clear thinking, and it is the core of a modern education.
  • 2.

    One of the drivers of this project is the view that math has much wider applicability than is generally recognized. In particular inference is a widespread need, and Bayesian probability is the core mathematics for thinking clearly about inference. Also computer programming is an increasingly important part of every aspect of life. Programming certainly requires clear thought and this activity is increasingly built on sophisticated mathematics.

    Before computers, if the mathematics led to the need for a significant amount of computation then it was impractical. This reduced the range of applicability of mathematics. That has changed and we need to integrate math with programming. If a student needs to find the area of a triangle given 2 sides and the angle, and they write a program that gets a result by approximate methods, then that is mathematics. Knowing the formula isn’t (unless you understand it and could derive it).

    Comment Source:One of the drivers of this project is the view that math has much wider applicability than is generally recognized. In particular inference is a widespread need, and Bayesian probability is the core mathematics for thinking clearly about inference. Also computer programming is an increasingly important part of every aspect of life. Programming certainly requires clear thought and this activity is increasingly built on sophisticated mathematics. Before computers, if the mathematics led to the need for a significant amount of computation then it was impractical. This reduced the range of applicability of mathematics. That has changed and we need to integrate math with programming. If a student needs to find the area of a triangle given 2 sides and the angle, and they write a program that gets a result by approximate methods, then that is mathematics. Knowing the formula isn’t (unless you understand it and could derive it).
  • 3.

    Graham Jones responded to my comment with this link: Dan Mayer's TED talk, which I can recommend. The speaker, Dan Meyer, looks at a text book question which is very well specified to make it clearly fit a previously provided mathematical treatment. This is the sort of thing that makes people think that math has limited applicability. He changes it to the sort of under-specified problem that we get in real life and challenges his students to think clearly about it, showing the role of math in that clear thinking process. Dan makes other good points.

    Comment Source:[Graham Jones](http://www.indriid.com/) responded to my comment with this link: [Dan Mayer's TED talk](http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html), which I can recommend. The speaker, Dan Meyer, looks at a text book question which is very well specified to make it clearly fit a previously provided mathematical treatment. This is the sort of thing that makes people think that math has limited applicability. He changes it to the sort of under-specified problem that we get in real life and challenges his students to think clearly about it, showing the role of math in that clear thinking process. Dan makes other good points.
  • 4.
    edited January 2011

    Aphorisms from (or reported by) David MacKay about clear thought are relevant: “Numbers not adjectives”, “Modelling not arm-waving” and “You can’t make deductions without making assumptions”. The last one is intended for statistical inference, but we can stretch it.

    Here’s a recent quote by an Economic pundit: “... a bias towards asset bubbles and overinvestment, while holding down consumption, until the system becomes top-heavy and tips over, as happened in the 1930s”. This sort of arm-waving with adjectives can’t be evaluated until it is given a mathematical form. It’s “not even wrong”.

    Comment Source:Aphorisms from (or reported by) David MacKay about clear thought are relevant: “Numbers not adjectives”, “Modelling not arm-waving” and “You can’t make deductions without making assumptions”. The last one is intended for statistical inference, but we can stretch it. Here’s a recent quote by an Economic pundit: “... a bias towards asset bubbles and overinvestment, while holding down consumption, until the system becomes top-heavy and tips over, as happened in the 1930s”. This sort of arm-waving with adjectives can’t be evaluated until it is given a mathematical form. It’s “not even wrong”.
  • 5.

    Mathematics needs to be up front about the fact that it shares the GIGO (garbage in, garbage out) problem with computer programming. Education should inoculate people against believing too readily in stuff just because of lots of complicated mathematics or impressive computer output.

    [I haven't even got to strategy, but I'll take a break for a while.]

    Comment Source:Mathematics needs to be up front about the fact that it shares the GIGO (garbage in, garbage out) problem with computer programming. Education should inoculate people against believing too readily in stuff just because of lots of complicated mathematics or impressive computer output. [I haven't even got to strategy, but I'll take a break for a while.]
  • 6.
    edited January 2011

    I think you might need to look for something similar to the English Language/English Literature pairing if you're making a really intense push for school curricula. It's a nice idea to dramatically de-emphasise the teaching of theories, and even more so methods and specific results, but given that not even in future will every student be hooked on academic learning, there are sets of things which it's worth trying to teach them even if they aren't interested in the higher level stuff, eg, teachers should definitely have a go at getting all students familiar with simple and compound interest and some elementary understanding of models like mortgages and credit card schemes even if the "most devoted to not paying attention" still don't learn anything. Of course, this could be in parallel to more general mathematics and clear thinking teaching.

    What does definitely strike me is how much school mathematics was taught "because it works out cleanly" even if no-one uses that particular topic. I'm probably relatively rare in the general population in that, working with cameras I do occasionally actually use the Euclidean geometry taught at school, but even I've never found an occasion where a cyclic quadrilateral has actually occurred in the world for me to reason about. Removing some of those things and replacing them with more general reasoning and less "clean" problems would certainly be a good thing. Of course, to some extent the best appreciation for general mathematics comes from working on actual examples, just as the best appreciation for how the English language has possibilities for metaphor and prosody comes from studying good pieces of existing poetry in English.

    Comment Source:I think you might need to look for something similar to the English Language/English Literature pairing if you're making a really intense push for school curricula. It's a nice idea to dramatically de-emphasise the teaching of theories, and even more so methods and specific results, but given that not even in future will every student be hooked on academic learning, there are sets of things which it's worth trying to teach them even if they aren't interested in the higher level stuff, eg, teachers should definitely _have a go_ at getting **all** students familiar with simple and compound interest and some elementary understanding of models like mortgages and credit card schemes even if the "most devoted to not paying attention" still don't learn anything. Of course, this could be in parallel to more general mathematics and clear thinking teaching. What does definitely strike me is how much school mathematics was taught "because it works out cleanly" even if no-one uses that particular topic. I'm probably relatively rare in the general population in that, working with cameras I do occasionally actually use the Euclidean geometry taught at school, but even I've never found an occasion where a cyclic quadrilateral has actually occurred in the world for me to reason about. Removing some of those things and replacing them with more general reasoning and less "clean" problems would certainly be a good thing. Of course, to some extent the best appreciation for general mathematics comes from working on actual examples, just as the best appreciation for how the English language has possibilities for metaphor and prosody comes from studying good pieces of existing poetry in English.
  • 7.
    edited January 2011

    Robert wrote:

    John has indicated that if the discussion is reasonably coherent he will use it as the basis of a blog post.

    I was trying to say that if you — or anyone — writes something that seems like a nice blog entry, I'll put it on the blog.

    Alas, I quite likely won't have the energy to take a discussion and synthesize it into a blog entry: I have a fair amount of energy for writing blog entries, but there are tons of things I desperately want to blog about. For example, I want to write blog entries about all the plans of action listed here!

    So, I hope you could use a discussion here to help you write a nice article explaining your point of view, with references to useful work.

    Comment Source:Robert wrote: >John has indicated that if the discussion is reasonably coherent he will use it as the basis of a blog post. I was trying to say that if you — or anyone — writes something that seems like a nice blog entry, I'll put it on the blog. Alas, I quite likely won't have the energy to take a discussion and synthesize it into a blog entry: I have a fair amount of energy for writing blog entries, but there are tons of things I desperately want to blog about. For example, I want to write blog entries about all the [[plans of action]] listed here! So, I hope you could use a discussion here to help you write a nice article explaining your point of view, with references to useful work.
  • 8.

    My tendency to react to things I disagree with is eventually constrained by my ability to see both sides of any argument, and a general belief in moderation. The whole process can lead in unexpected directions.

    Natural language is designed to be a good way to represent internal mental states. And internal mental states are where we exploit the brain's amazing capabilities to do parallel search for interconnections. So natural language has to be at the core of communication of clear thought (showing that I wasn't thinking too clearly myself in earlier comments). However when you get a real lot of natural language, like a large text book, I wonder how easy it is to get that into a good internal brain structure.

    Anyway this set me wondering whether one might try to copy the brains internal structures a bit. The idea is to have nodes that are connected in multiple ways and amenable to computer processing. The text is unambiguous (as far as possible) because the ontology and parsing is specified. Nodes can link to other nodes in various ways, including: (a) (parameterized) Bayesian network specifying the probability of a node given another (when meaningful); (b) software module interaction for nodes with associated software; (c) just links; ... The hope would be that you could put in a statement (like the economics one given partially above) and it would search around, find other relevant stuff, find data which might bear on the matter, code that might let you do relevant calculations on the data, and other useful stuff. This would be linked to information relevant to the individual. Individuals can specify how much they understand nodes, how much they agree with them. If you want to understand something new then it would lead you through other stuff you need to understand first. And it could do lots of other useful things to help you understand the subject...

    Comment Source:My tendency to react to things I disagree with is eventually constrained by my ability to see both sides of any argument, and a general belief in moderation. The whole process can lead in unexpected directions. Natural language is designed to be a good way to represent internal mental states. And internal mental states are where we exploit the brain's amazing capabilities to do parallel search for interconnections. So natural language has to be at the core of communication of clear thought (showing that I wasn't thinking too clearly myself in earlier comments). However when you get a real lot of natural language, like a large text book, I wonder how easy it is to get that into a good internal brain structure. Anyway this set me wondering whether one might try to copy the brains internal structures a bit. The idea is to have nodes that are connected in multiple ways and amenable to computer processing. The text is unambiguous (as far as possible) because the ontology and parsing is specified. Nodes can link to other nodes in various ways, including: (a) (parameterized) Bayesian network specifying the probability of a node given another (when meaningful); (b) software module interaction for nodes with associated software; (c) just links; ... The hope would be that you could put in a statement (like the economics one given partially above) and it would search around, find other relevant stuff, find data which might bear on the matter, code that might let you do relevant calculations on the data, and other useful stuff. This would be linked to information relevant to the individual. Individuals can specify how much they understand nodes, how much they agree with them. If you want to understand something new then it would lead you through other stuff you need to understand first. And it could do lots of other useful things to help you understand the subject...
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