tag:blogger.com,1999:blog-7718462793516968883.post7634661190956093730..comments2020-09-20T00:47:09.385-04:00Comments on MadMath: Teach LogicDeltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-7718462793516968883.post-22618217799629980472013-07-30T00:44:19.089-04:002013-07-30T00:44:19.089-04:00I totally disagree with the statement that "(...I totally disagree with the statement that "(m)ath students have no problem with logical thinking," a fortiori from my observation that seemingly ALL students have problems with logical thinking. I teach logic, and the difficulty with seemingly-obvious things like the distinction between "necessary" and "sufficient" baffles me. STEM or humanities, none of them seem to get it easily, and that's not the only example I could cite. On the contrary, it's usually only once I start showing them with specific examples that they understand, for instance, why material implication works the way it does.LWSCHURTZhttps://www.blogger.com/profile/06635573516962732975noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-26659456532795428802012-07-02T23:33:17.596-04:002012-07-02T23:33:17.596-04:00Interesting thoughts! mathfour, it seems all my ma...Interesting thoughts! mathfour, it seems all my math classes devolve into detailed-reading classes, so your daughter should be ahead of that curve. And Bill, I ADORE the idea of teaching limts early -- just tonight I was trying to grasp for the idea, a bit frustrating not to have it (e.g., central limit theorem)dcollinshttps://www.blogger.com/profile/04676035133383556920noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-35265352906673544052012-07-02T15:09:03.072-04:002012-07-02T15:09:03.072-04:00And, of course, there is no reason not to introduc...And, of course, there is no reason not to introduce the concept of limit right after kids learn rational numbers.Aaron Powellhttps://www.blogger.com/profile/06941521455903170510noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-31012622219724358842012-07-02T15:07:10.889-04:002012-07-02T15:07:10.889-04:00In math, it is quantification (not logic) the stud...In math, it is quantification (not logic) the students are bored and often disgust with. Add human dimension to a number -lecture will rock.<br /><br />Math students have no problem with logical thinking. It is the quantification they are bored and often disgust with. They have a problem of logic applied to quantities, numbers only. Logic applied to music will be ok for many of them.The point with math is to convince someone to take direction of quantitative thinking. Logic will follow, as it follows ordinary thinking.Aaron Powellhttps://www.blogger.com/profile/06941521455903170510noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-79462659070794994562012-07-02T14:42:22.644-04:002012-07-02T14:42:22.644-04:00From Reuben Hersh's book "What is Mathema...From Reuben Hersh's book "What is Mathematics Really".<br /><br />Any proof has a starting point. So a mathematician must start with some<br />undefined terms, and some unproved statements. These are "assumptions" or<br />"axioms." In geometry we have undefined terms "point" and "line" and the<br />axiom "Through any two distinct points passes exactly one straight line." The<br />formalist points out that the logical import of this statement doesn't depend on<br />the mental picture we associate with it. Nothing keeps us from using other<br />wordsâ€”"Any two distinct bleeps ook exactly one bloop." If we give interpretations<br />to the terms bleep, ook, and bloop, or the terms point, pass, and line, the<br />axioms may become true or false. To pure mathematics, any such interpretation<br />is irrelevant. It's concerned only with logical deductions from them.<br />Results deduced in this way are called theorems. You can't say a theorem is<br />true, any more than you can say an axiom is true. As a statement in pure mathematics,<br />it's neither true nor false, since it talks about undefined terms. All mathematics,<br />can say is whether the theorem follows logically from the axioms.<br />Mathematical theorems have no content; they're not about anything. On the<br />other hand, they're absolutely free of doubt or error, because a rigorous proof<br />has no gaps or loopholes.Aaron Powellhttps://www.blogger.com/profile/06941521455903170510noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-80494905401274614782012-07-02T14:40:49.576-04:002012-07-02T14:40:49.576-04:00This comment has been removed by the author.Aaron Powellhttps://www.blogger.com/profile/06941521455903170510noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-89379792545864925222012-07-02T13:45:40.586-04:002012-07-02T13:45:40.586-04:00Aha! You've just supplied me with part of my a...Aha! You've just supplied me with part of my argument for never letting my daughter into a math book. (She's currently 2.75 years old.)<br /><br />I believe her math education should start with learning to read instructions (DVD players, IKEA furniture, etc.) and then continue into good logical practices. After which, if she begs, she can be allowed a "math" book. This will then be consumed with vigor as she'll be able to follow everything. Because she learned logic first.<br /><br />Thanks so much!Anonymousnoreply@blogger.com