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Meaningful proof
Posted:
Oct 22, 2009 6:57 AM


I have been "lurking" on this list during this debate, and have found much of the discussion interesting.
It has been interesting to review my own definitions of triangle and isosceles triangle etc.
I am old enough to remember being taught my geometry from Euclid (but not by him), learning proofs by heart. It wasn't until many years later that I realised that the proofs were supposed to make sense.
Sadly, I am one of those who should never be let near a geometry class, because I don't think that teaching about ordered, labelled triangles is the best way to approach this in lower secondary at least.
At times on this list, I do have a sense of being lectured to by people who regard themselves as the cleverest kids in the room, whose word should be accepted on pain of ridicule. I am not picking out any one person as the biggest culprit in saying this. More than one person has behaved this way and as an educator, I don't find it helpful.
Teaching children about proof is not easy. Rigour is necessary, but we all need to start somewhere. Before demanding that we all start with the most refined definitions, we might remember Lakatos.
It is easy to criticise practice that is not rigorous in a way which is not helpful. I am currently concerned about a training video from "Teachers TV" in the UK that shows a teacher trying to move from "good" to "outstanding" (as graded by inspectors).
The video shows different ways of "proving" that the internal angles of a triangle add to 180 degrees. Most of the examples offered showed but did not prove and no distinction was made between them. I don't want to attack the teacher or trainer involved, but I want to make sure that the concept of proof is taught and learned effectively in our schools from an early age.
The Euclidean proof is within the capabilities of many 11 year old pupils, but it is not examined, so is often not taught. I don't want to go back to learning proofs of by heart without developing personal meaning. But I do want pupils to be challenged to think rigorously. It would be more productive to consider how we might achieve this.
Howard Tanner
Original Message From: Postcalculus mathematics education [mailto:MATHEDU@JISCMAIL.AC.UK] On Behalf Of Larry Copes Sent: 22 October 2009 05:07 To: MATHEDU@JISCMAIL.AC.UK Subject: Re: Human decency
Thanks, Jonathan, for sending along that quotation from an earlier email:
"But anyone who insists that* given a labelled triangle ABC* ABC and BCA are the same triangle should never be let near a geometry class."
I had tried to find the quote, but I've been dumping email like that under an extension of the principle of Ralph Waldo Emerson's, "The louder he talked of his honor, the faster we counted our spoons." The more adamantly someone claims something, using absolutes like "anyone" and "never," the more suspicious I am of the claim. I'm not surprised to see later evidence of the author's lack of intellectual integrity, though I withhold judgment on that claim pending his response.
My search for that quotation grew out of the belief that it initiated whatever unpleasantness has arisen in this discussion. I find the claim not only adamant but offensive. It's not a little ironic that its author is complaining about incivility.
larry
> > > > This is disturbing because I doubt that more than a few mathematicians > agree that triangle ABC and BCA are different triangles (none of the > mathematicians > I know best I have talked to about thisI mentioned no one's name > have heard > anything like this before). None of the mathematics books and other > sources > I've read that mention triangles use this definition, so I think it > is a > nonstandard definition of "triangle." However, it is a mathematical > definition > and such a triangle would still be a mathematical object. The > definition of > triangle I know and this alternate definition reflect the same > differences in > thinking about graphs versus digraphs or sets of n elements versus > ordered ntuples. > This is especially disturbing because he has presented no evidence > that teaching about > triangles using the definition I know as opposed to Tony Gardiner's > definition > causes more harm and confusion than teaching about triangles using > his definition. > > I wouldn't use the term "triangle" in his context but the term > "ordered triangle" > or "directed triangle" instead since these terms imply this kind of > structure on > triangles contained within Gardiner's definition. Perhaps students > would benefit > seeing both definitions? Perhaps so because real mathematics > classesassuming > a high school geometry class is supposed to be real mathematics > should reflect > what mathematicians think about mathematics. If there is no > universal consensus > about a particular definition, then I think we should be honest with > our students > and tell them that. We should be honest and tell them that either > definition may > be appropriate, and which is appropriate depends on the context. > Otherwise, > are we really teaching them the complete truth about what > mathematicians think? > When I think about this, I realize that I could do better about this > myself. > > > Jonathan Groves > > > > > Martin Tangora wrote: > >> Well, let's see. >> >> First of all, I did not send that to the list. >> I sent it only to T.G. >> It seems to me that to take a private message >> and quote it to the list is not the best etiquette. >> >> This explains why William McCallum had not seen >> my comments until he got them from T.G., >> and why Anne Watson had not seen them either >> (nothing to do with her junkmail filter). >> >> T.G. has an excuse. I originally thought >> I would send the message to the list, but >> I changed my mind; however, the long note >> from which he is quoting is phrased in places >> as if written to the list. So it is understandable >> that he took it as written to the list, >> and didn't check the header. >> >> However, that is an excuse for his mistake, >> but it is not an excuse for the way he quotes me, >> because he has deliberately made my quote >> into something vicious, to suit his purpose. >> >> Here's what I actually wrote to him  >> at the beginning of a long email to him: >> >>> In the issue of Mathematics Magazine that reached me >> yesterday >>> (October 2009) there is a review of a new book by >> Jason Rosenhouse, >>> "The Monty Hall problem [etc]." I quote: >>> >>> "Rosenhouse also points out 'an occupational hazard >> among mathematicians >>> ...the desire always to be the smartest person in >> the room...' >>> noting the 'relish' with which some mathematicians >> declare >>> others' arguments wrong. Let us all be chastened!" >>> >>> I might have said "the confidence that one is the >> smartest person in the room." >>> >>> Tony Gardiner is susceptible to this hazard to a >> remarkable degree. >>> And so am I. >> >> Notice that "Let us all be chastened" has been >> excised, >> and, most important, since T.G. is teaching us about >> collegial manners, he has omitted the "And so am I," >> which I am sure you all can see >> was intended to soften my comment about him. >> >> Notice also that I was not accusing T.G. of any >> special vice, >> but only of a general human failing to which I (and >> the reviewer) >> believe mathematicians (including me) to be >> especially susceptible. >> Therefore I think it is a bit of a stretch to call it >> "ad hominem." >> >> Having distorted what I said to him, and distributed >> it >> to the list, he then finds everyone on the list >> lacking in human decency and intellectual courage. >> >> Does this not support my point? >> >> Is it wicked to criticize one person, >> but professional to criticize the entire list? >> >> I apologize to the list for my comments, which I sent >> only to T.G., >> which I meant to be teasing, and which I had decided >> not to send to the list. >> >> I do not accept T.G.'s description of me as >> "delinquent," >> but would welcome the comments of the rest of you, >> on or off line. >> >> I guess T.G. has successfully framed the exchange >> so that if I show you his many thinly veiled insults, >> >> to me and to the entire profession, >> that would be an offense on my part and not on his. >> So if you desire clarification, please address me >> off line. If T.G. wants to know what I am saying >> about him off line, that would be understandable, >> and I will copy him if he likes. >> >> Yes, T.G., we have met. >> >> At 11:07 AM 10/21/2009, TONY GARDINER wrote: >>>> Rosenhouse points out [...] >>>> 'an occupational hazard among mathematicians [is] >>>> the desire always to be the smartest person in the >> room...' >>>> >>>> I might have said "the confidence that one is the >> smartest person >>>> in the room." >>>> >>>> Tony Gardiner is susceptible to this hazard to a >> remarkable degree. >>> >>> I have waited 5 days to see whether anyone on this >> list has the human >>> decency or the intellectual courage to repudiate the >> above kind of ad >>> hominem rubbish. >>> >>> I have had not one such message. >>> I leave others to decide for themselves what this >> tells us. >>> >>> I suspect I have never met the writer. >>> >>> Whilst it is irrelevant to the import of this >> message, I should >>> perhaps for the sake of truth assure the writer (and >> anyone else who >>> cares) that he could not be more wrong: progress >> depends on the >>> interaction between those unlikely twins  truth and >> humility >>> (including selfcriticism). >>> >>> I waited  but certainly not in hope that anyone >> would *disagree* >>> with the writer. We are all free to make such >> judgements; and any >>> teacher is bound to recognise the confusion which >> afflicts those >>> (even among "professors emeriti") whose pet >> assumptions are >>> challenged. >>> Rather I waited to see whether other members of >> MATHEDU understand >>> that, a list in which such opinions are allowed in >> place of reason  >>> without immediate comment and retraction  is beyond >> redemption. >>> >>> Our theme is profoundly important. >>> But we have brought it into total disrepute  not so >> much through the >>> original ad hominem nonsense (every community has >> its delinquents), >>> but by our subsequent silence. >>> >>> Sadly yours >>> >>> Tony Gardiner >> >> Martin C. Tangora >> University of Illinois at Chicago >> tangora@uic.edu



