Lockhart's Lament

[Domenico Rosa]

Yesterday, I received a message from the MAA with the link

http://www.maa.org/devlin/devlin_03_08.html

to Keith Devlin's column, which contains the link

http://www.maa.org/devlin/LockhartsLament.pdf

to the 2002 essay "Lockhart's Lament."

As I see it, Paul Lockhart's essay would be much more powerful if it were not written in such a complete historical vacuum. Although Lockhart decries the sterile formalism in which mathematics courses have been and continue to be taught, he makes absolutely no reference to the fact that the traditional mathematics curriculum was demolished by the excessive formalism and abstractions of the SMSG new math, as incorporated in the Houghton Mifflin series of books co-authored by Mary P. Dolciani. This apparent ignorance on Lockhart's part is likely due to the fact that he was educated with Dolciani-type books, and he may not be aware of the preceding textbooks.

The manner in which Lockhart ridicules Thales' Theorem (which he does not name), on page 19 of the PDF file, is utterly unacceptable--and it raises serious questions about the rest of his lament about Euclidean Geometry. When I studied 10th-grade Euclidean Geometry in 1963-64, at Everett High School, in the factory city of Everett, MA, we used the textbook by William G. Shute, William W. Shirk, George F. Porter, "Plane and Solid Geometry," American Book Company (1960). On page 25-27, the textbook contains a historical Note about Thales (640-546 B.C.), Thales' demonstration that all vertical angles are equal (considered to be the first theorem ever proved), deductive reasoning, and the components of a proof of a theorem. According to the Note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are!
 equal if one accepted some general notions such as:
1. all straight angles are equal
2. equals added to equals are equal, etc.  

At the top of page 10 on the PDF file, Lockhart writes: "So put away your lesson plans and overhead projectors, your full-color textbook abominations, your CD-ROMs and the whole rest of the traveling circus freak show of contemporary education, and simply do mathematics with the students!" Although this advice is quite sound, it is unfortunate that Lockhart conveniently makes absolutely no reference to the fact that all this rubbish has been produced and promoted by the self-styled math reformers of the past two decades.
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[Alain Schremmer]

I have not read Lockhart's essay in full but, I absolutely disagree  
that he "ridicules Thales' Theorem (which he does not name), on page  
19 of the PDF file". What he does ridicule is the manner in which  
geometry is currently presented/taught; he deplores what is currently  
being done TO Thales, Pythagoras, Eudoxus, etc. BY committees of  
educologists such as the committees which manufacture geometry books  
for Glencoe McGraw-Hill and which are not even listed on their web page.

As it happens, last summer, I helped my grand son go through a full  
"year of geometry" so that he could get into some program. However,  
after one look at the book and because he was going to be "tested", I  
had to tell him that what I would do would be to coach him for the  
exam and not really let him learn how to think like a geometer as I  
did. (Disclosure: I learned euclidean geometry long before  
Dieudonné's "A bas Euclide", i.e. from 1947 to 1954.) So, we started  
at the begining of Glencoe's Geometry and did everything the syllabus  
given by the school to my grandson said.

For someone who likes to think of himself as part geometer as I do,  
this was extreme, unadulterated horror. Every once in a while, my  
grandson would agree to put the Glencoe book aside for a while and to  
do a bit of geometry the way I had learned it, namely by looking at  
problems and fight them until something gave. But we couldn't afford  
to do this very often or very long.

What Lockhart describes on page 19 is absolutely true: "excessive  
notation", "a mountain being made of a molehill", etc. What I saw  
last summer essentially destroys what are normally considered to be  
tools towards attacking interesting problems, beating them to death,  
smothering them to death, without any interesting problem ever being  
considered without a shadow of what Pascal called "l'esprit de  
géometrie" (not the "spirit of geometry" but "the kind of mind you  
have to have to be good at attacking problems in geometry"). In other  
words, here we had a course in carpentry where nothing ever got  
built. And the fact is that although my grandson did pass and did get  
into the program, he will tell you that he has no idea of what  
geometry is about. I find this very, very sad.

[And now, my wife, a mathematician at the border of PDE and  
differential geometry, is coaching our grandson through Glencoe's  
algebra and has exactly the same type of reaction: she wanted him to  
thoroughly understand what he was doing but the book is constantly in  
the way.]

By way of another disclosure, my first published paper, although not  
dealing with geometry—which at the time I had completely repudiated—
dealt with "notational" issues (Des mots qui cherchent à être  
fonctionnels, Bulletin de l'APM, March 1965),  in a manner which  
today I would completely repudiate. My first book was also heavily  
Bourbakified to my great embarrassment today. (Fortunately, I  
realized what I was doing just in time to prevent the publisher from  
printing it.)

So, yes, at the time, (the sixties) it was easy to get into  
formalism, to see formalism as the issue, etc. But what happened  
afterwards is another story, one that still awaits being told.

Regards
--schremmer






On Mar 20, 2008, at 11:30 PM, Domenico Rosa wrote:

> Yesterday, I received a message from the MAA with the link
>
> http://www.maa.org/devlin/devlin_03_08.html
>
> to Keith Devlin's column, which contains the link
>
> http://www.maa.org/devlin/LockhartsLament.pdf
>
> to the 2002 essay "Lockhart's Lament."
>
> As I see it, Paul Lockhart's essay would be much more powerful if  
> it were not written in such a complete historical vacuum. Although  
> Lockhart decries the sterile formalism in which mathematics courses  
> have been and continue to be taught, he makes absolutely no  
> reference to the fact that the traditional mathematics curriculum  
> was demolished by the excessive formalism and abstractions of the  
> SMSG new math, as incorporated in the Houghton Mifflin series of  
> books co-authored by Mary P. Dolciani. This apparent ignorance on  
> Lockhart's part is likely due to the fact that he was educated with  
> Dolciani-type books, and he may not be aware of the preceding  
> textbooks.
>
> The manner in which Lockhart ridicules Thales' Theorem (which he  
> does not name), on page 19 of the PDF file, is utterly  
> unacceptable--and it raises serious questions about the rest of his  
> lament about Euclidean Geometry. When I studied 10th-grade  
> Euclidean Geometry in 1963-64, at Everett High School, in the  
> factory city of Everett, MA, we used the textbook by William G.  
> Shute, William W. Shirk, George F. Porter, "Plane and Solid  
> Geometry," American Book Company (1960). On page 25-27, the  
> textbook contains a historical Note about Thales (640-546 B.C.),  
> Thales' demonstration that all vertical angles are equal  
> (considered to be the first theorem ever proved), deductive  
> reasoning, and the components of a proof of a theorem. According to  
> the Note, when Thales visited Egypt, he observed that whenever the  
> Egyptians drew two intersecting lines, they would measure the  
> vertical angles to make sure that they were equal. Thales concluded  
> that one could prove that all vertical angles are!
>  equal if one accepted some general notions such as:
> 1. all straight angles are equal
> 2. equals added to equals are equal, etc.
>
> At the top of page 10 on the PDF file, Lockhart writes: "So put  
> away your lesson plans and overhead projectors, your full-color  
> textbook abominations, your CD-ROMs and the whole rest of the  
> traveling circus freak show of contemporary education, and simply  
> do mathematics with the students!" Although this advice is quite  
> sound, it is unfortunate that Lockhart conveniently makes  
> absolutely no reference to the fact that all this rubbish has been  
> produced and promoted by the self-styled math reformers of the past  
> two decades.
> **********************************************************************
> ******
> * To post to the list: email mathedcc@... *
> * To unsubscribe, email the message "unsubscribe mathedcc" to  
> majordomo@... *
> * Archives at http://mathforum.org/kb/forum.jspa?forumID=184 *
> **********************************************************************
> ******

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