對談講場

數學課外讀物書目
上期黃志華先生表達了一個愛看數學書者的願望。其實教育署輔導視學組數學科的
同工也曾彙集了不少數學課外讀物的書目，包括
Reference Books and Periodicals(1990) 
School Mathematics Club and Its Related Activities(1992)
小學數學參考書目(1992)
小學數學教學參考資料(1991)
透過小學數學課程推行公民教育(1991)

這些均可在北角百福道的數學教育資源中心找到。此外曾健華首席督學為香港課外
活動協會十週年文集《課外活動》（陳德恆編,廣角鏡出版社,1994） 撰寫的
"Organizing Mathematics Clubs in Secondary Schools"一文中亦
附有一份中英文的書目。香港數理教育學會於1986年亦編有《數學學會導師資料冊》。
筆者於《數學傳播》的「立體數學遊戲與空間想像力之訓練」（56期,頁78-96,1990）、
「數學與課外活動」（62期,頁96-109,1992）、「遊戲與數學教育」（66期,頁5
2-68,1993）等文的參考書目中亦列舉了不少有關書籍，有興趣者可作參閱。
	                                                黃毅英

「名數除名數等於不名數」
上期「ABCBCA?」一文見刊後, 續有追問格式問題, 特別涉及單位(名數)。小學
時若遇「$20可買每個$4的蘋果若干?」須寫「20元/4元=5」, 概「名數除名數
等於不名數」﹔而對於「$20可買每米$4的布若干」則寫「20元/4元=5米」。不
明所以之餘實須預先按題意定出「米」的單位。攷4實為4元/米, 前則為4元/個。
其實解答在於達意, 若格式不合規定首要追查背後思路是否紊亂, 拘執表面恐於事
無補。近閱Hall《代數學》一書, 其解答不拘一格, 卻旨在清晰表達思路。如159
節例三「二人相隔27哩同時出發, 若隨同一方向步行9時後相遇, 若反方向則需3小
時, 求二人速度」一問, 先設較快者每時行x哩, 慢者每時y哩。若同方向每小時拉
近x-y哩, 反向則為x+y哩。故有9(x-y)=27及3(x+y)=27。如是文字與算式交替, 
甚為清楚。Durell《中學代數》6章例3解答中先舉

         (x-1)/3.5 時 + 2/3時 + x/(10.5)時 = 4時 , 
再列        (x-1)/3.5 + 2/3 + x/(10.5) = 4  , 
亦可參考。於實際教學運作而言, 校內老師可先統一規定, 明確事前告訴學生, 以免
惹起不必要的爭執和混亂。
                                                                   黃毅英

More about "Directed Numbers"
In response to McClelland's (1994) and Wong's (1996) comments 
on "directed number" (see EduMath, 2, p. 55), I like to supplement 
the following:
1. The use of the term "signed number" is apparently correct since
   there is indeed a sign before the number.  Besides, it also points 
   to the algebraic notion of completing a commutative ring in which 
   we have, for  every  element,  an  additive  inverse  ( called the 
   'negative' of that element).
2. The use of the term "directed number" is even more forward looking, 
   though maybe less obvious, because the real numbers (or a bit more 
   sophisticated, the rational numbers) so formed by adding 'negatives' 
   constitute a one-dimensional vector space, an algebraic structure 
   in which we can talk about directions!  That comes the word 'directed.'  
   Of course, in this one-dimensional vector space, there are two 
   different directions only.
The study of the words invoked in mathematics quite often reveals 
meanings beyond surface interpretations and should not be taken light
-heartedly. It helps learners to categorize concepts, build up links 
between them, fit them into a grand structure at a more macroscopic 
level, and hence open the way to further learning.	
                                                           Fung Chun Ip  
                                       Hong Kong Institute of Education


A Visual Proof for  12 + 22 + ... + n2 =  (n(n + 1)(n + ( ) on the 
Backcover
Editor's Note: In selecting a diagram for the backcover of this issue, 
the editor came across a visual proof for the sum of squares formula 
which may not be familiar to many mathematics teachers. And it was 
later found that this proof was cited as due to our teacher and friend, 
Prof. Siu Man-Keung of the University of Hong Kong. When the editor 
wrote him to ask for permission to use this diagram , a history of it 
was revealed. Here it is.

Subject: Re: A visual proof by you for EduMath3 backcover
Date: Wed, 3 Dec 1996 15:27:52 GMT+8
Dear KM,
     You suggest to use a picture in one of my papers for the backcover 
of EduMath and ask if I agree. Of course I would not mind.  It feels so 
good to see one's work shown.  In fact that picture is not entirely my 
own invention.  It has a rather long story. The first time I drew it was 
when I published a paper with that as a diagram:
  M.K. SIU, Pyramid, pile, and sum of squares, Historia Mathematica, 8 
  (1981), 61-66,
with the idea taken from a paper by XU Chun-fang (documented in my 1981 paper):
  XU Chun-fang, From Chu Tong to Xi Ji and Si Yu Dou series (in Chinese), 
  Shuxue Tongbao, 2 (1965), 45-49.
  Xu surmised that YANG Hui did just that to discover the formula in his 
  book "Chengchu Tongbian Suanbao" of 1274.
     In a later volume of Historia Mathematica (vol.13 (1986)) the 
Russian historian of mathematics A.P. Youschkevitch added a comment on 
my paper, saying that the re-construction appeared in a 1937 Russian 
translation (by S.Ya. Lourie) of the book "Vorlesungen ueber Geschichte 
der antiken mathematischen Wissenschaften, Vol I" by O. Neugebauer (1934), 
and noted that the explanation by Neugebauer is not as simple.
     All these happened over a decade ago.  Then, just last week I 
received a copy of the translated "A History of Chinese Mathematics" by 
J.-C. Martzloff which just came out of press (Springer-Verlag, 1997) with 
the original edition in French published in 1987.  On p.303 I see that 
picture again!  According to Martzloff, this explanation appeared in 
"Shuxue Yao" (Key to Mathematics) written by DU Zhigeng in 1681.  I have 
not yet checked the original edition of the book, so I do not know whether 
the picture is really there in the 17th century book or not.
     In 1984 I offered that picture as a response to the call of the 
editor for "proofs without words" for fill-in in the Mathematics Magazine.  
It was published in vol. 57 (1984), p.92.  It was later collected into the 
book "Proofs Without Words" by R.B. Nelsen (Mathematical Association of 
America, 1993).

MK