> q` XmbjbjqPqP 5p::X+%DWWW8WX( tXX"XXX]]]{s}s}s}s}s}s}s$
uhrws9]]"]]]sXXs```]jXX{s`]{s```XXfXW'^``s0 t`x_dx`x` `]]]ss'`X]]] t]]]](((RW(((W"6DHumanity An Advanced Mathematical Topic
Derek Holton
I see mathematics as having two main pillars. One of these we either ignore completely or treat lightly as something that is slightly embarrassing. Personally I didnt realize that this pillar even existed until I was doing my Masters degree. I hope this is because Im old and things may be different now but I suspect that is not the case.
So what are these two pillars? One is content. That is the one that is universally taught and researched. Its the sum total of the facts, algorithm and theorems that have been established over human history. It contains counting, fast safe ways to add 3-digit numbers, the Cartesian plane, equations of straight lines, ways of finding gradients for functions, the Fundamental Theorem of Algebra, and , and all the theorems that exist in Analysis, Real or Complex. It s what we teach and it has affected the way we teach. Because this material exists we tend to tell it to our students and follow it with lots of routine practice examples so that it sinks in. It turns out that some of us are good at learning and regurgitating this material. In my youth it was enjoyable for me to find that the answer I got was the same as the answer at the back of the book. It was great to get good marks in the exam. And it was very good to have certainty and not flounder around as I did when trying to write an essay in English. In mathematics at school and university there is certainty but there was always mystery. I knew how to take 78 from 123 but Im not sure I knew why, for some time, I had to carry something. But I was good at following rules.
My second mathematical pillar is process and I find it difficult to differentiate that from content. Im also not sure if that is the best word to use to describe it. Process is sort of how you manipulate content when you are puzzled. I think that process doesnt really come in to play until you have a problem. And by a problem, I mean something that you cant immediately see how to do. Of course, the difficulty with this is that one womans problem is another womans algorithm. What I cant see how to do you may consider to be a routine exercise. Proving Fermats Last Theorem is still a problem for me. Put me in a glass cage for 100 years and I probably wouldnt be able to solve it. However, FLT is no longer a problem for Andrew Wiles, nor for a handful or so of other mathematicians. On the other hand, Goldbachs Conjecture is still a problem for us all.
Anyway, mathematical processes are things that come into play when we want to tackle a problem. At that point content is certainly useful but knowing all the mathematical results that exist is not enough to be able to solve every problem that exists. No dictionary has yet written a book.
Maybe using the analogy of pillars to support the mathematical enterprise is not a good one. Perhaps that makes content and process sound independent. Perhaps it would be better to think of content and process as being the weft and warp of the carpet of mathematics. This makes them sound much more interconnected and interactive, which is certainly how I see them. Clearly knowing only processes is not sufficient to solve a problem either.
Before I try to expand on process and its place in mathematical education. Let me look at a specific problem. You may feel that it doesnt involve an advanced mathematical topic but you might like to give it to an arbitrary set of mathematical majors and see how many are able to solve it. I guarantee that it is a problem.
Put 9 checkers in the bottom left 3 ( 3 array of squares of a regular chessboard. Now you can move a checker only by jumping over a neighbour onto a blank square either horizontally, vertically, or diagonally.
OOOOOO
The problem is to determine whether or not the 9 checkers can be moved this way into the 3 ( 3 arrays in the bottom right hand corner of the board or the top right hand corner. Of course, the top left hand corner is equivalent to the bottom right hand corner.
It seems to me that you cant do anything with this problem unless you experiment. Most of us, I suspect, have to get our hands dirty and move the pieces around. Experimenting is actually a very mathematical technique. I dont think you can call it content so it must be process. Its very mathematical because its the way that mathematicians work. Its the way that we begin to get a feel for how particular situations work. Its the way we play around with things, like children do, to gain an understanding of the way they work. But how often do we tell students that basic research involves play, and better still, how often do we encourage them, and give them opportunities, to play?
Experimenting leads to conjectures the way we think that things might work. And thats a fine part of mathematics getting to a conjecture. It still leaves us up in the air because we cant be sure, but its the first major step along the way.
So how are we going with the 9 checkers? Can we do one or both of the relocation assignments? What do you conjecture? What evidence do you base that on?
Getting back to our checkers, what content have we ever come across that is going to help us now? What does our play suggest? Are we just herding cats? Is there any hope of keeping the original square array together at all? Once its been broken up, can we ever put it back together again without going back to its original position? Surely we have to be able to put it together again if we are going to be able to get it to the other corners? If we cant do that, then we cant relocate the way weve been asked to. Has anyone any ideas?
OK. So look at this then. Heres one way of moving the entire 3 ( 3 array by following the rules of the problem.
9955578978978 .7834564 .64 .64 .612312312312 .
93.93.945454578378 .78 . . .6 . .6 . .612 .12 . .21
3.93.93.9454545. .87 .872 .8 .27 . .6 . .6 . .6 .21 . .1 . .1
3.938938945.45.456 .8127 . .127 . .127 . .6 . .6 . . . . . . . . . . . .
Fine. So it looks as if we can do both relocations. Are you convinced? Dont be. All we know is that we can move a 3 ( 3 array to another 3 ( 3 array. Why should we be able to move it to any 3 ( 3 array or, better, to the two specific 3 ( 3 arrays that have been assigned to us? Either then may still be impossible.
Right then, we now have a conjecture. Or do we? What do you think is going to happen? There are actually four conjectures: (1) both work; (2) neither work; (3) bottom right, yes and top right, no; and (4) bottom right, no and top right, yes. Which is it and why?
Its the why thats the problem. Proof is the essence of mathematics. Without proof we only have Physics. Is proof content or process? When weve found the proof we have content perhaps but before that can it only be process?
Supposing the true conjecture is conjecture i and weve been able to prove it. Then, even if we have had to have some help, that must give us a great sense of pleasure. Its fun to prove a result. Its even more fun if no one else has been able to prove it before us. Solving a murder mystery is fun. Being the first person on top of Everest is too. Are we largely taking the fun out of maths for our students because we dont let them loose on genuine problems?
But a mathematicians work is never done. Surely we cant stop there. What techniques have we found? How can they be useful in other problems? Can we extend or generalize what weve done?
Its probably obvious to take our 3 ( 3 array and look at n ( n or even m ( n chessboards. For what m and n can we take our 9 checkers to the other corners? But why stick with 9 checkers. How about 16? Or 25? Or r2? Or even rs in an r ( s array? Whats the real theorem thats governing whats going on here?
What has this got to do with humanity? One of my students is looking at maths and especially how girls react to mathematics classrooms. She found a student who had done very well at school in mathematics but at university this young woman majored in classics and did no mathematics. Why? Because there was no discussion in mathematics. She needed to discuss things. She found learning that way more enjoyable and more interesting. What discussion is there is mathematics? How did that come about? Didnt the tutorial system at Oxbridge encourage discussion in a dons study? Do mathematicians doing research together ever discuss what they are doing? Ive found collaborating with colleagues stimulating even when we havent been able to prove a theorem. For some reason when we show mathematics to the world we seem to have taken away its human face and only laid out the cold, stark bones for students to study.
Just recently I was editing material for a web site that provides professional development for New Zealand teachers. One of the units under consideration involved ratios of the form a : b : c. There was a weeks work for students on the various aspects of this topic. While I was editing I began to wonder why on earth students would want to do this. Where would they find applications? Further, the work required little teaching and was to be done by the students on their own. What would the girls in the class feel about that? Would they be able to work in pairs at least.
It made me think that similar things were true at university level. How much of what we teach is of any use to students? Unless they go on to become academics and teach it, is Zorns Lemma of any use to them? So would we be better to give them at least some problem like the 9 checkers problem in every course. Would it be better if they learnt content skills so that they could find the odd derivative when they needed it in business? Or would they be better off armed with an array of process skills that they might use to solve problems in their working life? And would it be better for the future of mathematics if more people saw the human side of mathematics?
I dont want to kill conte8 /0NOT!U!!!!!!!!!"","."5"6"g"h"q"r"{"|"""""""""""""""""""""""##
######"###/#0#2#3#9#:#<#=#D#E#G#H#T#U#]#^#`#a#g#h~05H*jh~0U jh~0 h~05h~0
h~0CJ$V*+89 fg $$Ifa$$a$$a$2mWmFf@ $$Ifa$
!Ff8
Ff $$Ifa$!"#$%&'()*+,-./0123456789:;<=Ff $$Ifa$=>?@ABCDEFGHIJKLMNPQRSTUVWXYZFf0 $$Ifa$Z[\]^_`abcdfhjklmnopqrstuvwxFf(Ff $$Ifa$xyz{|~TUN$a$Ff $$Ifa$NO!!!!!!!!!!!!!!!!!!!!!!!!Ff% $$Ifa$$a$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Ffd) $$Ifa$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Ff- $$Ifa$!!!!!!!!!"""""
""
""""""""""""Ff$2 $$Ifa$" "!"""#"$"%"'"(")"*","."0"1"2"3"5"7"9":";"<"=">"?"A"Ff: $$Ifa$$a$Ff6A"C"D"E"F"G"I"K"L"M"N"O"Q"S"T"U"V"W"X"Z"\"^"_"`"a"c"e"i"j"Ff`? $$Ifa$j"k"l"n"p"s"t"u"v"w"x"y"}"""""""""""""""""FfC $$Ifa$""""""""""""""""""""""""""""FfLFf#@#B#C#F#I#K#L#M#N#O#P#Q#R#S#Ff*^FfY $$Ifa$S#V#X#Z#[#\#_#b#d#e#f#i#l#n#o#p#q#r#s#t#u#v#w#x#y#{#}##Ffb $$Ifa$g#h#j#k#{#|#############################$$$$$ $
$$$$$$$$$$$$$$$$$$$$"%#%''((0*1*H*J*K*L*M*W*Y*Z*[*\*u*v*|*}*****+
++ h~0H* h~06 h~05 jh~0h~0h~05H*X#############################Fff $$Ifa$#############################FfJk $$Ifa$######################$$$$
$$Ff
tFfo $$Ifa$$$$$$$$$$ $!$"$#$$$%$&$'$($)$s%t%&&s't'I)J)
*$a$Ffjx $$Ifa$
**X+Y+..B1C133.m/m0m1m2m4m5m7m8m:m;m=m>mGmHmImTmUmh]h&`#$$a$++
+++$+(+//////////4l2m3m5m6m8m9m;mm?mEmFmGmImJmPmQmRmSmTmXmh~u0JmHnHu
h~00Jjh~00JUjh~0UU h~05 h~06 jh~0h~0'nt you cant do any maths without it. I do think that we need to encourage process though and show the human face of mathematics more often. I suspect that this is not easy. It may well be an advanced mathematical topic that is accessible from elementary grades through to tertiary mathematics. .
PAGE
PAGE 2
UmVmWmXm3
0&P 1F. A!"#$%>$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5///
4a:kd$$IflּMj 5Sp;Yw
tPPPP4
la>$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5///
4a:kd|$$IflּMj 5Sp;Yw
tPPPP4
la>$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5///
4a:kd$$IflּMj 5Sp;Yw
tPPPP4
la>$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5///
4a:kdt
$$IflּMj 5Sp;Yw
tPPPP4
la>$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5///
4a:kd$$IflּMj 5Sp;Yw
tPPPP4
la>$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5///
4a:kdl$$IflּMj 5Sp;Yw
tPPPP4
la>$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5///
4a:kd$$IflּMj 5Sp;Yw
tPPPP4
la>$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5///
4a:kdd$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kd#$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kd@($$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kd,$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kd1$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kd`5$$IflּMj 5Sp;Yw
tPPPP4
la0$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
/4a:kd9$$IflּMj 5Sp;Yw
tPPPP4
la0$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
/4a:kd.>$$IflּMj 5Sp;Yw
tPPPP4
la0$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
/4a:kdB$$IflּMj 5Sp;Yw
tPPPP4
la0$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
/4a:kd
G$$IflּMj 5Sp;Yw
tPPPP4
la0$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
/4a:kdxK$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kdO$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kdFT$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kdX$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kd]$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kdfa$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kde$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kd&j$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kdn$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kdr$$IflּMj 5Sp;Yw
tPPPP4
la"$$If!vh555555555 5
5
55
5
555555#v#v
#v
#v:Vl
t55
5
5/
4a:kdFw$$IflּMj 5Sp;Yw
tPPPP4
laD@DvxZNormalCJOJQJ_HmH sH tH DA@DDefault Paragraph FontRi@RTable Normal4
l4a(k(No List.)@.vxZPage Number4@4vxZHeader
9r ~+p~+*+89
$%+u u u u u u u u u u u u u u *+89
$%-.
!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstvwxyz{|}~z{tu67
!"#%')*,.0135789;=ABCDEFGHIJKMNOPRTVWXY[]_`abcdegijklmoqrstuwyz{|}~
!"#%')*+,-./24678;=?ABEGJLNOPQRSTWZ\]^adfhiloqrstuvwxy|~
!"#$%&'*-01258;<=@CFGHIJKLMNOop0 1 ~!!%%h'i'**T+U+V+W+X+Z+[+]+^+`+a+c+d+m+n+o+z+{+|++00000000000000000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00000000000000000000000000000000000000000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00000000000000000000000000000000000000000000000000000000000000000000000000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000000000000000000000000I0I0I0I0I0I0I0I0I0I0I0@0@0I0I0 $$$$$'g#+Xm/5!=ZxN!!!"A"j""""(#S####$
*UmXm !"#$%&'()*+,-.012347Wm '!!8
@X( nicahsotDH
h 9
3"Vn
b
#" WH2
#b/H2
#$bH2
#bH2
#
H2
#/H2
#,H2
# 9H2
#9:H2
#79B
S ?t~+
3 tW+X+X+Z+Z+[+[+]+^+`+a+c+d+l+o+y++FH X _
.4pyOTO U !
!!!!!'!""%%&'a'g'{((X+X+Z+Z+[+[+]+^+`+a+c+d+|++333333333333333333333333333W+X+X+Z+Z+[+[+]+^+`+a+c+d+o+y++X+X+Z+Z+[+[+]+^+`+a+c+d+|++~0~u
!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstvwxyz{|}~7
!"#%')*,.0135789;=ABCDEFGHIJKMNOPRTVWXY[]_`abcdegijklmoqrstuwyz{|}~
!"#%')*+,-./24678;=?ABEGJLNOPQRSTWZ\]^adfhiloqrstuvwxy|~
!"#$%&'*-01258;<=@CFGHIJKLMNW+X+Z+]+`+c+|++AW+@x
4&&&*~+@@@UnknownGz Times New Roman5Symbol3&z Arial3Times 1hFFx$Nx$N!4B+B+2P`|2)Humanity An Advanced Mathematical TopicuserJohann EngelbrechtOh+'0\6 $0
P\
ht|,Humanity An Advanced Mathematical TopicuserNormal.dotJohann Engelbrecht2Microsoft Office Word@F#@"bX@"bXx$G4Rs PrW."SystemW~9-j@Times- 2
* bfHumanity lKtCK*)M&
2
bf~J:2
bf An Advanced Mathematical Topic &lK%lKKCKCCK&B*JCtC**CB*%\KK*C
2
8bf ~C
2
bf ~D2
9bfDerek HoltonlC2CK&lK**KK
2
9Lbf ~C@Times-
2
bf ~-2
*MbfI see mathematics as having two main pillars. One of these we either ignore 6(,-6N,2,N,,'7,'63,2226H26N,262,!'66H2,63!62,(,6H-6,2,!6122",2
Vbfcompletely or treat lightly as something that is slightly embarrassing. Personally I ,2N2,,0'2!'",,'120','''2N,221'2,''''120',N2-!!,''31'(8,!'22,0(!A2
$bfdidnt realize that this pillar even222!!,,-,2,2'2,!,2,2\2
6bf existed until I was doing my Masters degree. I hope ,3',222I,'2221P0Y,'-!'2-2!,, 223-2
Vbfthis is because Im old and things may be different now but I suspect that is not the 2'!'!2,,,2(," !O!22!-22!221'!O.0!2-!2!",!-2!22H!22" "'2'2,,!2,!'!22!2-2
bfcase.s,,',
2
bf ~.
2
hbf ~-@Times--F2
'bfSo what are these two pillars? One is 82&H2,&,!,&2,',&H2&1,!&.%&H2,&'&-2
Z bfcontent ,28!,8!-F2
'bf. That is the one that is universally %&=2,&'%2,&22,&2,%'%222,!',1/2
Nbftaught and researched. ,212',22(!,'-,",2,2'(g2
N=bfIts the sum total of the facts, algorithm and theorems that !''3,''2N'2,'2!'2-'!,,'',12!2N',22'3,2!,N''2,2
Xbfhave been established over human history. It contains counting, fast safe ways to add 2,2,3,,3,',2'2,323,!22N,32'2#0 ,23,2',22221",'',",H.0'2,22
2
4 bf3~2
2
4 bf-~!2
4 AQbfdigit numbers, the Cartesian plane, equations of straight lines, ways of finding o21022N2,!'02,0C-!,',202,2,0,23,22'02!0'!,1202,'0H.0(02!0!2222/2
bfgradients for functions,1",2,2'!!2!!!22,22'D2
&bf the Fundamental Theorem of Algebra, !2,!7222,N,2,!=2,3!-N!2!!H1,2!,!!
@Times-..2
bf and , and all the ,*!!,22!!.!,22!,!2+.-2
Qbftheorems that exist in Analysis, Real or Complex. Its what we teach and it has o2,2!,N',2,,,3',2+H2,0'',C,,,2!,C2N3,3,,!',H3,,H,,,-,3,,22,,2+&2
Wbfaffected the way we teach. Because this material exists we tend to tell it to our ,!!-,,2$2,$H.0$I,$,-,2$$B,-,2',$2'$N,,!,$!,3''!$H,$,22$2$!+!$$1$22!m2
Abfstudents and follow it with lots of routine practice examples so s'22,2',22!22HH22'2!!222,2!-,,,,3,N2,''252
bfthat it sinks in. It turns 2,'22'22!2(2
sZbfout that some of us are good at learning and regurgitating this material. In my youth it 222,'2N,2!2',",2222,,,!231,22!-13"1,312'N,,!, 3O002222
Ubfwas enjoyable for me to find that the answer I got was the same as the answer at the rH,',240,2, !2!N,2 !222,2,,2'H,!! 13H,'2,',N,,'2, ,3'H,!,2,[2
X5bfback of the book. It was great to get good marks in 2,,22!2,2222 I,(1",,21,1222N,!2'2;2
XE bfthe exam. And it was very good 2,,3,NH22H,'2,#012222
Tbfto have certainty and not flounder around as I did when trying to write an essay in 2%2,2,%,,!,2/&,22%22%!2222,!%,!2222%,'&%22%H2,2%"031%2%H!,%,2%,''-0%22
>
SbfEnglish. In mathematics at school and university there is certainty but there was =21'2,, 2+O,2,N,-'+,+',222+,23+222,!'0+3,!-+'+-,!,20+22+2,!-+H,(a2
9bfalways mystery. I knew how to take 78 from 123 but Ih,H.0'P0',#0 22,I22H2,2-23"!2N32222 !42
!bfm not sure I knew why, for gN22'2!- 23-HH40!3!z2
$Jbfsome time, I had to carry something. But I was good at following rules.'2N,N,3,22,-,!"0-'2O,221B2 I,'1222,!22H21!2,'
2
$bf ~.
2
bf ~--=2
!bfMy second mathematical pillar is rZ0%',,222%O,2,N,,-%2,!%'%-2
xbfprocess 8,2,,''%-P2
.bfand I find it difficult to differentiate that ,22&&!22%%2!!,2%2%2!!,!,2,,%2,n2
}Bbffrom content. Im also not sure if that is the best word to use t!!2N,22,2 !O,'232'2!,"2,'2-2,'H2!222'-42
}bfo describe it. Process is g22,(,!2,8!2,,'('2
Qbfsort of how you manipulate content when you are puzzled. I think that process s'2!*2!*22H+022*O,222,,*,22,2*H2,2,022+-!,*22--,2*+*222*2,*!2!2,,'(!2
bSbfdoesnt really come in to play until you have a problem. And by a problem, I mean 22,'2! !-,0!,2N, 2 3 2-0 22"022 3,2, - 2"22,N H22 40 , 3!22-N! O,,2m2
Abfsomething that you cant immediately see how to do. Of course, tt'2N,221"2,$022#,,3!"NN,2,,0"(,,"23H"3"22""H!#,22!',"(2
bfhe difficulty with t2,"2"!,20"H32
HQbfthis is that one womans problem is another womans algorithm. What I cant see n2'&'&2,&22,&H2N,2!'&2!22,N&'&,222,!&H3N,2!'&,12!2N&&_2,'',,2!&(,-2
Tbfhow to do you may consider to be a routine exercise. Proving Fermats Last Theorem 22H222022N.0,22'2,!22-,!223,,3,!,',8!22228,!N,!'<-'=2,2!,Ow2
.Hbfis still a problem for me. Put me in a glass cage for 100 years and I'*'*,*2!22,N*!2!*M,**82)N,*2*,*1,''*,,2,*!2!**222*+0,-!'*,22+2
.
bf probably *2!22,212
Obfwouldnt be able to solve it. However, FLT is no longer a problem for Andrew H2222!)2,*,2,)3)'22,)))H3H,2-!*8<=*)')22)231,"),)3!22,N*"2!)H22",H}2
LbfWiles, nor for a handful or so of other mathematicians. On the other hand, _,';22!;!2!;,;2,22!2;2!;'2;2!;22,!;N,2,N,,,2';;H2;2,;22,!;2,22Y2
4bfGoldbachs Conjecture is still a problem for us all.H222,,2!'C22,,2",(',2!22,N!2!2',
2
bf ~-
2
bf ~-C2
m%bfAnyway, mathematical processes are th H30I.0*O,2,N,,,+2!2,,'',(*,",*2I2
me )bfings that come into play when we want to t31'*3-*,2N,*23*2.0*H3,2+H,*I,2*22
Qbftackle a problem. At that point content is certainly useful but knowing all the ,,2,3,32!22,N33H32,32223,22,23'3,,!,2032',!23223222H213,32,2
RUbfmathematical results that exist is not enough to be able to solve every problem that N,2,N,,,$!,'2'$2,$,3'#'$22$,22212$2$2,$,2,$2$'22,$,2,#0$2"22,N$2,P2
.bfexists. No dictionary has yet written a book.,3''H22,22,"02,(0,I!,2,2222
2
bf ~.
2
8bf ~-
2
bfM~Y2
GRbfaybe using the analogy of pillars to support the mathematical enterprise is not a -03,(2'31(2,),2,320)3!(2,!'(2('2222!(2,(N,2,N,-,(,2-!2"',('(22(-2
Qbfgood one. Perhaps that makes content and process sound independent. Perhaps it 1222'22,''8-!2,2''2,'N,2,'',22,2',22'3!3,,''''2222'22,2,23,2''8,!2,2''2
Rbfwould be better to think of content and process as being the weft and warp of the H222'2,'3,,!'2'223'2!',22,2',23'2!2-,''(,''2,31'2-'H-!',23'I,!2'2"'2,2
bfcarpet of m,,!3,42!4Nk2
@bfathematics. This makes them sound much more interconnected and ,2,N,,'54=2'4N,2,'42,N4'22224N2,24N2!,42,!,232,,,24,232
wWbfinteractive, which is certainly how I see them. Clearly knowing only processes is not 2,!,,2,H2,2',,!,2023H (,,3,NC,,!0222H322202!2-,'',''22C2
%bfsufficient to solve a problem either. '2!!,,22'22,,2!22,N,2,!
2
{bf ~.-ffbbeeaadd``cc__bb^^aa]]``\\__[[^^ZZ ] ]Y Y
\
\X
X
[[WWZZVV
Y
YU
U
XXTTWWSSVVRRUUQQTTPP՜.+,0hp|
UPNB+*Humanity An Advanced Mathematical TopicTitle
!"#$%&'()*+,-./012345678:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvxyz{|}~Root Entry F?-fXData
9{1TablewxWordDocument5pSummaryInformation(6DocumentSummaryInformation8CompObjq
FMicrosoft Office Word Document
MSWordDocWord.Document.89q