Note Assistants: Support for Solving Math Problems In this text I describe a method that supports think-and-write processes processes for the work on math problems.
Contents What are the basic ideas?.............................. ideas?..................................................... .............................................. .............................................. ..........................................1 ...................1 How can I make notes?................................ notes?....................................................... .............................................. .............................................. ...........................................2 ....................2 Note assistants: n example................................. example........................................................ .............................................. .........................................................! ..................................! How can I use the note assistant?............................ assistant?................................................... .............................................. ......................................................" ..............................." How can I produce and adapt m# own note assistants?...................................... assistants?............................................................ ...................... ...........$ ........... $ %riticism and responses............................ responses................................................... .............................................. .............................................. ..............................................& .......................& ppendix 1: Note assistants as a framework for problem sol'in(.......................................................) sol'in(.......................................................) ppendix 2: *ther elements in note assistants................................. assistants........................................................ ..................................................1+ ...........................1+ cknowled(ments.................... cknowled(ments........................................... .............................................. .............................................. .......................................................... ................................... ..11 ,ocument chan(es........................... chan(es.................................................. .............................................. .............................................. .....................................................1 ..............................11 1 bout the author................................... author.......................................................... .............................................. .............................................. ................................................ ......................... 12
What are the basic ideas? he method is perhaps best introduced b# the term paper software/ - a software/ that does not run on a computer0 but on sheets of paper0 as the most flexible hardware/ a'ailable in man# circumstances. Ima(ine on m# left a sin(le sheet of paper called the note assistant/. his acts as a kind of menu/ and contains ad'ice for crucial problem sol'in( situations how to start0 how to (enerate new ideas0 what to do when I3m stuck0 etc. *n m# ri(ht is an ! sheet where I make the actual notes on m# problem the editor/. s in real software0 I can choose a la#out suitable for m# notes. Whene'er I feel I could do with some problem sol'in( support0 I ha'e a look at the menu. his menu offers thin(s I can insert/ into the note sheet how to arran(e the notes for a special kind of in'esti(ation0 what ke#words to write down0 and abo'e all0 what thinkin( tools to use0 what 4uestions to ask or what ideas to tr#.
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he dia(ram indicates how su((estions from the note assistant for 'arious problem sol'in( situations can be taken o'er into the note sheet and that there is certainl# no need to use the note assistant at all if work is (oin( on well. Whether the items in the note assistant are actuall# helpful to me as the problem sol'er depends on m# le'el of expertise0 m# note-makin( preferences0 m# field of work and other factors - so note assistants should ideall# be adapted to me and e'ol'e to(ether with m# (rowin( experience. 5ater0 I will present ideas on how to customi6e note assistants.
How can I make notes? s mentioned0 I can choose between se'eral methods of note-makin( on the note sheet. Here comes m# fa'orite0 followed b# a number of possible 'ariations. I use a blank sheet of paper in landscape format0 si6e ! 7or lar(er8. I separate the sheet b# 'ertical lines into four e4ual columns. I or(ani6e the text in boxes/ that are separated b# hori6ontal lines. hese boxes are labeled 10 19 in column 10 20 29 in column 2 etc. in the upper ri(ht corner. corner. 7t this position0 the labels need less column space0 and it3s easier to add them later.8 In each box0 I can or(ani6e the text in hierarchies b# indentations. or a ma;or new idea0 I can start a new column. I can note sudden ideas at the bottom of column !0 in boxes !<0 != etc. o mark open issues0 I can add check-boxes like / at the ri(ht column border. It3s eas# to find them later0 examine the issue and tick off the check-box. If I want to continue the work from one box in a new one0 I can indicate this b# arrows between nei(hborin( boxes or b# references like see 2%/ or from :1,/ for box 1, on pa(e . I can add footnotes at the bottom of a column. I use a mechanical pencil and an eraser. eraser. he method works best if I write fairl# small. Ha'in( a non-smear pen is essential. In m# e#es0 this method of note-makin( has a number of ad'anta(es: s with other forms of note-makin(0 m# memor# is unburdened0 and I find i t easier to mana(e complex chains of thou(ht and trees of thou(ht. ># thou(hts are permanentl# documented. he method works well with usual math operations0 like manipulatin( e4uations. 9# switchin( between columns0 I can cope with chan(es between different lines of thou(ht0 at least to a certain de(ree. he same could be done b# usin( separate sheets0 but for me this is often a massi'e disruption of the flow of work. I can store awa# sudden ideas with ease and examine them later. rom m# experience0 writin( in narrowish columns encoura(es me to write neat notes0 and this neatness transfers to some extent to the entire work on the math problem. here are man# wa#s to alter the method: I can write on lar(er sheets like 0 or on ! double pa(es in a notepad. I can tr# a different number of columns especiall# if the columns seem too narrow. narrow. If labelin( the boxes with 10 29 seems too much trouble0 I can lea'e it out and address the boxes b# coordinates: I ima(ine the columns separated 'erticall# in e4ual parts a0 b0 c0 d and use references like b/. 7he printed nc#clop@dia 9ritannica had a similar s#stem.8 • • •
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If a lar(er dia(ram is needed0 I can use a la#out like this:
Notes on a problem will often stretch o'er more than one sheet. I ha'e found it easier to continue m# thinkin( when I ha'e pre'ious notes within si(ht0 so for me sin(le sheets work better than a bound notebook0 where I ha'e to switch between pa(es to read and to write. he problem of dealin( with a lar(e number of sheets is not within the scope of this text. ># ideas on this would probabl# ha'e to do with slip boxes or
akin( Btupid >istakes/ b# Cichard Cusc6#k0 founder of the rt of Droblem Bol'in(/ website 7http:EEwww.artofproblemsol'in(.comE8. 7http:EEwww.artofproblemsol'in(.comE8. he next pa(e shows a non-math example of a result of the t he note-makin( method. he sheet contains some aspects that ha'e not been mentioned in the text. Cemark: he tables on the followin( f ollowin( two pa(es should (i'e an impression of the actual la#outs. Bince the first table is basicall# an ! table on an F space and the second an table on an ! space0 the text is 'er# dense0 especiall# if 'iewed on a small displa#. displa#. Giewin( Giewin( the document on a lar(er screen or printin( it out will help. 7hanks to ,r. Houston for pointin( out this problem.8
What is this sheet about? |1A - it shows a way of note-making, a way of “thinking “thinking on paper” _____________________________________ What do you need? |1B - bank paper in A! - a non-smear pen - e"g" me#hani#a pen#i $ eraser _____________________________________ What%s the basi# ayout? |1& - use paper in ands#ape format - draw ines to form ! #oumns
ore things you #an try |2A - page numbers - date - footnotes at the #oumn bottom 13 - numbering - underine, #oour - tabes - diagrams - e.uations """ _____________________________________ +ow to dea dea with sudden sudden ideas |2B - you #an mark ideas for foow-up with a #he#k bo ( 'r( - work out a more detaied - try A) in ands#ape format #he#k-bo system with * #oumns - you #an ater ti#k off - ots of spa#e for your your ideas these boes( _____________________________________ - add some remarks on +ow to make tet boes |1 #ross-referen#ing - number the boes 4see 2&3 - in #oumn 1 with 1A, 1B et#" - you #an store unreated ideas - write the headine $ underine it at the bottom of #oumn ! - .uestions make - ook ook at the eampe/ good headines/ _____________________________________ - write down your thoughts +ow to #ross-referen#e |2& - short but inteigibe - there are eampes in 2B and )B - “outine” your tet - referen#ing bo !& on page 2( - indent your ines - see 2(!& - to show hierar#hies - ike this - when you%re finished with a bo( - draw a hori0onta hori0onta ine - start a new bo
When to start new #oumns |)A and new boes? - for important new ideas( - start start a new new #oumn - start start a new new bo - when resuming work from one bo in in a new one( - draw arrows '5 - use #ross-referen#es with “see )A” and “from 2&” et#" _____________________________________ 6ome ad7i#e on |)B “deep” thinking - work patienty - from one bo to another another - from one page page to another """ - 8 aways pursuing - open .uestions, - things you don%t understand - things you #an make better - ask basi# .uestions 13 What woud be ogi#a? 23 What%s What%s bothering me here? What%s the key probem here? )3 What #an 9 do now? now? - start a “:” se#tion - “:” stands for :uestions :uestions - use it at the end of a bo, of a #oumn to find open topi#s : - how to design a more refined system of thinking toos? - pros ; #ons of this method? 4see !A3
_____________________________________ 13 footnote for ater remarks and other things
- #ompare these notes with - mind maps - &orne =otes - digita note-making - other note-making systems 4they a ha7e their pros pros ; #ons/3 - ook at the foowing points( - #an you fo#us on your work? - no distra#tions from apps et#"? - #an you de7eop de7eop ong ong #oherent #oherent ines of thought? - #an #an you store away sudden ideas and eamine them ater? - #an you swit#h swit#h to to other other ines of thought without mu#h do#ument fidding? - do you ha7e ha7e an o7er7iew of your notes? - is is straightforward, organi0ed thinking en#ouraged? - are are your notes sti #omprehensibe after ) days, 2 months, 1 de#ade? - are tabes, diagrams, e.uations en#ouraged? : - digita 7ersion of this method? - tabe for abo7e #omparison? _____________________________________ image sear#h |!> on “note-making” _____________________________________ ate( 11"1"21@
Note assistants: An example he main task of the table on the next pa(e is to pro'ide concrete su((estions for ma;or problem sol'in( situations0 with re(ard both to cle'er la#out and to smart thinkin( tools. hese buildin( blocks should (i'e structure to the o'erall work on the problem. he problem sol'in( situations are hi(hli(hted in oran(e0 su((estions for ke#words that could be written down are in #ellow. he sheet is intended for an format. It contains 2 x ! columns0 numbered from 1 to &. he sheet is inspired b# a lot of authors0 especiall# >ason 7hinkin( >athematicall#80 Dol#a 7How to Bol'e It80 athematical Droblem Bol'in(80 ao 7Bol'in( >athematical Droblems8 and 9ruder 7Droblemlsen lernen im >athematikunterricht8. or alomitsines3 How to Bol'e Droblems: New >ethods and Ideas/ see ppendix 2.
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+ow to start?
6ome standard things to do +ow to try further approa#hes 9%m #onfused/ - 9%m stu#k/
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_____________________________________ +ere%s something #onfusing/ |!A +ere%s a diffi#uty C an obsta#e/ _____________________________________ &onfusion '5 'bsta#e |!B what things are #onfusing? why are they #onfusing? des#ribe the situation des#ribe the diffi#uties what is the #ore obsta#e? repeat that .uestion/ _____________________________________ What #an you do? |!& how #an the #onfusion be #eared up? is it possibe to make the obsta#e disappear? disappear? make a ist of options in7estigate the most promising ones 4see )A for ayout3 _____________________________________
9 think 9 ha7e su##eeded/
'ther Dsefu #on#epts
_____________________________________ 6ummary |@A write down the #ompete soution #he#k e7ery step _____________________________________ 5efe#tion |@B #an the resut be generai0ed C impro7ed? #an the method be generai0ed C impro7ed? #an 9 find a different soution? _______________________ what #an 9 earn from my work? what were the key diffi#uties? is there some probem so7ing beha7iour 9 shoud #hange? how #oud 9 impro7e the note assistant?
9%m frustrated C demoti7ated/ 9 ha7e a sudden idea/ &heer up/ |*A use supporti7e sef tak 4“Ehis probem ooks hard, but( - 9 #an pro#eed in sma steps, - 9 #an eamine one idea after the other” """3 remember su##esses from the past 4de#ide what to write down3 _______________ work on for Iust 1@ minutes ha7e a break and resume work ater
_____________________________________ Aha/ '5 9dea |JA write the idea into bo !> '5 start a new #oumn mark sudden ideas with a “ ” for ater #he#k-ups _____________________________________
ethods of
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=ote Assistant _____________________________________
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How can I use the note assistant? problem is (i'en0 and the I want to work on it. I ha'e m# note sheet as an editor/ in front of me0 prepared with ! empt# columns and perhaps a pa(e number and the date. I can access the note assistant as a menu/ with a sin(le (lance. Juided b# the list of problem situations hi(hli(hted in oran(e0 I find ad'ice on How to start/ in column 10 with some reasonable initial operations. I can fill the first boxes in m# note sheet with text0 math terms0 e4uations and dia(rams0 usin( the la#out0 the ke#words and the thinkin( tools su((ested b# the note assistant. fter this start0 columns 2 and pro'ide ideas on how to tr# se'eral approaches. %olumn 2 describes a number of standards approaches. approaches. It seems natural to check these first and turn to less direct approaches later. later. %olumn su((ests a two-step method. - irst step: I ma# look for inspiration in se'eral lists 7here: & - &,8 in the note assistant. I can write down a collection of approaches that seem worth closer in'esti(ation. - Becond step: I can examine these approaches in a suitable order and work out the details. If a first round of these steps did not lead to a solution0 I can repeat the process. Ksin( a reference from a box with (eneral content to boxes with more specific content 7as in box referencin( boxes & &,8 is a helpful desi(n0 especiall# when it comes to much more specific problem sol'in( tools for sin(le branches of mathematics. When difficulties arise0 column ! has a number of su((estions. 9ox ! has an interestin( function in the note assistant it illustrates the interpla# between existin( notes that lead to a certain situation0 and new elements in boxes !9 and !%. It seems important to me to ask for f or confusion and its causes this focuses on m# immediate experiences as a problem sol'er. 9ox !, su((ests to postpone an approach if further pro(ress seems not likel#. %olumn F su((ests some thin(s to do when the main work is done. >an# authors emphasi6e the ke# importance of this kind of reflection. here is a direct 4uestion on how to impro'e the note assistant. he contents of column " on dealin( with emotions are ar(uabl# experimental. I ;ust wanted to show in principle how this aspect could be included. he ideas on self-talk are inspired b# Cichard Nelson-Lones3 book ffecti'e hinkin( Bkills/. he item on sudden ideas in $ is perhaps of lesser 'alue0 althou(h a ke#word like haM/ . he ll Durpose ools/ ools/ in $9-$, contain a smallish number of 4uestions and su((estions that should lead to some pro(ress in almost an# situation. %olumn & forms a brid(e between (eneral processes and more specific math contents. he collection (i'en here is a bit arbitrar# and ser'es as an illustration. 9ox &, presents items to (enerate more ideas: In the spirit of creati'it# tools from other fields0 one mi(ht tr# to manipulate the problem elements b# a number of basic operators and see if this leads to interestin( insi(hts. he list in &, is lar(el# inspired b# the well-known B%>DC creati'it# tool. It seems reasonable to tr# this method after more direct approaches ha'e failed. 7Cemark: much more ambitious pro;ect would be to transfer concepts from the CI< approach to creati'it# in en(ineerin( to math.8 • •
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How can I produce and adapt my own note assistants? his 4uestion has two aspects: Which tools can I use to produce and adapt the note assistants ph#sicall#? How can I find the content I put into them? • •
word before. It seems a (ood idea to form note assistants from sin(le modules/. module module could consist of a headline that describes which problem situation or which math ob;ects the module is intended for0 followed b# a collection of note-makin( and thinkin( tool su((estions. xamples are (i'en in the note assistant abo'e basicall#0 basicall#0 the modules correspond to the columns in the note assistant. Ksin( modules offers a number of benefits: With modules it3s eas# to add la#outs and thinkin( tools for new problem situations and new math ob;ects. It3s possible to adapt and impro'e onl# one module without ha'in( to replace the entire collection. It ma# be stimulatin( to exchan(e modules within a communit# of students and teachers usin( the note assistant framework. •
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Here are some ideas on the tools I can use to produce and adapt m# own note assistants. 5ar(e paper sheets written b# hand. his is ar(uabl# the most strai(htforward wa#. wa#. In m# experience howe'er0 these sheets soon become rather mess#. Drintouts. In this essa# I prepared the note assistants with office software. he results are neat0 but the workflow of makin( some chan(es0 then printin( out the note assistant is cumbersome. o o a'oid this0 I can collect new ideas for the note assistants on a separate sheet and incorporate them from time to time into the note assistant. Dlastic sheets for filin( business cards. •
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hose sheets ha'e & or 1+ pockets for papers of business card si6e0 so b# usin( front and back side the# can store up to 2+ modules. hese sheets can form an entire librar# of note assistants for 'arious purposes and 'arious domains of math. Bin(le sheets or flash cards for the modules. In this case howe'er0 howe'er0 the one-sheet-one-(lance-mantra one-sheet-one-(lance-mantra has to be abandoned. ,i(ital and mobile de'ices. Here a(ain0 the one-sheet-one-(lance mantra has to be (i'en up.
Here are some ideas on the contents I can put into the note assistants. I can exploit the existin( literature on math problem sol'in(. I can ask people for ad'ice. I can adapt the note assistants accordin( to m# experiences with problems I3'e worked on. • • •
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In m# experience0 note assistants do not necessaril# become more helpful b# containin( more items. I found it more useful to focus on those situations where I am reall# stuck.
Criticism and responses here are se'eral points of criticism0 and I would like to address some of them. he four column la#out is a wildl# o'er-specified strait;acket./ In m# opinion it3s better to ha'e a clear concept of how helpful math notes could look like0 and then de'iate from that concept for (ood reasons0 than to ha'e no concept at all. s mentioned0 there are man# la#out 'ariations possible. he entire process is too formal and too complex./ I certainl# would not want to see the process followed in a dull routine. It is meant to pro'ide support - if the problem sol'er wants it. If the process is presented to a (roup of students0 it seems reasonable reasonable to ad'ance in suitable moderate steps0 presentin( elements of the process one b# one. 7hanks to Drofessor >ason for addressin( this point.8 bstract heuristic ad'ice like look at in'ariants/ is worthless to lots of students./ =es. I think that choosin( the ri(ht set of thinkin( tools to help an indi'idual student is a ma;or issue. r(uabl#0 r(uabl#0 these tools ha'e to be introduced0 illustrated b# examples and then be made a'ailable in the note assistant. What about knowled(e? What about experience?/ I think that there is an immense literature on math and on math problem sol'in( that will help readers to build up knowled(e and experience. 9ut there seems to be comparati'el# little information on the aspect of note-makin(0 so I concentrated on this. Is there an# e'idence that this actuall# works?/ pre'ious 'ersion of a problem sol'in( method based on mi nd maps was well recei'ed in seminars I3'e (i'en in the past participants seemed comfortable with the combination of a note-makin( method and thinkin( tools. rom m# personal experience0 I ha'e no doubt that the note-makin( method presented in this paper is much better suited to math problems. It is m# hope that readers ma# find some ideas useful and make them a part of their own problem sol'in( practices. If #ou ha'e su((estions on how to impro'e the method in (eneral or sin(le aspects of it0 I would be thrilled to hear from #ou. Dlease don3t hesitate to contact me under thomasteepeO(oo(lemail.com thomasteepeO(oo(lemail.com •
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Appendix 1: Note assistants as a framework for problem solin! rom a more abstract point of 'iew0 and without reference to math specifics0 we ha'e operate with two elements: irst: note-makin( note-makin( method we want to use for our work on a problem. his method can be - the four column la#out described abo'e0 - or a two column la#out with a main column and a reflection column0 - or mind mappin(0 - or ordinar# linear notes0 - or notes in a di(ital notebook0 - or notes with a math editor like 5#P0 - or somethin( else. Becond: set of thinkin( tools we could use for sol'in( problems. note assistant should pro'ide combinations of note-makin( elements and thinkin( tools combinations that can be applied directl# to the problem. In m# experience0 this is of special importance with re(ard to the handlin( of confusion and obstacles0 reflections and of dealin( with multiple approaches. With this (eneral perspecti'e in mind0 it is possible to construct 'arious note assistants b# combinin( thinkin( tools from texts on 7math8 problem sol'in( and note-makin( methods. In this spirit0 lan 2++"- 6ollmanEDDEN%>2++"-our-%orners.ppt our-%orners.ppt Be'eral books on math problem sol'in( hi(hli(ht the importance of note-makin(0 but pro'ide 'er# little details on how to do this in practice. In the past0 a combination of mind mappin( and problem sol'in( tools seemed 'er# promisin( to me. oda# I think that the four column la#out is much better much better suited for dealin( with math problems. 7he older ideas on mind mappin( can be found at https:EEwww.scribd.comEdoc https:EEwww.scribd.comEdocE$)2)")$E>athem E$)2)")$E>athematical-Droblem-Bol'in(-and->ind atical-Droblem-Bol'in(-and->ind->appin( ->appin( 8 *b'iousl#0 *b'iousl#0 the note assistant framework can be adapted to a number of fields besides math. •
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Appendix ": #ther elements in note assistants Here is a list of other elements a note assistant could contain. Items to prepare for problem sol'in( It ma# pro'e 'er# useful to ha'e an initial box in the note assistant 7and subse4uentl# in the note sheet8 to set the ri(ht attitude and the ri(ht mood towards problem sol'in(. Here are some possible items: - recallin( pre'ious successes0 - brin(in( to mind the importance of reflection etc. Ima(es Ima(es ma# ha'e more impact than words portraits of role models0 ima(es of (oals0 ima(es e'okin( clarit# or strai(htforwardness etc. *r: n ima(e of a rubber duck rom the wikipedia article on rubber duck debu((in(/: Cubber duck debu((in( is an informal term used in software en(ineerin( for a method of debu((in( code. he name is a reference to a stor# in the book The Pragmatic Programmer Programmer in in which a pro(rammer would carr# around a rubber duck and debu( his code b# forcin( himself to explain it0 line-b#-line0 to the duck. R...S >an# pro(rammers ha'e had the experience of explainin( a pro(rammin( problem to someone else0 possibl# e'en to someone who knows nothin( about pro(rammin(0 and then hittin( upon the solution in the process of explainin( the problem. In describin( what the code is supposed to do and obser'in( what it actuall# does0 an# incon(ruit# between these two becomes apparent. 9# usin( an inanimate ob;ect0 the pro(rammer can tr# to accomplish this without ha'in( to in'ol'e another person./ he method seems fit to be applied to problems outside debu((in(. 7Bome readers ma# find this and the pre'ious item o'er the top. he indi'idual problem sol'er ma# decide what works for her and what doesn3t.8 special mention must be (i'en to the book How to Bol'e Droblems: New >ethods and Ideas/ b# B. alomitsines0 published in 2++& and in central parts based on articles from the 1)&+s. ,r. rakali brou(ht m# attention to this book after I had written se'eral 'ersions of this paper0 see the document histor#. alomitsines proposes se'eral problem sol'in( methods with a stron( emphasis on combinin( writin( acti'ities and thinkin( tools0 in particular: a8 he description method: When first approaches to sol'e a problem ha'e failed0 take each sin(le part of the problem and write down a list of e'er#thin( #ou know about this part0 usin( short sentences. hen hen look at the list and see if the items ma# help to sol'e the entire problem. b8 he method of (ettin( out of loops: Write down a table: in the left column a list of all attempts to sol'e a problem #ou ha'e tried unsuccessfull# unsuccessfull# so far0 and in the ri(ht column a list of ne(ations to these attempts. hen tr# to construct new approaches to the problem from these ne(ations. c8 he spiral method for sol'in( problems: his is an al(orithm-like combination of the description method0 a means-end-anal#sis means-end-anal#sis and the method of (ettin( out of loops. In the means-end-anal#sis0 #ou look at the difference between what is (i'en and the (oal. he method is called spiral since #ou ma# (o throu(h se'eral loops of the process0 each time knowin( more about the problem. ,etails on these methods and man# examples can be found in alomitsines3 book0 which I stron(l# recommend. •
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Acknowled!ments >an# people ha'e contributed to the ideas presented in this paper. I would like to thank ,r. bhi;ith rakali0 rakali0 Werner Werner 9e(oihn0 ,r. strid strid 9rinkmann0 Hans-LTr(en Hans-LTr(en lschenbroich0 ,r. e'in Houston0 ,r. Lr( onopka0 ,r. rmin ramer0 Drof. ,r. imo 5euders0 Drof. ,r. Lohn >ason0 Hubert >assin0 Drof. ,r. >anfred Dren6el0 ,r. rauke Cademann0 Drof. ,r. Harold Bhapiro0 >artina eepe and %hristian Wolf. $ocument chan!es his document is a'ailable for free on https:EEwww.scribd.comEdoc https:EEwww.scribd.comEdocE2F1"&F"1!ENote-s E2F1"&F"1!ENote-ssistants-Bupport-for sistants-Bupport-for-Bol'in(->ath-Droblems -Bol'in(->ath-Droblems I upload re'ised 'ersions from time to time0 so here comes a list of chan(es made to the document.
%ersion
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+$.+1.2+1F dded: able of %ontents0 document chan(es table0 about the author. Note assistants and modules. Cemark on introducin( the process step b # step. >inor chan(es.
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+&.+1.2+1F %han(ed: Introduction with the paper software/ metaphor. metaphor. >inor chan(es. dded: *ther elements in note assistants.
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1!.+1.2+1F %han(ed: 9ox name in upper ri(ht corner 7instead of left8. Numerous impro'ements to the note assistant example. xpanded description of how to use the note assistant. Bome ideas on criticism.
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1$.+1.2+1F %han(ed: Ideas on producin( and adaptin( note assistants.
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21.+1.2+1F dded: Ideas from B. alomitsines3 How to Bol'e Droblems/
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About the author I was born in 1)$1 in IbbenbTren 7Jerman#8. 7Jerman#8. I finished m# doctoral thesis on (enetic al(orithms at the Kni'ersit# of >Tnster in 2++10 and since lea'in( uni'ersit# I3m workin( as an actuarial consultant. In the past #ears0 I ha'e spent a lot of time thinkin( about methods of math problem sol'in( and of problem sol'in( rele'ant to m# ;ob and0 occasionall#0 occasionall#0 to m# life. homas eepe lot6straUe 1 $+1)+ Btutt(art thomasteepeO(oo(lemail.com
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