Theory of Knowledge - Course Companion for Students Marija Uzunova Dang Arvin Singh Uzunov Dang
mathmatis was too important to gothator rmain inomprhnsil tomisundrstoodHr dotoral thsis, ntitld “Thmost.of Latti Shaps of Rings ofEquidistriutionof Cui, Quarti, and Quinti NumrIntgrsAn Artist’s Rndring” is prsntd atFilds:lvls. Th first lvl is a “laysap” withdiffrntusing languag appropriat and“laysplanations”,to th layprson or studnt ofundrstandalNxt, Harron prsnts thmathmatis.and “mathsplanations”, for th“mathsap”mathmatiian or prs in thamaturmathmatis ommunity. Sh invitsprofssionalto follow th argumnts up to thirradrsthrshold, and you an try to findmathmatialy following th link.yourstrms: Equidistan ofSarhshaps Pipr thsislattiis on of th aras of knowldgMathmatiswhih studnts ommonly hold limitinginaout thir aility. Tahrs anlifsths lifs or rat and rinforhallngBlow ar two artils rlvant to thisthm.of knowldg aquisition in mathmatis.issu1, y profssor of mathmatisSourJo Boalr, xplors how disovrisduationnurosin should rshap thfromand larning of mathmatis. Boalrtahingmathmatis lss as a mattr ofapproahsanswrs and provn mthods, and morrightan xris that ahivs undrstandingasollaoration and rativity. Follow ththrough1: Boalr, J. 2019. “Evryon Can LarnSourceto High Lvls: th Evidn fromMathmatisthat Should Chang our Tahing”NurosinMathmatial Soity Blogs)(Amriantrms: AMS BoalrSarhan larnEvryonprinipl, hr full argumnt should possilIngrasp at ithr lvl, and th thsis is intnddto “approximatly radal y approximatlyto(Harron 2016).anyon”an intrviw pulishd in The Hindu (DsikanInHarron idntifis th ida of th gnius2016)as on of th harmful mythsmathmatiianin th mathmatis ommunity, andprptuatdfor it to disardd. “Thr’s th ida thatallsis fixd, you gt it or you don’t; that thr’smathright way to think aout things. … W ndadiffrnt popl in math. W nd diffrntmorof thinking in math” (Harron quotd inways2016).Dsikan11.3 xplors two onptions of, andBoxpossiilitis rgarding, th aquisition oftwo2, y Barara Oakly, prsnts aSourut rlatd vision of mathmatis, ondiffrntmovs away from undrstanding, whthrthatat rativly or through rot larning.arrivdinstad xplors th ida of fluny inOaklyin lin with th onption ofmathmatis,as a languag. Oakly livsmathmatismathmatis is a hait of mind, a pattrnthatthinking that is dvlopd through sustaindofIn Oakly’s viw, mathmatialprati.is losr to know-how than toknowldgFollow th link to rad th dtails.know-that.2: Oakly, B. 2016. “How I Rwird mySourceto Bom Flunt in Math” (Nautilus,Braintrms: Oakly How ISarhmy rain Nautilusrwirdw think mathmatis is affts how wWhatand larn it. It affts what ass andtahto mathmatis larning looks lik, andarrirsit mans to fail and sud in aquiringwhatIII. Methods and toolsIII. Methods and toolsmathmatial knowldg.Box 11.3: The “maths brain” and mathematical fluencylink to rad th artil in full.15 Sptmr 2016)mathmatis to high lvlsknowldg in this AOK.343
III. Methods and toolsmathmatis is ommonly undrstoodKnowingth aility to rason through mathmatialasand symols. Is it possil to knowformulain othr ways, not as languag ormathmatisut as ativity or prforman—rprsntationas know-how? Exampls frommathmatisshow that th mind is notthnomathmatisonly hannl into th ralm of mathmatis,ourthat it an “known” through th wholutas an modid prati, and vn throughodyarrangmnt of othr ojts in th physialthIf w an do and find mathmatisworld.of th human mind, how far an w gooutsidfor mathmatis?lookingslugs do maths, electrons do maths, mineralsSeamaths. Rainbows do an incredible mathematicaldoEuler’s 18th-century formula, with its uniqueLeonhardproperties, is enacted here by a bird.mathematicalth link to rad th full artil: “How toFollowMathmatis”.Playtrms: Aon Thr’sSarhmaths in slugsmorcan we say that sea slugs and corals knowButgeometry? I want to argue here that inhyperbolicsense they do. … The world is full of mundane,someunconscious things materially embodyingmeek,complex pieces of mathematics. How canendishlymake sense of this? I’d like to propose that …wenatural systems are engaged in what wemodestcall the performance of mathematics. Rathermightthinking about maths, they are doing it. In thethanof their beings and the ongoing continuity ofbresgrowth and existence they enact mathematicaltheirand become mathematicians-by-relationships… Rather than being a remote abstraction,practice.can be conceived of as something moremathematicsmusic or dancing; an activity that takes place notlikemuch in the writing down as in the playing out.sodo ths raturs hallng what it mansHow“know” mathmatis? Wrthim liknstoto musi; th intrstion of th twomathmatishas a omprhnsiv disours indisiplinsaross many diffrnt ultural traditions.itslf,ragas of Indian lassial musi, for xampl,Thpassd down aurally from mastrtraditionallystudnt, ar rognizd as inhrntlytoTh Sanskrit word prastara mansmathmatial.study of mathmatially arranging ragas andthWhat is th diffrn twn trating1.as a languag vrsusmathmatis(a) Do you hav to know how to writ2.to do mathmatis?mathmatis(b) To what xtnt is this th as in thTo what xtnt is playing musi without3.al to rad musi omparal toingmathmatis without ing al todoingmathmatis?rad11III.4 In what ways can mathematicsbe known?(Wertheim 2017)… Next time you see a good rainbow,performanceand take a look … classical geometric opticsstopbegin to capture its complexity. A stunningdoesn’tof mathematical performance is enacted by apiecefalcon as it hurtles towards its prey; with itsperegrineheld straight so it can x one eye steadily on theheadat a constant angle of 40 degrees, it swoopsquarryat 200 mph in a perfect logarithmic spiral.downrhythms into plasing ompositions.(Wertheim 2017)For discussionLanguage, reading and writingtrating it as an ativity?othr AOKs?Figure 11.7 Sa slug344
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mathmatis was too important to go
that
or rmain inomprhnsil to
misundrstood
Hr dotoral thsis, ntitld “Th
most.
of Latti Shaps of Rings of
Equidistriution
of Cui, Quarti, and Quinti Numr
Intgrs
An Artist’s Rndring” is prsntd at
Filds:
lvls. Th first lvl is a “laysap” with
diffrnt
using languag appropriat and
“laysplanations”,
to th layprson or studnt of
undrstandal
Nxt, Harron prsnts th
mathmatis.
and “mathsplanations”, for th
“mathsap”
mathmatiian or prs in th
amatur
mathmatis ommunity. Sh invits
profssional
to follow th argumnts up to thir
radrs
thrshold, and you an try to find
mathmatial
y following th link.
yours
trms: Equidistan of
Sarh
shaps Pipr thsis
latti
is on of th aras of knowldg
Mathmatis
whih studnts ommonly hold limiting
in
aout thir aility. Tahrs an
lifs
ths lifs or rat and rinfor
hallng
Blow ar two artils rlvant to this
thm.
of knowldg aquisition in mathmatis.
issu
1, y profssor of mathmatis
Sour
Jo Boalr, xplors how disovris
duation
nurosin should rshap th
from
and larning of mathmatis. Boalr
tahing
mathmatis lss as a mattr of
approahs
answrs and provn mthods, and mor
right
an xris that ahivs undrstanding
as
ollaoration and rativity. Follow th
through
1: Boalr, J. 2019. “Evryon Can Larn
Source
to High Lvls: th Evidn from
Mathmatis
that Should Chang our Tahing”
Nurosin
Mathmatial Soity Blogs)
(Amrian
trms: AMS Boalr
Sarh
an larn
Evryon
prinipl, hr full argumnt should possil
In
grasp at ithr lvl, and th thsis is intndd
to
“approximatly radal y approximatly
to
(Harron 2016).
anyon”
an intrviw pulishd in The Hindu (Dsikan
In
Harron idntifis th ida of th gnius
2016)
as on of th harmful myths
mathmatiian
in th mathmatis ommunity, and
prptuatd
for it to disardd. “Thr’s th ida that
alls
is fixd, you gt it or you don’t; that thr’s
math
right way to think aout things. … W nd
a
diffrnt popl in math. W nd diffrnt
mor
of thinking in math” (Harron quotd in
ways
2016).
Dsikan
11.3 xplors two onptions of, and
Box
possiilitis rgarding, th aquisition of
two
2, y Barara Oakly, prsnts a
Sour
ut rlatd vision of mathmatis, on
diffrnt
movs away from undrstanding, whthr
that
at rativly or through rot larning.
arrivd
instad xplors th ida of fluny in
Oakly
in lin with th onption of
mathmatis,
as a languag. Oakly livs
mathmatis
mathmatis is a hait of mind, a pattrn
that
thinking that is dvlopd through sustaind
of
In Oakly’s viw, mathmatial
prati.
is losr to know-how than to
knowldg
Follow th link to rad th dtails.
know-that.
2: Oakly, B. 2016. “How I Rwird my
Source
to Bom Flunt in Math” (Nautilus,
Brain
trms: Oakly How I
Sarh
my rain Nautilus
rwird
w think mathmatis is affts how w
What
and larn it. It affts what ass and
tah
to mathmatis larning looks lik, and
arrirs
it mans to fail and sud in aquiring
what
III. Methods and tools
III. Methods and tools
mathmatial knowldg.
Box 11.3: The “maths brain” and mathematical fluency
link to rad th artil in full.
15 Sptmr 2016)
mathmatis to high lvls
knowldg in this AOK.
343