From: Ng Foo Keong <lefouque who-is-at gmail.com>

Date: Tue Jun 10 2008 - 10:42:32 PDT

Date: Tue Jun 10 2008 - 10:42:32 PDT

interesting comment about culture - it is a sad fact that

even in "maths strong" Singapore, maths teachers hardly know

anything about mathematical history, culture and philosophy and

do not bother to read beyond what they are required to learn

(when they were students passing those exams) and teach (when

they enter the teaching profession). Mathematics is viewed here

mainly as "content" and "skills" for "problem solving".

[Make no mistake about this -- our teachers and private tutors

are very good at the task of systematically teaching mathematical

processes, skills and concepts for examinations -- we have our

tried-and-tested training recipies. We're a society of hard-

nosed pragmatics, it seems.] Our syllabus (curriculum) does

mention things like attitudes and metacognition. Our teachers

at most pay lip-service to these, as these are not directly

assessed in examinations. and history? culture? philosophy?

Aw! Come on! Get Real! and by the way, Singaporean parents do

not give a toss about "low IQ" (if that means something) of their

children -- they $end their kids to tuition centre$ or pay private

tutor$. if they get low marks in maths, just practice harder.

Principals, teachers are expected to "produce results" in their

students. Schools are being compared one against another.

It's a pressure cooker culture over here. Oh, and our

publishers don't bother to produce thick and colourful textbooks

with diagrams are just space-fillers -- only those that are

pertinent to the thing being discussed at hand. [and, of course,

they have to make money too.]

i have said that i am a critical realist. the "realist" part is

a vestige of my enculturation in mathematics as practiced (or very

nearly so) by mathematicians, most of whom are platonists who

believe in an objective truth. So I still believe that, at least

in principle, a mentally-sound Cantonese can for example say "yee

ga saam dang yu mm" and get it perfectly translated to a German

"zwei plus drei gleich fünf" and faithfully translated back, and

for that matter, humans of any race, creed or tongue, can agree on

some transcendent truth (that we happen to codify as "2 + 3 = 5")

that is not a mere social construction. So i think there is

universal meaning for mathematics and there is this part about the

nature of mathematics.

However, the "critical" part of me, as a result of exposure to

post-modern indoctrination plus various learning theories, accepts

that mathematics meaning is mediated by tools (conceptual), symbols

and language. We are social beings that depend on dialogic processes

to create our own understanding. Our brain seems wired to be more

amenable to contextual learning. Abstraction comes later, only after

one is grounded and familiar with the processes in the various contexts,

so much so that these processes become "reified" (a la Anna Sfard) into

concepts that can be manipulated like real objects. That sets the stage

for higher and higher abstractions. I accept that it is a fact of human

existence that we rely on language. language point to meaning. Nevertheless

"meaning" cannot be merely "conveyed" by language - it has to reconstructed.

When learning is not grounded enough, the (objective, ontological)

mathematical 'meaning' is lost for the (subjective, epistemological)

learner. Learner feels a disconnect between meaning and symbol.

An example of this sort of would be when I was in my undergraduate days

listening to a renowned Fields medalist (the maths equivalent of a Nobel

laureate) giving a lecture on Group Theory. He was famous for coming up

with a 200+ page proof of a break-through that led to the classification

of all finite simple groups. [this is like the equivalent of completing

the chemist's periodic table for all the elements] He mumbled something

about the importance of "group actions" on "invariant members" while half

the audience was struggling to understand what a "group" meant. He knows

what he's talking about, but we were clueless. Many skipped his lectures.

So for him, he was connected to the meanings (having been heavily involved

in the practice, the culture, and part of history-creating process of

tackling the problem of "classifying simple groups". As undergraduates,

we were not involved and not engaged with the knowledge-making processes,

and approached the subject only via sets of worksheets and excercise. So a

lot of the 'meaning' (objective as well as subjective) is lost, because the

social and epistemological context was not conducive for students to be

connected to it in their (subjective) experiences. Maths to me is like

a mountain (which is there, whether you like it or not), but everybody

has to climb it for himself/herself in order to experience and know it - to

recreate the mountain in his/her mind, as it were; a good mountaineering

mentor is not one who merely waves pictures of the mountain in front of us

and give us nice talks, but someone who shows us how to climb the mountain

by climbing it with us. [this would be the ZPD part]

Perhaps this is the part that is missing is the vast majority of school

mathematics learning.

your M Ed student's case is interesting, as I would have asked her about

her own mathematical learning experiences, that might explain her anxiety.

however, your mention of "the steps of a maths/stats task" seems to suggest

that perhaps we (the non-professional mathematicians) tend to think of

mathematical problem-solving as linear processes. where did we get this idea?

In our learning, we almost always see the model answer perfectly presented by

the teacher in lock-step linear fashion, as if these "math geniuses" had

clairvoyance and bulldozered through problems (if there were indeed any).

In textbooks, we see all the proofs, all the "knowledge", facts, concepts and

procedures impeccably defined well-presented. We do not see the mathematicians

arguing back and forth over coffee, make conjectures, agonize, get refuted, hit

roadblocks, crush and throw their work into wastepaper bins, backtrack, re-work

their ideas over and over again to get to the final product. The novice maths

learner gets the impression that one has to memorize all the steps and present

it right from the first step to the last, and if one slips, one will

be condemned

by the almighty K12 school-teacher and laughed at by friends. The emotional

scars carry right on to undergraduate and post-graduate level. When

these people

become parents, school-teachers and lecturers, they tend to perpetuate this

linearized view of mathematical practice. This is my explanation of mathematics

anxiety from a social point of view.

Nevertheless, i believe mathematics is hard in its intrinsic nature; but some

of the difficulties in mathematics learning is ... er ... socially contructed

and avoidable (if we deconstruct and reconstruct our mathematical

epistemologies).

Hence the importance of the knowledge of culture, history and philosophy of

mathematics -- and getting one's own hands dirty with some real mathematical

challenges.

2008/6/10 Carol Macdonald <carolmacdon@gmail.com>:

*> Hi Shirley
*

*> I agree about being careful about people getting labelled as having a maths
*

*> disability. My understanding of much of what presents as a problem as a
*

*> phobic reaction or a block. We can survive a really bitchy language and
*

*> literature teacher, but there are lots of children who struggle or fail to
*

*> thrive with bitchy impatient maths teachers (who are good at maths and
*

*> intolerant of a lack of understanding). There is much else in the culture
*

*> to keep language and literature development going.
*

*>
*

*> But I would think from our perspective, we would have to regard maths as
*

*> being encoded in a different way, at least partially from a different
*

*> sociocultural history. The concept of number seems to be human-specific (I
*

*> don't mean visual estimation of small numbers). Then past a certain stage,
*

*> maths starts getting abstractions and syntactic embeddings that start to
*

*> look like the things that professional linguists love to use in analysing
*

*> language--and such linguists are very thin on the ground.
*

*>
*

*> I had a student do an M Ed on anxiety in studying statistics using CHAT, but
*

*> that was nearly 20 years ago, and we didn't have the conceptual tools that
*

*> we have now. I do remember very clearly when we did a role play and I asked
*

*> her to tell me the reasons for what she was doing in the steps of a
*

*> maths/stats task, and she got so anxious about it, that we had to leave that
*

*> out of the research.
*

*>
*

*>
*

*> But why does failure in maths carry so much more of a negative weight than
*

*> say, an inability to draw realistically or abstractly, for that matter? It
*

*> must have something to do with the status of maths and its pervasive use in
*

*> high status occupations; but there must be something about the nature of
*

*> maths itself, and as third order symbol system.
*

*>
*

*> What do you think?
*

*>
*

*> PS To get back to the original practical question, I think maths performance
*

*> is best enhanced by a good maths teacher in a one to one situation, building
*

*> up self-*confidence* at the same time as *competence*. Machines and
*

*> textbooks aren't likely to hone in the the zpd in the same way.
*

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Received on Tue Jun 10 10:44 PDT 2008

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