Re: Disability? Re: [xmca] Help in teaching and learning maths

From: Ng Foo Keong <lefouque who-is-at>
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
Hence the importance of the knowledge of culture, history and philosophy of
mathematics -- and getting one's own hands dirty with some real mathematical

2008/6/10 Carol Macdonald <>:
> 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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