[LogoForum] Re: logo and learning mathematics

The message below is being cross-posted from LogoForum.

Apparently I can't prevent Yahoo's space-saving optimizer from
removing spaces which were rather important to forming the
rectangles I attempted to portray in the text of the original, but
somehow reappeared in the text of this reply.
If anybody knows how to patch up this butchery in the orginal,
please let me know so that I can re-post the content to have it
appear as I intended.

Thanks.
Gene

--- In LogoForum@xxxxxxxxxxxxxxx, "logo_programmer"
<gene_sullivan@...> wrote:
>
> --- In LogoForum@xxxxxxxxxxxxxxx, papert@ wrote:
>>
>>> --- In LogoForum@xxxxxxxxxxxxxxx, "asumanw" <asumanw@> wrote:
>>>>
>>>> i am looking forward to hear views on how
>>>> logo helps in learning mathematics
>
>>> Okay.
>>> I suppose the `views' you receive may have
>>> to do with your present views.
>>
>> ---> I don't understand the reference to
>> "present" ..... as opposed to what?
>
> The state of one's view(s) either before or
> after a post. I don't know about you but my
> views are skewed before, after, and during
> posts in which I attempt to take another
> person's views into account. But I didn't
> `know where you were coming from' by posing
> the question. I didn't know if you were
> posing the question as a `teacher', a
> cognitive scientist, an epistemologist, a
> mathematician, a `concerned parent', an
> autodidact, etc. Each brings their own set
> of biases and `views' to the discussion. I
> was attempting to discover your biases and
> interests.
>
>>> How `fixed' are your distinction between
>>> `actual' mathematics versus mathematical
>>> notations/conventions?
>
>>> Logo, for example, tends to pander to
>>> infix notation of the sort (mis)used in
>>> "2 + 2 = 4" rather than often `better'
>>> deep-structural equivalents of the pre-fix
>>> and post-fix form
>
>> ---> "Pander" is the right word .... I
>> often think that it was a bad mistake to
>> encourage infix in Logo. For me the reason
>> was making Logo appear less "strange" to
>> the "math ed" community.
>
> Yes. There seems a self-latching silliness
> encoded in the 3-part canard, "Those who
> cando, do; those who can't do, teach; those
> who can't teach, administer." If one learns
> by mimicking those who `do' they are likely
> to replicate the `surface structural'
> notations and conventions of previous
> generations of mathematical doers. So both
> learners of mathematics and learners of the
> craft of `educating' vis-a-vis math are up
> against the often foolish mimicry of
> `standard' -- if not Procrustean --
> forms.
>
>> But it might have been (might still be)
>> better to educate them than to pander to
>> them.
>
> Pavel, the author or Elica
> <http://www.elica.net>, has provided ways to
> view Elica Logo programs as lisp and English
> as alternate forms of `surface structure'.
> Though I have yet to `see' any implementation
> which allows `functional equivalents'
> composed in either pre-fix `lispoid' notation
> or post-fix forth-like notation to be used
> interchangably.
>
>> BUT, surely this is a secondary issue since
>> once a learner has gone beyond the most
>> elementary level it should be possible to
>> understand infix as a convenient "surface
>> structure"
>
> Learner qua `learner' ... any hypothetical learner?
> `Possible' vis-a-vis any hypothetical learner?
> In previous posts
> http://groups.yahoo.com/group/LogoForum/message/11517
> http://groups.yahoo.com/group/LogoForum/message/12206
> I've speculated as to how `superstitious
> behavior' may result in a `learner' fixating
> or `imprinting' on an adopted-as-`orthodox'
> one-and-only-one `right' way of doing
> `addition', math, or whatever it comes to be
> beheld as. I've also speculated on how
> `functional fixedness' and the `Sapir-Whorf
> hypothesis' applies to so-called
> `conventions', standard forms, and
> `mathematical' notations which `function as'
> so-called `natural' languages.
> As for the `should' in `should be possible',
> there `should' be more than 3 types of
> religious practicioner in the form of
> shoulists, couldists, and is-ists. Though
> I'm not so sure I've ever noticed otherwise.
>
>>> So ... by way of `learning' mathematics,
>>> are you more concerned with fascilitating
>>> `actual' mathematics ... or superficial
>>> surface structural `notations' or
>>> `conventions'?
>>
>> ----> Most definitely I am concerned with
>> facilitating mathematical thinking.
>
> Do you have criteria or apperceivable themes
> which help you to spot it when you `see' it?
> What differentiates `mathematical thinking' from
> `logical' or `rational' thinking? Or `problem solving'?
> Or `philosophical' or `scientific' thinking?
>
>>> And which of `the maths', as the Brits,
>>> Aussies, and Kiwis' whould say?
>
>>> Some of `the maths' are more easy than
>>> others to `do' via/in Logo. If you want
>>> symbolic math you might be better off
>>> dabling in Lisp or Scheme, for example.
>>>
>>> And is this for `personal' use ... or are
>>> you looking to `teach' or `tutor' others?
>>
>> ---> Can we say "learning": rather than teaching?
>
> Sure. I've gone down on record in this forum
> as focusing on `learning' rather than
> `teaching':
> <http://groups.yahoo.com/group/LogoForum/message/12373>,
> Related point #4: Outcomes-based (re)framing
>
>> The point of Logo for me was to create
>> better learning paths into mathematics.
>
> `Paths' as a metaphor? I'd hate to be
> `trapped by a metaphor' to the extent where a
> would-be `path' came to function more as a
> `rut' in which I became stuck. I've heard
> some so-called `religious' folks claim that
> "All roads lead to God." I'm not sure that
> `all' roads lead to mathematical
> enlightenment but there are-qua-ARE both
> `additional' and `complementary' roads/paths
> vis-a-vis those meandering through Logo land.
>
>> I do use it "personally"... but this should
>> not be the dominant design criterion ....
>
> Okay. I can run with this. I've
> participated in discussions of design
> criteria and still have some interest in
> various levels of design. I'll start a
> SulliVenn Diagram with the `personal use'
> criterion represented first then add from
> there.
>
> Logo (re)design Criteria, version 1
>
> Legend:
> 1 - intRApersonal usability
>
> +------------------------------+
> | 1 |
> | |
> | |
> | |
> | |
> | |
> +------------------------------+
>
> note: if circles weren't so hard to form
> with ASCII codes I might have tried a Venn
> Diagram, but I trust you will grasp the
> rectangular equivalent of a circular area.
>
> It seems that at this stage we have a design
> which behaves as an idioglossia in that a
> person-scoped design which fascilitates
> `personal usage' while constituting `secret
> speech or language'. {ref:
>
http://www.tiscali.co.uk/reference/dictionaries/difficultwords/data/d0
> 006810.html}
> So, pursuant to manifesting a `social' or
> intERpersonal design, it might be permissible,
> if not desirable, to modify our design
> specification in process as follows?
>
> Legend for
> Logo (re)design Criteria, version 2:
>
> 1 - intRApersonal usability
> 2 - intERpersonal usability
> 3 - 1 AND 2
>
> +------------------------------+
> | 1 |
> | +---------------------------|---+
> | | | |
> | | | |
> | | 1+3 | |
> | | | |
> +--+---------------------------+ |
> | |
> | 2 |
> +-------------------------------+
>
> This results in a Venn-equivalent of 2
> overlapping circles depicting 3 explicate
> areas/regions as well as the implied 4th
> representing `neither'.
> I'm sure all posting to the LogoForum
> would agree that Logo as we know it --
> regardless of which quasi-unique
> implementation -- meets the criteria of being
> usable for both `personal' and
> social/interpersonal `things', projects, or
> activities.
> What happens if we try to factor in the
> `helps in learning mathematics' desiratum?
>
> Legend for
> Logo (re)design Criteria, version 3:
>
> 1 - intRApersonal usability
> 2 - intERpersonal usability
> 3 - 1 AND 2
> 4 - helps in learning mathematics
> 6 - 2 AND 4
> 7 - 1 AND 2 AND 4 ... aka (AND 1 2 4)
>
> +-------------------------------+
> | 4 |
> | |
> +-------+----------------------+ |
> | 1 | 1+2+4 | |
> | +----+----------------------|---+ |
> | | | 1+2+4 |2+4| |
> | | +----------------------+---+----+
> | | 1+3 | |
> | | | |
> +--+---------------------------+ |
> | |
> | 2 |
> +-------------------------------+
>
> Our desirata are increasing. But our notion
> of what constitutes or is-subsumed-by
> `mathematics' is not very refined. Nor is
> our notion of what features of our `Logo of
> discourse' might `help in learning'. So I'm
> fairly certain that most readers of this post
> can imagine what might happen if we add
> detail in both `the maths' and `types of
> learners'. Perhaps if I just flesh out the
> Legend at this juncture to save my decidedly
> lazy self the effort entailed in overlaying
> more and more rectangles?
>
> Legend for
> Logo (re)design Criteria, version 4+:
>
> 1 - intRApersonal usability
> 2 - intERpersonal usability
> 3 - 1 AND 2
> 4 - helps in learning mathematics
> 6 - 2 AND 4
> 7 - 1 AND 2 AND 4 ... aka (AND 1 2 4)
> 8 - Arithmatic (EG +, -, *, /)
> 16 - Algebra
> 32 - preschool children/learners
> 64 - gradeschool children/learners
> 128 - autodicacts
> 256 - ?
> 512 - ?
> 1K - ?
> 2K - ?
> etc.
>
>> at some stage of intellectual development
>> one can afford to be eclectic and I use all
>> sorts of other stuff as well.
>
> I'm afraid that some `ones' are too
> distracted and preoccupied with `assignments'
> part and parcel with being-qua-being regarded
> a `child' who functionaries/bureaucrats
> posing as `teachers' are tasked with leaving
> no such object of manipulation behind. And
> by the time these `ones' can -- are able to
> -- afford, in terms of precious time, to
> be-qua-be eclectic/catholic enough to engage
> in whatever activities are required to
> manifest intellectual development there may
> be precious little incentive or interest to
> do so. Though it seems that folks like you
> and I have spent what we could afford to
> compose and disperse our Logo-related
> aboutisms.
>
>> (Maybe instead of "can afford to be" I
>> should have said "ought to be")
>
> `ought' in a more-perfect world, perhaps.
>
>> BTW: Re: " which of 'the maths' " I like to
>> contrast "math" or "maths" with
>> "mathematics" ---- the short words refer to
>> the incoherent and certainly obsolete
>> grab-bag presented these days in schools.
>
> I'll agree to this distinction between the
> grab-bag phenomenon and whatever `real'
> mathematics are qua `are'. And `presented'
> is a rather agreeable word as well in the
> sense that a presenter presents `stimuli'
> which educe -- as in `educate' -- responses
> by/from/within a `learner' a_la a behaviorist
> model ... which does not `beg the question'
> that `mathematics' is being-qua-being
> `taught'. However the use of the term
> `mathematics' is still not sufficently
> disambiguated for many purposes.
>
>> A more serious case of pandering than using
>> infix is going along with the general
>> structure of the school math curriculum.
>
> `Curricula' -- plural -- I'd say.
> To use the singular form might promote the
> illusion that there is some grand-unified
> monolithic homogenious metaphorical `straw
> man', wouldn't it?
> And to use the term `structure' for something
> as variable from country-to-country,
> county-to-county, or village-to-village seems
> a stretch as well. How about `curriculum' in
> the sense of `curriculum vitae' as in `course
> of life' rather than the semantic synonymous
> with `Procrustean Bed' as used to `make
> uniform' the
> students-come-objects_of_manipulation
> institutionalized in so-called `institutions
> of higher learning'?
> {ref: http://en.wikipedia.org/wiki/Procrustes}
>
>> As an education activist my dilemma is how
>> far one has to do this in order to have any
>> influence at all.
>
> Please accept my condolences, sir.
> It seems that any activist who attempts to
> deal with `education' vis-a-vis the
> permutations of many so-called `educators'
> with many
> once_learners-come-institution_fodder_known_as_`students'
> has a tougher time of it than a `learning'
> activist who limits the scope of his or her
> interest at the individual-come-autodidact
> level.
> {ref:
> http://en.wikipedia.org/wiki/Autodidact}
>
> It seems that "Indian mathematical genius
> Srinivasa Ramanujan and Newton's contemporary
> Gottfried Wilhelm Leibniz were largely
> self-taught in mathematics." as per the
> article. So it appears that two such
> non-formally `educated' nobodies managed to
> have used -- something other than Logo,
> presumedly -- "what helps in learning
> mathematics". Which leads me to wonder
> ... if anybody has done `for'/`to' `math ed'
> what Bandler
> <http://en.wikipedia.org/wiki/Richard_Bandler>
> and Grinder
> <http://en.wikipedia.org/wiki/John_Grinder>
> did for/to `psychotherapy' via their studies
> of effective practicing therapists which were
> made public via their NLP (EG Neurolinguistic
> Programming) books. {biblio: The Structure
> of Magic Volumes I & II (1975, 1976)}
> Frankly, if I were interested in learning how
> to learn mathematics it would make more sense
> to study how great mathematicians have
> `done'/learned/used mathematics rather than
> how would-be `math educators' have `taught'
> math. So, if your `learner'-centered
> interest in the learning of mathematical
> precepts, concepts, methods, and such still
> holds then I'd recommend beholding Logo as a
> staged learning-facilitating `tool'/toy
> sandwiched between a child's `The Gears of my
> Youth' phase and Scheme, Jacal
> <http://swissnet.ai.mit.edu/~jaffer/JACAL.html>,
> proprietary/commercial computer-implemented
> math applications and such.
>
> Sincerely,
> Gene Sullivan
>

__._,_.___
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