language and math
roberto
who pursued research (practically and/or theorically) about the
possibility of learning mathematics as a human-like language;
an interesting old post in the python edu-sig list (i can't remember
it exactely, sorry) gives some light to the problem as far as regards
computer science;
but, which are the most affordable studies about math ?
thank you
--
roberto
Edward Cherlin
philosophy of language, and so on: The Sapir-Whorf hypothesis, that
language constrains the range of thought. This is hotly denied in much
of academia, but taken as a given by mathematicians who are aware of
the history of mathematical notations. Also,
Marvin Minsky (born 1927)
* You don't understand anything until you learn it more than one way.
The only serious attempt to make a significant part of math speakable
is Lojban, documented at http://www.lojban.org. There are said to be
three and a half fluent speakers of Lojban, some of whom are trying to
raise native speakers. There is a fairly extensive English-Lojban
translation program, including, for example la alis. cizra je cinri
zukte vi le selmacygu’e (Alice's Adventures in Wonderland).
Lojban has built-in grammar for elementary logic and set theory. It
has no built-in nouns or verbs, only relations with one to five
places. Noun phrases and verb phrases are easily constructed,
including proper noun phrases (la lojban.), mass nouns (lo ciftoldi),
and abstractions (le nu zutse). Verbs have an elaborate structure of
tenses and modes available. Any number of pronouns can be used.
Number, gender, tense, and other functions that are obligatory in many
natural languages are optional in Lojban, and there are optional
functions not common in natural languages.
Given the difficulties, I know of no serious studies of the issues you
ask about.
On Fri, Oct 22, 2010 at 09:35, roberto <robe...@gmail.com> wrote:
> hello, i'd like to ask to the list the following point:
> who pursued research (practically and/or theorically) about the
> possibility of learning mathematics as a human-like language;
>
> an interesting old post in the python edu-sig list (i can't remember
> it exactly, sorry) gives some light to the problem as far as regards
> computer science;
>
> but, which are the most affordable studies about math ?
>
> thank you
> --
> roberto
>
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>
>
--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://www.earthtreasury.org/
milo gardner
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Maria Droujkova
For my part, I take about ten times more time to multiply, say, two-digit numbers in English, compared to Russian...
Cheers,
Maria Droujkova
Make math your own, to make your own math.
Edward Cherlin
first language and algebra or calculus in their second language. Jill
Bolte Taylor reported losing all arithmetic for several years after
her stroke. She could dial phone numbers purely by shape, however.
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kirby urner
mathematics is nothing if not a human language. Who else
would it be for? Dolphins? ETs? Maybe, but not originally.
We might need to pause and ask what "mathematics" could
mean.
Keith Devlin reminds us that ants, bats, all manner of creatures,
are mathematical computations on the hoof as it were, with
Sir Roger (Penrose) butting in to explain that every so often,
thinking makes a logical leap through a non-computable space
to arrive at solutions that only have logical explanations post hoc,
if then (sometimes we give up trying to explain --
"explanations come to an end" as Wittgenstein so wisely put
it (or "justifications" depending on the translation)).
My mentor, R.B. Fuller, sought to change the mix in his
philosophical writings, going to the opposite extreme from
Bourbaki and including very little notation specific to the
mathematical literature, staying mostly in a kind of prose
not unlike Finnegans Wake by some reckonings. This was
a consciously taken decision and related to his project to
bridge the famous C.P. Snow chasm.
Indeed, his dedication to a prose-based approach to mathematical
topics caught him some cosmic fish, humanities types willing to
take the bait. Chief among them was Hugh Kenner, a famous
James Joyce scholar and author of 'The Pound Era', a book about
poets. He also wrote 'Geodesic Math and How to Use It' and was
a columnist for Byte Magazine (McGraw-Hill). Now *there's* a
polymath for ya!
If you crack the cover of Synergetics, you will just see a lot of
nonsense that tickles your fancy, combined with a wealth of pictures.
Is this mathematics? Some readers report having an easier time
with mathematical languages after steeping themselves in "Bucky tea"
whereas others experience no positive benefits and suspect it's
all snake oil.
Speaking of snake oil, what I've done is taken some of the geometry
one might distill from this "humanities drink" (like a tea), and coded
it up in Python, using the language of "math objects".
But then I'm quick to point out that "object oriented programming"
gained its impetus NOT by referring to some abstruse kind of
"object" known only to computer programmers, but to "things
in general" such as anyone might recognize: cats, elephants,
big wheels, hurricanes, camel caravans, Perl.
Some would argue that "processes" are not objects but this is
a grammatical objection, a reminder that we have different linguistic
forms depending on what type of object is under discussion. Python,
however, has a very simplified grammar wherein everything is treated
as an object. It's the ultimate exercise in "objectification" which
engineers are frequently castigated for doing, as if it were a crime.
http://worldgame.blogspot.com/2009/02/regarding-objectifying.html
Objects have attributes (properties) and behaviors, just like
people do (as demonstrated by sims in their sims doll houses).
Indeed, modeling objects with computer language objects is
what huge numbers of people do for a living (including me, in
some time slices).
So then the question is whether a computer language is "human-like".
At one point it was considered to be so, because machine language
was quasi-unreadable to most, but then Grace Hopper came along
with her compiler idea and realized dreams since Leibniz, as
channeled through Ada Byron in association with Charles Babbage.
APL (computer language) is human-like because it's like math, and
math is by and for humans. Iverson called APL an executable
math notation (MN).
When you look at it this way, there's no line between mathematics
notations and computer notations (in music either) other than the
fact that we need to administrate universities and that means
compartmentalization (entirely arbitrary in many cases, like
national boundaries).
FORTRAN, in its day, was touted as "human readable" just as
XML and SQL are so touted. But then people become spoiled
and say "that's not what I meant" and move the goal posts, once
again.
Nowadays we have a new language with its own idioms, called
"video" and/or "television". Mathematics is morphing thanks to
the new media being used to communicate mathematics.
Hypertext is also making a big difference, as the notion of
nodes and edges (hyperlinks) is now "in your face" like never
before (yes, I know brains are node-edge data structures too)
.... but I wander.
Kirby
Melanie
By 'human-like language', do you mean English, French, Chinese, etc.?
By 'learning', do you mean the natural immersion process of acquiring language, for example as children do it (as opposed to taking French 101)?
Are you interested in seeing how far you can get in communicating about what's for dinner by using mathematical chicken scratches (the things a monolingual Chinese speaker and a monolingual English speaker can put on a blackboard and both understand)?
Do you want to find parallels between how babies begin to point to the moon and say 'moon', and how children begin to make patterns and abstractions and so on, and then share their ideas with others?
-Melanie
Melanie Stein
Web Developer: html, css, wordpress
Accent Coach: helping non-native speakers use English with confidence
phone: (607) 256 7368
Yin Yang Websites
milo gardner
| Melanie, This discussion may need to be broken down into two parts. Part one can include the innate aspects of language and math, sometimes over lapping, and sometimes not. Part two can include conscious learning and awareness of verbal, and written math building blocks. Proponents of innate language skills publish interesting studies. The best one that I know of was reported by the U. of Washington. Very young children from wide ranging language groups were taught vowel sounds. Monitoring took place that implied children are born with an ability to speak any combination of vowel sounds. Learning of a mother tongue, say Swedish (with 13-15 vowels), or Spanish/English (5-6 or few vowel sounds) seems to take place by un-learning the vowel sounds that are not needed. I'll look for the one hour You-tube presentation of the vowel 'learning' by unlearning study. Aymara language and logic skills tend to show that certain logical skills are common to math and language, placing learners outside of conversations. As a general rule, the trivalent (outside of conversation) syntax does not exists in bivalent languages. Bivalent languages tend to teach children logical skills in mathematical class rooms. Proponents of innate mathematical skills seem not to publish equally interesting studies. I'd be please to read of studies that discuss innate math skills that tend not to overlap with language skills. Best Regards, Milo Gardner --- On Sat, 10/23/10, Melanie <ms.tra...@gmail.com> wrote: |
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Paul Libbrecht
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Melanie
Hopefully Roberto, the person who posed the query, understood you. I understood your letter even less than I understood his. I'm so in the dark I don't even know which deficiencies of mine are the problem. (math, linguistics, or both?)
Here are some of the things I didn't understand:
- "monitoring ... implied"
- I'll buy that Spanish has six vowel sounds (a, i, o, u and two variants of e), but English? I can only boil it down to 15 different vowel sounds (long and short a, e, i, o, u, oo, plus aw, oi and schwa). I believe you said 5-6.
- "Aymara ... skills show"
- "placing learners outside of conversations"
- In what country do they "teach children logical skills in mathematical class rooms"? I want to move there!
-Melanie
milo gardner
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LFS
Ah, Melanie and Edward. I was just getting ready to write that I so much enjoy your responses because I can understand them and find their content particularly relevant to how I think about things. I often reread them – not just here but in other discussions too.
1.
I actually often think about:
Is math an international language and How much of life is math?
There was a question on Pandalous – Is God a mathematician? My response
was that God is apriori a communicator and math is a language of communication
so yes - God is a mathematician, but that is not his totality. So my response
to Roberto would be indeed be exactly what you said Melanie. But perhaps I too
am not understanding the question.
2.
With regards to language and
math in general.
My son is bilingual in English and Macedonian and I often wonder how he does
math in his head because his skills come from school (Macedonian), but (I hope)
his abilities come from me (English).
On the other hand, I am like Maria – I teach in my second language, but
am not bilingual in the true meaning of the word. I know math in English, but
teach in Macedonian. I find that I always count in English, but I don’t
always solve problems in English. When I say “count”, I mean both
the verbal act of counting and visualizing the number 1. (I see a picture of
one frog with the number 1 underneath it and I “see” the letters “one”
– no kidding.)
Anyway, these are my thoughts and language and math.
Linda
From:
mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf
Of Melanie
Sent: Sunday, October 24, 2010 4:45 AM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] language and math
Roberto,
--
kirby urner
Ah, Melanie and Edward. I was just getting ready to write that I so much enjoy your responses because I can understand them and find their content particularly relevant to how I think about things. I often reread them – not just here but in other discussions too.
1. I actually often think about: Is math an international language and How much of life is math?
There was a question on Pandalous – Is God a mathematician? My response was that God is apriori a communicator and math is a language of communication so yes - God is a mathematician, but that is not his totality. So my response to Roberto would be indeed be exactly what you said Melanie. But perhaps I too am not understanding the question.
2. With regards to language and math in general.
My son is bilingual in English and Macedonian and I often wonder how he does math in his head because his skills come from school (Macedonian), but (I hope) his abilities come from me (English).
On the other hand, I am like Maria – I teach in my second language, but am not bilingual in the true meaning of the word. I know math in English, but teach in Macedonian. I find that I always count in English, but I don’t always solve problems in English. When I say “count”, I mean both the verbal act of counting and visualizing the number 1. (I see a picture of one frog with the number 1 underneath it and I “see” the letters “one” – no kidding.)
Anyway, these are my thoughts and language and math.
Linda
Bradford Hansen-Smith
| Kirby, you asked "...but maybe not Pi notation (using the capital Pi to mean "product of"). Why? Pure happenstance?" I have no way of dropping pics into email so you'll have to open attachment to see pic and my take on this. Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ --- On Sun, 10/24/10, kirby urner <kirby...@gmail.com> wrote: |
|
Bradford Hansen-Smith
| Disregard the two, image 001 attachments on previous email. I have no idea how they got there. The symbol pi attachment is what I posted. |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ |
| --- On Sun, 10/24/10, Bradford Hansen-Smith <wholem...@sbcglobal.net> wrote: |
kirby urner
Paul Libbrecht
The international language of mathematics is characterized by havinga lot of greek letters in it. The letters are not arbitrary, even thoughone might be taught the rule: use any "legal name" for a variable.In practice, theta is used for angles more than not, whereas lambdahas its own subcultures.
Sigma notation is learned rather early in one's career (using the capital Sigma to mean "sum of"), but maybe
not Pi notation (using the capital Pi to mean "product of"). Why?Pure happenstance?
Melanie
Do you dream in Macedonian or mostly in English?
I learned chemistry in Spanish, so now even though I'm back in the States, if I try to think about chemistry I have to turn the knob to Spanish first.
However, it's very feasible for your son to do math in Macedonian in his head, but still under the Linda influence. (At least the mother in me likes to think so.)
-Melanie
LFS
Hiya all,
First - short answer to Melanie. What language do I dream in? Like you, it seems to depends on the “language of the experience”. I know that if it is a “building” dream, it is in Macedonian as I have only ever physically worked on building my house here in Macedonia. I don’t think there is language at all in my nightmares though. I never thought about it. I will try to remember to check this next time I leave the electric blanket on too hot.
Kirby – first I need to know 1 thing and second to thank-you for good laugh.
1. (NYT) I understand the bicycle on the front of the bus. How is the bicycle on side of the bus hooked up? Did the bus run into it? It looks like it is both inside the bus, but its front wheel is on the curb iron thing.
2. You had 2 links together; I presumed that it was an accident of fate in the wording of the 2nd, but immediately clicked and that was the best “math” laugh I have in a long time. Thank-you.
I know that corporations are everywhere – I find it difficult to think about that. I know at 55 I should not be so naïve, but I often wonder when do these people say “enough is enough”. I remember that this summer in Ithaca we talked about the fact that ‘math is the new race’, i.e. the latest way to exclude the “undesirables”. It is funny (odd) that you use the word “proprietary” in your letter. Today I was struggling in my Basic Computing class first to remember that word, then to think of how to translate it into Macedonian and when that failed - trying to think of how to explain it other than proprietary = you’re screwed since this would not be appropriate in a class of 19 year olds where I am the responsible adult. (We use to only get English TV on the French channels. We do not know French, but we know “a suivre” and we still say ahsuvre to mean “proprietary”. (It means “to be continued”.)) This is our family “language”. But I digress.
I guess I have a more limited concept of mathematics; I don’t think everything is mathematics. It might be mathematical. I remember thinking that if the person had asked “Is God a logistician” that I probably would have said yes. I was watching Stephen Hawking’s Universe the other day and he said the everything exists because the big bang was not perfectly distributed and that these chance imperfections caused the law of gravity to form the universe. I would say that God is logic and logical – he is both the laws (mathematics) and the chance imperfections that make life interesting when you apply the laws. I planted 24 trees in a row. They all looked great for 2 years. Then one of them up and died. One could say that is mathematics, but perhaps we make the mathematics to fit life.
What a fascinating conversation and – as always – I am amazed at the stories we have hidden in us.
--
roberto
i have been definitely flooded by your replies :)
here my quotations
On Sun, Oct 24, 2010 at 4:44 AM, Melanie <ms.tra...@gmail.com> wrote:
>
> Roberto,
>
> By 'human-like language', do you mean English, French, Chinese, etc.?
>
> By 'learning', do you mean the natural immersion process of acquiring language, for example as children do it (as opposed to taking >French 101)?
definitely, "natural immersion" is a phrase perfectly fitting my thoughts
>
> Do you want to find parallels between how babies begin to point to the moon and say 'moon', and how children begin to make patterns and >abstractions and so on, and then share their ideas with others?
fairly so;
reshaping my point, this is the problem i'm interested in:
1. a child can learn one or even more languages, almost without worries;
probably, you know many perfectly double-native-speakers
2. as you have highlighted, math is a human language, by which means
we address practical or theoretical objects
3. so, the question was: is it possible for a child to learn
mathematics (up to a certain extent) by a natural immersion method,
with little or no guidance but simply "listening" to it (as they do
for their parents' language) ? please, forgive me, i know these terms
can lead to hundreds of interpretations;
3.1 a little fork out of the previous question (but it's not my main
concern now):
as far as regards math objects (numbers, variables, equations,
functions, etc.) would you agree to look at human languages as
super-languages who stay on top of math-language and are heavily
linked to it, such that you use them to manage and communicate math
objects to each other ? (of course you can do millions of other
different things with languages ...)
could you *speak* (i mean "read") a math equation *without* using a
human language ?
thank you again for your thoughtful comments
--
roberto
LFS
Divergence …
I really liked the picture pi with the spirograph. I wonder if I can make that in GeoGebra.
I would say – the international language of mathematics is characterized by having a lot of math-speak, which itself is characterized by having a lot of greek letters in it.
One of my main points in teaching math is to get my kids to not be afraid of math-speak – everything can be translated into people-speak.
Think of the definition of the identity matrix with kronecker deltas – or just say “1s on the diagonal, 0s everywhere else”.
Here is my contribution to the greek letter notation conundrum.
Theta is the angle name for 2D polar (r, theta), until one starts to work with 3D and then magically 2D is (ro, phi) and 3D is (r, phi, theta). And none of these is in proper (argument, function) notation.
Linda
From: mathf...@googlegroups.com
[mailto:mathf...@googlegroups.com] On Behalf Of Paul Libbrecht
Sent: Monday, October 25, 2010 8:30 AM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] language and math
This is a topic I am fascinated with but I'm afraid I'm diverging from the original poster's subject.
--
Bradford Hansen-Smith
| Linda-----Don't know if you were referring to my drawing of pi on the circle matrix when mentioning the "picture of pi with the spirograph" If you were then you will be interested to know that it was drawn with Microsoft Word and not a spirograph, which I do not think can do that. If you were refering to another drawing done on a spirograph I would like to see it. I am interested to see what you come up with using GeoGebra. So far I have not been impressed with the drawing capabilities of that program. |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ |
| --- On Mon, 10/25/10, LFS <alicet...@gmail.com> wrote: |
Algot Runeman
I've been following the discussion as a non-mathematician, lurking. The
acquisition of language by children is also not my study. However, I'll
toss in my comments.
Much early language is highly concrete. "A is for Apple" and such.
Concrete, hold in your hand concepts are easy to grasp (excuse the pun).
Later, "Walk, don't run" is reinforced by a parent holding a child's
hand to keep him/her from running ahead. "Lie down," is frequently
reinforced very early with a gentle/firm hand on the back while a child
is still in a crib.
Kid's do seem to comprehend the math concepts "and" and maybe "take
away" because of their concrete nature (another cookie for and, or
taking the plate away for the loss/subtraction of another cookie). Maybe
a parent might actually remove a cookie from the child's hand.
When things slip into the abstract, children begin to be challenged.
Just ask a parent how clearly their child understands "No".
I personally encountered serious abstraction issues when, as a high
school freshman, I took algebra. I struggled horribly all year and
passed only because the teacher recognized my struggle and effort, even
though success was absent. As a sophomore, I did well in geometry. I
suspect it was "easier" because it was concrete. I don't think my
difficulty was lack of intelligence, just an immaturity of abstraction
skills. Time passed...College required some math, but I avoided more
than I should have. In graduate school, I needed to take some statistics
which had algebra prerequisites I'd avoided. Here's the most interesting
point. I sat in on some classes and took the final exam, passing it
easily. In spite of avoiding most math classes for four years, my
abstraction "language" grew to equal or exceed my concrete skills. I
eventually even took and did well in Calculus.
It has seemed interesting to me that, at one point in history, algebra
was the topic of graduate level mathematics. Over time, educators took
it down the grades until it got to the ninth grade (US, freshman). I
suspect that students might have more universal success if their
"abstraction acuity" were considered before making it a requirement.
For what it's worth, I've also felt that I *might* have done better with
algebra if the graphing of equations had been concurrent with the
abstract concepts instead of being an add in for the final few weeks of
the year. I believe strongly that the chance to see the effects of the
changes in an equation would have tied the abstractions to the concrete
view of the equations. Maybe other students and teachers didn't need the
connection, but looking back makes me think getting the view of slopes
and then curves linked with the math would have helped.
--Algot
--
-------------------------
Algot Runeman
47 Walnut Street, Natick MA 01760
508-655-8399
algot....@verizon.net
Web Site: http://www.runeman.org
Twitter: http://twitter.com/algotruneman/
Open Source Blog: http://mosssig.wordpress.com
MOSS SIG Mailing List: http://groups.google.com/mosssig2
kirby urner
Hiya all,
First - short answer to Melanie. What language do I dream in? Like you, it seems to depends on the “language of the experience”. I know that if it is a “building” dream, it is in Macedonian as I have only ever physically worked on building my house here in Macedonia. I don’t think there is language at all in my nightmares though. I never thought about it. I will try to remember to check this next time I leave the electric blanket on too hot.
Kirby – first I need to know 1 thing and second to thank-you for good laugh.
1. (NYT) I understand the bicycle on the front of the bus. How is the bicycle on side of the bus hooked up? Did the bus run into it? It looks like it is both inside the bus, but its front wheel is on the curb iron thing.
2. You had 2 links together; I presumed that it was an accident of fate in the wording of the 2nd, but immediately clicked and that was the best “math” laugh I have in a long time. Thank-you.
I know that corporations are everywhere – I find it difficult to think about that. I know at 55 I should not be so naïve, but I often wonder when do these people say “enough is enough”. I remember that this summer in Ithaca we talked about the fact that ‘math is the new race’, i.e. the latest way to exclude the “undesirables”. It is funny (odd) that you use the word “proprietary” in your letter. Today I was struggling in my Basic Computing class first to remember that word, then to think of how to translate it into Macedonian and when that failed - trying to think of how to explain it other than proprietary = you’re screwed since this would not be appropriate in a class of 19 year olds where I am the responsible adult. (We use to only get English TV on the French channels. We do not know French, but we know “a suivre” and we still say ahsuvre to mean “proprietary”. (It means “to be continued”.)) This is our family “language”. But I digress.
I guess I have a more limited concept of mathematics; I don’t think everything is mathematics. It might be mathematical. I remember thinking that if the person had asked “Is God a logistician” that I probably would have said yes. I was watching Stephen Hawking’s Universe the other day and he said the everything exists because the big bang was not perfectly distributed and that these chance imperfections caused the law of gravity to form the universe. I would say that God is logic and logical – he is both the laws (mathematics) and the chance imperfections that make life interesting when you apply the laws. I planted 24 trees in a row. They all looked great for 2 years. Then one of them up and died. One could say that is mathematics, but perhaps we make the mathematics to fit life.
What a fascinating conversation and – as always – I am amazed at the stories we have hidden in us.
Melanie
Is it possible for a child to learn mathematics (up to a certain extent) by a natural immersion method, with little or no guidance but simply "listening" to it (as they do for their parents' language)?An interesting question! Before I contribute my two cents, I'll restate the question (I hope I've got it right):
Can children pick up mathematics by being surrounded by people talking about mathematics? In the same effortless way children learn to speak one or more regular languages (people-languages)?First, let's check if we agree about what we think is going on when children learn their parents' language(s). Here's what I saw in my toddlers when they were getting started with talking.
It seemed that at a certain point in his development timeline, the child was ready to learn 'moon'. And when he was at that point, he was receptive to learning that word in each of his languages. Say he learns 'luna' Tuesday evening when looking out the window with me before bed. The next evening, on the way to bed, he points to the window and says 'luna' to Papa, who's doing the pyjama-bedtime story thing tonight. Papa learned Latin in high school, and the pointing helps; he says 'Mond'. The child repeats it. Later, he hears the two adults talking to each other in English about the 'moon', and (if he can process English well enough) pulls that word out and adds it to his set of labels for that idea. In the bedtime stories that week, he notices the moons in illustrations and makes the connection with what he saw out the window. A one-word request to look at the moon gets incorporated into the bedtime routine. The adults respond by visiting the window with him to look for the moon. All three words have been learned in quick succession. Once the child is over the hurdle of learning the moon concept, adding the labels from the various languages does seem effortless. The third step, adding the English label, seemed the most effortless of all. But without the behind-the-scenes work done at English-speaking daycare, that step wouldn't have happened (and didn't, with my second child, whose daycare wasn't in English).
Another way in which learning a mother tongue seems effortless is that there's so much satisfaction and excitement built into the process, for both parent and child, because the learning goes at such a clip, and because there's such a strong collaborative feeling between the two. And what about when the child gets far enough along with understanding how the pieces can be put together that he starts making words up, effective words that you would like to use, too? And starts pointing out funny things, and explaining why it's funny, and you hadn't seen it that way before? I grant you that it is a miraculous process.
But here's what bothers me about the effortlessness idea. New parents are told to talk to their baby a lot. We're told that the more words a baby is exposed to, the smarter he'll be. And I don't buy it.
I had a friend who raised goats. She kept a radio on in the barn, tuned to a classical music station. She said it helped the goats thrive. I'm sure it did, but listening to the radio didn't make musicians out of the goats. And I don't think just having lots of words spoken in your vicinity makes you smart. It's the communicating that makes babies smarter.
And that's how I think it works for learning math, too. What good does it do my toddler to hear mama and papa talking over dinner about how to solve a certain type of PDE (assuming the interruptions allow us to have an actual adult conversation)? The child will learn math if he and I communicate meaningful mathematical things to each other. Let's make one sand 'birthday cake' for each person. (One to one correspondence!) Let's sing a birthday song. (Logic: if birthday cake, then birthday song!) Let's blow out the stick 'candles'. (We're making models!) Let's take pieces of 'cake' and 'eat' them! (We're partitioning!)
However, I do think it's fair to say that children of mathematicians will be immersed in math, because these parents will be interested in communicating things with their children that have mathematical meaning. And when their child makes a mathematical observation, the parents will recognize the math in what was said or done, and help the child go a little further.
-Melanie
Bradford Hansen-Smith
| Melanie.......thanks for your observations. We learn by observing what is modeled by our environment, making connections and modeling our own understanding of how it is, and then to try it out to see if it works, making what ever connective adjustment are necessary for alignment. This is the growing of our brains to be able to function and interact within the environment we find ourselves. It is extraordinary how unique this process is for each of us. Models are the informing experiences that communicate. When experience becomes abstracted and generalized away from personal communication conflict arises. "I wonder why you can't paste images into email. Let me know if you'd like me to try to help you work this out." I wonder to. I have yahoo mail and switched to the new att.net Mail that has a function to drop in images. It will not work for me. I am told I do not have the program necessary. Okay, I wasn't happy with the new att,net Mail anyway, and am back in the old att.net program and just send attachments. Brad |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ |
| --- On Tue, 10/26/10, Melanie <ms.tra...@gmail.com> wrote: |
|
|
Ute Frenburg
Here, here! I agree with you completely. As the mother of two and a licensed German and Math high school teacher, I am of the firm belief that it is the connections that we make directly with children that allows them to grasp the deeper meaning and in turn develop that greater intelligence of which they are capable.
I gave up teaching for ten years when my first was born and two years later my second came into this world, all the while I was not just talking at them, I was talking with them. We never used baby talk, but instead conversed about numbers and languages at their level and a little beyond. They were sponges. It was so exciting to find what they were naturally interested in and then expand their knowledge of that subject with its integration of math and German. My children are now 15 and 13 with a thirst of learning and a greater understanding of what is being taught in their respective schools...they even understand the teacher's humor when no one else in class does. :-)
Math is cross-curriculum and can be brought up in most any conversation. The more math is openly talked about, the more comfortable people become.
Thanks, everyone, for the great conversation. I have truly enjoyed following all input.
Blessings!
Rebecca Reiniger ~ reka...@opusnet.com
Ute Frenburg in SL ~ utefr...@gmail.com
Educational Coordinator, Dream Realizations
r.rei...@dreamrealizations.org
Co-Chair International Association for K-12 Online Learning (iNACOL) Special Interest Group (SIG) Virtual Worlds
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~ Dream Realization ~
An Educational Support Services Organization designed to facilitate colleges, universities, and other entities in need of virtual world math support.
We offer classes, training, and research design.
http://dreamrealization.ning.com
Working in coordination with ItOnlyTakes1, a math visualization research organization.
http://itonlytakes1.org
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Rebecca Reiniger ~ reka...@opusnet.com
Ute Frenburg in SL ~ utefr...@gmail.com
Co-Chair International Association for K-12 Online Learning (iNACOL) Special Interest Group (SIG) Virtual Worlds
http://twitter.com/Rekaloan
~ Dream Realization ~
An Educational Support Services Organization designed to facilitate colleges, universities, and other entities in need of virtual world math support.
http://dreamrealization.ning.com
Working in coordination with ItOnlyTakes1, a math visualization research organization.
http://itonlytakes1.org
roberto
>
> Thanks for the helpful explanation, Roberto.
>
> Is it possible for a child to learn mathematics (up to a certain extent) by a natural immersion method, with little or no guidance but simply "listening" to it (as they do for their parents' language)?
>
> An interesting question! Before I contribute my two cents, I'll restate the question (I hope I've got it right):
>
> Can children pick up mathematics by being surrounded by people talking about mathematics? In the same effortless way children learn to speak one or more regular languages (people-languages)?
>
> First, let's check if we agree about what we think is going on when children learn their parents' language(s). Here's what I saw in my toddlers when they were getting started with talking.
>
>
i was thinking about the starting math children usually learn in ages
0-5/6 or so
> However, I do think it's fair to say that children of mathematicians will be immersed in math, because these parents will be interested in communicating things with their children that have mathematical meaning. And when their child makes a mathematical observation, the parents will recognize the math in what was said or done, and help the child go a little further.
>
i share your view;
let me also quote a small part of the speech of prof. Frank B. Allen,
at NCTM 1988:
he said
"The teaching of mathematics should be regarded initially as an
extension of the teaching of language. Our efforts to develop an
awareness of the intimate relationship which exists between grammar,
mathematics and logic should begin with games of " How do we know?" in
the early grades (12) continue with the introduction of formal proof
not later than grade 9 (perhaps with the aid of flow-diagrams) and
culminate in the ability to read and write lucid essay proofs by grade
12."
learning logic may be a well-grounded bridge between learning language
and learning math
--
roberto
milo gardner
| Roberto, I agree with your points when logic is discussed outside of language, as well as inside of language. Languages differ greatly with respect to syntax and logic systems. One live language that I know of that forces speakers and listeners to logically think outside of the conversation, as abstract thought, is Aymara. Was not Jaime Escalante, the LA math teacher of "Stand and Deliver" fame Bolivian and Aymaran? Milo Gardner --- On Thu, 10/28/10, roberto <robe...@gmail.com> wrote: |
|
|
|
Edward Cherlin
answers you seek.
The various Unicode mathematical symbol blocks are intended to cover
all of published mathematics in European languages (including
Russian), in coordination with the American Mathematical Society.
There are a number of Unicode blocks for numbers used in various
writing systems: Aegean Numbers, Ancient Greek Numbers, Common Indic
Number Forms, Counting Rod Numerals, Cuneiform Numbers and
Punctuation, Rumi Numeral Symbols. This is in addition to numerals
included in the blocks of specific writing systems such as Devanagari
and others from India, or Egyptian Hieroglyphs.
I can provide much more detail to anyone interested.
On Mon, Oct 25, 2010 at 02:29, Paul Libbrecht <pa...@activemath.org> wrote:
> This is a topic I am fascinated with but I'm afraid I'm diverging from the
> original poster's subject.
> On 25 oct. 2010, at 00:12, kirby urner wrote:
>
> The international language of mathematics is characterized by having
> a lot of greek letters in it. The letters are not arbitrary, even though
> one might be taught the rule: use any "legal name" for a variable.
That is the programming languages rule, not the mathematics rule.
> In practice, theta is used for angles more than not, whereas lambda
> has its own subcultures.
And Hebrew for transfinite numbers, due to Georg Cantor.
> The vertices of a triangle... how do you name them?
> I just observed that ABC is used in Germany and Hungary whereas PQR is used
> in French.
In Euclid, Alpha, Beta, Gamma, Delta...ΑΒΓΔ.
http://farside.ph.utexas.edu/euclid/Elements.pdf
page 8, Book 1, Proposition 1
Κέντρῳ μὲν τῷ Α διαστήματι δὲ τῷ ΑΒ κύκλος γεγράφθω ὁ ΒΓΔ...
With center A and radius AB, draw the circle BCD...
I haven't seen an edition of Euclid in Arabic, although apparently 31
Arabic translations were made over the centuries, and one of them was
the source for the first Latin translation. Wait, I just now looked,
and here it is!
http://pds.lib.harvard.edu/pds/view/13079270
Alif, Ba, Ha...
http://pds.lib.harvard.edu/pds/view/13079270?n=25&imagesize=1200&jp2Res=.25
> Sigma notation is learned rather early in one's career (using the capital
> Sigma to mean "sum of"), but maybe
>
> Sigma because of S like sum, somme, Summe...
> Does anyone in this list have a language where sum does not start with an
> S-like letter?
> Unfortunately, Arabic seems to use the same sign, though reversed.
>
> not Pi notation (using the capital Pi to mean "product of"). Why?
> Pure happenstance?
>
> We are trying to collect these notations as observations in the mathematical
> notation census:
> http://wiki.math-bridge.org/display/ntns/
> One is expected to make observations from widely used books for much of the
> thinkable mathematical symbols.
> paul
>
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Mike South
first of all, thanks for your comments and sorry for my late post:
i have been definitely flooded by your replies :)
here my quotations
On Sun, Oct 24, 2010 at 4:44 AM, Melanie <ms.tra...@gmail.com> wrote:yes
>
> Roberto,
>
> By 'human-like language', do you mean English, French, Chinese, etc.?
definitely, "natural immersion" is a phrase perfectly fitting my thoughts
>
> By 'learning', do you mean the natural immersion process of acquiring language, for example as children do it (as opposed to taking >French 101)?
fairly so;
>
> Do you want to find parallels between how babies begin to point to the moon and say 'moon', and how children begin to make patterns and >abstractions and so on, and then share their ideas with others?
reshaping my point, this is the problem i'm interested in:
1. a child can learn one or even more languages, almost without worries;
probably, you know many perfectly double-native-speakers
2. as you have highlighted, math is a human language, by which means
we address practical or theoretical objects
3. so, the question was: is it possible for a child to learn
mathematics (up to a certain extent) by a natural immersion method,
with little or no guidance but simply "listening" to it (as they do
for their parents' language) ? please, forgive me, i know these terms
can lead to hundreds of interpretations;
kirby urner
<< ... >>
>>
>> 2. as you have highlighted, math is a human language, by which means
>> we address practical or theoretical objects
>
> I don't know if this is true...it's a tool expressed with language...it's a
> proper subset of language...I don't know if I would call it a language.
> Like a set of voice commands a computer understands...so severely reduced
> from what a real language can do...I'm just not sure it's accurate to call
> it that. Similarly with computer "languages"...they are more just a set of
> tools for making the computer do what you want. Could you really have a
> philosophical conversation or tell what you did today with them?
> Anyway, beside the point.
>
We could leave aside whether the assertion is true or not, taking
a page from the mathematicians in agreeing to premises, which
or not proved (not being proved theorems) but simply rules of the
road.
Coming from a background in philosophy, studying Wittgenstein
(whom I never met) at Princeton, I'm prone to think of the game of
chess as a language game, and I'm fine with calling it mathematical
to boot, if that serves some purpose.
Look at the number of notations used to communicate chess,
including the board itself, used in various styles. I played with
a fairly photo-realistic set of rendered pieces while waiting for
the bus 14 the other day. Is that immersed in a "mathematical
world"? I'd say so. And yet one could never order poached
eggs in the language of chess, unless bending the rules in
some way. **
>
> I do think that math shares a place with language in that it's something our
> brains seem kind of hard-wired to be able not just to do, but to enjoy
> doing. People learn language well because we don't try to teach it. I
> think in the situations where mathematics absorption/practice/natural gains
> in proficiency mimics the language learning environment, the acquisition is
> much closer to what happens with language. I would be willing to bet money
> on the outcome of any study done along this line--find a situation that
> mimics the three conditions above and look at the success rate of the
> learners compared to that of a classroom.
What I'm projecting as likely is the level of biochemistry in everyday
language appears to skyrocket, as people take a greater interest in
what they're consuming. We've all heard of mindless consumerism,
but once you really start thinking about it...
And for biochemistry to make sense, you need a lot of discrete
mathematics. It'll be how adults talk. Pictures of molecules, amino
acids... somewhat like chess. Kids two generations hence will
sound a lot more like medical doctors than we do (unless we're
already medical doctors, or talk somewhat that way).
Kirby
kirby urner
> Hey roberto, it's a twisted question that you ask, because
> mathematics is nothing if not a human language. Who else
> would it be for? Dolphins? ETs? Maybe, but not originally.
>
> We might need to pause and ask what "mathematics" could
> mean.
>
More along these lines, posing questions:
An important concept I wish to get across to students involves
surrounding a ball with more balls of the same size. Were
these circles, I might use coins (again, all the same radius).
The verbal descriptions (example below) plus various visuals,
plus an actual tactile model (something to hold), is about all
that I'd need to anchor the concepts. But then a student has
to be fluent enough in the first place, to connect the words
to the pictures and vice versa.
Example:
"""
Six balls on a table surround a central nuclear ball, all
have the same radius. Above and below this plane of
seven balls (one in the center, six around), we have
valleys where more balls will nest. I am able to fit three
more balls above, and three more below, for a total of
12 balls around the nuclear ball.
"""
Then comes a picture, and an animation:
http://www.rwgrayprojects.com/synergetics/s02/figs/f2201.html
http://4dsolutions.net/ocn/xtals101.html (top right)
"""
Then comes more elaboration, fine tuning, even caveats.
There's more than one way to orient those additional
triangles of three balls (one above, one below) relative
to each other. One way gives the vertexes of a shape
called a cuboctahedron (depicted above).
The cuboctahedron may be described in terms of four
intersecting hexagons. The pattern of six balls around
a nuclear ball may be found in four planes.
"""
Making the above clear might involve more pictures,
more hands on. So many students would just glaze
over trying to read paragraphs like this. Part of it is
not knowing how to subvocalize, which for many
readers interrupts their reading comprehension.
How to pronounce "cuboctahedron" then?
So are these "mathematical" paragraphs? Some (including
me) would probably call this "descriptive geometry". There's
a vocabulary, words such as "polyhedron" and "sphere
packing". Developing familiarity with this terrain might
involve looking at closest packed fruit stacked up in
supermarkets. The textbooks often go there for imagery.
On the other hand, coughing up a lot of XYZ coordinates
or vector notation might *look* more like what people
think of as mathematical, but if the aim is to get mental
imagery across, from human A to human B, then is all
this notational claptrap really necessary?
On the contrary, if you're trying to program a computer
to generate these animations (cartoons), then yes, you'll
want your XYZ (or other) coordinates. But if it's just one
storyboard artist talking to another... words and sketches,
a few models, should do the job, no?
A lot of "mathematical notation" serves no purpose (yet).
How might we continue with our descriptive geometry then?
"""
A second layer of balls around the first layer of twelve
will conform to the same cuboctahedral shape, consisting
of 8 triangular facets and 6 square ones (for a total of 14
in all). The next layer of balls after the layer of 12 will
contain 42 balls. The next layer after than will contain
92 balls, then 162. Picture a growing cuboctahedron with
more and more balls, organized in successive layers.
The number of balls in layer L, starting with L = 1, is
given by the equation N = 10 * L * L + 2.
"""
We now have a sequence of numbers associated with a
geometric shape. No XYZ coordinates have been introduced.
The sequence is 1, 12, 42, 92, 162.... 10*L+L + 2. There
seems to be some caveat about L=0, as then we'd like
1, not 2. A computer program might involve conditional
branching. Lots of ways to go. We could look up the above
sequence in the On-Line Encyclopedia of Integer Sequences.
Continuing...
"""
"All of these numbers are in fact found in actual viruses,
12 for certain bacteriophages, 42 for wart viruses, 92 for
reovirus, 162 for herpesvirus, 252 for adenovirus and 812
for a virus attacking crane-flies (Tipula or daddy-long-legs)"
- The Natural History of Viruses
by C.H. Andrews (W.W. Norton R Co., 1967).
"""
http://www.4dsolutions.net/synergetica/synergetica2.html
We have clearly entered the domain of biology and/or
biochemistry somehow, with this number sequence. A
connection to make is from the cuboctahedron, the shape
we've been dealing with, and the icosahedron, which is
the shape of the viral sheath, the housing for the RNA.
The passage above refers to the number of capsomeres
in various viral "capsids" (these containers).
Going back to the On-Line Encyclopedia of Integer
Sequences, we find that the "cuboctahedral numbers"
and "icosahedral numbers" are the same sequence.
Why is that? Again, we resort to a cartoon, an
animation, not to a lot of squiggles / notation of a
cryptic nature. We're talking human-to-human, not
human-to-machine. There's a way the balls rearrange.
To do this, you need to gut the internals (take out
the interior layers) and just work with one hollow
shell at a time.
Time for another picture (scroll down to Fig. 466.00a ):
http://www.4dsolutions.net/ocn/numeracy0.html
(plus there's an animated GIF right next to it)
So here we've made a lot of links, starting from ball
packing, moving through some geometric concepts
(e.g. hexagons), introducing an algebraic formula
(in terms of L) and another geometric transformation
(cuboctahedron <--> icosahedron).
Students able to communicate with one another in
terms of sketches, descriptions, hyperlinks to related
topics: are they communicating in a "math language"?
They are communicating mathematical concepts by
means of language, I think we could say.
I hope the above provides some useful grist for the
mill. We're talking about what constitutes a mathematical
language.
In this thread, we've touched on the idea of "immersion"
i.e. what would it be like to be surrounded by people
"speaking math".
I'm suggesting that when it comes to conveying concepts,
ordinary language may well serve a critical role, however it
tends to be abetted with specific shop talks. Machinists,
people who work with machinery, are quite familiar with this
pattern. They've learned to visualize, based on a shared
vocabulary. The need for precision is met.
Adding a lot of XYZ coordinates with accuracy to nine
decimals might be all well and good for programming
a computer, but lets not confuse these possibilities with
necessary requirements.
If we just had one Turing Complete computer language
(pick one, any one), would we have all we need to express
mathematical ideas?
No, I don't think so.
Nor is any one logic notation or set of symbols in itself
sufficient, in my estimation, to glue our thinking together
into coherent and systematic views.
We actually require human languages to make sense of
our mathematics, to provide context. Human languages
are not ancillary, not second banana. They remain core.
Which is why I think exercising one's ability to
write / read / speak a kind of descriptive geometry
language is critical. It's a fluency we're in danger of losing
and/or not developing.
People know what "mental arithmetic" means (the ability
to do some simple computations "in one's head"). What
would "mental geometry" mean in contrast? Should we be
developing this ability in parallel with mental arithmetic
capabilities?
Having students collaborate on "geometry cartoons" at whatever
level of proficiency, would seem a great way to go. Just learning
what all the Platonics have in common, versus the Archimedeans,
is a great way to approach the idea of "filtering on criteria" (an
idea with very wide application).
Learning to count edges, faces and vertexes is its own set of skills.
Deducing some facts about a shape, given other facts, is a bridge
to the idea of "proofs". Trigonometry is like this: from parts of
the triangle, and generalizations about all triangles, derive other
parts of the triangle. That's a metaphor for thinking in general:
from parts, plus a sense of the whole (semi-intuitive), get to
other parts. Non-computational leaps may occur. Sometimes
the reasoning is post hoc, after flashes of insight (Penrose).
In my view, we've gotten side tracked by a lot of the squiggles,
the cryptic / dense notations, at the expense of the visual
imagination (blame Bourbaki if you like). From traditional
schooling, we learn that "flatland" geometry is somewhat
computationally intensive (using XY analytic coordinates)
but that "spatial" geometry is even more difficult and intensive.
We have become overly intimidated by our own projections
of what is required to "talk sense" about space.
On the other hand, we naturally live and breath as spatial
creatures. No one has ever taken that "flatlander" point of
view such as we're asked to imagine in Abbott's 'Flatland'.
Just describing spatial stuff in a form closer to ordinary
prose (examples above), abetting with animations (such as
Youtubes) could do wonders for math-science fluency I'm
thinking.
The algebra (vectors, matrices etc.) comes into the picture
over time, but with this more sophisticated visual imagination
already a fait accompli in more cases.
Spatial geometry doesn't have to be supported from below
with all kinds of notational claptrap. On the contrary, it's
the most natural thing in the world for language to wrap
our minds around scenery, vistas, architectures -- and
that's what mental geometry (and/or geography) is all about.
Kirby
Bradford Hansen-Smith
| Well said Kirby, I might add that all those " ...squiggles, the cryptic / dense notations,..." are the results of people looking at spatial geometries, making notes about their observations and generalizations, then forgetting they saw anything. Just because people discovered this stuff a long time ago does not mean students today can't discover it for themselves; that would make it far more meaningful than what can be found in any book. |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ |
| --- On Tue, 11/2/10, kirby urner <kirby...@gmail.com> wrote: |
|
|
|
Maria Droujkova
reshaping my point, this is the problem i'm interested in:
1. a child can learn one or even more languages, almost without worries;
probably, you know many perfectly double-native-speakers
2. as you have highlighted, math is a human language, by which means
we address practical or theoretical objects
3. so, the question was: is it possible for a child to learn
mathematics (up to a certain extent) by a natural immersion method,
with little or no guidance but simply "listening" to it (as they do
for their parents' language) ? please, forgive me, i know these terms
can lead to hundreds of interpretations;
There is an indirect proof: a child who has NOT done math until a certain age will not develop certain types of reasoning, such as algebraic. Savants can do math without much outside intervention, though. Something to think about!
3.1 a little fork out of the previous question (but it's not my main
concern now):
as far as regards math objects (numbers, variables, equations,
functions, etc.) would you agree to look at human languages as
super-languages who stay on top of math-language and are heavily
linked to it, such that you use them to manage and communicate math
objects to each other ? (of course you can do millions of other
different things with languages ...)
could you *speak* (i mean "read") a math equation *without* using a
human language ?
The answer is probably, "It depends." There are certain problems that aren't language-based for me, at least I don't think so. There was a problem about loops in a grid during GeoGebra-NA in Ithaca which I solved visually, and it did not feel like any language was involved. Equations like x+3=2 on the other hand go through language for me. Preferably Russian :-)
Working on my accent with Melanie helps to see a lot of connections to surprisingly mathy brain parts. For example, order is of much importance - so things that have to do with commutativity are probably language-mediated. I can say "Max and Dan" much better than I can say "Dan and Max."
I am planning a family experiment for inviting parents to experiment with their babies, making their speaking and the environment more math-rich. I wonder what it will do.
Cheers,
Maria Droujkova
Make math your own, to make your own math.
kirby urner
My friend and colleague Glenn Stockton (whom I've mentioned before as a "Neolithic Math teacher") discovered the Mobius strip for himself, complete with all the astonishment and wonder that comes from realizing it just has the one side.
But when he showed it to his teacher, the response was cold: this had already been discovered by someone else.
Another friend, Steve Holden was really interested in chemistry and had been studying concepts on his own.
When he went up to the teacher after class one day, with some questions, he was informed those topics would not be touched upon this semester, implying he should stick to the assigned readings. He gave up on chemistry and went into electronics instead (today he's chairman of the Python Software Foundation).
People have many ways to kill off the excitement of discovery in kids, and one is to keep reminding them they're not the first.
This may sound cruel and unprofessional, but then the discipline we call mathematics has this competitive aspect, where only new discoveries get rewarded.
Kirby
Bradford Hansen-Smith
Here is a real life math problem. http://xkcd.com/135/?ref=nf |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ |
|
|
Costello, Rob R
>
> reshaping my point, this is the problem i'm interested in:
> 1. a child can learn one or even more languages, almost without worries;
> probably, you know many perfectly double-native-speakers
> 2. as you have highlighted, math is a human language, by which means
> we address practical or theoretical objects
> 3. so, the question was: is it possible for a child to learn
> mathematics (up to a certain extent) by a natural immersion method,
> with little or no guidance but simply "listening" to it (as they do
> for their parents' language) ? please, forgive me, i know these terms
> can lead to hundreds of interpretations;
>
similar question posed by Seymour Papert. Made the point that if you look at poor results overall of US students learning foreign languages, say French, you'd conclude those kids didn't have much aptitude for that learning ... yet take the same children and raise them in France, and it happens more naturally; and with fluent results
so he posed the possibility of 'Mathland' - ie a culture that does for math learners what living in France does for learning French; ie supporting the maths learning to occur somewhat parallel to language acquisition
(i notice that the Suzuki method makes the same claim for music learning - perhaps with more traction; more methods that are achieving impressive results)
a key part of Papert's proposed culture of mathland, was using computers to explore interesting mathematical ideas and approaches : making abstract concepts more intuitive and more concrete, and allowing explorations in different ways ...
his Mindstorms book captures a lot of this : and for me, its a powerful argument ... has the role of the computer changing both the mode of learning, and the content of what we think of as school level maths ... a often quoted example being that a circle can be defined as the process of taking 360 little steps forward and to the right :
repeat 360 [fd 1 rt1] (in logo style)
maybe more intuitive than
x^2+y^2=r^2 etc
and perhaps closer to vector calculus models, without needing to spell that out !
makes a lot of sense for me .. explorations with spreadsheets, or with programming, always feel richer to me ..
although i'd say the impact of the Papert ideas on curriculum have, in the end, been negligible; the Logo era seems to have ebbed...
(he picked up on the 'grammar of schooling' to refer to the inertia of schooling : its tendency to reproduce itself and swallow promising innovations..)
yet there is a reprise of 'computational thinking' : eg here
http://www.google.com/edu/computational-thinking/resources.html
outside formal schooling - interesting
: the papert paper quoted there (An Exploration in the Space of Mathematics Educations) is a typically provocative examination of what could be
> 3.1 a little fork out of the previous question (but it's not my main
> concern now):
> as far as regards math objects (numbers, variables, equations,
> functions, etc.) would you agree to look at human languages as
> super-languages who stay on top of math-language and are heavily
> linked to it, such that you use them to manage and communicate math
> objects to each other ? (of course you can do millions of other
> different things with languages ...)
not my area, but i gather many think this way : that the common meanings of language we use and develop underpin the more formal and precise usage in mathematics
> could you *speak* (i mean "read") a math equation *without* using a
> human language ?
not sure that question makes sense : if a human is doing the reading, are we sure that ruling out language is possible? eg even in an apparently graphical mode of perception, or an appreciation of underlying maths structures rather than surface words or symbols; does it make sense to rule language out of all that..
Important - This email and any attachments may be confidential. If received in error, please contact us and delete all copies. Before opening or using attachments check them for viruses and defects. Regardless of any loss, damage or consequence, whether caused by the negligence of the sender or not, resulting directly or indirectly from the use of any attached files our liability is limited to resupplying any affected attachments. Any representations or opinions expressed are those of the individual sender, and not necessarily those of the Department of Education and Early Childhood Development.
Edward Cherlin
Sugar education software.
I wrote about the use of OLPC XOs in literacy, as a substitute for the
parent reading storybooks to children. The child should still sit on
someone's lap with the computer "reading" the stories.
Now imagine an interactive play world where each game has a
mathematical point. I'm thinking about how to teach fractions with pie
slices. In principle, I know how to teach the invert and multiply rule
for dividing fractions, but I'm not ready to create the software and
test the idea. I have to finish a "book" on discovery first. You can
look at a partial draft at
http://www.booki.cc/discovering-discovery/
although I don't think it will make much sense if you don't have an XO
or some other version of Sugar at hand. Possibly not even then. ^_^
> --
> You received this message because you are subscribed to the Google Groups "MathFuture" group.
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--
LFS
I so think this is a cool idea.
Can't try your link (no xo), but was interested in the "invert and multiply".
Today I was trying to explain why 1/(1/2) is 2 without this rule so I said and drew
(a) 8/2 means "How many pieces of size 2 in 8?" Answer: 4.
(b) 1/(1/2) means "How many pieces of size 1/2 in 1?" Answer: 2.
It is so far back in the dark ages I cannot remember how my teachers explained dividing by fractions. Is this the way you explain this?
Linda
-----Original Message-----
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Edward Cherlin
Sent: Monday, November 08, 2010 9:41 PM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] language and math
Mike South
Hiya Edward.
I so think this is a cool idea.
Can't try your link (no xo), but was interested in the "invert and multiply".
Today I was trying to explain why 1/(1/2) is 2 without this rule so I said and drew
(a) 8/2 means "How many pieces of size 2 in 8?" Answer: 4.
(b) 1/(1/2) means "How many pieces of size 1/2 in 1?" Answer: 2.
It is so far back in the dark ages I cannot remember how my teachers explained dividing by fractions. Is this the way you explain this?
Bradford Hansen-Smith
Talk about "...far back in the dark ages" |
"Ours is not to wonder why |
Just invert and multiply" What better way to kill the joy of wonderment.Even if I don't remember what to do, I will know to do what I am told and not to question. Can't we do better than this? |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ |
| --- On Mon, 11/8/10, Mike South <mso...@gmail.com> wrote: |
Sue VanHattum
>I think the most pedagogically sound approach is to teach them this couplet:
Maria Droujkova
In case it's of any help, here is my take on fraction division, with three different models that make sense:
- Rectangle model
- Common denominator
- Ratio to one
Different people like different models better, and preferences are rather strong.
http://naturalmath.wikispaces.com/Divide+a+fraction+by+a+fraction
Cheers,
Maria Droujkova
Make math your own, to make your own math.
milo gardner
|
|
|
|
Mike South
"Ours is not to wonder whyJust invert and multiply"What better way to kill the joy of wonderment.
Even if I don't remember what to do, I will know to do what I am told and not to question.
Can't we do better than this?
Maria Droujkova
I would like to separate "models" and "algorithms." Models explain WHY and HOW, but algorithms only show HOW. That's why "invert and multiply" isn't in the list of models. It does not matter what names we call things, though - as long as both WHY and HOW are addressed!
kirby urner
abstract algebra games, which might be actual games,
on a computer or game board.
Please forgive my somewhat prolix expansion on this
idea as I explain it to myself again. These are dots
I've frequently connected but it doesn't hurt to spiral
through again and teach myself some math.
One might have Add World where two colored blobs
meet and Add ( + ) giving a new colored blob. Then
the Neutral character is introduced, with a special
sign or color. N always leaves a colored blob as it
is: N + b = b; a + N = a.
Then one might have Multiply World where any two
colored blobs meet and Multiply ( * or x), giving a
new colored blob. This time, the Neutral character
is likewise special. M * b = b; b * M = b and so on.
Then we talk about blob combinations in both worlds
that give rise to these respective Neutrals, N and M.
Any pair a, b that adds to give N is called a pair
of inverses in that a and b are considered inverses
of one another. Additive inverses in Add World,
Multiplicative inverses in M world.
Only at this point might we reintroduce subtraction
and division into these respective Worlds. Considered
as a unary operator, - will "flip" a blob into its own
inverse. A second - will flip it back. --a = a. We
could consider / as a unary operator as well. //b = b.
*(/b, b) = M. *(b, /b) = M. +(a, -a) = N and so on.
How we're ready to talk about a / b as shorthand
for a * (/b). Likewise a - b is shorthand for a + (-b).
a/a = a * (/a) = *(a, /a) = M, b - b = b + (-b) =
+(b, -b) = N. We typically use the symbols 1
and 0 in place of M and N.
Written out lexically like this, the above has the
look of somewhat high level math, definitely algebra.
Putting an operator outside parentheses and going
+(b, -b) = N looks somewhat LISP-like. I'm not that
attached to any particular notation. Rather, to keep
minds nimble and flexible, we should keep the
door open to multiple possibilities. At some turn
through the spiral, actually boot Scheme and show
[+ 3 2] as a machine-implemented math notation.
The animations with colored blobs would look more
like particle physics. The operator for the World could
appear once in the corner of the screen, as the colored
blobs zing around and occasionally collide, at which
time the magic happens and a new blob appears.
We could also have blobs spontaneously splitting
apart, e.g. 5 -> 3 + 2 or -3 + 8.
An advantage of starting early with such "particle
physics" cartoons is that we keep coming back
to them even when the nature of the blobs changes.
These could be matrices or polynomials or
permutations (mappings). Or perhaps addition
and multiplication are modulo some modulus Z.
What's fun in the latter case is to gather up all
the "totatives" of the modulus (i.e. positives < modulus
with no factors in common except 1) and put those
in the collision chamber where they multiply
modulo Z. They turn into one another, i.e. the
resulting blob is always a member of the original
set. Each blob has an inverse as well. M and N
are each their own inverses. If Z is prime, even
addition is closed (results are within the original
set).
The above makes sense against a backdrop where
(a) we show a lot of math cartoons and (b) we start
getting into stuff like modulo arithmetic relatively
early. The latter is sometimes called "clock arithmetic"
because time measurement is indeed cyclic, consists
of wheels within wheels. Actually, so are the
individual columns in place notation cyclic i.e. in
base 10 we go 0..1..2... 8..9.. then back to 0. The
idea of "carrying" applies to time keeping as well:
we start the next 24 hours, but have added 1 to
the passing of days "column". The computer
function giving the number of days since Jan 1 1970
is what again? I'm thinking of Unix time, given in
seconds. Yikes this stuff gets complicated (time
measure gets messy):
http://en.wikipedia.org/wiki/Julian_day
My apologies again for going on at such length.
I have an investment in this "blobs" approach because
of my "math objects" approach using an object oriented
paradigm (as distinct from the more functional
Scheme/LISP based paradigm -- or compare with
APL/J). I have this whole curriculum worked out
where Python classes look like little snakes, or
creatures, given they have __ribs__ and all. So
these blobs become somewhat biological, with
instances being "full of shared internals" one might
say.
Kirby
Kirby
Peter Tierney-Fife
fraction by another really mean, and why does "invert and multiply"
work, what are types of models that can represent division of two
fractions, etc.) is the book -Knowing and Teaching Elementary
Mathematics: Teachers' Understanding of Fundamental Mathematics in
China and the United States- by Liping Ma. (See chapter 3.)
As for an *algorithm* that can help explain why the invert and
multiply rule works, one way is to multiply each fraction by 1 in the
form of a new compound fraction that is, in both the numerator and the
denominator, the reciprocal of the divisor fraction, then simplify:
original problem: 3/4 ÷ 1/2
multiply each fraction by 1 in the form (2/1)/(2/1): (3/4)([2/1]/
[2/1]) ÷ (1/2)([2/1]/[2/1])
set up as the original fractions each times the numerator of the
inserted fraction, over the common denominator 2/1: ([3/4]*
[2/1]) / (2/1) ÷ ([1/2]*[2/1]) / (2/1)
simplify to just the numerators of the above, since dividing by the
same non-zero denominator value (2/1): ([3/4]*[2/1]) ÷ ([1/2] *
[2/1])
simplify the second part, since 1/2 multiplied by 2/1 is 1, to:
([3/4]*[2/1]) ÷ 1
simplify by removing division by 1, leaving: (3/4)*(2/1), which is
the result of the "invert and multiply" rule
I think.
;-)
-Peter Tierney-Fife
On Nov 9, 12:09 pm, Maria Droujkova <droujk...@gmail.com> wrote:
> Milo, I put your historical overview in a separate page linked from the
> I would like to separate "models" and "algorithms." Models explain WHY and
> HOW, but algorithms only show HOW. That's why "invert and multiply" isn't in
> the list of models. It does not matter what names we call things, though -
> as long as both WHY and HOW are addressed!
>
> Cheers,
> Maria Droujkova
>
> Make math your own, to make your own math.
>
>
> > My view is that the history of mathematics offers students several ancient
> > doors to understand modern arithmetic operations, including fractional ones
> > that introduce algebra.
>
> > Best Regards,
>
> > Milo Gardner
>
> > From: Maria Droujkova <droujk...@gmail.com>
>
> > Subject: Re: [Math 2.0] language and math
> > To: mathf...@googlegroups.com
> > Date: Tuesday, November 9, 2010, 4:27 AM
>
> > Edward, great work!
>
> > In case it's of any help, here is my take on fraction division, with three
> > different models that make sense:
> > - Rectangle model
> > - Common denominator
> > - Ratio to one
>
> > Different people like different models better, and preferences are rather
> > strong.
>
> >http://naturalmath.wikispaces.com/Divide+a+fraction+by+a+fraction
>
> > Cheers,
> > Maria Droujkova
>
> > Make math your own, to make your own math.
>
>
> > I'm working on a version of Mathland in Squeak/Etoys, as part of the
> > Sugar education software.
>
> > I wrote about the use of OLPC XOs in literacy, as a substitute for the
> > parent reading storybooks to children. The child should still sit on
> > someone's lap with the computer "reading" the stories.
>
> > Now imagine an interactive play world where each game has a
> > mathematical point. I'm thinking about how to teach fractions with pie
> > slices. In principle, I know how to teach the invert and multiply rule
> > for dividing fractions, but I'm not ready to create the software and
> > test the idea. I have to finish a "book" on discovery first. You can
> > look at a partial draft at
>
> >http://www.booki.cc/discovering-discovery/
>
> > although I don't think it will make much sense if you don't have an XO
> > or some other version of Sugar at hand. Possibly not even then. ^_^
>
> > On Sat, Nov 6, 2010 at 10:10, Costello, Rob R
>
> read more »
Ihor Charischak
Maria Droujkova
Another model that makes some sense involves playing
abstract algebra games, which might be actual games,
on a computer or game board.
Please forgive my somewhat prolix expansion on this
idea as I explain it to myself again. These are dots
I've frequently connected but it doesn't hurt to spiral
through again and teach myself some math.
One might have Add World where two colored blobs
meet and Add ( + ) giving a new colored blob. Then
the Neutral character is introduced, with a special
sign or color. N always leaves a colored blob as it
is: N + b = b; a + N = a.
I've done something like this with my daughter, when she was 2-4 years old. She called multiplicatively generated worlds "spirit worlds" - for example, the world with 2, 4, 8, 16 in it, "the world of the spirit of two." It was all visually done, of course, through something similar to Fractal Abacus: www.atm.org.uk/journal/micromath/mm181gibson.pdf
The corresponding fractions were there too. I think she was roleplaying some Miyazaki movie. Additive worlds were "fake" (not real), e.g. 2, 4, 6 wasn't a real world. She also had a special way of talking about 0 and 1. I think I have a conference report about it somewhere.
Cheers,
MariaD
Tim Fahlberg
should say that she first shared your
couplet:
Ours is not to wonder why
Just invert and multiply"
often used to teach math in such a hilarious way!
Now that Linda has introduced me to this wonderful group I'll have to
come in and visit more often.
Thanks again!
-- Tim
On Nov 8, 4:17 pm, Mike South <mso...@gmail.com> wrote:
> > .
> >http://groups.google.com/group/mathfuture?hl=en.
>
> > --
> > Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
> > Silent Thunder is my name, and Children are my nation.
> > The Cosmos is my dwelling place, the Truth my destination.
> >http://www.earthtreasury.org/
>
> > --
> > You received this message because you are subscribed to the Google Groups
> > "MathFuture" group.
> > To post to this group, send email to mathf...@googlegroups.com.
> > To unsubscribe from this group, send email to
> > .
> >http://groups.google.com/group/mathfuture?hl=en.
>
> > --
> > You received this message because you are subscribed to the Google Groups
> > "MathFuture" group.
> > To post to this group, send email to mathf...@googlegroups.com.
> > To unsubscribe from this group, send email to
> > .
Maria Droujkova
The spirit of four: A case study of metaphors and models of number construction
Objectives
Researchers have noted the need for investigation of relationships between different types of reasoning and number construction models (Confrey & Smith, 1995; Olive, 2001; Pepper & Hunting, 1998; Steffe, 1994). The goal of this study was to look at the development of number construction through the lens of metaphor.
In particular, the study investigated the interplay between a largely multiplicative environment and the development of reasoning within this environment that was significantly different from scenarios from other studies.
Conceptual framework
Observing young children makes a strong case for viewing mathematical thinking as fundamentally metaphoric (R. Davis, 1984). Metaphor is the recursive movement between a source and a target that are structurally similar, both changing in the dynamic process of learning (B. Davis, 1996; R. Davis, 1984; English, 1997; Lakoff & Johnson, 1980; Lakoff & Nunez, 1997, 2000; Pimm, 1987; Presmeg, 1997; Sfard, 1997).
For analyzing number construction, I used the counting scheme (Olive, 2001; Steffe, 1994) and the splitting conjecture (Confrey & Smith, 1995; Lehrer, Strom, & Confrey, 2002). The metaphor that connects sources of sharing, folding or similarity, and the target of multiplicative one-to-many actions can be considered the basis of splitting as a cognitive scheme. The metaphor that connects the source of counting and the target of the number sequence is the basis of the counting scheme. In the splitting world multiplicative reasoning develops via grounding metaphors with sources such as sharing. In the counting world multiplicative reasoning is based on the linking metaphor which connects interiorized, reversible counting with iterable units (Olive, 2001; Steffe, 1994).
Modes of inquiry
This paper presents a longitudinal case study of reasoning in a child up to the age of five, whose home environment was restructured to incorporate more multiplicative activities. Researchers often consider metaphor to be private, unformulated and difficult to study (Presmeg, 1997). Additional access issues came from the need for a very young subject necessary to trace the beginnings of number concept development, and from the longitudinal nature of the study. These considerations pointed to the necessity of a close relationship between the subject of the study and the researcher, and I invited my daughter “Katya” to be the subject of the study. As a parent, I was in a privileged position of access to the majority of the details of Katya’s day-to-day life, as well as to the meaning of her utterances and gestures.
Data sources and evidence
Data for the study came from fieldnotes of observations as a participant-observer; videotapes and audiotapes of unstructured and semi-structured interviews; photographs of activity settings; and a collection of artifacts used in activities.
Results
The non-sequential order in which conventional number names first appeared in Katya’s speech corresponded to multiplicative, rather than counting, actions. For example, the utterance “two twos” appeared about eight months earlier than the word “four,” and also earlier than the word “three.” Appearance of “two threes” in games preceded the use of the words “four,” “five” and “six,” and appearance of “two fours” preceded the use of numbers greater than four.
In constructing numbers from one to four, Katya used individual (Presmeg, 1997) metaphors based on instant recognition of the quantity. In these metaphors, the source was an image with a quantity intrinsically embedded in it, such as “dog’s legs” for “four.” Katya mostly used mixed references for multiplicative situations, for example, “two dogs” to signify “two times four.” This availability of two systems of signifiers provided a language necessary to address the asymmetrical nature of the multiplication models Katya used. For example, in the case of “two dogs” the words underlined the distinction between sets and set members in the set model of multiplication. Lack of signifiers for this asymmetry of multiplication models may be problematic and may hinder development of multiplicative reasoning. Confrey and Smith (1995) note that “a counting number is typically used to name the result or outcome of a split” (p.75, italics mine).
If learners see the splitting and counting worlds as isomorphic (Confrey & Smith, 1995), they can understand structures of one world by making parallels with the corresponding structures of the other world. Children’s structure transfer attempts become especially visible when they differ from accepted standards. For example, researchers often focus on children inappropriately applying additive strategies to multiplicative situations (Post, Behr, & Lesh, 1986). Katya frequently tried to use multiplicative relationships instead of additive. For example, when asked to continue a pattern of arrays made out of circles: 2 by 1, 2 by 2, 2 by 3, ___ she attempted to iterate the previous array twice, drawing a 2 by 6 array instead of the expected “2 by 4”. Upon my explanation that a pair of circles is added to the array in each step, Katya said, somewhat angrily, that these pictures “are not real.” Multiplicative relationships were more “real” to her.
In another unexpected example, a square was split into four equal squares, and then each of the small squares was split into four tiny squares. Katya used words signifying size gradients, such as “large, small, and tiny,” and “babies and adults,” consistently across different multiplicative worlds. This metaphor of “growth” united different multiplicative worlds and allowed Katya to compare their structures, working on what a mathematician would call “powers” or “base systems.” Katya used the word “spirit” to denote the action in each world, for example, talking about “the spirit of four” in the split square above. She claimed that if we cut the 4-square piece in four, the result would be zero. Upon cutting, she was surprised that the result was one square. However, in repeated activities with the same picture, or with pictures based on other powers from other split worlds, Katya consistently said that the result of splitting the power base picture would be “zero”, or “nothing,” even after observing again and again that it turned out to be one.
I hypothesized that these names were expressions of metaphors for the origin, and I told Katya that researchers call the entity in question “the origin.” We compared the origins of additive and power-based structures, and Katya felt validated to discover a “real” zero at least at some origin. This instance of isomorphism between additive and multiplicative worlds helped Katya to build her idea of the origin as a “superordinate construct” (Confrey & Smith, 1995), whereas the idea was problematic while she stayed within the multiplicative world.
Deeper understanding of connections between additive and multiplicative reasoning can benefit further theory construction in areas such as number construction, ratio and proportion, or exponential functions. Practitioners can draw on possible uses of metaphors for working with deep mathematical ideas throughout the mathematical curriculum. Since the majority of studies of young children are done in additive environments, research of cases developed in a predominantly multiplicative environment can provide a valuable vantage point for theory development.
Reference
Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66-86.
Davis, B. (1996). Teaching mathematics: Toward a sound alternative. New York: Garland.
Davis, R. (1984). Learning mathematics : The cognitive science approach to mathematics education. Norwood, NJ: Ablex.
English, L. D. (1997). Analogies, metaphors and images: Vehicles for mathematical reasoning. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors and images (pp. 3-20). Mahwah, NJ: Lawrence Erlbaum.
Lakoff, G., & Johnson, M. (1980). Metaphors we live by. Chicago: University of Chicago Press.
Lakoff, G., & Nunez, R. E. (1997). The metaphorical structure of mathematics: Sketching out cognitive foundations for a mind-based mathematics. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors and images (pp. 21-92). Mahwah, NJ: Lawrence Erlbaum.
Lakoff, G., & Nunez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being (1st ed.). New York, NY: Basic Books.
Lehrer, R., Strom, D., & Confrey, J. (2002). Grounding metaphors and inscriptional resonance: Children's Emerging Understanding of mathematical similarity. Cognition and Instruction, 20(3), 359-398.
Olive, J. (2001). Children's number sequences: An explanation of Steffe's constructs and an extrapolation to rational numbers of arithmetic. The Mathematics Educator, 11(1), 4-9.
Pepper, K. L., & Hunting, R. P. (1998). Preschoolers' counting and sharing. Journal for Research in Mathematics Education, 29(2), 164-184.
Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. London, England; New York, NY: Routledge & K. Paul.
Post, T., Behr, M., & Lesh, R. (1986). Research-based observations about children's learning of rational number concepts. Focus on Learning Problems in Mathematics, 8(1), 39-48.
Presmeg, N. C. (1997). Reasoning with metaphors and metonymies in mathematical learning. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors and images (pp. 267-280). Mahwah, NJ: Lawrence Erlbaum.
Sfard, A. (1997). Commentary: On metaphorical roots of conceptual growth. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors and images (pp. 339-372). Mahwah, NJ: Lawrence Erlbaum.
Steffe, L. P. (1994). Children's multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3-40). Albany, NY: State University of New York Press.
Maria Droujkova
Make math your own, to make your own math.
Bradford Hansen-Smith
| Sorry Mike it escaped me, but let's face it, internet humor has got to be more than just smileys. |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ |
Bradford Hansen-Smith
| Maria, If you can find it I would be interested to know what your daughter thought of the 0 and 1early in her life. I find confusion in ways grownups talk about these two symbols. When the 0 is folded and reveals 1 diameter, there is no confusion. Brad |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ |
| --- On Tue, 11/9/10, Maria Droujkova <drou...@gmail.com> wrote: |
|
|
Bradford Hansen-Smith
| Thanks Maria, |
| "Katya said, somewhat angrily, that these pictures “are
not real.” |
| I could not have said it more clearly myself, except she said it when she was very young, it took me about fifty years before expressing the same thing; but then we teach math by giving reality to these symbols and it is difficult to hold one own position against the math "experts". I find the question of origin is on the minds of those closer to their experience of origin than most of us that have been years removed. There is a reason for many people in the last stages of dying reverting back to childhood. Math language has made a zero of the circle, meaning nothing, origin then has no meaning except for as we give unit meaning in mathematical context. Sometimes words get in the way of meaningful metaphors of understanding. That may be the only way we can make sense to rid the confusion of ideas and just go back to find meaning in the metaphor of origin. |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ --- On Tue, 11/9/10, Maria Droujkova <drou...@gmail.com> wrote: |
|
roberto
>
> Now imagine an interactive play world where each game has a
> mathematical point. I'm thinking about how to teach fractions with pie
fans but also for many others
take a look at the activities in the beginning where it explain
divisions and multiplications using lego bricks
--
roberto