Cardinality vs. Ordinality
kirby urner
Maria Droujkova
I visitied the SubQuan wiki page @ MF and have been followingthe thread. Glad to learn you count on your fingers Maria. Myco-teacher Glenn reminds me you can use a hand for base 12if you divide each finger into 3, I'm forgetting what the thumb isfor then. Base 15 anyone?
The thumb is for pointing. Another system based on division of fingers into areas is base 10 - it was used by the merchants to secretly show prices to one another without others in the marketplace overhearing. The merchants would hide hands under flaps of their coats, and read one another's finger position by touch. At least that's how the story went in one of my old books. I am attaching the picture.
A young child's version of cardinality and ordinality is the indexing of sequences, which becomes especially apparent in multiplicative sequences. Here is what I mean, for base two:
Cardinals (and their order in the sequence), additive sequence: 2(first), 4(second), 6(third), 8(fourth), 10(fifth)...
Cardinals (and their order in the sequence), multiplicative sequence: 2(first), 4(second), 8(third), 16(fourth), 32(fifth)...
For toddlers, the quantities are visual rather than symbolized, naturally. For example, I usually work with family trees, with 2 being parents, 4 grandparents, 8 great-grandparents... And the ordinal numbers reserved for generations. See the attached screenshot from an interactive.
Note that coordination between counting and "fractal units" (powers, splitting, subquaning) is necessary whether you start from counting or from fractal work.
Cheers,
MariaD
kirby urner
On Thu, Dec 30, 2010 at 2:34 PM, kirby urner <kirby...@gmail.com> wrote:I visitied the SubQuan wiki page @ MF and have been followingthe thread. Glad to learn you count on your fingers Maria. Myco-teacher Glenn reminds me you can use a hand for base 12if you divide each finger into 3, I'm forgetting what the thumb isfor then. Base 15 anyone?
The thumb is for pointing. Another system based on division of fingers into areas is base 10 - it was used by the merchants to secretly show prices to one another without others in the marketplace overhearing. The merchants would hide hands under flaps of their coats, and read one another's finger position by touch. At least that's how the story went in one of my old books. I am attaching the picture.
I've read that exact same story, I think in this History of Mathematics volume I have on my shelves, or is it History of Numbers?
As another aside, what we're taught as young Quakers some of the time, for example in "Sunday School" (some call it "First Day School") is that Quakers were among the first to introduce a "fixed pricing system" i.e. same rate for everyone, no matter your social station. This was somewhat revolutionary and helped galvanize the concept of a catalog with published fixed prices, not always re-negotiated depending on this or that, and on which job title etc. (half off for the King's cook etc.). They probably still made the distinction between wholesale and retail though. Some Quakers were more like Costco.
I'm sure such warehousing practices were not unique to any one religious sect. The point is you could hardly have modern Economics as a theory, with is Pareto-Optimum idealizations, without this "transparent price information" aspect. Once prices disappear, become part of a multi-dimensional set of linear equations (or even non-linear), you're kinda sunk as far as modeling goes. Or it least it becomes a lot more difficult.
Through Oregon Curriculum Network I've got this four-component approach to STEM with one sector labeled Supermarket Math. This is where we get more into the "money stuff" and the idea of inventory levels, shares, cooperatives, employee owned, other business forms. In Oregon, we might study the case history of the Tillamook company, a cooperative (agricultural collective) that's still active to this day. We might also study Burley, the bicycle cart manufacturer, and Nike, for their different business models.
You need graphic approaches, more like comic books. This is already a well established genre, but the content needs continual reworking. Economics is not the E we're interested in, as much as Engineering (in STEM).
A young child's version of cardinality and ordinality is the indexing of sequences, which becomes especially apparent in multiplicative sequences. Here is what I mean, for base two:
Cardinals (and their order in the sequence), additive sequence: 2(first), 4(second), 6(third), 8(fourth), 10(fifth)...
Cardinals (and their order in the sequence), multiplicative sequence: 2(first), 4(second), 8(third), 16(fourth), 32(fifth)...
Where I might start with Cardinality is with names of people or objects that single them out. This gets to something many children think about in school, which is how they may or may not have the "same name" as a classmate. Discussions of identity and the grammar surrounding it makes sense at all levels.
What I try to avoid is giving the sense that the only objects we care about are numbers. Naming the species, coming up with a taxonomy for animals and plants, is a way of distinguishing them. Before you can have "two or more of a type" you need to have types, but before you can have types, you really need to have multiple specimens.
Connecting a sense of cardinality to a sense of needing to categorize, pigeon-hole, identify, as a member of some set (e.g. venomous vs. non-venomous snake types, or shell fish types) then sets us up for ordinality e.g. counting "how many". If you haven't yet identified the different species or brands, then you won't be able to tally "how many" of each type.
Cardinality precedes Ordinality in this sense: you need to have sorting categories before you can tally "how many" of each (each what?).
For toddlers, the quantities are visual rather than symbolized, naturally. For example, I usually work with family trees, with 2 being parents, 4 grandparents, 8 great-grandparents... And the ordinal numbers reserved for generations. See the attached screenshot from an interactive.
Note that coordination between counting and "fractal units" (powers, splitting, subquaning) is necessary whether you start from counting or from fractal work.
In the namespace I'm designing here, cardinality and ordinality float freely from "numbers" quite easily. Fingerprints and retinal scans provide a kind of cardinality for identifying things, as to UPC codes (bar codes) at the supermarket. That latter are numbers of course, but the idea is "unique pattern" (for which numbers may be unnecessary -- how a face looks, its feature, is part of its "cardinality" in this sense (what distinguishes it from other faces)).
Kirby
--Cheers,
MariaD
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 mathfuture+...@googlegroups.com.
For more options, visit this group at http://groups.google.com/group/mathfuture?hl=en.
milo gardner
| Kirby, Your elevation of Quaker unit pricing may be out of place. The type of unit pricing that you ascribe to Quakers ("... were among the first to introduce a "fixed pricing system" i.e. same rate for everyone, no matter your social station" ) was is use 4,000 years ago in Egypt. Note the pesu, an inverse proportion unit, used to value a glass of beer, loaf of bread, and any product (domesticated geese, ducks, quail, dove ... that were fattened grain) for wage payments per: http://www.nytimes.com/2010/12/07/science/07first.html?_r=1&ref=science Happy New Year to all, Milo Gardner --- On Thu, 12/30/10, kirby urner <kirby...@gmail.com> wrote: |
Edward Cherlin
>
> Kirby,
>
> Your elevation of Quaker unit pricing may be out of place. The type of unit pricing that you ascribe to Quakers ("... were among the first to introduce a "fixed pricing system" i.e. same rate for everyone, no matter your social station" ) was is use 4,000 years ago in Egypt. Note the pesu, an inverse proportion unit, used to value a glass of beer, loaf of bread, and any product (domesticated geese, ducks, quail, dove ... that were fattened grain) for wage payments per:
>
> http://www.nytimes.com/2010/12/07/science/07first.html?_r=1&ref=science
The idea of prices for beer and bread determined by quantities of
ingredients is quite different in its effect from the idea of stating
your price for everything in the store, and not bargaining. Nobody
prices beer according to its nutritional value today, but the Quaker
influence is still with us, even though in considerable decline.
The Quaker revolution in business goes much deeper than set prices. It
includes everything from giving correct change, so that one could send
a child to a store with a list, to actually delivering the value
agreed on in a contract, even years afterward. This was a matter of
great importance to sailors, among others. (See the scene in Moby Dick
where a Quaker is signing up sailors and harpooneers for shares in a
two-year voyage.) The effect was so strong that to this day the gray
business suit modeled on the "suit of Quaker gray" is called a
"closing suit" by salesmen, because it was such an important symbol of
trust.
--
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
|
|
|
|
|
|
|
|
|
Mary O'Keeffe
kirby urner
kirby urner
To unsubscribe from this group, send email to mathfuture+...@googlegroups.com.
Edward Cherlin
> Quakers have this ethnic lore just like any group that needs to be cross-checked against reality.
If only Quakers told these stories, and it were not a matter of
historical scholarship, you would have a point.
> In this case you had a persecuted UK-based sect (1600s) that then became frequented by some elite figures in the UK, which turned tables, yet the reputation for honest and plain speech persisted and, with the backing of royalty, meant lots of businesses grew from Quaker practice, also some committee-based styles of government (especially when melded with NavAm brands (ideas lifted from Iroquois etc.)).
> Anyway, for awhile there, Quakers were enjoying living standards that would have shocked their oft-imprisoned plain-dressing ancestors, and you could see where there might be some guilt and soul searching to go along. A lot of the "invented fairness" mythology is self-serving in this sense, i.e. "yeah, we were doing well for ourselves, but it's because we invented quality customer service" (how often do we hear that I wonder -- every time someone reinvents a wheel, right?).
> Quakers lost a lot of their wealth and clout in the New World because they were not wanting to pay war taxes to fight the local population, having become friends with same. It only got worse with the underground railroad. Very few Quakers left by this point, not to be confused with the Amish, a different endangered species.
This is a very strange rant, Kirby, especially for one who calls for
fact-checking. What are your sources for these claims? Some of them
look as though you made them up yourself.
Why don't we stick to something we know about, in this case, math.
--
kirby urner
On Fri, Dec 31, 2010 at 15:03, kirby urner <kirby...@gmail.com> wrote:If only Quakers told these stories, and it were not a matter of
> Quakers have this ethnic lore just like any group that needs to be cross-checked against reality.
historical scholarship, you would have a point.
This is a very strange rant, Kirby, especially for one who calls for
> In this case you had a persecuted UK-based sect (1600s) that then became frequented by some elite figures in the UK, which turned tables, yet the reputation for honest and plain speech persisted and, with the backing of royalty, meant lots of businesses grew from Quaker practice, also some committee-based styles of government (especially when melded with NavAm brands (ideas lifted from Iroquois etc.)).
> Anyway, for awhile there, Quakers were enjoying living standards that would have shocked their oft-imprisoned plain-dressing ancestors, and you could see where there might be some guilt and soul searching to go along. A lot of the "invented fairness" mythology is self-serving in this sense, i.e. "yeah, we were doing well for ourselves, but it's because we invented quality customer service" (how often do we hear that I wonder -- every time someone reinvents a wheel, right?).
> Quakers lost a lot of their wealth and clout in the New World because they were not wanting to pay war taxes to fight the local population, having become friends with same. It only got worse with the underground railroad. Very few Quakers left by this point, not to be confused with the Amish, a different endangered species.
fact-checking. What are your sources for these claims? Some of them
look as though you made them up yourself.
Why don't we stick to something we know about, in this case, math.
Edward Cherlin
outside? Why are you talking about folklore needing to be checked, and
not checking it yourself, and making up unsupported conjectures?
I have a great affinity for Friends. The first, best book on Buddhism
I ever read, at age 12, was The Teachings of the Compassionate Buddha,
by Friend Edwin Burtt. It is still the best introduction to Buddhism
in English. Friends also do much of the best work on peace and other
global issues.
kirby urner
> Being a Friend is one thing. But do you know the Society from the
> outside? Why are you talking about folklore needing to be checked, and
> not checking it yourself, and making up unsupported conjectures?
>
But you haven't established that I'm making unchecked or unsupported
conjectures.
I gave you at least one of my sources, written by outsider Catholics. If
you clicked on the journal link, you'd have seen a picture of the source,
direct quotes and everything.
"""
Penn signed a treaty with the Indians and sought to establish a
just and peaceful commonwealth in Pennsylvania. The Quakers
dominated the political life of this colony, the wealthiest and
most populous in America, until 1756, when they refused to
vote a tax for war against the Shawnee and Delaware Indians
[sic]. Others, less concerned about fair treatment for the original
inhabitants, took over the reins of government. (pg. 23).
"""
I also substantiated my claim that Cadbury was an influence on fair
treatment of workers (a more utopian "company town" concept). George
Bernard Shaw was influenced by this example.
Here's from IMDB on that (other citations available):
"""
Shaw based the central theme on the life of John Cadbury, the founder
of the Cadbury chocolate business. Mr. Cadbury was a Quaker who
spent much of his life working for social reform.
"""
http://www.imdb.com/title/tt0033868/usercomments
As to whether Quakers invented "fixed pricing", I said that's some of the
lore that gets shared in Sunday school or whatever, and I'm in a position
to know that, regardless of whether it's true or not.
It's in the second paragraph of the Wikipedia article on "fixed pricing"
which is only proof of what I claimed: that's it's part of the story
Quakers tell themselves (no I didn't put this paragraph in):
"""
In the United Kingdom, "fixed price" has a similar meaning, and commonly
indicates that an external party (often the government) has set a price level,
which may not be varied by individual sellers of a good or service. As part
of their rule of honesty and plainness, Quakers set a fixed price for
their wares.
"""
So far, I'd say I'm batting 1000 and you're just being a curmudgeon.
> I have a great affinity for Friends. The first, best book on Buddhism
> I ever read, at age 12, was The Teachings of the Compassionate Buddha,
> by Friend Edwin Burtt. It is still the best introduction to Buddhism
> in English. Friends also do much of the best work on peace and other
> global issues.
>
I may well know more about Buddhism than you supposed as well,
having lived in the Tantric Buddhist Kingdom of Bhutan, where my
parents were working (not as missionaries). I've also studied it
at Princeton. My autobio is on-line.
Whether Burtt wrote the best intro is of course your personal
opinion.
>>> > Quakers have this ethnic lore just like any group that needs to be
>>> > cross-checked against reality.
>>> If only Quakers told these stories, and it were not a matter of
>>> historical scholarship, you would have a point.
They do tell these stories and they are a matter of historical scholarship.
>>> This is a very strange rant, Kirby, especially for one who calls for
>>> fact-checking. What are your sources for these claims? Some of them
>>> look as though you made them up yourself.
>>>
As for some Quakers feeling uneasy about their period of financial
success (Lloyds of London, Barclay's Bank...), that's a matter of
record as well.
You'll see some nostalgia for getting back into exercising financial
powers. It's not just about assuaging guilt for having done so well
for themselves.
http://nancysapology.blogspot.com/2006/08/quaker-intellectuals.html
http://www.hampshirequakers.org.uk/businesspolitics_ans.php
http://www.leveson.org.uk/stmarys/resources/cadbury0503.htm
http://www.westhillsfriends.org/QVWreputation.html
This "socially responsible investing" concept ("doing well by
doing good") has drawn many Qs into the micro-lending and
other models championed by some of the Islamic banks.
Qs and Islam have had pretty good relations over the years (our
former Western Friend editor practiced Ramadan, wrote about
these good relations).
Islamic intellectuals may find more of intrinsic interest in my
Coffee Shops Network business model, a topic in Casino Math.
These are casinos wired to serve nonprofits (NGOs) along
the "church bingo" model, with the added twist that you
may self promote (sounds like a sin) by chronicling your
exploits and commitments of winnings. Lots of Jungian
psychology here (capitalizing on our desire to be heroic).
Lots of math to discuss.
Kirby
Edward Cherlin
misunderstood, or that I am drawing implications from what you wrote
that you did not intend.
Or simply that we are in violent agreement.
> --
> 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 mathfuture+...@googlegroups.com.
> For more options, visit this group at http://groups.google.com/group/mathfuture?hl=en.
>
>
--
Linda Fahlberg-Stojanovska
I always considered counting to be addition since one counts and points at each object "one, two, three,..., six" and then says "There are six objects."
So it seems to me that you are saying "(I have) one, (and then plus one is ) two, (and then plus one is ) three, ...
To me counting as addition is the first process (function) we learn.
However, I have no background in education or working with small children.
Is this how one thinks formally of counting?
BTW: I tell my kiddies (students) here in Europe that if the Egyptians hadn't counted thumbs we would all have been perfectly happy with base 8 (the imperial system) and when they say the metric system is "better", I give them a piece of paper and say "divide this into tenths". Of course, the foot having 12", the yard having 3', etc. does not help my argument. :)
Maria Droujkova
Back to Cardinality ... I have a question.
I always considered counting to be addition since one counts and points at each object "one, two, three,..., six" and then says "There are six objects."
So it seems to me that you are saying "(I have) one, (and then plus one is ) two, (and then plus one is ) three, ...
To me counting as addition is the first process (function) we learn.
However, I have no background in education or working with small children.
Is this how one thinks formally of counting?
I like this explanation. I think you can view addition through counting or counting through addition. Usually both come at the same time, and yes, usually their combination is learned as the first math process. However, splitting (exponentiation) can be learned as the first math process, as well. Multiplication is a hybrid of the two, depending on their coordination, and probably should not be learned before them. On the basis of my observations I believe it's pretty important to learn counting and splitting together, as TWO "first things" that appear in the first year or two of each baby's life. Then multiplication can be learned on the basis of these two processes.
BTW: I tell my kiddies (students) here in Europe that if the Egyptians hadn't counted thumbs we would all have been perfectly happy with base 8 (the imperial system) and when they say the metric system is "better", I give them a piece of paper and say "divide this into tenths". Of course, the foot having 12", the yard having 3', etc. does not help my argument. :)
Origami axioms support folding into any number of division. Kids finding the simple challenge of folding paper into three equal parts pretty hard! Five is even worse. These are excellent playful tasks which I use often, as well. Mastering them moves thinking ahead quite a lot.
Cheers,
Maria Droujkova
Make math your own, to make your own math.
kirby urner
I'm not doing this to have an argument. It may be that I
misunderstood, or that I am drawing implications from what you wrote
that you did not intend.
Or simply that we are in violent agreement.
kirby urner
On Tue, Jan 4, 2011 at 12:45 PM, Linda Fahlberg-Stojanovska <lfah...@gmail.com> wrote:
Back to Cardinality ... I have a question.
I always considered counting to be addition since one counts and points at each object "one, two, three,..., six" and then says "There are six objects."
So it seems to me that you are saying "(I have) one, (and then plus one is ) two, (and then plus one is ) three, ...
To me counting as addition is the first process (function) we learn.
However, I have no background in education or working with small children.
Is this how one thinks formally of counting?
I think the notion of Cardinality vs Ordinality I'm advancing in the
context of the Oregon Curriculum Network writings (Wikieductor etc),
somewhat differs from the more established meaning in that I'm not
considering these solely and exclusively to be the properties of
"numbers". The meanings are close enough though, such that
I'd be hard pressed to find better words, so I'll go with these two.
Coming from a philosophy of mathematics background (Princeton,
PSF etc.), I'm looking for terms teachers might use with each other
as shared heuristics. This imagined (projected) faculty is pretty
much free to develop its own curriculum, importing raw material
from elsewhere, and it always helps to have shared maps. Maria
has been good at providing these, lots of simple yet meaningful
diagrams.
By Cardinality, I mean the property of being both a unique instance
or specimen of a thing, such as a human being with a name, birth
date (if you know it), particular parents (may not be known either)...,
and a member of a class or category (e.g. humans, mammals,,,
planets).
Mars, Earth, Jupiter, Mercury have cardinal identities irrespective
of how we choose to order them. By orbital period, distance from
sun, size... these things go together. But one could also rank for
specific elements or compounds, in which case Earth comes first
for H20, then Mars second (has water), Jupiter third maybe (on
one of the moons?).
The "Cardinal sense" is what tells a child that the cookie placed
behind the curtain, and then restored to view, is the same cookie.
It may not be of course (a stage magician's trick?). One may
be wrong about whether something is the same or different.
Maybe "Chuck" is an alias for this other guy, so where you
thought a conversation was about two different people, it
was actually about just one. The surprise of finding out what
you thought were two, is really one (happens a lot in detective
work) is at the core of the Cardinality concept.
And yes, Ordinality comes in right away, because of counting,
because of the temporal sequence. You have 10 individuals,
sortable by weight, volume, age, other attributes, but then
there's just the random order of which you happen to count
first, second, third... any sequence will do, there's still "the
order in which they were counted" (or "entered into the system"
why might say w/r to parts in inventory, patient, client or
membership roster).
The importance of "random" will be start to become clear,
right at the outset of the "ordinality" discussion, our shared
investigation into what this term means (or "guided discussion"
if you prefer).
Sometimes you assign random identifiers to individuals that
are nevertheless sortable. Having a sort is so necessary
to speedy storage and retrieval! That's why we have
alphabetizations, or, more generally "collations". What does
it mean to sort all titles "alphabetically" across multiple
languages?
Librarians have had to work on that problem.
The math I'm sharing does not shy away from this problem
as "non-mathematical" simply because titles are not number
strings. We have as much interest in unicode strings, other
encodings. They may be stored in binary format at the
end of the day, but the point is our math is more about
"alphanumeracy" than simply "numeracy". I have an article
in FoxPro Advisor about that ("our math" meaning "that of
the Silicon Forest" in this context):
http://www.flickr.com/photos/17157315@N00/5126606082/in/photostream/
And yet you're not trying to attach much "significance" to
the sort (i.e. the identifier is "random and meaningless" -- but
also unique (we hope)).
In some language games, it's quite important if you're at the
head of the pack by some criterion, like if you're one of the
first ever of such and such a dollar bill. But is there a way to
tell? Do postage stamps have serial numbers? As a rule they
do not. You can identify postage stamps by issue, but it gets
really hard to tell them apart at a more granular level.
We're straying back into cardinality again, just wondering
if two things might be the same or not.
Anyway, that's just a sense of what I'd like teachers to
internalize, when teaching for this particular play book
(maybe just for a day, not looking for slaves). It helps
when designing lesson plans to have these concepts in
the foreground.
How might we emphasize the difference between
distinguishing and sorting, between identifying and
ranking? That's Cardinality vs. Ordinality and it comes
up a lot in explaining "how things work" in our various
branches (Casino Math, Supermarket Math,
Neolithic Math and Martian Math are the four).
In Biblical terms, the Cardinality vs. Ordinality distinction
is a lot like the difference between naming (the creatures
in the garden) and judging them (knowing which are
naughty and nice). Cardinality came first. Ordinality
(judging, playing god) seemed sinful (an error of some
kind). You could segue to the sin of pride if you need to,
e.g. if teaching on some "morals" track (many schools still
have those).**
I like this explanation. I think you can view addition through counting or counting through addition. Usually both come at the same time, and yes, usually their combination is learned as the first math process. However, splitting (exponentiation) can be learned as the first math process, as well. Multiplication is a hybrid of the two, depending on their coordination, and probably should not be learned before them. On the basis of my observations I believe it's pretty important to learn counting and splitting together, as TWO "first things" that appear in the first year or two of each baby's life. Then multiplication can be learned on the basis of these two processes.
I do like this emphasis on splitting as relating to
multiplication. This helps with the biological
understanding of cellular mitosis and the progressive
doublings. It's important in STEM that we don't
completely sever the connection between the
mathematical and biological connotations of
"multiply".
BTW: I tell my kiddies (students) here in Europe that if the Egyptians hadn't counted thumbs we would all have been perfectly happy with base 8 (the imperial system) and when they say the metric system is "better", I give them a piece of paper and say "divide this into tenths". Of course, the foot having 12", the yard having 3', etc. does not help my argument. :)
Origami axioms support folding into any number of division. Kids finding the simple challenge of folding paper into three equal parts pretty hard! Five is even worse. These are excellent playful tasks which I use often, as well. Mastering them moves thinking ahead quite a lot.
I'm leaning towards the idea that having multiple mensuration
systems keeps us on our toes and everyone using metric
and only metric, 24/7, would represent a loss of biodiversity.
Yes, it's inconvenient, but "convenience" is not the only
variable in the tradeoff. Sometimes one accepts diversity
on other grounds.
At this point, there's simply "backward compatibility" to
think about. People still need those parts, with those
given measurements.
The Japanese language is a good example of a hybrid
set of standards co-mingling. No one is trying to
"disentangle" the Kanji from all the rest of it (or maybe
there is such an effort?). Lots on Romanji have come
in, and now the computer languages... So that's a lot
of biodiversity. Makes Japanese hard to learn though.
A tradeoff.
Why try to purge the world of "imperial units"? I don't
really see the need. Just keep a lot of conversion
constants going, like we have today. Spreadsheets
are great at this.
Kirby
** "morals track" on my mind recently because I've
been learning a little bit about a certain school in
Rangoon from awhile back (its still there), run as
a Methodist academy. You could take the straight
Christianity track, in which case you had so-and-so
for a teacher, or you could put a little more distance
between yourself and the Christians and take
"morals" instead (in which case you still got a lot
of the same Methodist stuff, but hey, it's a
Methodist school so whaddya expect?). My friend
was a "morals track" teacher.
Here's the school I mean:
http://en.wikipedia.org/wiki/Basic_Education_High_School_No._1_Dagon
(hmmmm, my Firefox is not displaying Burmese,
only these unicode mahjong tiles...).
Sue VanHattum
Warmly,
Sue
Cooper Macbeth
Working with Greeno and Droujkova on another thread and simply not having the time to clarify some concepts, I would like to attempt to see how succinct I can be. (hmmm, not a good start. 
)
1) Cardinality refers to the 'quantity' of a group. It is the answer of the question, "How many?" Quantity is the sticky word here. The best scientific explanation I have found is that neuroscience research has shown small quantities to be instinctual in birds and mammals. The term 'subitize' from the Latin for 'sudden' is used to indicate sudden recognition to the answer of "How many?" Though research shows infants as young as three weeks recognize what we later label as one, two, and three 'items' and even later with the symbols 1, 2 and 3, it is clear that it is not learned. Other research shows that with a practice, adults can subitize past ten and toddlers past twenty. I will discuss quantities greater than nine later.
2) Ordinality (Order-nality ) is a sequential label, or alphabetizing, of a group with quantity greater than one. It answers the questions: Which one was first (or any specific location in the sequence)? Which came next? Which came before? Which came after? Our Arabic numerals, as well as the our alphabet, can be used for ordering (see Peano axioms). The ordering sequence must be memorized, it is not instinctual; "F" following "E" must be taught. Herein lies an early source for dyscalculia. A student who does not make the connection between the ordinal sense of a number and the cardinal sense of a number might be looking for the answer of "What is 13-5?" the same way we would look for the answer to "What is M - E?" The answer "H" does not instantly come to mind does it? Now imagine "BD+GM?" or even better, "DF.RS x KW.ORJ?" Suffering yet?
3) Before I go to counting there is something else that scientific research has found but, for some odd reason, is never explicitly stated: birds and mammals have the ability to know when a subitized value is NOT correct. In other words, it is just as instinctual to know when a quantity is not anticipated. In fact this is exactly how research is conducted on infants. A small quantity of objects are moved behind a screen and then the screen is removed. If the quantity is greater than or less than the original quantity then the infant focuses on the ERROR longer than E does when the quantity is the same. Error detection is a critical aspect of number sense and life, maybe the most important! [FYI: E is the generic pronoun.]
4) Counting involves selecting an object from an uncounted group, assigning it to a number/alphabet (or other Peano like line), and then identifying it as belonging to the counted group (whether by sticky, moving it, whatever). Now the interesting part, if the objects are placed in such a way as to prevent subitizing then the ability to detect counting errors is removed. Unless one knows what the count should be, as in a deck of cards, or quantity of dimes in $2.80, there is no way to definitively count without error. It becomes probabilistic to count without error. Why is this? I will make two analogies to the ubiquitous computer. A major error in computer data is caused by cross-talk. This occurs when two electrical channels (think wires if that helps) are 'too close' to one another and the information on one crosses over to the other. Another error comes from noise, that is, too much electrical noise is near the channel and the data gets changed. [FYI: this is why we use parity bits/error-detection and correction bits in computer design.] In the mind, we have two channels that are active in most humans, a visual channel and an audio channel. Both these channels reference the vocabulary part of our brain. Most, but not all people, will have trouble counting if someone nearby is saying numbers too loud. This crosstalk causes errors that do not always result in 'counting over'. Noise occurs when vision is obstructed (items are too small, too cloudy), music might be playing (one unconsciously starts 'singing' the words to the song rather than the numbers, and so on.) Noise is a big problem for most counters. Another error that occurs when counting are mechanical errors. Not only a major problem with man-made devices but also for our own fingers. If you grew up with a deck of cards in the house you are familiar with recounting caused by sticky cards. Even a mechanical clicker used for counting may not be pushed hard enough to catch the ratchet mechanism.
5) Is counting the same as adding by one? It depends on whether subitizing is involved or not. Again, research on infants showed that when adding one item to items already behind a screen, if, when the screen was removed, the quantity did not 'add up' correctly, the infant noticed. They detected the error. If the subitizing ability is removed then errors can go undetected and adding by one is just like counting.
6) What about cardinal quantities greater than nine, greater than what we can subitize? This is where vector algebra and subQuan come into play. Vector algebra provides a mathematical explanation for extending 'quantity' notationally beyond nine (really beyond one) and subQuan provides the organization for subitizing and error detection to operate up to the resolution of the display media or human eye, whichever fails first. Basically, these two concepts enable the construction of a naming convention and visualizing ability of numbers much greater than nine and lay the foundation for other number systems and mathematical concepts. See http://bit.ly/g0afnn and http://dreamrealization.ning.com/profiles/blogs/four-steps-to-polynomial-1. I believe these two theories form the foundation for number sense!
--
kirby urner
Cooper Macbeth
Edward Cherlin
The clear distinction between cardinality and order type comes from Georg Cantor's papers and his book, Abstract Set Theory. The essence of cardinality in this theory is the ability to put two sets into one-to-one correspondence, creating equivalence classes of sets. The essence of ordinality is the trichotomy rule, that no matter which elements A and B we pick from the ordered set, one of A<B, A=B, A>B is true. Well-ordering requires the additional rule that there are no infinite descending sequences. It follows from this that well-ordered ordinal types form a well-ordered sequence.
Finite cardinals and ordinals match one-to-one, starting at 0. We count by matching items against the well-ordered set of the natural number ordinals. Mathematicians and computer scientists typically start counting at 0, and everybody else at 1. The two notions of number diverge at countable infinity, the cardinal aleph-null and the ordinal omega. Aleph-null + 1 = Aleph-null, but omega + 1 is the second countably infinite ordinal. There are uncountably many ordinals between omega and the first uncountable ordinal, but no cardinals between aleph-null and the first uncountable cardinal, whichever that is according to your views on the Continuum Hypothesis.
Working with Greeno and Droujkova on another thread and simply not having the time to clarify some concepts, I would like to attempt to see how succinct I can be. (hmmm, not a good start.
)
1) Cardinality refers to the 'quantity' of a group. It is the answer of the question, "How many?" Quantity is the sticky word here. The best scientific explanation I have found is that neuroscience research has shown small quantities to be instinctual in birds and mammals. The term 'subitize' from the Latin for 'sudden' is used to indicate sudden recognition to the answer of "How many?" Though research shows infants as young as three weeks recognize what we later label as one, two, and three 'items' and even later with the symbols 1, 2 and 3, it is clear that it is not learned. Other research shows that with a practice, adults can subitize past ten and toddlers past twenty. I will discuss quantities greater than nine later.
2) Ordinality (Order-nality
3) Before I go to counting there is something else that scientific research has found but, for some odd reason, is never explicitly stated: birds and mammals have the ability to know when a subitized value is NOT correct. In other words, it is just as instinctual to know when a quantity is not anticipated. In fact this is exactly how research is conducted on infants. A small quantity of objects are moved behind a screen and then the screen is removed. If the quantity is greater than or less than the original quantity then the infant focuses on the ERROR longer than E does when the quantity is the same. Error detection is a critical aspect of number sense and life, maybe the most important! [FYI: E is the generic pronoun.]
4) Counting involves selecting an object from an uncounted group, assigning it to a number/alphabet (or other Peano like line), and then identifying it as belonging to the counted group (whether by sticky, moving it, whatever). Now the interesting part, if the objects are placed in such a way as to prevent subitizing then the ability to detect counting errors is removed. Unless one knows what the count should be, as in a deck of cards, or quantity of dimes in $2.80, there is no way to definitively count without error. It becomes probabilistic to count without error. Why is this? I will make two analogies to the ubiquitous computer. A major error in computer data is caused by cross-talk. This occurs when two electrical channels (think wires if that helps) are 'too close' to one another and the information on one crosses over to the other. Another error comes from noise, that is, too much electrical noise is near the channel and the data gets changed. [FYI: this is why we use parity bits/error-detection and correction bits in computer design.] In the mind, we have two channels that are active in most humans, a visual channel and an audio channel. Both these channels reference the vocabulary part of our brain. Most, but not all people, will have trouble counting if someone nearby is saying numbers too loud. This crosstalk causes errors that do not always result in 'counting over'. Noise occurs when vision is obstructed (items are too small, too cloudy), music might be playing (one unconsciously starts 'singing' the words to the song rather than the numbers, and so on.) Noise is a big problem for most counters. Another error that occurs when counting are mechanical errors. Not only a major problem with man-made devices but also for our own fingers. If you grew up with a deck of cards in the house you are familiar with recounting caused by sticky cards. Even a mechanical clicker used for counting may not be pushed hard enough to catch the ratchet mechanism.
5) Is counting the same as adding by one? It depends on whether subitizing is involved or not. Again, research on infants showed that when adding one item to items already behind a screen, if, when the screen was removed, the quantity did not 'add up' correctly, the infant noticed. They detected the error. If the subitizing ability is removed then errors can go undetected and adding by one is just like counting.
6) What about cardinal quantities greater than nine, greater than what we can subitize? This is where vector algebra and subQuan come into play. Vector algebra provides a mathematical explanation for extending 'quantity' notationally beyond nine (really beyond one) and subQuan provides the organization for subitizing and error detection to operate up to the resolution of the display media or human eye, whichever fails first. Basically, these two concepts enable the construction of a naming convention and visualizing ability of numbers much greater than nine and lay the foundation for other number systems and mathematical concepts. See http://bit.ly/g0afnn and http://dreamrealization.ning.com/profiles/blogs/four-steps-to-polynomial-1. I believe these two theories form the foundation for number sense!
On Tue, Jan 4, 2011 at 1:21 PM, kirby urner <kirby...@gmail.com> wrote:On Tue, Jan 4, 2011 at 9:58 AM, Maria Droujkova <drou...@gmail.com> wrote:
On Tue, Jan 4, 2011 at 12:45 PM, Linda Fahlberg-Stojanovska <lfah...@gmail.com> wrote:
Back to Cardinality ... I have a question.
I always considered counting to be addition since one counts and points at each object "one, two, three,..., six" and then says "There are six objects."
So it seems to me that you are saying "(I have) one, (and then plus one is ) two, (and then plus one is ) three, ...
To me counting as addition is the first process (function) we learn.
However, I have no background in education or working with small children.
Is this how one thinks formally of counting?
Very formally, as in Peano arithmetic and Gödel recursive functions, yes.
I think the notion of Cardinality vs Ordinality I'm advancing in the
context of the Oregon Curriculum Network writings (Wikieductor etc),
somewhat differs from the more established meaning in that I'm not
considering these solely and exclusively to be the properties of
"numbers". The meanings are close enough though, such that
I'd be hard pressed to find better words, so I'll go with these two.
Coming from a philosophy of mathematics background (Princeton,
PSF etc.),
Yale in my case. Double major, Math and Philosophy.
I'm looking for terms teachers might use with each other
as shared heuristics. This imagined (projected) faculty is pretty
much free to develop its own curriculum, importing raw material
from elsewhere, and it always helps to have shared maps. Maria
has been good at providing these, lots of simple yet meaningful
diagrams.
By Cardinality, I mean the property of being both a unique instance
Now you are in Humpty-Dumpty territory. Yes, you can define terms to mean whatever you wish in your own context, but nobody else is bound to recognize them. The conventional mathematical name for your property is Identity.
or specimen of a thing, such as a human being with a name, birth
date (if you know it), particular parents (may not be known either)...,
and a member of a class or category (e.g. humans, mammals,,,
planets).
Mars, Earth, Jupiter, Mercury have cardinal identities
Cardinal identity is redundant. These are just identities.
irrespective
of how we choose to order them. By orbital period, distance from
sun, size... these things go together. But one could also rank for
specific elements or compounds, in which case Earth comes first
for H20, then Mars second (has water), Jupiter third maybe (on
one of the moons?).
The "Cardinal sense" is what tells a child that the cookie placed
behind the curtain, and then restored to view, is the same cookie.
It may not be of course (a stage magician's trick?). One may
be wrong about whether something is the same or different.
Maybe "Chuck" is an alias for this other guy, so where you
thought a conversation was about two different people, it
was actually about just one. The surprise of finding out what
you thought were two, is really one (happens a lot in detective
work) is at the core of the Cardinality concept.
And yes, Ordinality comes in right away, because of counting,
because of the temporal sequence. You have 10 individuals,
sortable by weight, volume, age, other attributes, but then
there's just the random order of which you happen to count
first, second, third... any sequence will do, there's still "the
order in which they were counted" (or "entered into the system"
why might say w/r to parts in inventory, patient, client or
membership roster).
Now you are approaching the boundary of combinatorics.
The importance of "random" will be start to become clear,
right at the outset of the "ordinality" discussion, our shared
investigation into what this term means (or "guided discussion"
if you prefer).
Sometimes you assign random identifiers
Not actually random, in common practice. The essential property of database key identifiers is uniqueness. It is quite commonly achieved by assigning identifiers in numeric sequence.
to individuals that
are nevertheless sortable. Having a sort is so necessary
to speedy storage and retrieval! That's why we have
alphabetizations, or, more generally "collations". What does
it mean to sort all titles "alphabetically" across multiple
languages?
Librarians have had to work on that problem.
Research Library Group joined the Unicode Consortium for just that reason. Knuth sets a modest version of this (ASCII-only) as a fairly hard problem in Volume 1 of The Art of Computer Programming. For example numbers may have to be treated as if spelled out in the source language of the document, or at least of its title.
Judging is nothing like ordinality. The relative concept is a set and its complement.
I can argue at great length, via alternate paths of reductio ad absurdum, why attempting to discuss monism and duality as a duality is a fundamental category error, and worse.
I like this explanation. I think you can view addition through counting or counting through addition.
There are a number of other methods. The farthest out there at the moment is Conway's Numbers and Games theory, which derives both in a transfinite sequential process that goes on as long as possible according to the particular axioms of your preferred set theory, that is, typically, until you have passed beyond all members of the set of Ordinals to the Class at the very end of the mathematical universe. It results in the real numbers, a vast class of infinitesimal numbers, and an even larger class of games, each of which is infinitesimally close to some ordinary number.
Usually both come at the same time, and yes, usually their combination is learned as the first math process. However, splitting (exponentiation) can be learned as the first math process, as well. Multiplication is a hybrid of the two, depending on their coordination, and probably should not be learned before them. On the basis of my observations I believe it's pretty important to learn counting and splitting together, as TWO "first things" that appear in the first year or two of each baby's life. Then multiplication can be learned on the basis of these two processes.
I tend to follow Gattegno, Montessori, and Piaget in these matters.
I do like this emphasis on splitting as relating to
multiplication. This helps with the biological
understanding of cellular mitosis and the progressive
doublings. It's important in STEM that we don't
completely sever the connection between the
mathematical and biological connotations of
"multiply".
BTW: I tell my kiddies (students) here in Europe that if the Egyptians hadn't counted thumbs we would all have been perfectly happy with base 8 (the imperial system) and when they say the metric system is "better", I give them a piece of paper and say "divide this into tenths". Of course, the foot having 12", the yard having 3', etc. does not help my argument. :)
Origami axioms support folding into any number of division. Kids finding the simple challenge of folding paper into three equal parts pretty hard! Five is even worse. These are excellent playful tasks which I use often, as well. Mastering them moves thinking ahead quite a lot.
I'm leaning towards the idea that having multiple mensuration
systems keeps us on our toes and everyone using metric
and only metric, 24/7, would represent a loss of biodiversity.
We have calculators now that can work in picolight-years per fortnight or whatever else might strike your fancy. I for one don't miss British Lsd money outside of historical novels.
Edward Cherlin
i. 5
0 1 2 3 4
# i. 5
5
#'01234'
5
Similar considerations apply to nested or boxed arrays in various APL-like languages, where elements can be of different type and shape.
LISP makes everything a tree, a list of lists. Enhanced APLs use forests, multi-dimensional arrays of arrays. Both are Turing-complete, and the choice is a matter of taste or convenience.
I believe expanding cardinality to include anything beyond quantity will introduce problems in other fields. Perhaps a viewpoint from programming will clarify my refusal to adopt such a definition as you propose. [That's ok, we can disagree, we'll just have to remediate those students that cross our programs. :-) ].
One of the most important concepts in mathematics, computer science, law, and many other fields is contextual meaning. This is closely related to the idea of bound variables and function definitions that have a particular significance only within the region where the binding or the definition applies. It leads to the question of namespaces, as well.
Learning to contextualize is not remediation, but a central part of understanding anything.
kirby urner
kirby urner
On Fri, Jan 7, 2011 at 12:28, Cooper Macbeth <cooper...@gmail.com> wrote:
I believe expanding cardinality to include anything beyond quantity will introduce problems in other fields. Perhaps a viewpoint from programming will clarify my refusal to adopt such a definition as you propose. [That's ok, we can disagree, we'll just have to remediate those students that cross our programs. :-) ].
One of the most important concepts in mathematics, computer science, law, and many other fields is contextual meaning. This is closely related to the idea of bound variables and function definitions that have a particular significance only within the region where the binding or the definition applies. It leads to the question of namespaces, as well.
Learning to contextualize is not remediation, but a central part of understanding anything.
kirby urner
kirby urner
Just to add, since the discussion has turned to syntax and notation (mathematics is also about other stuff):My students learn to distinguish data structures based on paired ( ) " " [ ] and { }, which are pretty obvious delimiters in Latin-1, like you'd be crazy not to use them (which could mean "good crazy" like in J).So when you eyeball a math script, objects jumping around, and the teach asked "how many lists do you have", the student immediately knows not to count tuples or dicts or whatever types (call them "ducks").That's exercising a student's cardinal sense: what type of thing is this? Is it a list? How many lists do we have? That's close to the original meaning.Ranking the lists, by size, time created (birthday), language used... these are what we call "collations" in library science, and that's where we get a lot of our ordinality sense. Librarians need to store and retrieve quickly, a non-trivial problem. Many great minds have gone into it, John Dewey especially. We should look at DNS and URLs as continuing in the same tradition, i.e. cyberspace = digital library (how my students might see it, yours too right?).The URL (or URI) is another fantastic example of a "thing" (noun) i.e. a file or resource of some kind. Does it make sense to order all URLs top to bottom? Probably not.Kirby
Cooper Macbeth
"... 'Warriors of the Net' ... is quite about having a peaceful world, where ideas are shared freely and people are not chastised for sharing with one another. "Kirby
"Learning to contextualize is not remediation, but a central part of understanding anything."
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
kirby urner
As the thread's initiator under this 'Cardinality vs. Ordinality' thread, I do feel some obligation to at least sweep the namespace, some guy with a broom.
Ed put his finger on "namespace" as a really important concept (relates to "context").
So much "conversation" in this world is just people talking past one another, thinking they know the meanings of their interlocutors, but really just projecting their own meanings. There's this assumption that we know what another is saying just because we ourselves have meanings for another's words as well.
Partly where math gets some of its mystique is people get tired of the "mushiness" of "ordinary language" and want to retreat to something more "rock solid", which is where logic enters, and a sometimes superstitious belief in the power of symbols (any number of arcane notations; each a namespace).
However, a deeper understanding of mathematics comes with the awareness that "namespaces" grow here is well, which is why I somewhat prefer the English tendency to "read maths" (in the plural) versus "math" (singular).
In the philosophy of mathematics you have this writer Ludwig Wittgenstein, a name often batted about in some circles. He came up with the idea of "language games" to help with his investigations, not only of "ordinary language" (where he grounded his later philosophy) but in the philosophy of mathematics. There's a somewhat intimate relationship twixt "language games" and "namespaces" if one likes (I like).
Although difficult to find in bookstores, I highly recommend this graphic novel (historical fiction) centered around Bertrand Russell's life in which WItt. also features, along with other thinkers of that day -- you get back to time lines, biographies, the stuff of history, and this is important. The novel is called Logicomix and originates from Athens, Greece. Here's a blog link showing the cover:
http://coffeeshopsnet.blogspot.com/2010/05/buzz-about-shops.html (might wanna skip the text, full of local variables)
Divorcing mathematics from its anthropological context is one of the mistakes teachers make (in my view -- others like math precisely because it seems, to them, to transcend ethnicity).
However, in getting back into anthropology, we're empowered to take a more mathematical approach.
For example this idea of namespaces has a logical implementation in computer languages (machine logics) and so students will maybe have that as background when looking at the various:
* belief systems
* world views
* frames of reference
* paradigms
* ontologies
in which the maths figure.
Thank you for sharing some autobio as well. I find that helpful and useful, to a get a sense of your angle.
You also focus on brain anatomy and physiology and look for links between neuroscience and learning / teaching effectiveness. I find this a buzzing subculture among many I rub elbows with, a vibrant and active meme pool. My daughter is quite interested in neuro-everything as well and keeps bringing relevant literature to my attention in this domain.
When it comes to namespaces, I've invested a lot of time and energy in deciphering some pretty obscure and difficult namespaces, some of which contain original and useful teachings that should not go to waste, should not be lost to history. I see Milo as engaging in similar activities, although we have only partially overlapping areas of focus (he's the Egyptian Math expert).
In order to preserve that which appears to me to have value (quite a bit going with Polyhedra), I do a lot of my own curriculum writing and designing, making use of photography, web pages, animations, computer programs, physical models, other media.
Opportunities to share this material with other teachers, and/or to test curriculum segments in the field, are often of great value to me. However, for others to get much value from what I do, I need to keep my meanings clear and definite enough to satisfy that craving for certainty people have, when exploring logic.
Kirby
--
Edward Cherlin
I will do what I can, and I can recruit higher-level mathematicians than myself.
> Not my glory but Yours...Cooper Macbeth
>
> --
> 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
> mathfuture+...@googlegroups.com.
> For more options, visit this group at
> http://groups.google.com/group/mathfuture?hl=en.
>
--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Edward Cherlin
belief, necessary truth, permissions, obligations, and others. It's
even more of a zoo than the usual varieties of logic and set theory.
One of the most fundamental theorems in the usual realm is that truth
cannot be defined in a consistent and useful mathematics. Otherwise we
could use Gödel's construction to create a sentence claiming "This
statement is neither true nor false."
I maintain, but others will argue against, the proposition that you
might not believe something, at the same time that you believe that
you believe it. We get a lot of that in religion and politics, and
occasionally in science and math.
Thanks for the link to Logicomix. It's going on my wish list ASAP.
--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Edward Cherlin
Covering a span of sixty years, the graphic novel Logicomix was
inspired by the epic story of the quest for the Foundations of
Mathematics.
Maria Droujkova
Do you know how to fold paper into three or five equal parts? If you ever figured it out (and if not, try it before reading further)...
The brain tends to think in twos, because of body symmetry and some built-in mechanisms. Many sacred ideas deal with three, because three is "many" and it's complex. Asking kids to draw radial symmetry with three parts, fold a strip or leaf of paper into three, or separate an interval into three parts by marking is a hard task for many. When they finish, they gain new understanding on symmetry and equality and balance.
It sounds a bit esoteric, because I don't quite know what this exercise does, or why. I offer it together with certain topics, such as non-linearity or multiplication or Euclid's constructions, and it moves thinking ahead. I have a small but growing collections of such exercises that somehow "hack" understanding. For example, turning upside-down (I hold little kids upside-down if they can't stand on their head or do hand-stands) is wonderful for when we work with inverses of any sort, e.g. inverse operations or functions. Doing "live mirror" games (when we mirror one another) is great for symmetry and commutativity.
Cheers,
Maria Droujkova
Make math your own, to make your own math.
kirby urner
> I took a course in Modal Logic in college, covering concepts such as
> belief, necessary truth, permissions, obligations, and others. It's
> even more of a zoo than the usual varieties of logic and set theory.
> One of the most fundamental theorems in the usual realm is that truth
> cannot be defined in a consistent and useful mathematics. Otherwise we
> could use Gödel's construction to create a sentence claiming "This
> statement is neither true nor false."
>
I've grown suspicious of analytic philosophy, as have many
in my generation.
> I maintain, but others will argue against, the proposition that you
> might not believe something, at the same time that you believe that
> you believe it. We get a lot of that in religion and politics, and
> occasionally in science and math.
>
I like the bumper sticker: don't believe everything you think.
Science is subject to meme viruses (both positive and negative)
as is religion and politics, if the history of the last century is to be
taken seriously.
Many white coated experts in their scientific fields, with top degrees,
embarked on projects .
I've been reading lots of Edwin Black stuff recently. My daughter
is also reading 'State of Fear' which is especially importance for
its appendix.
One needs to include the anthropology when studying math
and science. Where is the funding coming from and what
findings would benefit whom? This needs to be a part of the
analysis *especially* if there's a goal to stay "objective".
> Thanks for the link to Logicomix. It's going on my wish list ASAP.
>
Russell comes off as quite tormented. The role of philosophy
and logic as a counter to madness, both individual and cultural,
is a core theme of this book. Wittgenstein comes off as a bit
of a nut and his second philosophy is not discussed (it's not
really about him).
Kirby
Sue VanHattum
> For example, turning upside-down (I hold little kids upside-down if they can't stand on their head or do hand-stands) is wonderful for when we work with inverses of any sort, e.g. inverse operations or functions. Doing "live mirror" games (when we mirror one another) is great for symmetry and commutativity.
Warmly,
Sue
Maria Droujkova
When I try to fold a piece of paper in 3, I'm bringing one end over, and visually trying to get that two-layered side equal to the other flat side. I never get it just right. I tried 5 just now, and figured I'd fold both ends toward the middle for 2 two-layered parts and a flat part in the middle. If all 3 of those were equal, I'd have 5. But that didn't come close.
I overlap two end parts over the middle part. It's kinesthetic plus visual: as the paper wiggles around, it looks and feels right when the fold is into three equal parts. It takes a while the first time, sometimes as long as five minutes for a kid, but then you can get pretty exact with it.
Here is a very formal way to construct a three-fold based on origami axioms. It's a part of a demonstration why origami axioms are more powerful than Euclid's, namely, "doubling the cube" problem. The intersection of the diagonal and the fold to mid-point makes a third. There is an algebraic proof there.
http://plus.maths.org/content/power-origami-1
Alexander Bogomolny
--
Bradford Hansen-Smith
|
Sue & Maria, folding a piece of paper (assuming you are
using a quadrilateral of some kind) in 3 and 5 equal parts is not difficult
once you have folded the circle. The symmetries of 3, 4, 5, are fundamental
divisions expressed fully as 3-6, 4-8, and 5-10 symmetries. You can see they
are derivatives of the ratio 1:2, the first fold in the circle. 3, 4, 5 are the only three ways a circle can be folded in a ratio of 1:2. Fold the circle three times 1:2 and get three diameters;
fold circle four times in ratio 1:2 and get four diameters; and fold 5 times
1:2 and get five diameters. Each division will fold the circle into 3, 4, 5 diameters
of 6, 8, 10 equal divisions. The radial measure is the same to different
proportional divisions of the circumference; each consistent to the first fold
of 1:2. Understanding 1:2 proportional divisions make it easy to fold any piece of paper in to 3, 4, and 5 equal divisions; you are folding the ratios of 1:2 each time to the number of times to each symmetry. This is not difficult for 5 and 6 year old students to do once you have done it with the circle first. It is difficult to approach as an isolated problem. Accuracy in folding is to see the 1:2 proportion and when all the folds are made, you simply look to see if all edges are in alignment, if not make adjustments and then crease. This is not so much about thinking as using your eyes. Our eyes are made to see proportionally; we take that away from children when we teach them to measure before giving them proportional experience. I never use this as a separate activity; it is always embedded in the larger exploration of the circle. There is no formula to learn, it is simply how the circle works, thus applicable to all other shapes. Brad Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ facebook.com/wholemovement --- On Sat, 1/8/11, Sue VanHattum <suevan...@hotmail.com> wrote: |
|
Maria Droujkova
Sue & Maria, folding a piece of paper (assuming you are using a quadrilateral of some kind) in 3 and 5 equal parts is not difficult once you have folded the circle. The symmetries of 3, 4, 5, are fundamental divisions expressed fully as 3-6, 4-8, and 5-10 symmetries. You can see they are derivatives of the ratio 1:2, the first fold in the circle. 3, 4, 5 are the only three ways a circle can be folded in a ratio of 1:2. Fold the circle three times 1:2 and get three diameters; fold circle four times in ratio 1:2 and get four diameters; and fold 5 times 1:2 and get five diameters.
I am not getting these results. When I fold circle once, I get 1 diameter and 2 parts. When I fold it again, I get 2 diameters and 4 parts. Folding it again (three times altogether) yields 4 diameters and 8 parts, and so on. Can you show a picture of where 3 and 5 come from?
Linda Fahlberg-Stojanovska
Hiya
When you say into 3 –does that mean we folding to get 1/3 of a circle – like cutting a pie into thirds?
Linda
--
Bradford Hansen-Smith
| Maria and Linda, in regard to 3 and 5 folding you are approaching it from a traditional mind
set about 1, 2, 3, 4, 5, ... and by cutting circles apart. When
folding quadrilaterals you are limited to only that symmetry and can fold to only that many sides. See attachment to see correlation of folding the circle and folding a polygon. This is not about measurement as much as proportional relationships. |
Brad Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ facebook.com/wholemovement |
| --- On Sun, 1/9/11, Maria Droujkova <drou...@gmail.com> wrote: |
Maria Droujkova
"You can see they are derivatives of the ratio 1:2" - in what sense? Is it because you need to fold in half first?
"3, 4, 5 are the only three ways a circle can be folded in a ratio of 1:2" - what does it mean? I can fold into 7 in a very similar manner.
Juan
My name is Juan. I am a math tutor. The paragraphs quoted below from a
post of yours strongly drew my attention. It makes sense to me that
animal brains should have specific regions devoted to dealing with
cardinality. Numbers are all around us in nature. I believe it is
important for an animal to survive in the wild, to be able to
recognize, process, and respond to different cardinalities with
different actions. Abstract numbers (cardinality, cardinalities, or
"numerosities") show up in specific, concrete, real-time, real-world,
practical instances all the time, all over the place.
I remember one time a student asked me, with deep curiosity: "Why are
there so many numbers? Where are they?"
I answered: "They are everywhere. How many tables are there in this
room? How many people? How many windows? How many days ago was your
last vacation? How many cars are there in the city?"
This was a good enough answer for the student. She said: "Right, OK,
yes, I can see that. It makes sense."
Recognizing, handling, and reacting to cardinality in everyday life
situations does not necessarily require, or produce, the ability to
philosophize about cardinality, or to define it in abstract terms.
In my own, personal, individual namespace (to use Kirby's
terminology), the simplest definition of cardinality comes about when
you are willing to let go of logical consistency.
Cardinality is one of those things, concepts, or ideas, that our mind
creates out of the constant stream of input data from the outside
world, by noticing that any two given objects are both equal and
different at the same time. This is the very reason you can count the
items in a collection, because they are equal and different at the
same time. You include them in the count because they are equal,
otherwise you wouldn't count them together. You do not count windows
when you are counting chairs. Two chairs are equal because they are
chairs, so you include them both in the chair collection. However,
they are different, that's why you assign one number to one chair and
a different number to another chair. You count: 1, 2, 3, and so on.
You don't go around pointing your finger to all different chairs while
repeating: "1, 1, 1, 1, ..."
So, again, in short, you count them because they are all "equal" but
you also count them because they are all "different". Here, to
maintain sanity and common sense, I assume both words "equal" and
"different" mean so just "to some extent." They are relative terms,
relative to context, and to what you are doing in the moment, and to
your particular goals when considering the objects your brain is
counting in the first place.
In my view it does not get any simpler than this, and simplicity is
valuable enough for me to get comfortable with the seeming
contradiction implied in saying that objects are equal and different
at the same time. The devil is in the details. When trying to rule out
contradiction, to make room for some logical consistency, we
inevitably get into hair-splitting, philosophical arguments, and then
the complexity of our increasingly nuanced terminology (namespaces
galore) runs amok all the way up to infinity and its infinite
varieties of more or less infinite infinitude.
So, to keep it simple, I sum up my definition of cardinality as: The
multiplicity of unity, and/or the unity of multiplicity, as necessary
or convenient.
I hope this particular viewpoint may be of some interest to you.
Juan Castaneda
On Jan 8, 5:32 am, Cooper Macbeth <coopermacb...@gmail.com> wrote:
> A specific area, the inferior parietal
> region, seems designed just for cardinality. It is predominantly this
> finding that I am working with regarding subQuanning. As such, I am
> constantly seeking a definition of cardinality that is general enough that
> all can understand, that does not include any additions or interpretations,
> and that does not interfere with any concepts built upon it as there seems
> to be a one-to-one correlation between cardinality and a specific part of
> the brain. I understand cardinality is fodder for mathematicians but I do
> not believe it is their domain of expertise, no insult meant in anyway.
> Cardinality is appearing to be a fundamental instinct in all birds and
> mammals and am not sure to which domain 'instincts' belong. As a cognitive
> engineer, I merely try to correlate my understanding of computer hardware,
> microcode, interfaces, subsystems, systems, and languages to recent and
> current developments in neuroscience looking for nuggets where 'hardware'
> meets 'software'.
>
> currently my professional belief that this instinct to recognize the oneness
> in things is the foundation for building context. As our internal map of
> reality builds, based predominantly on vision and hearing, our ability to
> discriminate within the 'ones' lays the foundation for our adjectives and
> adverbs. This foundation enables us to move from just seeing 'things' (ones)
> to seeing them in there similarities, in their uniqueness, and in their
> togetherness. These 'things' become a limitless supply of input upon which
> to build our understanding of the world and others like us.
>
Bradford Hansen-Smith
|
I am having trouble with this conversation about folding. First, it is getting mixed up with "Cardinality vs Ordinality" . Second, all the suggested video and demonstrations about folding are referring to squares and rectangles and are used to formulate some proof or establish a math concept. Lacking the math and origami background many of you have I am at a conceptual disadvantage, just as all of you are at a disadvantage of not having spent as many years folding circles as you have drawing pictures of them.
John, I think we are back to our original conflict from years ago about the definition of a circle. Your uncomfortable position is yours not mine. I do not construct by folding the circle, (only when joining them is there construction) nor is there intention to prove mathematical bases for folding. Information I get from the circle and the first fold forward generates information to know what options will follow. This is not “guesswork” You say; “He may be showing ideas but I can see no way to achieving them in practically,” that may be the limitation of the mathematical system we have accepted and our inability to see beyond it.
“The question arises about education as to whether this should be taught without mathematical rigor, if necessary pointing out the error or approximation. I think to do so is to put doubt into a student's mind for the future.”
What doubt does folding circles bring to mind about the future when it is suggesting there is more to gain from folding circles than drawing images of them? What math is not filled with approximations?
Linda, You have started out with a goal of to do to the circle (make the largest triangle possible in the circle) which leads to construction using what we know limiting our observations. This is not wrong, just limiting the comprehensive nature of the circle and all the other information inherent in a principled process that comes from the circle itself. Yes there is a lot of information in folding circles. If you are really interested I have just published a book “OneFoldCircle” that discusses over 120 math concepts and relationships in the first fold in half. These are only my observations and I am not a mathematician.
Yes Maria, before anything can happen there must be the alignment of folding the circle in half; 1:2, then 3:6, 4:8, 5:10. The 3, 4, 5 refer to number of diameters. Two diameters are inherent in the first fold by choosing the two most opposite points and touching them together making the square a special case quadrilateral. Any two points less than opposite will form a kit relationship of four inscribed points with one diameter. This is a dynamic process of folding, not a linear counting process. I have stopped short and missed something if you can fold seven equal divisions in the circle using the 1:2 ratio of development. I am interested to see how you have done this. |
|
Brad Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ facebook.com/wholemovement --- On Sun, 1/9/11, Maria Droujkova <drou...@gmail.com> wrote: |
|
|
Linda Fahlberg-Stojanovska
Hiya Bradford,
Each of us is looking at things from our perspectives and our needs. But that is what I love about this mailing group. I would never have thought folding anything was interesting until you started these discussions. It doesn’t matter that I adapt them – it matters that the ideas are shared and each of us learns to broaden our thinking. And cardinality and ordinality – I took from that discussion that even at an early age we need to think not just about counting, but also about infinity (and various infinities).
kirby urner
<< snip >>
This was well put Juan, very clear, thank you.
Counting is one of the standard and important things we do
when faced with a collection.
Sometimes that uniqueness, that makes all dogs different,
requires fetching of that one individual dog from the set.
This is where ordinality comes in, or order. Do the dogs
have identification tags? How do these help? If you have
just a few horses in stalls, you simply remember which
horse in where. How about sheep. If it's a large flock,
do shepherds actually know one sheep from another in
every case? Or just the gross numbers?
Each box of cereal on the supermarket shelf has a UPC
code (bar code) which some lookup table relates to
price and to inventory. Whenever a box goes out the
door through a checkout lane, the money comes in
and the inventory gets decremented (time to re-order?).
Does each box of cereal have a serial number (OK a
little bit of a pun)? Can one box be distinguished
from another? In some products, yes. But once
you open the box, the individual cornflakes will not
be tagged....
And so on. What I like about this discussion is you
may easily involve young minds in the conversation
because of the wealth of experiences. Lets think
about how we know to put these things together (like
on the same shelf). Now lets think about how we
tell them apart.
If someone came to you and said "get me this book
from the library in a hurry" how would you do that?
How do you find things? Lots of math here.
We're talking about the basic concepts of sorting
and counting, recognizing and differentiating, and
organizing.
This is all mathematical content in a way, and quite
apropos to discussion to come later, about databases,
about maps, graphs, partial ordering, networks...
data structures.
In the New Math of my childhood, the only data
structure of apparent keen interest was "the set".
Given decades of machine logic, I think a more
heterogenous assortment of "data structures"
makes sense, including in the early years.
Now that we have sophisticated computer languages
that are free of charge, and rather inexpensive
equipment, it makes sense to interact with these
data structures live. Interpreters that show up in a
"chat window" and permit interactive interrogation
have a lot of promise, though here we're probably
talking about keyboard skills, so maybe we're
several turns in the spiral, working with teens.
When studying maths, we learn how to "index" into
an array. Why not teach this at the same time as
we "index" into a book using the index in the back.
These aren't identical concepts, but they're close
enough to study in tandem. Then we have paired
associations, where you don't go in with a sequential
index (as with an array or list) but with keys, unique
identifiers.
For those using Latin-1, or any alphabet with a known
collation that young kids are expected to learn, the
examples are ready at hand. Is "R" > "M"? We may
give that meaning, and say "yes, because R is
further along towards 'Z' so 'Z' is the greatest of
all" -- language games of that nature.
Pretty soon, we're talking about phone numbers,
Facebook ID numbers, account numbers of all kinds.
The real world applications of grouping and sorting,
storage and retrieval, are endless. Building up the
nomenclature, including the concept of "data structure"
(such as an array or list, or a set), is not a wasted
investment.
Eventually, we'll analyze documents for their
structure. A book, with a table of contents and index,
is likewise a data structure with a recognized format.
We learn how to consult reference materials. There's
library science in this picture, as well as computer
science. Pretty soon, we're looking at "the document
object model" (DOM) and poking around in web pages,
using Javascript. Welcome to the digital math track.
Etc.
Thanks again,
Kirby