Re: Do you design games, interactives or videos about multiplication?
m...@ms.lt
I really love your poster of multiplication models. As you know, I mapped
out mathematical ways of thinking:
http://www.selflearners.net/ways/?d=Math
I'm fleshing out with examples from Paul Zeitz's book. But I also want to
illustrate them with examples from the most basic mathematics.
Your poster makes clear, as I've learned from my tutoring, that
multiplication (and related counting, adding) is a very sophisticated
activity that can be approached in many different ways.
I share below a preliminary sorting of those ways that might correspond
with part of my "house of knowledge". In particular, there is a part that
explains how two things (systems, sets, viewpoints) can be related and the
geometry (visualization, restructuring) that arises from such a relation.
I think that the models of multiplication that you're documenting
illustrate various of the Zermelo-Frankel axioms of set theory.
For example, there is an axiom that, given several sets, we can consider
their union. That's the kind of thinking that allows us to count up items
that have been grouped.
There is another axiom that says that if two sets A and B have the same
elements, then they are the same set. That's the kind of thinking that
allows us to rescale a shape (like a teddy bear) by expressing it in new
units.
There is another axiom that says, given a set A, there is a "powerset"
P(A) consisting of all of the subsets of A. That's the kind of thinking
that says, given two conditions (rows and columns) we can consider the
various ways of satisfying both conditions.
I share below my notes on how I'm cataloging the ways from the poster.
Some of them seem to use essentially the same kind of thinking. One way
that I'm not sure is on the poster is "circular counting" as with
symmetry, where you may count round and round until you are done. For
example, given three cookies placed at each hour of a clock, you could
take one cookie each time you get to the hour, and so you will get them
all in 36 hours. This is similar to how children (and adults) count out
money, divide out money, they give a bit to each, and if there's more
left, they give out a bit more, until it is all given out.
Maria, your images are so informative (and I use so much text), that I've
included one here:
http://www.selflearners.net/ways/?d=Math#1392
along with the copyright notice from your poster. And I link to a related
page at your site. I would like to likewise include the other images.
I spoke recently with several professors at the University of Chicago
about my "house of knowledge" and will be writing it up for them to take a
look at. If there is any way to get funding for related research or
development of math learning materials (in the Public Domain) please think
of me and let me know!
Andrius Kulikauskas
http://www.selflearners.net/ways/
m...@ms.lt
(773) 306-3807
Blue Island, Chicago
VARIATIONS - the total number of differentiations
Fractal - iterations x iterations - Evolution 10
RESCALING - the reexpression in new units - multiplication as action?
Scale and Stretching - scaling factor x size - Adjacency graph, Atlas
TOTAL ORDER - the total number of units moved
Skip Counting - skips x skip size - Total order? because of ordinal
Repeated Addition - repetitions x repeated number - Total order
Number Line - steps x step size
POWERSET - the number of ways of satisfying two conditions
Combinations - types x types - Powerset (conditions)
Array - rows x columns - Powerset (2 conditions)
Area - side x side - expression of multiple units x expression of
multiple units
UNION - the total number of items across all groups
Sets, per each - sets x items in each set - Composition
Folding and Splitting - splits x parts per split - Composition
Symmetry - regions x objects in each region -
COUNTING IN CIRCLES - the total number of steps made
Symmetry
Collecting taxes (or counting) incrementally (by taking from each) (or
giving out to each)
Time and Money - time x money - Directed graph
---------------------------
Wouldn't it be nice, I ask myself frequently, to have a collection of open
and free virtual manipulatives and videos sorted by particular
multiplication models? Steve Thomas and I are going to aggregate such a
collection over the next few weeks, and you are invited to participate.
If you have open and free interactive software, videos or articles about
multiplication models, would like to prototype some models rapidly, or
even have software development ideas, please join our asynchronous seminar
here:
http://mathfuture.wikispaces.com/MultiplicationModels
We will use better tagging and sorting tools, but this is the starting
page for now.
The project is in part based on an SBIR grant Natural Math had for
developing game prototypes. We hope to use it for sustainable research and
development, including grant and publishing efforts, in the near future.
Cheers,
Maria Droujkova
Make math your own, to make your own math.
m...@ms.lt
Thank you for your reply, and thank you again for your intellectual
insight and leadership in highlighting multiplication's diversity and
centrality!
I attach a diagram in which I organize six ways of thinking about
multiplication. I'm approaching it as a philosophical problem. As your
models show, there are many ways of thinking about multiplication. I'm
trying to understand their origin in the mind. Here are my notes to my
diagram.
My conclusion is that "Multiplication is reproduction of internal structure".
Multiplication is an action (reproduction) upon an object (and its
structure).
The object that we multiply may have more or less structure:
* It may be a "whole", like a basket which we can't see inside.
* It may be a "multiple", like a basket with apples, all identical.
* It may be a "set", like a basket with apples, all distinct.
* It may be a "list", like a basket with apples that are all placed in order.
The action likewise projects more or less structure:
* It may "retain" the object, keep it unchanged, as with multiplication by 1.
* It may "recopy" the object, keeping it identical.
* It may "rescale" the object, keeping it distinct.
* It may "redistribute" the object, keeping it in order.
My conclusion is that the action must project more structure than the
object manifests. (Multiplication is "recounting" and I think if the
object has too much structure then the action can't count it in a new
way). This yields six possibilities:
* A whole can be recopied (however many copies), then again, then again.
This is like fractal multiplication, as with your five-legged starfish
whose each leg holds another five-legged starfish. It is like multiplying
by powers of 10.
* A whole can be rescaled. This is proportion, as with your teddy bear
projected on a screen. The rescalings are actions that can be composed,
magnifying and shrinking. They can be reorganized and canceled away. I
sometimes talk to my students about "magnifying drops" (each drop
multiplying by 10) and "shrinking drops" (each drop dividing by 10) and
ask what happens when we add one drop after another drop.
* A multiple can be rescaled. This is like skip counting or repeated
addition. Note that here the numbers added are cardinals, which is to
say, we don't care in each subgroup what order they had, it's not
relevant, we're simply adding up the sums.
* A set can be redistributed. A set is anything which can be thought as
multiple units, where each element is distinct. A set can be the rows of
a chessboard or an array. It can be the breakdown of a length, for
example, 2 feet and 1/2 foot. We multiply the set by applying the
distributive law and multiplying each element of the set separately. And
that multiplication can be noncommutative, which is to say, we can make
sure that the element is followed by the action. Such multiplication
typically looks like a set matched with another set, yielding "box
multiplication", as when we multiply (3 feet + 1/4 foot)(2 feet + 1/2
foot) and get 4 products which we then sum up. This is also the standard
computation of multiplication, for example, multiplying 2 digit numbers
together.
* A multiple can be redistributed. This is similar to skip counting but
it is counting collections, and so it is counting everything up as if we
were counting up each item in each collection. Thus it is recounting
ordinals.
* A whole can be redistributed. This is long division. We can focus on
cases where the remainder is zero, or we can simply keep dividing forever.
This is like children (or pirates) "dividing out" money, "counting out"
money ("one for me, one for you, ..."). Most of the whole is divided out;
then more of it; then more and so on.
In rescaling, we think in terms of "actions". In redistributing, we think
in terms of "multiple units", such as running a marathon in 2 hours + 9
minutes + 32 seconds.
Recopying has its own intrinsic scale. Rescaling involves two scales, one
for the input and one for the output. Redistributing involves three
scales: one for each factor and one for the product. (For example, one
for rows, one for columns, and one for cells).
The philosopher, logician, semiotician Charles Sanders Peirce spoke of
three kinds of signs: symbol, index, icon and I also add the thing
signified, making for why-how-what-whether, Aristotle's four causes. I
think recopying is reproducing the icon or pictorial representation,
rescaling is reproducing the index or causal representation, and I imagine
redistributing is reproducing the symbol.
There's nothing interesting about "retaining" because it's simply
multiplying by one. Likewise, "lists" are not relevant for multiplying
because they are too determined. I think of a list as a sum with
noncommutative addition whereas a set is a sum with commutative addition
and so the latter can be reorganized.
In my earlier letter, I conjectured a connection with the Zermelo-Frankel
axioms of set theory. Each of the ways of thinking about multiplication
manifests a different axiom of set theory:
* The pairing axiom (that a pair of sets is a set) is manifested by the
fractal system of recopying the whole.
* The axiom of extensionality (that two sets with the same elements are
the same) may be manifested by the rescaling of the whole, in that it is
the same whole and elements but renamed, relabeled, rescaled.
* The axiom of union (the distinct elements in sets can be united in a
union set without redundancy) allows us to recount ordinals.
* Whereas the well-ordering principle (that an order can be imposed)
allows us to recount cardinals (where there may indeed be redundancy).
* The power set axiom (given a set, there is a set of all its subsets)
arises if we think in terms of conditions so that conditions A and B are
given by AxB = "A and B", and likewise AxBxC = "A and B and C" and so on,
as in redistributing the set.
* Redistributing the whole may relate to the "axiom of regularity"
because, in long division, you don't want the remainder to be what you
started with, you don't want such a loop.
Overall, this may give a way of choosing a set of axioms for set theory
that is less arbitrary then the usual Zermelo-Frankel axioms of set
theory.
I think the value of my taxonomy of multiplication models can be:
* To provide an overall framework, what is multiplication?
* To show a complete variety of possible models for multiplication.
* This includes finding models that may have been missing, and grouping
together models that are essentially the same.
* Allow comparisons with this same cognitive structure as it appears in
other fields of life.
The four levels (whole/retain, multiple/recopy, set/rescale,
list/redistribute) and the six pairs together are ten of the twenty-four
rooms in my "House of Knowledge". I have now added many "ways of figuring
things out" from Paul Zeitz's book and I invite us to take a look at:
http://www.selflearners.net/ways/?d=Math and also to compare with other
fields, such as Gamestorming business innovation games, see:
http://www.selflearners.net/ways/ I'm currently studying the Gospels for
ways that Jesus figured things out
http://www.selflearners.net/ways/?d=Jesus and I'm crowdfunding for that
http://ms.lt/I In general, I appreciate any work that I might have the
chance to do!
Andrius
Andrius Kulikauskas
m...@ms.lt
(773) 306-3807
--------------------------------------
On Wed, May 18, 2011 at 2:20 PM, <m...@ms.lt> wrote:
I think that the models of multiplication that you're documenting
illustrate various of the Zermelo-Frankel axioms of set theory.
For example, there is an axiom that, given several sets, we can consider
their union. That's the kind of thinking that allows us to count up
items
that have been grouped.
There is another axiom that says that if two sets A and B have the same
elements, then they are the same set. That's the kind of thinking that
allows us to rescale a shape (like a teddy bear) by expressing it in new
units.
There is another axiom that says, given a set A, there is a "powerset"
P(A) consisting of all of the subsets of A. That's the kind of thinking
that says, given two conditions (rows and columns) we can consider the
various ways of satisfying both conditions.
What is the utility or beauty of bringing these axioms into
multiplication? I don't immediately see it, but I believe it may be there!
In particular, P(A) is not directly multiplicative, and neither are any of
other axioms, so bridges will have to be made somehow.
I share below my notes on how I'm cataloging the ways from the poster.
Some of them seem to use essentially the same kind of thinking. One way
that I'm not sure is on the poster is "circular counting" as with
symmetry, where you may count round and round until you are done. For
example, given three cookies placed at each hour of a clock, you could
take one cookie each time you get to the hour, and so you will get them
all in 36 hours. This is similar to how children (and adults) count out
money, divide out money, they give a bit to each, and if there's more
left, they give out a bit more, until it is all given out.
"Circular counting" happens when you play with mirror books or make
snowflakes. I never connected it to "round robin" sharing before, but it
makes sense. Especially given the fact young kids find it so natural.
And it makes me think of pirates. They had one of the first democracies
out there, and mathematically interesting ways of sharing.
Maria, your images are so informative (and I use so much text), that I've
included one here:
http://www.selflearners.net/ways/?d=Math#1392
along with the copyright notice from your poster. And I link to a related
page at your site. I would like to likewise include the other images.
You are welcome to use images for non-commercial projects. I need to
change Copyright to Creative Commons license, anyway.
I spoke recently with several professors at the University of Chicago
about my "house of knowledge" and will be writing it up for them to
take a
look at. If there is any way to get funding for related research or
development of math learning materials (in the Public Domain) please
think
of me and let me know!
You can either work for hire as developer of materials; or make some
yourself and sell them; or do workshops around your or other people's
materials. These are three ways I know for making money around curriculum
development.
Maria Droujkova
Maria Droujkova,
Thank you for your reply, and thank you again for your intellectual
insight and leadership in highlighting multiplication's diversity and
centrality!
I attach a diagram in which I organize six ways of thinking about
multiplication. I'm approaching it as a philosophical problem. As your
models show, there are many ways of thinking about multiplication. I'm
trying to understand their origin in the mind. Here are my notes to my
diagram.
My conclusion is that "Multiplication is reproduction of internal structure".
Multiplication is an action (reproduction) upon an object (and its
structure).
The object that we multiply may have more or less structure:
* It may be a "whole", like a basket which we can't see inside.
* It may be a "multiple", like a basket with apples, all identical.
* It may be a "set", like a basket with apples, all distinct.
* It may be a "list", like a basket with apples that are all placed in order.
The action likewise projects more or less structure:
* It may "retain" the object, keep it unchanged, as with multiplication by 1.
* It may "recopy" the object, keeping it identical.
* It may "rescale" the object, keeping it distinct.
* It may "redistribute" the object, keeping it in order.
Andrius, thank you for the inspirational work you did on multiplication. I like the style of your diagram, in particular. I think we can use the above categories as a start, for structuring interactives and models we've collected.
I suggest we use the second list, actions, as top-level categories. Tags like "set, list" for objects can be applied within categories.
We need to have two approached to multiplication. The first, action-on-objects or change, is represented above. The second approach, correspondence (between objects) still needs to be addressed. I am attaching a paper comparing the two approaches in the context of computer science.
One of the multiplication models not in the poster is based on gears and their ratios. Tony Forster made an applet about it based on the discussion Dmitri Droujkov started, which you can see here: https://groups.google.com/group/mathgamedesign/browse_thread/thread/dc15f0c47277ab86
Where in the above action list would gears go?
Imagine a stack of cheese being sliced in two. Same question: what action is it?
Imagine a paper being repeatedly folded. What action is that?
Let us keep building!
Cheers,
Maria D
milo gardner
Multiplication also is means to study math history within related cultures. Cultures across time have used the properties of multiplications differently. In the Western Tradition related cultures seem to have palced a large emphasis on geometry as a means to abstract shapes and math thinking as paired actiivities. But what forms of multiplication(s) did Greeks use, and for what purpose(s)?
Scanning available Greek texts it is clear that rational numbers were scaled to unit fraction fraction series for business purposes. The unit fraction aspect of the Greek texts were adopted by Arabs and medieval scribes in everyday business transactions. These related cultures over 3,000 years created commodity units for sale in local and international markets. The final saleable commodity units required an understanding of multiplication on three of the four levels mentioned in this thread:
1. It may "r*ecopy" the object, keeping it identical.
Egyptians, Greeks, Arabs and medievals loved identity statements. Without identical restatements commodityunits and transactions therewith could not be conducted.
2. * It may "rescale" the object, keeping it distinct.
Egyptians, Greeks, Arabs and medievals rescaled abstract versions of saleable commodities within weights and measure systems, topics that college students can easily study. In lesseer forms high school and elmentary students can study the same math and business threads. Businss at any time breaks up a large inventory of a given commodity into smaller saleable units while maintaining profit margins for each level of the distribution chain.
3. * It may "redistribute" the object, keeping it in order.
Egyptians, Greeks, Arabs and medievals distributed abstract versions of saleable commodities (and objects) before, during and after business transctions were conducted.. Accountants at any time create expected and actual inventories of each commodity for each level of the distribution chain.
That is, stating abstract versions of the four properties of multiplication are easily understood when applied to real cultural situations, in almost any cultural era. Each cultural era may place a different emphasis on the four multiplication properties ... as Egyptians, Greeks, Arabs and medievals created identity statements, rescaled identity statements, and redistrubed actual products within business chains that crossed national boundaries.
After 1585 AD the scaled rational number aspect of business transactions ended in its traditional forms. New European business units were created that used the same four multiplication threads. The new business units used base 10 decimals ... as the metric system emerged over the last 300 years ... a point that Europeans love to stress ... without giving proper footnotes to the parent Western Tradition businessmen and mathematicians that made today's mathematics possible.
Best Regartds,
Milo Gardner
From: "m...@ms.lt" <m...@ms.lt>
To: mathf...@googlegroups.com; multiplica...@googlegroups.com
Cc: mathgam...@googlegroups.com; living...@yahoogroups.com
Sent: Thu, May 26, 2011 11:18:50 PM
Subject: [Math 2.0] Re: Do you design games, interactives or videos about multiplication?
Maria Droujkova,
Thank you for your reply, and thank you again for your intellectual
insight and leadership in highlighting multiplication's diversity and
centrality!
I attach a diagram in which I organize six ways of thinking about
multiplication. I'm approaching it as a philosophical problem. As your
models show, there are many ways of thinking about multiplication. I'm
trying to understand their origin in the mind. Here are my notes to my
diagram.
My conclusion is that "Multiplication is reproduction of internal structure".
Multiplication is an action (reproduction) upon an object (and its
structure).
The object that we multiply may have more or less structure:
* It may be a "whole", like a basket which we can't see inside.
* It may be a "multiple", like a basket with apples, all identical.
* It may be a "set", like a basket with apples, all distinct.
* It may be a "list", like a basket with apples that are all placed in order.
The action likewise projects more or less structure:
* It may "retain" the object, keep it unchanged, as with multiplication by 1.
You received this message because you are subscribed to the Google Groups "MathFuture" group.
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Maria Droujkova
2. * It may "rescale" the object, keeping it distinct.
Egyptians, Greeks, Arabs and medievals rescaled abstract versions of saleable commodities within weights and measure systems, topics that college students can easily study. In lesseer forms high school and elmentary students can study the same math and business threads. Businss at any time breaks up a large inventory of a given commodity into smaller saleable units while maintaining profit margins for each level of the distribution chain.
Milo,
Some of your examples use multiplication more as a metaphor for general actions. I'd like to pull some of the more direct and literal examples for the multiplication model collection. I think there were some interesting examples in the measure and weight system. The interactive calculator we made for cubits-palms-fingers etc. comes to mind, as well as the rest of the dictionary you put together: http://mathfuture.wikispaces.com/Egypt+Math+Glossary
milo gardner
Ancient cubit and fingers scaling standards have not be decoded in clear ways. Bruce Friedman spent a year on such a product related to the Torah and Old Testament uses of the cubit and so forth.
What we do know about the multiplication side of this cultural issue, Egyptians scaled rational number n/p by LCM m to mn/mp such that the divisors of mp were selected (in red) that best summed to numerator mn thereby created the best unit fraction series.
Greek and/or Arabs modified the older Egyptian multiplication method to a division context by scaling rational number by an LCM m in a subtraction context per"
(n/p - 1m) = (mn - p)/mp, with (mn -p) set to unity (1) whenever possible.
When (mn -p) could not be set to unity (1), a second subtraction step was connected to the older Egyptian method, ending the calculation.
Example from Fibonacci's Liber Abaci
(4/13 - 1/4) = (16 - 13)/52 such that
(3/52 - 1/18) = (54 - 52)/936 = 1/468
meant 4/13 = 1/4 + 1/18 + 1/468
There were several cubits. Knowing which one was in use at any given times makes the data base at time unreadable.
Yet, we can learn a great deal about the ancient multiplication scaling methods that were used by Egyptians, Greeks, Arabs and medieval scribes by studying this class of text.
Milo
From: Maria Droujkova <drou...@gmail.com>
To: mathf...@googlegroups.com; multiplica...@googlegroups.com
Sent: Fri, May 27, 2011 7:00:06 AM
Subject: Re: [Math 2.0] Re: Do you design games, interactives or videos about multiplication?
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.
kirby urner
One futuristic math I work on encourages students to
seize control of the multiplication operator, designated
* (star) and to script it themselves. The primitive grammar
comes preloaded with the four operators, plus a bunch
of other symbols, but the syntax is open to molding
new meanings.
__mul__ is the behind the scenes "__rib__" along
with __add__, __sub__, __truediv__ and these stack
up in a "__rib__ cage", suggesting a neuronal spinal
chord with autonomous methods:
head: type of Creature (lineage / inheritance):
b __behavior node__ # like __mul__ ( * )
a __behavior node__ # like __sub__ ( - )
c __behavior node__ # like __add__ ( + )
k __behavior node__ # like __truediv__ ( / )
b __behavior node__ # etc.
o __behavior node__ # etc.
n __behavior node__ # __call__ gives Creature a "mouth"
e __behavior node__ # __repr__ is for representation
the Creature goes to our types hierarchy and inherits
capabilities from ancestors, permitting the usual kinds
of encapsulation associated with this family of math
notations. One may also delegate responsibility for
various logical tasks (filtering, concatenation,
comparison, introspection), such that one's exoskeleton
wraps internal components, much as nerve cells wrap
mitochondria.
To take a more concrete example, a PermCell creature
might store "dna" in the form of a storage of key:value
pairs, with a bijection of keys to values such than an
inverse mapping exists. A classic Dolciani approach
might come in handy here, with domain, range, co-domain
and all the rest of it (surjection, injection...). Most of my
students are adults, and therefore have that base API
inculcated by the primitive infrastructure, still considered
real schooling in some circles.
Multiplying PermCells in a binary fashion begets a
demonstration of closure, associativity without commutativity,
inverting, and neutrality ("like Switzerland" might be a
mnemonic here). How this looks in practice, once the
notation is sealed and fed to silicon circuitry (humans
compute along -- follow the bouncing ball, anticipating
what the computer will say as it evaluates successive
instructions from the mathematician). I'm leaving in the
tracebacks, indicating the kinds of errata students get,
with the messages controllable from within the same
notation. The actual screen session makes use of
>>> from ost import permworld
>>> cellzz = permworld.PermCell(permworld.permDNA)
Traceback (most recent call last):
File "<pyshell#158>", line 1, in <module>
cellzz = permworld.PermCell(permworld.permDNA)
AttributeError: 'module' object has no attribute 'permDNA'
>>> cellzz = permworld.PermCell(permworld.PermDNA)
Traceback (most recent call last):
File "<pyshell#159>", line 1, in <module>
cellzz = permworld.PermCell(permworld.PermDNA)
File "C:\Documents and Settings\HP_Administrator\My Documents\OST\workspace\courses\Python4\software\site-packages\ost\permworld.py", line 41, in __init__
raise ValueError("Not recognized DNA")
ValueError: Not recognized DNA
>>> cellzz = permworld.PermCell(permworld.PermDNA())
>>> cellzz
PermCell: (('a', 'j', 'c', 's', 'u', 'x', 'h', ' ', 'z', 'b', 'y', 'n', 'i', 'e', 'd', 'm', 'p', 'w', 'r', 'v', 'k'), ('g', 'l', 'o', 't', 'f'), ('q',))
>>> inv_cellzz = ~cellzz
>>> inv_cellzz * cellzz
PermCell: (('a',), (' ',), ('c',), ('b',), ('e',), ('d',), ('g',), ('f',), ('i',), ('h',), ('k',), ('j',), ('m',), ('l',), ('o',), ('n',), ('q',), ('p',), ('s',), ('r',), ('u',), ('t',), ('w',), ('v',), ('y',), ('x',), ('z',))
It's not that hard for people to understand bijections of a-z (plus space) to a-z.
kirby urner
We've seen a lot of writing at the Math Forum adopting this approach,
in part to break the hold of "math as repeated addition" -- which I haven't
seen grappled with here as much.
( Sometimes "breaking the hold of" is just the opposite of a given
lesson plan's agenda: making multiplication mean "repeated addition"
is often what's up. )
In the approach I most favor, we put a lot of emphasis on the Polyhedrons
early on, with attention to criteria and sets (what's a Platonic? what's
an Archimedean? what's a Johnson?) and of course on Euler's
V + F = E + 2 (which we hint might have been Descartes' had he
not been so afraid -- insert Monty Python skit here).
Polyhedrons participate in three basic operations we want to go into:
rotation around axes (polarity, 2 of an aroundness kind); translation
through space; re-sizing (re-scaling). With those three, used in combination,
you have a lot of animation possibilities (ala Blender -- blender.org).
Re-sizing is multiplication because of 10 * (F*F) + 2 i.e. the 2nd
powering associated with surface area, attaches to a multiplying
even numbered coefficient, in this case 10 for the cuboctahedron of
1, 12, 42, 92, 162 balls in successive layers (check OEIS). The
volume of said cuboctahedron is just 20 * F**3 (F to the 3rd power
-- not saying "cubed" as that interferes with mathematical thinking).
I'm abbreviating quite a bit, as I don't want to recap everything.
Associating addition with +2 and linear motion (the number line
equator), and multiplication with x2 and surface areal rates of
growth, is the core mnemonic here. Multiplication takes on a
resizing and/or subdividing role, as the Frequency of whatever
object is raised to respective powers, for area and volume
respectively. The Polyhedrons themselves are rendered in a
"geometry of lumps" (Menger) ala Blender and/or POV-Ray
and/or VRML or whatever format. Students do lots of
string substitution, going from one namespace to another.
More info:
http://4dsolutions.net/presentations/connectingthedots.pdf
Kirby
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> http://groups.google.com/group/multiplicationmodels?hl=en
>
Edward Cherlin
Bieberbach groups. There are pictures available of crystals of all 230
types, plus pseudo-crystals with 5-fold symmetries and other
structures of interest.
http://en.wikipedia.org/wiki/Space_group
On Fri, May 27, 2011 at 15:40, kirby urner <kirby...@gmail.com> wrote:
> I liked this emphasis on rescaling.
>
> We've seen a lot of writing at the Math Forum adopting this approach,
> in part to break the hold of "math as repeated addition" -- which I haven't
> seen grappled with here as much.
>
> ( Sometimes "breaking the hold of" is just the opposite of a given
> lesson plan's agenda: making multiplication mean "repeated addition"
> is often what's up. )
On the other hand, repeated anything is the essence of Gödel's
recursive functions, which are at the foundation of computability and
decidability theory. This is also the essence of LISP programming. See
The Little Lisper or The Little Schemer for a tutorial.
> In the approach I most favor, we put a lot of emphasis on the Polyhedrons
> early on, with attention to criteria and sets (what's a Platonic? what's
> an Archimedean? what's a Johnson?) and of course on Euler's
> V + F = E + 2 (which we hint might have been Descartes' had he
> not been so afraid -- insert Monty Python skit here).
and, I assume, figurate numbers.
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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://wiki.sugarlabs.org/go/Replacing_Textbooks