Re: [Math Game Design] Re: Do you design games, interactives or videos about multiplication?
Maria Droujkova
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.
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
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
--------------------------------------
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
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
Cheers,
Maria Droujkova
Make math your own, to make your own math.
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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