Re: Do you design games, interactives or videos about multiplication?
m...@ms.lt
Thank you for the additional multiplication models. Bicycle gears are
given by the formula:
number of front teeth
---------------------- x circumference of wheel
number of back teeth
= distance traveled per revolution
That's an example of the proportional scaling ("rescaling the whole", like
with projecting a teddy bear ). You could imagine starting out with two
gears (the gear fixed to the wheel, and the gear you are pedaling) each
the size of the back wheel. Replacing those giant gears with smaller
gears is an action of magnifying or shrinking the distance traveled. You
could add more gears to further magnify or shrink that ratio. Adding
teeth to the gears is changing the units (and that aspect could be
considered "rescaling the multiple", as with skip counting, but that is
the correspondence of the number of teeth to the circumference of the
gear, NOT the relationship between the two gears). The fact that
multiplication is taking place at two different levels makes it
challenging to think about because the gear ratio is "abstract" and not
concretely, directly related to the size of the wheel.
If I have a stack of 10 slices of cheese and I slice them all in two, then:
* if I'm simply changing the units, so that I'm thinking now of 20 small
slices rather than 10 large slices, so that 1 large slice = 2 small
slices, then I'm skip counting, "rescaling the multiple".
* if I'm thinking of each large slice as consisting of a left piece and a
right piece, (a first piece and a second piece, distinguishable or
"labelled" with a child's ID 10x2), then I'm "redistributing the
multiple", as with "sets, per each".
* If a rectangular paper is folded in half, and half again, and yet again,
and so on, and they are all considered repeatedly applied transformations
of the same whole, then that is fractal multiplication, "recopying the
whole", as with your paper snowflake.
* If a paper is folded once, and then again, and again, but those actions
are thought as taking place separately, and especially, if I'm focused on
the labelled components (rather than the repeating whole), then it is
label multiplication, "redistributing the multiple", as with your pie
halves sliced in five slices each.
(I am separating your "folding and splitting" examples into two different
kinds of multiplication.)
I'm thinking now that:
* "Whole" means "an unspecified unit" as when we are copying or scaling or
distributing, acting on an "unknown"
* "Multiple" means that we have a recurring structure (labeled or not),
but the instances of that structure are not labeled, thus they are all the
same "unit".
* "Set" means that we have distinct "units" as is the case when we have an
expression in multiple units, all distinct.
* "List" is when those terms come in a definite order.
I've started reading the paper you sent me. I think of "Whole - Multiple
- Set - List" as talking about objects and, analogously, "Retain - recopy
- rescale - redistribute" as talking about actions. Multiplication is
restructuring through an action on an object. I will try to understand
the author's point of view, it sounds interesting.
Maria, I would use six different categories, such as:
* Fractal: http://www.selflearners.net/ways/?d=Math#1526
* Proportion: http://www.selflearners.net/ways/?d=Math#1527
* Tally: http://www.selflearners.net/ways/?d=Math#1528
* Box: http://www.selflearners.net/ways/?d=Math#1529
* Label: http://www.selflearners.net/ways/?d=Math#1392
* Divide out: http://www.selflearners.net/ways/?d=Math#1530
Whatever names you think best, and then tags for the "whole-multiple-set"
and "recopy-rescale-redistribute".
Thank you all for feedback! It's great and I keep improving these ideas.
I want to crystallize them and collect classic illustrative examples.
One example of fractal multiplication that comes to mind is multiplication
by 10. I noticed this year that the decimal point is not logically
positioned. It should be at (under or over) the "ones" place. Then the
system would be symmetric. The space to the left would be the "tens" and
the space to the right would be the "tenths". A little (video) essay
could explain that when we multiply or divide by 10, it is the number that
actually moves, not the decimal point. It's like the fact that the Sun is
at (or near) the center of the solar system and the Earth revolves around
it, not the other way around, as it may seem. The way that the decimal
point is placed is very destructive pedagogically, it makes people (like
me) think that there is something between the digit; that there is some
mysterious difference between the whole numbers and the fractional
numbers; and most sadly, that (jaded) teachers don't see that there's
something wrong with the system, as I remember thinking as a child; and
that there's no reason to care. I suspected that the decimal point is
where it is for typographical reasons; maybe because it arose along with
printing? or derived from accounting notation? My point being that an
adult who appreciates what I've just written (and more along these lines)
would appreciate some thing very deep about math (including how to
calculate 10% discounts by shifting numbers to the right).
I believe that 24 such essays would be enough to teach all of math. (They
should be accompanied by many examples and illustrations). The
multiplication models account for 6, and the whole-multiple-set-list for
another 4.
I'll try to join you for your meeting this evening on multiplication
models, Wednesday, June 8th, at 8pm ET. Is it online? Where can I find
details?
Andrius
Andrius Kulikauskas
m...@ms.lt
(773) 306-3807
http://www.selflearners.net
On Fri, May 27, 2011 at 2:18 AM, <m...@ms.lt> wrote:
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
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