Martian Math summer camp

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kirby urner

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Jul 9, 2018, 8:57:40 AM7/9/18
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The Martian Math curriculum has a "summer reading" flavor, meaning we suggest students use some free time over the summer to consume books, not just movies. We start with science fiction as a genre.  My version of the curriculum points to such classics as Red, Blue, Green Mars (a trilogy) and of course to War of the Worlds by H.G. Wells.


The model here is playing with toys as a gateway to fantasy e.g. kids don't just move dolls around, they "play house" giving personalities to the dolls and structuring narratives. Math toys and kits imbued with fantasy or narrative elements help these artifacts gain traction in the imagination, which is a goal. 

Ideally, one daydreams about math, instead of daydreaming to escape from math.

One example story line I show and tell about involves ETs (we can call them Martians) coming to Earth and wanting to collaborate on a project. Having visited a dam under construction in Lesotho, where engineers from around the world came together, I was inspired to imagine Martian and Earthling base camps on either side of this canyon, where the dam would be built.

Storyboard pictures: 
https://flic.kr/p/8thDHW  (pouring concrete)
https://flic.kr/p/zqhPtk    (Martian Base is a tetrahedron)
https://flic.kr/p/LwqV6h  (Martian Base as classroom model)

Where the mathematics comes in is these Martians multiply differently such that our terminology around "two squared" and "two cubed" seems rather alien.  Their 2 times 2 is an equilateral triangle, all sides 2.  Instead of vectors at 90 degrees, they're at 60 degrees. 2 x 2 x 2 is an 8-volumed tetrahedron.

Storyboard picture:
https://flic.kr/p/25MGEDT  (multiplication)

Given 1 to the 3rd power is a tetrahedron instead of a cube, the Martian-Earthling ambassadors need to come up with some conversion convention.  You see where I'm going with this. I'm teaching 1900s humanities. Understanding the literature of the recent past requires some decoding.

Martians and Earthlings agree on a canonical unit sphere, radius R, diameter D.  The Martian unit of volume is a tetrahedron T of edges D.  The Earthling unit of volume is a cube C of edges R. 

Question:  volume T > volume C or the other way around?  Answer:  T < C.  Ratio:  sqrt(9/8)  <-- C is larger by this conversion constant.




More humanities reading:

Kirby

PS: Happy Birthday Bradford Hansen-Smith if you're still subscribed!

Bradford Hansen-Smith

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Jul 9, 2018, 11:55:29 AM7/9/18
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Thanks Kirby,

My birthday gift to all that are still looking to mathf...@googlegroups.com is:

WHY?


Why don’t we learn to fold circles the same time we learn to draw circles? Is it because we have been taught the circle is a picture of a unit concept by cutting the sphere apart? What prevents us from cutting out the circle we draw on paper and folding it? Why don’t we fold the circle disc, a spherical compression of unity? There is no reason to get stuck at the arbitrary boundary to the center or to stop at the circumference, both determined by the compass used to draw circles. Don’t we know concentric circles go infinitely in as well as infinitely out where scale is relative and direction is minimum two? The circle is the center. Folding the circle is both compass and straight edge, 2-D & 3-D, generalized math relationships; why don’t we know all this is in the circle first? Do we know the way out of the box is to put the square back into the circle where it belongs? Every fold in the circle is right angle movement about change and connection, not containment. Is it for comfort in using formulas to fold squares or is it ignorance about the circle that prevents folding it? Why do we embrace fragmented units and separation rather than unity of the whole circle? Why do we put up with conscious misinformation, mistaken ideals, and purposeful malfeasance? Is it too difficult to admit we are out of alignment with intelligent healthy living?
I’m not looking for answers; you have your own. I’m sharing a few questions.



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Bradford Hansen-Smith
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kirby urner

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Jul 14, 2018, 9:12:08 AM7/14/18
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Hi Bradford --

Thank you for your birthday rant about circles.

For all we know, some lucky students do get more time folding circles; there's no way we would know.  Like hardly anyone knows they're teaching Martian Math in Portland.  We're flying under the radar here.

However I'm thinking you're correct that few fold circles, and furthermore, few do any kind of cutting and folding, of circles, squares, or anything else.  Paper and scissors are a hassle.  Teachers might need to spend from their own pockets (like I did -- for the photocopying, not the scissors).

I brought scissors and plane net outlines of the Martians' A-modules, which fold into tetrahedral wedges.  I decided to make this an optional activity for my summer campers and only one of fifteen kids, maybe two, showed any interest.  We were in a computer lab and the computers were like magnets to their attention. 

A different kind of maker space would have been more conducive to practicing manual skills maybe.  I did an all-6th-graders assembly at the daughter's school over a decade ago where kids seemed pretty into it.  We managed to get a whole tetrahedron out of the deal, and that was just a small part of the event.

https://flic.kr/p/LmzoK9  ("A" module template)

https://flic.kr/p/28yphNH  (what every Martian kid knows -- a canonical hierarchy of volumes)

I like that you start with paper plates a lot of the time, which saves time.  No need to cut out the circles to begin with. 

So there's no branch of origami aside from yours that starts with circles?  Nothing Japanese? 

I do find some hits when I searched, which is encouraging:


You'd *think* there'd be a fork or branch that took it in the same direction you've explored.  Like, c'mon!

The A-module templates need to be folded in one of two ways:  bend the creases one way and it's "left handed" (which is which?); bend creases the other way and it's "right handed".  Like with gloves (left and right). 

There's no way to translate or rotate in space to turn left into right, yet the concept "congruent" is such that left and right are congruent.

https://flic.kr/p/LwqVZb  (Martian base, tetrahedron = 24 A mods, 12 left, 12 right)

When we convey geometric thinking, I think we need to talk about chirality (not in spellchecker) and congruence together.  Chemistry is full of chirality talk.  Physics too.  Just focusing on congruence alone is going to dumb ya down vs-a-vs the whole "handedness" discussion.  Anyone who plays Tetris knows a left handed L can't turn into a right handed L.

Since I only photocopy the template to one side of the construction paper, I find it easiest to fold in a way that keeps the lines showing, to make the creases, then reverse those creases on some of them.  I just use tape to make the wedges.  A fancier cutout might use tabs.

We also have B, T, E and S modules (all tetrahedrons).**  Martian kids learn this stuff early.  Earthing kids are mostly clueless, which is why we need summer camps like mine.  How do they expect to collaborate with Martians if they don't have any appreciation for their culture right?

Kirby


PS: The Martians have no concept of infinity, at least not like ours, and they get by just fine (technologically they're ahead).

** the T & E have exactly the same shape but area:volume ratio differs.  Since Galileo at least, people have understood that scale matters e.g. insects that carry their own weight and then some, would not be practical if the size of humans.  Volume grows much much faster than surface area, both as a function of linear dimensions.  The newish book Scale by Geoffrey West is clear on this concept.  But do we share it with kids?  Is this power law connecting linear, areal and volumetric change part of "common core"?  I forget.

Bradford Hansen-Smith

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Jul 14, 2018, 10:49:44 AM7/14/18
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Hi Kirby,

It is no surprise you only got 2 of 15 kids interested. If you give them more than just a circle; scissors to cut pieces and templates to follow then you are inherently instructing them to methods of construction and not a process of revealing through their own exploration.  I do not put down construction, only that it must be balanced by observing their own discoveries in what is revealed by what they do. To get anything out of a circle you only need a circle.

I always start out and end with the circle, nothing but the circle. But I have a suggestion since you give them scissors. Have them first draw a circle on some paper, cut the circle out without cutting through the edge of the paper so there is a circle and a circle hole in the paper. Then look at the properties of both and discuss the similarities and differences. The properties will be the same for the whole circle and the hole circle except for the difference of 2 open planes and 2 solid planes. Explore both folding them the same way in half and looking at what happens to each. Now a process is revealed without trying to mathematically construct anything.

Yes computers are magnets, a one way suck-up of curiosity, but does not satisfy creative needs, so we end up creatively violating people in the most horrendous and subtle ways.

If you will notice all that you find about origami circle folding is using the circle shape as if it were a square with no appreciation for the unique comprehensive or information generating nature of the circle. I find it amazing that nobody else is folding circles for the information that is there, but so it appears.

As you point out the congruent nature of chirality in the first fold of the circle in half is reason enough to fold the circle and have a discussion, not to mention the reformations with that one crease and other math concepts it reveals that can be explored as they are discovered. It is not the math information we present but more importantly in what context do we give it that allows students the space to be curious and explore without being right or wrong or that they should even know it at all.

You have lots of tools, examples, and history to substantiate construction both of Earthing and Martians systems, when in fact both are found in the folds of the circle.

I start a workshop by having students fold the circle in half, then discuss what they see. Kids see far more than adults. I just ask question for clarity and deeper observation, that is mostly what I do, letting the students come up with the information through their own observations. It is only from that information we know what to do next, and there is always more than just one direction. It is open country after that depending on where you want to go with it.


Earthing kids are clueless because Earthing adults are clueless. Try folding circles first, then to go to constructing polyhedra as needed. It is all good for a balanced understanding of what to do with units within the context of unity. Summer camp should be a time of fun, curiosity, and exploration, not the other.

Brad


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Bradford Hansen-Smith
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