Trying to Understand the First Paragraph of “A Concise Course in Algebraic Topology”

Screenshot 2018-01-21 at 5.31.44 PM.png

I. 

Wikipedia is here, and quite helpful in expanding on this definition. Here is my attempt to rewrite it in my own words, fleshing it out in ways that are helpful and natural to me.

What is a topological space?

Imagine that you have some collection of points that are in the Cartesian plane. Maybe it’s a circle that’s centered around the origin. Maybe it’s a line, like y = 2x + 3. Maybe it’s a grid of points falling perfectly along the chassis of the plane.

In a way, we can also imagine the city that you live in or the state or country as a collection of points. My apartment is one of these points in space, and I’m part of a neighborhood — Washington Heights. But there are other ways of talking about where I am. I’m not just in Washington Heights, but in NYC, in NYS, in USA, on Earth. There are points, arguably, south of where I live that are a part of two neighborhoods — Washington Heights and Harlem. These neighborhoods aren’t coextensive, but they do have some overlap.

Maybe there are some folks living out in the sticks that are part of no neighborhood. They’re off the grid, so to speak.

Let’s imagine a computer program that knows how to assign every point in a map of NYC to the neighborhoods to which it belongs. Let’s call that program N. You give N a point, and it gives you a list of neighborhoods.

Imagine if we let N loose on any of those spaces that we were talking about before — the disk, the grid of integer points, the portion of a line — and then let’s say that any of those spaces is a topological space if when N is applied to the space, nothing weird happens. Here is a checklist of four weird things that could happen:

  • You apply the computer program N, but there’s a place that gets assigned neighborhoods that it’s not in. Like, maybe the program assigns a spot in the middle of the Hudson River to Washington Heights, even though a random spot in the river isn’t really a part of our neighborhood.
  • Imagine that N says that my apartment is in Washington Heights, but not in NYC. That’s messed up. Bad N. If you can imagine a place like that, it’s not a topological space.
  • If I lived a bit further north, I might be a part of two neighborhoods: Washington Heights and Inwood. N should assign me not just to Washington Heights and to Inwood, but to a new, hybrid neighborhood: Inwoodton Heights. If N can’t really invent such a neighborhood, then I don’t really live in a topological space. This one starts moving away from my regular intuitions about neighborhoods, clearly…
  • …and so does this last test. As I keep on mentioning, my apartment is in a neighborhood Washington Heights. If N assigns my apartment to WaHi, then it should also report that there’s another neighborhood that I’m a part of that’s more local than WaHi, like my block. My block has the property that (a) I’m in it, it’s my neighborhood and also (b) everywhere on my block is also part of Washington Heights. If there isn’t such a place, I don’t live in a topological space.

Moving away from apartments and cities, let’s think about the x-axis. Real estate agents who are trying to get people with $ to move to a place decide what neighborhoods mean in the real world. But what does a neighborhood mean on the x-axis? Let’s define a ‘hood of a point (like 3.1) to be any open interval (like between 3 and 5) that includes the point.

Let’s make sure that our computer program N would work alright in such a place:

  • If you’re a neighborhood of a point, that point is in it. Yay!
  • If you’re in an interval (like between 3 and 5) and that interval is in a bigger interval (like between 1 and 6) then you’re also in the bigger neighborhood.
  • If you’re part of two neighborhoods (like between 3 and 5 and also between 3.05 and 7) then you’re also part of the overlapping neighborhood (between 3.05 and 5).
  • If you’re part of a neighborhood (like between 3 and 5) there is a smaller, more local neighborhood that you’re part of (like between 3.05 and 4.95) and, more to the point, there always will be.

Huzzah — it’s a topological space!
II.

One more twist: this way of thinking about topological spaces is not standard any longer. People prefer to talk not about neighborhoods but instead about open sets. So we have to make sense of that, even though it’s been nearly a decade since we took Real Analysis.

Another wise wikipedia page says, “an open set is an abstract concept generalizing the idea of an open interval in the real line,” which makes sense. It tells you the delimit of a set, but it doesn’t include its boundary.

This isn’t enough, though. Since we defined a topological space in terms of neighborhoods, we want a definition of “open set” that relates “open set” to neighborhood. We need to connect these two concepts.

[Here’s where I got a little bit lost, so I went looking for help from another source. Google google, ended up at the Math Stackexchange.]

It doesn’t take much to merge the two concepts, as far as I can tell from what I’m reading. To fit the rules of neighborhood assignment, your potential neighborhood has to pass the following test: if a point is assigned a particular ‘hood, the point actually has to be in that ‘hood. (Fancy talk: if N is in N(x), then x has to be in N.)

An open set adds one slight additional requirement: if you’re in N, then N’s your neighborhood. All this eliminates is the possibility that a neighborhood is “too big,” including not just Washington Heights but also a random stretch of the Hudson River where nobody lives.

This is a definition that will do the trick for us, and allow us to connect the old-fashioned (but apparently useful for beginners like myself) definition of topological space to the new and trickier (but apparently useful for topology pros) definition that is couched in terms of open sets.

Here’s the open set version: still imagine a map of some space (like NYC) and still imagine that the map contains a bunch of points (like my apartment and other peoples’ apartments) and still imagine that there are ways of grouping those points that come with the map (like neighborhoods)…

…or don’t, and instead imagine some section of the x-axis and the Cartesian plane, and imagine that there are ways of making subsets of that section of the line or plane…

And now there are three rules about the neighborhoods (or subsets):

  • The empty set is in it — i.e. the neighborhood of nobody is a neighborhood
  • You can’t escape the space via the union of subsets — i.e. mush together any of the neighborhoods (subsets) and you’re still in a neighborhood (subset)
  • You can’t escape the space via the finite intersection of subsets — i.e. be in as many of the neighborhoods (subsets) as you wish to at once and you’re still in a neighborhood (subset)

Not entirely obvious to me yet why these two definitions of topological space are equivalent. I see that they both have the intersection, I see how the intersection of disjoint sets would imply that the empty set is in the topology (but what’s wrong if there’s only one set in the topology according to the neighborhood definition?), and I guess the superset/subset axioms must cover the “closed under union” axiom of the open set definition? I’m going to let that slide for now.

III.

What are some examples and non-examples of a topological space? We already mentioned the x-axis, equipped with open intervals.

In fact, for things like the x-axis — metric spaces — the textbook page already tells us that we can think of topological spaces as things where the neighborhoods are like open intervals. They’re open disks, or open spheres, that act precisely in the way you’d expect them to. No funny business, no weird stuff.

As the textbook also says, we should think of this all as an attempt to point at what it takes to have a space that captures our feelings about “nearness.” So if something’s near, there’s always something nearer, that seems to be the most important part to me.

But whenever there’s a new idea, we need some examples and contrasting non-examples to set our heads on straight, and this is no exception.

The x-axis — i.e. the real line — is an infinite collection of points, but there’s no reason why a topological space needs to be infinite. And if our map of points is finite, there’s no reason why our computer program — our function — N has to be anything but a list of neighborhoods.

So, suppose you have four points in your space: 1, 2, 3 . What neighborhood assignments would result in this being a topological space, and which wouldn’t? Here is a helpful image from wikipedia:

360px-Topological_space_examples.svg.png

  • If you care your space into two sets — {} and {1, 2, 3} — then you haven’t really done much carving, but that’s a topological space, the trivial topology. That sort of captures the idea of nearness in an absolute way — everything is near, nothing is not, like a terrible party. (No idea if that analogy makes mathematical sense, by the way.)

Thinking about the function N and the first set of axioms: the same and only neighborhood has been assigned to the three points and they’re part of it, there are no supersets or subsets, so trivially we’re done.

  • In contrast, if you carve your space into {}, {1, 2, 3}, {2} and {3} (bottom left) that’s not a topological space. From the point of view of the second definition, the issue is that the union of {2} and {3} isn’t included, so it’s not a topology. From the point of view of the first, the problem is that {2, 3} is a superset of {2}, and every subset containing a neighborhood must itself be a neighborhood. (Is this right?? It doesn’t feel right.)
  • What if your topology contains {}, {1}, {2}, {1, 2}, {1, 2, 3} (middle left)? The second definition is indeed sort of easier to use for these discrete examples. It’s pretty clear: the union of {1}, {2} is {1, 2} and that’s in.

I need more examples, and wikipedia has a few more:

  • Let the space be all the integers, and let the collection of subsets be all the finite subsets of the integers, any list of them, plus the set of all the integers. So, for example, if you have a given point like -5, there would be all these open intervals that we’d want to say are the neighborhoods of -5, like “integers between -10 and 2” and “integers between 0 and -10000,” and we’d also say “all the integers.” That wouldn’t work, though, because you the union of a bunch of these subsets isn’t necessarily part of the topology. Take the union of all the finite sets that don’t contain zero — {-2, -1, 1, 2}, {1, 5, 6}, etc. — that’s an infinite set, but there’s only one infinite set in the collection (all the integers) and this can’t be that infinite set. Therefore, it can’t be in the collection, and this can’t be a topology.

The tricky thing for me is relating this all to the talk that a topological space preserves our notion of “nearness.” Is it possible to relate each of these axiom failures to a failure of our notion of nearness? Personally speaking, my notion of nearness no longer operates when our space has just four points. Is there a good way to think about this that I’m missing?

TO BE CONTINUED, WHEN I TRY TO READ THE SECOND AND THIRD PARAGRAPHS OF THIS TEXT AS I ATTEMPT TO UNDERSTAND A PROOF OF BROUWER’S FIXED POINT THEOREM. STAY TUNED

 

Little Shoes, Never Worn

For sale:
baby shoes,
never worn.

Machine washable,
slight stain,
probably pee.

Baby fine;
shoes not.

In addition
baby has
huge feet.

Truly big,
flippers really.

Small head,
small hands,
small nose,
giant feet.

Never worn
baby shoes
for sale,
or just take them.

Posted in Fun

My Answers to “Six Questions Math Educators Need to Answer Honestly”

[Questions can be found here.]

1. What is mathematics?

Don’t know. Pass.

2. Where should we learn mathematics?

Don’t know what “should” means here. Pass.

3. Should mathematics be a compulsory subject?

Question: What would the immediate effect of changing of making math non-compulsory tomorrow would be?

Answer: Worse teaching.

We’ve had decades (centuries?) to learn a bit about how to teach what we teach. I’m not saying it’s easy to teach systems of inequalities well; I’m saying that if I’m going to teach systems of inequalities there are dozens of resources I can turn to, curricula that have thought carefully about how to put the pieces of the course together to reach many students.

Have you ever taught a new course or an elective? It’s hard, isn’t it, not knowing how to go about the work. A world where we radically change the curriculum or where we eliminate mandatory classes is that world. It’s not like number theory and graph theory is easier to teach than polynomial division. Yeah, number theory is really cool, but it’s hard to turn cool things into learning. That’s the whole art of teaching.

So go ahead, make math non-compulsory. Honestly, I don’t care, what we ask kids to study in school is mostly arbitrary anyway. In the 1920s it was unclear whether math would be a compulsory subject in high schools — NCTM was created, in part, to protect math compulsory math offerings. If they’d lost that fight, who knows what hoops we’d be asking students to jump through before graduating. Mandatory home ec? Mandatory workshop? What reason do we have to think that these mandatory offerings would be better taught?

And if we make most of the curriculum non-mandatory? That’s like college, and if you don’t like how k-12 is taught then, wow, get excited for higher ed.

The fact is that there are tremendous pedagogical benefits from having a lot of people teaching and learning the same thing. School requirements are arbitrary, but there are benefits to keeping things the way they are.

4. How should we learn mathematics?

Too broad, pass.

5. Who should learn mathematics?

Don’t know what “should” means here, pass.

6. What is the purpose of mathematics education?

Tempted to pass, but I think there’s something interesting to think about.

The purpose of math education depends on the purpose of math, and the purpose of math has varied across history, across cultures, across math departments, varied so much in the time of Fermat that there wasn’t even a single thing called “mathematics.” From The Mathematical Career of Pierre de Fermat:

With due allowance for exception, one may sort those practitioners into six broad categories: the classical geometers, the cossist algebraists, the applied mathematicians, the mystics, the artists and artisans, and the analysts. Although only one or two of these categories constitutes what one might call a “school” of mathematics, and although the work of many individuals falls into several categories, nonetheless each category distinguishes itself from the others by characteristic attitudes toward the nature and purpose of mathematics, its problems and methods of solution. Each category has a distinctive style, and the different styles often conflict in essential ways.

When people talk about the “values of math” or the “purpose of math education,” know that the question is ill-formed. Math education emerges from the competing needs of students, parents, communities, our government. It’s inherently contradictory, and given that the conflicts are genuine it probably should be sort of a mess. Teaching is not hampered by these conflicts, teaching is the job of making sense of these conflicting needs. That’s the job.

Math in 2018 is used for joy, pleasure, getting rich, pursuing justice, attacking foreign governments, passing tests, teaching, and dozens of other purposes too. When educators say “we have to get back to the values of math” that’s a lateral move; that’s no clearer. Math has as many purposes as math education.

“Math education is a very young child of mathematics,” Singh writes, a completely incorrect statement. One thing I learned from Learning Modern Algebra is that pedagogy is a major source of mathematical innovation. Part of why you’d need a way to find Pythagorean triples is to help you come up with nice problems for your students.

Singh calls math education “a spoiled and rotten child” of mathematics, as opposed to mathematics, whose purposes are clear and pure:

Mathematics is not up for debate. It is what it is, and it has been tattooed in so many civilizations and cultures for many millennia. Its purpose and mandates, might run parallel to math education at times, but in reality, they operate on a higher plane of truth, justice, beauty, play and love 

Historically speaking, this is nonsense. Math changes, math has different purposes. Math is contradictory, math education is contradictory. Math is beautiful and ugly, so is math education. In math education we’re not the spoiled child of math — we’re the grown sibling, sometimes in touch, sometimes not.

So I’m entirely against Singh’s idea that we can look to the values of math and measure math education against them. In sum: Stop pooping on math education and educators. The end.

The Bible According to My Son, Who Likes Both The Exodus and Dora the Explorer

CHAPTER 3

1     When Pharoah learned what Moses had done he rose to kill him, but Moses fled to the desert.

2     The LORD sought Moses in the desert, but could not find him.

3     He could not find him wherever he looked. The LORD looked over here, and also over there.

4     And the LORD said, “I can’t find Moses no matter where I’m looking.” So he went to ask Dora.

5     Dora spoke unto the LORD, “I know how to find Moses. Check the map.”

6     And the MAP spoke unto the LORD saying, “If there’s a place you got to go, I’m the one you need to know,”

7     “I’m the map. I’m the map. I’m the map.”

8     The LORD thanked Dora very nicely and then he found Moses. “There you are,” He said.

9     “You’ve got to go back to Pharaoh, Moses. Otherwise you’re going to be in big trouble. I’m going to get mad at you, I’m going to yell and scream at you. And you have to tell Pharaoh: ‘Pharaoh, you’re a bad guy.’ Tell him that.”

10     And then the LORD went with Dora and they had arroz con leche with abuela and they sang songs and had fun.

Posted in Fun

In Love with Being Lost

Sunil Singh writes:

What is mathematics? If I would have answered this question before I began my teaching career, I would have been provided some cliched, stock answer revolving around its importance to many disciplines like science, engineering, economics, etc. — basically it being some practical workhorse. Now? Well, since I wrote a book about it, it is simply about happiness. Finding sporadic moments of bliss learning something new and wonderful about mathematics. Just playing with numbers, puzzles, brain teasers, conundrums, algebraic ideas, etc — mucking around in the sandbox of math. The more I know, the less I know. The less I know, paradoxically, the better I feel about my ideas about mathematics.

There’s no doubt that mathematics involves happy moments. But does Andrew Wiles’ description of mathematics sound like happiness?

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room…

Not that this is necessarily inconsistent with joy. Maybe you get your kicks wandering around anonymous dark mansions — there’s something for everybody. But you’ve got to admit, it sure doesn’t scream out FUN.

So which is it: is mathematics all about happiness or about stubborn frustration? Well, why not both? If, as Singh suggests, we’d best think of mathematics as an “art” then we get to ask, what other art is only about happiness?

I think educators sometimes emphasize feelings of joy and experiences of beauty in math at the expense of all the other things you can feel or see. For a while, I’ve been interested in visualizations of ugliness in math — is such a thing possible? And, as the Wiles quote shows, mathematics often involves far more than happiness.

“Of course math involves more than happiness,” you say, “but the point is that happiness is the goal, the purpose, the carrot that mathematicians are always chasing.”

Here’s an idea though: what if what we’re really in love with is the feeling of being lost in a dark mansion?

Stopping by Ft Tryon Park, Way the Heck Uptown, on a Snowy Afternoon

So you’ve got woods? Yeah, we’ve got those.
Well, not exactly, but something close.
A park with hills and rocks and trees
That people flock to whene’er it snows.

Your woods are dark, deep, and lovely?
I think our park you’ll find quite comfy.
Your little horse would love it here
I’ve heard the woods get quite lonely.

He’ll ask if he can stay and play
Romp with kids and ride their sleigh.
When we get cold we can go home
To watch snow fill our alleyway.

Woods do sound nice – save your pity,
The park’s also very pretty,
And no one’s lonely in New York City.
And no one’s lonely in New York City.

Posted in Fun

Generalizing Circles and Ellipses

[Reading and thinking about: n-Ellipses and the Minimum Distance Sum Problem, a paper that I was able to stick pretty well with until about the last third.]

1. Generalizing Circle Definitions

There often comes a point in my geometry classes where I ask students to come up with a definition that describes a circle. It’s always interesting to see the ideas they land on. Some students try to capture the perfect curviness of a circle; others talk about the circle’s symmetry. When I share the “textbook” definition (which kids rarely come up with) there are sometimes sighs of relief or yelps of excitement in the room:

Circle: Take all the points that are a certain distance away from a center point — that’s a circle.

It’s a beautifully simple definition, and it’s one that leaves plenty of room for generalization.

If we’re looking to generalize the circle, a great next step is to analogously define an ellipse. I like to imagine the center of a circle splitting in half and turning into the two foci of the ellipse.

gifsmos (1)

(Sorry for the glitchy gif.)

We can define an ellipse in a way that is analogous to the “textbook” definition of the circle:

Ellipse: Take all the set of points that are a certain distance away from two focal points — that’s an ellipse.

Just to clarify, what I mean by “certain distance away” is we’re looking for the sum of the distances to the two focal points. So if our “certain distance” is 10, then a point that is 3 away from one focal point and 7 away from the other would be part of the ellipse.

2. Generalizing Circle Constructions

Sometimes you see geometry problems about goats on a rope. The point being that the rope-stuck goat is restricted by a circle. That goat could trace out a circle, and you can also trace a circle with a pin, string and paper.  You can also make a lovely ellipse with two focal points with some string and some pins:

F99FRTL0DPEPORTHE1.MEDIUM.jpg

Why stop here, though? The definition of ellipse (let’s call it a 2-ellipse) can be expanded to include 3 focal points, 4 focal points, 176 focal points, really focal points, no matter what integer is.

n-ellipse: Take all the points whose distances to each of n focal points sum to a certain number. This is the n-ellipse.

James Clerk Maxwell was sixteen years old when he invented a contraption that generalized the pin and string method for constructing an ellipse. He figured out a clever way to wrap the string around so that you could use pins and string to produce a 3-ellipse.

Screenshot 2018-01-03 at 10.08.25 PM.png

From the collected papers of Maxwell. He presented this at the Royal Society of Edinburgh. The three Fs are the focal points. 

3. Generalizing Equations

The equations generalize nicely too.

Circle (a.k.a. 1-Ellipse): \sqrt{(x-a)^2+(y-b)^2} = D

Screenshot 2018-01-03 at 10.17.17 PM.png

2-ellipse: \sqrt{(x-a_0)^2+(y-b_0)^2} + \sqrt{(x-a_1)^2+(y-b_1)^2} = D

Screenshot 2018-01-03 at 10.22.13 PM.png

3-ellipse: \sqrt{(x-a_0)^2+(y-b_0)^2} + \sqrt{(x-a_1)^2+(y-b_1)^2} + \sqrt{(x-a_2)^2+(y-b_2)^2} = D

Screenshot 2018-01-03 at 10.27.14 PM.png

Alas, no tex.

n-ellipse: \sum_{i=0}^{n}\sqrt{(x-a_i)^2+(y-b_i)^2} = D

(Most students learn a different equation for an ellipse, defined in terms of minor and major axes. It’s not exactly a load of fun to analytically derive that from this equation for the 2-ellipse, I’ve found.)

4. Smallest Ellipses?

Each of the definitions above leave room for a “certain distance.” But are there any restrictions on what that distance can be?

For a circle, that “certain distance” is the radius, and the radius can be whatever magnitude that you like. Given a center, make your radius huge, make it tiny: there’s always a circle of those points just that distance away.

gifsmos (3).gif

For a 2-ellipse, though, this isn’t so. Imagine one focal point that’s one million miles away from the other, and imagine that we’re interested in the points whose sum of distances from the focal lengths is one inch. That’s simply impossible to achieve. Being one inch away from one focal point would necessarily involve being nearly a million miles away from the other.

At some point, an ellipse might be too small for its focal points.

So: what is that least distance, given two focal points? What is the least amount of distance that you can sum to, given focal points? What are these critical points of the distance sum function?

For a 2-ellipse, the minimum distance should just be the distance between the two focal points.

gifsmos (4).gif

But what about for a 3-ellipse? or a 4-ellipse? an n-ellipse?

I’m still having trouble with the analytic solution in that paper, but graphically you can just graph a bunch of different distances and try to see where things are headed. Here is a snapshot from this Desmos graph, which graphs a collection of distances for three foci (which you can move around the graph — try it, it’s fun!).

Screenshot 2018-01-04 at 7.27.07 AM.png

This is a contour plot of the distance-sum function. We could also visualize this as slices of a 3D graph, where the “z” axis is the varying distance-sum.

Here is a 3D graph of a 2-ellipse, showing how it bottoms out at a very narrow ellipse (practically a line segment) at a height of 2 — the distance between the two foci:

Screenshot 2018-01-04 at 7.30.54 AM.png

And here is a 3D graph of a 3-ellipse, which also bottoms out at a certain distance-sum above 0:

Screenshot 2018-01-04 at 7.36.11 AM.png

In contrast, here’s the 1-ellipse, a circle. It has no minimum radius:

Screenshot 2018-01-04 at 7.37.17 AM.png

5. A cool problem

Say that you have three cities that are special to you, for whatever reason. Maybe you grew up in one of them, went to college in another, and have some family in the third.

If your only priority was to be as close as possible to each city, where should you live?

This is really, really similar to the question of where the minimum distance-sum is for an n-ellipse.

Screenshot 2018-01-04 at 7.40.06 AM.png

(It makes sense that it starts getting circle-ish as we get farther away from the focal points. From a million miles up in the air, those focal points might as well be on top of each other.)

Go to this graph and play around with your cities. You can also add more cities.

7. Now what?

I’m not sure. I tried to write this a few different ways, but in the end all I could figure out was the info dump you see above. Is there a way to rewrite this so that there’s a stronger narrative? Should I come up with problems for each section to give readers a chance to stop and think about stuff?

I think this needs to go on the back-burner, and I just need to write more about math and hope that eventually I figure a few more things out.

The toilet is going to eat my pee

“I’m going to tell you that the toilet is going to eat my pee,” he said. It was my son, nearly three, standing with his pants over his ankles. He held himself, umm, firmly.

“I’m going to tell you that the toilet is going to eat my pee,” he repeated.

“OK.”

“The toilet is going to eat my pee.”

“Got it.”

“The toilet is going to eat my pee. The toilet is going to eat my pee. Water is going to come out and it’s going to eat my pee.”

Flush.

“The toilet — it ate my pee.”

***

This is more or less where my head is these days. The last six months have been ridiculous in our tiny apartment. Minus the random and intense freakouts, our kid is becoming a kid — out of diapers, into a big-kid bed, counting and talking all the time. And our baby is teething, sitting, eating, even maybe sleeping.

But most of all there’s been a lot of poop, pee, yelling, family time at 3 AM, etc.

In the meantime, I’ve been feeling a little bit lost lately. I’m feeling stalled out on a bunch of projects that I was excited about. I’m having trouble getting stuff done. I’m tired, and I feel dumb a lot of the time. This sometimes happens, I’ve learned (even before having little kids) and there’s nothing to do about it, just work it out.

Part of that working out is happening on this blog. I don’t like it, but the only way I know to get out of a rut is to find a low-stakes place to just make an ass of myself, and that’s what I intend to do in this space.

Without getting too whiny, I’ll say that I’m struggling with the same things that I’ve struggled with since the start: what does it mean to be a teacher who writes? what do I know? what can I learn? where can I publish? am I headed anywhere with all this?

I need to figure out if there’s a way to do what I do, but in a way that feels a bit bigger than how I’m currently doing it. I’m not sure what “bigger” means yet but that’s part of what I’m working on.

(Oh god, kid: go to sleep! This bedtime is dragging on and on.)

So: follow my Problems With Teaching blog for things that are a bit more put-together than what I’ll be trying out in this space. There’s an automatic IFTTT trigger so that posts here get posted on Twitter, so you can subscribe to this blog or just follow casually via my @mpershan feed. If I end up blogging a lot here, I’ll try to cut down on my tweets so the internet isn’t over-Pershaned.

Posted in Fun

“A liquid moon”

Winter Trees William Carlos Williams
All the complicated details
of the attiring and
the disattiring are completed!
A liquid moon
moves gently among
the long branches.
Thus having prepared their buds
against a sure winter
the wise trees
stand sleeping in the cold.                                                                           

What of the moon can be said to be liquid? Not the rock itself, but rather its light, which flows from surface to reflective surface (or from branch to branch).

This reminds me that we only know of the moon as reflected light — save for a few of us who have actually gone and touched the darned thing. The moon can appear huge or small, white or orange, but all this is really a trick of the light, which over time flows into various shapes and colors.

And that reminds me that literally all that we see is only reflected light, and is liquid in that same sense. Some things we touch most things that we know, for most of us, we know through seeing (or our other liquid senses — smelling, hearing). The entire visual world itself is liquid as well, in this sense, and all these images and reflections flow like the moon does. The “wise trees” tell us this truth, about the flowing nature of our perceptive worlds — that’s my read, at least. Your’s?

Chimps are better than humans at some working memory tasks

Amazing. From the research report:

Chimpanzee memory has been extensively studied. The general assumption is that, as with many other cognitive functions, it is inferior to that of humans; some data, however, suggest that, in some circumstances, chimpanzee memory may indeed be superior to human memory. Here we report that young chimpanzees have an extraordinary working memory capability for numerical recollection — better even than that of human adults tested in the same apparatus following the same procedure

Watch that video for details of the test. The biologists think this might be connected to eidetic imagery:

These data showed that the chimpanzee subjects can memorize at a glance the Arabic numerals scattered on the touch screen monitor and Ayumu outperformed all of the human subjects both in speed and accuracy. Our results may be reminiscent of the phenomenon known as ‘eidetic imagery’ found by Jaensch. Eidetic imagery has been defined as the memory capability to retain an accurate, detailed image of a complex scene or pattern. It is known to be present in a relatively high percentage of normal children, and then the ability declines with age…The results fit well with what we know about the eidetic imagery in humans.

To me, this raises the tantalizing possibility that our working memory limitations are a core part of what it means to be human. We’ve been designed to hold fewer things in our head at once.

Other things I want to read, in relation to all this. I’ll add more if I find (or if people find for me!) other related reads.

Evolution of working memory

Does this connect to dual process theories? Dual process theories of higher cognition