Monthly Archives: September 2016

slopes

Vi Hart Weekly, Sept 27, 2016

Welcome back to Vi Hart Weekly! As you know, every week has a theme, and sometimes when the theme is less obvious people enjoy drawing their own connections and trying to guess it. Spoiler: last week’s theme (or at least the one I intended) was tiny things / scale.

Now on to this week!

1. Improvisation of the Week:

Some piano + voice improvising I did after work on some day I forget. I forget what it sounds like, because getting it out of my brain is what recording and sharing things is all about. Bye piano thing!

I like Audacity for sound recording because it is so very simple. It doesn’t do hidden secret things or try to be smarter than the user. But when recording this I had a problem where occasionally Audacity would insert a bunch of zero values into the samples. All the information is there on either side, just these extra bits create an awful click sound.

I was hoping to be able to quickly search the samples for strings of zeros and remove them, but unfortunately they’re not quite zero, making it a little harder. Just getting rid of strings within a certain range could false positive on soft sections too. Really the thing to do is look for sudden changes in slope that go to/from zero. I managed to get rid of most of the clicks!

2. Vacation of the Week:

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I’m feeling good and am later than usual for this weekly because I managed to take three days of vacation!

Says my brain. Really two of those days were the weekend, brain, and those don’t count. Ok, so one day of vacation, says my brain. Except I worked until 2pm before leaving for said vacation, so.

But pretty good, doin’ pretty good on the actually-take-time-off-and-don’t-burn-out thing. I only filmed one video while on vacation and I didn’t even edit it on vacation. Taking time off is important for being able to continue doing work, but it’s easy to look at that time and regret that I got so much less done in the past week than in weeks when I work constantly! A reason not to judge productivity on a weekly basis. Need larger sample sizes to smooth over the zeros.

3. What’s up at eleVR:

screen-shot-2016-09-27-at-4-26-21-pmWe’ve been spending some time lately organizing our many projects and blog posts, and I’m constantly amazed at how much we’ve done. So many experiments forgotten and dug up as we go through old bits of research. Our projects page is awesome.

One of these old experiments we never documented was when we tried making 360 video with just a normal smartphone and a cheap clip-on wide-angle lens, to see how inexpensive simple consumer VR creation could be. So I wrote up a post about that, and cut our old documentation footage into a video (in the post too).

But the big news is that Evelyn Eastmond is officially joining my team! I’m really excited about this because she is exactly the kind of deep-thinking world-aware artist-programmer I love working with, and I’ve been wanting to work with her for a while. So I’m just thrilled that she’s agreed to join. And now she’s in the regular eleVR blog rotation too, so you can check out her posts.

4. Vi Hart Video of the Week:

I posted about this in my previous blog post because videos sometimes get their own posts, but it’s still the Vi Hart video of the week.

5. Paper of the Week: Inventing Graphing: Meta-Representational Expertise in Children, by diSessa, Hammer, Sherin, and Kolpakowski

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Recommended to me by Paula Te in relation to my work on Real Virtual Physics, this is a fun paper that describes a 5-day activity with a group of sixth-grade children where they are encouraged to invent and improve on different ways of representing motion.

It’s interesting to read through the thought processes of the kids as they critique various invented representations, and move towards things resembling traditional representations of motion (distance over time, speed over time, etc). It was also interesting that a big sticking point was how to represent the duration of a stop. Ah zeros, always messing things up.

I think there’s something very important about understanding what kind of thing a representation-of-motion is, and what information it should contain, that can only be gained by working with more than one kind of representation-of-motion (even if some of them aren’t very good). Once one knows what kind of thing it is, then looking at traditional representations makes more sense.

It’s also interesting to read transcripts of bits of students’ discussion, and how critique of peers—who are indeed the world experts in their own invented representation—can motivate deeper thinking than discussion of an existing standard representation (which even the teacher might not know the reason for various choices).

Lookin’ At Slopes: The Calculus of Bad Driving [part 1]

I made a video! Again! It’s just a thing that happens sometimes.

This one is a true story about when I got mad at a taxi driver that skidded through the crosswalk in the rain, and why they were DEFINITELY DOING MATH WRONG.

Part 2 will be out next week, maybe?

Also, obviously I started writing this script a long time ago because it hasn’t rained in ever. But I still think if California focused more of its policies and funding towards higher quality math education, it would make our streets safer.

Transcript:

Today, I was walking across the street and this taxi comes zooming up to the crosswalk at a red light and slams on the brakes and skids across the crosswalk a foot in front of me and I’m like, dude, it’s raining, do you not realize how that affects the derivatives of your position over time, my goodness, I was one foot away from being turned into a pancake by your bad math education.

Now, every driver knows that speed is what happens when you change your position over time (such as the changing position of your car as it approaches the position of the crosswalk) and that acceleration is a change in speed over time (such as, y’know, how even if you change your speed by decelerating as much as your brakes and road conditions allow, you will still be going a positive speed by the time you reach the crosswalk, ). What many drivers don’t seem to know is that these changes are related by mathematical laws. Some drivers are like, you’re here, and then later you’re there? No one can explain this.

And, I can understand that. The idea that anything ever goes anywhere is kind of tricky if you’re Zeno and calculus hasn’t been invented yet. I mean, say you’re 20 meters from the crosswalk. How does one hit pedestrians if before you can get to the crosswalk you have to drive halfway to the crosswalk and then you have to drive halfway between there and the crosswalk and then halfway between there and the crosswalk? and so on. If each of these steps took the same amount of time, then that would be quite an interesting deceleration, you’d never hit anybody like that. But say you drive those first 10 meters in one second, and those next 5 meters in half a second, and the next 2-and-a-half meters in a quarter of a second, it doesn’t matter how many infinite bits of distance you’re adding up, you can break apart those 2 seconds and 20 meters in whatever way you find interesting but 2 seconds later you’ve still gone 20 meters and 2.1 seconds later you’re still trying to ruin my day.

There’s this stereotype about California drivers that whenever it rains, which is rarely enough these days, traffic stops because all the drivers are freaking out like what is this substance all over the ground, we don’t know how to do math to it.

The common wisdom seems to be that when it rains, you should just drive slower. A classic error of calculus, because it’s not really the speed that’s the problem with rain, but how it affects acceleration.

It’s like this: you’re goin’ along at a constant speed, uh, this is time and this is speed and this line is nice and flat so no change in speed is occurring, you’re just driving at 50 miles an hour. But then, oh no, there’s something in front of you, so you slam on the brakes. Now your speed is decreasing, decreasing, until you hit a speed of zero and stop. If you’re at a slower speed to begin with, then this line intersects zero earlier, you can stop faster. So far so obvious.

The slope of this line changes depending on your car and on road conditions: maybe you come to a stop real quickly, or maybe your brakes are bad or the road is icy and you just kinda glide for a while until finally you hit zero. Your car might be able to decelerate real fast when it’s dry, but not so fast when it’s raining, and then even if you start out slower it might take longer to actually stop. You can’t just drive slower, you have to leave more distance between you and the car in front of you, and start braking earlier when you’re coming up on a light, which is why that’s what they tell you in driver’s ed. Whereas if you’ve got lots of room and can decellerate for a long time, you can start at a greater speed even if it takes a while to decelerate to zero. Which is the part they don’t want to tell you in driver’s ed.

Of course, this is graph is kind of misleading because it’s not like the crosswalk is here, this axis shows time, not place. And when we need to stop, we usually don’t care about when to stop so much as where to stop. This graph shows the speed of a taxi that needs to stop at a crosswalk, but let’s overlay the position graph in red, same time axis, different y axis, so we can show where the crosswalk is . Here’s where the Taxi is when I see it coming towards the crosswalk, here’s the crosswalk, 20 meters away. So the driver is going along at a constant speed, that’s this nice linearly increasing distance, realizes it’s a red light and slams the brakes here.

It’s slowing down, and the distance over time starts this nice deceleration curve. Of course, in my case it doesn’t reach the flat zero slope of a stopped car until it’s gone through the crosswalk. Wish it could stop sooner, but once you decide to stop, there’s a max decelleration, you can stop faster if you have better brakes or less momentum or if the ground is dry but there’s always a max slope your speed can drop, which means a max curve your position can take so there we are in the crosswalk. 

Accelleration, speed, and position, these things are related so don’t run me over in the rain. Lookin’ at slopes.

But the story doesn’t end there. We’re leaving off with the taxi driver stopped in the crosswalk but what happens next will surprise you. Or, not really so much, but I wanna talk about hover cars? Anyway see you next time for part 2.

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Vi Hart Weekly, September 18, 2016

Welcome back to Vi Hart Weekly! I am glad you are here!

1. Designer of the week: Otto Zenke

In particular, his miniature rooms, of which there are several in the Greensboro history Museum, which I was at last week. Curator Jon Zachman was kind enough to let me put a spherical camera inside one of these tiny rooms and tell me a little bit about history behind them. It’s a pretty compelling use of spherical video:

2. Geometric space of the week: Hyperbolic Three-Space

This week I finally got around to documenting a project I did with eleVR, Henry Segerman, and Mike Stay, back in 2014. Or at least, I documented a little bit of it. We made hyperbolic virtual reality.

In this video I show an unusual property of an unusual tiling of three-dimensional hyperbolic space. Hyperbolic space gets really big really fast, but not fast enough to fit six cubes around an edge without something very interesting happening at the vertices:

3. Book of the week: The Tiger’s Wife, by Téa Obreht

The Tiger’s Wife is a subtle piece of literature. In some ways it is very small and personal: one young woman recollecting stories surrounding her grandfather, in the days following his death. It only very subtly connects to bigger themes and bigger events, in a way I very much appreciate. It is well-written with evocative imagery, and just the occasional hint of that sort of literary cruelty à la Infinite Jest, if you know what I mean.

4. Pint of the week: Organic Sugar Plum Grape Tomatoes

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Tiny tomatoes are, in general, the best food. This is because, being tiny and numerous, a pint of them has more surface area than a pint of larger tomatoes. Sometimes cherry tomatoes are tiny enough. This time, grape tomatoes and the best tomatoes. Because of their size and elongated geometry, they are almost entirely made up of outside.

Because they have so much outside to eat, and that’s where pesticides tend to sit, the organic component is extra important when eating a pint of tiny grape tomatoes.

5. Paper of the week: Assessing student learning of Newton’s laws: The Force and Motion Conceptual Evaluation and the Evaluation of Active Learning Laboratory and Lecture Curricula, by Ronald K. Thornton and David R. Sokoloff

This is related to my work on Real Virtual Physics. It describes the assessments they did with students using tracked physical objects to learn about speed, acceleration, and visualize them as graphs, etc.

My favorite thing about this paper is that when discussing student answers it doesn’t refer to answers as being right or wrong, but instead refers to answers as either aligning with Newtonian physics or not. It mentions some other conceptual frameworks students use to think about motion, and the idea that having a model of physics in your head, even if it’s wrong, is it good sign of student ability to learn. It reminds me of some some of the stuff Papert talks about, that memorizing laws of motion is pointless if one doesn’t know what kind of thing a law of motion is. Papert recommends students get experience with microworlds that have different kinds of laws of motion, before trying to learn what the laws of motion of the actual world are. Of course, Newtonian physics is not our most accurate way of modeling real motion, so I appreciate this paper’s emphasis on teaching students to think in a Newtonian way, rather than teaching them the right answers.

My least favorite thing about this paper is that it is evaluating a curriculum and learning tools at the same time that it is evaluating the evaluation used to evaluate those tools. The results could mean nothing more than that teaching to the test yields higher test scores. They do address this issue by correlating responses to different question types, and it sounds convincing and everything, but it certainly leaves more work to be done.

Paper is online here.

6. News Introspection of the week: explosion in Chelsea

Whenever a breaking news story is going viral on twitter, I take the opportunity to introspect by asking myself what I want to be true. This kind of awareness is important to limiting the effects of confirmation bias and becoming more able to understand the world as it is.

It’s easy, on twitter, to judge others and get outraged about what they want to be true, but that’s less helpful. Some people want whatever truth supports their politics. Some people like big news because big news is exciting, and our human brains can’t help wanting exciting things sometimes. Me, I’ve learned I like tiny news. Maybe I have enough excitement in my life.

I’ve practiced news-introspecting enough to recognize, when I saw Chelsea trending, that I wanted the explosion to be minor and accidental, that I would be drawn to things that support that wish, and more skeptical of reports that it was an organized terrorist attack that caused serious harm (and more reactive against anyone speculating that might be the case).

It’s looking right now like the truth is somewhere between those things, an intentional act of terror that is not connected to outside groups, and that we are very lucky in how minor the damage was. And I’m aware that now I am happy to stop following the story, while those who are hoping a connection to terrorist groups will be discovered are more likely to keep following, so if evidence in that direction is discovered (even if it’s not certain yet) their beliefs will be confirmed, and then they can stop following too, so we all get what we want. Hooray!

7. Pumpkin Spice of the Week: Pumpkin Spice

Documenting this thing because, well, because:

 

The last screenshot is from this site that analyzes headlines for clickbaitiness: http://coschedule.com/headline-analyzer

I did that bit because it is my headline art.

Seriously though, pumpkin spice seems to be one of those cultural things that secretly broadcasts some secret group affiliation. Also I would actually like to have some, but it turns out that not all places are forcing pumpkin spice into everything. Instead I have no pumpkin spice but lots of over-analysis.

THE END

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Vi Hart Weekly, September 11, 2016

Welcome to Vi Hart Weekly! As you’ve probably noticed, a common thread between each Weekly is that each one has a theme, a common thread. This week’s theme is Common Threads.

1. Art of the Week: Where We Met

I live in SF and travel a lot, so I regularly get to pass under Janet Echelman’s “Every Beating Second” in the San Francinco airport. Last weekend I got to see one of her newest installations in Greensboro, NC, “Where We Met”, and it’s something else entirely. Outside, subject to the wind and weather and changes in light, it has a life and presence unlike any public art I’ve ever seen. Day and night, the park is full of people lying in the grass underneath gazing upward.

Perhaps because the sculpture is still relatively new, I overheard many conversations from people discussing the sculpture, an exercise in noticing, inventing and asking questions to each other. What does the shape have to do with the tension of the strings? Is the color at night because of the color of the art piece or the color of the lighting? What material is it made out of, how strong is it, how heavy do you think the entire sculpture is? Does the shape mean anything or is it just random?

It’s beautiful to overhear such focused noticing happening not in a gallery, but in a public park, by all sorts of groups of people. And quite wonderfully, answers to many of these questions can be found in the Greensboro History Museum right next to the sculpture, in an exhibit titled “Weaving Wonder with Historical Threads”. I was pleased to be able to touch some of the material the sculpture is made out of (hooray tactile learning!), as well as learn the secret behind the sculpture’s design, and its connection to the textile industry.

A related mini-exhibit, right inside the entrance to the museum, asks visitors to weave threads according to their own answers to a few questions (and the color thread depends on the visitor’s home location). I love the simplicity and tactility, and that the result is a sort of infographic data visualization thing.

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2. Book of the Week: Dr. Montessori’s Own Handbook

Continuing the theme of education and active learning, and realizing that while I knew a lot about Montessori education but had never read anything written by Maria Montessori herself, I read “Dr. Montessori’s Own Handbook”. It’s quite interesting to read the very precise curriculum descriptions and motivations for every detail of every activity and how it contributes in subtle ways to a child’s future learning. While I was familiar with the overarching ideals and structure of Montessorian education, I hadn’t realized how detailed, thorough, and intentional a curriculum existed, one which apparently most schools labeled “Montessori” follow only in part, if at all.

Many of the educational materials and concepts she invented are familiar to me, but some seem to have been gotten left behind, including exercises involving feeling different materials, a children’s haptics library. Recall Margaret Minsky’s haptics library, which we discussed a couple weeks ago! Now I want to play the Montessorian game where I gather a bunch of material samples in pairs, mix them up, and have to pair them back up blindfolded, by developing and using various sorting strategies, as well as developing my ability to focus attention and achieve completion of a task, as well as learning the delicate touch and physical delight in my environment and all those things she describes so eloquently in her curriculum.

3. Noticing Math in Things of the Week: Jump Roping

I went to the National Folk Festival in Greensboro NC last weekend, and while I saw many excellent performances from all over the world (the festival is national but the folk is international), one particularly caught my mathematical interest.

I’d seen some competitive jump roping before, and it’s pretty impressive. But at the festival the Bouncing Bulldogs did some moves I’d never seen, involving multiple people holding more than one jump rope, for example three girls with three jump ropes between them, weaving themselves over and under the shared jump ropes in a way that must require a great deal of experience and intuition with certain topological patterns. Here’s the closest thing I could find on YouTube, by another group:

I’ve heard of jump ropes as an active educational tool for things like counting and number patterns (jumping rope while counting up by twos, down by odd numbers, or even just using the string of the jump rope to measure, braid, or… perhaps weave?), but there’s some deeper mathematics going on in these multi-person jump-roping tricks that I believe have yet to be vigorously studied. I think there’s a lot of fertile ground here, and the art of jump roping is ripe to be revolutionized by mathematics in the same way juggling was. Also, the world could always use more great mathematicians, so it would be good if we could figure out exactly how their intuition for these forms could transfer to existing mathematics.

4. Snake of the Week: This One Particular Rat Snake

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This is my new snake friend! I didn’t realize they made them so big in North Carolina. 

Not for jumping rope with.

5. Presidential Chocolate of the Week: Obama Kisses

Before North Carolina, I went to Washington DC to meet with some folks from the Office of Science and Technology Policy about active learning in mathematics. There’s not much to say about the meeting itself, besides that they seemed to pick a good group of people to meet with them, and if that meeting is the start of a conversation rather than a full one, it won’t have been a waste of time. Also it was nice to be recognized by the Chief Technology Officer of the United States, Megan Smith, who I think is pretty great. But the most tangible result of the meeting is that apparently when you have a fancy White House Meeting you get Obama Kisses. Every liberal’s dream. I’ve got to stick with this education policy thing so that I can dream of Hillary Hugs someday.

Each kiss is threaded with a paper ribbon that I was hoping would say something like “thanks for visiting, sure wanna hear more of your policy ideas later”, but it’s just the usual message of love:

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I’ve got a lot of thoughts on the meeting, and education policy in general, that will take me some time to sort through and write up.

6. Never Forgetting of the Week: 9/11

The bulk of my family lives in North Carolina, and so while I was in the state, last night we had dinner together. Given the date, we got to remembering 9/11. Where we were when we heard, what our day was like, and of course, the family stories, which are not mine to tell.

As I was telling my own recollections of the day, which I haven’t thought about in many years, I couldn’t help but notice the difference in technology and communication between then and now. I was in school, and this was back when kids generally didn’t have cell phones, much less smart phones, much less laptops. Most classrooms didn’t have a computer, much less an internet-equipped computer. There was a loudspeaker announcement that told us about the plane crashing into the tower, and that announcement didn’t tell us why or how big the plane was or how bad the crash was and we were left to speculate, with no other way of gathering information, while the teachers were hearing the same thing we were, expected to go about the day as usual.

Oh, I should mention: this is in New York. I grew up in New York, and almost all my family lived there at the time. We have family and family friends who were in and around those towers. But by the sound of the initial announcement, it wasn’t serious, it was probably a small plane that accidentally crashed because of an instrument failure or something. So if I were to tell you where I was when I heard the news, well, I heard the news for the first time slowly, in many places, across the day, with increasing horror. I heard the news without twitter (which didn’t exist) or YouTube (which didn’t exist), I heard it without an iphone (which didn’t exist) or facebook (which didn’t exist). I heard it in bits and rumors, it was forbidden knowledge, the teachers weren’t supposed to be telling us any news, and generally the administration was trying to get the school through the day.

But our 2nd period teacher turned on the radio (a radio, an honest-to-goodness radio!) and we sat listening quietly all period. I was listening when the first tower went down, and when the second tower went down, and those of us who cried did our best to cry quietly, and then the bell rang and we went to the next classroom and tried to pretend everything was ok so that our rule-breaking teacher wouldn’t get in trouble for not keeping us in the dark.

As the day went on, in rumors and speculations not unlike twitter today, we tried to sort out what was real. The pentagon was attacked? Really? Surely that’s something a younger student made up?

It was a long school day. It ended. I went out to the pay phone and called home and finally learned that everyone was, whether by accident or skill, alive. And then at home we sat by the tv and watched the news, the same footage over and over and over and over

I learned not to obsess like that over horrific news footage ever again.

Today I’m coming home from visiting my family. I wrote this Weekly on my flight back to the west coast, and we’ll be descending back into SFO with its lovely Echelman sculpture soon, “Every Beating Second”. In the last airport, eating a sandwich before my connecting flight, I looked up and was shocked by the sight of the footage on the tv, the smoke billowing from the tower, the clear blue sky in the background. It hit me with a visceral force.

I remember smelling that smoke for a long time after. Our perfect blue skies never quite returned.

7. Child On A Plane of the Week

Despite that I’m not that into children or planes, I actually quite like children on planes. They’re a controlled and restrained breath of fresh air in a crowded box full of adults desperately pretending not to notice each other. The little girl in front of me is asking “Would you rather be a solar system, a hedgehog, or Hermione?”

It does sound very nice to be a solar system, but I think I’d most like to be Hermione, who with all her power and intelligence dedicated her life to doing what is right rather than what is easy.

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Vi Hart Weekly, September 4, 2016

Welcome back to Vi Hart weekly! This week we’ve got the robots of the week, embodiment, education and beer, turtles and snails.

1. Robots of the week

During a group discussion of collective intelligence, one of my colleagues brought up the work of Radhika Nagpal, which brought us to watching YouTube videos of the termite-like Termes robots she developed at the Self-Organizing Systems Research Lab at Harvard.

One thing led to another, and we started to act out robot algorithms on the floor. We were all aware of previous work using embodied acting-out-of-algorithms to help teach and understand them (examples include Andrea Hawksley’s Binary Dance workshop, folk-dance sorting algorithms, and even some of my own work with Mike Naylor on Human Geometry). But rather than follow up on any of that, we quickly filmed what we were doing and went home, and then I proceeded to edit together a ridiculous video to convey my full joy at the endeavor in which we had decided to partake.

2. People noticing patterns of the week

After seeing “Radhika’s Robots”, artist and collaborator Evelyn Eastmond sent me a message noting the connection to her latest video (above), which in fact I had seen just a couple days before. I’m glad Evelyn noticed this connection in form, because I completely missed it! We followed the form of the termite robots to amuse ourselves, while forgetting that this form also displays the weight and burden which the robots carry for the collective good.

A commenter on “Radhika’s Robots” noticed another connection:

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In fact, a few other commenters referenced snails, saying, “Snail snail snail” and “Snail snail snail snail” and such. It’s a thing.

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3. Paper of the week

The paper of the week is “Teaching Children to be Mathematicians vs. Teaching About Mathematics“, by Seymour Papert. No, wait, I changed my mind. The book of the week is going to be by Papert, so let’s have the paper of the week be “Children Learning by Doing“, by Alan Kay. It’s labeled “rough draft”, but it collects a lot of previous material he’s written about Squeak Etoys and learning into one place.

I’ve talked a lot with Alan about this particular work, and a couple months ago I started to make a prototype bringing some of these ideas to virtual reality. This week I started to do some more serious work on this prototype, so I wanted to dig deeper into a lot of the stuff we talked about, and get more context for the conversations we’ve had about learning and math education in general.

Alan has a lots of faith in children’s ability to understand deeply complex ideas, if they have the right support. I haven’t done much educational material on purpose, but as long as the things I make tend to often fall into that category by accident, it’s good to get a deeper background into what’s already been done.

4. Book of the week:

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The book of the week is Mindstorms by Seymour Papert. This is a classic on computers in education (really THE book), yet somehow I hadn’t read it until now. After reading some of Papert’s other papers this week though, and realizing just how good he is, I couldn’t resist!

Above is a picture from the cover of one of the editions, showing the “Turtle” robot that can be programmed by simple instructions in LOGO, the language Papert developed with children in mind.

Mindstorms is great. I love the emphasis on the culture and context of education that allows children to be immersed in mathematics, the same way languages are best learned when immersed. I love that he highlights debugging as being a powerful intellectual tool for emphasizing process and progress over right or wrong results. I love that he talks about how a large part of what mathematicians do happens before equations get written down, that mathematical thinking is not just about logical processes but includes this other thing that lets us know when and where to apply those logical processes, the creativity and intuition that point our logic in the right direction.

And of course there is the common thread with my current research group about tools-to-think-with, and when applied to education this means putting children in the context where mathematical tools are the tools they need to think with to reach their goal. Learning mathematical tools is a means, not an end.

There’s a lot I can say about this book, and it’s easy for me to see many ways in which it has influenced the thinking of people I know and of the education system at large. I recommend it. 

5. What’s up at eleVR

The prototype of Real Virtual Physics is at a place where I’ve started sharing and writing about it. And being kept up at night by new ideas for toys to add to it.

There’s something profound about having real-time feedback based on your body’s motion that can give you a feel for how things behave. That’s part of how I’ve always thought about mathematics: the art of understanding how abstract objects behave.

In Mindstorms, Papert likens understanding mathematics to understanding a crowded room of strangers. It might take a while to get to know them, and introductions help, but in the end it’s all about your own personal connection with those people and understanding how they behave, how they react, what they do and why. It’s a good metaphor. When giving mathematical objects abstract inputs that must be computed by hand or one by one through a computer, it’s difficult to get information on how things behave. But real-time feedback based on your own body’s movements makes it so easy to explore, if only we had the tools to do so. So I guess I’ll keep working on making those.

6. Beer of the Week

No Hero, by Evil Twin Brewing. It’s a simple dark and lovely oatmeal stout in a world of overly-ambitious too-many-flavours stouts, and it comes in a can. It’s got a bitter finish without the bother of being all hoppy, chocolatey, or coffeey. No Hero works behind the scenes to save the day.

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