Posts tagged metaphor

Example spaces and the hedonic change

Oct 21st

Posted by MariaDroujkova in Life notes

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I think about instructional design as an art. So its rules must be open and few, to allow emerging systems. One of the top rules is The Rule of Many – namely, “The Harvard rule of three” since three is many. Any math entity children experience needs an example space of three or more things, some of them made by children.

For example, if you are offering kids operations, don’t just stop at addition – show several different ones and invite kids to make up their own. Exponentiation is fun – here’s a hands-on example of it: http://youtu.be/TR_8SDNQ0ks

I just read a blog post by Seth Roberts illustrating the importance of The Rule of Many.  Seth writes:

The Willat Effect is the hedonic change caused by side-by-side comparison of similar things. Your hedonic response to the things compared (e.g., two or more dark chocolates) expands in both directions. The “better” things become more pleasant and the “worse” things become less pleasant. In my experience, it’s a big change, easy to notice.

I discovered the Willat Effect when my friend Carl Willat offered me five different limoncellos side by side. Knowing that he likes it, his friends had given them to him. Perhaps three were homemade, two store-bought. I’d had plenty of limoncello before that, but always one version at a time. Within seconds of tasting the five versions side by side, I came to like two of them (with more complex flavors) more than the rest. One or two of them I started to dislike. When you put two similar things next to each other, of course you see their differences more clearly. What’s impressive is the hedonic change.

I sent this post to Dor Abrahamson, who recommended an article that indicates “connoisseurship” may be more complex: Learning Concepts and Categories: Is Spacing the ‘‘Enemy of Induction’’? by Nate Kornell and Robert A. Bjork

ABSTRACT—Inductive learning—that is, learning a new concept or category by observing exemplars—happens constantly, for example, when a baby learns a new word or a doctor classifies x-rays. What influence does the spacing of exemplars have on induction? Compared with massing, spacing enhances long-term recall, but we expected spacing to hamper induction by making the commonalities that define a concept or category less apparent. We asked participants to study multiple paintings by different artists, with a given artist’s paintings presented consecutively (massed) or interleaved with other artists’ paintings (spaced). We then tested induction by asking participants to indicate which studied artist (Experiments 1a and 1b) or whether any studied artist (Experiment 2) painted each of a series of new paintings. Surprisingly, induction profited from spacing, even though massing apparently created a sense of fluent learning: Participants rated massing as more effective than spacing, even after their own test performance had demonstrated the opposite.

Kornell.Bjork.2008a

There is probably a flow channel between massing and spacing, but it’s not clear what features of learning this particular balance involves. It can be noticing similarities vs. differences, or inductive vs. deductive thinking, or more esoteric balances within emotional and beauty responses (“the hedonic”).

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How I imagine change

Oct 19th

Posted by MariaDroujkova in Life notes

3 comments

The discussions, collaborations and collective actions aimed at change are continuing to increase in intensity, depth and scope. It makes me want to clarify how I imagine change. In my mind, it’s a two-step process.

Step 1. Deep inside, say good-bye and get detached from the system you want to replace. Withdraw creative and social currencies such as attention from it.

  • “One who should inspire and lead must be defended from traveling with the souls of others.” So, progressors need to share a “magic circle” such as a game (D&D?), abstract math, spiritual practice (40 days in a desert?) or a sci-fi universe (Babylon V?). Their third place has to be sufficiently out-of-this-world.
  • “Scales, times and places are declared largely irrelevant.” It matters little anymore if the old system involves millions of people, has been going on for decades and is happening where one lives. This makes it somewhat easier to be brave.
  • “Don’t seek legitimacy from dominant institutions.” We can seek resources from them, if it does not mess up the emergent economy of the new systems. Just don’t make that grant from an old system one of the main promotion points.
  • “Reject the act of labor required for everyday production.” Well, not all of it – don’t starve – but a lot of labor should go into creating new systems rather than participating in the old ones.
  • “Nothing is created until something is destroyed.” The destruction here is purely informational. Shift attention to new systems, and stop caring for and discussing the old ones. Attention is an incredibly strong currency. Criticizing a system only makes it stronger, by investing the attention currency into it.
  • “Don’t equate the detachment with disappearance.” Be visible and welcoming to people participating in any system, especially during Step 2. As an aside, this increases the personal safety, since it’s harder “to disappear” visible people.

Step 2. Build.

  • “Tell me, do you stand up and speak out when you encounter a moment of unexpected joy, warmth, beauty or compassion in your life?” Be a support activist for good people. Pay attention (place the attention currency) in new systems, and spread the word.
  • “Don Quijote didn’t ship.” Take on tasks that can finish successfully and quickly. Don’t discuss redoing all k-20 curriculum “come the revolution” – help a next door family appreciate a math topic, today, and share the know-how with colleagues.
  • “Think globally, act locally.” Make working prototypes that work well and grow. This means making a lot of prototypes that don’t work or don’t grow, and discarding them quickly.
  • “Ragtag bunch of misfits” is a trope about unlikely heroes winning the day. It’s heartening to believe it.
  • “Loving one another in the context of Perl.” Don’t fight within the bunch, just because fighting big old systems is too frustrating and hopeless. Actually, don’t fight, love. In the context of the new systems.
  • “Do small things with great love.” Because new systems will be small at first. Love is the engine of growth. Don’t let that love stand in the way of discarding prototypes as needed. It has to be a non-attachment love.
  • “Unlimited self-generated morale.” As well as other self-generated and emergent entities: DIY structures and sustainable economies. Collaborations within networks are fine, but dependencies on the systems being replaced are problematic.

When in this process do we fight the old systems? In my picture, never: the new systems just recruit and grow until they are as strong as necessary. Old systems can then die from the lack of recruits, continue to support those who love them, or evolve into something new. It’s none of my business what they do. The only type of fighting I support is the immediate defense of projects from hostile takeovers. This is, for example, the tactic of homeschoolers. Homeschooling, the second-fastest growing education system, normally pays little attention to school systems. But news of any action that would restrict homeschooling freedom spreads through the networks within hours, and meets a very strong response that is usually enough to prevent it. Online education, currently the fastest growing method, does not directly fight any old systems either. It just grows by hundreds or thousands of percents a year.

So:

  1. Let go of old systems, in the heart and in the mind
  2. Build new systems
  3. …
  4. WIN!
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Curriculum as a platform

Oct 16th

Posted by MariaDroujkova in Life notes

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I spent a good part of this weekend reading and talking about discussions of Steve Yegge’s escaped internal letter to his colleagues at Google, called “Stevey’s Google Platforms Rant.” You can find one of the discussions here, for example: http://news.ycombinator.com/item?id=3101876 It has been spreading in programmers’ circles.

It provoked a revelation for me – words and images I can use to explain what my work is all about. Natural Math motto: “Make math your own, to make your own math” means curriculum as a platform, rather than a product. The platform has curated content created with an open API, so to speak. The community of practice co-producing the system needs a flat structure – a rather distributed, fractal network.

Features of curriculum as a platform:

  • Materials are extensible, so users – students, study groups, developers – change them continuously
  • User groups are peer-to-peer partnerships or co-ops, helping everybody to contribute
  • Contributions are transparent, acknowledged, honored and commented upon
  • Groups have tools for sustaining the flow by tracking individual tasks, time, and progress, possibly in playful ways
  • Tracking tools help creative, social and monetary economies of the system to stay sustainable
  • The platform has starter high-quality content: “killer apps” created on the platform
  • Ways to contribute are simple, open and creative: neither rocket science, nor worksheets
  • With special tools, users curate the content based on shared values within user groups: they make collections, distill most useful parts, sort, and tag

For example, we are building “Moebius Noodles” as a platform for advanced young math. During the crowd-funding campaign in September, we announced “Moebius Noodles” as an extensible, live and open system. This invited a very heartening stream of content offers, both from excellent veteran educators and authors, and from parents who wanted to share, for the first time, what they are doing with their kids. I consider this fact an early proof of concept. I can’t wait to see the system in action.

Games and revolutions

Oct 8th

Posted by MariaDroujkova in Life notes

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Games devoted to fluency rather than creative mastery or exegesis of the subject matter are tools of continuity. They attempt to maintain a firm, secure grip on the generational transmission of information. The grip is slipping in the digital, co-produced, post-book world.

Kids These Days (TM) write and otherwise author several orders of magnitude more than kids used to author.

Can drill games maintain the continuity? Is this one of the reasons it’s sort of hard to find financing for games that can teach kids to lead interesting lives through developing a subversive attitude toward the status quo?

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Metaphors explaining Khan Academy

Jun 23rd

Posted by MariaDroujkova in Life notes

1 comment

Me dearest, in comments at Action-Reaction:

There is a need for an ecosystem of different tools and communities for mathematics education. I doubt any one community (or person, in Khan’s case, originally) can build all tools for all needs.

The best each of us can do is create our own OER materials, or remix other people’s materials. There is a natural tendency in social media (and of course broadcast media) to aggregate attention on a few people. Despite this tendency, there are thousands of OER content creators for mathematics (the field I know), and there are more appearing online every week. Let Khan do his thing, and do your own!

In comments at Teaching your Middle Schooler:

My daughter likes Khan Academy for reference. Basically, she uses it as a dictionary of algorithms. She does not like badges, gold stars, or any of that (Alfie Kohn’s poster child, lol) so she does not feel like trying the tracking system.

Alexandre Borovik, in comments at Mathematics under the Microscope:

As a revision tool for standard tests the Khan Academy could perhaps be VERY good — I do not question that. But you in effect compare KA with a microwave: not very convenient for serious cooking, but great for re-heating ready meals.

Mike South, in Living Math forum (members only):

So I would think of it like this–suppose Joe Trailblazer had blazed walking trails throughout a vast stretch of wilderness that previously was only utilized by people who went in with expert guides.  Joe realized that a great deal of what the guides contributed was simply knowing their way around, and said, I can solve this once for everyone, and did.
Then Clarence Crybaby comes along and says “A bunch of paths is no replacement for an expert guide!  Is a PATH going to point out to you the sounds of the mating call of [insert exotic animal]?  Is a PATH going to point out the beauty of [insert really interesting thing some particular plant does that you would never notice unless you knew what to look for].  And all these people are praising him like he’s some kind of god!  This just makes things worse!  People will think we don’t need guides any more, and all this is doing is reinforcing the mistaken idea that all there is to the wilderness is knowing your way through it and back!”
Clarence actually has a very good point–there is much more to getting value from the wilderness than navigating your way through it.  It’s also a good point that many people completely do not understand what is available in the wilderness, and it is an accurate observation that many think (and they are wildly incorrect about this) that what Joe Trailblazer has done has basically “replaced the expert guide”.

Update: Carol Cross, in Teaching your Middle Schooler:

Forrest Gump taught us that life is like a box of chocolates.  I would say that Khan Academy is like a can of soup.  Education, however, is like a family dinner.

Khan may be a master teacher (maybe…there are certainly lots of master teachers), and the Internet is a vehicle by which he can can himself (or other master teachers) and make it easily available.  And canned soup is certainly handy to have.  You can get canned soup from Master Chefs–for example, Wolfgang Puck sells canned soup–that probably tastes pretty good and that is pretty healthy (although I don’t think it can match the homemade chicken, barley, and vegetable soup that I make weekly for my son’s lunch and that takes a minimum of about 30 hours, since I use my friend Laura’s recipe for making super-healthy 24 hour bone broth as the base for the soup).

But canned soup does not a family dinner make.  The family dinner is about the other people, and the relationships, and tablecloths and silverware and candlesticks, and the conversations, and all of that, even if the family is eating canned soup for dinner.

So, Maria, there is my analogy.  Khan Academy is like a can of soup.  I might occasionally give my son a can opener and tell him to go heat one up, but I would never confuse it with a meal.

Michael Paul Goldenberg, in Math Circle Salon:

Khan represents mathematical fast food: doctored to smell like actual food, to hit craving buttons in people’s reptilian brains, but devoid of real nourishment.

Bonus metaphors:

Content is hard currency. Aggregated content is capital.

Dig and fill: The shadow scholar

Nov 22nd

Posted by MariaDroujkova in Math Accent

2 comments

Unable to display content. Adobe Flash is required.

One of the  most-discussed articles this month in Chronicles of Higher Education, The Shadow Scholar introduces “Ed Dante” who writes student assignments for hire. Here’s how he describes his daily work…

“In the past year, I’ve written roughly 5,000 pages of scholarly literature, most on very tight deadlines. But you won’t find my name on a single paper.

I’ve written toward a master’s degree in cognitive psychology, a Ph.D. in sociology, and a handful of postgraduate credits in international diplomacy. I’ve worked on bachelor’s degrees in hospitality, business administration, and accounting. I’ve written for courses in history, cinema, labor relations, pharmacology, theology, sports management, maritime security, airline services, sustainability, municipal budgeting, marketing, philosophy, ethics, Eastern religion, postmodern architecture, anthropology, literature, and public administration. I’ve attended three dozen online universities. I’ve completed 12 graduate theses of 50 pages or more. All for someone else.

You’ve never heard of me, but there’s a good chance that you’ve read some of my work. I’m a hired gun, a doctor of everything, an academic mercenary. My customers are your students. I promise you that. Somebody in your classroom uses a service that you can’t detect, that you can’t defend against, that you may not even know exists.”

One of the most heart-wrenching parts of this story: what took Dante over the moral event horizon was his college experience. He wanted to do some real work that OTHER PEOPLE WOULD FIND USEFUL.

“I was determined to write for a living, and, moreover, to spend these extremely expensive years learning how to do so. When I completed my first novel, in the summer between sophomore and junior years, I contacted the English department about creating an independent study around editing and publishing it. I was received like a mental patient. I was told, “There’s nothing like that here.” I was told that I could go back to my classes, sit in my lectures, and fill out Scantron tests until I graduated.

I didn’t much care for my classes, though. I slept late and spent the afternoons working on my own material. Then a funny thing happened. Here I was, begging anybody in authority to take my work seriously. But my classmates did. They saw my abilities and my abundance of free time. They saw a value that the university did not.”

Some prisons and armies use a form of punishment:  first, you dig a trench. Once you are done, you fill it back in. Doing work that nobody will ever appreciate reminds me of this psychological torture.

Different people require different use/practice ratio to find their practice (learning tasks) reasonably meaningful and motivating. Everybody understands that doing work that is useful for others requires some amount of practice tasks. However, many college courses and whole program set this ratio to zero.  The prospect of current tasks being useful years into the future is way too distant to motivate most humans.

Have more tasks that have immediate use for some currently living people and communities. Students can write for collaborative open resource projects, review papers for conferences, program and distribute needful software, and otherwise pitch in where work is needed. This way, even if they hire someone to do their work (which will be less likely this way), at least the work will be useful to the society!

A few examples of immediately useful learning tasks that proved successful in my teaching

  • Write a Wikipedia article
  • Compose music for a clip based on an essay
  • Illustrate a book
  • Comment on an active popular blog or forum
  • Answer questions at a topic help forum

Another big huge motivator is play. In a twisted perverted way, grades provide a game mechanics that supply a sort of motivation. But this is another story.

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All you can eat soap

Sep 15th

Posted by MariaDroujkova in Math Accent

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The math clubs started today. Oh, the things we can do together!

There is a lot of writing going on in my life just now, so in lieu of a blog post, here are a couple of old Russian jokes.

A guy walks into a village store and buys a bar of soap, thinking it a food item. He takes a bite, makes a face, chews and swallows.
Shopkeeper: But sir, this is soap!
The guy, biting off another piece: Soap or not, if you paid money for it, you finish it!

Dmitri said, just before I ran across this picture: “All you can eat SOAP!” It’s a great metaphor – but for what? I think for automatically generated boring math exercises. Thousands of them!

“Lecturer remote control” is a Russian meme I can’t find in English. The remote control is usually hand-drawn on desks university lecture halls.

This contributes more to the theme of  “Lectures that rock.” Here is what students want to be able to do to the lecturer…

Four large buttons: Off, Self-destruct, Fetch the beer, Smoke break

Two sliders: Volume, Tone

Three-positional regulator can switch to: Fairy Tale, Lecture, Joke

The display shows the time left till the end

The small button simply says, “Failure.” This is for when something miscellaneous is wrong with the lecture.

Translated Lecturer Remote Control

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Problems and their camp followers

Sep 7th

Posted by MariaDroujkova in Math Accent

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What is a problem for one person can be a puzzle or an exercise for another. A smart teacher can turn an exercise into a problem by being “less helpful.” Many curricula and educators also work hard on turning all problems and puzzles into exercises by attaching a step-by-step guide to every one, and by formalizing math before solving as such begins. Lockhart comments on the results:  “I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.”

Video: Dan Meyer on being less helpful.

  • Problems are mathematical questions for which the solver has no readily available methods of solutions, but has ways and the intrinsic intellectual need to figure them out. In particular, solvers do not know all mathematical concepts or formulas they will use.
  • Open-ended problems have multiple correct solutions, often infinitely many of them. The correctness of open-ended problem answers is determined by the mathematical qualities of the solution, such as rigor, logic, definitions, and aesthetics.
  • Exercises are mathematical questions for which the solver knows what methods, concepts and formulas will lead to the solution.
  • Closed-ended problems and exercises have answers known exactly ahead of time. For example, multiple-choice questions are always closed-ended.
  • Puzzles in mathematics are similar to problems in that the solver does not know which methods, concepts and formulas to use. The difference is that problem solving involves eventually developing a mathematical method for solution, whereas puzzles require intuition, finding a trick, or guessing, and may not involve any methods at all. There is no clear line between puzzles and problems, and most collections have a mix of both.

Relationships between problems and puzzles are complicated. Consider the nine-dot puzzle, a classic. Join all nine dots, using four straight lines, without lifting your pencil from the paper.

nine dots in three rows

The difficulty of this puzzle is psychological: most people assume that lines have to start and end inside the assumed square. Once solvers start to experiment with lines that go beyond the square, the solution usually presents itself soon. There is no general math method involved. Unlike another classic, “the wolf, the goat and the cabbage,” or the nine-coin puzzle that open the door to the whole class of interesting math problems and investigations, the nine-dot puzzle stands by itself.

The nine-dot puzzle helps to make a powerful problem-solving point about assumptions. Puzzles may have a powerful role in mathematical problem-solving, similar to “the Mozart effect.” Puzzles support mathematical values and develop the mathematical sophistication, on a meta-level. I do not consider puzzle-solving the same activity as problem-solving. Evidence: quite a few people like one and hate the other, or are good at one and not the other.

What to make of pastimes like Sudoku and KenKen? Their combinatorial complexity quickly overwhelms human processing capacities, such as memory and attention. Human solvers are forced to develop and use intuition-based strategies, which is a puzzle trait. Yet there are also consistent, formulated strategies and rules solvers develop and use, which is a problem-solving trait. Finally, individual steps are routine exercises. The balance between puzzle, problem-solving and exercise traits in Sudoku, in particular, hits the sweet spot for millions of people. Chess and several other abstract games are similar in their balance.

I have a gut feeling, not yet supported by readings or experiments, that there is some sort of 3d flow channel among problems, puzzles and exercises. Each person has particular needs for the balance between the three to develop mathematically. The balance shifts with time. I have no idea how to visualize 3d flow channels.

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Its own context?

Aug 31st

Posted by MariaDroujkova in Math Accent

1 comment

The following is a part of the Flow Channels brainstorm in the Natural Math wiki.

Paul Lockhart, in his “Lament” (the book version) talks about the importance of math as its own context, without trying to motivate it through any applications. I think it’s not about applications – it’s about metaphors! You can use “math as its own metaphor” so to speak, or approach it non-metaphorically, as autists supposedly do, which works great – for those who are willing to, and can, follow that path. In my observation, a tiny minority of people self-initiate or choose this self-contained approach to math, given the choice among multiple contexts as metaphor sources.
Metaphor_source_and_target.jpg
Metaphor_death.jpgHere is how metaphors work. At first, there is a single entity, which only retrospectively can be named a metaphor. For example, a kid can think that division IS fair sharing. People frequently feel uneasy and even offended if you call their metaphors “metaphors” in this stage. It’s a defense mechanism, allowing the metaphor to support enough “roleplay” for the person to develop rich images that can later sustain formal structures. Forcing early formalization breaks the play, disrupts the natural learning rhythm, and leads to despondent feelings average math classes currently invoke.

If we manage to sustain rich image making in the all-important early stage, and gently help students move on to noticing some patterns in what they do, the patterns become math, and the metaphor turns into a simile. Patterns at this stage can just as easily become science, philosophy, or any other pattern-based discipline, but let’s focus on math patterns for now. In our example, once the kid starts noticing, say, that you can’t share certain quantities fairly without splitting, she is doing math, namely division, and sharing becomes LIKE division. When the context of sharing becomes unimportant (though the vocabulary may remain), and the focus fully shifts on quantities and their properties, the metaphor “dies” (Lakoff) and the new math structure, now self-sustained, is born.

To recap the life of metaphor in the context of the model of mathematical learning created by Pirie and Kieren:

  • Metaphor promotes Image Making and supports Image Having
  • Metaphor turns into a simile during Property Noticing, when its source and its target visibly separate
  • The source of the metaphor dies, and the newly born math structure stands alone, in Formalizing

If you happen to love a context other than math – dragons, marine biology, car racing, fashion design – using it as a source of your math metaphors can be as powerful as using math as its own context. However, using math as its own context allows for mathematical elegance and depth not available otherwise. It has to happen, as well.

Flow_Channel_Context.jpg

Bonus: Madison’s poem

know what i figured out?

the meaning of words isn’t a fixed thing!
any word can mean anything!
by giving words new meanings, ordinary english can become an exclusionary code!
two generations can be divided by the same language!
to end that,i’ll be inventing new definitions for common words, so we’ll be unable to communicate.
don’t you think that’s totally spam?
it’s lubricated!
well, I’m phasing.
marvy.
fab.
far out.
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Mobius metaphor

Aug 29th

Posted by MariaDroujkova in Math Accent

1 comment

Jos Leys makes great math movies, including this gem of a metaphor:

I can’t wait to use his Mobius Bloom’s footage to strengthen the point I’ve been trying to make – that the Creating tasks are not for advanced students, gifted people, or graduate school only! Actually, struggling students, young kids and noobs need Creating tasks much more.

There are always jobs for kid, teen and grown-up interns at Natural Math. You help the world, develop valuable skills, and enhance your portfolio. What’s not to love?

Yesterday Karen Mellendorf of MonarchBooks alerted me to a math t-shirt from Woot, with this design:

I was too slow to buy the limited edition, but I really admired the thoughtfulness and deep knowledge of “who’s who in math entities” that went into designing it. It’s a list of topics for two-three years of a good math club meetings, right there! Including the Mobius strip, of course.

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