Flow Channels

"Flow Channels" is a term from the psychological theory of flow, calling for the balance of the opposite forces. It's a metaphor for achieving powerful learning experiences through strengthening both opposites in each pair: skills and concepts, guiding and following, chaos and order and so on.

The original Flow diagram by Mike C.

by Alan Kay
Many of these ideas were used in the earlier designs of the GUI.

The connections are probably explained in several of my (videoed) talks that various people have put on the web.

25 years ago I used to do a 90 talk on user interface design that incorporated all the ideas that we drew our designs from.

I probably said something in the chapter I did for Brenda Laurel's Human Computer Interface Design Book.

The diagram I made was to have "Challenge" be the vertical axis, and Skill Level be the horizontal. The simple way to look at this is that when these are roughly equal, "Flow" happens along the 45 degree arrow drawn from the origin. But this arrow is pretty narrow, so if you have more skill than challenge you get easily bored, and more challenge than skill you get easily anxious.

And right around the origin is pretty uninteresting in general.

The way we used this was to ask "How can we widen the flow arrow to be more tolerant of disparities between Challenge and Skill Level?

For more challenge than skill, one way to stave off anxiety is to make the environment safer -- so safety lines for climbers, nets for acrobats, and in a computer user interface, put in a very comprehensive UNDO, which allows the user to make mistakes but to always recover. This encourages exploration.

And for more skill than challenge, one tries to raise the level of attention and interest. Tim Gallwey had 20 or more ways to help his tennis students see what seemed to be the same old yellow ball, as a new kind of thing each time (what is the shadow like on it, how fast is it spinning, etc.?).

In cooking prep, it's all about the satisfaction of tuning in on your muscles and how it feels to slice and chop and peel. Similarly, one of the things you can do in a computer interface is to supply small reactions to everything the user does, so there are things that are like tactile feedbacks (e.g. dragging, etc.) We even made a mouse at Apple that gave enough force feedback so you could feel the "mass" of the stuff you were dragging around.

This is all "Xerox PARC 101" but very little of this has made its way into the rest of computing (or education).

In any case Mike's work is really worth delving into.

Strong Guidance, Strong Following

Maria Droujkova

I consider myself a follower of the Russian school of thought about math ed - Vygotski, Luria, Davidov - who advocate very strong social (group, mentor and material) scaffolding as necessary for learning.

At the same time, I strongly believe in following each student's interests and learning styles, student's choice of learning activities, helping students make their own mathematics, and overall consensual decision-making - in short, unschooling.

It may seem you can't guide and follow at the same time, but in fact the two sides work very strongly together. When students choose their own activities, they are very likely to be very interested in accepting much help and guidance - as long as mentors don't hijack the activity altogether. The later is a mistake I make occasionally. When students trust the consensual tools in the group, they also trust offers of topics and tools more, because, after all, they said heart-felt "Yes" to every single previous offer that was actually implemented.

Nobody is easier to teach than a kid who fondly remembers a long string of math activities you selected and loved together.

I find that I provide strong guidance in these three ways:
- math tools, such as software, geometry construction methods, and problem-solving methods
- small techniques that assure success and quality, for example, ways of folding origami neatly, how to hold a ruler, how to line up digits using graph paper, the necessity of software version control, steadying hands to take a project photo
- math values, such as creating your own math, seeing patterns, attention to minute similarities and differences, reuse of algorithms, rigor, precision, and curiosity

I follow students in:
- choice of topics: deep math is everywhere, after all
- choice of activities: if they want to do their math in drawing-based roleplay, or on the trampoline, or with Legos, so be it
- mathematical behaviors, for example, the necessity to celebrate after each discovery, the individual need to switch activities frequently or finish one before starting another, the desire to draw and build before measuring and computing, and the need to make and display collections of examples

Bad things happen when you guide students too much, as people like John Holt, John Taylor Gatto, and Ivan Illich remind us. But following students too much also breaks the flow; Alan Kay talks about it in terms that make a lot of sense to me (loss of powerful ideas).

Brenda Weiss

"Small Techniques that assure success and quality"
I use this all the time to create a successful experience that puts my children (or other young children) in a leadership position. Simply watching for what I can do to facilitate success creates confidence and a sense of ownership in the youngster. I don't recall intentionally applying it to math, and didn't connect it in my mind with strong guidance. I think of it as a leadership technique that encourages growth and taking responsibility before all the pieces are in place to truly "do it" without the presence of a pro. Kind of like an early apprenticeship.
We use this in the kitchen, with housework, gardening, cleaning out the car ... and so on. Providing chunks of carrots the right size to cut safely, instruction on how to use a knife without cutting fingers, a stool for comfortable work, regular snacks for good energy, remembering to put water on to boil so that it will be ready when they remember they need it.

"providing strong guidance"
In Maria's description of providing strong guidance, I recognize the elements we use to scaffold activities so they work across several grades: tools, techniques and values. My children toss out an idea they want to explore, then trust me to search out necessary materials. Most recently, space and astronomy has led to a simple telescope, cool summer nights, late mornings, and perusing books from the shelf, the store and the library.

Structure and Freedom

This is a slide from Steve Hargadon's opening keynote at Reform Symposium 2010:

Steve made a point that great unleashing of energy (characteristic of flow) historically happened when new opportunities for freedom opened up. The internet makes possible many new "freedom tools" for education, in particular. In the last five minutes of the event or so, we discussed the need for case-by-case balance between freedom and structure for individuals and communities.

This was Question of the Week in September 2010.

Created and Discovered Math

Too much "created" is like the famous spherical cow in a vacuum.
external image SphericalCow2.gif
Karim L. from Mathalicious:
It reminds me of Econ 1: assume that all people are rational. I guess it's easy to pull rank like that when everyone's a freshman.

Fail Quickly

Natural Math email group discussion August 2010
  • Level 0: cloud filtering - Fail to pay attention to most educational activities (out of hundreds of thousands existing and those you can make up), because there was nothing and nobody to recommend a glance.
  • Level 1: bookmarking - Very few activities do recommend themselves through a correlation, human- or machine-made: a word from the trusted people of the PLN (personal learning network) or "Ten million viewers of this 5-star YouTube video can't be wrong" or Amazon's "If you liked this book, you will love that book" or the next game in a series from a beloved developer. In short, look for an idea that clicked with those ideas that previously worked well. Among such ideas, select a few to try, based on how rewarding are the process and the results of the activity.
  • Level 2: quick trial - Rewards have an approximate timeline for reaping them. I am bold, maybe reckless, in my child-like rejection of, "You don't see any point in the exercise now, but you will appreciate the results ten years hence in college." Some short-term rewards should be apparent to all participants within minutes or hours of starting an activity. This filters out about 95% of existing methods. Try remaining Level 2 activities for long enough that they "work as intended." If this means going too far out of the personal or group flow channels, reject the activity or send it back to Level 1 to try it later.
  • Level 3: love and growth - Out of activities that worked as intended at Level 2, select a few you want to do again. That's where investigations and research of long-term benefits can happen. Use these activities as the basis for correlations at Level 1.


The infographic is based on:
"PLN" by Alec Couros via Terry Eberhart (link)
Tag cloud from Denise's "Let's Play Math!" (link)

Hassle/Joy and Calls from The Other Side

I am pasting the entire Seth Godin post here, from: http://sethgodin.typepad.com/seths_blog/2010/08/how-big-is-your-red-zone.html
It relates to the recent "fail quickly" discussion: http://groups.google.com/group/naturalmath/browse_thread/thread/e671c9a945d246a8
And in general to flow channels.

How big is your red zone?

Every activity worth doing has a learning curve. Riding a bike, learning to read, using Facebook... the early days are rarely nothing but fun.
Take a look at this three part chart. The first shows how much joy someone gets out of an activity. Over time, as we discover new things and get better at it, our satisfaction increases. At some point, there's a bump when we get quite good at it, and then, in most activities, it fades because we get bored. (In the top graph I've also added the Dip, showing the extra joy from being an expert, but that's irrelevant to this discussion).
The second graph shows the hassle of that same activity. Riding a bike, for example, is horrible at first. Skinned knees, bruised egos. Twitter is really easy to use the first few times, so not so much red ink there.
The third graph is just the two overlaid. That zone on the left, the red zone, is the gap between the initial hassle and the initial joy. My contention is that the only reason we ever get through that gap is that someone on the other side (the little green circle) is rooting us on, or telling us stories of how great it is on the other side.
The bigger your red zone, the louder your green dot needs to be. Every successful product or passion is either easy to get started on or comes with a built-in motivator to keep you moving until you're in. This is so easy to overlook, because of course you're already in...

Creating and Understanding

Bloom's taxonomy should not dictate the order of tasks in a learning experience! Creating level tasks are not just for graduate students, or advanced/gifted learners, or those who spent a long time with a topic. In fact, struggling learners, young kids and noobs can benefit from Creating level tasks much more.


"Math as its own context" and "Another context as a metaphor source"

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_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.


Vertical and Horizontal Mathematics

submitted by Krishna Subedi

This relates to "math as its own context" flow channel. Quote from the Realistic Mathematics site:

Treffers (1978, 1987) formulated the idea of two types of mathematization explicitly in an educational context and distinguished "horizontal" and "vertical" mathematization. In broad terms, these two types can be understood as follows.
In horizontal mathematization, the students come up with mathematical tools which can help to organize and solve a problem located in a real-life situation.
Vertical mathematization is the process of reorganization within the mathematical system itself, like, for instance, finding shortcuts and discovering connections between concepts and strategies and then applying these discoveries.
In short, one could say — quoting Freudenthal (1991) — "horizontal mathematization involves going from the world of life into the world of symbols, while vertical mathematization means moving within the world of symbols." Although this distinction seems to be free from ambiguity, it does not mean, as Freudenthal said, that the difference between these two worlds is clear cut. Freudenthal also stressed that these two forms of mathematization are of equal value. Furthermore one must keep in mind that mathematization can occur on different levels of understanding.

Content and Meta-discussion

On one hand, we need to develop math content. But then we also need to discuss HOW we do math: philosophy, pedagogy, education theory, parenting - in other words, meta-discussion. Michael Nelson at p2pu suggest to start content-heavy and then move to more meta-discussion later, for good flow:


Do you like to create a shared vision first, or just get going and doing things?

Joe Corneli brought up an interesting review Content, Social, and Metacognitive Statements http://www.springerlink.com/ index/L26M2X6U11694083.pdf
Here is what Joe wrote:
  • A higher percentage of contentful talk is correlated with higher learning gain.
  • Getting students to self-explain improves learning.
  • Metacognitive statements ("I get it" or "Oh, O.K") are often *negatively* correlated with learning gain (possibly because they often express falsehoods!).
  • Social dialog ("Hi, how are you doing?") is often negatively correlated with learning gain (chit-chat).



Reference: http://headrush.typepad.com/creating_passionate_users/2005/06/featuritis_vs_t.html

More is better? Gradual/abrupt

The flow channel model assumes the Western "more is better" value. The Featuritis curve above is an example of a different approach. In learning, we want to expand the Happy User Peak and turn it into a Happy User Plateau. I am not sure how to draw this as a flow channel. The following is my reply to a LinkedIn discussion at the Math, Math Education, Math Culture community (members only). People who brought up these ideas were Victoria Kofman and Michael Friedberg.

Students need to learn to deal with BOTH gradual and non-gradual situations. Everybody has a natural "window of comfort" about how gradual learning has to be. If you go more gradually than that, you are bored. If you skip more than that, you will be anxious. Both lead to (different) failures to learn.


http://groups.google.com/group/naturalmath/browse_thread/thread/1d3fb6bbc37854cd More discussion there.

Simple exercises and complex problems

by Don Cohen - from a Natural Math group discussion and a private communication
also relevant this Math Future discussion

From a local parent whose 2 children came to work with Don for a number of years:
"Typical approach studies one simple concept at a time- boring- isolated, irrelevant. Instead- have a more interesting, complicated problem, that uses these concepts in finding the answer. This leads the student through math concepts, seeing them in their natural context and usefulness. Also, when the problem is finally solved, the "Look what I can do!" feeling spurs further exploration of math."

In trying to solve the complicated problem, the student needs to use the simple concepts to understand the whole thing.

An example
I give a 4th grader a quadratic equation like x^2 - 5x + 6 = 0
Before I do this I make sure they know 5^2=25 and 3-4 = -1
Then we guess a number (need the rule for substitution and order of operations). So they give me a guess number, say 1.

1^2 - 5*1 + 6 =???
Is it equal to 0?

So they are squaring a number, multiplying, subtracting,adding negative and positive numbers, and deciding if the sentence is true - all without thinking about it. When they find the 2 numbers that work, they really feel like they have accomplished a great feat! And when they see a pattern of how the 2 numbers relate to the adding number and coefficient of x WOW - this is not busy work! Then I give them a few others to solve... Then they have to make a quadratic equation for Mom and Dad and one for me. And they make up really hard ones for me! "Look what I can do!" feeling spurs further exploration of math." That's the real payoff.

In this video Jenny, one of Don's students, talks about quadratics.

Divergent and convergent thinking

from a Natural Math club discussion

The vast majority of kids test at genius levels on divergent thinking. Many educational methods discourage divergent thinking, and as a result, most adults are quite mediocre at it. Here's a cool animation made out of TED lecture that mentions it:

Carol Cross, an old time Math Club member, is leading a discussion about convergent vs. divergent thinking at her Peer To Peer University parent course on learning psychology that I am taking: http://p2pu.org/math-future/ node/15191/forums So far, people think it's important to develop both convergent and divergent thinking.

You will see a lot of Math Club activities asking kids for a variety - a large collection - of ideas and answers. Drawing infinity is another divergent activity. This is hugely important for problem solving, where you need to generate many different ideas, try and try again before a solution emerges.

If a group of kids has a good flow, more divergent and more convergent thinkers help one another. Seeing a few examples of other people drawing infinity helped convergent thinkers to make up some ideas too. But seeing convergent thinkers organize different types of drawing, others started to organize their collections as well. We will do more organizing of infinity representations today.

When you design group activity, there is a need for a rhythm inviting kids to be divergent and to be convergent.

Productivity vs. friction of desk chair

From XKCD, of course.
As the CoKF approaches zero, productivity goes negative as you pull OTHER people into chair-spinning contests.

Empty and full environments

Zucchermaglio (1993) broadly categorizes learning environments into two groups: empty and full. Full environments are full of content for the learner to absorb. In contrast, empty environments do not contain explicit content. Instead, the message of the empty environment is only realized when it is engaged by the learner.
Toward a cognititve ergonomics of educational technology. In T. Duffy, J. Lowyck, & D. Jonassen (Eds.), Designing environments for constructivist learning (pp. 249-260). Heidelberg, Germany: Springer-Verlag.

Vision and Routine, Vision and Action, Inspiration and Routine

via Carol Cross

"Vision without action is a daydream. Action without vision is a nightmare."

An article from iJourney.org

"All human activity can be viewed as an interplay between two contrary but equally essential factors -- vision and repetitive routine. Vision is the creative element in activity, whose presence ensures that over and above the settled conditions pressing down upon us from the past we still enjoy a margin of openness to the future, a freedom to discern more meaningful ends and to discover more efficient ways to achieve them. Repetitive routine, in contrast, provides the conservative element in activity. It is the principle that accounts for the persistence of the past in the present, and it enables the successful achievements of the present to be preserved intact and faithfully transmitted to the future.

When one factor prevails at the expense of the other, the consequences are often undesirable. If we are bound to a repetitive cycle of work that deprives us of our freedom to inquire and understand things for ourselves, we soon stagnate, crippled by the chains of routine. If we are spurred to action by elevating ideals but lack the discipline to implement them, we may eventually find ourselves wallowing in idle dreams or exhausting our energies on frivolous pursuits. It is only when accustomed routines are infused by vision that they become springboards to discovery rather than deadening ruts. And it is only when inspired vision gives birth to a course of repeatable actions that we can bring our ideals down from the ethereal sphere of imagination to the somber realm of fact. It took a flash of genius for Michelangelo to behold the figure of David invisible in a shapeless block of stone; but it required years of prior training, and countless blows with hammer and chisel, to work the miracle that would leave us a masterpiece of art."

Inspiration and routine - other words relevant here.

Massing and spacing

via Seth Roberts, Dor Abrahamson, Nate Kornell and Robert A. Bjork

"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."

"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."