That’s why it takes incredible effort and systematic, constant measures of all sorts to keep discipline in any situation with low adult:child ratio. On the other hand, parent coops, homeschool classes, work-study programs, volunteer groups that welcome kids and other places with high adult:child ratios typically extend almost no efforts on discipline, and yet have wonderfully disciplined kids – naturally.

This calls for pretty profound changes in how the society runs, and I fully realize this. A large minority of parents (4% overall in the US, 7% among college-educated parents) take measures within families and local communities to make this happen for their kids:http://blog.p2pfoundation.net/family-educator-commons/2010/08/09 Other measures that may work, especially for teens, are work-study programs, apprenticeships with professionals, and overall integrating kids more into the grown-up world, rather than segregating them. It will take some doing, surely! Meanwhile, a good short-term measure is to open classrooms to multiple parent and community volunteers to raise the adult:child ratio. There are a lot of retired, unemployed, studying to be teachers or childhood researchers, vacationing and working-from-home people who would welcome the opportunity to help.

In particular, check out these four presentations from Math Future people:

Friday, February 3
1pm ET Math game development, by Maria Droujkova http://www.wiziq.com/online-class/701943-math-game-development-communities-and-networks
Mathematics educators need to create excellent learning games, which is a hard enough task. But even more challenging is the task of helping everybody – millions of kids, parents, teachers – design or remix their own games. Communities, peer groups and cognitive tools such as taxonomies of games can make these two tasks possible and sustainable.

2pm ET Mathematical art: Learning mathematics by doing mathematics, by Dani Novak http://www.wiziq.com/online-class/701953-mathematical-art-learning-mathematics-by-doing-mathematics
We will present the MuMart “Music Math and Art” wiki and the computer language APGS and give examples of how to learn and teach math in an intuitive way using computers.

9pm ET Place shape vs. place value: A visual foundation for math, by the Dream Realizations team http://www.wiziq.com/online-class/707754-place-shape-vs-place-value-a-visual-foundation-for-math
What do decimal place values actually mean: thousands, hundreds, tens, ones? Let us show you their shape and you can determine the value. Four- and five-year-olds can do it, how ‘bout you? Get a glimpse into the fantabulous payoff of learning to subQuan. Come enjoy a math topic that doesn’t involve much thinking because your eyes do most of the work.

Saturday, February 4
1pm ET Numbered notes music notation, CO12 presentation by Jason MacCoy http://www.wiziq.com/online-class/701971-numbered-notes-music-notation
We will be introducing a revolutionary new form of music notation called Numbered Notes. It uses numbers instead of letters and is so easy to learn that people can play in just minutes. We will be explaining a brief summary of the history of music notation, how numbered notes is the next step forward from what we currently do and how Numbered Notes is an ideal tool to show the connections between music and math. Free sheet music will be available and participants will be able to try it out for themselves on our free website keyboard.

See you there!

I am leading a MOOC (massive open online course) this Spring. The sign-up is open January 17-22 at P2PU School of Math Future:
http://p2pu.org/en/groups/ed218-developing-mathematics-the-early-years/

The course is offered for credit to Arcadia University students, and for School of Math Future completion certificate to everybody. It has the following overarching themes:

- Personally meaningful and relevant mathematics achieved through projects, games, problem-posing and problem-solving.
- Computer-based mathematics, including interactive simulations, modeling tools, solvers, and children programming platforms.
- Lifelong learning for teachers, with the focus of online communities and networks for teacher support, and building your personal learning networks

You can learn more about MOOCs here: http://en.wikipedia.org/wiki/Massive_open_online_course

Join the adventure, and spread the word!

• Solve an exercise
• Solve a problem
• Solve a class of problems
• Use non-standard solutions
• Create new problems
• Create new classes of problems, with their solutions

Modern mathematics including paper folding offers distractions from the central dual multiplication definition conflict. Multiplication defined as been repeated addition and scaling of rational numbers co-existed as main stream Western Tradition ideas 4,000 years ago, and maintained the tradition for 3,500 years.

Math historians report Egyptian fraction cultures formally used the paired multiplication definitions by 2050 BCE. Specifically, the Egyptian Middle Kingdom. Ahmes, a 1650 BCE scribe, recorded a 2/n table that scaled 2/3, 2/5, 2/7, …, to 2/101 to concise unit fraction series that followed a dual multiplication method.

Modern scholars scratched their collective heads during the 20th century when only reporting the additive side of the paired dual set of multiplication definitions. Ahmes 2/n table introduced 87 arithmetic, algebraic, geometric and weights and measures problems that required a dual understanding of the multiplication definitions.

Both sides of the multiplication definitions were needed by Ahmes, and Egyptian scribes, as scribes as late as Fibonacci in 1202 AD used to record the Liber Abaci, Latin speaking/writing Europe’s arithmetic, algebra, geometry and weights and measures instruction book for 250 years.

Of course, with the death of Egyptian fractions, and the birth of modern base 10 decimal arithmetic in 1600 AD, the ancient dual definition of multiplication conflict seemed to disappear. But has it?

I think not. Modern mathematical physics reports the same dual conflict in ways that would have made ancient Egyptian fraction scribes shake their heads.

Can you explain how using this virtual zome tool will give students a better understanding of polyhedra than actually building it from scratch for themselves? I have the same question about any virtual experience when compared to actual experience of doing something. I assume you have done a lot of model construction and it is easy for you to understand having the experience, but what understanding do students get with only virtual experience?

Here was my reply:

It is better to have both experiences. The reason is that they are different. In particular, and to answer your question, there are three major features of virtual tools that physical tools don’t have.

1. EASY SHARING
Virtual constructions can be uploaded to the web and emailed around. I can’t directly email you the construction of the lopsided origami dragon I made yesterday, though I am attaching a photo of the end product (and I could take a video, for sure). But it’s not as easy as with virtual objects, and you don’t get the perfect copy of the real thing, but a representation of it. I remember our exchange of many emails about me trying to replicate one of your constructions. It took quite a lot of work to share.

2. EASY STEP REVIEW & UNDO
Speaking of the dragon, I would love to rewind the construction step-by-step and find where I made the extra fold: the wings look different. It’s somewhere around step 9 of 21. I don’t feel like finding the mistake in my paper version: it will ruin the dragon completely, and I am not sure I will trace the mistake anyway. Repeatable step-by-step review, analysis and changes are hard to do by hand, especially for young students whose memory works differently and has fewer registers than adults have.

Step review works wonders with sharing. A student can send the whole construction (often animated, or a screencast – easily made!) and ask peers or mentors to analyze steps, or post questions like, “What would you do differently in Step 5?” With some environments, they can then all share their fully interactive constructions that are answers to that question.

3. EASY DYNAMIC LINKS AMONG REPRESENTATIONS
You can dynamically link formulas, graphs and constructions, which support depth of mathematics. It provides a certain holographic view on the essence of math, metaphorically speaking. GeoGebra is probably a better-known example of this, with algebraic representations linked with geometric constructions. Check out DGS (dynamic geometry software) systems in Paul Libbrecht’s i2geo series (more coming up, stay tuned) at Math Future for beautiful examples:
http://mathfuture.wikispaces.com/JSXGraph_DGS
http://mathfuture.wikispaces.com/CaRMetal_DGS
http://mathfuture.wikispaces.com/i2geo

The word “easy” here is the difference between thousands and millions doing the three activities I described above.

—

I posed the same question to Katherine, my daughter, who added two items to the list:

In virtual constructions, you can see infinity. (In particular, I am thinking of fractals – MD).

It really helps to change a variable and see what happens to the construction as a result. It is very hard to do in physical space.

I am adding another key item that came to mind: modularity. Once you build a module in a virtual space, you can copy and paste it whole. In physical space, you have to repeat all constructions step-by-step at all times.

Make your own

Many Eyes

Fractions

Fractions are easy visually (not in other representations), so they are a frequent target:

From xkcd

From OccupyGeorge.com

Timelines

There are two main reasons people may want to visualize timelines. First, something like this is too much for our short-term memories. Second, the inner structure of events becomes apparent visually.

From xkcd

By Charles Minard, 1869

Stop-motion animation

From NY Times
Watch the video around 7:00 to 7:50 for an excellent use of stop motion animation. Note the use of color, size, location and symbols.

By Hans Rosling at Gapminder via Carol Cross

Bonus

1-hour video on data journalism

Algot Runeman linked this article: http://www.freakonomics.com/2011/11/02/the-way-we-teach-math-and-language-is-wrong/

I said that the phrase that learning must be idiomatic caught my eye. And Sue Hellman responded:

I decided to take a look at the article referenced in tkosan’s response to Linda’s question (link– http://206.21.94.61/misc/permalink/procedural_vs_functional.html).

I think ‘math idioms’ might be related to ‘functional units’ in this article. They would be stand-alone skills that allow one to perform real tasks but which can also be nested in/scaffolded into larger tasks in a way that makes them also meaningful and uesful.

Using meaning or function, instead of procedure, as the starting point is akin to the way I was taught to teach a second language. The initial phrases learned enable a person to do something real (eg. to say good morning, or introduce yourself, ask for a glass of water). The vocabulary and structures one learns in these ‘functional’ units in turn become the framework upon which more complex functional units are built. Everthing links to everything else and always to something that was simple, comprehensible, meaningful, and useful in its own right.

I’m in the process of creating a unit on fractions for a school in the Caribbean. If I follow this process, I have to start by asking myself what a kid there would want to be able to do that having some ‘fraction language’ would make possible or easier. That becomes the starting point rather than what I think of as the easiest skill or normal first step. What would they want to be able to do?

What an excellent question to ask for curriculum development!

I am adding this as a task to our upcoming School of Math Future fraction seminar: http://p2pu.org/en/groups/fraction-interactives-seminar/