Cards, corals and kids

For almost a year, we have been meeting with a local group of parents and kids to work on an arts and sciences projects about coral reefs. We have 5 days left on the crowd-funding campaign to produce our final product: a set of playing cards based on the models of ocean creatures we’ve been creating. The models themselves are now on exhibit in the local community art gallery.

Please visit our project’s campaign – clicks help to raise its visibility on the site – and spread the word!


Here is the latest campaign update:

Hello, my dear campaigners! In this update, I talk about two examples of modeling coral with crochet and paper.

The most interesting feature of the wire coral is the dramatic spiral. How do you model this? Very easily with crochet! Just make a chain of stitches. In the second row, put two stitches into each one stitch from the chain. The 1:2 ratio will do the work for you! Experiment with other ratios, as we do all the time in this project, and see what happens.

We highly recommend you explore a coccolithophore single-cell plant under the microscope – or use wonderful open images from others who did. The funky plant looks like a sphere covered in rings. How do you model this with paper folding? Use the fact that polygons approximate circles. Start with a paper circle. Our student used two-colored cupcake liners. Inscribe a polygon, such as a hexagon, into it. You can draw the hexagon or just imagine it there. Fold along the straight lines. You will form a very organic-looking ring!

I love to see children developing their mathematical imagination!

Good calculation games

Here are games that work well with arithmetic tasks. By “work well” I mean:

  • Mechanics provide players enough agency and freedom to develop tactics, so that the overall activity is still a game (rather than a quiz)
  • Mechanics produce strong, pleasant flow, as evidenced by popularity
  • The overall feeling of the game is vaguely mathematical (this one is hard to define and requires human judgment with some artistic license)
This gets two and a half out of three on my “quality math game” definition (attached).

Long-term solution to school discipline

I think the only long-term solution to child discline is to raise the adult:child ratio to about 1:5 or more, in all situations. Humans are, among other things, group animals with powerful instincts dictating behavior. Any situation where there are too many kids will instinctively feel threatening to kids, on the ancient basis of being interpreted as “not enough providers in the group.” The instinct is, so to speak, to push extras out of the nest.

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: 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.

Math presentations at Connecting Online 2012 February 3 and 4

There are forty-five fine presentations at this weekend’s web conference, CO12:

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

Friday, February 3
1pm ET Math game development, by Maria Droujkova
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
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
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
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!

Sign up for the open online course “Developing mathematics: The early years”

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:

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:

Join the adventure, and spread the word!

Join my math game design online event February 3rd


I summarized some of my thoughts on math game design for the upcoming online event February 3rd, which is a part of CO12 (Connecting Online 2012) conference. In particular:
- Defining intrinsic math game mechanics, and why we want them
- Taxonomies for math game designers
I would very much like comments about my slides, which are here:
The presentation will happen online on February 3, 1pm Eastern Time, and is open and free. You can log in here:

The degrees of creativity in math

This is a list from Charles Fadel‘s presentation at this year’s Computer-Based Math Education Summit in London. It can easily be adapted to activities other than problem-solving:

  • 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
I am eager for all the videos from the Summit, which should be up on the site soon.

Multiplication as (not?) repeated addition… in ancient Egypt

Milo Gardner wrote something I just want to quote as a holistic take on a modern hot topic: a dual definition of multiplication. This comes from a thread in “Math, Math Education, Math Culture” on LinkedIn.

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.

Virtual constructions and physical constructions

In a Math 2.0 email group conversation about screencasting and tools like vZome, Brad Hansen-Smith of WholeMovement posed this question:

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.

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.

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.

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:

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.

A few of my favorite visualizations

This is for the Oceans homeschool coop meeting we are having today.

Make your own

Many Eyes



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

From xkcd



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


1-hour video on data journalism