Sign up for the open online course “Developing mathematics: The early years”
Jan 18th
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/
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/
Join the adventure, and spread the word!
Join my math game design online event February 3rd
Jan 15th
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: https://docs.google.com/presentation/d/10gG23rJdWKwdDh4sWgj9uYptx4XSiFKmYbeAk0YUT94/edit
The presentation will happen online on February 3, 1pm Eastern Time, and is open and free. You can log in here: http://www.wiziq.com/online-class/701943-math-game-development-communities-and-networks
The degrees of creativity in math
Dec 27th
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
Dec 16th
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
Dec 9th
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.
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.
A few of my favorite visualizations
Dec 1st
This is for the Oceans homeschool coop meeting we are having today.
Make your own
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
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
Math idioms
Nov 21st
This comes from the Math Future email list discussion: http://groups.google.com/group/mathfuture/browse_thread/thread/71711320007c2f47/67e4e32417b09153
Algot Runeman linked this article: http://www.freakonomics.com/20
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/
The future of Math Future
Nov 20th
We started the event series in 2009: http://mathfuture.wikispaces.com/events
As of today, there are 113 events in the main series, as well as more special events such as meetings of working groups or P2PU classes and seminars. Here are next steps I would like the series to accomplish in the next quarter or so. Please suggest computer platforms and human actions that will help. Please add, subtract (or multiply) the list.
- Pre-event discussions can help interested people find one another and form preliminary topics and questions to be discussed live during the event. This means…
- …advance schedule: announcing events at least 2-3 weeks ahead of time. Together with the list of following events coming up, and possibly past events on related topics.
- Open multi-community discussion points. That is, event announcements are prompts to discuss the event topic, right there. I suggest this group plus the LinkedIn group “Math, Math Education, Math Culture” plus Twitter hashtag #mathchat plus any groups associated with the event (for example, the blog of the host, or their community forum). We can also frame this as a one-week seminar at P2PU’s School of Math Future.
- Curation and aggregation. We will aggregate contents from all these platforms before the event starts, and summarize threads – in a place with a “reply” button for post-event discussion. We will also have a taxonomy of events by topics and other qualities, such as math game design, computer-based math, family math and so on (the same event can have all these tags and more).
- Announcements by topic. We need an email-based list, separate from discussion groups, that allows people to monitor upcoming events by specific topics, based on event taxonomy, and without any other email traffic whatsoever.
- Research paper. We ask every event host, “How can people collaborate with you and help you?” The answers are excellent data for a study. Seeking co-authors.
- Conference. This January, we can have a Math Future strand at the Learning 2.0 conference, with the goal of organizing a blended (face-to-face plus online) conference soon, as well.
Mozilla Festival’s mathematically auspicious location
Nov 4th
Greetings from the Penrose Way! #mozfest is about to start. Meanwhile, check out the mathematically rich building, called Ravensbourne.
How lectures help my understanding
Oct 30th
Richard Hake wrote today:
It is 40 years since the first publication of Donald Bligh’s classic work “What’s the Use of Lectures?” (London, Bligh, 1971). It was a devastating critique, based on thorough empirical research, of the use of the lecture as the main method of teaching in higher education. It had been established that the only educational function lectures were capable of achieving was the transmission of factual information, and even then they were no better than other methods, and lecturers wildly overestimated the amount of information students were capable of remembering.
I use lectures to get a holistic representation of the lecturer. The voice, the style, mannerisms – everything! For some strange reasons, I understand texts and other media of people I’ve seen live much, much better than just texts. Because of this effect, I try to catch presentations by people whose books or articles I frequently read. Recordings work for the same purpose; interacting, such as asking a question in person, works significantly better. That’s why I always recommend my kid to approach any presenter after the presentation and ask something significant.
This may explain the popularity of webinars or YouTube videos of popular book authors, for example, the Future of Education series. I think people are intuitively trying to achieve this bonus to understanding books!
It’s not clear how many minutes of a lecture (and how many conversations) one needs to achieve this effect. Speaking for myself, about 15 minutes of a lecture and a five-minute live exchange provide a good start for me. I strongly prefer longer live interactions (hours) with people with whom I collaborate, for example, to write an article together. Though I do collaborate with people I’ve never seen and heard, too.
Here is a quick webcam video of me reading a few sentences from this post – see if it helps to understand where I am coming from…
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