MariaD’s blog

Because there are two legs on each side!

At a Math Club, we were working with paper punches and folding. Symmetry of all sorts was on everybody’s minds. A. (4) drew a unicorn with two legs. We all started to ask about it, and she said: “It has two legs on this side and two on the other side!” Then she turned the paper over and drew two legs on the other side - on the other side of the paper!

This is a sketch for an Early Algebra activity from Math Clubs. Kids can draw their own family trees, use photographs, or clipart of their favorite characters.

Once the tree is built, it can be used for several activities. Start from common words, gradually moving to mathematical terms:

• How many grandparents are there? (point to the “grandparent” level on the tree). What about great-grandparents?
• We have one child, and we have two parents in the first generation from the child, and four grandparents in the second generation, and eight great-grandparents in the third generation… How many people are in the fourth generation? Fifth? How do you know?
• Mathematicians use the term “power” here. For example, we can say “grandparents” or “the second generation from the child” or “two to the second power.” Two to the first power (parents) is two. Two to the second power (grandparents) is eight. What is two to the third power? There is a symbol for it:

  2³=8

• What generation has eight people? What power of two makes sixteen?
  Figuring out which generation each quantity means is a lot like logarithms.
  We can say, “What generation has sixteen people?” or we can write:
  log₂16=4
• Add up all generations up to a certain level, say, “grandparents”. Compare to the number in the next level. What do you observe? Is it always the case?

What is interesting for me, as an educator, are the many educational innovations developed and (by now) perfected within homeschool communities. How does “post-school” educational system look like? Here is a partial list of educational practices that are quite widespread, accepted and well developed:

(by CommLab )

- Rapid prototyping of everything, short cycles of evaluation and change, and correspondingly short educational experiences are the norm. Families have moved from “package deal” of whole set curricula (”this is what you do for middle school”) to hand-picking books, teachers, and methods for each child for each 2-4 months of each subject. A kid can stay with a program that works for years, or drop one that does not in a few weeks. This leads to increased quality of programs.

(by Oakland Community College)

- High value is placed on engagement, love for subjects and personal relevance of activities both for activity leaders and for all participants. It is expected that participants and especially leaders of activities CARE. Children are much more likely to be learning topics and subjects that are meaningful for them personally, in ways they personally find engaging. Much discussion happens, and much know-how is accumulated about ways of finding and developing meaningful activities for particular subject areas.

- Deconstruction of “age” and shift to ability levels and styles is frequent among homeschoolers. One often sees age spreads of 3-6 years within each homeschool group activity. Grouping by age is rare and loose (e.g. “teens and tweens” rather than “fourteen year olds”). Correspondingly, friendships and informal communities form across ages, based on common interests and activities.

(by iTunes U)

- Barter economies, gift economies, network economies, coops and other innovative (or age-old) alternative forms of education financing are widespread. Homeschoolers value and often use open and free software and open educational resources, as well as the culture of exchange and communal use of resources. Interestingly, the largest benefits of homeschooling as far as standardized tests and college admissions go happen in the poorest families with lowest-educated parents.

(by Turno)

- Co-production models of learning, where learners and teachers are curriculum co-creators, project learning, unit studies and other active learning models are prevalent among homeschoolers.

- Homeschoolers often form “nakama” groups, small, local tight friend and family groups getting together to achieve their goals, and tied personally as well as educationally. High value is placed on friendships, and day-to-day educational decisions come from these personal ties.

- There are active, robust local communities and global support networks for homeschooling families, for anything from finding an appropriate math program for highly gifted ADHD Asperger kid who likes computers, to helping a family through tough economic times. Homeschoolers are some of the most socially networked demographics, which include lightning-fast spread of politically relevant news, such as proposed laws.

I think of homeschoolers as a distributed think tank and early adopters of education practices of the future.

My comment to USA Today article at http://www.usatoday.com/news/education/2009-05-28-homeschooling_N.htm

Here are some ways I use Twitter for education:

1. Hashtag aggregation for topical conversations at online events, conferences and so on
2. A way to communicate about face-to-face events as they go on, with other attendees, without breaking up the oral presentations flow
3. Ongoing conversations about particular topics with “whoever may tune in” - check out #math for example or #educhat
4. Aggregation of tweets for web sites or blogs using widgets
5. News: all global and most local events are trending, and often you can learn personal information from participants
6. Learning what colleagues are doing, such as events, conferences, projects, meetings, communal documents. Following people I like to their events, which tend to be interesting because these interesting people choose to participate.
7. Quickly publishing my short thoughts before they go away. This promotes creativity, somehow. People should seriously look into the effect.
8. The feeling of being close with colleagues. It’s like living in a small town and bumping into someone at the grocer’s and the drug store and the park. You communicate with the same person on LinkedIn and their blog and then Twitter too, and it strengthens the neighborly feelings.

Here is an excellent presentation by Tom Barrett, or @tombarrett:

And here is a cute cartoon on some of Twitter dangers:

And here is a great blog post on using backchannels, including Twitter, for your events, from TwitTip.

Update: a clever way to use Twitter to support doctoral dissertation writing!

In an overview of the Natural Math project, an audience member asked why we are doing such diverse things. Why work, all at once, on mathematical art, programming-based math, algebra for toddlers, meta-cognition and quite a few other directions we pursue at the same time? These are necessary tools for a gradual, gentle cultural change we are making.

The future is already here, it’s just not evenly distributed. The future where toddlers and their parents play with algebraic ideas, where kids contribute to real work as apprentices, where everybody is able to create or improve mathematical conjectures, definitions and metaphors. We are working on inviting more and more people to work and play in this future, now. This mind map shows some of our tools.

Full screen view.

This Saturday, I participated in a wonderful workshop on real time collaboration. Here is the mind map summarizing it, with links to appropriate sites and details:

It is shared under Creative Commons license by Change Management Community.

Some mathematical definitions seem ridiculous to the general public. Consider a set operation called Inclusion Map. Here is one silly-looking definition!

Definition: Inclusion Map
Given a subset B of a set A,
the injection f: B –> A
defined by f(b)=b
for every b that is a member of B.

This definition seems to describe a function that does not do anything whatsoever, in fancy terms. What is it for?!

March is the Disability Awareness Month, which I did not know until the good people in CVS announced their Caremark program and blogging contest. The way a society treats every member, in all the diversity of styles and abilities, is the reflection of its humanistic advancement.

Math is often used inhumanely. Many people see math as an abstract torture tool, because it repeatedly causes them mental anguish. No less frequently, math is used as a gatekeeper, a way to prevent large groups of people from access to careers or programs that often have nothing to do with the entrance test’s mathematics. This is not right, and we need to solve these problems.

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I thought more people hate math than love math. However, Google search says otherwise! The phrase “love math” returns 449,000 hits and the phrase “hate math” returns 145,000 hits. So, about three times as many math love as math hate! More love than hate is good, but how does math compare to other human endeavors?

Love math = 449,000 Hate math = 145,000 Love/hate ratio = 3.09

Love physics = 42,400 Hate physics = 15,500 Love/hate ratio = 2.74
Fewer people express their emotions about physics, and also the emotions are slightly more even than math emotions. How about chemistry?

Love chemistry = 83,300 Hate chemistry = 14,800 Love/hate ratio = 5.63
Among those who expressed their feelings, chemistry has a stronger popularity ratio than math or physics.

However, sciences lag incredibly far behind some of the arts. Consider:

Love music = 14,000,000 Hate music = 268,000 Love/hate ratio = 52.24

About thirty times more people confessed love to music than love to math, while there are about the same number of math and music haters.

“Mathematics love” by *keng

This is a growing slide show with examples of social mathematics web projects. It goes with a study of children participation in social math. If you know more good examples, please comment. Click on the “menu” in the lower right corner of the presentation to embed it into your blog or site.

Trackbacks

Connectivism in Education ning: “How about creative writing? We were taught to write creatively even when we were young. So can we have creative Maths?”

Great Expectations blog: “An excellent collection of math-rich sites built around user-generated content,are found in the form of a slideshow presentation,compiled by Maria D of natural math blog.”

Bizmo Diaries blog: “If you think math is boring, maybe you’re not being “social” enough. Think of math as a contact sport — doesn’t have to mean rough, although when life gets that way, math can help sometimes (or call it computing).”

LETSI - learning, education, training and systems interoperability blog: “Social Mathematics. I mean, that’s just one of those areas that makes my head turn in ways I never thought it could turn. Maria’s got me pegged — even as a former math teacher, and a person who “sees the Matrix” with regularity ( // I nerd out when it comes to programming, logic and math), the picture in my head of social learning is largely driven by practices in social media — and they are almost entirely language/narrative-based scenarios.”

Mathematics 24×7 ning: “Earlier I was also under this impression that children cannot do or learn Math outside Math classroom, but after experimenting with them and with selected projects I really found it useful . On students network they ask queries, answer to assignments by uploading their presentations/files etc. I have seen a positive impact on students who are shy in asking problems in a class. I have used blog/wiki/podcast features in my Math class for not only teaching learning Math but also eradicating a phobia of learning the subject.”

Algebra is the study of patterns, structures and changes in them.

The language of algebra includes representations such as symbols, graphs, tables, and words that describe mathematical models of situations.

You can edit this map freely: it’s a wiki-map. Click on the MindMeister icon to edit. I have no idea what people will do with it.

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