What Computer-Based Learning of Math Should Look Like

Walter Russell Mead (and his commenters) highlight two articles (here and here) on Virginia Tech’s excellent computer-based learning setup for their mathematics classes. Most of the work is done on the computer, either at home or in a shared Math Emporium where teachers are available to help when necessary; which, except for the computer work, is very much like how my class works. It seems close to the ideal way of using technology to allow flexibility in learning and assessment, and is in many ways similar to New York City’s School of One program.

VT’s approach requires some self-motivation on the part of the students — students are able to use the 24 hour a day Emporium at any time — but the model should fit be a good fit for Montessori middle and high-school students who have much independence in managing how they use their time during the day.

Their assessment method also nice as it allows students to pace themselves and take their tests when they’re ready. It is based on students proving that they’ve learned the material — how they learned it is not important, nor is how many practice tests they took before they get to the test.

Each course is broken up into a series of “modules,” available on Emporium computers or the Internet, that students are required to complete within a certain amount of time. Each module outlines a specific set of mathematic principles and concepts. These are translated into specific examples to review and problems to solve.

Once the module materials are completed, students can take randomly generated practice tests that draw on a central bank of thousands of potential questions. If they get questions wrong, the computer refers them back to the appropriate materials, and there’s no limit to the number of practice tests they can take. When they decide they’re ready, students come to the Emporium to take an official, proctored test that’s generated in exactly the same way as the practice quizzes. Then they move to the next module. Instead of marking progress by time—the number of hours spent in proximity to a lecturer—Emporium courses measure advancement by evidence of learning.

— Carey, K., 2008: Transformation 101 in Washington Monthly

↬ Tyler Cowen ↬ Walter Russel Mead ↬ Ms. Douglass

Pictures from the Royal Society

Knap-weed or matfelon and cornflower or bluebottle, by Richard Waller (1689) from The Royal Society's Picture Library.

The Royal Society’s Picture Library is now available online. It contains images from some of the seminal scientific works of the last four centuries. It’s an excellent resource for teachers and students, who, with registration, can get free high-resolution images for presentations and unpublished theses.

I’m particularly attracted to the biological drawings at the moment because I’m trying to get students to practice their scientific drawing and diagramming.

Control your Destiny: How the Adolescent Brain Works

During your adolescence, which lasts from your early teens into your 20’s, the brain changes rapidly, you develop new abilities and capacities, and the habits of mind and skills you develop will last long into adulthood.

Abilities: The last part of the brain to develop is the Frontal Lobe. It’s responsible for reasoning and judgement — aka Executive Function. So, it’s somewhat understandable that teens often have poor impulse control — their Frontal Lobe (the prefrontal cortex in particular) is still developing.

The parts of the adolescent brain.

However that’s not an excuse. It is essential for adolescents to be held to account, because it’s only by practicing responsibility that they get to learn how to use their Executive thinking skills.

Because that’s how we learn — by practicing.

When we’re learning something new, brain cells, called neurons, reach out and connect to form networks. As we practice and focus on specific things — certain patterns of movement or certain ways of thought — some of the unused connections get pruned away, while others become stronger. The axons that connect the most-used pathways get coated in myelin, which acts as an insulator to make sure signals can pass quickly and efficiently.

Neurons in the brain transmit information to each other along long axons and across the synaptic gap.

By reorganizing the connections between brain cells, the brain learns and becomes better at what you’re practicing. Thus we gradually transition from novices to experts.

However, there is a cost.

Making strong pathways makes for quicker thinking about the things we’ve practiced, but makes our brains somewhat less flexible at learning new things. We develop habits of mind that stay with us for a long time.

Some of those habits we might not actually want to keep; and there’s also the possibility that we might not develop some habits of mind that we really would like to have.

The development of the frontal lobe during adolescence opens a window of opportunity for learning good judgement/executive function, but it does not mean we actually will learn it. We need to actually practice it.

So, if you would like to know yourself, want to be able to control yourself, and, especially, want to shape the future person you will become, then you’re going to have to figure out: which habits of mind you want to be practicing and which ones you don’t.

Longer School Days?

Peter Orszag advocates for increasing the length of the school day by about 2 hours.

As a teacher, I know I would appreciate a little extra time in all my subjects. Based as my experience as the sole teacher in a middle school classroom, I think about how much more we could have done with the extra time to round out the curriculum. But I think it only makes sense to add those two hours if they’re used properly. More of the same — like sitting at desks — is unlikely to help a lot.

Orszag points out that there’s some evidence (see Dobbie and Fryer, 2011 (pdf) and Fryer, 2011 (pdf)) that longer school days have improved student performances. But it’s crucial to note that the longer days are part of extensive changes in the curriculum that I don’t think can be separated from the other changes: “frequent teacher feedback, the use of data to guide instruction, high-dosage tutoring, increased instructional time, and high expectations” (Dobbie and Fryer, 2011); “a more rigorous approach to building human capital, more student-level differentiation, frequent use of data to inform instruction, and a culture of high expectations” (Fryer, 2011).

When I think of longer school days, I tend to think of a more apprenticeship model. Giving students time to work on personalized projects, interacting with experts as they need.

↬ Ezra Klein

Ultra-Violet Vision: Seeing like the Butterflies and the Bees

Visible light (what we see) versus including ultra-violet light (what the bees see). Images by Klaus Schmitt: http://www.pbase.com/kds315/uv_photos

Dr. Klaus Schmitt has some utterly amazing photographs that simulate what bees and butterflies can see. They can see ultra-violet wavelengths of light, which we can’t.

Schmitt maps the ultra-violet in the image to blue to make it visible to our eyes.

His site (Photography of the Invisible World (updated)) has a lot more pictures and information about the process.

Monet’s Ultra-violet Vision

Monet's two versions of "The House Seen from the Rose Garden" show the same scene as seen through his left (normal) and right eyes.

The eye’s lens is pretty good at blocking ultra-violet light, so when Claude Monet (whose works we visited earlier this year) had the lens of his eye removed he could see a little into the ultra-violet wavelengths of light.

Monet’s story is in a free iPad book put out by the Exploratorium of San Francisco called Color Uncovered (which I have to get). Carl Zimmer has a review that includes more details about Monet and how the eye works.

↬ Joe Hanson

P.S.: All of Monet’s works can be found on WikiPaintings, a great resource for electronic copies of old paintings (that are out of copyright).

Painting the Universe: How Scientists Produce Color Images from the Hubble Space Telescope

The images taken by the Hubble Space Telescope are in black and white, but each image only captures a certain wavelength (color) of light.

The Guardian has an excellent video that explains how the images from the Hubble Space Telescope are created.

Each image from most research telescopes only capture certain, specific colors (wavelengths of light). One camera might only capture red light, another blue, and another green. These are captured in black and white, with black indicating no light and white the full intensity of light at that wavelength. Since red, blue and green are the primary colors, they can be mixed to compose the spectacular images of stars, galaxies, and the universe that NASA puts out every day.

Three galaxies. This image is a computer composite that combines the different individual colors taken by the telescope's cameras. Image from the Hubble Space Telescope via NASA.

The process looks something like this:

How images are assembled. Note that the original images don't have to be red, blue and green. They're often other wavelengths of light, like ultra-violet and infra-red, that are not visible to the eye but are common in space. So the images that you see from NASA are not necessarily what these things would look like if you could see them with the naked eye.

NASA’s image of the day is always worth a look.