Cross Cutting Concerns

The last entry talked about providing multiple illustrations of overriding themes within a subject. A closely related concept is providing views of the same or similar material, but when viewed through a different lens. In most cases the math, art, history and science curricula are developed in isolation. In reality there is significant overlap and meaningful interaction among these fields. Was Brunelleschi an artist, an architect, or an engineer? Was Leonardo da Vinci an artist or a scientist? Especially with da Vinci, how did these interests interact to produce a greater whole? In a more modern setting, Paul Graham's Hackers & Painters "points out that the similarities between hackers and painters are endless". James Burke's Connections has as a central theme of unexpected connections among technological advances, science, and economics. The Metropolitan Museum of Art even has a science department.

I have, for example, a strong memory of visiting the Fogg Museum at Harvard and seeing connections between studies and sketches to design patterns and iterative development development in computer software.

Remember that the recurring theme for this set of posts is making education more effective in the sense of making it more likely to be employed when encountering real world problems outside of the context of the classroom. Viewing material from multiple viewpoints is highly effective in this cause. Most directly, repetition is known to improve learning and recall of the subject matter. Presentation of material from multiple viewpoints and in multiple contexts increases both the ease with which it is incorporated into the student's existing knowledge base, and the depth of their understanding. It is important that the instruction emphasize that these are related presentations of the overarching theme. As always the students should be deeply involved in discussions about the relationships among the different contexts.

It is best if the repetition is separated in time. For example covering the development of the steam engine from an engineering perspective, then a couple of months later covering it from an economic perspective.

In addition to promoting both a deeper and broader understanding, this approach also strengthens motivation for understanding fields not obviously related to a student's central interests. A student interested in economics might ask why they should study physics. Now this style of teaching would show that economic concerns frequently set the tone for scientific progress, and that scientific progress frequently enables entire new economies. High speed communications enables corporations to be distributed over the entire planet, and brings many cultures closer together. Efficient transportation allows the distribution of manufacturing, and also the distribution of resources - especially in a time of crisis.

Another technique that I have had good success with is presenting the same abstract principle in significantly different contexts. It is important that the link between the examples and the general principle be explicit, understood by, and ideally discussed by the audience. The best results I have seen is when some of the audience provide the explanation themselves, perhaps with a little bit of coaching. This has additional benefits such as engaging the audience as an active participant in their own learning, and demonstrating to the audience that the material can be understood by their peers. This has been a subject of some interesting recent research^1,2 – indeed, reading this is what got me started writing about this topic. Relevance makes an appearance again – examples that are relevant to, or memorable for your audience are more effective.

In physics conservation principles such as the conservation of momentum and conservation of energy are stressed early with multiple examples from mechanics and electromagnetism. After four of five years these and other fundamental principles are ingrained into the learner's thought patterns.

Most disciplines lack such overarching principles so more effort is required to identify and incorporate abstractions into the learning process. Computer science, for example, has my favorite abstract principles, the concept of abstraction, and its child, the layered architecture. However, it is easy, perhaps even common, to focus to tools and algorithms while losing site of these important principles. To teach them in a way that is usable throughout the learners career requires that the abstractions be revisited with a clear discussion on how the principle manifests in each specific situation.

1) Analogical Encoding: Facilitating Knowledge Transfer and Integration, Proceedings of the twenty-sixth annual meeting of the cognitive science society
2) Learning and Transfer: A General Role for Analogical Encoding, Journal of Educational Psychology 2003, Vol. 95, No. 2, 393– 408

Putting knowledge – and education – to use.

This same principle has fascinating implications in the hiring process and in ethics.

People's ability to bring knowledge to bear on a problem as strongly dependent on the context of the problem. In software engineering I frequently raise an issue only to be countered with “I already know that”. Yes, but you are not employing that knowledge in your work. Knowing something about a topic, and indeed being able to discuss it fluently, is not the same as being able to employ that knowledge in a real world situation. Here, Joe Kraus, the founder of Excite, currently with Google Ventures, describes experts and "the mismatch between what they say is important, and what they actually do". This same phenomenon is commonly referred to in instruction as Transfer of Learning or Generalization. Understanding and addressing this is critical for effective instruction, instruction that can be employed out of the classroom in the real world, or even to disparate problems within the classroom that are manifestations of an underlying principle.

Of course I expect that many of you are saying "I already know that". But are you actively putting your knowledge to work? Are you measuring the effectiveness of your techniques?

What might some of those techniques be? Consider for example priming the pump – setting students into the frame of mind where they can see where the lessons will go, and how the material is relevant to the real world, especially their specific real world interests. Take a few minutes out of the last lesson of the week to give some pointers on what the next week will cover. Include comments guided by the known interests of the students. Yes – this means getting to know what drives your students. They will be thinking about the questions or applications from the preview all throughout your lessons. Also, cast the preview in a form that makes it clear that these goals are achievable by them. When I give a lecture I provide an abstract designed to get your attention and show the value of the talk. At the beginning I will provide an overview and touch on the relevance of the talk to your real world concerns. During the talk I will talk about applications, and if possible engage some of the audience members in a discussion about issues and applications important to them. I can remember when I was teaching and I employed these techniques then to good effect. OK, I admit I didn't generate any abstracts for my students.

More soon...

Expanded The Catalog To Include Charged Planes

The electric field from the positively charged plane converges towards a negative charge.

The growth of the efield toolkit and associated visualization catalog continues. I have added the ability to represent charged planes and also the ability to automatically generate field lines. The visualization catalog now includes several examples of charged planes, parallel charged planes, and point charges and planes.

Of course both the toolkit and the catalog are released under the popular open source Apache License (V2).

Clearly the choice of positions for the field lines needs some work. It is not particularly bad, but the hand positioned lines from the earlier examples present superior illustrations of the physics. The old method will continue to work, but at least for the case of the charged plane the automatic generation of the field lines is excellent.

It is also clear that some time should soon go into optimizing the representation and drawing of the field lines. Each line is currently represented as one vertex array, and drawn with one drawArrays call. It seems I should be able to batch these and generate several field lines with one drawArrays call.

Renovated The Web Site

The original web presence was put up to give me a little bit to show around while I was at SXSW. Recently, I had the opportunity to take a bit of time and implement some upgrades. It includes a much more modern style overall, some additional exposition about the importance of visualization in learning mathematics and the sciences, and perhaps the most significant upgrade, a catalog of prepared visualizations. Each visualization is contained in a small web page hosted on GitHub, and is suitable for inclusion in content as an iframe. Each one has a built in help, and if the browser does not support WebGL, the visualization loads a PNG image of the visualization with a yellow warning icon indicating that WebGL is not active on the browser. The warning icon links to instructions for enabling WebGL on your system. This makes it very easy to get started and determine that visualization really does have an impact on your learners.

A Gaussian surface within a charged distribution.

The addition of charged distributions and Gaussian surfaces provides for much more sophisticated visualizations. This example includes a a charged sphere with inner radius a=20 and outer radius b=50. At the center of the charged sphere is a Gaussian surface of radius 10. Zoom in on the image to see that the electric field does not penetrate the Gaussian surface, and that the Gaussian surface is completely within the hollow center of the sphere.

Working through the details to get a reasonable appearance for nested translucent objects without complex rendering was definitely challenging.

SXSW Feedback

I was able put together a proof of concept and show it off to a lot of people at SXSW, and the feedback was overwhelmingly positive. Based on that feedback I have expanded the documentation, made the project easier to use and more modular, and put it out as an open source project on GitHub.

Next I will be adding the ability to specify charge densities, such as a charged sphere, line, plane or ring, and also the ability to automatically generate the field lines without needing to specify a starting point to trace the field lines.

I also spent some time covering some fascinating material on the interplay between neuroscience and learning. One of the talks drew an interesting car analogy. To paraphrase, pedagogy and psychology are like learning to drive fast, where neuroscience is like opening the hood and understand how the car achieves it's speed. This would undoubtedly make you a better driver.

Some of the more interesting items included

• The Neuroscience of Language and Learning
• Neuroscience, Games & Learning
• Neuroscience, the Science of Learning, Educational Reform
• Educational Neuroscience: How Education Shapes Brain Development

Well, after digesting all that I am going to work out the code to display a charge distribution on a web page – that means a volumetric rendering that will function well on highly limited devices such as a cell phone.

Illustrating the concept

This interactive rendering is generated by
HTML5 JavaScript and WebGL.

A quick internet search for electric field will show a large number images. Most of which are similar to the initial appearance of the example on this page. However, the electric field is an intrinsically three dimensional phenomenon. Making the leap from the standard two dimensional representation to a three dimensional understanding is difficult at best. So start with a clear three dimensional representation. With OpenGL ES on Android or iOS, or with WebGL as done here, a much better representation can be presented. This image rotates and zooms in response to many of the standard browser actions such as a mouse or a finger drag. Though the touch interaction has only been tested under Chrome and Firefox on Android. This provides a differing three dimensional view that shifts in direct response to the user or learner action. This allows a very fast comprehension of the three dimensional nature of the field. This is an early proof of concept to show the general class of examples that can be generated, and with the right choices in the software design, it is easy to modify the presented example and incorporate it into existing content.

Why Visualization

  I can remember the first program I ever wrote when I was about 11 or 12. It solved a physics problem from one of my father's books: m=m₀(1-v²/c²)^½. The program produced a lot of numbers for v and m, but this table of numbers did not provide any great insights. So I decided to graph them. Seeing the the curve of m running to infinity rapidly as v gets close to c immediately gave me a deeper understanding of what the equation really means.

  A few years later in my first real job – the same thing but on a bigger scale. Doing computational physics at the Princeton Plasma Physics Lab. This code produced massive sets of numbers, and once again, simply looking at the numbers was not the best way to digest the data. So we graphed them. Of course we checked the numbers to ensure that the calculations were correct, but the real understanding came from the visualizations. And all of the publications and presentations used the visualizations.

 A few more years later and I was visiting the San Diego Supercomputer Center, and I saw a large screen displaying an interactive three dimensional model of a molecule. I can remember having the flash – wow this would be great for teaching. I knew immediately that many of the advanced mathematics and physics concepts I had struggled with would be understood much clearer much faster if quality visualizations were available.

  Now, graphics capability which was once the exclusive province of large institutions can be carried in your hand. We have a strong focus on STEM education, we have the Internet as a distribution medium, and we even have a graphics API embedded in most browsers. It is a great time to bring together all of this knowledge and experience with the desire to make a difference and build first class visualizations targeted to instruction.

A 30 Second Pitch

 VizIT Solutions produces high quality visualizations for instruction in mathematics and the sciences.

 First rate instruction clearly presents material to the learner, further it engages and invites them to question and explore. Interactive visualizations are one of the most powerful, rapidly evolving, and underutilized methods to achieve this. We need to do better.

 Mobile and games technologies have placed powerful graphics capabilities literally in our hands. We will exploit this to generate a new generation of instructional content particularly well suited for intrinsically three dimensional material such as electric and magnetic fields.