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Simon Khuvis

Student B.S. Engineering, Cooper Union for the Advancement of Science and Art

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How can computer models help us build intuition?

The use of visual diagrams to explain and understand difficult concepts is as old as history itself, but in the twentieth century, for the first time, engineers and scientists were able to enlist the help of computational tools to represent systems with greater clarity and detail. While computers, with the right peripherals, are able to present data to all the senses, in two or three dimensions and through time, perhaps their greatest pedagogical virtue is their interactivity. People learn by doing: young children internalize Newton's Laws long before their first formal physics class by manipulating the world around them. Computers offer the promise of similar interactivity for systems which are less readily accessible, or even entirely esoteric. In my Bioelectricity class, for example, we have been using computer simulations of the complicated Hodgkin-and-Huxley membrane equations to gain insight into neural reactions to various experimental stimuli. How can computer models be used to learn, understand and ultimately build intuition about systems in nature and science?

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    Feb 15 2012: A challenge inherent in using computer models to build intuition is that often the computer models themselves can be built on faulty intuition. For example, a simulation like SimCity is based on a set of assumptions about what motivates people and makes them happy. It's difficult to build a successful city in that game without embracing and internalizing those underlying assumptions. This can be dangerous because of the power of these interactive computer models as learning tools. While computer models can be a powerful way to build intuition, there are generally no safeguards to ensure that the intuition they help build is actually valid.
    • Feb 15 2012: That is an excellent point. Alan Wolf, a professor of physics at my college, published an article about electromagnetic field line diagrams and the faulty assumptions on which they are based. These often computer-generated graphics are used to teach countless numbers of students electromagnetics every year.
      You can find the paper here: http://www.physics.emory.edu/~weeks/lab/papers/wolf-ajp96.pdf
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        Feb 16 2012: cool! go alan, go alan...its ya birthday
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      Feb 16 2012: I agree James. Often someone designing a simulation has to make simplifying assumptions, and it can lead to a misalignment between what actually happens, and what is shown to happen in the simulation. However, if you can compare the simulation against raw data and there is a strong correlation, then the construction of a simulation may give you some insight into the mathematical properties of the phenomena you are simulating, even if you can't articulate those phenomena explicitly using a mathematical model.

      However, you do really need to ensure that you are thoughtful in your use, and you don't assume that what you have provided is an accurate measure of reality; you need to check.
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        Feb 21 2012: David,

        I agree with your point and would like to expand on it. As you state, "you don't assume that what you have provided is an accurate measure of reality." I believe one would need a pretty solid basis of certain material before using simulations. The solid basis would allow one to have an intuitive understanding by interpreting the material "with a grain of salt".

        Sometimes these models aren't the best way to have an intuitive understanding. For example, when talking about having an intuitive understanding with science, lab work and hands on work lets you see exactly how much of an assumption you must make. Lab work lets you see how a real life application can lead to different results than expected. I believe the struggle of hands on work can lead to a better understanding than just a computer model.
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      Feb 17 2012: On another note, intuition on any level may be bad. More times than not, one would have to dive deeper and understand the roots of what one is trying to make a conclusion from intuition, and by then, the conclusion is not based on intuition anymore, it's based on experience and knowledge.

      For example, I've had someone ask me where they can get an ethernet "y-splitter". They wanted to share their network connection by using a y-adapter for ethernet. Intuition from other devices (rca, bnc, stereo, etc) tells us we can do this. One may argue, "of course!, networking is complex and it requires two way communication, so of course you can't split the cable that easily". But, before ethernet, there was thicknet, which was essentially one giant coaxial cable shared by multiple computers, yet they can do two-way comm. (so with thicknet, you can use a y-splitter). Rather, one needs to know how ethernet protocol works. One would have to understand how it works before they can conclude that these mystical ethernet y-splitters do not exist. However, even after having that knowledge, one cannot use intuition to construct their own network hub. Ok, say they learn how network protocols work, they still cannot construct their own network hub, they need to learn about signal propagation to know that certain long wires need to be in twisted pairs. One can find similar kinds of issues with any topic of study.

      We may say intuition is useful, but to me, it's not any better than a coin toss (that is unless you have more knowledge, but by then it's not intuition)
    • Feb 17 2012: What is needed is some level of wikilike interaction with the model, then, it sounds like, a 'backstage' or a way to tag stuff where people can communicate their doubts or maybe in a more formal circumstance discrepancies between what is observed and what's modeled, so that this data can drive the creation of new models.
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      Feb 21 2012: James,
      While what you and Simon are saying about simulations incorrectly portraying the information may be somewhat correct, the overall point of them is to simulate, not give a 100% accurate representation. It's this simplification that allows us to develop intuition for such things, since our minds can have a hard time building such complicated models on their own. Using the example of Professor Wolf's paper on electric field lines, we all admit that the current diagrams are flawed. But it's these flawed pictures that help us gain intuition of how electric fields act in general, and we can then extend this understanding to a full three dimensional picture of electric forces and fields. I think as long as we remember not to take a simulation to be the reality, they can be an invaluable tool.
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    Feb 16 2012: I hate a love/hate relationship with the fact that when you code/program anything, the program will do exactly what you told it to, even if that's not what you meant it to do.

    what's exciting about this is that if you've done everything syntactically correctly and you still see something you didn't initially expect, you're forced to change up your understanding of whatever it is you are modeling, thereby developing (correct) intuition for how some system or mechanism works.

    for say, the hodgkin huxley model which we just learned, the equations that govern it may not mean anything to us in terms of visualizing how the equations behave and how they act dynamically, but when you turn that into a model, you can actually get a clear picture of what's going on. also if you have a generally correct intuition for the characteristics of a dynamic model, turning that into a computer model can help to point out nuances that you didn't initially think of...
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    Feb 15 2012: When it comes to using computers to build intuition and learn about our world, I completely agree that the key factor is interactivity. The young child who learns about gravity by knocking a vase off of a table is essentially performing a science experiment. She has a feeling that maybe something interesting will happen if she hits the vase, and then gives it a try and observes the results. The key thing that makes this type of experiment so effective is the continuous, immediate sensory feedback we get from our environment.

    One of the most powerful things about interactive computational simulation is the ability to perform these same sorts of experiments on all kinds of systems that we can't normally directly manipulate with our hands. But for this type of interactivity to be most successful as a learning tool, its feedback has to be immediate and engage the senses in a rich way. WIthout this rich, immediate feedback, people have a much harder time understanding and internalizing causal relationships.

    Here's something I built to explore how these ideas could be used to teach kids about chemistry:
    http://pattenstudio.com/projects/chem
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    Feb 15 2012: I heard about this on NPR the other day. I think it's really cool. It's one way computers are learning about us, and in turn they are teaching us about ourselves:

    http://www.affectiva.com/affdex/#pane_overview

    This might not be exactly what you're looking for, but maybe it will spur some ideas...
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    Feb 21 2012: It seems to me that the best applications of computer models are when people use them to compliment their own cognitive strengths to solve problems. Take the structure and and function of proteins, for example: Humans are able to sequence proteins, and with this sequence, have developed a number of tools for solving for the structure. All of these tools require massive numerical optimization problems, which computers are best at. But once you have an idea of the structure of the protein, how do you figure out if it makes sense biologically, or how it works?

    This is where many researchers see the problem as one that people must solve, with imagination and intuition based on years of experience and knowledge of how atoms and molecules tend to move. At Weill-Cornell Medical college, Prof. Jason Banfelder has build "The Cave," a room outfitted with screens covering the ceiling, floor, and three walls, used in conjunction with a supercomputer. Large amounts of funding have been poured into this project which serves simply as an advanced visualization tool, with which researchers can visualize protein structures to gain intuition about reactivity and function of different biological entities.

    This is this first really cool use of advanced visualization that I can think of. What other cool uses can you think of?
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    Feb 20 2012: Just like Simon described the use of the Hodgkin-Huxley membrane equation simulator in our Bioelectricity class, I have been using computer software such as Autodesk Robot and Revit in my Civil Engineering classes in order to build intuition on models of buildings and bridges. Although these structures are not necessarily “systems in nature” as Simon described, and are instead man-made, using computer models are invaluable tools to analyze the forces and loads in the structure. By applying seismic, wind, dead or live loads on the beams and columns of the model, we can correctly simulate the effect of these loads on the building as if it was actually standing. Computer models have been able to oversimplify our everyday lives, especially as engineers, whether it is simulating equations - as Simon stated, or simulating the effect of an earthquake on a building. The most useful aspect of computers is that they can give us insight on many physical properties whether we encounter them in real life or not. Gravity might be easy to experiment with in real life, while something like membrane potential is not. As technology and modeling techniques advance, we can further our knowledge of science and nature around us.
    • Feb 21 2012: Thank you for your insight, Steven. I think that even though engineering applications are not natural per se, we could still call them "systems of nature" because the governing principals of physics (i.e. the equations which your CAD programs are simulating) are the ones that exist in nature. You are just providing a specific set of starting conditions.

      It is interesting that you say that this software is oversimplifying our lives as engineers, and that programs can simply give you the right answers when it comes to simulations of load-bearing structures. Do you think that perhaps, in this case, computers have replaced the need for human intuition in the field?
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        Feb 21 2012: Computer modeling has become a requirement of almost all walks of life and as Steven mentioned, you expect to have this software interaction as an engineering student. I don’t think the software undermines or replaces the intuition in the field but rather helps build it (I’ll add a caveat here and say, if used correctly. What I mean by this is actually sitting down to understand why it is that a program might give you the results it is giving you.). For example, once we were asked to design an exercise machine for a class and model it in Solidworks. I started out with pen and paper to help clarify an idea and the program helped further, even just with the visualization of objects in space (Oh, you can’t attach this limb here because when it rotates it will impact the main frame, etc). Just like ball-and-stick models for organic chemistry were introduced as tools, computer modeling is simply another tool that people can implement in both trying to explain a concept or solve a problem.
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          Feb 21 2012: I have mixed feelings towards the use of computer simulations because I feel that the oversimplification idea that was mentioned leads to the plugging in numbers into a black box and receiving a number at the other end which is meant to have some kind of significance. However, for me, if one doesn't know the specifications of the computer model exactly, it could lead to drastic mistakes especially when considering a building. For example, I had to do energy calculations on an organic molecule in different 3-dimensional geometries using a program called Spartan. There were several different energy calculations that the program could perform each with its own assumptions and each used preferentially for specified conditions. However, in that case we were just told which calculation to run and therefore I gained no insight into how exactly the calculation was performed. On the other hand for a Neuroscience class that I took I had to write a program modeling a firing neuron. In this case I had to personally input and understand each line of code and all of the mathematics which did indeed give me a greater understanding and intuition for how the system worked.
  • Feb 17 2012: On the notion of intuition: this is troubling in that it implies a layer of reasoning that is hidden from consciousness and is often associated with metaphysical connotations.

    By definition, intuition is knowledge without inference or reason yet from a neuroscience perspective it follows that such knowledge does not spontaneously manifest but rather is the product of cognitive processes, which are themselves a product of neural circuit function. I would think that rather than propagating and being reliant on "intuition" and it's perceived benefits we should aim to understand the underlying cognitive processes that yield superior understanding compared with traditional reasoning approaches.
    • Feb 17 2012: I tend to agree, although I think that having a part of the mind which you're not conscious of is probably necessary to reduce overhead in thinking about things - like the concept of data hiding in computer science... I think when people talk about intuition they're really talking about things like inferring from pattern recognition or by analogy, as well as rules which they've learned and then internalized and incorporated into their perceptions and judgements and are no longer aware that they're incorporating into their decisions, but which ineffably seem to make their decisions "better" than someone else's on a subject. I've heard people talk about "developing your intuition on a topic" and a lot of times they bring up analogies to think about (like the water pump analogy with electricity). I also remember a study in which a complex formula predicted the next position of a dot on a screen, and you'd get ten dollars if you could reliably predict where the dot would land next after a while, and $100 if you could explain how you were able to predict it. Participants were able to learn where the dot would land next, and so could a neural net, but they were unable to put their understanding into words - they didn't have privileged enough access to their own brains to trace their thought processes. Being conscious of your thoughts seems to be a higher level mode and I think a lot of the details are hidden from us because it was simply never to our adaptive advantage to be aware of them.
    • Feb 17 2012: I would definitely agree with November's assessment, but specifically to address your concern over becoming overly reliant on intuition to understand complex processes, I would say that an intuitive grasp of a certain concept can help immensely in the creative process. For example, I am currently working on a device to harness energy from ocean waves. While equations exist describing hydrostatic and hydrodynamic situations of all imaginable kinds, I've found it much more useful to rely on my intuitive grasp of fluid physics as a jumping-off point and then to go back and quantitatively analyze any new improvement. While I do know some people who like to fiddle with equations as their preferred means of understanding a given subject, I would wager to say that they are in the minority when it comes to their chosen method of learning.

      The brain is an inherently stochastic organ -- in fact, computer models show that without a healthy dose of noise, neurons would only be able to respond in a nearly on-off fashion to graded stimuli. That noise, in combination with population-averaging effects, allows for the generation of a graded, smooth output in response to varying stimuli. While it might be uncomfortable to accept any level of randomness in our thinking, I would argue that that this is necessary for any real innovation to take place. As Albert Einstein said: "imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world."
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    Feb 16 2012: I think the key here, is the interactivity and pertinently, the instant feedback a computer modelling can give. However, a computer program is only ever as good as the programmer that wrote it.
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    Feb 16 2012: Computer models are a great tool to visualize abstract information or theory. Scientists with computer modeling tools can make bold assumptions and let the computer to run the simulation to see the assumption's feasibility. Science continues to progress, its complexity has developed in such great depths that it is almost impossible to detach oneself from utilizing any computer modeling.
  • Feb 22 2012: I am thinking that intuition can help build computer models.
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      Feb 21 2012: Hi Bridget

      I completely agree with you that we need to trust our intuitions provided that we figure out how to increase our intuitions. Increasing our intuition can come from rigorously struggling and studying concepts and making familiar connections through analogies and simulations. This should be a constant learning and growth process.

      I feel like intuition is something that should be developed and nurtured early on in life through education in schools and homes. Families and educators should feel compelled to insure that children grasp full understanding of concepts whether it is through computer models or through some other means of teaching concepts so as to increase intuition.
  • Feb 18 2012: There is no question as to the benefits of intuition (and imagination) as a faculty for connecting dots in non-canonical ways that reveal novel and/or deeper understanding. What I think is important and useful is the ability to model the process of intuition and this we would have trouble performing without a grasp of its underlying mechanisms. As November suggests, this process involves parsing and comparing the abovementioned dots with historical knowledge that may not be necessarily related - possibly entirely unrelated - in order to achieve a superior understanding. I am personally doubtful of the brain's stochasticity and the signals that presently are interpreted as noise may in the near future be found to exhibit an ordered pattern.
  • Feb 17 2012: (i get to the point at the end - guaranteed)
    you ask "how can we gain insight" and later you mention intuition as the cognitive skill you seek to acquire. ill assume you mean the action of grasping something .in some languages the word learning also means taking or catching. if this is indeed what you meant i would like to define it in another way as well. intuition is what we have when we are committed to the whole. it is when our perspective of a system is as broad as possible.

    the model should always be a system because reality is. but what would be a model worth studying? one idea could be a machine that is man made, but than we would miss out on all the juicy reality we left outside of it.
    i think the best way to have intuition for something is to be immersed in it, and than fully communicate it through another system that is rich. the distance between the two is always mind blowing like learning a new language. the two systems are the same yet different.
    to be more concrete, as for science class, i know mostly this. i wouldnt start with asking what a computer can do, i would ask what is the thing that needs grasping and also "what is it like to grasp it". i would treat the model like a game and different parts of the system like "stories" or characters with a biography. "game" and "story" should be regarded broadly.
    if you know the system well enough to teach it you will get the tone of the stories right and the feel of game. if you are only discovering the system this would make a thinking exercise in empathy. im sure there is a special place in the world for ppl who identify with scientific factors. anyway thank you for this question.
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      Feb 20 2012: Hey Shani,

      I like the analogy you have made between model and story: we really need to understand each "character," and their "biographies" in order to create a successful "story."

      The model is helpful because it allows us to put the story on paper. Because some models can have so many variables, it often gets confusing, and difficult to remember relationships between things. The model helps us build this intuition because it organizes all of our thoughts - its like writing the story on paper, as opposed to recalling it from memory all of the time.

      We no longer risk getting lost in our thoughts as often because we have put them all into the model. The program helps by automating one step of our thinking process so that we may focus on another, and further our intuition or understanding of the problem at hand. Assuming that we have put the proper work and preparation into the model, I believe that models are certainly beneficial in developing intuition.
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    Feb 17 2012: Not sure if this question is appropriate but why would we want computers to have intuition? Why can't computer advancements aim at being better tools to us, humans who have intuition and not try to supersede us? Isn't the goal of making tools (including technological ones) is to make our lives and our existence on earth better and less destructive?
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      Feb 20 2012: Hi S Lam,
      The issue isn't that we need computers to have intuition. It is so that we can use technology in order to develop our own intuition. If you watch this TED talk about how complexity leads to simplicity: http://www.ted.com/talks/eric_berlow_how_complexity_leads_to_simplicity.html , you'll see that breaking down complex models is actually less cumbersome than originally thought. However, what is important is that Eric Berlow has had multiple dealings with such issues so that he has developed an intuitive sense as to where patterns may emerge. This intuition definitely wasn't something that he one day woke up; it is something that was developed over time. And he wasn't alone on stage; he used technology to map out the web of complexities. We do not need computers to have intuition; we simply need them to provide us with models that simplify complex ideas so that we develop a certain feel, an intuition, regarding the topic in order to make our lives like you said better and less destructive.
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    Feb 16 2012: I never finished it, but this is the beginning of a classroom simulation, relying on the notion that knowledge is transferred between different students in the room (or between the teachers in the room and the students).

    http://davidwees.com/classroomsim/

    There is a lot of work that would need to be done to work on this simulation (it has some pretty serious bugs at this point). For example, it ignores the idea that given the right conditions, students can spontaneously discover ideas on their own (without relying on outside information necessarily) and the variables related to transmission of knowledge are incomplete. It is an unfinished attempt to look at information transfer theory as it would apply to education, and to see if we can gain any insights as to appropriate arrangements of classes (and pedagogical styles) based purely on information theory.

    Obviously, I'm sharing this example as a sample of something we could attempt to use the power of computing to solve (whether we would be successful is entirely a different matter).
    • Feb 18 2012: Hey David, I think this is an interesting way to try and apply information theory using a computer model. I can imagine a simulation like this helping a teacher build intuition in how they should position themselves and students in a classroom. In the related talk about complexity leading to simplicity, Berlow talks about use of good visualization tools and embracing complexity to try and find better, simpler answers to problems. I see a simulation such as yours as an example of what Berlow is talking about: trying to define simpler, adjustable variables to solve the complex problem of how to best teach in a classroom.
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      Feb 19 2012: This is a really neat simulation, thank you for sharing. I have studied information theory mostly from a coding theory point of view and have never really thought about it in a social setting. Personally, this simulation definitely helps me gain intuition about the topic of information theory. When studying error-correction techniques (for a communication channel) I often lose sight of the overall goal of trying to increase throughput and quantify the transmission of information and get too caught up in the math. This model has shown me that a mathematical model just as complex as the ones I've studied can be explained to somebody with no a priori knowledge of information theory through a computer simulation of students absorbing knowledge in a classroom setting.
  • Feb 16 2012: Intellect and intuition seem diametrically opposed.
    Intellect is limited.
    Intuition is boundless.
    Left brain/Right brain in simplistic terms.
    Intuition largely (if not entirely) depends on dropping any rational/analytical/linear approach.

    It seems to me that computer models (or technology in general) can help us test the validity of our hunches or intuition.
    The phone rings - "Oh hi, how are you? I was just thinking about you."
    Technology confirms a connection between individuals that seems to be based on intuition.
    Perhaps this validation of our intuition will allow us to trust and use it more.
    Do you choose between your head or your heart? Do we have to choose? Maybe we can have both.
    We live in a fundamentally left brain/right-handed world. This has lead to a 'civilization' in crisis, in a severe state of imbalance.Perhaps technology can bolster our innate intuitional abilities and help us as a species to become more human again..

    That said, I realize I may be misinterpreting the crux of the ambiguously worded premise of this debate.

    I am interested in technology in the service of human kind - in using these models to help in building intuition in human beings, in human hearts.
    Looking at some of the comments below (a few of which admittedly went right over my head) perhaps you are talking about building intuition in machines.
    If that's the case then we are more lost than I thought.
    • Feb 21 2012: Hello Mark,

      You bring up a good point about computer modeling giving validity to our intuition. My first inclination was to disagree with you and say that computer simulations can, in fact, build our intuition. However, I then started thinking about the process of coding and running a simulation from scratch (as an engineer or scientist might do).
      Being an engineering student, this process is something that is familiar to me. In order to program such a simulation, it is necessary to have insight into the subject. Hence, the simulation is not a means of building intuition, but rather a means of testing intuition.

      There is a TED talk by Eric Berlow called “How complexity leads to simplicity” in which he argues that trends and patterns are easier to discern in larger amounts of data. The beauty of the computer simulation is that it lets us generate large amounts of data in a very short time. Again, though, in order to generate useful data, we have to go into the simulation with some pre-existing knowledge (or, at least, an intuitive guess) of the final outcome.
  • Feb 16 2012: The term "interactivity" and what it implies is probably as primitive as calling the phenomena we call black holes, black holes. They are surely more than that and what is going on in the generality we call "interactivity" is surely more than just interactivity between persons and their media devices--it's shifts in control of time in which we change from passive subjects of linear streams of other people's information into objective relevance discerners whom must add to and subtract from our own changing picture of understanding before being caused by circumstance to assemble our own linearly compromised messages to other persons regarding our issues or communicate with ourselves in the form of "notes". If the educational/motivational model does not address and accommodate all of this properly and also factor in that human beings are always engaged in a neuroplastic dynamism in which their minds either physiologically grow new capacity or atrophy in the opposite direction, we carry on a very shabby fraud which probably has a 1% versus 99% split between masters and mere acquiescers to process. Gee, that percentage seems familiar.

    I have labored long to develop a grip on how to "model" new relationships with computers, with each other using the full spectrum of non-linear digital media devices including computers, and we with ourselves with as well using such technologies--and yes, incorporating formal sensitivity to neuroplastic dynamism that if, put into a supplanting system could force the establishment and people to choose between continued dysfunction and new function. But trying to convey the details has proven a exercise in exercise. Now if I could just dictate and have artists and scientists do what I say, we could reform education and change the future. Tough deal when everyone else wants to change the world to in their own way.
  • Feb 16 2012: expert systems stuff could help too, since it could do a lot of the logical symbol-pushing and answer questions or suggest relevant information.
  • Feb 16 2012: Technologies like Augmented Reality and Mixed Reality coupled with computer simulations can actually change the face of traditional education.
  • Feb 16 2012: A lot of this is already done, really, but:
    Augmented reality seems like a promising approach, its already being used a bit in interactive visualisations. Being able to overlay internal views of things onto the things themselves, or deal with models of things interactively with gestures or occular control or neuro control seems useful.
    Being able to modify or tag models with meta-data collaboratively might help people build different visualisations based on the tags. Like people might tag all the possible tumors they see in some high resolution scan of a patient, and a doctor might check the spots that are getting multiple hits by having them stand out in some way, as a heat map skinned over the model or something.Some of this kind of stuff, after you've gathered enough human data you can start training neural nets to identify features based on identifications made by people.
    I like the idea of using data hiding in 3d models a lot. Being able to open up a thing and dig into it with more complexity or pack it down to a simpler form means you can kind of tweak which aspects of the representation you need to see in how much detail. This is also a pretty good way to simulate and blueprint stuff, since the highest level blocks of your simulation can be dummy modules until their details are filled in.
    Since lots of different kinds of systems interact with each other, I'd suggest a kind of universal modular simulation system, where users could get on and collaboratively apply detail, that was in some way tagged or connected to physical scans or models if possible, with a lot of different views that a person can arrive at, and with the ability to take the model part of it and hide it completely, or make it as abstract as possible, or dig into specific details as deep as necessary. modules might also be linked to media that can explain how that component works. And if all the people using the model can communicate with each other, they can all share intuition.
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    • Feb 15 2012: Mathematical models can often be frighteningly abstract. Oftentimes this is one of their greatest advantages, but to the prospective student, this can pose a roadblock to comprehension. A common analogy in the world of physics is the description of space-time as a rubber sheet. General relativity is, itself, a highly mathematical discipline which involves tensor calculus in four dimensions, but with the rubber sheet analogy it is possible to build a kind of "physical intuition" with regard to space-time by relating it back to a system with which the student is already familiar. In that way, it is no longer necessary to churn through endless calculations to arrive at qualitative results -- the student can simply apply his or her intuition to conclude what would happen, for example, to light as it passes by a massive object.
      See: http://www.youtube.com/watch?v=XkAPv5s92z0&feature=related
      I wonder if there is a more accurate way of visually representing four dimensions with a computer simulation?
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        • Feb 16 2012: I don't think that one necessarily needs an understanding of the physical rules of a system to model it. Simply a set of mathematical equations will suffice. If you're willing to entertain the notion of separate categories of knowledge, understanding and wisdom, a knowledge of the mathematics of a system is necessary to construct the model, but that model can help build understanding and, in turn, wisdom.
          One can know how to work a computer -- that is, if one presses a series of keys, one can perform a certain action, but that knowledge is of limited value. If someone understands the workings of a computer, he or she can explain the purpose of each of those keystrokes. Finally, if one is wise, one can do things on the computer that one has never seen, and even extend that expertise beyond the world of computers.
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    Feb 15 2012: There are so many ways. In mathematics a student can check his intuition as to how an equation might be described in a graph by predicting first and then verifying. by using technology.