Abstraction and Synthesis in Education

LatticeKnowledge Transfer and Acquisition

What is the immediate goal of our education system? At face value it is to transfer knowledge…right? To be specific, it is knowledge transfer by the system, and knowledge acquisition by the student. We do not teach intelligence; we assume intelligence as a prerequisite. We even perform intelligence tests if the student shows scores outside the bell curve, in order to understand why. If we thought we were responsible for producing poor learners we would not look at those students as aberrations in the system. This goes the other way as well. If we thought we could produce exceptional learners we would not look at them as aberrant either.

But what if the system could produce exceptional learners? What is the long term goal of the education system? To produce effective learners we assume. Not necessarily exceptional, but effective in the sense of being able to function in the world. In addition to filling their heads with knowledge, we want to send them into the world, productive, able to solve problems, run a business, create art, make good decisions, appreciate life, and of course, continue learning…maybe even transfer some knowledge of their own. All this is within the realm of knowledge transfer, whether we are transferring an academic subject or imparting a set of skills such as playing an instrument, or providing a framework to facilitate acquisition of knowledge.

So education can be described as a system of knowledge transfer and acquisition. The idea of knowledge transfer also presumes that the system has the knowledge collected so that it can be transferred to the student. We do not question that component usually. We also do not question the abstract nature of the knowledge that the current system attempts to transfer. Nor do we question if what the system transfers is actual knowledge; no definitions of knowledge are offered to the student in order that they may recognize it as such. But we should question all of this if we want to improve our perspective on the education system. We need to validate if the system has captured actual knowledge and if it is organized and presented in a way that is useful, practical…linked to the real world.

Knowledge Metrics

So if the educational system’s goal is to transfer knowledge, our student’s scores are the direct measure of knowledge acquisition. But have we defined knowledge? What is it that we transfer? To answer that, look at what we currently measure. The “how” is less important for the moment, because if we do not agree that “what” we are measuring is knowledge, then the “how” is irrelevant.

Testing scores are not seen as direct measure of knowledge transfer, meaning the effectiveness of the system. We put the responsibility on the students directly to learn, and only measure the system’s effectiveness at an indirect, aggregate level, such as the percentage of students graduating, or the percentage of students with an A average, etc.

Since we normally do not measure if the system is effective for a given student, we have no way of improving the system at the level of the individual. So measures of the system’s effectiveness stay in the realm of the random distribution; the bell curve. In any new system, we should think about how to measure at the level of the individual.

In terms of  effectiveness of the system for individual students, the fact that learning requires so much struggle on the part of the student and the system shows that we do not understand the inherent leverage. Watch a kid learn something they love and you will get the idea of leverage. There is no struggle…there is effort, but not struggle. We need a way to capitalize on this natural drive,  and even to protect it. If we do not we risk having a less effective system and permanently impacting the individual learners.

The Bell Curve (Random Distribution)

Overall the current approach is to measure the student’s command of abstract concepts within a subject. We either show them a problem and they have to apply the concept using an operation (as in math and science and spelling) or we ask them to analyze something and extract an example of the concept from it (as in language arts), or we ask them to create something that shows understanding as in a school project. Although much of this comes close to appearing as practical, we keep the student in the realm of the abstract because that is the only way we can point to a knowledge “bit” and show learning has taken place. So we measure understanding of concepts. The better your understanding, the better your score, in theory.

As long as those scores follow the bell curve, a random distribution, we know that the system is not influencing learning directly. The system casts the seeds of knowledge out into its realm, and lets them fall where they may. If they fall on fertile ground (competent teacher, intelligent student, good transfer process), the result is good. If not, the result is not good. It is as simple as that, and the random distribution is the proof.

This means broadly, that students only realize their potential by some measure of chance. Scores are NOT necessarily representative of intelligence. In fact it is possible to cheat, so high scores are not proof of high intelligence.

Power Law Distribution

So what would it mean if we saw a power law distribution of scores? Then we would know that the system is influencing knowledge transfer directly. It would mean the results are ordered and non-random.

Power laws are a characteristic of networks. They indicate a transition from a random state to an ordered state. The different levels represent different phase states, and the progression from one to the next is sudden, not gradual. If we see scores follow this curve it would mean the results are not randomly distributed. It would mean that something in the system is triggering wide scope integration of knowledge.

Progression via Phase Transition

Yes, it would mean the distance separating the most learned from the least would be greater. But it would also represent something else…that differing scores truly reflect a level of development; not test taking ability or memorization, or cheating. And each development stage would be a dramatic leap forward in understanding. The current grade levels like K-12 would not make sense, because they are a linear progression of ability. In a world of educational phase transitions everyone would progress at the rate they need to, because the leap to the next phase state (level of knowledge integration) would happen in its own time.

What is Knowledge?

Abstraction and Synthesis

Our look at metrics defined that our current perception of knowledge is the number of concepts we acquire. We consider knowledge an abstraction from the real world. Abstractions are developed via analysis. Synthesis of those abstractions back into the real world occurs based on how the student is able to generalize via analogy and metaphor. But it is somewhat random and the more abstract the topic, the more difficult this is. “When am I ever going to use this?” is a common complaint from students about abstract subjects like math.

Knowledge Networks

Phase states along a power law distribution would represent super-learning, with each phase state resulting from a non-linear leap in understanding. How would you measure these phase states? You would need a network model of knowledge. First, define knowledge as the ability to make connections between topics within an area of learning (node to hub), and intelligence as the ability to make connections across different areas of learning (hub to hub). These connections represent the knowledge network and a measure of learning or acquiring the network connections.

Teaching Intelligence

The more connections within a subject or a hub in network terms, the broader and deeper your understanding of that subject. The greater your connections from hub to hub, the greater the speed you can learn at (theoretically). This is due to the shortening of the number of nodes you need to go through to get from one subject to the next…the degrees of separation between subjects.

Of course, this is tantamount to heresy…are we saying we can teach…intelligence? In a word, yes. It is not that crazy though. It just means that over time, you can increase your IQ, which means that you can create structures in your mind that allow you to learn faster, which means you will do better on IQ tests. Of course this is theoretical and is not proved or disproved at this time. But it is not crucial to justifying this overall approach to the education system. Scores on IQ tests do not predict success in the real world, and here we are more focused on practical, usable knowledge. This is still something we should look into, but as a related area of inquiry.

Testing for Knowledge Acquisition

So if we measured the number of connections a students could make rather than their ability to memorize, we would immediately see a new way of organizing the curriculum to teach to that. Certain knowledge elements within an area of learning would form hubs that other knowledge would connect to. The difference with this approach is that learning would not follow a linear progression. It would remain at low levels until knowledge acquisition reached a threshold level of complexity and then understanding would leap forward to the next phase state.

An approach to actually doing this would be to compile concept maps for each student, showing the connections within their brain on all the topics. You could then easily count the number of connections, but you could also go further. You could identify gaps in their knowledge network, and you could identify unique connections that no one else has.

How would you accomplish the creation of all these concept maps? The students would create them, and this activity would take the place of today’s testing. The benefit would be lifelong, or at least as long as the person remains a student and kept expanding their concept maps. Knowledge transfer would be very customizable then, for each individual’s strengths and weaknesses.

The “how” in terms of the technology is not important for now. If we can all agree that this is the goal, we can get there using some combination of technologies, including pencil and paper.

What Knowledge Looks Like

Start with a topic like a bridge…say the Brooklyn or Golden Gate. Organize the topics to support every aspect of the bridge. You would teach history, including local, national and global. You would go back even into ancient history, as you bring in math, physics, and chemistry. Materials, their composition and strength brings in engineering topics to link the math and science. The historical topics also link you to politics and economics.

English could be taught as the need to learn new vocabulary and language structures increases, which in turn provides more tools for thinking and supports further knowledge acquisition. In fact, what other reason is there for teaching language if not to use it…to apply it in support of real thinking and communicating? If there is no need for me to have a certain level of language development, then it will be just another abstract academic subject.

But what about stages of development? You can make the argument that we need to supply certain concepts at the proper developmental stage, or else we lose something…time, efficiency, learning potential, or whatever. But does the conceptual academic learning drive development, or does our development through experience or empirical understanding drive the ability to learn the concept? If the latter, then postponing acquisition of that new concept will not eliminate that potential. It is more about the sequence of concepts we acquire, than the actual year we learn them in.  If it is the former, then we simply need to structure this curriculum to teach the appropriate knowledge at the correct stage. All we are really talking about here is a different way of organizing the curriculum. But the expected results are powerful.

Knowledge Structure

So you begin to see how the hub topic links to smaller hubs and nodes, just like a network. At some point, the student would gain insight (phase transition), due to the integrated manner in which you impart the knowledge. The knowledge is presented, not just in abstract form, but integrated as it is in the real world. No longer would students struggle to see the relevance of what they are learning.

But we would also be teaching a new skill: synthesis. We have no shortage of teaching analysis, but we do not teach our students how to take that abstract knowledge and synthesize it, integrate it with other knowledge. So it is incomplete in a way that can stifle creativity (the making of new off-topic connections) and even understanding. But based on we have shied away from such thinking in our education system, might mean that we do not look at wide scope integration as teachable. Maybe we look at this capability as representing intelligence. And you can not teach intelligence right? but then again, what if we could? What would the impact of that be? How would it change us as a society? What would it mean for our economy?

One of the greatest potential arguments against such a Connectivist approach to curriculum would be the sheer effort to create these integrated concept maps that we would impart to the students. And this argument shows the sheer lack of integrated knowledge within our educational system. If our teachers can not integrate the abstract concepts of the current curriculum, then how can we ever expect our students to do so? This becomes the greatest argument FOR this approach, because it points directly at the degree of non-integration produced by our current educational system.

Keep in mind that nothing we have outlined here involves such radical changes to the education system that you would need to scrap it and start over. You can even use current teaching methods. All we are talking about here is moving from a curriculum consisting of abstract subjects, to an integrated curriculum arranged as a knowledge network, and the way we measure and test that curriculum. The idea of “K-12+ grades” could remain, but you would most likely need to remove the age-based correlation…or at least smudge it. But this could be based on the students scores and dealt with over time. What would need to change is the “business rules” around grade promotion more than anything.

You could still look at things as a Behavioralist, or Constructivist, in terms of how learning occurs. Demonstration of knowledge acquisition (testing) would need to evolve. In addition to the idea of showing the connections between topics, you could also have project-based demonstration of knowledge acquisition as we do today.

Still, we would need to open our minds to how we assess the results, which could be extremely creative; to the point of showing competence greater than the teacher. This possibility is a consequence of this approach to learning. So while we could assess a student’s progress based on this, we would not want to have this situation reflect negatively on the teacher.

All in all, would it be a lot of work? In a word, yes. But it would be worth it.

Next Steps

If we understand how we got here, it might help us define a strategy for moving forward. This proposal still allows the use of abstract, Aristotelian thinking in terms of how we describe topics; it does not preclude that form of analysis. But it does ask that we use it only to augment our systems view of the curriculum rather than the basis for everything that we do.

It is the dedication to abstract thinking that has imprisoned us in our current problems. It has continued since ancient times, at least in the west. The Roman and Greek gods are examples of abstract entities. The eastern way was different, non-abstract. In that view there is only one thing and it is either all us (everything is part of us) or it is simply all (we are part of everything). From there you can start drawing circles around subsystems if you want, but there is always the “set of all sets”, the “one”. The Yin Yang symbol is a perfect starting place. Do not just look at it as the duality of opposites (although that is a profound realization to come to). It also illustrates the interaction of systems that eventually end in the one system; the “System of all Systems”.


So how do we start? We would work backwards. Broadly, we would proceed by designing the assessment approach first. This is where learners would need to show the connections they understand between the knowledge topics. Having the learner construct concept maps, possibly with links to source material or other concept maps (such as lower level concept maps of concepts), might be the most straight forward method. We would evaluate the number of connections, and the organization of them into “hub” concepts (those with significantly larger number of connections), as well as the specific hubs, nodes, and connections. Since the hubs shorten the path between nodes, their existence is a critical part of the structure, and would need to be measured. The analysis of the structure would also need to exhibit the power law distribution. So these are all measurable, achievable, and relevant. Learners might score extra for adding their own nodes or hubs, as that is part of the path to creating the phase transition to the next state or level of understanding


Understanding the assessment methods would be key to designing the curriculum; it establishes the criteria. With a clear understanding of what the finished state would look like, we would begin assembling the curriculum into nodes, connections, and hubs. Again, we would work backwards by starting with the end state. The example we used was a topic like a specific bridge, but it could be the Grand Canyon, the Arizona Meteor Crater, the Amazon, or anything that could act as a non-abstract structural knowledge element. As long as it is a system (like the Amazon), or is a structure linked to systems (like the Grand Canyon), it will work nicely. The difference is that you are not teaching is as an abstract topic like Geology, Climatology, or Science, but as all topics related to a given structure, including topics that support your ability to integrate the specific knowledge (math, language arts).

You begin to see that you are taught from the outside in with this approach, not bottom up or top down. So does this not simply change the problem? Instead of starting with abstract topics, and struggling to integrate them into practical knowledge, you are starting with integrated, practical knowledge, and will struggle to separate them into abstract topics such as math and language arts, right?

Not so fast. Babies learn through the toddler stage into the kindergarten stage all without these abstract concepts. In fact, they learn much more effectively and at a faster rate than any other time in their lives. and it is ALL, practical, integrated knowledge they acquire. When they enter the education system, is when it all comes to a grinding halt. That is when the disintegration into abstract subjects becomes a problem, because it is important to that education system, but not particularly important to the student.

As long as we think this way, then yes…we could argue that we are replacing one problem with another. But the wide scope integrations that will result from this method will far surpass any issues with pulling information into abstract topics. Doing this would actually be fairly simple for someone who already has this broad knowledge structure, and the existing awareness of the abstract concept. Further, math is still being taught as math…it is just not separated from how you apply it. So the rate of acquisition is driven by the relevance to the hub topic.

We would just need to avoid measuring this new method by old standards, because the old standards place an emphasis on learning, not application. So students might be behind or ahead of current standards for their age with this approach. But that in itself should not be cause for praise or criticism of it. It is a different view

Phase States

We would then take the assessment and curriculum and assemble them into Phase States to replace the current age-based grade levels. You could indeed have multiple grade-steps between each phase state, but there should not be a need for that. While there is some social benefit for having same age kids together, there is justification for having same knowledge level students together (not counting sports). To be continued…

© Copyright 2008 Keith Sherwood


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