Conceptuaal hierarchy and proper learning

[Onar Åm]

Hi,

I am interested in the notion of epistemological hierarchy and proper learning. A major point in objectivist epistomology is that an unbroken hierarchy is needed to avoid floating abstractions. This is a brilliant insight and a cornerstone of rationality. However, does this imply that knowledge must also therefore be learnt hierarchically? This seems to be a dominant view among objectivists today. The view seems to be that you need to learn e.g. history chronologically and you need to learn simpler concepts before you can proceed with more advanced concepts to avoid floating abstractions.

I disagree. I believe it is perfectly possible to properly acquire and integrate new knowledge even though it is not learnt in proper sequence and learnt hierarchically. Not only is this possible but is the normal mode of *discovery*. In other words, this is how we form completely *new* and unknown concepts by the process of induction. My argument is that it is more mentally challenging to learn out of order and out of hierarchy, but certainly possible. The only requirement for proper integration is that all the learnt concepts *eventually* are properly placed in a hierarchy.

Let me give an example of how learning by induction (out of order) is harder than learning through tutoring hierarchically, namely the whole word method versus the phonic method. The phonic method (which I am a strong advocate of) starts with the names and sounds of letters and from these compose word chunks and words. That's the hierarchical method. As opposed to this is the whole language approach which starts with the words and leaves it to each child to *induce* the rules of spelling to the child. This can certainly be done, but is far more mentally challenging than the hierarchical method. In fact, we can think of it as an IQ-test with a higher IQ threshold of achievement. This means that the children with an IQ significantly above the threshold will be able to learn the rules of spelling (through induction) equally well as the children that learn them hierarchically through instruction. However, all children that are near or below the threshold will struggle and generally perform much poorer using the whole language method than the phonic method. In the end the smart kinds will learn the rules of spelling equally well through whole language as the ones that learn through the phonic method: the end result is the same. However, for the kids that are not sufficiently smart they will not be able to induce the rules of spelling from examples and therefore their learning will be significantly impaired.

Because of this I am an staunch advocate of the phonic method, particularly for weaker kids. Now, this in my view is a special case of learning out of order versus learning hierarchically, and the result also holds for other areas. It IS perfectly possible to learn a topic higher up in the conceptual hierarchy *before* one learns one further down, but in order for them to not end up as floating abstractions they must later be connected with their more fundamental concepts.

Let me give you an intriguing example of such higher level learning. It is for instance perfectly possible to learn a significant amount of vector mathematics before learning numbers and counting. The vectors can be visually represented using arrows, which then ground them perceptually directly, thereby bypassing the need for learning numbers. A child can visually inspect the length and direction of a vector by looking at the arrow. Also vector arithmetics can be learnt intuitively this way. Vector adding and subtraction is simple without having to learn numbers. Later vector arithmetics can be informed by numbers in order to *numerically* understand the concept of length.

Now, I hold that the natural state of acquiring new knowledge is through the process of out of order induction. What this entails in real life is that people live their lives making out of order observations, and then through memory connects these observations to form an ordered whole. In fact, I would argue that the primary evolutionary function of consciousness is to enable to the organism to process new and unknown information and integrate them. I would also argue that language has allowed us to greatly speed up this process of learning by tutoring, i.e. bootstrapping. This bootstrapping process is most efficient when done hierarchically. Essentially this lowers the IQ-threshold of the problem greatly and allows learning many orders of magnitudes faster than by the process of induction. This is evidenced by the fact that an advanced philosophical or mathematical theory may have taken centuries to develop and strained the most brilliant minds to the brink of their capacities, whereas the people who follow can easily learn the concepts in a very short time through instruction and tutoring.

[DavidOdden]

I believe it is perfectly possible to properly acquire and integrate new knowledge even though it is not learnt in proper sequence and learnt hierarchically. Not only is this possible but is the normal mode of *discovery*.

Based on your examples, the first question I have is what you mean by proper sequence and hierarchically. Let me apply this latter point:

The phonic method (which I am a strong advocate of) starts with the names and sounds of letters and from these compose word chunks and words. That's the hierarchical method. As opposed to this is the whole language approach which starts with the words and leaves it to each child to *induce* the rules of spelling to the child.

This would be a good application of the realization that knowledge must be learned hierarchically. Phonics starts with the lower level concepts, such as the letter and the sound, and builds hierarchically on that. In general, "a" is [a], "e" is [e], "d" is [d] and so on. General phonetic rules can be induced; later, more complicated special rules can be induced on top of that foundation. While I don't understand the current practice in phonics vs. whole language method, as I understand it, the fundamental difference is that with whole language, the child isn't given any conceptual framework for understanding language and writing, and isn't even given a hint that the concept of "rule" is valid or useful; but phonics encourages a rule-governed view of language, even to the point of explicitly telling the child what the rules are.

The issue then is whether children are capable of inductively reaching a higher order generalization given just raw data, and doing so in a reasonable period of time. The issue is not one of learning out of order vs. learning hierarchically, it is one of learning everything on your own as though you were lost in the woods (the whole language method) versus a method that organises and presents reality in a way that the child can learn based on logical order. Word pronunciation is logically based on letter pronunciation, therefore you should learn letter pronunciation first, before you learn word pronunciation (and the attending "irregularities").

Let me give you an intriguing example of such higher level learning. It is for instance perfectly possible to learn a significant amount of vector mathematics before learning numbers and counting. The vectors can be visually represented using arrows, which then ground them perceptually directly, thereby bypassing the need for learning numbers.

I imagine children can learn certain basics of vector physics without knowledge of numbers, but I don't know what it would mean to learn vector mathematics without learning numbers and counting -- the question is, is it mathematics? Do you mean learning that if you lay two sticks end-to-end, you cover more of the ground than you do with either of the individual sticks? I don't think children are learning the same thing at all when they learn numeric addition vs. stick-concatenation, since there are no units with stick-concatenation. For instance, the perceptually evident, in the form of a 6 cm stick added to a 12 cm stick, is totally different from a distance of a half a meter as opposed to 2 meters. But 6+12=18 carries over to 6+12=18. A closer analog would be addition via marbles in a bowl.

Now, I hold that the natural state of acquiring new knowledge is through the process of out of order induction.

I don't understand what out of order induction is. If you're saying that children cannot efficiently induce the correct generalizations from unorganised reality, I agree.

[KendallJ]

LIsa Van Damme, in her Objective Standard essay "Learning in Heirarchy, the Most Neglected Aspect of Modern Education" makes just this case I think. She advocates teaching subjects in heirarchy, for example learning science in the heirarchical method in which it was discovered.

[y_feldblum]

This seems like "M in education" - intentionally teaching floating abstractions without integrating them from, or even connecting them to, concretes.

[DavidOdden]

LIsa Van Damme, in her Objective Standard essay "Learning in Heirarchy, the Most Neglected Aspect of Modern Education" makes just this case I think. She advocates teaching subjects in heirarchy, for example learning science in the heirarchical method in which it was discovered.

Okay, but let's apply this to reading. The first methods of writing were iconic -- drawing pictures of objects, so a picture of an ox looks like an ox. This got formalized so that the drawings stopped looking like the things, and the pictures started to be used for sounds of particular words, and that got streamlined into sounds of a syllable (the first syllable) in a (common) word, and then letters came to represent just one sound. There are a lot of hierarchies that you could use, and sometimes the hierarchy of discovery / development is not appropriate. I also don't think it's particularly sensible to start with Fire, Earth Air and Water as the elements, nor do I think that phlogiston is an essential part of a science curriculum. My shtick is that concepts should be taught in the order of logical dependence: if you have to understand concepts A and B in order to understand C, then teach A and B first. I hope that's what she's saying as well.

[IdeaSave]

Onar:

You imply that induction is learning "out of hierarchical order". In fact, however, induction is exactly the opposite. Induction begins with what one already knows, adds new observations, and then _integrates_ that into new knowledge. That process is inherently hierarchical, specifically, building up to more abstract knowledge from less abstract knowledge.

Giving a child an abstraction whose foundation he does not know, and then expecting him to figure the foundation out on his own is certainly not "teaching". Perhaps in some simple cases a child could succeed starting like that, but nothing he has in his mind qualifies as knowledge until he grasps the whole hierarchical chain from perceptual reality to the abstraction in question. Give a child something very abstract, and he likely will never learn it on his own (e.g., try giving a child "quantum physics").

Perhaps you're confusing this issue with what adults can do to make implicit knowledge explicit? In that case, one can start with an abstraction, and then break it down, step by step by asking "What is it?" at each step, ending when the perceptual level is reached. That isn't induction though, it's deduction.

Even in the case of a child who starts in the wrong place, but eventually reduces the topic back to the perceptual level, that is a bad way to do things. This is because when starting at the perceptual level and building upwards, the child learns things that are the basis for many concepts _other_ than the specific one he is trying to learn. Doing the reverse establishes only the basis for the starting concept.

Mark Peters

[Onar Åm]

Based on your examples, the first question I have is what you mean by proper sequence and hierarchically.

By the method championed by my many objectivists such as Lisa VanDamme. You learn science in the order it was chronologically discovered. Or the phonic method where a word is built on top of letters, learning to count before learning vectors etc. I hold that this usually is a very *efficient* means of learning, but not the *only* one that leads to proper integration.

While I don't understand the current practice in phonics vs. whole language method, as I understand it, the fundamental difference is that with whole language, the child isn't given any conceptual framework for understanding language and writing, and isn't even given a hint that the concept of "rule" is valid or useful; but phonics encourages a rule-governed view of language, even to the point of explicitly telling the child what the rules are.

I think is is about right. I'm not too familiar with whole language either so I don't know how much hints they give of the underlying structure. As a didactic method I think the whole language approach is reprehensible. If the child is to learn through instruction, rather than exploration then why on Earth teach using anything but the optimal method!?

The issue then is whether children are capable of inductively reaching a higher order generalization given just raw data, and doing so in a reasonable period of time. The issue is not one of learning out of order vs. learning hierarchically, it is one of learning everything on your own as though you were lost in the woods (the whole language method) versus a method that organises and presents reality in a way that the child can learn based on logical order.

My point is that in the real world (i.e. not in the artificial learning environment of a school) this is precisely how you learn many new things. You cannot properly integrate all the new information at once because you don't know yet how to classify the new information and how it relates to other information. This is what I refer to as discovery or exploration. Let me give you an example out of order induction: learning new roads by the driving method. Suppose you drive around in a city in an unknown part of town. Since it is unfamiliar to you there is no way that you can integrate this new experience into your existing network of knowledge. Then you may suddenly find yourself entering a familiar road and you go: "oh, *that's* where that road leads to." Then you are able to integrate floating new data with existing knowledge. In an unstructured environment where you cannot follow a textbook or for some other reason cannot learn in a highly efficient order you need to rely on your ability to induce out of order.

I imagine children can learn certain basics of vector physics without knowledge of numbers, but I don't know what it would mean to learn vector mathematics without learning numbers and counting -- the question is, is it mathematics?

Absolutely. With vectors you can a very good visual understanding of e.g. the commutative law. With vectors you can perceptually inspect that A+B=B+A. Likewise with the associative rule. Also with vectors you can get a perceptual grounding of negative numbers. -A is simply A with a reversed direction. This also gives the opportunity of learning the sign rules (++=+,+-=-+=-,--=+) in terms of rotation. "+" means zero revolution. "-" means half a revolution. Anyone you learns this will never ever again forget that --=+ because that's just completing a full revolution. Using vectors one can perceptually acquire a sense of proportion without learning to count, and thereby get a perceptual grounding of the basic rules of arithmetic. Even if they have not yet learnt about units they can still learn a lot of properties. Obviously you can *start* with units, but it is actually simpler to start with vectors of arbitrary lengths to teach associativity, commutativity, adding, subtracting and negative numbers before learning any units.

I don't understand what out of order induction is. If you're saying that children cannot efficiently induce the correct generalizations from unorganised reality, I agree.

Out of order induction is precisely unorganized learning, and because of this it is far, far less efficient than learning in an organized manner where someone has already gone through the painful process of discovery and concept formation, organizing them and then teaching them in a particular, structured way. But even though it is far less efficient doesn't mean that learning by this method is impossible. Indeed, any new scientific discovery is essentially unorganized leraning. For a human being learning in real life is not simply going to school learning things in a preordained order. There are often strong elements of random in acquiring new knowledge. In other words, out of order induction is not only possible but sometimes necessary.

[Onar Åm]

Onar:

You imply that induction is learning "out of hierarchical order". In fact, however, induction is exactly the opposite. Induction begins with what one already knows, adds new observations, and then _integrates_ that into new knowledge. That process is inherently hierarchical, specifically, building up to more abstract knowledge from less abstract knowledge.

Yes, this is organized induction, which typically occurs when information has been structued by humans -- post induction. But this is not the only way that induction may proceed. One may acquire information in an unorganized fashion and then only later discover that they are actually related and integrate them appropriately. Notice that I am not saying that "out of order" induction means that hierarchy is violated only that the hierarchy is constructed out of order, i.e. where you may learn floating abstractions first only later to ground them properly in lower level concepts on which they depend. Or opposite: you may acquire information which remains disintegrated for a long time until you discover their relationship and integrate them.

I am arguing that learning in a particular sequence may be far more efficient and less cognitively demanding than other paths, but that it is still possible to acquire properly integrated concepts by non-optimal means. In a sense, tutoring is also a kind of "bootstrapping" where the child is exempted from being exposed to tons and tons of real data. Induction in the form of discovery is really, really hard work, and sometimes even the greatest minds are not enough to crack the problem.

By organizing data, giving names and presenting them in a particular order, explaining relationships discovered and induced by others, one is in a sense "cheating." The child does not discover the relationship on her own, but is actually being given an instruction, call it a temporary floating abstraction if you like. The child then has to interact with this abstraction, this symbol she is being taught, and ground in concepts she already knows. In this sense, even hierarchical instruction is not fundamentally different from out of order instruction, but much more efficient.

Giving a child an abstraction whose foundation he does not know, and then expecting him to figure the foundation out on his own is certainly not "teaching".

Agreed, but suppose that you have one concept B which builds on A. Then a teacher first, for some reason teaches the child B. Obviously it now is a floating abstraction, but its mechanics can still be learnt. Now the teacher may teach A afterwards and then the concept will be properly integrated, even if the order of learning was reversed. I am not saying this is a good thing, although in some instances it may be. I for one often like to get an "overview" of the knowledge I am about to learn. So rather than delve into the details first and then organically and gradually realize why on earth I am learning all these details I often start with quickly running through the curriculum seeing what's ahead and also seeing where we are headed. Even though I may not know sufficient details to *use* the tools properly I know their functions and what they are for. This then organizes my learning. This corresponds to looking at a map of where you are going ahead of time rather than discovering the road as you go. The map is obviously a floating abstraction to you if you've never been in a place before. Nevertheless, the map is still a useful tool for you to orient. Once you do get to the place on the map, you integrate the perceptual data with the map abstraction.

Perhaps in some simple cases a child could succeed starting like that, but nothing he has in his mind qualifies as knowledge until he grasps the whole hierarchical chain from perceptual reality to the abstraction in question. Give a child something very abstract, and he likely will never learn it on his own (e.g., try giving a child "quantum physics").

Obviously, because the task is way, way too cognitively demanding in this case. The more unorganized a field is, the more cognitively demanding it becomes. That's why even the brilliant Aristotle made several important errors in his metaphysics.

Even in the case of a child who starts in the wrong place, but eventually reduces the topic back to the perceptual level, that is a bad way to do things. This is because when starting at the perceptual level and building upwards, the child learns things that are the basis for many concepts _other_ than the specific one he is trying to learn. Doing the reverse establishes only the basis for the starting concept.

I am not sure I agree with this. It may be true, but I don't quite see why. If you have B build on A, then why does it matter (for the purpose of integration) that you learn A then B as opposed to you learn B then A? Granted one is likely more efficient than the other, but in both cases you have learnt both A and B. The difference is that by first learning B you *inform* your learning of A. I.e. you know why you are learning A and focus your mind accordingly. This is not always a bad thing.

[IdeaSave]

You're right that somebody could have a floating abstraction in his head, and later connect it to reality, but until the connection to perceptual reality is made, the child is confused, which not a comfortable or desirable state to be in. Once a child is confused, effort must be spent to undo the confusion before it can be replaced with actual knowledge. The human mind is oriented towards connecting things, and giving a child an opportunity to make _wrong_ connections is a mistake.

An adult in that position may be able to withstand that discomfort, effectively by saying "Okay, in this context I want to learn G, but right now I only know A through C ... I'll put up with being in the dark for a while if I start with F." In this example, perhaps it is tolerable, since F has to be retained only long enough to get D and E. But if the missing hierarchy is very much deeper, the adult will get lost and confused due to the crow epistemology.

When teaching in hierarchical order, a teacher needs to provide a wide range of concretes that when integrated, lead to the abstraction that unites them. One student might only need subset X of those concretes, while another needs subset Y. Having the whole range available to begin with, though, means that there is a broader base of concretes available for later inductions. There is so much pressure from "the crow" when going in the opposite direction that I doubt the child could retain as many of those concretes, thus robbing him of knowledge that he could've used later.

If you haven't read Lisa Van Damme's essays on hierarchy in teaching, I highly recommend it. Note that everything I've written is my view only - I don't speak for Ms. Van Damme (or Objectivism, of course).

Mark Peters

[Cogito]

the adult will get lost and confused due to the crow epistemology

...

There is so much pressure from "the crow" when going in the opposite direction that I doubt the child could retain as many of those concretes, thus robbing him of knowledge that he could've used later.

What is this crow?

[KendallJ]

Okay, but let's apply this to reading. The first methods of writing were iconic -- drawing pictures of objects, so a picture of an ox looks like an ox. This got formalized so that the drawings stopped looking like the things, and the pictures started to be used for sounds of particular words, and that got streamlined into sounds of a syllable (the first syllable) in a (common) word, and then letters came to represent just one sound. There are a lot of hierarchies that you could use, and sometimes the hierarchy of discovery / development is not appropriate. I also don't think it's particularly sensible to start with Fire, Earth Air and Water as the elements, nor do I think that phlogiston is an essential part of a science curriculum. My shtick is that concepts should be taught in the order of logical dependence: if you have to understand concepts A and B in order to understand C, then teach A and B first. I hope that's what she's saying as well.

This is exactly what she is saying. I'll quote the basic thesis,

There is a necessary order to the formation of concepts and generalizations. A child cannot form the concept of “organism” until he has first formed the concepts of “plant” and “animal”; he cannot grasp the concept of “animal” until he has first formed concepts such as “dog” and “cat”; and so on. The pedagogical implication of the fact that there is a necessary order to the formation of abstract knowledge is that you must teach concepts and generalizations in their proper order. An abstract idea—whether a concept, generalization, principle, or theory—should never be taught to a child unless he has already grasped those ideas that necessarily precede it in the hierarchy, all the way down to the perceptual level.

So her basic techniques include recognition of the heirarchy, and not teaching abstractions that have not had proper context set, and also at each level, tightly integrating abstractions back to the perceptual level. She covers the applications to science, history, literature, and the proper limitations of Montessori method. In science it is not so much that one should follow the scientific trail, through all the ugly mistakes, but that the development of science in general follows from the less abstract concepts to the highly abstract.

It should be a fundamental rule in science education, and in education in general, that students are always taught material they can thoroughly grasp for themselves. If a teacher is presenting a principle about the physical world, the students should either be able to point to the evidence in reality—or to recreate the process of observation, reasoning, and generalization—that supports the principle. This means that science should be taught in a proper order: beginning with the simplest, most easily observable facts about the physical world, and proceeding systematically to more complex, abstract theories. It demands that the approach to science be essentially historical in progression, since the simplest discoveries are necessarily the earliest. This is the approach we take at VanDamme Academy.

Howzat? More, and I'd have to suggest you buy a subscription. Which you should have anyway...

[KendallJ]

What is this crow?

This is the "crow epistemology" or the idea that any organism's cognitive capabilities allow it to hold only a certain number of concretes as separate entities simultaneously. Beyond that, the mind necessarily starts grouping them. It is the conceptual faculty that allows man to reduce inumerable concretes into concepts and thus gain unit economy so that he can keep from overwhelming his cognitive capabilities.

I believe the hypothesis is that this number is in the single digits even for man. Think about a counting exercise. How many objects does it take for a person to have to count them rather than directly recognizing them as four or five.

The referent is to an experiment conducted with crows, whereby one sees how many entities entering a clearing it takes for a crow to cognitively switch from individual identification to seeing only "many", and be scared away.

An adult in that position may be able to withstand that discomfort, effectively by saying "Okay, in this context I want to learn G, but right now I only know A through C ... I'll put up with being in the dark for a while if I start with F." In this example, perhaps it is tolerable, since F has to be retained only long enough to get D and E. But if the missing hierarchy is very much deeper, the adult will get lost and confused due to the crow epistemology.

And another effect of this is that information taught out of heirarchy is necessarily boring. It takes much concentration, and it appears necessarily dense or arbitrary. Even if an adult could withstand it, there is no reason to do this since it is much more enjoyable when taught correctly.

[DavidOdden]

Howzat?

That's good. The point was that you should follow the logical development of a subject, regardless of the historical development.