#19 Rasch Versus Norm Referenced Tests

At IttyBitty Thoughtswe are discussing Rasch analysis and norm referenced tests. It was triggered by Rob's political opposition to the state tests that raised the issue the quality of the state tests. The quality of tests depends on two distinct aspects—their validity or in what the tests say they are testing, and their technical properties or the test calibration techniques. Is a Rasch based test better than a norm referenced test?

Also, we must distinguish between the state's core program of quantitative tests and the ODE math specialist's abortive attempt to fabricate tests of problem solving.

In a nutshell, the state quantitative tests are conceptually bankrupt and technically brilliant. The state has never conducted any validity studies of their tests but Oregon is a national leader in its technical use of the Rasch model. If you want to learn more about the Rasch model, go to our discussion of the Rasch model and follow John's responses.

#18 Place Value as Groupings

Len made the observation that in school, we teach kids that place value is a grouping of counters into inclusive groups, not an algebraic operation. The common technique is to have kids make groups of beans on a bean stick or put Unifix cubes together, or use some other base ten manipulative materials. Isn’t that a simple operation of making inclusions much like classification that children easily master in the primary grades?

Place value is more complicated that simple groups of groups. Remember, it has mathematical properties but let’s look at these math properties from the point of view of making groups of groups.

We want to look at the place value operations themselves so we might think about a numeral as a representation of a system of quantified inclusions. This quantification of the classes must include the idea that 1 ten could only be considered as a class that includes 9+1 ones. I can't draw it here but we could show this as a hierarchy like a biological classification system in which the class of hundreds includes 9+1 subclasses of ten. Ten includes the ones. The numeral tells us in essence how much of a hierarchy is present and how many separate hierarchies there are. So 234 would tell us that there are 2 hierarchies of the hundreds classes both with their complete subclasses; there would also be 3 more hierarchies each of the tens classes though neither included yet in a hundred class but each having complete ones included in them; and finally there would be 4 individual classes of ones not included in any super class of ten.

Thus, place value operations could be represented as the creation and transformations in the hierarchies. For example with the hierachy I just described for 234, with the addition of more ones, the 9+1 ones could produce an inclusive class of a ten class. The same with the tens. The digit positions in 234 essentially name the highest classes and the number of hierarchies. in each hierarchy showing in the hierarchy. (so the 3rd position position of the 2 means there are hierarchies with three levels going up to the hundreds level and there are two of these hundreds hierarchies). Altogether, there would be 2 hierarchies up to hundreds level, 3 up to the tens level, and 4 up to the ones level.

But a mathematical requirement is that the size of any group at any level must be 9+1. In other words, the student must understand this constancy of the size of grouping that we call the "base." So to understand place value, the child has to quantify the groups, insure they are always exactly 9+1 in every group, and think in terms of how many hierarchies the numeral represents.

Making bean sticks doesn’t capture these mathematical properties very well unless somehow the child figures out that the base has to be constant for the place value operations to work. Grouping when he gets to 9+1 remains just an arbitrary rule whereas borrowing and carrying require the base concept for the operations to work.

#17 The Importance of Strands as Developmental Pathways

Once we can follow the development of a cognitive system in students’ activities, we can see that the continuous construction of something like number is composed of a series of re-workings and re-organizations of an existing system. These step-by-step constructions create long sequences of developmental pathways students must follow. Once we can see the essential nature of these developmental pathways, then these strands take on huge importance as strands. The ability to see the development of a cognitive system is what transforms a concept that named as something to be covered in the curriculum to a long series of constructions forming the developmental pathway leading up to the completed concept.


TRADITIONAL CURRICULUM STRANDS

When knowledge is more traditionally viewed as learning various ideas that make up a topic with some ideas are harder than others, then strands are simply a listing of easier to harder ideas. Since the ideas that are the goals of a curriculum plan are ideas that come from teaching by the teacher or learning from objects or from transfers of something from another area of thought, the strands of the curriculum are merely listings of the various ideas without any necessary order or constructive steps. One comes before another simply because it appears to be more difficult. Student must learn single digit numbers first beginning with 1, 2, 3, 4 and so on. So primary teachers have students make number cards in order to teach them each number. Teaching number in this way of having students learn each number indicates there is no construction of the concept of number as a system of thought that encompasses all numbers. If the student sees and names five things as 5 then he knows 5 and can move on to 6. But the construction of number is something far different.

Even when teachers begin to understand that students’ understandings of number may be different from learned effects due to direct teaching, they may often believe number is somehow based on knowing some other idea that students then use to understand number. For example, some teachers believe that students must understand the part/whole relationships of classification before they can understand number. But inherent in this idea seems to be the notion that somehow classification is transferred to number. Apparently they believe that part/whole inclusion relationships are first learned in classifying plant and animals, and then transferred and applied to number. This transfer notion is not based on the principle of the construction of number and does not explain the formation of understanding as the result of a developmental pathway of number having its own constructive process that proceeds independently beginning very early in sensory-motor coordinations. But for primary students there is no such thing as transfer. They can only put toggether an understanding of number by constructing it out of an earlier understanding of number which in turn comes from a yet earlier understanding. Nowhere is there a transfer from classification or a transmission from objects or the teacher.


WHY DEVELOPMENTAL PATHWAYS, NOT STRANDS ARE IMPORTANT

The reason developmental pathways are extremely important is that knowledge cannot be transmitted. Instead it can only be constructed and that means it must always follow its developmental pathway. It means that the development of number follows its own constructive pathway as does the development of class. It means every concept can only arise in students’ thought by each following its own developmental pathway.

It is true these separate pathways seem to correspond in a loose way with curriculum strands. But there is a great difference between the two. What distinguishes a developmental pathway is its unique and continuous constructive sequence of modifications and re-organizations out of its previous forms. A developmental pathway describes how a system of meaning changes as it develops. It is a real, factual sequence that can be discovered by research.

Strands, on the other hand, do not correspond and are not derived from any psychological facts of development. They are simply collections of topics to tell schools what all students are to be taught. The ideas making up a strand are often unrelated, and they specify arbitrary grade level assignments to teachers for what they are to teach "at" their grade level. They have no factual basis since they are not derived from any research nor are they coherent since they are not the constructions of any valid theory; they result simply from the assertions of curriculum writers and committees who think certain idea might go together and should be lumped together. They tell teachers what teachers should "cover" at their grade level without regard to any placement within a developmental sequence of constructions. Some topics in a strands apparently are thought to be primary ideas while others are taught later since they appear to borrow something from a primary topic or transfers something from another area. So, for example, measurement appears to borrow the idea of number and apply it to various properties of objects such as weight, volume, density. But there are no verified developmental sequences specifying the pathways for the construction of weight or volume or density. So measurement is considered a strand in which all kinds of diverse and unrelated concepts are thrown together simply because they all involve some kind of quantification. To take another example, some writers think number borrows the idea of part/whole relationships from classification. Number also has order to it so seriation also is thought to have to come before number. Classification and seriation are simpler ideas so they must be taught before number so that number can borrow set inclusions and order relations from classification and seriation.


WHY THERE IS NO TRANSFER FOR PRIMARY STUDENTS

But the research shows number does not borrow and transfer anything from classification or from seriation or counting. Even though for the adult, the inclusion relationships of classification appear to correspond directly to the cardinal relationships of number sets and subsets and ordinal properties of number correspond to seriation, there is in fact no transfer. And aside from these research facts which show this not to be true, it is also theoretically not possible for students to take what they learned from classes or relations to apply them to number. The idea that primary students might borrow from constructions they have already made has no validity.

There are three reasons for this theoretical inconsistency that makes the idea invalid. First, to borrow the inclusion relationship from classification would mean student would have to be able to first see and abstract the relationships of inclusion distinct from the specific classes forming their classifications. They would have to differentiate between the form or operations of classification from what he has classified. It is the whole system they must abstract, not a single operation. Abstracting the form of the system of classification without its content would mean the student would have to somehow substitute an algebraic symbols for the concrete classes, they would have to think of classes without the content of cows and animals so that they could transfer just the form of the system, the A + A¢ = B along with its other operational forms, its reversible, associative, and tautological forms, that together operate with mathematical properties of a grouping structure.

Second, students aren’t remotely aware of the nature of the form of mental operations they are using, only the content of their thought, the product of mental operations. Students see and organize classes cows, pigs, animals, etc. but they are totally unaware of the mathematical forms or reasoning they are using.

Third, to know that the inclusion properties of classification can be transferred and applied to number would mean students could already see the correspondence and appropriate use of inclusion relationships to number, in other words, they could already sees in number that number systems have inclusion relationships. But if they do not understand number, they do not have the structure of number in their thinking, and they have no way of knowing what they is looking for, what structure would actually work for number. Even if students did somehow know inclusion relationships could be appropriately applied to number, they would have no way to apply the relationships since they don’t have number yet, only a preconception of number which doesn’t include understanding what numbers are or how they are composed. The transfer of knowledge from one area to another area is actually a very abstract process always involving abstraction and is never simply an application. To see that two areas have the same operational form in their make-up is to already be able to understand and see these exact same forms in the two different areas. Analogies are an example of this advanced form of abstraction, and they are very difficult for young children. Not until students are in high school and able to make abstractions and compare them are they able to successfully understand analogies. Thus, students cannot use analogies to transfer a form from one area to another until they are able to abstract the possible forms in both areas of thought.

The fundamental law of primary development is that knowledge does not result from transfer, and therefore there is no shortcut for the student. He must follow the necessary constructive sequence step-by-step in order to put together an understanding of each area of thought. This law of primary construction means that number is not simply the use of inclusion relationships of classification applied to number, the measurement of weight is not simply an application of number to weight, nor is any quantification of a quality due simply to an application of number.


IF KNOWLEDGE IS CONSTRUCTED, THEN THE DEVELOPMENTAL PATHWAYS ARE IMPORTANT

Since all the diverse concepts developed by primary students to describe the world are each the result of their own sequence of constructions, it means that our curriculum objectives must be organized specified in strands that correspond to these developmental pathways. Thus, the curriculum strand for number must specify the sequence of constructions in the development of whole number. It must describe how number begins by successive abstractions starting from the coordinated forms of early sensory motor activities with understanding successively abstracted and reflected at higher levels of thought. Number becomes more and more abstract as a system of understanding. This movement from the plane of activity directed to organizing objects to the reflection of the coordinations of these activities onto successively higher planes of thought describes a developmental pathway

Developmental pathways occur for all concepts, as for example, in the development of classes in which students’ organize and coordinate their grouping of things into classes, the naming of these classes, and the properties of objects that form each class such as cows as a class with certain properties, pigs with their distinguishing properties, animals with theirs, living objects, and so on. Each concept must be constructed. There is no innate or classification or number ability that can be transferred or recalled to be then applied to a new situation. Before they can see common forms among various ideas, students must first structure each situation so as to understand each situation and then abstract only that aspect from each that forms the similarity. But of course seeing this similarity depends on some degree of abstraction from both situations in order to see what they have in common that can be distinguished from the differences in the two situations.

In fact, almost all learning problems appear to derive from teachers asking students to use concepts which they have not yet constructed. Attempting to present rather than have students construct an understanding results in pseudo-learning, that of a memorization without full understanding. More appropriately, teachers can take up a previous activity students can successfully do such as making number cards in order to have students refine and abstract a bit more understanding of number from that activity. Or teachers may take up a new area to begin a series of constructive activities by beginning with what students can do and organize of the new topic. But primary teachers cannot make students transfer or apply some previous to something new. Nor can they present students with completed knowledge and expect them to somehow figure it out. They cannot make student learn a new idea they present unless students have some way to return to what they already understand and which they can use to develop into the completed form.

To the extent students construct an idea is the extent to which they learn an idea. The new must be structured in its own series of constructions following its own developmental pathway in order to be understood even if that construction of understanding of a new problem or situation happens to occur rapidly and almost instantaneously or through a longer, more laborious process. But once teachers intend on teaching a concept they must include in that intent the method by which they will help students develop the concept. Their task is always to go back to the earlier form of a concept by finding out what understanding, what cognitive system students’ are using and using those earlier forms of understanding to help students construct the next more advanced form. Teachers can only trigger an existing understanding; they can only use students’ existing system of understanding to operate on objects. They can only challenge students to make the next doable construction, and to help students abstract and represent this next level of understanding. They cannot ask students to bypass this construction by some method of transfer or by giving them the outlines of the new understanding to follow.

Significant learning always involves the development of a richer understanding out of an existing one. It means an understanding of number always comes by another small step of new abstract which make a little more advanced and leads to formulating a bit more of an abstract representation by students. Each step is one which students must make even if the teacher attempts to do it for the student. The danger in doing it for the student is of course that the student never will do it.


CONCLUSION

Thus, the developmental pathways or strands should not mix and confuse the various concepts that are being developed. Number mustn’t be confused with numeral, or geometry, or measurement. In the curriculum, the developmental pathways of each must be distinguished and protected. It is of course teachers’ task to promote student constructions of the representation of other objects in addition to number. Numeral is important as are the geometrical concepts of space. The task of helping student construct the representation of diverse objects is made much easier once of the developmental pathways such as the construction of number is understood. The students face the same general problems of construction for each concept. Initially numerals are figural objects tied to their perceptual forms. Spatial objects are perceptions and neither are yet composed of relationships of parts composing the whole. Thus, students remain tied to images just as they are with their early notions of number. Students cannot yet represent the numeral or spatial object as an object of thought.

Students must be able to take apart a numeral or a spatial object so as to discover the parts that make it up. A face must be taken apart to identify iuts parts that can be initially defined by boundaries that enclose the whole and that include the subparts. The face has a line around it defining the whole and inside this boundary are smaller parts like eyes that also are defined by boundaries. Things go together if they are inside the whole or if they are placed in an order in which two elements go together because they are in close proximity or are touching. So students, by acting on objects and transforming them are adding together partws to make the whole and taking the whole apart to identify its parts just as they add numbers to make a whole or take a whole number apart to identify its subsets. From these actions they can they can then attempt to reconstruct the constructions and de-constructions of their activities. For example, they perceptually guide their motor activity of tracing around shapes like faces and then abstract this form of coordination so as to reproduce the parts and the whole it produces through drawing activities as a form of representation. Following a boundary becomes an abstracted ability (a very simple, topological ability) which can be used to reproduce a face by imagining its boundaries.

So students are constructing and reconstructing all kinds of objects by first operating on them in some coordinated fashion using physical actions of transformation and then, by attempting to abstract and redo these same forms of activity in thought, to re-present the object in thought without direct perceptual contact with the object and even in its total absence. Math systems (structures) compose the mental objects we call number. Geometric systems compose the spatial properties of objects, the shape of objects and their location in relation to other spatial objects. Physical systems compose mental representations of the physical properties of objects, their weight, volume, density, color, etc. Physical systems also represent causal changes in the properties of objects as objects act on one another such as weight pulling down one end of a lever. Biologic systems compose mental representations of living objects, first their classification based on their properties and then their causal processes of change. Social/psychological system compose social objects, systems of government, economic systems, relationships due to inter-individual interaction like cooperation or bossy one-way relationships, and the systems of personality that make up the psychological subject including all the cognitive systems (structures).

We must have the developmental pathways of each of these objects carefully described in curriculum specifications. We cannot teach having only a vague reference to outcomes. A curriculum is inadequate if it only makes reference to procedures, concepts, or topics without describing their developments even if these collections are grouped together to give the appearance of a strand. Knowledge is the development of mental representations of the world and these representations depend on mental system that must be developed. Mental systems are means whereby we reconstruct and represent objects to ourselves in our thought—physical, mathematical, biologic, psycho/social They are the ends, the means are developmental pathways of construction. Therefore, an adequate curriculum must fully describe the developmental pathways of these different objects if it is to specify an adequate curricular intent for teaching.

#16 Researching Place Value Understanding

NUMBERS VERSUS NUMERALS

Some folks have generated some interesting conversations about place value, interesting because of the problems of both the theory of place value and the facts regarding when understanding is achieved. A bit on the theory of place value seems in order.

Developing better instruction in numeration, specifically, the place value system of numeration, would be greatly helped if we made clear the distinction between number and numeral. Kids develop an understanding of number by the end of 1st grade such that they can understand the composition of number and as a result, how to put parts together to form the whole and take a whole apart into its constituent parts. They understand the composition of numbers such that they can add two numbers to find the number that includes the two as parts, and they can likewise subtract out one part from a number to find the other part. But they don’t necessarily add to this understanding an understanding of place value numerals. The numerals they use are only figural symbols representing numbers, not place value numerals. What our research in Molalla showed was that place value subtraction that involves the borrowing and carrying operations is based on understanding the algebraically more complex concept of place value which comes much later than the relatively simple concept of number.


RIGHT ANSWER DOESN’T MEAN UNDERSTANDING

Investigating students’ understanding is quite different from that of checking for students’ correct use of procedures to get an answer. Merely using a procedure doesn’t mean the student understands the procedure. So, before we can conduct experiments into students’ understanding of place value, it is essential that we define clearly up front exactly what this object of study we call understanding is that we are addressing. That attempt to precisely define place value in our Molalla research resulted in a paper.

This formal description of place value then formed the basis for what must exist in students’ thought as necessary and sufficient conditions for an understanding of place value. This formulation made it possible to formulate some of our experimental methods for investigating the psychological development of place value in students. It seems to me that much research suffers because it is not clear as to psychologically exactly what the object of study is. People think they can tell when student understand place value or when they don’t. Failures in using place value operations correctly obviously are indicative of lack of understanding but the converse isn’t true. Just because students apparently are using place value operations correctly doesn’t mean they understand. Correct performance may only be a pseudo understanding that appears because itg has not been tested by a "foil" that challenges it to test it, or it may exist because there has been no careful questioning into the students’ place value logic.

For example, problems of place value often test little more than the production of correct answers. These are behavioral responses to calculation problems, not displays of the cognitive system of understanding. Of course we can teach calculation procedures which students can then learn to perform even without understanding. But as soon as we assert understanding is important, then the problem of how to define it becomes important. For that, we need to think in terms of how to define a cognitive system. This system language is necessary and useful both in mathematics and in psychology but because it seems to be lacking in ed research, there seems to be much confusion resulting.


FROM LEARNING BY TRANSMISSION FROM THE OUTSIDE OR BY THE DEVELOPMENT OF A SYSTEM OF UNDERSTANDING?

Of course I may be wrong about understanding. It may simply be the result of learning. Information about place value or other mathematical ideas may actually be no more than all the bits and pieces learned by students. This knowledge may move directly in all its little bits and pieces from teachers or from objects, or text to the learner who experiences them directly as first facts. The gradual accumulation and storage of all these discrete bits of learning and experience may be the sum total that adds up to place value understanding.

But that position of some kind of privileged prior data that is transmitted and received untainted by the learner has never been shown to be a coherent nor empirically verifiable phenomenon. It simply is a naïve conception of learning by most people. Once we understand how to conduct research into mental systems we can find methods to confirm or disprove the notion that knowledge is transmitted. We find from this research that the transmission theory of knowledge is simplistic and hugely lacking, and from a scientific point of view substantially wrong. It is not, then, an adequate basis for instruction. Teachers need something grounded in research that describes who understand does form in the human mind.

So since the direct transmission from the outside is not the source of place value understanding, the problem always comes back to asking how exactly does knowledge arise and form in the learner if not directly received even in its most primitive and simple parts? What exactly is the source and origins of place value in the human mind?


A DEFINITION OF UNDERSTANDING IS ESSENTIAL

So to investigate how it arises in thought, we must begin with a clear definition of exactly what place value understanding consists. Thus our first task is to understand the system of place value. Then we can look to see whether it is present in students’ thinking in its valid form or in an earlier and weaker form. Then we can investigate how the completed form arises out of its weaker form of understanding. Looking for this development is much different than attempting to measure whether external factors affect the rate of development. Certainly the quality of instruction or early mathematical experiences of home life do affect the rate of development but it is the actual development that we must understand if we are to powerfully affect development. So to begin, we must tackle the problem of how to describe the place value system as a valid system of thought.

#15 The Axioms of Ideology

An invesitgation of thinking boxes that control thought and positions will have to dig deeper into them than to only recognize that they exist. We have to try to describe their essential forms.

One way to do this is to do something like what mathemativians do when they begin their investigation into a mathematival system by laying out a minimum of basic definitions and axioms upon which the system is based. I noticed an example of a natural form of this process at work in idenitfying the essentials of a thinking box by Hindrocket at Power Line. These bloggers demonstrate considerable powers to get behind the words of various writers to see what kind of epistemological stuff they are made of. The sometimes very naturally pull out the basic defining notions upon which an ideology is based, a process of abstraction akin the mathematical process of abstraction and formalization of a math system.

In the example of Hindrocket at Power Line, the thinkig box he analyzes is the anti-Iraq war ideology that also expresses dislike for President Bush. He identifies three axioms that form its basic definitions and control its reasoning. First, the war was a mistake; second, the puported absence of WMDs means there can be no good justification for the conflict; and third, all casualties are, therefore, a waste.

What is relevant in this example is not the political issue and its merits but the idea that we might be able to abstract certain definitions and axioms that would adequately define the system. Notice that this process means the writer isn't simply dismissing or distorting the anti-Iraq war ideology; the writer is in fact demonstrating a deeper understanding of the ideology. What Hindrocket identifies are facts regarding how the thinking box operates, not how Hinrocket's thinking box operates. We may disagree, of course, but the important point is he demonstrates the attemtp to factually (honestly) describe the essentials of the anti-Iraq war ideology.

So Hindrocket demonstrates a valuable epistemologic tool for analyzing thinking boxes. The fact that het provides a demonstration of how this process of analysis and abstraction can be done so efficiently and easily suggests this process of abstraction might be of general use and value in analyzing education ideologies and in paticular, the thinking boxes of those we wish to understand who prominently advance certain reform programs.

#14 Putting Hubris to the Test

From my perspective, the Chalkboard Project suffers a bit from hubris, the idea that it will ride in to solve the problems we educators have, and the idea that it will have some way to leverage change even it should find answers. CP has a very long way to go to get up to speed on the huge number of ed issues surrounding their overly broad and vague goals.

Funding aside with its presumptions, the issues among educators of what the best practices are and how we should implement them would require a massive amount of rersearch and development by CP. Otherwise it is merely another voice in the cacophony harping at educators. Second exactly what leverage over schools does it think it will have? It doesn't own the schools. It's no expert, so what will it do? Go the legislature to mandate more nice things.

If CP really thinks it has answers, it should do the hard work of developing and proving its answers. Start some lab schools, CP, and show us your notions correspond to reality. For years many of us have advocated lab schools as a way to develop and demonstrate what might actually be possible as to educational performance and economic efficiency. If you want to be a change-oriented leader, will you step forward and put your ideas to the test?

#13 Chalkboard Lessons?

The article in the Oregonian reporting the Chalkboard Project's latest four ideas for ed reform once again triggered exasperation among some of our friends. The ideas so far have been disappointingly trivial—reduce class size? create an education plan for each student? raise teachers' qualifications? n ed rainy day fund? re-do school budgets?—and a bit condescending to educators.

The folks I've talked to think it is turning into a waste of time, money and opportunity because it avoids talking about the underlying causes of the system design—a protected, government monopoly. Richard Meinhard thinks it needs help and discussion from folks who have studied and understand the problem of why change and improvement is so difficult in a system of government schooling—not "input" from a bunch of citizen meetings. A superintendent friend of his said, "I've given up on the Chalkboard Project."

So…what's wrong with it. What should it do? Is the process all wrong? Will it help or hurt reform efforts?

#12 Epistemologic Access to Our Principles

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WHERE DO PRINCIPLES COME FROM?

The problem of the source of the foundations of our principles, what I call epistemologic access, is nicely raised by the BrothersJudd BrothersJudd Blog: FROM THE ARCHIVES: BULWARKIANISM:BrothersJudd locates the differences between Conservatives and Libertarians in their fundamental difference in the creative source for Man. Conservatives believe "that Man is Created, that it is from the Creator that right are derived, and that Man is bound by the laws of the Creator."

They cite Edmund Burke http://www.brothersjudd.com/index.cfm/fuseaction/reviews.detail/book_id/741 as an exemplar: "There is but one law for all, namely, that law which governs all law, the law of our Creator, the law of humanity, justice, equity—the law of nature, and of nations."

"Libertarianism in its most extreme, or most immature, form rebels against the notion that even God can limit liberty. That rebellion is the point at which it turns into mere license. Like all extremisms, it envisions itself as uniquely pure and uncompromised, but it must be obvious that having abandoned the idea of a Creator and rights as issuing from Him, this kind of libertarianism becomes incoherent. For if Man is not endowed with dignity by virtue of being Created, then what is the basis for saying that the individual should be inviolate? It can only be that you say it should be so--but if I believe otherwise, necessarily with equal validity, then what may I not do to you? And so a doctrine of liberty that has no other basis than the individual and his capacity to imagine freedom will descend into anarchy."

Now I agree that this Libertarian warrant for their epistemologic access to the truth is insufficient but I think the appeal to God won't do the trick either. It's clean, easy, but at heart still an assertion, an ipse dixit since even if God did create us, we still don't have access to Him. And the grip on our thought that a certain idea has often prevented progress both at the psychological level and at the broad level in the history of science.

It's easy to show how naive conceptions among children compels them to see facts differently than adults and to reason differently. In fact, their facts and reasoning are simply wrong when compared to scientific understanding that has been verified and demonstrated as valid. The fascinating problem is why and how this incorrect form of thought arises and maintains itself in children's thought only later to be gradually replaced by more factual and valid forms. The same occurs at the level of the history of science.

Perhaps the most spectacular example of wrong thought persisting in the face of evidence and reason was Aristotle's physics. It's conceptions of mass and motion had long been disproved, the scientific method had even long been around yet physical science was only conducted within its framework up until the time of Galileo. Why then did it persist in the face of contradictory evidence and why and how did it come to so quickly become supplanted by Newtonian physics? Again, we find certain axiomatic foundations of Aristotelian science controlled scientific thinking. These principles were held as obvious, inviolate, and foundational.

IT HAPPENS IN ALL FORMS OF THOUGHT

The problem of epistemologic access arises in our principled reasoning in morality or political philosophy, economics, and, sooner or later, or perhaps never for some people, we must go beyond asserting the obvious nature of our foundational principles no matter how gripping is our feeling of certitude or logical necessity, to figure out where or how these moral principles and logical certitudes we rely on for the whole of our world view come from or arise. The problem is important because the basic principles with which we begin control everything that follows just as in mathematics, the aximatic principles we adopt, and they can be any as long as they are non-contradictory, determine everthing that follows. We don't have to adopt the foundational axiomatic basis for Euclidean geometry. We could lift some of this principles to create a weaker geometry. Or we could change some to create an alternative geometry. But once we have chosen, then everything else follows. Since they are controllers of our thought, we must demand a warrant of sufficiency for them.

WE MUST JUSTIFY OUR EPISTEMOLOGIC ACCESS

For those principles that are not derivatory for which we assert are absolutely pre-givens then this assertion means we believe we have gained epistemologic access to some truth outside our system of thought upon which our system rests. If so, those principles requires some kind of warrant of sufficiency. Otherwise the whole of our thought lapses into a simple an article of faith. It's true that most truths can be derived from other truths just as in geometry we can derive all kinds of theorems we can prove come from more basic notions previously proven true. Of course for our starting point we take certain axioms or undefined concepts as obviously true only later to realize they aren't necessarily obviously true and we actually have a choice. The problem is that if we admit the choice of axiomatic bases is in fact a choice, how then can we keep it from being an arbitrary and egocentric choice deep down driven by our needs and wants? To hold our principles demands we answer the question of epistemologic access or suffer the charge they are rather arbitrary.

After the opening day of geometry class in which the basic axioms and undefined elements are introduced, the whole of the semester is predetermined by the initial framework and is devoted to using them to build the edifice of geometry. But should the opening day principles be blindly accepted? Must they remain unquestioned? Obviously we need them or we can't continue with geometry but should we not justify them somehow? Before we start, someone must question them. We can do this by comparison to show other possible principles that could be made foundational. Since they also could be used as a starting point but would produce a different system, the problem is to find some basis for choosing between the two.

When someone is attempting to impose their system on us, we can determine our acceptance of their system and do our choosing by demanding to see the warrant of sufficiency to see exactly what the epistemology basis is for the system. What are their foundational principles and how do they come to these? At heart, are their principles based on intuition, on divine authority we have received, on some notions of empirical demonstration or induction, what?

Our teacher must have some warrant of sufficiency for accepting the geometry principles upon which she bases the study of geometry. Does she take the system unquestioningly or does the teacher sees the epistemologic problem and respond? Without this foundation we cannot compare and evaluate the system as a choice among systems, or, if we are naïve, we may have considerable difficulty in accepting the assumptions of the first principles. Either way, for the non-believers, it becomes difficult to "get into" the system. It becomes simply a choice relative to the other person's wants and demands. So how do we know when foundational principles are sufficient as a basis for the edifice of thought we have or intend to construct?

Now back to the Libertarianism and Conservatism example. It seems to me the problem raised by the BrothersJude rests in how to establish a basis for first principles for establishing a political philosophy of freedom. What are the choices? Natural law or Divine Creator? Neither for me are sufficient. I think we must turn to a third alternative, that these principles themselves are a product of the development of thought and are not simply pre-given by divine intervention or natural pre-existence. For me, the question is simply a return to scientific epistemology which returns us to the question of how thought arises in the human species, a problem which has been satisfactorily answered for many instances of human thought and knowledge. We simply need to treat it as another problem for investigation by scientific epistemology.

#11 Defining Place Value

As part of the Molalla Math Project, we put together a definition of the place value system of numeration that I want to summarize here. Place value is a rich system of mathematical operations for creating and transforming numerals such as borrowing, carrying, column-based computions, algorithms, and so on. It’s a basic and very important math idea, but unfortunately students encounter considerable difficulty in understanding it. Our research showed that by middle school, only about a fourth have a handle on the concept.

So it seems important that before we do anything, we be clear about exactly what this thing is we call place value. We’ve got to understand the nature of this numeral system with its particular operations that is distinct from the whole number system and its operations. Without this precision in our own understanding of place value, it seems unlikely we can evaluate and work with students’ place value understandings and misunderstandings, or conduct objective research into the development of students’ understanding of place value.

But we found something strange. When we turned to expert sources that we would expect to provide us with a clear description of the place value system itself, we could not find help from any of the math textbooks and teachers’ manuals, the state curriculum objectives or materials, or even the math education literature we searched. All we could find about place value were the common procedures everyone uses with place value but not definitions or explanations of the place value system itself. It appears to us that the math ed field thinks that is all there is to place value. The filed describes nothing but various procedures and problems involving place value but not the place value system of numeration itself. If the experts in the field take for granted any definitions and explanations of the place value system and refer only to calculation problems and directions for performing various procedures and algorithms, then it is seems likely that students’ knowledge of place value will be reduced to only whether or not they can perform these place value procedures and problems correctly to produce correct answers. Without knowing what understanding the system of place value is, of course teachers and students must rely on memorization of place value procedures.

Obviously if we don’t know what place value is, then we can only work with students’ behaviors and perfomances in following place value procedures and not students’ understanding. If we cannot know whether students understand place value then obviously we cannot teach for understanding. So teaching and research both appear to be based on getting students to follow and remember procedures rather than the underlying place value system itself. So we need to be honest with ourselves. How can we teach place value, insure students understand it, or conduct research into its psychological development in student thought if we don’t have some precise definitions of what place value numeration is?

Since we could not find help from the math ed field, we decided to tackle the problem ourselves. We began by thinking about how mathematicians and scientists define things, and in particular, math systems. One approach we found was listing the necessary and sufficient conditions. These properties are important but necessary and sufficient conditions define valid properties of place value system but not its transformations themselves. A definition of a math system has to include more than properties. It has to include the transformations as well. It’s the overall system we’re after, not just its properties.

We find that the more formal methods by mathematicians specify both the elements of the system and the transformations of the system. We could then use this idea to describe the elements of the system and then describe how its operations work. The operations operate on the elements to add them, subtract them, multiply and divide them. So we have a method for defining place value system, define its elements and then define its operations.

The elements of place value system are the numerals. Numerals turn out to be a bit more complicated than they first appear. The numeral 16 appears to be two digits that represent sixteen counters. But numerals actually have some key aspects that necessary if numerals are to work properly in a place value system. We listed six aspects that make up numerals.

Place Value Numerals

First, numerals are composed of one or more digits placed together; these are the digits 0-9 of our common place value system.

Second, numerals use the places as signaling indicators. The ordinal position of the places occupied by the digits stand for certain place values. The ordinal position is a representational element indicating the place value the digit is counting. The one in 16 is a count of the base value of ten, that is, a count of the number of groups of 9+1 elements.

Third, numerals have a constant base that is the fixed size for any group that can be formed. The group size is not usually explicitly represented unless we move into another base other than our standard 9+1 grouping. Groups are either of single elements or other smaller groups but any grouping is always exactly the size of the base; it is a constant. In the standard system, the base value is always 9 + 1. When we count one more than 9, we always call the next number of things 10 or one group. The base value of what we call ten (one group) is actually one more than 9. So we need only enough unique digits to count up to the size of the group or base since then we start over by counting whole groups until we get to 9 and start over again counting groups of groups. A place value system of numeration always has one less digit than the standard size of the base, plus it has the digit 0.

Place Value Operations

We use five forms of place value operations in organizing numerals. We use two operations inside a single place and one inside a single numeral. We also have two operations that operate across places or numerals to move values among places.

First, inside a place, the repeated multiplication of the base represents a place’s value. To determine how many times the base is multiplied times itself, we use the ordinal value of the place’s position minus one. Thus the place value corresponds to the placed because both the base the and position are involved in composing it. For the first position, we use the base 0 times for a place value of 1. For the second position we use the base once for a place value of 10. For the third we multiply the base value times itself twice for a place value of 100. (The more advanced way to represent the repeated multiplication of a single value is by exponents. The first position is 10 to the 0 power or one. The second position is 10 to the second power or ten squared and so on.)

Second, inside a place, the multiplication of the digit in the place times the place’s value represent the place’s total. In 26, for the tens place there is a total of two tens or twenty.

Third, inside the numeral we combine, add, each place total to form the numeral total. In 26 we add the place totals, the two tens and the six ones, to form the numeral total.

Fourth, we have operations among the places that move values up the places such as from the ones place to the tens place. We use this operation of moving value up the places in the form of regrouping such as when we combine two numerals in column addition, e.g., 26 and 16. The total value of the two numerals combined remains constant after the regrouping so that after adding up the ones to get the total of 12 ones and 3 tens, we can regroup to get 4 tens and 2 ones without changing the overall value.

Fifth, besides regrouping which moves values up the places, there is its reverse operation that moves values down the places as in borrowing.

The Place Value System of Numeration

The place value elements and operations form a valid system, that is, a coherent whole within which we can use the operations to transform the elements "move around" the elements without contradiction. We can change a numeral by moving values up or down the places by "borrowing" or "regrouping" in any way and as many times we like and the overall total value will remain constant. We can exactly undo every change we make by reversing the borrowing operation with the regrouping operation. These two operations are the exact reverse of each other and the produce a null or zero change when combined. Furthermore, the order in which we do these operations does not matter, the end result will be the same. And we can use 0 in a place as a null or no change operation when we combine it with other place totals. Thus the place value system provides us with all kinds of operations we can use to transform a numeral while keeping the overall total constant. The place value operations produce an infinite number of numerals each of which is always another numeral. The system has a rich, mathematical logic.

The System Itself

This rich mathematical logic having the kinds of properties we described tell us the place value system is truly a mathematical type of system and not just a set of symbols or a language or figural representations for number. It is a distinct system with its own operations and elements that are not to be confused with the number system. Place value numerals are generated mathematically; they have a mathematical composition, not a figural one like words. These properties of place value make it a kind of math system we call a mathematical group. A mathematical system with group properties have a direct and reverse operation, associativity among its operations, a null operation, and closure such that the operations always produce some new numeral.

Conclusion

Now that we see the rather complicated make-up of the parts of a place value numeral and the mathematics of the place value operations, it makes a little more sense why this mathematical richness causes 3/4s of our students to have so much difficulty understanding place value.