#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.