#9 Subjectivity and Objectivity in Knowledge

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The Purpose of Intelligence

We are intelligent organisms. That means we can plan and solve problems regarding activities we carry out either as they are happening or as represented in thought as future or past events. Because we have re-constructed the world and actions as objects of thought we can think about reality in our mind. We can re-present it; we can think back in time about the objects and activities of our previous experiences, and we can think about future possible actions and objects. By organizing groups of objects and activities in as representations of thought, we can compare and explain them, transform them, derive their properties, solve problems, compare values, make decisions, and all the rest whether we are performing the actions on actual objects or not. Awareness of actions means constructing more and more complete representations of objects and activities by our reflective thought, and as a result giving our thought increasingly more power to regulate and organize our activities. We understand better what we are doing and its effects; and we become more thoughtful about what we are doing. Because the mind can represents reality and actions in thought, it serves as an amazing cognitive organ of human adaptation

How does the mind achieve this remarkable ability to ‘see’ and think about objects and events even though they may not be present or may no longer be present? How is it possible that the things we can ‘see’ in our mind’s eye are often located in another place, or in another time? And where does the world of possibilities we visit in our minds come from? If intelligent behavior depends on the mental capability to represent and replay either real or imagined activities and objects and activities, how does this representational world arise in human thought? Even more marvelous, the knowledge and thought we achieve can be shared and passed along to others through language. It too is a remarkable feature distinguishing humans from other species. It is a vast and complex world we are able to reconstruct in thought.

Considering the new born infant, the construction of this vast system of mental capabilities to represent things is an amazing accomplishment of every human being. How then do we explain where this wonderful mental machinery comes from? Does it come through the language used to teach young students in school? Does it come from experience with objects? Perhaps there is some kind of innate reasoning abilities that account for the ability to learn, or perhaps the innate ability actually comes from brain development and maturation. Perhaps its source is social interaction and language. Or perhaps it is seen in the ideal forms that exist and which are apprehended by seeing their approximations as they appear in the concrete world.

Understanding how mental capabilities arise, whatever their form, is the essential basis for professional knowledge upon which educational programs are designed. Fulfilling the intent of education to actively promote the development of intellectual capabilities depends on the practical value of our scientific knowledge of student development. Educators’ professional knowledge base is not based on common sense but on scientific research of how knowledge develops within students.


Number and Amount

Let’s take the simple example of number. How exactly does it arise in young students’ minds such that many of them develop complex patterns of computations and relationships about numbers? How is it they can divorce a number from the group of objects that represent a number? When we show 12 counters to the kindergartner, most can count them and match another set of 12 directly side by side. But if we disturb the arrangement, the young students think the number has changed. Even somewhat older students if the arrangement is altered will be unsure if the number has been retained and will recount the counters. And even if young students match two rows in one-to-one correspondence and insist they have the same number, if we then spread out one row they change their answers and assert one row then has more. Even more astonishing, some students a bit older actually count the two rows when one is spread out, tell us both rows have 12 counters, but then go to assert that the number is different, and one row still has more.

Or take the example of kindergarten students’ understanding of amount such as the amount of clay in a ball. Here’s is an instance of physical knowledge in which nothing could be simpler than to know that if we squish a ball of clay a bit, the amount of clay certainly hasn’t changed. We don’t add any or take any away. How could young students not see that the amount is the same? Yet that is precisely what young students sees. If we ask them how they know one ball of clay now has less, they point out that it is not as tall. It is pushed down to make it less. So they point out what is a direct fact for them, the observation that the ball clearly and simply has less clay. But, if we point out that it is wider, amazingly they simply change their views regarding the amount and say it means it has more. The contradictions bothers them not one bit.

The facts these young students see are so compelling to them that most will not believe us when we show them the amount is the same. They don’t see it that way although some will dutifully accept the adults’ ideas (but questioning or posing the problem later in time shows these students still hold their ideas that the amount changes). The evidence for these students is irrefutable; they observe that the squished ball, in fact, is shorter, and they ‘see’ that it has less. Nothing could be more obvious to them than this direct evidence they see in the objects.

The same is true of young students’ notions of number. They see that the number of counters in the line that is spread out has more than another row equal in number that is compressed. They point out that some of the counters of the long row stick out past the end of the other row of counters; that shows, they assert, that the spread out row has more counters. So these raw data the young students observe directly in the objects, the facts they see, are not only incorrect, but since they are incorrect, the facts directly observed from their experience with these objects cannot lead them to a more advanced and correct knowledge of number and quantity. This difficulty in seeing even the most basic of facts directly shown by the objects presents a serious problem. How can it be explained?


What Students See Depends on What They See With

What the developmental research reveals to us is that knowledge doesn’t arise in the students’ minds by directly receiving information the outside such as through more experiences with counters or with the amount of clay in the ball nor does it come from explanations or discussions. And it is not a product of maturation or brain development. The problem is that the mind must use its existing mental machinery, as incomplete as this understanding is, to form the meaning that they see in the situation of the clay balls or the counters. The mind relies on its mental capabilities, its conceptual buckets, by which to see and record information from outside the mind. Since information cannot be transmitted through the air, the mind has no way to directly receive information from vibrating light or air waves directly through the senses. These stimuli are not even raw data for the mind until the mind acts to single out certain stimuli out of the mass and transform them into data. To transform an interaction with objects into a representation by thought, means there must be some way, some means whereby information corresponding to the objects comes to be reconstructed, organized, and represented as a mental activity within the mind.

The mind must make its contact with the objects appear to itself as observations of the object but the mind doesn’t make contact with physical objects, only with mental objects. Since physical data cannot be directly received by the mind, these observations by the mind can only be reconstructions of the object, not direct copies. The fact that what the mind of these young students observe is actually as yet only a partial reconstruction of the objects means that these initial data that are observed are very poor, very misleading, very fleeting, and very fragmentary.

The reason for this poor quality of these data is that they must depend on the weak mental capabilities the young students’ minds are using to organize and represent the clay balls in thought. The young students mentally have nothing else available. They need the fully developed concepts of amount or of number as mental capabilities that allow them to see amount and number properly but their notions of quantity and number are still weak, incomplete, and not yet able to organize things into a coherent, rational wholes on the place of representational thought. These students don’t have the mental capabilities these two concepts of number and quantity could provide, and as a result they must still largely rely on their simple, direct perceptions of the appearance of the objects. And as these appearances change as the objects are moved or reshaped, their perceptions change. Without a mental capability that would replace perception and correct its misleading impressions, their mind has no way to reconstruct, represent, and organize the changes in the counters or clay ball; they have only an almost non-existent capability to follow the changes. All they can must is to see separate, isolated actions on the clay balls or counters.

So the appearances they perceive as observations in thought easily mislead them. They can only mentally represent what they think they see. Their minds do not yet have mental capabilities for organizing mental activity directed to reconstructing and representing the object as an object of thought. They see only simple, isolated changes in a series of separate, disconnected centrations on the changing situation so that as one row is extended past the other or the ball is flattened; these are narrow, isolated centrations on a single aspect that produce misleading observations of number and quantity. The overall impression is that the whole is getting larger, or perhaps smaller. The mental machinery of young students is undeveloped and weak, and what they see with that mental machinery is distorted.


The Students’ Facts Are Not Our Facts

The problem is that from our point of view the outside information students are receiving is wrong because of the meaning the students are attributing to the objects. What they think they are observing is erroneous. They are seeing the facts differently and distorting them in spite of the careful demonstration of amount in the clay balls. They can’t see what we see as objective facts no matter what we do. The problem lies with the internal mental capability students are using to form their understanding of the amount of clay and what they observe in the changes of amount.

Their internal mental machinery is already a complex system of all kinds of organized patterns or methods that organize what they see in their developing minds and how they think but it is still not adequate. They must further develop their mental machinery because there is no other way to objectively record the facts from the objects. The mind can’t directly receive meaning from the outside because meaning can’t fly through the air. It is not transmitted so all these young students have to rely on is their inadequate understanding and the erroneous facts they see in the clay balls. How then is better knowledge of amount even possible? It can’t come from more experiences with the objects since the facts from experience are actually misleading. And they can’t use some kind of rational processes of induction to correct errors since they do not yet have a rational logic available that would allow them to understand and reason about changes in amount. Their machinery is giving them false information, but what is worse is that they do not know their facts are wrong. Thus, development is the key.

Examples from developmental research show over and over again how powerful the distorting effects of an undeveloped internal capability are in shaping what young students see. It is not possible for young students to see simple facts directly without the prior development and use of an internal, mental capability. To see objective data, we must use a corresponding fully developed concept which makes the observation possible. The concept, such as quantity or number, shapes the kind of data we see in the situation. We use our concept of amount as a mental capability by which to make the data we see regarding amount appear to be objectively coming directly from the objects. It seems to be a result of direct observation as if it is coming directly from the properties of the object without any subjective involvement. In fact, what we observe is due to applying a concept to reconstruct a property and represent it in thought.

And over and over again, careful investigation of young students reveals that what they see in the objects as true data appear to adults to be distorted and subjective data. But in adults, their data to them appear objective and as directly due to the object. The adults observations appear to be objective without subjective distortions. So the tendency of lay people is to think our knowledge starts out as subjective and gradually become more objective as if somehow we rid ourselves of subjectivity in our knowledge. But if they don’t get rid of subjectivity, how do adults manage to see things objectively whereas young students do not? Either adults must get rid of the subjective aspect of their knowledge or their subjective aspect somehow must develop into a form that is less intrusive, distorting, and one that can produce more objective observations.

The research is clear, but only if we actually follow the development of concepts. By tracing them back and following their formation, we find that the second solution is clearly the true explanation. Mental capabilities do develop into more rational forms that lead to less distortion in the observations that make. What happens in development is not that adults gradually shed the subjective aspect in their observations and knowledge but that their subjective aspect develops into rational forms that no longer lead to distortions in the observations. The subjectivity seems to disappear because thought operates in an obviously necessary and objective manner. Adults obviously can not get rid of their notions of amount and number since these subjective instances of mental machinery are necessary in making observations about amount and number we see in objects. What happens is the simpler and distorting notions of quantity and number we held as young students develop into rational concepts that not only produce our logical reasoning about the changes we observe in amount and number but also fit more objectively with the actual physical transformations of the objects. Now since it fits so well and is so logical, our data about number and quantity seem to come directly from the objects. But this objectivity is only possible because of the development of the subjective aspect, not to its disappearance.


The Parallel Developments of Subjectivity and Objectivity

Thus, the two aspects of every act of knowing, the subjective and the objective, develop in parallel with each other. As the subjective, internal mental capabilities of concepts develop, what they ‘see’ also develops. What the mental machinery sees in objects as observations is due to the more extensive fit of the mental machinery with the objective properties of external objects. The transformations of objects are not irrational. A mind that has available rational concepts can thus make rational attributions of these transformations to the objects and thereby ‘see’ objective facts in objects. Subjectivity and objectivity develop together as the two poles of an act of knowledge.

The essential point is that subjectivity and objectivity are not static entities or states and one does not arise out of the other. They are twin processes that result from the activity of the mind as it applies its organized forms to its mental objects. It transforms objects in order to know and represent them. The subjective aspect of activity is due to the developing organization of mental activity, and the objective aspect is due to the results imposed on the activity by the objects as activity applies itself to the object. The distortions we adults see in young students’ thought comes from the subjective property of their pre-conceptual forms of organization that they use to record their observations. The distorted facts they see comes from the pre-conceptual forms they use to record those observations of the objects. Facts cannot come from objects without applying an action to the objects.

As soon as we do something to objects we can see things, their properties and forms. We can physically manipulate them but to reconstruct and represent objects in thought, we must do these actions mentally in there more form. In thought we outline objects, divide them into parts enclosed in large wholes, sequenced their various elements, put various aspects into correspondence, color them, weigh them, locate them in space in relation to other things or to space as a container, attribute meaning by actions we can use them for, etc

Knowledge of objects is composed of the totality of all the things we can do either in a practical sense or in a conceptual sense to the objects. To classify is an activity that transforms objects into classes. To enumerate is an action that transforms objects into counters or units. To enclose is and activity that outlines and transforms objects into a spatial form. To multiply two series is an action that transforms two numbers into a product. Everything we observe depends on an organized mental activity that is applied and fitted to observation it makes possible. It is true that there can be no mental activity applied to mental objects without there being mental objects, and it is also true that there can be no mental objects without there being some kind of organized mental activity that makes the mental object appear. Thus, knowledge is an activity reconstructed from sensory motor activity applied to objects whose twin subjective and objective poles fit together. They are two sides of the same activity, the one appearing as the organization due to the subject and the other as the observation that results due to the object. Thus the content and process of in an act of knowledge are simply the result of analyzing mental activity in its two aspects, the one inward directed to the subject and the other outward directed to the object of thought.

Thus, human thought does not get rid of its subjectivity, it organizes it to make less intrusive and thus to disappear, and that re-organization is the role of development. In this regard it means young students are no different from adults or even scientists in their attempts to reflect on and develop their subjective mental organization so they can better observe and represent the world. The only difference between adults and young students is in the degree of development of the subjective mental capabilities that are employed. Every act of knowledge depends on a subjective aspect but in adults that subjective aspect has achieved a form that is much more rational and logical, and so it seems not to distort the observations it can record. It produces objectivity.

In adults, the completed concept of amount and number functions in the same manner as an incomplete concept does in young students the only difference is in the organizational form that different between the simpler notions of amount and number and the more advanced completed concepts. The adult concepts produce far less distortion in the observations these concepts make possible but both adults and young students employ subjective methods in making observations of the ball of clay; it cannot be otherwise. The differences that appear are due to the fact the adult subjective capabilities of amount and number have a much longer developmental history such that they have formed into more rational, stable capabilities that better fit to their observations they can record off the objects. These subjective concepts also organize adults’ reasoning about amount to make it logical whereas the incomplete notions of young students can organize only irrational bits and pieces of thought. Thus, all knowledge results from the universal functioning of both what we see and what we see with, but "what we see with" varies in its development from simple notions into the conceptual systems of adults both of which more or less fit themselves to the properties of objects. It is the more advanced concepts of number and amount that form the subjective mental capabilities which fit more objectively to the objects and the transformations of their amount and number that we can observe.


The Subjectivity of Adult Thought

Adult thought, even though controlled by subjective mental capabilities, seems to exist only as objective entities separate from the mind that somehow are transmitted into the mind. But the existence of the facts and the logical coherence of our understanding of the facts is not due only to the objects to which we have applied our concepts, they are due to the concepts we apply to the objects and therefore the facts we see depend on the level of development of our concepts. The facts may seem so obvious as to suggest we used no subjective machinery to produce them. The objectivity of our observations seems to have no subjectivity in them. They seem to be ‘raw’ data undistorted by interpretation.

But subjectivity is involved, and it is its development into rational concepts that makes its ‘subjectivity’ disappear by no longer being intrusive into the observations or organizing a faulty logic. The subjective capabilities have not left the scene. These subjective capabilities are what correct our perceptions to see the objective facts, and, by comparison, they also let us see our differences with students. They allow us to observe students’ mistakes as subjective distortions of erroneous facts, and they let us observe the funny kinds of irrational reasoning the young students use. The students’ subjectivity shows; ours doesn’t.

Since our subjectivity doesn’t show to us (at least to most people), the kind of data we see and record as ‘raw’ un-interpreted data does not appear as dependent on the level of understanding we bring to the situation in the form of our concepts. The facts we can see and learn depend on the prior development of the concepts that are necessarily involved in the learning. The source of concepts is development, not learning. Thus, the important conclusion is that development is not the result of learning; instead, learning depends on development. There is a huge difference between educational methods which intend to promote development and those whose primary aim is learning. Since learning is not possible until there is the development of the intellectual capabilities, development is the more fundamental goal of education.

#8 The SAGE Math Course Objectives

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TRADITIONAL AMERICAN-STYLE RESEARCH AND INSTRUCTION

When researchers look at how students respond to missing addends in an equation, for example:

8 + 4 = __ + 5

expressing a simple math fact, they find that most primary students, more than 90%, don’t understand the equation or the equality sign. What’s worse, in spite of mathematics being taught continuously from Kindergarten to 6th grade, students make no progress, none, in their understanding of the problem. Over 90% of 6th graders fail to use 7 for an answer. They use either 12 or 17.

One group of researchers after finding such dismal failure of understanding among primary students believe the solution to student misconceptions lies in

"engaging them in discussion in which different conceptions of the equal sign emerge and must be resolved. In these discussions students can be encouraged to clearly articulate their conceptions of how the equal sign is used and to make them explicit so that the other members of the class can understand the different perspective represented in the class. It is necessary for students to be clear about their conceptions of the equal sign and what supports those conceptions in order to attempt to figure out how to resolve the differences."

But is the origins of mathematical knowledge social interactions, especially in this manner? And have these researchers found the cause or only a symptom for lack of the development of student understanding? Should they rush to their favored instructional solution of social interaction or would it be useful to dig into the causes of how students develop understanding? If we understood how students construct an understanding and how they develop ways to represent their understandings, would that knowledge help us redesign instruction? A lot of education research admittedly is useless. For example, some researchers ask, "What are the causes of students’ failure, and they tend to focus on factors outside the student—how to present or teach equations such as whether there should be student interaction or not, whether manipulatives should be involved, whether it should be broken down into smaller steps and more directly taught and learned, that sort of thing.

But other researchers ask, "How do students develop mathematical understanding? How do they develop their representations of their mathematical thinking?" This kind of developmental research is useful. The traditional research questions tend to be based on questions about what we do to students, and they produce traditional answers about what things teachers should do to students differently. The researchers who ask how students develop their understandings, they are asking a much deeper, more profound question. How does mathematical thought arise in human thought?
In the traditional research question, researchers are asking about what external factors make students learn. In the developmental question researchers are asking, given all the stimuli students experience, how do they organize mathematical thinking? The developmental researchers are asking about the internal developmental factors instead of the traditional external learning factors.

Believing learning is the source of mathematics has never gotten us very far in promoting student growth simply because knowledge doesn’t fly through the air to the student. It doesn’t matter what source we are trying to use—mathematics expressed clearly in our words, demonstrated with objects, shown by patterns in the world, given through students’ experiences with manipulatives, transmitted through social interactions and discussions, or from practice in procedures we give them, or we should add, waiting for brain growth. None of these are the source of mathematics. Mathematics doesn’t exist outside the human mind in the world of objects or in some kind of ideals of the world located outside students’ thinking. Mathematics is an organization of our mental activities of thought that organizes itself into systems, systems such as the whole number system that produces number facts, the classification system that produces biological and other kinds of classes, order systems that produce the relations of difference in the weight or length of objects and so on. We can’t make students organize their thinking into these systems. We can develop methods which stimulate and make them efficient in their development work.

The fact that American education is based on transmission notions of math development means that developmental instruction is not used. The result is that students never have the opportunity to put together the mental systems for understanding whole number, or fractions, or place value, and so on. It produces the math achievement gap and the miserable performance we get overall in mathematics education. The missing developmental instruction produces the lack of understanding of students we find in every area of mathematics. If we show students a numeral and then show him the borrowing operation, only 1/5 of middle school students know the value remains constant. All the rest believe the new numeral has a different value. For them 47, 4 tens and 7 ones, represents a different number than 3 tens 17 ones. The tests of learned procedures show many student performing computational problems but the tasks measuring mathematical understanding show many of the same student do not understand the problems they are computing.


THE POINT OF THE SAGE MATH COURSES

The point of SAGE math is to help teachers transform their instructional methods so they can help their students develop an understanding of mathematics. The change from traditional American-style instruction to developmental instruction depends on the course helping teachers investigate and see the systems of understanding as they appear in students thinking. The transformation of instruction to a promote student development rather than learning depends teachers developing the ability to see and work with cognitive systems. To help teachers develop the ability to see systems at work, the SAGE math course uses problem solving tasks that give teachers special problems to pose to students, questioning techniques, and a scoring system directly representing the level of development of the student’s system of reasoning that is at work and displayed in his problem solving. The course attempts to turn teachers into researchers investigating mental systems in students. From this research-based understanding, the course then shows teaches how to organize developmental instruction by giving teachers examples of developmental activities for students and teaching methods for interacting with students. The course has to more teachers through these objectives in order to help them develop their math instruction to make it more powerful in affecting the development of student understanding.

If I were to summarize the SAGE math course objectives, I would say the course begins with helping teachers see the fundamental problem of understanding how math knowledge arises in student thought. If teachers believe that mathematics comes by learning from a source outside the student—manipulative experiences, direct teaching, social interaction and discussion, seeing patterns in things, etc—then a course about developmental instruction will make no sense. But once teachers see the basic developmental problem that the student faces in putting together an of the math equation with the missing addend—since he doesn’t understand it or rather understands it in a different way, his different method of understanding makes him see things differently than the teacher does, and he has no way to make sense out of explanations or demonstrations or experiences of how it works differently than what he thinks and sees—then a course that focuses on students’ constructive activities rather than on teachers teaching begins to make sense and fill an important need.

Next the course begins to help teachers see the systems at work in students’ thinking by showing them the two aspects of every student thought—its methods or system for making meaning out of the situation and the things the system lets students see and represent. Obviously the content of what is experienced comes from the outside world, the things presented to students, but the internal meaning maker he uses to makes sense out of the stimuli is of internal origins. The meaning maker is the intellectual tool or conceptual bucket the student uses to take in information. Learning depends on the development of these conceptual tools or systems. Thus, every act of knowledge is composed of both an internal attempt to impose meaning on something and on the observable features of the object of the student’s attention that the mental activity produces. So analyzing both the internal system and the external observable facts—what the student sees with and what the student sees—is a difficult step.

The previous objective is difficult, almost impossible, without having some kind of analytic tools or models that give teachers an idea of what to look for in students activity, what the possible forms are. These analytic models are the third objective of the SAGE math course; they are the systems or structures of thought that we can find in students thought. The course attempts to give teachers curriculum and videotape examples to help them see which of the systems is at work. Seeing a classification at work in producing the nouns a student uses is perhaps the easiest. Ordering systems are also relatively easy to see as they are at work producing words like "heavier than", "colder", and all the comparatives and adjectives students use to express the relations of difference they see among objects. Seeing the underlying whole number system that is at work producing the numbers and relations students see is a bit more difficult. Mathematics has been so emphasized as a vast collection of procedures, computations, and math facts to be learned that it is hard for teachers to switch to systems thinking. The course attempts to help teachers see these various systems at work so that when teachers see the missing addend problem we started with, they will recognize it as the vicariance system of numbers rather than the simple additive system.

But even the ability to see and describe the systems in students’ thinking doesn’t explain how they develop. It’s one thing to be able to see and measure the level of student understanding but quite another to explain exactly how a weak, incomplete system somehow within itself has the processes necessary to development, that is, to re-organize and reflect itself onto a higher plane of thought.

Interacting with these three objectives of understanding the development of cognitive systems is the last objective of understanding and using developmental instruction. Once teachers can see development of systems, they can see the construction of mathematical systems being recapitulated over and over again in every one of their students. It is this sequence of constructions that the developmental teacher follows in her instructional cycles with students. Each constructive problem the students tackle and overcome opens up; the need for the next construction. This course objective helps teachers insure each student puts together each step of the construction of the mathematical edifice so that no student is faced with trying to understand a problem beyond what he has constructed, can understand, and can represent. Thus the course helps deconstruct the missing addends problem back to its earlier forms which students are able to produce. The teacher then works with the students authentic productions of these earlier forms to help students develop and represent them step by step until they understand vicariance relationships and their representation in equations. The fact that students cannot understand the mathematical meaning of the missing addends equation tells the developmental teacher that instruction shouldn’t begin with the equation; it should begin with what students can understand, produce as facts, and represent symbolically in some earlier form.


THE SAGE MATH COURSE OBJECTIVES

So to summarize the SAGE math courses, we can list four main objectives:

1. The basic problem of the source of students knowledge—Theories of math development and the factors of development; development versus learning. At the end of the course, we want teachers to be able to explain why math isn’t and can’t be due only to learning.

2. Analyzing the two aspects of knowledge (mental activity)—The internal assimilatory system and the accommodations it sees due to the effects of the external object of attention. At the end of the coursee, we want teachers to be able to see that in every example of learning, there is an assimilatory system at work inside the students’ minds shaping what they see and represent due to accommodation to the external object.

3. The various models of systems and their development—The part/whole systems, the ordering systems, the multiplicative systems that use more than one order or part/whole systems simultaneously, the number system which combines part/whole and order into cardinal and ordinal properties of number, and measurement systems. At the end of the course from examples of curriculum or learning activities we want teachers to have the ability to see and describe which system is at work in organizing the meaning for students.

4. Developmental instruction—Transforming instruction so it begins with students’ understandings and representations and then directly promotes their development. Developmental instruction is the main goal of the course, and it is far more complex than the other objectives.

Developmental instruction transforms:

a) the teaching method of discrete lesson into constructive learning cycles,
b) single answer questions into problems of method and understanding (replacing the emphasis on learning information with problem posing),
c) simple learning activities into constructive learning,
d) experiences demonstrating and experiencing carefully designed manipulatives into student organized demonstrations beginning with "raw" objects,
e) "what" type of curriculum objectives of what is to be investigated and learned into developmental goals defining constructive sequences beginning with existing student understandings,
f) the presentation of symbols and words students must learn into beginning with and developing students’ forms of representation,
g) tests of recall into problems solving demonstrating methods of understanding.

These courses are about teacher development; they are not about learning new classroom activities or some new information about math education. Development involves fundamental change in understanding and method and therefore does not occur suddenly. The course, if it is successful, at least opens the door so that teachers see a new and much more powerful direction for the development of their instructional program. If we are successful in hooking teachers into investigating student development and developmental instruction, then that new direction is evidence they have mastered enough of the tools and knowledge of the course to continue to grow and develop in this new direction as self-directed learners.

#7 Developmental Instruction

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In general, we can group the differences between developmental and non-developmental instruction into two categories. We will use the example of lessons surrounding the equal sign to demonstrate these two differences in instructional programs. The first difference concerns whether the instruction works with completed knowledge or whether it begins with activities and understandings students generate. The second concerns whether the instruction works with procedures, symbols, or words or whether it works with the wholes of thought, the systems that form students’ meanings. Using tasks to investigate students’ thinking helps make these two ideas clear even though these assessments are not the basis of instruction.

ASSESSMENT VERSUS INSTRUCTION

We use the carefully designed task method of developmental assessment to see and measure the developmental level of mental systems at work as they organize students’ thought and produce the students’ understandings. The same idea works much the same way in instruction but with some modifications. If we intend to promote the development of cognitive systems of understanding, we do much the same in posing problems, obtaining displays of students’ understanding, and abstracting the system they are using just as we do formally with task assessments. But the intent of instruction is not to test kids, and there are important differences.

Two differences between teaching and assessment are key. First, assessment is teacher driven, and second, assessment simply takes a snapshot and then ends the activity whereas developmental instruction repeats and extends activities in cycles in order to promote the construction and development of students’ understandings. Assessment of students by the teacher traditionally has not been intended to be part of an educational activity and for the benefit of students. If assessment was to be of benefit to students in their educational activity, then the instruction would at some point in the instructional cycle help students get and give their own feedback as to their current level of understanding of something. Teachers would help students organize and perform their own self-assessments, something seldom done probably because so much of current programs design activities to be teacher driven, not student driven.

Teaching, in contrast to testing, takes the kind of understanding students are working with and helps students use those systems in cycles of investigation into the object of study and thereby to develop them. Developmental instruction therefore can’t be a one lesson affair; it has to continue student activity by organizing it around cycles of problem posing, investigation, and reflection which in turn lead back to refinements and extensions of the problem students’ see and new student investigations. Learning should lead to more learning, not be a one-shot, closed affair. So it is instructional cycles that, by their design based on the development and enlargement of students’ systems of understanding, distinguish developmental instruction from non-developmental instruction.

Tasks, then, are models for how we can pose problems with objects that allow us to see an authentic display of the level of meaning students can put the objects. We could, of course, show them words and symbols but to uncover the meaning, the mental systems they have at work, we need to return to activity on objects.


DEVELOPMENT VERSUS DIRECT INSTRUCTION

We can distinguish developmental programs from those programs based on direct teaching and receptive learning. Programs based on presenting procedures and having students learn the procedures do not have the constructive cycles of developmental instruction that follow and promote the constructive sequence in building understanding nor are they based on the students’ productions expressing their cognitive systems at work. If we present the equation 4 + 3 = __ to young students, we can teach them a procedure to follow to find the correct answer. We can start by showing them how to use counters to add the two numbers and then count the sum and write it in. We can have them practice these ‘facts’ until they have them memorized. But we have no idea what kind of meaning the facts have for students since we don’t have a display of the mental system of understanding. And, as a result, there is no student construction involved.

In direct instruction, once the material being presented is learned to mastery, the lesson boundary ends the activity since it has achieved its objective. The teacher then moves to the next objective and presents the next piece of new material to be learned. Good direct instruction attempts to sequence the step by step learning of students and to insure each step is mastered but it is not necessarily constructive of the mental systems students must use in the learning activities. It tends to reduce itself to simple, repeatable facts and procedures such as in learning to phonically decode. It does not recognize or by intent work with students’ systems of understanding. Learning, of course, always depends on the student having some prior mental system by which to take in information, to interpret its meaning, to understand it, to remember it, and in general, to generate an ability to form the knowledge.

Whereas the objective of learning is for the learner to take in and retain something presented externally, the objective of development is to help the learner construct internally a better system of understanding. In developmental instruction, the intent is to help the learner put together an understanding of the important concepts, not learn small pieces of sequenced information. In developmental instruction, everything depends on begin able to see and work with students’ mental systems. In direct instruction, the teacher works with facts and procedures. Student understanding is conceived as a result of the accumulation of many acts of learning. In contrast, developmental instruction turns the relationship between development and learning around. The development of some ability to learn, such as developing the concept of number, must occur before learning using that concept can occur such as learning math facts. The research is clear on this point. Without the conceptual understanding, the facts students see are distorted and erroneous, and so learning based on these distortions cannot be the source of objective knowledge. The ability to learn is the product of development, not the other way around.

Carefully designed direct instructional methods are useful when the development of understanding is not at issue. For example, most children have the ability to hear various sounds and parts of words. They also have the perceptual ability to see differences among figural symbols such as letters and words. Teaching them phonics so they can retain and use the sound symbol relationships in reading does not depend on the development of some ability to learn that is not yet present. Children develop auditory and perceptual discriminatory abilities very early. Direct instruction works well in the case of beginning reading. In those areas of the curriculum where the development of the intellectual systems is at issue such as in mathematics, direct instruction cannot continue to ignore developmental problems and remain effective. Thus, seeing and working with the intellectual systems of students is key for developmental instruction in all the disciplines.


AN EXAMPLE OF NON-DEVELOPMENTAL INSTRUCTION

From the literature we find this latest example of non-developmental instruction that is not of the direct instruction variety. Researchers presented the equation to 8 + 4 = __ + 5 to students; they found that students put 12 in the blank. In fact, at all grades up through 6th, less than 10% put in 7. The conclusion of these writers is that students don’t understand equality and the equal sign. That finding, of course, is obviously true. And these writers believe that it is most helpful to challenge students’ misconceptions and to have students discuss the meaning of the equal sign. This technique is a non-developmental approach.

These writers, and they seem to be typical, believe that attempting to use class discussions and challenges of student misconceptions will help students learn the proper meaning of the equal sign. However, despite appearances, this technique is decidedly not a developmental method of instruction. It relies primarily on social interaction. The reason for being non-developmental is that it does not begin with and work with students’ mental systems, what students’ understanding can produce and represent as math facts as abstracted from their activities in organizing the counters. It begins with the advanced symbolic representations of the adults—with the vicariant cardinal relationships of sets and subsets of number that are abstracted and symbolized with numerals, +, -, and =. The students are supposed to find some way to understand the ideas without going back and constructing the understandings leading up to this advanced level.


CHANGING INSTRUCTIONAL ACTIVITIES TO MAKE THEM DEVELOPMENTAL

What we can do to make the instruction more developmental is to pose a more developmental problem, that is, to move the problem down from its lofty abstract, symbolic level to the level of students’ activities organizing objects, questions that are also more of the ‘how’ than the ‘what’ variety. From the actions students generate with the objects, they can then abstract and represent the meaning their results have for them. For example, we could give student 12 counters and ask them if they could somehow show how to combine two groups of counters into one overall whole set and then represent that whole action. Now once they have the idea (and they will first focus only on the answer the produce and not the groups they started with), we can then ask them to develop more combinations and share how they represented their findings on their papers.

Young students do not produce mathematical equations for the math facts they demonstrate so the development of representation is just as important as the development of the mathematical meaning, the relationships. To find more relationships we can ask them to make some conjectures about relationships among facts. Again, we want them to demonstrate them, share, and discuss them with others. Almost any child above the first grade level can push two piles together to make a whole and then count. There is high success with this activity since it is within the conceptual range of all students. And surprisingly, all primary students can also represent on paper their activities of combining two sets into 12, they just don’t use the formal mathematical symbols as yet. But they can develop their representations of their math facts by beginning with drawing-like signs that are closer to their actions as they so far understand them. These rebuses usually take the form of little drawings, arrows, circles, and pictures. Soon, students can move to replacing a set of hand drawn counters with numerals. Their representations seem to follow a pattern of development although the "development" of representation actually depends on and derives from the corresponding development of the mathematical meaning students have become aware of and are attempting to represent and share.


CONSTRUCTIONS SHOULD LEAD TO MORE CONSTRUCTIONS

Now using students’ activities to generate math facts with 12 counters and to represent them can lead to all kinds of interesting new problems. We can pose problems about how many different math facts there are for 12, how they could prove their conjectures, whether 0 and 12 is a new fact or not, whether 5 and 7 is the same as 7 and 5, whether having 12 is the same or different in number from having 5 and 7, whether we can undo 12 into parts, whether all the math facts are equal ( 5 and 7 are equal to 8 and 4?) and so on with each solution needing the student to demonstrate with some method that their conjecture is valid.

These kinds of activities working with mathematical relationships as facts that students can organize and represent, all concern the developing meaning the whole number system has for students, not the representational symbols of math. Each constructive step follows from the previous accomplishment. For example, there comes a point when students will want to be more efficient in their written expressions than laboriously drawing counters and arrows for movements. The teacher can raise the problem of how to represent certain actions more efficiently. If students draw 5 counters together to form a group, how can that be replaced with a representation, like a symbol, that works better than drawing 5 counters? The teacher can push on students to look at how to represent the action of combining. Is it a one-way action? If 5 and 7 make 12, are the 5 and 7 gone? If the 12 still has the parts of 7 and 7, wow would students replace the arrows and represent the idea better with a different symbol? If students arrive at understanding that 5 and 7 are the same as the 12 instead of ‘making’ 12, how would the represent this relationship better?

There are dozens of constructive activities that help students derive and represent all kinds of relationships, the math facts if nothing else. These constructions can help students put together all the internal structure of the whole number system that memorization of math facts bypasses. Besides constructing the whole number system, these methods help students put together corresponding methods of representation for the various mathematical meanings they construct. Representation is the naming of our meaning. Once students understand something, they are then receptive to learning how to represent it. How does society represent the fact 5 and 7 are 12? Who would students ask for this information? What sources do they have available? How could they compare how they represented a math fact with how the experts represent math facts?

Of course the example we gave of the traditional method of instruction presenting the equal sign in equations, e.g., 8 + 4 = __ + 5, for students to attempt to understand bypasses all the rich developmental constructions that finally lead up to the understanding the relationships making up the system of understanding for this problem. The point is that to be developmental, instruction must begin with what students generate and represent and it must follow cycles of construction. It does not begin with completed knowledge presented to students in symbolic form.


DECONSTRUCTION

We can fairly easily "deconstruct" the math meaning hidden in the example of non-developmental instruction to see what kind of system is at work in a completed understanding. In the 8 + 4 = __ + 5 problem is the idea that a whole number can be broken down in two different ways and that these two different subsets are both equal to each other, a basic system of whole number system called a system of vicariances. Whenever we talk about whether substituting one way of breaking a whole down into its parts with another way to break it down into parts and with the understanding that the two ways still represent the same whole and can substitute for each other, we are thinking about vicariances, substitutions, in which parts equal parts. This system is different from thinking about how the total made by the parts is the same as the whole such as 5 + 7 = 12. Thus, young students can understand fairly quickly the vicariant relationship that 5 and 7 are equal to 4 and 8. But they must have an opportunity to put these meanings together before representing them with advanced math symbols. Removing one of the parts thus can be easily solved as a simple procedural problem of getting an answer once the meaning of vicariant substitutions is constructed.

Students also must put together the meaning and representation of combining objects. To do this, they must abstract the operation of addition out of their activities of combining counters to form wholes. They can learn the use of the + symbol to represent the action of combining when they have constructed the meaning of addition. The same is true of the = sign. When students put objects together, they perform a temporal sequence of activities of counting the two subsets, pushing them together, then counting the whole set, and finally arriving at the end point, the answer. This functional use of activity to get an answer is better represented by an arrow as students naturally do than with an = sign. There is no equality for students. There is a process by which they ‘make’ something, namely 12. The fact that they "make" something with their activity leads them to focus on the goal or end result. They leave the beginnings behind because they changed them into 12. So the = sign they are required to use doesn’t mean equality; to them it means "get an answer."


AUTHENTIC MEANS WHAT STUDENTS CAN ORGANIZE AND PRODUCE

It would be much better to see how students naturally express the meaning of finding different ways of making parts of 12 than to give them a symbol they don’t understand. It is not the symbol they should be investigating. They should be investigating mathematical relationships that they then represent. The example of instruction presents a fairly advanced system of mathematical meanings and symbols as the objects which students are to investigate. These advanced conceptual objects cannot be found in students thought. As a result, students struggle with the problem and miss out on the constructive steps leading up to this level of understanding, not the mark of a developmental approach. The non-developmental instruction in the example doesn’t begin with students’ understandings and representations to then help them to construct more understandings from their initial systems.

We needn’t throw out our math programs since can always change simplistic learning activities into developmental ones by deconstructing them and then fitting them more honestly to students’ actual level of understanding. The technique for changing traditional learning activities means going all the way back to students actions on objects, things they can do, and from th organized activities they produce, have them abstract facts and conjectures along with their representations. We make activities developmental by adapting them to students by to allowing students to use their natural representational methods. We avoid making students begin with the completed adult understandings as abstracted and expressed in symbolic representations or in manipulatives that demonstrate mathematical relationships. We want students to construct these.

Students should construct their manipulatives; in place value instruction, for example, they should figure out how to make base ten blocks as a way to represent their understanding. The manipulatives shouldn’t be designed for them. It is that organization that is constructive. To let students be constructive allow students to return to the level of their actions on objects, posing problems of how to organize the objects into wholes and parts, and how to represent these in written signs or symbols. These activities are "authentic" because they allow students to begin with their own mental systems and the activities they can organize as they respond to problems they see and can understand and thereby generating, demonstrating, and representing their constructions.

What system is the student currently using and how can we develop that system, that is, what is the next step in the student's constructive sequence?

#6 Tasks vs Tests in Measuring Growth

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THE SUPPORTING FACTORS OF MATHEMATICAL DEVELOPMENT

The theory of learning we hold determines what kind of assessments we value. If we believe mathematical knowledge comes from objects, we may believe that experience with manipulatives is the essential source of mathematical development. If we believed in the power of mathematical experiences with objects that let students find answers to problems using objects, we might value specially designed manipulatives that students could use. These specially designed objects would model and demonstrate the mathematical relationships so that students could use them to find answers to problems. We could create base ten blocks so students could use them to solve place value problems.


Perhaps we believe practicing mathematical procedures reinforces concepts and teaches students to retain ideas in long term memory. We could design a curriculum in a series of steps that would carefully insure the student had learned each of the mathematical procedures and their refinements that were necessary for mathematical knowledge.

Perhaps we believe in a visual mathematics in which images capture relationships that can be seen and investigated by students. Then our mathematical models will be designed to create a visual model of relationships.

Perhaps we believe there is an innate mathematical ability produced by the maturation of the brain that students can then use to think about what is being presented and taught

Perhaps we believe in the social origins of knowledge in which mathematical development is produced by the teacher providing scaffolding for students to reproduce mathematical ideas that are above their current natural level but still within their zone of proximal development. These factors are all to some degree necessary and important for mathematical development but the research shows clearly that none of them are the source of mathematical development.


THE SOURCE OF MATHEMATICAL DEVELOPMENT

Mathematics does not come from a source outside students. Mathematics isn’t located in objects, images, or language. Nor can new mathematical knowledge be transmitted or discovered. Nor is there even a primitive form of mathematical knowledge that is due to heredity. There are innate schemes of sensory motor activity but not mathematical knowledge. And mathematics is not discovered in patterns supposedly seen in the world. Nor is mathematics invented since it always comes out of some pre-existing forms of thought and presents itself as a matter of logical necessity and universal timelessness. How could whole numbers be invented if everyone arrives at the same idea throughout the world and throughout history?

We cannot transmit meaning, only trigger it. The source of mathematics lies within the activity of students, within their ‘meaning makers.’ The source arises in the forms of coordination students put into their activities and that are successively organized, grouped together, and reflected on the plane of thought into non-contradictory and logically necessary systems. It is their rational organization that gives them their stability and gripping power over thought. Activities begin at the sensory motor level as direct actions on physical objects organized by innate schemes that immediately adapt and grouped together. These coordinations developing in activity and eventually become reflected and reconstructed as primitive mental operations. They then form into rational wholes, the deductions of concepts, relations, and numbers.

For words, images, objects—experience of any kind—for any of these to "transmit" knowledge, they must trigger mental activity by students. All these representational stimuli can do is to trigger mental activity that is already available within the thinking of students. The meaning students see in any transmission from objects and words is exclusively controlled by the internal organization and grouping together of mental operations into systems, not the transmissions of knowledge from outside sources which the systems make possible. This internal control over meaning things have for students means at ealier stages of development they see facts differently than they do when they achieve full understanding. Students cannot see facts correctly, and so they cannot use these distorted facts to construct a more objective meaning of the situation.


MENTAL SYSTEMS MAKE LEARNING POSSIBLE

Certainly outside stimuli can provoke and challenge internal systems of mental activity but the essential fact is that what students ‘see’ and ‘know’ in their response to a situation or the stimulus of words and objects depends on what they see with, the current state of their mental systems. The organization of their mental systems has its own source of development, a mechanism that is built into living systems. The mechanism appears as the self-regulation of these dynamic wholes as they attempt to reproduce their activities and compensate for problems encountered in operating on the object of attention.


THE DIFFERENCE BETWEEN LEARNING AND DEVELOPMENT

The essential idea that distinguishes development from learning is that development of new forms of organization consists of a process of grouping together, not of receiving more information. Learning is the reception of information from outside. Development is the organizing and adapting of mental activities to the object of thought. The different transformations of mental activity, the various forms of reasoning of thought, fit themselves together so that they group into a coherent system or whole forms. The system formation isn’t instantaneous but develops out of students’ attempts to make their thoughts work without contradictions and instabilities as they operate on objects in their thought. They go back over what they just did to try to make conscious what they just did and why their actions produced the results they did. They are attempting to collect together the actions back to the start so as to understand. It is a process of grouping together and reflective abstraction.

So the essential ability that exists before learning occurs and that makes learning possible is precisely the mental system students use. The mental system defines what they see and understand in the situation. Now for some people it makes no sense whatever to suggest that some internal system that has its own mechanism of development could somehow control thought and that it could develop and fit itself so well to the outside world without there being some transmission of information from the outside world. In fact for most people this belief in the necessity for the transmission of at least some prior sense data out of which to then form knowledge may be true.


IF WE CAN’T SEE SYSTEMS AT WORK, THEN WHAT?

Is it not, then, just a matter of choosing what theory of knowledge seems most plausible to us? Actually, no. We can withhold judgement and base our interpretation on investigation into the existence of these mental systems. We can turn ourselves into scientists and study for ourselves the nature of mental systems. But seeing systems at work is a huge challenge and is not attained immediately. The understanding of living systems is constructed step by step by testing a refining our best notions of these systems with objective methods that allow verification or denial or our conjectures. We can see if students really do see the same facts differently than we do. We can see if it is a matter of language or if there really does seem to be something real at work in students thought that is regulating and organizing it.

Tasks provide a powerful and objective tool for these developmental investigations. If we intend to find out if cognitive sytstems are the mental machinery at work behind the behaviors and performances of students and that these systems are the essential meaning makers, then we will want to develop the ability to see and work with these systems so that we can actively promote their development. Since the meaning students understand depend on the development of cognitive systems, the primary goal of teacher development is to see these systems at work in students’ thought. If we can see these at work, we can then subordinate learning to the development of the specific cognitive systems needed to understand and make learning possible.

If the meaningful learning of math facts depends on the prior development of the cognitive system that organizes thought so it understands number, then we will want to work with math facts in a different way. We will want to help student combine and take apart the set of 7 to see that the total doesn’t change, that number is conserved while reasoning with it and making various combinations and relationships. Math facts are taught differently so they become avenues of constructions rather than things to learn. If students do not have the opportunity take 7 apart into 4 and 3 and then see if they still have the 7, they do not have a means by which they can put together an understanding of 4 + 3 = 7 and 7 – 4 = 3, a logic that allows students to transform numbers in order to understand them.


TASKS—A TECHNIQUE AND A KNOWLEDGE OF SYSTEMS

To see and work with mental systems means we must have two things—some technical language that allows us to describe a mental system in its various stages of development and some techniques for obtaining the data necessary to see which system is operating in students thought. Knowing the technical aspects of a systems lets us design tasks that produce the adequate display of reasoning we need in order to investigate and see the system at work. We need some knowledge of systems in order to administer tasks but we can begin by following tasks in a rather procedural manner and combining this with examples and descriptions of the system. Tasks are much more advanced forms of assessments that single answer tests.


TESTS—RETENTION OF LEARNED MATERIAL

In contrast, assessments that are tests, mostly assessments based on single answer responses, give us a display of the output of cognitive systems at work but not the systems themselves. If we stand outside the factory door and view the finished products coming out, we have no way to describe the machinery inside that is constructing the products. We need a way to ‘see’ students mental machinery at work.

Tests compare students one to another in terms of grade level ‘norms.’ But they do not isolate single concepts to provide us with an objective description of how the student understands a particular important piece of knowledge. Even though they count right answers, single answer tests are less objective than tasks since produce descriptions and a direct correspondence of actual cognitive systems the make up the knowledge students possess. The object being measured is students’ knowledge, and the actual mental machinery students are using to understand a topic corresponds more directly to the meaning and understanding they have than does a single answer which obviously could be produced in any number of ways, memorization, guessing, incomplete understanding, visual memory, etc.

Tests combine many different concepts into a single assessment to produce scores representing students’ mathematical knowledge. These scores represent a notion of global wholes or aggregates such as strands indicating students’ knowledge of content, measurement, statistics, computations. Student knowledge is conceived as an overall aggregate of atomistic pieces, a body of knowledge sampled by a test. But such student scores only represent a supposed state of students’ knowledge that is a global, undifferentiated whole. It is not composed of rational parts making up the whole and it cannot be taken apart into its simpler elements. We cannot know what each test item indicates as to a particular form of knowledge nor how the different items and the parts they might represent are organized into larger wholes of a structure of the discipline. If mathematical knowledge is constructed step-by-step, then for instructional purposes we will want assessments that allow us to disaggregate mathematical knowledge into its components and sequences of development.


TASKS MEASURING THE STATE OF DEVELOPMENT OF VARIOUS SYSTEMS

If we believe mathematical knowledge is not simply learned from some outside source, transmitted through language, or due to internal brain maturation but is in fact due to the successive constructions of mental systems that make learning possible, then we will want an assessment method that allows us to see and describe the current mental system at work in students’ thinking. We will want concept specific tasks. We will want its constructive steps, We will want its relationship to other concepts in the overall sequence of construction of a mathematical area. Since it is the organization of reasoning into a system that we want to measure, the assessment tasks must reveal and assess the patterns of reasoning, not the answers produced by the mental machinery. And the developmental assessments by using tasks instead of tests must be be anchored to the actual point in which understanding is achieved and not to average scores achieved by students.

Developmental assessments are not anchored to student grade level norms or ‘cut’ scores set by a committee as a standard for student performance. Instead, tasks establish objective descriptions of levels of understanding up to the point of a completed understanding. Because of these objective benchmarks (‘objective’ because they correspond to something real in students’ thinking) the tasks allow us to diagnose each students understanding of any given concept, and they can enable us find out what percentages of students at each grade understand any given concept. Rather than norms defining students, students define the norms.


CONCLUSION

Tasks serve two broad purposes. One is the diagnosis of students in instruction. The other is the study and development of a professional understanding of systems. If we wish to find out if there is more to student development than learning and the transmission of ready made mathematical knowledge from an outside source or through experience with objects, then tasks are an essential tool for investigating and seeing the cognitive systems at work organizing mental activity. Tasks are not explanations of the development of systems, they are only descriptions of the various forms of systems. They don’t tell us how or why mental systems re-organize themselves into more advanced forms at higher levels of thought but they do give us objective descriptions of the current level of students’ understanding. The essence of mathematical knowledge is the meaning it has for students, not the learned procedures and facts that can be copied. Hence, we want assessments that directly capture the development of meaning, that is, mental systems, not learned facts and procedures.

In education, our professional challenge is to understand and develop the ability to see mental systems. Developmental tasks give us an essential tool for investigating systems and eventually ‘seeing’ systems at work. Besides providing powerful diagnostic tools for instruction, tasks provide the teacher with the advanced professional knowledge and techniques that sets them apart from folks who hold common sense and simplistic notions of the transmission of knowledge.

#5 Number—Where Does It Come From?

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POSSIBLE SOURCES OF NUMBER

When we want to teach students about physical properties such as weight, color, shape, etc., we have real objects with the properties that we can show students. But where is number located. How can we show students number?

Can the student find number in objects? Can we show the student a number? No. The research is clear. Students don’t see number in objects until they have the concept and can read it into the objects. If we show young children 7 objects, they see what is in their perceptual field but their perception does not construct the concept of number as a perceptual field. They see size for number; they can perceptually recognize shapes, color, movement, size, boundaries, touching and separation, but they cannot see the concept of number in a visual scene no matter how carefully the objects may be arranged. Number is not a visual property inherent in objects. What students see are physical objects, not number as an abstract concept.

Nor is number innate. It does not come from brain growth. The brain wires itself as a result of activity. The organization that results eventually in a brain with complex organization is not due to an innate blueprint that contains number. The explanation of why 4 + 3 = 7 will never be reduced to the brain.

Nor is the concept of number contained in words. Number cannot be transmitted verbally either by explanations or reading about it. Words are representational elements but they do not contain meaning. That are attached to meaning but they do not contain meaning. The meaning of number must be organized in thought in order to ‘see’ its meaning in words or in a set of objects. Students must have some concept they can use if they are to understand the words the teacher uses.

Some writers have even suggested that number is a pre-existent idea that we are all born with or that students discover. But again the research shows clearly that there is innate concept and no ideal form of number that pre-exists for students to discover. Even if number did somehow exist in some ideal form floating apart from human thought as a form of truth, how would students’ minds make contact with this idealized realm of existence?

Perhaps number is contained in the math symbols, the numerals we teach students to say. But recognizing a symbol and saying a sound carries nothing more with it than a learned response. It has little meaning until it is invested with something far richer to give it the meaning of number. The word can be socially transmitted but meaning cannot. Words can only trigger pre-existing concepts in the child.

Perhaps students find number in the repetition of the counting experience. Perhaps students learn a pattern from the physical activity of counting objects. But the pattern doesn’t exist for the student until they have the ability to recognize patterns in repetitions such as following a set of seriated counters. An experience remains an experience on whatever plane of activity it occurs. A sensory motor activity, by being repeated, doesn’t produce thought about the activity simply by its repetition. No matter how many times the child repeats a math fact, the meaning of the fact does not increase due to the repetition. Experience remains experience until it is interpreted and given meaning.

None of these factors can account for the development of mathematical thought but all these factors do affect the development of number. If they are not the source but only auxiliary processes for development, what is the source. These factors ‘feed’ students’ mathematical activities but none of them can show or demonstrate number to students who do not yet have the concept. Students must somehow construct the concept of number before they are able to see number in the activities, in objects, in words, in patterned iterations.


THE SOURCE OF NUMBER—REFLECTIVE ABSTRACTION

The research clearly shows that the origin of number lies within an abstraction of the organization of activity directed to objects. It is a construction in the sense that it is not discovered as a pre-existent entity and it is not an invention since it derives from something pre-existing which students’ find in their own activity.


A NECESSARY FIRST STEP—ORGANIZING THINGS

The research shows how students more or less unconsciously use patterned behaviors to organize objects. The ‘do’ something to objects such as gathering toys together into separate groups or they may successively put toys into a line one behind the other. At early stages of development the ability perform organized actions on things operates without conscious intent and awareness. The actions remain subconscious as sensory motor behaviors. But once children begin repeating a behavior such as placing an object next to another object, they can repeat this little organized activity to produce something larger than isolated elements. They can produce a line of cars. The toys are no longer randomly arranged. The child has created an organization, a larger whole, that may appear as a line or as a group.


THE OTHER PART—THE ABSTRACTION

The crucial part of the process occurs after they have exercised a scheme to produce some kind of organization. That process consists of a reflective abstraction in which children begin to think about how they produced what they see. They have performed some patterned activity such as placing each object next to previous objects. They see they have produced an organized whole, perhaps a line of objects. By abstracting the method of placing objects next to each other, they can produce the organized whole once again in another situation and time. The little schemes of finding an object and putting it next to a previous object allows them to reproduce the line. They can use this behavior to construct an object they can observe as a mathematical objects, a line or serial placement of toys. This mathematical series has nothing to do with the physical properties of the objects, and it has everything to do with the organization the child put into the toys.


THE MATHEMATICAL OBJECT

Nobody taught the child to create this little organized whole out of toys (although some insightful parents may actively encourage this kind of organizing behavior by their child) nor did he see a picture in a book or read directions which he could copy. He cannot copy since he has no conceptual bucket by which to record, remember, and transport the form shown in a picture or in words. When viewing a completed organization of toys someone else has formed, the child cannot see the method of composition by which the form was produced. He even finds it difficult to think simultaneously about each of the objects as individual elements at the same time he thinks about the overall whole he makes. If he focuses on the appearance of the whole, he cannot hold in his mind the individual elements. If he focuses on individual elements, he loses track of the overall whole being produced.

None-the-less, by grouping together schemes of placing toys, he has developed the ability to form something larger than each of his separated centrations on individual objects previously allowed him to do. This whole he produced is a mathematical object. Most importantly, since he used objects, he has something to observe that he has produced. He can reproduce this ‘something’ by performing the little series of repeated, special activities on the objects, the placing. He can do it again to produce the effect.


CONSTANT ADJUSTMENT MEANS THERE IS ACTIVE REGULATION OF ACTIVITY

But when he attempts to use his schemes to re-produce the larger organized whole, he must constantly attend to them to adjust the actions as he applies the schemes according to the situation and objects in order to make the patterned whole appear again. He cannot blindly, randomly repeat something. He must locate a toy, grasp it, place it, adjust its position so it looks right. He must give at least some thought to making sure each action is successful in forming the organized whole he intends on re-producing. This natural attempt to fit and adjust his activities to make them successful is the work of the function of the regulators that are inherent in the dynamics of all living activity. Humans aren’t passive storage containers for knowledge. They are the seat of self-regulated activities.

Regulations are the little adjustments of the activity as it compensates for small disturbances or problems it encounters and must overcome in carrying out the project of arranging the toys. The regulations compensate for differences among the toy, their size, whether they belong, the area needed to extend the arrangement, and so on. Regulation is the essential property of intelligence. An unusually large toy might be rejected because it doesn’t seem to fit to the notions of what toys can used. To compensate for its size that makes it difficult to use, the child simple rejects it. To successfully use the toy, the child might have made it smaller in some way so it could then fit in. Or the child might anticipate what kinds of toys will work and select them before starting. This anticipation is the mark of a more advanced form of activity.


MATHEMATICAL FORM PRODUCED BY THE CHILD

The essential mathematical idea is to see that a mathematical form was put into the objects by arranging them. The child has some simple coordinations of activity, some little schemes, that he can use to produce an arrangement. As soon as he produces some kind of organized form with the objects, he can attempt to reconstruct the activities he used to produce the form. This attempt to re-use his coordinated activities and his beginnings of thinking about these activities as he must adjust and adapt them is the process of reflective abstraction.


AGAIN—REFLECTIVE ABSTRACTION AS THE SOURCE OF CONSTRUCTION

Reflective abstraction is the process of abstracting the form or method of activities that is used to produce a repeatable, organized whole. The child must first use an existing scheme, pattern of activities, or method to organize objects. The abstraction comes from attempting to think back on how the actions were performed. It attempts to reflect on the plane of thought the unconscious form of the schemes used in the behavioral activity. It moves activity a tiny step higher in making more conscious. But reflection is also the process of re-organizing all the individual schemes into a thought that can replace them. Thought is simply activity occurring on a different plane, the plane of consciousness, as mental activity begins to disconnect itself from physical activity. Just as the physical activity employed an organizer, a scheme, for initiating the activity, thought also must organize activity on the plane of thought. Thought reflects and reconstructs what occurred at the behavioral level thereby making the child more aware of the operations used to produce the mathematical form of the line of toys.

Reflective abstraction—this fundamental process of drawing out the implicit coordinations of our activities to reconstruct and reproduce them on a higher and more explicit level of mental activity—is the essential constructive mechanism that is the source of the development of new mathematical thought at every level from the new born infant up to the most advanced mathematician. The child does not see number in a set of objects shown to him but he can organize the objects and abstract his methods of organizing the objects to develop the idea of number. Mathematics doesn’t come from verbal explanations or drill and practice. Counting is a rote activity imposed on the child by the sequence of numeral names but it is a behavioral patterns that is learned and does not allow a reflective abstraction by which to pull out the concept of number.


TWO FORMS ABSTRACTED AND COORDINATED INTO NUMBER

To construct the idea of number, the child must use his existing methods of organizing objects to do two things. He must mark off objects to turn them into a set of counters. He must distinguish among them by putting them into some kind of order. He can then use the learned sequence of numeral names to follow the order of the counters and to stop when all counters of the set are enumerated. The simultaneous putting into a set and ordering the counters are the twin operational schemes that form the concept of number, not all at once, but by gradual step-by-step constructions from activities and reflective abstraction.

For example, imagine a child playing with toy car and a toy auto parking garage. At an earlier point in development, the child can only apply a scheme of assimilation—like putting the car into the mouth, turning it over, pushing it, throwing it—to each object but without relating objects and schemes one to another. At a later point, when schemes are organized into larger wholes and can incorporate several steps, the child plays with the toy cars and auto ramp as a whole object. He may more or less think about which group of cars he is placing into the auto ramp and which groups he is putting outside or in a different part of the ramp. Or he may more or less think about lining the cars up as they enter the parking ramp. He may move the first car and create a gap which he then fills with the second car creating another gap which he fills. He may try to think about how he moves the cars forward and the gap move backward. These patterned activities are the sources of grouping into sets and ordering in series, the two properties of number.

Thus to form the concept of number, the child must coordinate set making of objects simultaneously with ordering the objects to produce the cardinal and ordinal properties of number. The child doesn’t receive number from outside sources, from counting, from words, from teachers, from books, nor does he discover number as pre-existing in objects, from being show a set of objects that are supposed to demonstrate a number, from ‘seeing’ patterns in things, or from experiences practicing procedures given to him such as counting. for there to be development, the child must do the organizing work and must do the reflection. Each child constructs the concept of number from a process of putting organization into objects and then reflecting on the methods he used Thereby he abstracts and reconstructs the behavioral operations on the plane of mental operations.


MATH EDUCATION

Student activity-based mathematics doesn’t mean students invent or discover. It means there is a disciplined adherence and careful attention to requiring students to organize objects, to abstract their methods, and to represent their methods. Certainly the instructional process can employ objects, discussions, coaching, direct telling, practice, and all the rest but all at the appropriate point in the cycle and all in the service of promoting the process of organizing objects and reflective abstraction. When students are required to organize objects and then to pull out and represent the form of the operations they used, these student constructions and representations will reflect how they actually understand mathematics concepts and that allows authentic instructional assistance because the teacher can follow the development of students’ constructions. Students’ organization of activity and their products of reflection depends on how the student understands, not what they have copied and learned. And that process of organization, reflective abstraction, and represesntation is the essence of math education and indeed, all of mathematics itself.

#4 The Logic of Seriation

THE TWO DIFFERENCES BETWEEN CLASSIFICATION AND ORDERING

A classification of things does not have a necessary corresponding spatial arrangement that can be perceived whereas a series of ordered differences does have a spatial sequence that corresponds to the seriation. We can perceive a relation, a difference between two ordered objects, whereas classes cannot be perceived. Since we can perceive the form of an ordered series of sticks according to their differences in length, does that make teaching or understanding seriation easier?


THE SERIATION PROBLEM

Does the understanding of seriation start with the perception of a series of things from which understanding is then abstracted? Or does the understanding of seriation depend on the development of an operating logic the student uses? If there is a logic in students’ ordering activities that does not come from seeing an ordered series, then what is the difference between perception of a series and the logical seriation of the same series?

A prevalent misunderstanding of learning by lay people is that perception can be the source of knowledge; this seems particularly compelling as for example, in trying to show students an ordered series of things from which they are expected to understand seriation. One of the main explanations for the origins of knowledge is that it begins with sensory input.


RESEARCH SHOWS A LOGIC DEVELOPS

The researchers studying the growth of knowledge have used a variety of methods to investigate how seriation develops and whether it derives from activity or from perception. For example, we can show primary students an ordered series of sticks and ask them draw what they have been shown. We can use a box and watch how students use tactile seriation to explore the sticks and then draw the configuration that he intends to construct. Or we can give unordered sticks and ask students to put them in order.

The experimental research shows clearly that the development of the logic guides activity such as ordering, drawing a series, inserting sticks, and describing a series. The logic always begins with a simple, isolated activity of comparing two sticks to see one as big and the other as small. The student at this level can only organize isolated duples of big and small, or two groups of several long lines and then extremely short lines when drawing. Then as the student attempts to coordinate two separate duples another difference appears, and the student produces big, medium, small. The students still seems apparently unconcerned with the graphic character of the unconnected duples or triples they produce.

But at the next stage, the graphic character of a series emerges for the student. They attempt to make the whole series look right, and they use trial and error to give the perception of the whole series a good form. Now the set of elements that belong to the series (the extension) includes more than duples or triples, and the order (the intension or property of the sticks) begin to emerge. But the order is tied to perceptual comparisons of proximal sticks and the overall extension depends on a perceptual good form. Neither is yet logical or coordinated with the other. As a result, the student has difficulty inserting a stick or composing a method of always choosing the largest stick of the remaining ones to add to the smaller sticks already starting the series.


THE LOGIC OF ORDERING

To understand the seriation, students must be able to identify the specific ordered differences they are combining and coordinate these into an overall series or whole. The student logically must add together the ordered differences into the whole by isolating differences and adding the relations together to make the series ‘get’ larger and larger. Likewise, the student must be able to take apart or undo the differences by isolating each of the relations and undoing the difference it added to go back down the series to get smaller and smaller.


THE GROUPING TOGETHER INTO A COMPLETED SYSTEM

But there is one more step in the development of seriation. It is one thing to add together the relations of difference to compose a series and it is another operation to take apart or undo the series. In both cases, students may not ‘remember’ exactly what they started with so they are not sure if they can exactly undo things. Both the operations of putting the relations together into a series and taking apart to undo the series have to be coordinated for there to be a non-contradictory, logical understanding. Simultaneously coordinating both operations requires that the student group them together into a mental system so that each action is controlled by the mental system. The system started out as a small, local system that coordinated only the relations of big and small relations of two objects and ceased to operate as soon as attention shifted to the next objects. Then it grouped together little, medium, big. Then a more continuous but one way series. Finally it completes itself by grouping together both the addition and subtraction of relations making up a series.

This addition of combining individual relations between elements and taking them apart to undo the series operates in logic like the addition and subtraction of numbers operate. Numbers form a system which logically controls the operations. For example, in student language, "if you add 3 and then take away 3, you haven’t changed anything." The result is 0, and adding 0 means exactly the same as you started with. So the operations fit together into a system that insures the operations always work out logically, that is, so they aren’t contradictory.

Thus, the action of inserting a stick can be accomplished logically by simultaneously coordinating the relations of smaller and larger to find the position in which the stick is larger than the one preceding it and smaller than the one following it. Rather than perception, the student is using thought to organize activities, solve the problem, and explain it. This logic did not come from perception; it came from the successive organization of activities. Logic corrects perception by enriching it with relations that let the student ‘see’ the order. This seeing is not perceptual but is an mental object of thought.


CONCLUSION

Understanding seriation does not come from perception. We can’t show students a series of sticks or two different stick and expect them to abstract an understanding of differences in length and the seriation of differences. Understanding seriation depends on the development of an operating logic students use, and that logic evolves out of the students’ increasing coordination and grouping of their activities.

To the layperson it appears easy to teach since all one has to do is simply show students what they need to know about seriation. They don’t see that there is a logic in students’ ordering activities that takes a long time to develop and does not come from just seeing an ordered series and copying it in thought. There is a huge difference between perception which is a sensory motor contact with objects and the abilityh of logical thought to compose and take apart ordered differences of a series. A perception cannot be composed and taken apart. It doesn’t allow reasoning or problem solving, the contribution of logical understanding of seriation.

#3 Classification—Extension and Intension

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MEANING AND SET MEMBERSHIP

Classes are concepts; all nouns refer to concepts, or potential concepts not yet fully understood. The two properties of classes, their extension and intension, are more important than the obscure terms suggest. Extension simply refers to what objects belong in a class. Intension refers to the characteristics or properties that define membership in a class. Intension is the meaning of a class. It’s important for teachers to understand how challenging the problem of classification is for young children. The challenge comes from developing the coordination between what concepts object s belong to, their set membership, and the what properties of the objects are essential to defining set membership. A child may put cows and pigs together and call them animals but only because they look different. They may be unaware of the distinguishing features that make cows different from pigs or that define them both as animals. Snakes may not be animals to them. And they almost never classify insects as animals. Perceptual appearances don’t form a solid basis for conceptualizing things that children need as a foundation upon which to build their academic knowledge.


NOUNS REFER TO CLASSES AS CONCEPTS

Classes are concepts that apply to separate objects. Concepts like volume, straight line, time apply to a continuous pat/whole aspect of reality. Classification is important since every noun, every reference by name to a group of similar things, depends on the classification system at work. Instruction in classification begins in the primary grades with helping students form the concepts of living and non-living things. Teachers help students with the concepts of shapes, numbers and numerals, perhaps various occupations, weight, quantity, and so on. At the primary grades, few nouns are yet conceptualized at the level of being true classes. Helping students conceptualize the important nouns in academic instruction forms the basis of the descriptive knowledge of the primary grades. Primary students tend to see only isolated categories of things that go together, not inclusive classes. Their nouns tend to be unstable categories tied more closely to activities with the objects. "Cows are for giving mild and living on a farm." "We ride horses." That kind of thing. These categories aren't related logically to other categories. For example, cows may not also be animals.


EXTENSION—THE HIERARCHY IDENTIFYING CLASS MEMBERSHIP

Classification seems simple enough, just putting things together that go together, but understanding the unseen psychological system itself turns out to be the challenge. We can't just do classification if we are to teach it. We must be consciously aware of the underlying system that remains hidden to most people. As teachers, we have to understand how the hidden mechanism of classification we are unconsciously using itself works to coordinate the features of classes with their inclusion into a classification hierarchy of inclusions to produce classes. Not only that, we have to understand the earlier forms of classification; these are psychological systems we may no longer be using but which are the forms primary children are using.

Classification is a system of logic because to represent cows and animals means cows and all other animals are combined into the class of animals. The relationships of a classification hierarchy are formed by combining sub classes into more inclusive super classes. Or the reverse also is available. We can take apart animals into cows, pigs, and other animals. This combining and taking apart mathematically could be represented as the addition of classes and the subtraction of classes. It expresses itself in language as we talk about things. "All cows are animals. Some animals are cows, some are pigs but there are other kinds of animals as well. There are more animals than cows." We can tell whether this operational system is present as a specific ability in a student by giving him special problems and listening to his reasoning.


INTENSION—CHARACTERISTICS DEFINE MEANING

But classification is more than forming and combining classes or taking them apart. All this has to be done based on the characteristics or properties that the objects have such that they belong to a class. Obviously, a cow doesn't have some of the characteristics of pigs but it has all the characteristics of being an animal. So a class has two aspects, one has to do with which objects go in which classes. Do all cows belong in the class of animals? Do all animals belong in the class of pigs? When we see an object standing out their in the field, what classes can we put it in? We haven't mentioned the properties of the object that we are using to classify it. We are only referring to which objects go in which classes. The technical name for that is the extension of a class. The extension of a class is the identification of the class membership, which objects go in which classes.

Obviously, we can't put objects into classes without seeing some kind of characteristic features that let's us know it belongs in a class. We look at certain characteristic features of an object and see these are characteristics every cow has and no pigs have. We also see characteristic features of the object that every animal has including all pigs. So the characteristic features are what the object means to us. If we refer to an object as a cow, we can imagine features that we know all cows have, all mammals, all animals, all living things, etc. But seeing the object as a cow means we don't see it as a pig. The meaning of the object is also that it is an animal. The meaning of things are those properties that distinguish it from others and that identify its similarities to other objects having the same name. The technical name for the properties that form the meaning of a noun is comprehension or intension of a class. Of course, all this logic and meaning are not yet present in young children's representations of things.


THE DEVELOPMENTAL PROBLEM—COORDINATING INTENSION AND EXTENSION

Young children face a difficult problem, a developmental problem, not a problem of learning. The developmental problem means there is no direct way to solve the problem except by a series of attempts at organizingthe objects into classes, checking them out as to the properties, and then trying to go back and readjust the classification method. This developmental problem for children needs to be clear. Since representing things as concepts with names means children have to know what characteristic features an object has. They have to identify only those features that are the key and common for every member of a class. What properties make all these objects of this class similar? Then these properties have to be used to define the objects that belong to a class.

But which comes first—identifying the properties all members of a class have in common or forming the class of all members that belong? To figure out what characteristic features all cows have in common requires that the child get a collection of cows together to see what characteristic features they have in common. But to gather together a set of objects that are all cows requires knowing what characteristic features to use to identify objects that are cows. The child's problem is how to coordinate the features that seem to define similarity with the names applied to objects such that the resulting classes form a hierarchy of inclusions each with its defining properties. Thus, the researchers label this developmental problem as "the coordination of intension and extension"—the simultaneous identification of properties with the grouping into nested nouns, i.e., classes.

If we go deep into this developmental problem of how to form classes, we find we can't really show or tell students how to form classes. They can only make an attempt to group things and name them in a certain way and then test out how they tried to organize their understanding of the objects by comparing to what others say. If others are using inclusive classes but the child has only categories, the child has no means by which to understand what is said as classes. The child can only form categories. So what the teacher thinks she is saying and what she knows she means is not the same meaning the child hears in the teachers' words—the child has no other choice. The child can only use whatever system he has developed so far.

Once the student can coordinate the characteristic features of objects with a hierarchy of naming and he understands classification, then he can receive and understand the teacher's classification talk. Her words have conceptual meaning. But until then, all he hears and has available to understand with are simple categories and a variety of perceptions from experiences with the objects. The objects he calls cows may mean that the object for him has many features and is tied to his experiences with it, most of which are irrelevant to classification in order to conceptualize and thereby understand the object. The students particular experience may not have much to do with the class of cows as such as adults conceptualize them. Adults know that "we drink milk from cows" and "cows have horns and tails" but those characteristics are not relevant to conceptualizing the class of cows (except inferentially such that they are mammals and vertabrates but of course that's not what the child means). The properties, perceptions, and meanings the child attaches to cows come out of his experiences and activities with cows. So the child has to identify just which characteristic features are the proper ones to define the class of cows and which characteristics belong to other classes as well and which ones are irrelevant.


CONCLUSION

The primary teacher’s task is to help students learn about the properties of things and to develop the ability to classify things. When we ask someone to define what they mean by a term, we are asking for their classification. A definition always has both the extension and intension. "A fork is an eating utensil with tines." From this we can see the extension, the hierarchy of inclusion relations—forks are included in the class of eating utensils. We also have the specific defining property, the tines as the distinguishing property from other eating utensils such as spoons. This "is a" "has a" definition is the goal for every important noun in the primary curriculum since every noun carries with it the possibility of being conceptualized as a class with defining properties and therefore properly understood. That conceptualization coordinating the properties with class membership is an essential development of primary students as they construct an understanding of the world .