#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.