#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.