#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?