#16 Researching Place Value Understanding

NUMBERS VERSUS NUMERALS

Some folks have generated some interesting conversations about place value, interesting because of the problems of both the theory of place value and the facts regarding when understanding is achieved. A bit on the theory of place value seems in order.

Developing better instruction in numeration, specifically, the place value system of numeration, would be greatly helped if we made clear the distinction between number and numeral. Kids develop an understanding of number by the end of 1st grade such that they can understand the composition of number and as a result, how to put parts together to form the whole and take a whole apart into its constituent parts. They understand the composition of numbers such that they can add two numbers to find the number that includes the two as parts, and they can likewise subtract out one part from a number to find the other part. But they don’t necessarily add to this understanding an understanding of place value numerals. The numerals they use are only figural symbols representing numbers, not place value numerals. What our research in Molalla showed was that place value subtraction that involves the borrowing and carrying operations is based on understanding the algebraically more complex concept of place value which comes much later than the relatively simple concept of number.


RIGHT ANSWER DOESN’T MEAN UNDERSTANDING

Investigating students’ understanding is quite different from that of checking for students’ correct use of procedures to get an answer. Merely using a procedure doesn’t mean the student understands the procedure. So, before we can conduct experiments into students’ understanding of place value, it is essential that we define clearly up front exactly what this object of study we call understanding is that we are addressing. That attempt to precisely define place value in our Molalla research resulted in a paper.

This formal description of place value then formed the basis for what must exist in students’ thought as necessary and sufficient conditions for an understanding of place value. This formulation made it possible to formulate some of our experimental methods for investigating the psychological development of place value in students. It seems to me that much research suffers because it is not clear as to psychologically exactly what the object of study is. People think they can tell when student understand place value or when they don’t. Failures in using place value operations correctly obviously are indicative of lack of understanding but the converse isn’t true. Just because students apparently are using place value operations correctly doesn’t mean they understand. Correct performance may only be a pseudo understanding that appears because itg has not been tested by a "foil" that challenges it to test it, or it may exist because there has been no careful questioning into the students’ place value logic.

For example, problems of place value often test little more than the production of correct answers. These are behavioral responses to calculation problems, not displays of the cognitive system of understanding. Of course we can teach calculation procedures which students can then learn to perform even without understanding. But as soon as we assert understanding is important, then the problem of how to define it becomes important. For that, we need to think in terms of how to define a cognitive system. This system language is necessary and useful both in mathematics and in psychology but because it seems to be lacking in ed research, there seems to be much confusion resulting.


FROM LEARNING BY TRANSMISSION FROM THE OUTSIDE OR BY THE DEVELOPMENT OF A SYSTEM OF UNDERSTANDING?

Of course I may be wrong about understanding. It may simply be the result of learning. Information about place value or other mathematical ideas may actually be no more than all the bits and pieces learned by students. This knowledge may move directly in all its little bits and pieces from teachers or from objects, or text to the learner who experiences them directly as first facts. The gradual accumulation and storage of all these discrete bits of learning and experience may be the sum total that adds up to place value understanding.

But that position of some kind of privileged prior data that is transmitted and received untainted by the learner has never been shown to be a coherent nor empirically verifiable phenomenon. It simply is a naïve conception of learning by most people. Once we understand how to conduct research into mental systems we can find methods to confirm or disprove the notion that knowledge is transmitted. We find from this research that the transmission theory of knowledge is simplistic and hugely lacking, and from a scientific point of view substantially wrong. It is not, then, an adequate basis for instruction. Teachers need something grounded in research that describes who understand does form in the human mind.

So since the direct transmission from the outside is not the source of place value understanding, the problem always comes back to asking how exactly does knowledge arise and form in the learner if not directly received even in its most primitive and simple parts? What exactly is the source and origins of place value in the human mind?


A DEFINITION OF UNDERSTANDING IS ESSENTIAL

So to investigate how it arises in thought, we must begin with a clear definition of exactly what place value understanding consists. Thus our first task is to understand the system of place value. Then we can look to see whether it is present in students’ thinking in its valid form or in an earlier and weaker form. Then we can investigate how the completed form arises out of its weaker form of understanding. Looking for this development is much different than attempting to measure whether external factors affect the rate of development. Certainly the quality of instruction or early mathematical experiences of home life do affect the rate of development but it is the actual development that we must understand if we are to powerfully affect development. So to begin, we must tackle the problem of how to describe the place value system as a valid system of thought.