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Selected artifacts from the in-service program (B2)
"What does it mean to study a geometric figure mathematically?"
(Borasi, 1992) (6 pages)
(unpublished manuscript prepared for inclusion in the multi-media package "Introducing math teachers to inquiry: Framework and supporting materials to design professional development" [Borasi & Fonzi, 1998])
There seem to be at least five important elements to take into consideration if our ultimate goal is to "know" a specific geometric figure:
1. DEFINITION -- i.e., we need to be able to articulate (and, first of all, to agree about) what the figure/ shape actually is.
2. EXAMPLES -- i.e., we need to be aware of what are "significant" examples of the figure.
3. PROPERTIES -- i.e., we need to become aware of properties that are common to all instances of the figure, and are somewhat "characteristic" of it.
4. USES -- i.e., we need to be able to put to use our knowledge of the figure in various applications, within mathematics as well as in other domains (this will also involve becoming aware of potential applications of the figure).
5. GENERAL NOTION OF GEOMETRIC FIGURE -- i.e., our knowledge of geometric figures more generally can guide our study of a specific one and, viceversa, engaging in the study of a specific figure may contribute to our understanding of geometric figures more generally.
All these five elements are clearly interconnected, and have been separated in this list only to allow us to look in more depth into each of them and the role each can play in our study of a geometric figure. In what follows, I will try to elaborate on these points, with the goal of helping us all realize how very complex and PROBLEMATIC the study of a geometric figure is, how it involves a good deal of creativity and judgment and it is far from being strictly objective and straightforward, and finally how it can provide each of us with a "micro-experience" of what it means to "do mathematics as a mathematician".
1. DEFINITION
Agreeing about "what a specific figure is" seems an obvious starting point if we want to further study that figure, since it seems a crucial premise to be able to:
Having a rigorous "mathematical" DEFINITION of the figure would seem to resolve these problems, as such a definition would enable us to (a) examine each ``potential example" and verify whether or not it satisfies the definition, and (b) verify whether what we ``think" is a property of the figure can be logically derived from the definition. (Is this also your sense of what a definition should do in mathematics? are there more/less criteria that you would like to require from a mathematical definition?) Thus, having a definition for the figure in question may seem a good starting point for studying it further.
But, if the definition is not given to us a-priori (as it is certainly the case when mathematicians develop a ``new" figure), how can we create a good definition of the figure we are trying to study?
And, once we have created one, how can we decide whether it is indeed a GOOD definition for that figure?
Furthermore, there may even be cases in which there may be disagreement about what the figure really is, that is, there may be COMPETING interpretations of the figure (consider, for example, our experience with DIAMOND, and historical examples such as the concept of INFINITE NUMBER). In these cases, how do we/ mathematicians decide between these competing interpretations, and come up with a definition?
For sure, in order to resolve "reasonably" rather than randomly debates of these kind, at the very least we will need to have more information about what are the IMPLICATIONS of assuming each interpretation. More specifically, we may need to:
Occasionally, this kind of exploration can even make us realize that BOTH alternative interpretations may be important, and decide to "split" our initial concept of the figure into two separate ones, with two distinct definitions (as it was historically the case, for example, with the concepts of INFINITE NUMBER and PRIME NUMBER).
It is also important to realize that the final decision of what to assume as the DEFINITION of the figure will have been made by the community of mathematicians of the time, and perhaps it may even be revised at a future time, if further study of the figure at some point were to reveal some unexpected results that challenge the validity of the initial decision (see, for example, what happened historically with the concept of POLYHEDRON).
In sum, in cases when alternative interpretations of the figure are possible, it becomes clear that a DEFINITION of the figure is by no means the starting point of its study, but rather almost the culmination of a preliminary and tentative study of the same. The definitions we end up accepting also lose an absolute and objective connotation, and become more TENTATIVE and "revisable". We may also start wondering if this is true of ALL DEFINITIONS, since we can never guarantee that at a future time we will not encounter some new results or applications that will make the mathematicians reconsider their initial decisions (as it was the case historically, for example, with PRIME NUMBER.)
(Do you find this conclusion at all surprising? What does it tell us about the nature of mathematics and how mathematical results are constructed? Do you think these consideration have any impact on why and how middle school students should approach the study of geometric figures?)
2. EXAMPLES
Even if we had a mathematical definition we all agree upon, it would be very difficult (and quite unnatural) to study a geometric figure just by deriving its logical consequences. Rather, it is quite obvious that a RICH SET OF SIGNIFICANT EXAMPLES would play a crucial role in becoming aware of the characteristics of that figure and its potential uses. More concretely, we could use this set of examples to:
But, how do we make sure we have such a rich set of SIGNIFICANT examples to start with? In other words, since the set of examples of any given figure is infinite, how can we make sure that we are working will all the examples we really need, and thus make "good" observations? (For example, as an extreme, imagine how many wrong conclusions could we reach about TRIANGLE if all our examples were right triangles!)
The truth is, we CANNOT EVER BE SURE! Historically, changes in definitions have in fact occurred most times when new, unexpected examples that "created trouble" were discovered (as it happened, for example, with POLYHEDRON and PRIME NUMBER). At the same time, there are a number of things we can do to try to minimize the problem.
First, we can try to "double check" our observations by trying to derive them logically from the definition. This approach would allow us to eliminate one type of error (i.e., attributing to the figure a property that not all its examples may share), but not another (i.e., "missing" to recognize some important properties of the figure).
To address the second type of error, we have instead to try to come up with as many as possible SIGNIFICANT examples. There are no fool-proof rules to do that, but some consideration may help us here. First of all, note that no specific example per-se is or not significant; rather, what we need to do is to assure that the COLLECTION of examples we work with is varied and representative enough. This is still not a straightforward process, and what makes two examples significantly different and therefore worthwhile to add to the set may differ depending on the figure considered. (For example, what does it mean for two examples to be significantly different in the case of CIRCLE, POLYGON, or HEART?).
It is also important to realize that looking only at accepted examples of the figure may sometimes not bring us very far in the study of the figure. To better appreciate what DISTINGUISHES the figure from others, for example, we may need to contrast EXAMPLES of the figure with COUNTEREXAMPLES. A very important role may also be played by those cases that we are not sure whether to consider at first as examples or not -- what I'd like to call BORDERLINE EXAMPLES -- as they can help us articulate our image of the figure, and what we really want to consider as belonging to it.
3. PROPERTIES
First of all, what do we mean by PROPERTIES of a geometric figure? We can respond to this question in two different, but compatible, ways: (a) they are all those characteristics that are COMMON TO ALL THE EXAMPLES of the figure, and (b) they are all the statements that can be LOGICALLY DERIVED FROM THE DEFINITION of the figure. (By the way, another occasion for questioning the appropriateness of our definition or image of a figure, and the need for making some changes, occurs whenever these two criteria become contradictory for a proposed property).
We have already mentioned how we can identify properties of a given figure: derive them logically from the definition; look at patterns in a rich set of significant examples and then verify them with the definition. An additional source for conjecturing potential properties of a figure could also be looking at APPLICATIONS of the figure: most applications make use of some characteristic property of the figure, or of subcategories of the same (example: diamonds are preferred shapes for kites: why? could it be the stability created by the fact that their diagonal are perpendicular?).
Notice how, once you "think" you have identified a property by looking at examples or applications, you still need to verify whether that property is indeed characteristic of the figure or not -- in other words, you need to check whether it can be derived from the DEFINITION. Yet we said earlier that sometimes we need to explore what are desirable properties of a figure before we can decide among alternative definitions; and a definition is often created by selecting a minimal set of properties that allow to characterize the figure in question and distinguish it from others. Is this a chicken/egg kind of problem? How can it ever be resolved?
Another puzzling fact we have to live with is that, in most cases (maybe even all cases!), we can never be sure that we have derived ALL the properties of a figure. Though all the properties are POTENTIALLY contained/ captured by the definition (since they must be logically derivable from it), we may not have yet been able to make them explicit. Or, we may have a hunch that a certain characteristic belongs to all instances of a figure, but not be able to prove it from the definition (think, for example, of the fact that the circle has a constant curvature, but this could be discovered and proved only after the relatively recent advent of differential geometry). The list of properties of a complex figure may even be infinite, and therefore impossible to articulate completely. An interesting consequence of this is that we may not ever be able to say that we have COMPLETED the study of a figure (as one might have initially thought possible when ALL its properties had been identified and proved).
At the same time, we could argue that a thorough explicit list of all the properties we know about a specific figure may not be that useful in any case. At the other extreme, however, it is also true that even if the definition contains implicitly all the information about a geometric figure, we would not feel that we KNOW a figure just because we know the definition. When do we then feel satisfied that we know enough about a given figure? what kind of properties would we decide are most SIGNIFICANT and important to add to its definition if we want to be able to make use of the figure itself? what would we want to communicate to other people as a result of our study of that figure?
4. USES
When we study geometric figures in school, we tend to do it in abstract, without ever considering their possible uses and applications. This is not, however, what happens when mathematicians engage in the study of a specific geometric figure; such a study is always motivated by some interest in that figure, i.e., by the expectation that knowing more about it would be worthwhile and USEFUL -- and in most cases this has to do with existing or expected applications of that figure either within or outside mathematics.
What we know about a geometric figure would not do us much good if we cannot put it to some use when appropriate. And this requires often, besides an understanding of the properties of that figure, also awareness of what these possible applications could be.
Less obvious, but equally important, is the fact that being aware of some applications and uses of the figure can also contribute to its "mathematical study". As we have already mentioned before, this can in fact help:
In addition, looking at the usage of the NAME of the figure in non- mathematical applications may also help realize important differences between the MATHEMATICAL CONCEPT of the figure and the more INTUITIVE IMAGE and usage of the same in natural language and real-life.
5. GENERAL NOTION OF GEOMETRIC FIGURE
Even if, at a particular point in time, we may be focusing on studying a specific geometric figure, such a study is never really done in isolation. What we already know about other specific geometric figures and about geometric figures in general, is likely to inform the way we approach our study and contribute to it in a variety of ways:
Viceversa, it is also important to realize that there is a lot that we can learn about the notion of geometric figure, and geometric figures in general, by engaging in the in-depth study of a specific figure. This is especially true if we consciously seek such more general understanding, and facilitate it by reflecting on the process and trying to generalize from the specific experience. (Does that change the role/value of engaging students in the in-depth study of a specific figure and the time we would be willing to devote to it within a course?)
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