# EXPLANATION OF TAXONOMIC TABLES

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## BASIC STRUCTURE

The Level I and Level II taxonomies generated by the American Mathematics Metadata Task Force start with words and phrases. These words and phrases are intended to describe the content of mathematical resources. They are NOT keywords that necessarily appear in the resources themselves. Words and phrases in the taxonomy can be combined in several ways to select a desired set of resources from a collection of available resources in different ways. One way is to combine words and phrases using and, or, and not. For example, if we wanted resources that discussed proofs and discussed geometry, we could look for resources simultaneously labeled with proof and geometry. This would not, however, be a good way to search for resources concerned with geometry proofs. A calculus exercise might, for example, be a proof with an application to geometry but not be a geometry proof in the sense understood by a teacher or student in a geometry class.

### Narrowing

To get around this problem we must add structure that allows us to look for proofs in the context of geometry. Another way of saying this is that we can use the word proof to narrow the word geometry. In doing so we can describe the notion of geometry proof as distinct from the intersection of the notions of geometry and proof. Note that this differs from using the word geometry to narrow the word proof. Resources labeled as proofs in the context of geometry are things like "two column proofs" or proofs using axioms of projective geometry, whereas geometry in the context of proofs refers to proofs that use geometry, such as a geometric proof of the arithemetic-geometric mean inequality. In order to distinguish between proofs in the context of geometry (geometry proofs) and geometry in the context of proofs (geometric proofs), we could use notation borrowed from computer science and write geometry.proof for the former while writing proof.geometry for the latter. In a formal sense this is correct, but the phrases geometry proof and geometric proof are more meaningful to the audience for which these taxonomies are intended. For this reason, we include the more natural phrases and indicate that geometry proof is a narrowing of geometry.

The opposite of narrowing is broadening. Thus if rectangle narrows polygon, we at the same time say that polygon broadens rectangle. This is useful for changing searches. If a student has run across the concept of perimeter in the context of rectangles, she will not find many explanations of the more general concept of perimeter if she only looks at resources that discuss rectangles. A new search using the broadening of rectangle to polygon is likely to be more fruitful. Broadening is also useful for resolving the difference between words used simply as keywords and words that appear in a structured taxonomy. The word pi, for instance, is often used as a keyword in searches. In our taxonomies, pi does not really stand alone. Rather, there is pi in the context of geometry and pi in the context of number. These two meanings of pi, as used to describe the content of mathematical resources, are related but different. If a user searches for resources described by the word pi, she could presumably be asked which pi she wishes to use.

### Alternates and Alternates with Possible Broadening

These taxonomies are meant to address the the practical problem of labeling resources for user populations that include students, teachers, parents, and the general adult public. Given this goal, it is desirable to include multiple words or phrases that describe approximately the same set of resources but that are derived from different user groups or different usage. For example, times table and multiplication table both refer to the same thing in primary and middle school mathematics and both should be included in the Level I taxonomy. We call these alternates. Times table is an alternate to multiplication table and multiplication table is an alternate to times table. In general, alternates are not fully symmetric. For example, in the context of geometric transformations dilation is sometimes used as a synonym with magnification and sometimes used to include both contraction and magnification. The words argument and proof could be quite distinct or functionally identical depending on the educational context and level. Most of the time, asymmetry between alternates involves a possible broadening and narrowing. Dilation is an alternate but possible broadening of magnification; argument is an alternate but possible broadening of proof. We include this structure in the taxonomies.

## LABELS AND COLUMNS IN THE TABLES

Root Terms (Used to Label the Tables)

Each taxonomic table is labeled with a root term. Abstractly, root terms are terms which have no broadenings and which represent rather general contexts for labeling resources. Concretely, the root terms for Level I are derived from the NCTM standards and the root terms for Level II represent either standard courses in high school or college or large topics within such courses. This is for organization only and does not imply that a resources labeled with a specific root term are not relevant to many other mathematical contexts.

Main Terms (Words and Phrases in the Leftmost Column of a Table)

The terms that appear in the leftmost column of a table are called main terms. Main terms are narrowings of the root term and have narrowings themselves. Main terms are printed in a bold faced font regardless of where they appear. Thus if you see a bold faced term in the narrowing column, you know that it also appears as a main term somewhere else in the table and has further narrowings.

Broadening (Words and Phrases in the Second Column)

A broadening is a more general context which can define the meaning of a given word or phrase. For example, geometry is a broadening of pi because geometry is a context which helps define a particular meaning of pi, namely the geometric as opposed to numeric or other interpretation. Terms that appear in the broadening column are broadenings of the main term to their left. We do not list the root term, which is always a broadening.

Narrowings (Words and Phrases in the Third Column)

A narrowing is a way of modifying the context defined by a given word or phrase so as to specify a substantially smaller set of resources than would be defined by using the conjunction of the two words or phases. Thus pi is a narrowing of geometry because saying that a resource involves the "geometric meaning of pi" gives more information than simply saying that the resource involves geometry and pi. Terms that appear in the narrowing column are narrowings of the main term to their left.

Alternates (The Meaning of the <| , |>, and <|> symbols)

An alternate word or phrase is one that should be considered as an alternate search term because it often describes a similar set of resources. There are three flavors of alternates Pairs of words or phrases that are almost always equivalent. This relationship is denoted by the symbols <|>. As an example, times table <|> multiplication table means that the phrases times table and multiplication table are virtually interchangeable. Pairs A and B of words or phrases in which B is an alternate but possible broadening of A. This is denoted by A <| B. As an example, magnification <| dilation means that there are some situations in which dilation and magnification define the much the same set of resources, but there are others in which dilation defines a substantially larger set. This comes about because sometimes dilation is synonymous with magnification and other times dilation is used to mean either a magnification or contraction. Note that we sometimes use A <| B when B is always broadens A but a searcher might nonetheless want to consider using the word or phrase B instead of A in many contexts. An example is contraction <| dilation. Pairs A and B of words or phrases in which B is an alternate but narrower version of A. This is denoted by A |> B and is the same as B <| A.

## TOWARDS FORMAL DEFINITIONS

We have defined the notions of narrowing, broadening, alternate, and attribute by giving examples. For some purposes it is helpful to give a more formal framework and more formal definitions. This is done in Taxonomies for School and College Mathematics, a PDF file which is also available as a Word document. The important thing to note here is that the relations of narrowing and broadening are not transitive: for example, pi is a narrowing of geometry, and calculation is a narrowing of pi in the context of geometry. Yet calculation does not appear as a narrowing of geometry. For this, and for many other reasons, the taxonomies given are not trees.

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