*Dedicated to the
research, development, implementation, and standardization of metadata for educational
and research mathematics.*

**AMS Panel discussion: Wednesday, January 19. Ballroom
Balcony A, Marriott Wardman Park Hotel. 2:15 - 5:15. Immediately followed by
American Mathematics Metadata Task Force Meeting. **

The Proposal. The MMWG decided to simply name the three taxonomies Level 1, Level 2, and Level 3. Each of these will consist of a list of controlled vocabulary together with a set of relationships among these vocabulary. This applies to the MSC, which will be used for Level 3, as well as to the new taxonomies proposed. The taxonomies are described as follows. Comments on these descriptions are welcome.

Level 1. Controlled Vocabulary: Controlled vocabulary as used and encountered by students and teachers in early mathematics including number sense, arithmetic, and basic geometry and other subjects taught in traditional elementary and middle schools. A rough description of the mathematical level might be "pre-variable mathematics." Relationships. Each term in the Level 1 controlled vocabulary will be associated to those terms which are specializations, generalizations, and equivalents of the given term. A specialization of a term is one that is more specific, that serves as an example, or that could reasonably be viewed as a subtopic. Examples of specializations are: square and triangle are specializations of polygon pi and negative number are specializations of number equation and variable are specializations of algebra A generalization of a term is the opposite of a specialization and consists of a topic under which the given term might fall. Thus the generalizations of pi might include number, circle, geometry, and measurement. An equivalent term is a term that in many contexts means the same thing. At Level 1 we might want to include both the terms "multiplication table" and "times table", which would be equivalent.

Level 2. Controlled vocabulary: Controlled vocabulary as used and encountered by students and teachers in mathematics typically encountered after 8th grade and through sophomore level college mathematics courses in North America. Topics include those used for the last year of postsecondary education in the TIMSS benchmarks. Relationships: As in Level 1. For each term in the Level 2 controlled vocabulary we will identify those terms which are specializations, generalizations, and equivalents of the given term.

Level 3 Controlled vocabulary: Controlled vocabulary reflecting accepted profession practice and typically encountered in advanced undergraduate, graduate, and research mathematics. Relationships: As in the MSC, Level 3 will be a two-layer tree structure with topics at the first layer and sub-topics at the second layer. Each leaf of the tree will include a set of related topics. Level 3 will be identified with the MSC. Lack of Transitivity. It should be noted that the relationships used in Level 1 and Level 2 are not intended to be transitive (or possibly even anti-symmetric)! As an example, pi might be considered a specialization of circle and number. Circle, in turn, might be considered a specialization of shape, and number might be considered a specialization of counting. But pi would not be considered a specialization of shape or of counting. A reason for this non-transitivity is that the structures being defined contain both semantic and axiomatic associations. The link between pi and circle is axiomatic; the definition of pi involves a circle. The link between circle and shape is semantic. Another reason is that of we think of a term as defining a region in document space, neither generalization nor specialization require strict containment. If one region is "mostly contained" in another, then it can be considered a specialization. Note that anti-symmetry may also be violated by number and counting. These may not be considered equivalent, but it could be argued that each is a specialization of the other in the context of searching for educational material. This again may be attributed to the use of semantic associations which are not constrained by the rules of an axiomatic structure and the fact that containment of associated regions need not be strict. Consequences of the Lack of Transitivity. Search engines like Infoseek and Yahoo can display paths through a browse structure that lead to a topic being searched. Without transitivity, this is not possible. The MMWG did not view this as an issue since the relationships defined are sufficient to answer the underlying question of where to look to find out more (or more specific) information. Moreover, the metadata associated to an object will be primarily for the use of intelligent engines and agents and not for display. (Note: What might create a difficulty is the notion of a taxonomic stairway in the IMS metadata specifications. This notion does not appear in LOM, so will be ignored for now. -RR-) Comment on the Names "Level 1", "Level 2", and "Level 3". The MMWG gave some serious thought to this. Words like "elementary", "intermediate", and "advanced" were rejected because they are likely to mean different things to different people. Using educational levels to name the taxonomies suffered from even more problems: meanings differ with geographic region, educational levels focus on subject matter rather than vocabulary, and educational levels are inaccurate in many of the non-traditional learning contexts. The MMWG settled on neutral terminology with no connotations other than the vague notion that the three levels are ordered in terms of a natural progression through mathematics.

Overview of Work. At the August meeting the MMWG saw a non-enabled set of screens that will be used by NEEDS to associate metadata with objects in their database. NEEDS has agreed to extend these screens to include information needed for tagging mathematical objects. To make this work, it is necessary to decide:

Which LOM tags will be used "as is" and which will be extended.

Which metadata tags will be mandatory for mathematical resources.

Which metadata tags will be optional but available for mathematical resources.

For each tag requiring a "best practices" list, which list will be used or, if necessary, created.

For tags with generic values (e.g. "very low" to "very high") what, if any, definitions or further explanations will be associated with these values in the context of mathematics.

For Classification, which taxonomies (other than content classification) will be used. It might be necessary to invent some!

Other LOM elements that will need to be considered in the future. "Optional" means there will be a place to input this data but need not be defined for all documents. Discussion of Individual Elements. The following is an example of the kind of work that must be done on some elements. The element in question is Educational.SemanticDensity. In mathematics, this might be interpreted as describing the density of specialized notation and jargon but is difficult to define in absolute terms. Yet it must be defined, if even only roughly. If left undefined, a high school teacher might consider a college text to be high in semantic density whereas a research mathematician might consider the same text to be low. The following is a draft attempt by RR.

Cooperation and commitments from digital library projects NEEDS has been developing a system for tagging data in their database. Release is imminent. NEEDS agreed to modify their software to include the categories and terms in those categories deemed necessary by the MMWG. This will provide a prototype for labeling mathematical resources with metadata. Required from the MMWG are suggested elements and tags (as in this report, but with details.) The ENC and Math Forum will map their categories to IMS metadata elements. NEEDS is working on a SMETE portal. Mathematics will be a community within that portal. The ENC and Math Forum have agreed to port data to the SMETE portal in order to provide test data for the above prototype. In addition to the above, NEEDS will provide a prototype of a metadata-enabled search engine that will work on their database. The Math Forum has agreed (prior to the August meeting) to implement a prototype of an "invisible" search. A user will self-identify some parameters such as grade level and discover resources in the Math Forum appropriate to a user with those parameters. The ENC has agreed to provide assistance in developing lists of controlled vocabulary and associated relationships for the proposed Level 1 content taxonomy. The Math Forum and the ENC will cooperate on doing the same for Level 2. The ENC expressed interest in providing long-term "secretariat" services for Level 1. These involve maintaining and disseminating the classification scheme. Other commitments: Punaho school will review Level 1 and Level 2 taxonomies. Mary Craven will attempt to involve the Hawaiin Council of Teachers of Mathematics. Linda Yamamoto will post news of this effort to the Physics, Astronomy, and Mathematics section of the Special Libraries Association. RR will work further on the LOM elements external to the content classification.

The MMWG August meeting was attended by the following persons in alphabetical order. Participation varied from one hour to two days.

Don Albers (Mathematical Association of America)

Joe Buhler (Deputy Director, Mathematical Sciences Research Institute)

David Collinge (Senior Product Manager for Central Media, Pearson Education)

Mary Craven (Department Chair, Punaho School, Honolulu, HI)

Charles Drucker (Headlund Digital Media)

Wade Ellis (West Valley College, Saratoga, CA)

Ann Jensen (Mathematics Librarian, UC Berkeley)

Gene Klotz (Director, the Math Forum)

Brandon Muramatsu (Director, NEEDS project)

Ruth Radetsky (Middle School Teacher, San Francisco)

Robby Robson (Oregon State University)

Len Simutis (Director, Eisenhower National Clearninghouse)

Linda Yamomota (Mathematics Librarian, Stanford University)

Acknowledgements. The Eisenhower National Clearinghouse generously supported the August MMWG meeting through travel grants to some participants. NEEDS and UC Berkeley hosted the meeting. Brandon Muramatsu did a wonderful job of making local arrangements. Brandon Muramatsu and Greg Paschall provided technical assistance as well. Gene Klotz helped greatly in the pre-organization. A note of thanks goes to Mike Hodges at Pearson Education for taking an active interest in our efforts.

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