Mathematics Metadata Working Group

This document reports on work done and decisions made at a meeting held August 10 and August 11, 1999, in Berkeley, California. Please see the last section for acknowledgments and a list of attendees. This document was prepared by Robby Robson.

Comments and Suggestions: Comments and suggestions should be posted to the forum http://forum.swarthmore.edu/dicsussions/math_metadata or sent by email to robby@orst.edu.

Issues covered: The issues covered in the main part of this report are

1. a mission statement of the Mathematics Metadata Working Group (MMWG)
2. a brief report on IEEE Learning Object Metadata Standards
3. the decision to classify mathematics using three separate taxonomies
4. the detailed structure of these taxonomies
5. a discussion of other LOM tags and their relevance to the MMWG. Includes rough proposal for which tags to use.
6. Implementation, work, and cooperation commitments
7. timelines and commitments for the completion of a first public release of mathematical metadata standards

Attachment: Attached is an annotated version of the LOM 3.5 base scheme.

Requests for comments and notes by the author. Comments on any and all aspects of this document are welcome.

The following mission statement is proposed for the MMWG. It is important to have such a statement for setting future agendas and for explaining MMWG work to potential partners, contributors, and funding agencies. Comments are welcome.

Until recently there have been several organizations developing potentially completing standards for pedagogic metadata. Over the past six months the situation has changed dramatically. The IMS project and ARIADNE have both agreed to use the IEEE Learning Object Metadata, or LOM, as their standards. The latest LOM specifications, together with annotations, appear as an attached document.

It should be pointed out that there are some difficulties in mapping IMS metadata to LOM and that, at least in the opinion of the MMWG, there are portions of the LOM base scheme that are too restrictive or have not anticipated the full range of learners and learning environments. The MMWG struggled with this and eventually decided to follow the lead of NEEDS and to judiciously define the necessary extensions to LOM. These decisions will be explained and communicated to the IEEE Learning Technology Standards Committee and to the IMS project.

The structure of LOM. The LOM base scheme is divided into nine categories defined as follows (source: LOM v3.5):

The MMWG has decided to work only on those portions of this scheme that need special attention for the purpose of cataloging and searching mathematical materials. But other parts of the base scheme must be included in software that implements metadata, e.g., that tags documents with metadata or that uses metadata to retrieve documents. The most immediate implementation will be as part of the SMETE portal under development by NEEDS, and most of the more general tags have already been addressed by NEEDS software that will be modified to handle mathematical resources.

The MMWG proposes to classify the content of mathematical resources using three taxonomies corresponding to three different levels of vocabulary. Whereas it might be tempting to view these taxonomies as correspond roughly to American grade levels, they should be thought of in terms of vocabulary usage rather than in terms of subject matter.

History of This Decision. The first idea for classifying mathematical content was to extend the MSC to encompass grade school, high school, and college mathematics. This was quickly seen to be a poor option for a number of reasons:

• The meaning of identical vocabulary can change both subtly and radically across levels of mathematics. It would be very hard, if not impossible, to fit lists of vocabulary used by existing digital libraries into the MSC in a consistent fashion.

• There are vocabulary terms, such as "times tables", that decidedly would not fit in with the MSC but which might be important to a large user group interested in finding mathematical resources.

• It is not clear that the two-level tree structure used by MSC is the appropriate taxonomic structure, and in fact it is not the one proposed by the MMWG.

• There is no organization which has as its natural responsibility of maintaining, disseminating, and updating a classification spanning all of mathematics.

During its June meeting the MMWG abandoned the idea of extending the MSC and proceeded to propose a division of mathematics into four or five separate taxonomies based on progress through the traditional American educational system. Several issues were left open for further consideration: should some of the levels be merged, what were appropriate names for the levels, and what should their internal structure be.

Further Considerations. At this August meeting the MMWG tackled these issues. The following points were made:

• The audience for mathematical resources extends well beyond the traditional student taking a traditional class at a traditional stage of schooling. Today’s cadre of learners includes large numbers of adult learners, returning students, students with disabilities, distance learners, and self-directed learners. Neither the taxonomies nor the names for the taxonomies should assume otherwise.

• Metadata-enabled searching and learning is of interest to students, teachers, corporate trainers, researchers, editors, and the general public. It is by no means limited to mathematicians or to people studying mathematics as a discipline.

• It is better to have fewer rather than more taxonomies. Simpler is better as well. It is not the responsibility of a classification scheme to imbue intelligence into search engines. Rather, it is the responsibility of a classification scheme to provide a basic structure which increasingly intelligent search engines and agents can use and extend.

• There are other metadata tags, primarily in the Education category, that can be used in conjunction with any content classification to further describe a resource. Good search and retrieval schemes will rely on a rich combination of metadata tags to define the context in which a classification scheme will be interpreted.

• There is nothing precluding labeling a resource with multiple taxonomies. This will be necessary for resources that naturally fall near the boundary between two levels and perhaps will be done as a matter of routine.

• Metadata tags will likely be invisible to the general user. They have to make sense to authors and catalogers. This observation should guide the way in which tags, taxonomies, and terms are defined.

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.

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.

Controlled vocabulary: Controlled vocabulary as used and encountered by students and teachers in mathematics typically encountered after 8^th 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.

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.

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:

1. Which LOM tags will be used "as is" and which will be extended.
2. Which metadata tags will be mandatory for mathematical resources.
3. Which metadata tags will be optional but available for mathematical resources.
4. For each tag requiring a "best practices" list, which list will be used or, if necessary, created.
5. 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.
6. For Classification, which taxonomies (other than content classification) will be used. It might be necessary to invent some!

The following is a first cut at 1 - 3. Much of this was done after the August meeting and represents the opinion of the author and not necessarily of the MMWG.

IMS Core Metadata Elements. The IMS best practices document identifies a set of elements as "Core" elements. It is suggested that these be included as mandatory or optional as follows. Some suggestions/explanations are given. See the appendix for more.

Recommended Additional Mandatory or Optional Elements. Most of the following elements were discussed during the August meeting. A few have been added by the author of this document. Only those elements not in the IMS Core (see above) are included.

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.

This is the only example of LOM definitions, other than content, given in this report.

• 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.

Robby Robson

August 17, 1999

See http://math-classes.orst.edu/metadata/docs/LOM35_annotated.html