One lesson that I have

*not*been very successful at teaching involves a lot of vocabulary of both kinds, so for one assignment I focused on this lesson on classifying numbers. I've been puzzled at how poorly my classes have done on this topic. Some of the problems, such as the inability to remember what "integer" means, would be addressed with the usual bag of vocab learning tricks of flash cards, personal dictionaries, repetition, and repetition - this would be a word of the first kind, where the learning task is simply to memorize a letter combination and to link the word to its referent. However, words such as "equivalence," "inclusion," and even "all," "some" and "none" are also necessary, and maybe such content area concepts are all so unique that no list of teaching approaches can be made - on the other hand, I'd be surprised if the field of linguistics does not provide some broad, general insights on acquiring such words. However, whatever these insights might be, they have not trickled down to the teacher ed classes I've taken, at least not while I've been paying attention.

All Algebra I and Algebra II textbooks I've seen introduce the topic of classifying numbers by presenting a Venn Diagram of the usual sets, and provide no further clarification about how Venn Diagrams work, assuming, it would seem, that this visual is self-explanatory and can replace a verbal discussion of basic ideas of sets. However, students do not generally know how Venn Diagrams work. In trying to clarify the idea I have asked multiple Algebra I and Algebra II classes to draw Venn Diagrams of statements such as

- All high school students study math.
- Some high school students study music.
- No high school students are senior citizens.

After class discussions and multiple examples of Venn Diagrams for situations where one set is included in the other or where the sets are disjunct, a majority of students will get the point and draw accurate diagrams for natural language statements. However, many students do not get to this point, at least not in Algebra I, and will continue to diagram "All x is y" by drawing the circle for y inside the the circle for x, and I am wondering why. I have not yet tried to find out whether it is mainly a matter of not matching the appropriate visual representations with the words "all," "some" or "none," and whether they would be able to correctly answer questions about sets of numbers based on sentences alone. I have basically emphasized translation between Venn Diagrams and statements. Whatever the problem might be, I have a greater appreciation for how hard it must be to learn high school math without a pretty strong grasp of pretty abstract words for referents that can not simply be pointed to.