On 26 July, Donna originally asked what evidence we would like to see that she has been "doing her job" with students in the upper elementary grades. I will respond assuming that upper elementary includes grades 7 & 8 from the middle school, but I will focus specifically on the geometry strand. [perhaps others could respond to other strands]
I believe that, by eighth gd, "average but well-founded" students should be operating comfortably at Level 2 on the van Hiele scales* [they go from Level 0 thru Level 4]. At Level 2 (Informal Deduction), the relationships between classes of shapes are the objects of thought. Shapes can be classified by using only minimum characteristics. Students can follow an informal deductive argument but would probably not be able to construct such a proof;however, they begin to focus on logical arguments about properties.
Concerning appropriate activities at Level 2, Van De Walle says: "Continue to use models with a focus on defining properties. Make property lists and discuss which properties are necessary and which are sufficient conditions for a specific shape or concept; Include language of an informal deductive nature: all, some, none, if-then, what if, and so forth; Investigate the converse of certain relationships for validity; for example, the converse of 'if it is a square, it must have four right angles' is 'if it has four right angles, it must be a square'; Using models and drawings as tools to think with, begin to look for generalizations and counterexamples; and Encourage hypothesis making and hypothesis testing."
Thus, at this level a student might use a drawing in order to follow a deductive argument where the latter is supplied by the teacher [such as a proof of the Pythagorean Theorem]. Models might be used to find counterexamples or to test conjectures.
Such students should be able to define shapes and explain how different classes of shapes are related to one another. They should be able to do geometric "constructions" with tools such as the MIRA. They should be able to independently explore relationships with computer tools such as the Geometer's Sketchpad or the Geometric Supposer and with the computer language LOGO.
Now to have achieved all this with your students, YOU clearly need to be familiar with at least the first four van Hiele levels [0-3];you must be able to diagnose where each student is on the van Hiele continuum;you must be familiar with activities that will move kids along from one level to the next;and so forth. You should also be comfortable with the various computer tools and language mentioned.
And you must understand that the van Hiele levels are sequential;they are not age dependent;advancement results from exposure to appropriate experiences;and you must teach at the level where students are at the time, even as you challenge them to operate at the next higher level of thought.
If you can do all these things, and your students can do the things mentioned earlier, then I'd say you've done a GREAT job in geometry at the upper elementary level. :)
*There are many resources concerning the van Hiele levels of geometric thought. See, for example, the NCTM 1987 Yearbook on Geometry, the 1991 Math Teacher article by Teppo, The January 1986 article by Burger & Shaughnessy in the Journal For Research in Mathematics Education, and the very readable Geometry chapter [filled with activities at Levels 0,1, & 2] by John Van De Walle [Longman, 1994]
Ron Ward/Western Washington U/Bellingham, WA 98225 firstname.lastname@example.org