"Hypotheticals"

As promised, I've been working on a paper addressing the NCTM Curriculum and Evaluation Standards for Geometry 9-12. However, there's a particularly appalling practice in which the NCTM engages which needs to be addressed separately. Chapter 7 of the PSSM (on Geometry) on p. 310 gives the following example of geometrical reasoning, which is presumed to follow from teaching geometry using NCTM principles.

The following proof demonstrates an ability to select and focus on important elements in the diagram, and it shows a solid understanding of the concepts involved and how they can be assembled to solve the problem.

First, I noticed that since AB and DE are parallel, angles B and E must be congruent. Also, angles ACB and DCE are congruent, since they are vertical. So now I know that the two triangles (ABC corresponds to DEC) are similar by angle-angle similarity. But that tells me that their corresponding sides are proportional. Since DE = 4(AB), I know that all the sides of triangle DEC are 4 times as large as the corresponding sides of triangle ABC, so CD = 4(15) = 60.

Now I just need to find the other side of triangle DEC to find its perimeter. But DF makes it into 2 right triangles, so I can use the Pythagorean theorem on each of those. FE^2 + 48^2 = 52^2, so FE is 20. (Actually, I just noticed that this is just 4 times a 5-12-13 triangle, but I saw that too late.) Then looking at CDF, this is 12 times a 3-4-5 triangle, so CF must be 36. (I checked using the Pythagorean theorem and got the same answer.) So the perimeter is 52 + 60 + 56 = 168.

Once I find the perimeter of ABC, I'm done. But that's easy, since the scale factor from DEC to ABC is 25%. I can just divide 168 by 4 and get 42. The reason that works is that each of the sides of ABC is 25% of its corresponding side in DEC, so the whole perimeter of ABC will be 25% of DEC. We already proved that in class anyway.

Wow! That’s the most impressive display of geometrical reasoning I’ve ever seen. I taught geometry for 6 years, and not even students who earned an A in my honors geometry/trigonometry class could have written this explanation. If this is an example of what NCTM math can accomplish, I’m sold. But wait a second… “Note particularly how the fictional student finds different connections to be sure her reasoning is sound.”

Incredible! They illustrate how their methods are supposed to work with a “fictional student”!? Has any REAL student ever written anything remotely like this? Can you imagine the outrage if doctors described how experimental cures would work on “fictional” patients?! And sadly this is hardly unique in NCTM “research.”

P. 311 – “The following hypothetical example illustrates how students might investigate relationships in a dynamic geometry environment and justify or refute conclusions.”

P. 342 – “Consider the following hypothetical classroom scenario”

I must admit I find it very difficult to take "research" like this seriously. How can I craft a logical response to a book that violates the most basic tenets of valid scholarly research so flagrantly? Maybe I should instead just concoct some "fictional" students who write dissertations in algebraic geometry after completing a traditional curriculum.