Tuesday, February 16, 2016

Common Core math: Fix or Replace

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Recall that each real number has two square roots. 
The above statement is taken verbatim from page 260 of the Precalculus and Advanced Topics Module 1: Teacher Materials (see it here) as posted on the for engageny website.

That statement, as written, is false. The number 0, which is a real number, has one square root, 0 itself.

The error in this statement could be explained away as due to the occurrence of a root with multiplicity two, but I have to take exception to that.  This module is a 431 page document, and the term "multiplicity" is not used anywhere in that document. Since the document appears as a "cookbook" designed for someone who is otherwise unable to teach this topic, it would seem that such a detail should be addressed, or accounted for in some other way. A very simple way would be to rephrase the statement:
Recall that each non-zero real number has two square roots. 
Back to the module itself, take note that it is 431 pages long. 

Module 1: 431 pages
Module 2: 489 pages
Module 3: 398 pages
Module 4: 291 pages
Module 5: 254 pages

Thus the Precalculus and Advanced Topics modules total 1863 pages. That makes for a very long cookbook. With that many pages, I presume that not many people will read it cover-to-cover.

So who would read it? Could we presume, for a moment, that those in need of guidance are the ones most likely to pick it up? (Or should I say, "link it up"?) Does it really do a service to those in need if it contains easily fixable mistakes?

I notices this little error (be it of commission or omission) solely because I wanted to see how NYSED was going to deal with what I used to call "math 4", and I had a problem due to this as the link to what I wished to see was missing. (If the image below is unclear, the original is currently here):
Nowhere on this page was there a link to Pre-Calculus curriculum. The only links were to the side, and they were links to the 5 modules.

Intermission: please NYSED, is it "Precalculus" or "Pre-calculus". I can go either way, bu please let me know your choice.

So I began to look at the modules, and opened up Module 1.

While in Module 1 I looked at page 210. That page asked a question about a reflection across the line \[y =  - x\].   I would hope that by the time they get to math 4 the students would be aware enough to know that this reflection is the same as a reflection across \[y = x\] followed by a reflection through the origin.

A reflection across \[y = x\] is \[(a,b) \to (b,a)\] and a reflection through the origin merely negates the coordinates , so this rule should be simple to ascertain. The module does indeed connect complex numbers to the concepts of transformation of the plane.

Alas! This page ultimately uses the "new" topic to validate the "old" topic, using complex numbers to verify what was previously known about the composition of basic planar transformations. How about using previously known mathematics as a foundation for new mathematics? This module seems to do the reverse. (Hey teach, this new stuff is just a more complicated way of doing the old stuff!)

For those who are unaware, transformations of the plane are dealt with (well, I hope), in Module 1 from Geometry. See Topic C here.

And we wonder why students get turned off from math?

The plotting of complex numbers as points in the plane is commonly referred to as an Argand diagram, due to the work of amateur mathematician jean-Robert Argand. I searched this module for a reference to Argand, but found none. That is like discussing relativity without mentioning Einstein. (the module soes mention video games (See Lessons 22 and 23 in the module.)

Since this Precalculus module does involve square roots of complex numbers, I would feel remiss in not commenting on another statement I found  within  engageny , 
There are other numbers on the number line between the integers.  They are called square roots.
This statement is from  page 28 of Grade 8 Module 7 Lesson 2 . The second sentence in this quote is downright misleading. (Every number is a square root of some number!) Unless a dissertation on rationals, irrationals, and reals is to be had, the sentence should just be omitted.

We have to do better.

The more I dig into Common Core the more I wish it would be discarded.  Then maybe we could start it over and do it right.

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