Friday, December 4, 2015

Lesson 6-5: Congruent Figures (Day 65)

The next lesson in the text is Lesson 6-5, on the definition of congruence. I will post the same worksheets as last year, except that I decided to change one of the questions in order to add an extra review question from yesterday's Miniature Golf activity.

This is what I wrote last year for Lesson 6-5. Notice that last year's post includes some comments by Dr. Ze'ev Wurman, a major traditionalist -- therefore, today's post counts as our traditionalist post for this week. I decided to add some more recent information from Dr. Wurman for this year:

Lesson 6-5 of the U of Chicago text is on congruent figures. Congruence is one of the most important concepts in all of geometry, especially Common Core Geometry.

As I mentioned many times on this blog, the word congruent is defined very differently in Common Core Geometry than under previous standards. We all know what it means for two segments to be congruent -- that is, that they have equal length -- or for two angles to be congruent -- that is, that they have equal measure. The new definition ofcongruent appears to be original to the Common Core, and yet, it isn't. Years before the Core, the U of Chicago text used the following definition of congruent -- indeed, it is mainly because of this definition that I chose the U of Chicago as the textbook on which this blog is based:

Two figures, F and G, are congruent figures [...] if and only if G is the image of F under a translation, a reflection, a rotation, or any composite of these.

And there we have it -- this definition of congruent predates Common Core. But many opponents of Common Core do not like this new definition. One such opponent is Ze'ev Wurman, a member of the commission here in California that reviewed the Common Core standards. Although his views are posted at several websites, one of the best Wurman articles I found is at this link:

Skipping down to the discussion of the math standards -- since, as Wurman himself points out, math is his area of expertise -- the author begins with some elementary school standards. For example, Wurman gives this standard:

1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Wurman states that this standard should have stopped after the first sentence. Instead, it goes on to prescribe some nonstandard algorithms for addition. I already discussed much of this in many previous posts. Since much of what Wurman writes about grades 1-3 echo what I wrote about the lower grades, I am in full agreement with Wurman for the lower grades.

But then Wurman moves on to Common Core Geometry. Here's what he writes about it:

A true content standard would simply say “Students prove triangle congruence” or, perhaps, “Students understand triangle congruence,” leaving the method of instruction to the teacher. Instead, Common Core not only dictates how to teach congruence, it insists on a specific experimental method of instruction that has an established a track record of failure where it was invented [...]
[emphasis Wurman's]

He then gives a link to a PDF file about the Russian mathematician and scientist A.N. Kolmogorov, whom the PDF credits as the creator of geometry based on transformations. Wurman implies that this geometry was tried out in Russia (i.e., the Soviet Union) and was a big failure.

The link between Kolmogorov's axioms and Common Core, to me, is uncertain -- but it is possible that both U of Chicago and Common Core derived their ideas from Kolmogorov.

Why should we use the transformation approach? In pre-Core Geometry, we must define the word congruent three times -- first for segments, then for angles, and finally for figures. But in some ways, this is an ad hoc approach. In the Common Core, we only definecongruent once, and it applies to segments, angles, and figures all at once.

In the Common Core, congruent means "identical up to isometry" (and later on, we see that similar is defined as "identical up to a similarity transformation"). There are many concepts in college-level mathematics that are defined similarly -- such as topologically equivalent ("identical up to homeomorphism"), equinumerous ("identical up to bijection"), and so on. Furthermore, the Lebesgue measure of a set is defined so that two sets such that there is an isometry mapping one to the other have the same measure.

So in some ways, the Common Core definition is more rigorous than the pre-Core definitions. Also, in some ways, the Common Core definition predates Kolmogorov by a wide margin -- Euclid himself used it as the Principle of Superposition in his proof of SAS (Proposition I.4).

In Hilbert's formulation of Euclid's axioms, congruence is a primitive notion -- that is, it is undefined just as pointline, and plane are. Actually, it's two undefined terms, since Hilbert considers segment and angle congruence separately. As I mentioned before, we can't define an undefined term, but instead we know what it means through the use of axioms or postulates. Hilbert provides six axioms of congruence -- these cover the Equivalence Properties and Segment and Angle Congruence Theorems as given in this section, some of the Point-Line-Plane and Angle Measure Postulates, and SAS. We notice that Hilbert's congruence is completely nonmetric -- there is no notion of distance or angle measure anywhere.

So which formulation should we use? This is a Common Core site and so I use the Common Core definition of congruence, but in the long run, which is best for the students? The usual guiding principles is that if a concept is easy for the students to understand and leads to a higher concept, then the students should learn how to prove it. But if the lower concept is difficult for the students, it should be made into a postulate and not proved in class.

So we can see a full continuum, from more proofs to more postulates:

  • Common Core: SAS, ASA, SSS all proved (using transformations)
  • Hilbert (supported by David Joyce): SAS postulated, ASA, SSS proved (using SAS)
  • Status quo: SAS, ASA, SSS all postulated (most texts)
  • Minimalist: SAS, ASA, SSS not mentioned (isosceles/parallelogram properties postulated)
The argument from Wurman and other Common Core opponents is that proving SAS, ASA, SSS from transformations only confused students (which would be the reason why this would have been a big failure in the Soviet Union) and that they should be assumed as postulates. But then, we wonder, why not go one step further and state that all proofs confuse students, so that all proofs involving SAS, ASA, SSS should be dropped, and the properties of triangles and parallelograms assumed? Why is the status quo, where SAS, ASA, SSS are assumed and used in proofs, exactly the right level of complexity for the students?

Well, this is what I hope to find out through this blog. It could be that these Core opponents are correct, and that the status quo level of complexity is exactly appropriate for high school students taking geometry. To me, this is not as clear-cut as elementary math, where the standard algorithms for addition and subtraction are clearly superior to the nonstandard algorithms. This is the reason that I agree with the traditionalists for K-3 math, but not high school math yet.

As for the other theorems proved in this chapter, the Equivalence Properties of Congruence is proved in a way that is standard for many types of transformations -- by using the identity, inverse, and composite functions. The Segment and Angle Congruence Theorems are proved using reflections only, since the text states (in the "Shorter Form" of the definition of congruence) that only reflections, or a composite thereof, are needed to establish congruence. But sometimes it's easier for students to visualize other transformations -- for example, in the Segment Congruence Theorem, one can simply translate X to Z, so that X' and Z coincide. Then one can rotate W' to Y, so that W" andY coincide. In the text, both of these are reflections instead.

Notice that this lesson, 6-5, is the first lesson in which the word congruent appears. The U of Chicago text is careful to use phrases such as "of equal length (measure)" instead of congruent.

I've mentioned before that many people -- both teachers and subjects -- use the words equal and congruent interchangeably. There are two distinctions to make -- one is that numbers (including lengths and angle measures) are equal, while segments and angles are congruent. The other is that we don't know that any figures are congruent until we know of an isometry mapping one to the other, which the Segment and Angle Congruence Theorems provide.

In this course, the latter distinction has priority. I admit that I myself have called angles "equal" (when it's their measures that are equal) on this blog -- because I don't want to call them "congruent" until reaching the Segment and Angle Congruence Theorems. I am especially guilty of this when I write phrases such as "Angle A = Angle B" because it's so much easier than trying to write an angle symbol in ASCII. Occasionally, I would underline a slash: m  A = m  B is the best I can do. Of course, I can't really draw a congruent sign at all, unless I write ~= and you just imagine that the tilde is directly above the equal sign.

Now all of this was last year. So what does Wurman have to say this year? Well, here's an article that he and Dr. Bill Evers wrote earlier this week. The article discusses the state of Massachusetts and its plan to drop the PARCC test. I already discussed Massachusetts back in my Thanksgiving post:

In this article, Wurman repeats what he wrote last year about Common Core math:

But, first, the Common Core is not “world class.” All serious studies have found Common Core academically mediocretrailing behind foreign high-achieving countries in the achievement it expects from students.  Indeed, the sequence in which topics are taught in high-achieving countries varies widely. Singapore’s curriculum differs from Japan’s, which in turn differs from Hong Kong’s. Pretending that Common Core has succeeded in finding the unique and perfect combination of content and sequence is both foolish and arrogant.
The Common Core says to teach Algebra I in ninth grade (although high-performing countries teach it in eighth). It dictates how proof of the congruence of triangles should be taught in the classroom and requires that a method, known as “rigid motions,” be used – although that method has never been taught successfully in K-12 education in America. Parents have filled Facebook with examples of unnecessarily complex math methods using lattices and making tens. The Common Core English standards dictate the amounts of literary text (novels, poems, essays) versus informational text (Federal Reserve newsletters, edicts of the Environmental Protection Agency) that students should read in different grades.

I agree with some of what Wurman writes here about elementary math -- but I still think that the lattice method is just as rigorous as the standard algorithm for multiplication. But regarding Geometry using transformations ("rigid motions"), we've found out that actually, there aren't as many transformations on the PARCC as Wurman and others fear. Most congruence questions on the PARCC are based on SSS, SAS, etc., as usual -- the transformation questions mainly involve the coordinate plane and are similar to what I posted this week.

Finally, here's one comment to the article -- written by Bruce William Smith, another traditionalist I sometimes mention here on the blog. Smith's main concern is about testing:

That death knell could be accelerated if the federal annual testing mandate were brought to an end; however, unfortunately, a congressional conference committee has just approved that Bush-era mistake's renewal, the Every Student Succeeds Act, which should be defeated unless it respects the right of families to opt out of the test-obsessed state schooling that the children of this century have suffered under, to little positive effect.

I'll have more to say about the Common Core and overtesting some time next week.

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