On the other hand, the only way I would sub today is for my old district. Of course, this was very unlikely, and in the end I didn't sub today.
Here I go again, making a traditionalists' post out of schedule. Last week on the scheduled day, I didn't write much and declared that the real traditionalists' posts are the "Sue Teele" posts, as her multiples intelligences are the other side of the debate.
But over the weekend, our main traditionalist Barry Garelick posted. And he's responding to an article written by another author whose ideas we've seen recently -- Jo Boaler:
https://traditionalmath.wordpress.com/2018/10/27/spinmeister-dept/
San Francisco’s Unified School District decided to eliminate access to algebra for 8th graders even if a student is qualified to take such a course. The latest article to justify the action is one written by Jo Boaler (whose self-styled approach to math education in my opinion and the opinion of many others in education who I respect has been ineffective and damaging) and Alan Schoenfeld, a math professor from UC Berkeley whose stance is consistent with math reformers. I.e., “understanding” takes precedence over procedure, among other things.
Garelick and the other traditionalists have mentioned San Francisco Unified in the past. It's almost always to criticize the district's eighth grade Algebra I policy. He quotes Boaler's article:
“The Common Core State Standards raised the level and rigor of eighth-grade mathematics to include Algebra 1 content as well as geometry and statistical topics previously taught in high school.”
And the traditionalist disagrees. In his article, he ultimately mentions senior-year AP Calculus -- the class that the traditionalists really care about. Raising the rigor of Math 8 to include a little more algebra is irrelevant if it doesn't lead to seniors taking a class called AP Calculus. Oh, and by the way:
Translation: For those students who wish to take calculus in 12th grade, they can double up math courses in 11th grade, so they can take Algebra 2 and Precalculus. As far as what they mean by “conceptually rich courses that benefit everybody”, it’s anybody’s guess.
In other words, it doesn't count as a true path to Calculus unless students can take math for only one period a day, with no summer school, with AP Calculus as the capstone class. (I do agree with Garelick that Algebra II and Pre-Calc together is a very tough load for a junior.)
Garelick writes:
A high school level course includes rational expressions (i.e., algebraic fractions), polynomial division, factoring, quadratic equations, and direct and inverse variation. The 8th grade standards do not include these. I teach an 8th grade math class as well as high school algebra for 8th graders.
I agree only in part. I usually consider Common Core 8 to line up with the first half of Algebra I. So factoring and quadratics are Algebra 1B topics that don't appear in Common Core 8. As for rational expressions and polynomial long division, these appear in some Algebra I texts, but many high school teachers save these for the end of the year and ultimately skip these topics. Since Garelick states that he teaches eighth Algebra I, I wonder whether he teaches these topics and if he does, what letter grades his eighth graders earn on the tests. (As for direct variation, Garelick addresses this later in his post.)
Also, notice that Boaler never claims that Common Core Math 8 is more rigorous than Algebra I. She means that Common Core Math 8 is more rigorous than pre-Core Math 8 in most states (other than California) for which Math 8 does not equal Algebra I. Common Core Math 8 is more rigorous than non-Cali pre-Core Math 8 in that some (not all) Algebra I content has been added. But it's easy to get confused because this is an article in a Cali newspaper about a Cali district, and so Garelick thinks that she's trying to compare Common Core Math 8 to Cali pre-Core Math 8 (= Algebra I).
Garelick compares Common Core Math to his favorite Dolciani text from 50 years ago:
I supplement freely with a pre-algebra book by Dolciani written in the 70’s and other materials. The emphasis on ratio and proportion in 7th and 8th grades is rather drawn out and can be done more concisely, rather than harping on what a direct variation and proportional relationship is. Traditional Algebra 1 courses present direct variation in a much more understandable way, rather than the “beating around the bush” technique that defines such relationships as straight line functions that go through the origin, and whose slope equals the “constant of variation/proportionality”.
Recall that I found a copy of a 1970 Dolciani text at a library book sale. But I could find no mention of direct variation in the text. (My text is called "Course 2" -- I suspect that Garelick actually uses "Course 3" in his classroom.)
I look at my other 1960's-era text, Moise's Geometry, and I notice this in the preface:
"In recent years, there has been extensive discussion about the content of the geometry course ordinarily taught in the tenth grade."
So to Moise, Geometry is a sophomore course, yet to the traditionalists who prefer texts from his era, Geometry is a freshman course. The idea that eighth graders should be in Algebra I or seniors in Calculus is a fairly recent one. It never occurred to Moise, Dolciani, and other textbook writers of the 1960's and '70's that Calculus should be taught in high school.
Garelick writes:
But the real goal of San Francisco’s elimination of algebra in 8th grade is to close the achievement gap as evidenced by the last paragraph in the article.
Traditionalists don't wish to close the achievement gap -- instead, they favor tracking, which is the exact opposite. If most of the students in Garelick's eighth grade Algebra I class or the AP Calculus class are members of privileged groups (with "privilege" as defined by Eugenia Cheng), then so be it.
SteveH returns to comment in this thread:
SteveH:
Astounding. Why not eliminate the leveled groups in K-6 they use for differentiated instruction? It all doesn’t make any sense to the most casual observer. CCSS has a slope that leads to no remediation in college algebra – they say this! – but Jo Boaler, etal. claim that it’s normal to magically change that slope in high school to get to calculus, a level difficult even for those who get algebra in 8th grade.
Recall that Boaler wrote the preface to the Number Talks book, so I assume that she supports the methods used in that book. One of its authors, Cathy Humphreys, actually does mention Calculus in her book (in the chapter on fractions). Let me provide the full context:
"One day, as Cathy was working with sixth graders to help them find different ways to compare fractions, the class was unusually passive -- and almost sullen. Finally she stopped and asked what was wrong. After a minute or so, Anthony spoke up, and, even though it was some years ago, his words are still etched in her mind: 'Mrs. Humphreys, we had fractions in third grade and fourth grade and fifth grade. We didn't get them then, and we don't get them now -- and we don't want to do them anymore!' Not being able to 'get' fractions made Anthony feel unsuccessful -- and who wants to work on things that make us feel like that?
"But for success in high school, there is no avoiding fractions. Students who are successfully learning complex concepts in algebra, trigonometry, and calculus can become confounded by a fraction in the middle of an equation."
Notice that Anthony, the sixth grader, doesn't want to do fractions. He'd much rather leave a problem set blank than answer Question #1 if it contains a fraction.
We know what solution Cathy Humphreys and Jo Boaler would recommend -- start doing Number Talks on fractions. But to Garelick, SteveH, and other traditionalists, anything other than traditional math would block Anthony from getting to AP Calc.
OK, then, so I'd like to see what the traditionalists would do with a student like Anthony. He makes it clear that he doesn't want to do fractions, so assume that he'd refuse to answer Question #1 on a traditional p-set with fractions. Go on, traditionalists, show us your magic!
Lesson 5-2 of the U of Chicago text is called "Types of Quadrilaterals." In the modern Third Edition of the text, quadrilaterals appear in Lesson 6-4.
This is what I wrote two years ago about today's lesson:
Lesson 5-2 of the U of Chicago text covers the various types of quadrilaterals. There are no theorems in this section, but just definitions. The concept of definition is important to the study of geometry, and in no lesson so far are definitions more prominent than in this lesson.
The lesson begins by defining parallelogram, rhombus, rectangle, and square. There's nothing wrong with any of those definitions. But then we reach a controversial definition -- that of trapezoid:
Definition:
A quadrilateral is a trapezoid if and only if it has at least one pair of parallel sides.
(emphasis mine)
Just as with the definition of parallel back in Lesson 1-7, we have two extra words that distinguish this from a traditional definition of trapezoid -- "at least." In other textbooks, no parallelogram is a trapezoid, but in the U of Chicago text, every parallelogram is a trapezoid!
To understand what's going on here, let's go back to the first geometer who defined some of the terms in the quadrilateral hierarchy -- of course, I'm talking about Euclid:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI22.html
Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
Of course, the modern term for "oblong" is rectangle, and a "rhomboid" is now a parallelogram. The word "trapezia" is actually plural of "trapezium." In British English, a "trapezium" is what we Americans would call a trapezoid, but to Euclid, any quadrilateral that is not a parallelogram (or below on the quadrilateral hierarchy) is a "trapezium." But the important part here is that to Euclid, a square, for example, is neither a rectangle (oblong) nor a rhombus. He makes sure to say that a rectangle (oblong) is "not equilateral," and that a rhombus is "not right-angled." And of course, neither a rectangle nor a rhombus is a parallelogram (rhomboid).
These are called "exclusive" definitions. For Euclid, there was no quadrilateral hierarchy -- each class of quadrilaterals was disjoint from the others. But since the days of Euclid, more and more geometry texts have slowly added more "inclusive" definitions.
One of the first inclusive definitions I've seen was the definition of rectangle. It was mentioned on an episode of Square One TV, when a Pacman parody character named Mathman was supposed to eat rectangles, and then ate a square because "every square's a rectangle." I would provide a YouTube link, but I haven't found the link in years and it doesn't come up in a search. (I even remember someone posting in the comments that just as for me, his first encounter of the inclusive definition of rectangle was through watching that clip when it first aired so many years ago!)
But many of my family members were also teachers, and one relative gave me an old textbook that still mentioned some exclusive definitions. In particular, it declared that a square isn't a rhombus. A little later, my fifth grade teacher then taught the inclusive definition of rhombus. I then blurted out that a square isn't a rhombus, then actually brought the old text to school to prove it! She replied, "Wow!" but then, if I recalled correctly, told me that this definition was old, and that by the new definition, a square is a rhombus. And so all modern texts classify the square as both a rectangle and a rhombus, and that all of these are considered parallelograms.
So we see that there is a tendency for definitions to grow more inclusive as time goes on. (We see this happening in politics as well -- for example, the definition of marriage. But I digress.) And so we see the next natural step is for the parallelogram to be considered a trapezoid.
One of the first advocates I saw for an inclusive definition of trapezoid is the famous Princeton mathematician John H. Conway. He is best known for inventing the mathematical Game of Life, which has its own website:
http://www.conwaylife.com/
But Conway also specializes in other fields of mathematics, such as geometry and group theory (which is, in some ways, the study of symmetry). Twelve years ago, he posted the following information about why he prefers inclusive definitions:
http://mathforum.org/kb/message.jspa?messageID=1081135
The preference for exclusive definitions arises, I think, from
what I call "the descriptive use". Of course, one wouldn't DESCRIBE
a square table as "rectangular", since that would wantonly use
a longer term to convey less information. So in descriptive uses,
there's a natural presumption that a table called "rectangular"
won't in fact be square - in other words, a natural presumption
that the terms will be used exclusively.
But the descriptive use is unimportant to geometry, where the
really important thing is the truth of theorems. This means that we
should use a term "A" to include "B" if all the identities that
hold for all "A"s will also hold for all "B"s (in the way that
the trapezoid area theorem holds for all parallelograms, for
instance).
You might worry about the consistency of switching to the
inclusive use while other people continue with the exclusive one.
But there can be no consistency with people who are inconsistent!
I've seen many geometry books that MAKE the exclusive definitions,
but none that manage to USE them consistently for more than a few
theorems.
Indeed, Conway advocated taking it one step forward and actually abolishing the trapezoid and having only the isosceles trapezoid in the hierarchy! After all, there's not much one can say about a trapezoid that's not isosceles -- just look ahead to Lesson 5-5. There's only one theorem listed there about general trapezoids -- the Trapezoid Angle Theorem, and that's really just the Same-Side Interior Angle Consequence Theorem that can be proved without reference to trapezoids at all. All the other theorems in the lesson refer to isosceles trapezoids. In particular, the symmetry theorems in the lesson refer to isosceles trapezoids. (Recall that Conway's specializes in group theory -- which as I wrote above is the study of symmetry.) I suspect that the only reason that we have general trapezoids is that they are the simplest quadrilateral for which an area formula can be given.
This is now another digression from Common Core Geometry, so I'll just provide another link. Notice that here, Conway also proposes a hexagon hierarchy based on symmetry. There's also a pentagon hierarchy, but there are only three types of pentagons -- general, line-symmetric, and regular -- just as there are for triangles. It's easier to make figures with an even number of sides symmetric.
http://mathforum.org/kb/message.jspa?messageID=1074038
Another advocate of inclusive definitions is Mr. Chase, a Maryland high school math teacher. (And no, in yesterday's post and today's I'm referring to two different Maryland teachers.) I see that he is so passionate about the inclusive definition of trapezoid that he devoted three whole blog posts to why he hates the exclusive definition of trapezoid:
http://mrchasemath.wordpress.com/2011/02/03/why-i-hate-the-definition-of-trapezoids/
http://mrchasemath.wordpress.com/2011/02/18/why-i-hate-the-definition-of-trapezoids-again/
http://mrchasemath.wordpress.com/2013/08/12/why-i-hate-the-definition-of-trapezoids-part-3/
One reason Chase states for using inclusive definitions is that it simplifies proofs:
When proving that a quadrilateral is a trapezoid, one can stop after proving just two sides are parallel. But with the exclusive definition, in order to prove that a quadrilateral is a trapezoid, you would have to prove two sides are parallel AND the other two sides are not parallel.
Regarding some of the others to whom I refer regularly, Dr. Wu uses the inclusive definition:
"A quadrilateral with at least one pair of opposite sides that are parallel is called a trapezoid. A trapezoid with two pairs of parallel opposite sides is called a parallelogram."
while Dr. Mason uses the exclusive definition:
"A trapezoid is by definition a quadrilateral with precisely one pair of parallel sides."
(emphasis Dr. M's)
So which definition should I use for trapezoid? Well, this is a Common Core blog, so the definition favored by Common Core should have priority over all other definitions. The following is a link to the information that will appear on the PARCC End-of-Year Assessment for geometry:
http://www.parcconline.org/sites/parcc/files/ES%20Table%20Geometry%20EOY%20for%20PARCC_Final.pdf
And right there in the column under "Clarifications," it reads:
i) A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”
And that plainly settles it. The PARCC Common Core assessment uses the inclusive definition of trapezoid, and so it's my duty on a Common Core blog to use the Common Core definition. Of course, we notice that this is the definition given by PARCC -- but so far I've seen no information on what definition Smarter Balanced is using. It would be tragic if PARCC were to use one definition and Smarter Balanced the other. But as I can't say anything about Smarter Balanced, I will use the only definition that's known to be on a Common Core test, and that's the inclusive definition. The fact that this definition is already used by the U of Chicago is icing on the cake.
There is one problem with the inclusive definition of trapezoid, and that's when we try to define isosceles trapezoid. The word isosceles suggests that, just as in an isosceles triangle, an isosceles trapezoid has two equal sides -- the sides adjacent to the (parallel) bases. But in a parallelogram, where either pair of opposite sides can be considered the bases -- the sides adjacent to these bases are also equal. This would make every parallelogram an isosceles trapezoid. But this isn't desirable -- an isosceles trapezoid has several properties that parallelograms in general lack. The diagonals of an isosceles trapezoid are equal, but those of a parallelogram in general aren't. But the diagonals of a rectangle are equal. So we'd like to consider rectangles, but not parallelograms in general, to be isosceles trapezoids.
[2018 update: According to the old Moise text I mentioned above, all parallelograms are isosceles trapezoids while no kite is a rhombus. Otherwise his definition of trapezoid matches U of Chicago's.]
This dilemma is mentioned in the comments at one of the Chase links. It's pointed out that there are two ways out of this mess -- we may either define isosceles trapezoid in terms of symmetry, as Conway does, or we can use the U of Chicago's definition:
"A trapezoid is isosceles if and only if it has a pair of base angles equal in measure."
Some don't like this definition, because it violates linguistic purity -- the word isosceles comes from Greek, and it means "equal legs," not "equal angles." But as it turns out, it's a small price to pay to make the quadrilateral hierarchy and other theorems work out. And besides, any geometer who calls a nine-sided polygon a nonagon should just shut up about linguistic purity!
Note: Mr. Chase has made only one post in 2018. It's all about using pure Geometry to prove identities in Trig.
END
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