As I reflect on the lessons that I have posted so far, sometimes I think about how important Geometry is as compared to the other high school math classes. Many traditionalists talk about the importance of Calculus and getting as many students into AP Calculus classes as possible. Obviously, the main purpose of the class called Precalculus is to prepare students for Calculus, but of the courses before Precalculus, many would argue that it's Algebra I and II that matter the most for Calculus preparation, not Geometry. This is despite the fact that many problems in Integral Calculus involve the calculation of areas and volumes.
The push to get eighth graders into Algebra I classes is mainly so that these students would be on pace to take Calculus as seniors without having to take a class in the summer. If a student doesn't take Algebra I until freshman year, then a popular choice is to take Geometry the following summer to get back on the path to Calculus. As much as Geometry is my favorite math class to teach during the school year, I'd recommend that a student wanting to accelerate take Geometry the summer after freshman year, rather than take it as a sophomore and Algebra II the summer after that. Summer school classes move rapidly, and there is a huge jump in difficulty from Geometry to Algebra II. I once tried to tutor an Algebra II student in the summer, and the class didn't go well for him -- even though he had a C+ about halfway through the course, he was hoping for a B or better and ended up dropping the course. From that point on, I told any student considering taking Algebra II in the summer to wait until the fall to take the course.
So to me, Geometry in the summer is not a bad idea. But some schools trying to accelerate students go one step further -- they just jump directly from Algebra I to II and skip Geometry altogether. One school system well known for accelerating students is BASIS Charter Schools. This charter network does not exist in California -- its schools are in Arizona, Texas, and DC.
http://basisschools.org/basis-model/basis-5-12-curriculum.php
One notices that there is no Geometry course in the BASIS system. But the Saxon text is mentioned. I have mentioned the Saxon high school texts earlier on the blog -- we discovered that Saxon is actually an integrated course where the three traditionalist courses Algebra I, Geometry, and Algebra II are consolidated into two courses, which Saxon calls "Algebra 1" and "Algebra 2." So BASIS students actually do receive Geometry instruction after all. According to the link, the goal at BASIS is actually to get the students to BC Calculus by junior year:
5th grade: Saxon 76 or 87
6th grade: Pre-Algebra
7th grade: Algebra 1
8th grade: Algebra 2
9th grade: Pre-Calculus
10th grade: Calculus AB
11th grade: Calculus BC
12th grade: Post AP or "Capstone" Course
This is definitely a very accelerated pathway. Notice that the one Saxon text that I own -- Saxon 65 -- would be considered a fourth grade course at BASIS -- and even kindergartners are assigned the first grade Saxon text. Then again, notice that BASIS is a charter school -- that is, that parents choose to send their students there. So students go in fully aware that all classes, not just math, are well above grade level.
Personally, I'm not sure whether I could have handled the BASIS course load. I might have been able to survive BASIS math, but I'm not sure about other courses. For example, AP English appears in sophomore year, and students take three science classes -- biology, chemistry, and physics -- beginning in sixth grade! I know I'd struggle in those classes -- and that's assuming that I could even survive the elementary school, where every student takes Mandarin. It's hard enough to learn another language that uses the familiar letters A, B, C, but it's so much more difficult to speak a language that doesn't use our alphabet.
Getting back to Geometry, I point out that there are schools that don't require students to take Geometry before (traditionalist, not Saxon) Algebra II. Here in California, many of our community colleges have only Algebra I (or a placement test score) as the prerequisite for Algebra II, which in turn is the prerequisite for transfer-level courses. So this opens the door for schools to go from Common Core in 8th grade to Algebra I in 9th, skip over Geometry (or offer it as a non-required course in summer) to Algebra II in 10th, and then on to Precalculus and Calculus. As a Geometry fan, I am not sure whether I like this or not -- but it does satisfy the SteveH criterion of getting students to Calculus by senior year.
As I reflect upon my Geometry course so far, I notice that by view count, my most popular post by far, with 48 views, is Section 5-2 of the U of Chicago text, "Types of Quadrilaterals," posted back during the second week in October. I'm not quite sure why this lesson has the most hits -- I wonder whether it's due to readers searching for the inclusive definition of trapezoid, which we are starting to hear about more and more often.
I've heard that the PARCC uses the inclusive definition of trapezoid, so I decided to check the posted PARCC practice tests to find out whether this is true. There are only two mentions of trapezoids on the Geometry End-of-Year (EOY) Assessment, but neither trapezoid is a parallelogram. Instead, the inclusive definition is mentioned at the following link:
http://www.parcconline.org/sites/parcc/files/ES%20Table%20Geometry%20EOY%20for%20PARCC_Final.pdf
I give the inclusive definition because that's how the word is defined in the U of Chicago. The above link implies that PARCC also uses the inclusive definition. But what really matters is whether there are any questions on the actual PARCC test such that whether an answer is correct or incorrect hinges on whether the inclusive or exclusive definition is used. Then again, just because no such question appears on the practice PARCC, it doesn't mean that one won't appear on the real PARCC.
Meanwhile, I've been thinking about Dr. Katharine Beals again, and her criticism of Common Core and Cavalieri's Principle. Well, it's only fair to mention another post she wrote a few weeks ago mentioning Cavalieri. She writes that inclusion of Cavalieri's Principle is an example of a much greater fallacy, which she calls "Labels vs. Concepts":
http://oilf.blogspot.com/2015/03/when-it-matters-whether-1-is-prime.html
Even the best of multiple choice tests risk occasionally testing knowledge of labels rather than knowledge of concepts, and therefore, so that she wouldn’t feel stumped for stupid reasons, I found myself drilling her on things I really couldn’t care less about. Things like:
Is 1 a prime number?
Is 0 a natural number?
What is the “median” vs. the “mode”?
What does the expression |X| mean?
What is scientific notation?
So we see that Beals is strict as she wants to avoid questions that test for "labels," -- not only does she find Cavalieri's Principle unacceptable, but even "Is 1 a prime number?" is too much "label" and not enough "concept" for her liking.
Beals has an 8th grade daughter and is rightly concerned that the girl might get a low score on a test, only because she doesn't know all of the labels. That is, she would have otherwise received a high score had the test been 100% about content and 0% about labels.
On one hand, Beals does have a point here. In many of our classes, there are many English learners, and they might understand the concepts but not the labels. Back when I was student teaching, I met a transfer student from Vietnam. We know that many Asian nations are ahead of the U.S. in math, and so this student already knew most of the Algebra I that I was teaching. But the school didn't move him up to Geometry because he might have had trouble understanding the lessons -- not because of the alphabet (as Vietnamese, unlike Mandarin, uses the same many labels that appear in Geometry.
My problem is that this opens up a slippery slope. "Is 1 a prime number?" is a label question, but would "Is 7 a prime number?" be a label question? Would "Give the prime factorization of 24" be a label question? Can any question mention the word "prime" without being a label question? Likewise, "Is 0 a natural number?" is a label question, but would "Is 7 a natural number" be a label question? "Is sqrt(2) an irrational number?" -- is that a label question? At some point, we wouldn't be able to mention any mathematical term on a test, out of fear that doing so will make the term a "label."
And of course, in Geometry, the entire course is about labels. One thing that I often tell students is that understanding definitions -- of labels -- is half the battle. If one is trying to complete a proof and a long word appears in either the Given or Prove steps, then it's almost certain that "Definition of ..." that term will appear in the "Reasons" column of the proof.
It's easy to give an arithmetic test that's purely content -- just give a list of math facts and ask students to answer them -- and in Algebra a list of equations to solve is plausible. But is it possible to write a Geometry test that's 100% content, 0% labels? I notice that the questions that I've been giving from the U of Chicago text come from the "Questions on SPUR Objectives" -- where SPUR stands for Skills, Properties, Uses, and Representations. Surely the questions from the "Skills" section would be Beals-approved content questions. As it turns out, five chapters -- Chapter 3 (Angles and Lines), Chapters 8 through 10 (Area and Volume), and Chapter 15 (Circles) have the majority of the SPUR questions from their respective "Skills" sections -- and certainly we would expect all of these measurement-heavy chapters to have many questions on skills and content (and they are most closely related to Calculus to boot). On the other hand, Chapter 11, on Coordinate Geometry, doesn't have any Skills questions at all.
My most popular lesson is on types of quadrilaterals -- a label-dominated section, to be sure. Beals specifically mentions "kites" as an unnecessary label (along with Cavalieri's Principle and Common Core transformations like dilation, etc.) -- and I'm sure that she'd consider the question "Is a trapezoid a parallelogram?" to be just like "Is 1 a prime number?"
Then again, we notice that Question 3 of the "Skills" section of Chapter 5 asks students to draw a kite that isn't a rhombus. Even if identifying a kite is a "label," drawing a kite is a content skill, especially if one constructs the kite using straightedge and compass -- no matter what Beals says.
This is the time of year when schools are making hiring decisions for the next school year, and as a credentialed substitute teacher, I want to be ready. As many of you know, one part of the process is the demonstration lesson. Since my quadrilateral lesson is the most viewed lesson, I choose to prepare this lesson as a potential demo lesson. I have decided to change this lesson in order to make it a little more interesting.
What used to be a "bonus" question about a hierarchy in biology -- students are to classify the words man, cat, animal, mammal, primate, chimpanzee, lion, feline, plant into a hierarchy -- is now given as an opening question in order to get students thinking about hierarchies. I want to get the students thinking about something they hopefully already know -- a biological hierarchy -- and apply this prior knowledge to something new, the quadrilateral hierarchy. Notice that the debate about inclusive vs. exclusive definitions applies to biology as well -- for example, is man an animal? Many people use the word "animal" to mean non-human, but in biology, Homo sapiens is a member of Kingdom Animalia, so biologists are using an inclusive definition of "animal."
In giving this lesson, is there anything I can do about Beals's biggest fear -- that students who are strong at content but weak at labels will struggle with this lesson? Will English learners -- especially those from strong math countries like China and Vietnam -- be able to understand? I decided to drop "primate" -- a word that may be unfamiliar even to native English speakers -- and replace it with "canine," as more students should know what the dog family is. Other than that, I consider classification to be an important skill -- it's just as much "content" as it is "labeling."
I also like my lesson on Section 5-2 because it leads to my Section 5-3 lesson, on conjectures -- and this is the basis for the game that I often play as a substitute teacher. For lack of a better name, I've decided to name this game "Who Am I?" -- since the first two questions I ask are "What's my age?" and "What's my weight?" before moving on to the mathematical questions.
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