Introduction: Traditionalists and the CBEST
I've stayed away from the traditionalists for most of the summer. But there's an article over at the Joanne Jacobs website, first posted back on Pi Approximation Day, and I simply can't ignore how some of the traditionalist commenters responded.
https://www.joannejacobs.com/2021/07/would-be-teachers-fail-licensing-tests/
Only 45 percent of would-be elementary teachers pass state licensing tests on the first try in states with strong testing systems concludes a new report by the National Council on Teacher Quality. Twenty-two percent of those who fail — 30 percent of test takers of color — never try again, reports Driven by Data: Using Licensure Tests to Build a Strong, Diverse Teacher Workforce.
And even though this refers to licensing tests in many different states, California is specifically mentioned later in the article:
California, which refused to provide data for the NCTQ study, will allow teacher candidates to skip basic skills and subject-matter tests, if they pass relevant college classes with a B or better, reports Diana Lambert for EdSource.
The California Basic Skills Test (CBEST) measures reading, writing and math skills normally learned in middle school or early in high school. The California Subject Matter Exams for Teachers (CSET) tests proficiency in the subject the prospective teacher will teach, Lambert writes.
Anyway, I mentioned the CBEST in one of my COVID What Ifs earlier this summer -- and as I pointed out in that post, I failed the CBEST on my first attempt. Of course, I passed the math section with flying colors, and I did well on the reading section. The problem I had was with the writing section.
For the writing section, we had to write two essays in the allotted time. We were given the topics for the two essays. I had no problem with one of the two topics, but I struggled with the other. This was over a decade ago, so I don't remember the exact topic, but it was something like "write about a time when you disagreed with someone famous." And I simply couldn't think of anything to write about.
There was about half an hour left in the testing period, but I didn't know what to write about. I ended up stalling for most of the time, then wrote a few halfhearted sentences about something -- nowhere near the length of a proper essay. I knew that I'd have another opportunity to pass the writing section.
My first attempt was in December, and my second attempt was in February. This time, I found both required topics easy to write about -- and I could use the entire testing period on writing, since I'd already passed the math and reading sections. So this time, I finally passed the writing section, and so I was ready to proceed with my credential.
But let's see what the traditionalists have to say about those who don't pass the CBEST on their first attempt -- and yes, this includes some familiar names.
GL:
Passing the California CBEST is about the equivalent of earning. 900 on the English/Math parts of the SAT (a pretty low bar for a college grad.).
Bruce William Smith:
To avoid our students meeting such teachers, those of us in schools should examine the equivalent of CSET (CBEST should have been tested far earlier, around the end of junior high school) subjects before allowing any teachers’ college trainees to access our classrooms for internships: if states like California refuse to safeguard academic standards, schools offering internships will have to do so instead.
Darren (Right on the Left Coast):
A high school graduate should be able to pass the CBEST •without trying or practicing•. A potential teacher, a college graduate, should pass that test as easily as swatting a fly. If it were up to me, those who fail the CBEST would not be allowed to retake it for a year.
And the other comments are similar. They are all basically saying the same thing -- because I didn't pass the CBEST on my first attempt, then I'm not smart, I'm unable to write at an eighth grade level, and, most importantly, I'm unfit to be a teacher.
And this is all because one of the two particular writing topics at that December examination simply didn't resonate with me. The February topics did resonate with me, which is why I ended up passing the test in February. If only I hadn't taken the test at all in December, so that February was my first attempt instead, then I would have passed it on my first attempt.
Suppose your CBEST topic were something like "What is your favorite color and why?" and you don't have a favorite color. OK, I grant that you might be able to name any random color and then give enough "reasons" why you like that color to pass the test. Maybe there's something else to fill in that blank, "What is your favorite __________ and why?" in a way that it's not so easy to answer on a timed essay unless you already have a favorite "blank."
Or how about this -- since traditionalist Darren Miller is "Right(-wing) on the Left Coast," suppose the CBEST topic were something that right-wingers might struggle with, such as "Explain why equity is more important than equality." I wager that Miller wouldn't pass that test "as easily as swatting a fly."
Ironically, if I had to write an essay on that December topic today, I could do so easily. I can easily write about someone I disagree with -- namely the traditionalists.
On the other hand, notice that the solution here in California is to eliminate the CBEST and CSET as requirements for a teaching credential. I can't agree with this -- and my reason for disagreeing is the bottom line. After all, I've already passed the CBEST and CSET, and so if those tests were no longer required, I'd lose my advantage. I'd suddenly be competing for teaching jobs with candidates who haven't passed that exam.
This forces me in the middle, defending the status quo. I'd rather keep the tests and allow multiple chances to pass them. I oppose both requiring students to pass on the first try (or force students to wait a year before retrying) and dropping the tests altogether.
A Rapoport Geometry Problem
It's also been some time since I posted Geometry problems from the Rapoport calendar. Once again, it's not that she hasn't included Geometry on her calendar -- it's just that most of the Geometry problems keep landing on non-blogging days. This time I finally catch some Geometry.
Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:
Find x.
As usual, all of the given info appears in an unlabeled diagram, so I must label all of this myself. In Triangle ABC, D is on AC so that ray BD is an angle bisector. AB = 68, CB = 40, AD = 51, CD = x.
As it happens, it's a straightforward example of a theorem that doesn't appear in the U of Chicago text -- nor in most other high school Geometry texts -- but it does appear on Rapoport calendars:
Angle Bisector Theorem:
If a ray bisects an angle of a triangle, then it divides the opposite side of the triangle into segments that are proportional to the other two sides.
Once we know this theorem, it's easy to set up and solve the proportion:
51/x = 68/40
x = 30
So the desired length is 30 -- and of course, today's day is the thirtieth.
But if we don't know this theorem -- and most high school Geometry students don't -- then this problem is difficult. So let's try to prove the theorem. The fact that it involves proportional sides of a triangle is a big hint that it has something to do with triangle similarity. And dividing a side proportionally reminds us of the Side-Splitting Theorem in particular.
Keeping this in mind, here is a two-column proof of the Angle Bisector Theorem:
Another Rapoport Geometry Problem
There were several other Geometry problems on the Rapoport calendar this week. This one was particularly tricky for me.
Yesterday on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:
Find AD + AE.
This time, the diagram is partly labeled -- only A, D, E are specified. I'll fill in the rest:
In Triangle ABC, D lies on AC, E lies on AB. AD = CB = 16, DC = 10, Angle A = C, Angle AED is right.
Well, we are asked to find AD + AE and we're given AD, so the main task is to find AE. But I was completed stumped as to how to find AE, and unfortunately I had to go to Twitter to see how others are faring on this problem. They came up with several different ways to solve it, but all of them involve drawing an auxiliary line that never occurred to me. This is the altitude to AC through B. Let's label the foot of that new perpendicular line F.
We are given that Angle A = C, and we also have Angle AED = CFB = 90 and AD = CB = 16. Thus Triangles AED and CFB are congruent by AAS, and so AE = CF by CPCTC.
But what is CF? We notice that for Triangle ABC -- which is isosceles since its base angles A and C are congruent (Converse Isosceles) -- the altitude BF that we drew in is also a median. So CF is exactly half of AC, which we know to be 16 + 10 = 26. Thus CF = 13, and so AE = 13.
Now we can finally add the two sides we need -- AD + AE = 16 + 13 = 29. Therefore the desired sum is 29 -- and of course, yesterday's date was the 29th.
OK, that's enough of Rebecca Rapoport for now. Let's get back to Eugenia Cheng's book.
Molly's Mathematical Mysteries 7: Mixed-Up Library
Let's begin Cheng's seventh adventure:
"Molly squeezes through the exit door and finds herself in a gigantic library. Polished wooden bookcases are bursting with books in all shapes and sizes."
There is another note for Molly on this page:
"These are special books that you can read in any order you like, to make a different story. Can you work out how many different possible orders there are? Turn the wheels to make this number, and you'll be able to reach the key to the next room."
There are five books that fell onto the floor, so the question is asking us how many different ways we can order these five books. Before we return to the math, I admit that this is quite interesting -- here we have a five-part story, and we can read the parts in any order yet they still form a coherent story.
For example, Molly's adventures can almost go in any order without disturbing the story -- yes, her room turning inside-out has to be #1, but she can go through 2-6 in any order before reaching the library here in #7. So let's say she goes 4-5-6-2-3 -- to the hall of endless doors, the boiler room, and the carpet room before going up the impossible staircase and through the garden maze. This is an example of an interchangeable story, but there could be a much more interesting one where, say, there could be either a happy or a sad ending based on which of the five stories is last. (This seems like the sort of thing Douglas Hofstadter might have attempted with one of his Godel, Escher, Bach stories with the Tortoise and Achilles.)
OK, so let's get to the author's explanation:
"There are five books, so there are five possibilities for which one you read first. After that there are only four to choose from for the second book, then three left for the next book, then two, and then there's just one left to be the last book. We can multiply these together to get the total number of possible orders: 5 * 4 * 3 * 2 * 1."
Even though Cheng doesn't use the term, this number is 5 factorial. And so our young readers can figure out that the total number of orders is 5! = 120 -- and that's the number that Molly should enter on the combination lock.
And there's also a challenge for the reader:
"The green bookcase is organized by the patterns on the book's spines. Can you spot the odd one out in each row?"
Which one doesn't belong? This one is too visual for me to describe on the blog -- suffice to know that the extra book has an extra symbol on its spine -- two of some symbol instead of one.
Molly's Mathematical Mysteries 8: Symmetry Garden
Let's begin Cheng's eighth adventure:
"The bookcase creaks open to reveal a luscious green garden. Molly has made her way right through the house!"
There is another note for Molly on this page:
"To open the gate, find one butterfly that is not symmetrical. It should match the butterfly on the gates -- and make them open!"
Symmetry is, of course, very important in Geometry. On this page, all the butterflies are folded up, and so the young reader must unfold the butterflies to determine whether their right halves match up with their left halves.
The author explains the symmetry of the butterfly:
"There are several types of symmetry. One kind is where you can fold something in half and it's the same on both sides -- this is called reflectional symmetry."
She goes on to define rotational symmetry -- as well as a third symmetry that we often overlook:
"There's also a kind where you can move something sideways or in another straight line direction and it looks the same, like a fence or the pattern of a brick wall -- this is called translational symmetry."
Objects that have true translational symmetry are necessarily infinite. Tessellations -- which we've seen earlier in Cheng's book -- have translational symmetry.
And so there is a symmetry corresponding to each of the Common Core isometries. Indeed, an object has T-symmetry for some transformation T if its image under T is itself. Technically, this means that "glide reflectional symmetry" is possible. But since the composite of any glide reflection and itself is a translation, any object with glide reflectional symmetry must also have translational symmetry -- and hence is an infinite object.
And there's also a challenge for the reader:
"Can you find all three types of symmetry in the garden?"
We already know that the butterflies have reflectional symmetry. Some of the flowers have rotational symmetry, and the gate itself is translation-symmetric. We'll go through the gate in my next post.
Lemay: More on Chapter 17
In each of these posts, I'm continuing to return to some of the Lemay chapters that we passed over. I now want to look at Chapter 17.
In this chapter, Lemay introduced exceptions. The problem was that all of her examples were just snippets of code, with no complete executable programs.
Let's try a simple Pythagorean Theorem program. We have a method that takes three integer sides and determine whether they are the sides of a right triangle. But this method can throw exceptions, such as some of the sides being zero or negative, or the three sides not being the sides of a triangle (that is, violating Triangle Inequality).
public boolean IsARightTriangle (int a, int b, int c) throws NonPositiveSide, NotATriangle {
if (a<=0 || b<=0 || c<=0)
throw new NonPositiveSide();
if ((a+b<=c) || (a+c<=b) || (b+c<=a))
throw new NotATriangle();
return ((a*a+b*b==c*c) || (a*a+c*c==b*b) || (b*b+c*c==a*a));
}
public static void main (String args []) {
PythagoreanTest x = new PythagoreanTest();
try {
if (x.IsARightTriangle(3, 4, 5))
System.out.println("3-4-5 is a right triangle");
else
System.out.println("3-4-5 is not a right triangle");
if (x.IsARightTriangle(1, 2, 3))
System.out.println("1-2-3 is a right triangle");
else
System.out.println("1-2-3 is not a right triangle");
} catch (NonPositiveSide np) {
System.out.println("Not all sides are positive");
} catch (NotATriangle nt) {
System.out.println("This is not a triangle");
}
}
}
class NonPositiveSide extends Exception {
public NonPositiveSide () {}
public NonPositiveSide (String msg) {
super(msg);
}
}
class NotATriangle extends Exception {
public NotATriangle () {}
public NotATriangle (String msg) {
super(msg);
}
}
Output:
This is not a triangle
Conclusion
Track and Field at the Olympics have finally started, and I'm definitely enjoying it. At the time stamp of this post, I've watched many of the preliminary heats in the sprint events. The first final with medals awarded is in the 10,000 meters. But the 10K final first aired on Peacock streaming, which I don't have, and are airing on delay on NBC, starting just after the timestamp of this post.
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