In Chapter 14 Mandelbrot focuses on a famous fractal -- the Sierpinksi triangle -- and several related fractals of various dimension. The classic Sierpinkski triangle can be doubled by taking three copies of it, and so its dimension is log_2(3) or log(3)/log(2), which is 1.5850.
Mandelbrot goes go on to describe other fractals -- for example, the Menger sponge, which is somewhat like the Cantor middle-thirds set but starting with a cube rather than a segment. The dimension of the Menger sponge is log(26)/log(3) or 2.9656.
In the "Scaling Fractals" section, since scaling is based on dilations, we can use the formulas for similarity dimension to determine the dimension for each fractal. The "mathemusician" Vi Hart -- whose videos I've linked to on the blog before -- created three videos about fractals -- in particular, the dragon fractal. I only provide the link to the first video here:
If you watch all three videos, see what Vi Hart does here. At the start of the first video, Hart says that she started doodling fractals because she was bored during an Algebra II lesson on logarithms. But then, at the end of the third video, she wonders what dimension the fractals have, then gets stuck because she can't solve the equation for similarity dimension -- because it requires logarithms!
Meanwhile, today is the last day of Unit 2 and the day of the test. I keep mentioning that the test is today because it's the last day before the three-day Columbus Day weekend. Officially, of the two districts where I work, only one of them actually observes Columbus Day. The other holds a Staff Development Day on Monday, so that there are no students and no lessons. So technically speaking the district does not observe a Columbus Day Holiday -- but in practice, many students and teachers alike just say that there's no school on Monday because it's Columbus Day. And so on the blog I will continue to call it "Columbus Day." Notice that this is the year that the holiday falls on October 12, referring to the actual date (October 12, 1492 on the Julian calendar) when the explorer first landed in the Bahamas. (No, it's not the date of his birth, which is in fact unknown. Also notice that the Canadians always celebrate Thanksgiving on the same Monday as Columbus Day.)
Because I'm giving the test today, this is a good time to return to the concerns of the traditionalists and other opponents of the Common Core. First of all, earlier this year, during the "How to Fix Common Core" series, I mentioned the "Consistency Core" proposal. Consistency Core is defined to be the educational curriculum of the children of the president, or another high-ranking official in his cabinet, like Arne Duncan or his interim successor John King. (Note: From what I've heard about Dr. King so far, he is strongly pro-Common Core. His children attend public schools in Maryland, which I believe is a Common Core state.)
I bring this up again because in the news this week, there has been much speculation about First Daughter Malia Obama and where she will attend college -- remember, I mentioned that she will graduate before her father leaves office. It was actually part of my speculation about on which of the three math tracks at her school (Sidwell Friends) the First Daughter has been placed. Schools that she is considering include Stanford and two Ivies her parents attended -- Harvard and Princeton. Recall how the three math tracks differ on how Geometry is taught -- the lowest track uses Michael Serra's Discovering Geometry with proofs saved until the end, the middle track intersperses proofs through the year, and the highest track emphasizes proofs strongly. One would think that Malia would be on the middle or higher tracks if she's applying to the Ivy League -- but then again, some people point out that no school would turn down a First Daughter regardless of her academic record.
I begin by mentioning a line that may have been spoken by the world's most famous scientist, Albert Einstein, that is frequently quoted in Core debates:
Everybody is a genius. But if you judge a fish by its ability to climb a tree, it will live its whole life believing that it is stupid.
Whether Einstein actually said this or not is immaterial. What matters is that this line is invoked to be a criticism of the Common Core -- especially the tests. So in particular, the tree-climbing test is supposed to be the PARCC or SBAC, and the fish represents a student who doesn't perform well at the tested subjects (tree-climbing), but is great at a subject that isn't tested (such as swimming). This quote is often accompanied with a picture of not just the fish, but a monkey, elephant, dog, and various other creatures being asked to climb the tree. Obviously it's unreasonable to expect any of the animals other than the monkey to climb the tree, and so likewise it's unreasonable to expect most students pass the Common Core tests.
The problem with this analogy is that it's incomplete. So let's finish this analogy by filling in some of the missing details.
First, we ask ourselves, why would anyone want to test for climbing a tree, anyway? Well, let's say it's because tree climbers are known to make ten times as much money as non-climbers. Now a picture is starting to form, as in the real world, STEM majors are said to have more money-making potential than non-STEM majors. And so the tree climbing test corresponds to a Common Core exam that tests for possible STEM skills -- let's say it's the junior-level math test for PARCC or SBAC, which goes up to Algebra II (the most commonly failed Common Core tests). Just as Einstein's fish believes that it's stupid because it can't climb a tree, students feel that they are stupid after failing the junior-level math test, even if they have talents other than Algebra II. Yet anyone who can't pass an Algebra II test is unlikely to be successful at a STEM career with its higher salary potential.
So now we can see the problem. If we give the tree climbing test, then we are anti-fish because a fish can't climb the tree and so will fail the test. But if we don't give the tree climbing test, then we are still anti-fish because now we're saying that we don't want fish to make ten times as much money! So no matter what we do, we're anti-fish!
Now let's extend the analogy at little further, and ask ourselves, why would tree climbers make ten times as much money anyway? Here's an idea -- we say that tree climbers are able to pick the apples from the top of the tree and sell them. All the other animals benefit because they can all eat the apples that are picked -- including the fish, in this analogy. If we get rid of the tree climbing test, the fish won't feel stupid anymore, but they'll feel hungry.
Of course, this analogy isn't exact because some animals are carnivores and can't eat apples. But in the real world, everyone does benefit from the fruits of STEM -- which include technology that makes our lives easier and fun, as well as improved medicine (such as a cure for cancer). We can imagine that the apples correspond to Apple products (as in the Silicon Valley Apple).
So our goal isn't to make the fish feel stupid -- it's to make the monkey feel good enough to climb the tree and pick the apples. It's the apples that matter, not feelings. But now here's another problem -- let's say that the tree climbing test is designed to pass anyone who can climb five feet. This allows for other animals -- let's say the cat -- to pass the test. But the average apple tree is about ten feet tall, not five feet. So the result is that no one can eat any apples, because no one climbs the ten-foot apple tree.
In the real world, this corresponds to a major traditionalist concern. Here a five-foot climb denotes the ("pseudo") Algebra II that appears on the Common Core test, while a full ten-foot climb corresponds to the Calculus that one would need to take in order to be well-prepared to produce apples and Apple products.
We see now that the five-foot tree climbing test is a compromise between having the ten-foot test that would allow the monkey to pick the apples and just getting rid of any test that the fish may fail. But we see that there are still no apples and the fish still feels stupid. The compromise ends up benefiting no one except maybe the cat.
Here's what I say about how to solve this problem -- if tree-climbing benefits all the animals in that they can all eat the apples, then the test should demand climbing a tree high enough to pick them. It doesn't matter if 90% of the animals fail the test as long as the 10% who pass it can feed the 90%.
But the key word there is if. The monkey may pick all the apples from the tree, but if the fish, after having failed the tree-climbing test, finds that it has no job prospects, then it won't be able to buy any of the apples and still goes hungry. What we need is a test that allows the monkey to pick the apples yet still allows the fish to use its special skills to get a well-paying job so that it can buy the apples.
Similarly, we want to prepare students for Calculus so that they can work towards a STEM career and create more Apple products. But it must be done in a way so that the 90% who can't pass a Calculus test still have some job prospects so that they can purchase more Apple products. I'm talking about the students who don't do well in math -- the ones who'd rather doodle during math class than learn about logarithms. How will they be able to succeed in life?
Now suppose our fish is allowed to take a swimming test, at which it excels, and is able to get a good job as -- we might as well say the obvious job that requires swimming -- a lifeguard. But in our animal world, lifeguards make only a third as much as apple pickers. So although the fish is better off now than it was when tree climbing was the only way to pass, it still makes only a third as much money as the monkey. So are we still anti-fish because the monkey is making three times as much money as the fish, even though each animal is working at a job that's tailor-made for it? This is a complex question that transcends education and enters the realm of economics and politics. (For example, replace the animal names with "male" and "female" human and suddenly this becomes a discussion of the gender wage gap.) I do not know the best answer to that question.
We can keep coming up with more and more analogies until the cows come home. Analogies are nice to understand the problem, but what we need are solutions. And even though I've spent all summer trying to come up with solutions (like Consistency Core), they are only partial solutions. How can we get as many students as possible into Calculus without alienating the students who can't do Calculus?
I've been discussing the traditionalist Dr. Katharine Beals and her century-old Wentworth text for algebra many times here on the blog. Well, today she finally posted an example for Geometry:
And this time, she doesn't even have to go as far back in time to find a good Geometry text. Dr. Beals chooses Weeks & Adkins A Course in Geometry, published in 1970, and compares it to the New York State Regents exam. She writes that the only proof that appears on the Regents exam is to show that the sum of the exterior angles of a triangle is 360 degrees. But in the Weeks & Adkins text, there are hundreds of pages involving proofs.
We know that many New Yorkers complained that the Algebra I Regents test was too hard. If we go back to the Stop Common Core New York page that I linked to earlier -- the one where so many parents were upset the Algebra I test -- Geometry was only mentioned thrice, and two of those concerned how the freshmen who failed last spring's Algebra I Regent will fare in Geometry as sophomores this year. Only one comment is about the Geometry Regents test itself:
“My daughter took the CC geometry regents June 2, she said the wording was terrible, that you need to be an English major to understand.”
And I doubt that this refers to the one question that appears on the Beals blog. There's nothing unclear about "Prove the sum of the exterior angles of a triangle is 360 degrees." (But then again, I wonder whether the terrible wording concerns a question about a "label" that Beals would criticize as well.)
Let's compare Beals and her Weeks & Adkins text to what's posted here on my own blog. She writes that the page of proofs that she posted is about one-sixth of the way through the book. That is, it would be taught about one hexter after the start of the year -- in other words, right now according to the blog calendar (which is probably why she posted this particular page). We see that this page mostly concerns the Isosceles Triangle Theorem -- Lesson 5-1 in the U of Chicago text. According to Beals, the Corresponding Angles Postulate hasn't appeared in the Weeks & Adkins text yet, and so all the proofs on this page must be valid in neutral geometry.
Now we're finishing Chapter 4 today, so it might appear that Lesson 5-1 is around the corner. But recall that according to my plan at the start of the year, I'll cover Unit 3 on Rotations before moving on to Chapter 5 (which is my Unit 4). And according to that plan, my Rotations unit is when I'll cover SSS, SAS, and ASA Congruence. I assume that like most pre-Common Core texts, Weeks & Adkins has covered at least SAS Congruence by now, and used it to prove Isosceles Triangle Theorem. And so I must admit that the course posted on this blog -- while it may be well-ahead of the Geometry classes at the schools where I'm subbing -- is still behind the Weeks & Adkins text.
The proofs that appear on this page may be nearly equal in difficulty with the proofs that appear in Dr. Franklin Mason's text. The Isosceles Triangle Theorem appears in Chapter 3 of Dr. M's text. (The level of rigor that Beals proposes is certainly more in line with the middle or higher proof-based Sidwell tracks mentioned earlier in this post.)
Some parents may complain that there shouldn't be any proofs on the Regents -- that is, making their sophomores complete the proof is like forcing a fish to climb a tree. But Beals is upset that there are not enough proofs on the test -- like climbing a five-foot tree instead of a ten-foot tree.
There's another post I want to discuss on from the Beals blog. In this post, Beals asks, why do some history teachers hate history? History class ends up being the boring memorization of dates, such as "In 1492, Columbus sailed the ocean blue." But in the comments, the discussion ended up turning into math yet again:
Some traditionalists keep pushing Calculus that they forget that some students don't need to learn the subject at all. One anonymous commenter wrote:
I am a recently retired community college math teacher who does NOT hate math. However, I do have a disagreement with the continual increase in the level of algebra-based math required for a student to get a four-year degree. I believe that all students who are capable of college-level work in any subject (except for a very few with specific disabilities in math) are also capable of learning Elementary Algebra, or first-year high school algebra. This should be required of the student BEFORE starting at a college of any level. However, the requirements of many four-year colleges do not stop with first-year algebra. They include algebra II, college algebra, and sometimes precalculus. I don't understand the necessity for any of these subjects for students whose degrees are in non-technical areas. There are many areas of mathematics besides algebra which can be just as rigorous in the sense of requiring thought and concentration, while providing more applicability for the students. I am thinking particularly of probability and statistics, in which most Americans are woefully under-educated. A rigorous probability and statistics course can reinforce the algebraic material from Algebra I while giving students a tool they can apply in their lives outside of education as well as in most of their chosen fields. Do not denigrate a so-called "no algebra" degree track without looking more carefully at the details.
(Actually, if I were in charge of the world, I would also roll back the more recent requirements that all high school students must pass Algebra II to graduate from high school. Talk about putting a requirement in place using an argument that confuses correlation with causation. You see, the people in charge of education policy were not adequately educated in statistics.)
Yes, it's correlation vs. causation that's at the heart of the Common Core debate. Since learning how to climb trees is correlated with making more money, it's easy to think that it's the cause and thus we would teach everyone, including the fish, how to climb a tree.
The commenter implies that perhaps non-technical (non-STEM) majors only need to take Algebra I in order to be admitted to a four-year college. If we look at the replies, we see that this leads to a debate as to what constitutes "Algebra I." Are quadratic equations considered Algebra I? Are conic sections considered Algebra I? (When I took algebra, the former was the very last thing we learned in Algebra I, while the latter was in Algebra II.)
I was considering showing up Beals here and demonstrating that the Geometry that I post on my blog can be just as rigorous as that which she posts on her blog. If I decide to do that, it will have to be a little later, as Chapter 5 is scheduled for November. But today is already reserved for my Chapter 4 Test -- oh, that's right, the test! This is what I wrote last year about the test:
Here is my version of the Chapter 4 Test. In this post I explain the answers.
1. Students should draw a point P' on the other side of line m, the same distance as P, but in the opposite direction. In other words, P' should be drawn so that m is the perpendicular bisector of
2. Students should draw the perpendicular bisector of
3. The measure of the angle is 48 degrees -- exactly double that of the given angle.
4. The symmetry line is the line containing the angle bisector of the given angle. This follows directly from the Angle Symmetry Theorem. The hard part about writing this quiz is that I couldn't include every single property or theorem on the Quiz Review. Students should be familiar with all of the properties of reflections (in other words, the Reflection Postulate) as well as of the results derived from these properties (the theorems).
5. The orientation is counterclockwise, of course, since reflections switch orientation.
6. Reflections preserve distance.
7. There are two pairs of angles -- angles B and C, and angles BAD and CAD.
8. The conclusion follows from the definition of symmetry line and the fact that reflections preserve angle measure. Technically, we need the definition of symmetry line -- we can't just use the Reflection Postulate directly because nowhere in the picture is any mention of a reflection. So we need the definition of symmetry line to get from "line AD is a symmetry line" to "the reflection image of triangle ABC is triangle ACB." But I'd accept it as a correct answer if a student only mentioned that reflections preserve angle measure, since a full two-column proof isn't required.
9. A square has four symmetry lines -- one horizontal, one vertical, two diagonal. This is actually not proved until later on -- but once again, a full proof isn't required.
10. The reflection image is (-a, b). There were actually examples of this in the Exercises, but only where the coordinates were numbers, not variables. If the students are confused, a teacher can change this question to something like the reflection image of (1, 2), and even encourage the students to draw in a quick graph to demonstrate it.
Hopefully I didn't make this quiz too difficult for the students. I hope that all the readers of this blog enjoy your three-day weekend -- if your school observes it. My next post is Tuesday, October 13th.