"It is this self-similarity in all its part, however small, that makes this curve so wondrous. If it appeared in reality it would not be possible to destroy it without removing it altogether, for otherwise it would ceaselessly rise up again from the depths of its triangles like the life of the universe itself."
-- Ernesto Cesaro (Italian mathematician 1859-1906)
This is the second page of the section "Fractal Worlds." Once again, the first page is blocked since it landed on Sunday, a non-posting day. But that doesn't matter -- we already know what fractals are because we read Benoit Mandelbrot's book two years ago. (Oh, and during this pages on fractals, let's include the "Benoit B Mandlebrot" label.) In the quote above, Cesaro is in the middle of describing a famous fractal, the Koch snowflake curve.
Here are some more excerpts from this page:
"This is the essence of fractals. So what is a fractal? Perhaps mathematicians have purposely avoided giving a definition to not restrict or inhibit the creativity of fractal creations and ideas that are formulating in this very new field of mathematics."
There are two pictures on this page. One is of the Koch snowflake described above. The caption is:
"The first four stages of the Koch snowflake. The Koch snowflake is generated by starting with an equilateral triangle. Divide each side into thirds, delete the middle third, and construct a point off that length from the deleted side."
This is the time of year that we often thing about snowflakes in our Christmas designs. And the second stage of the Koch snowflake even looks like a star of David, which fits the other holiday currently being celebrated, Hanukkah. So let's enjoy both holidays by linking to a picture of the Koch snowflake as it's being formed:
(Yes, perhaps there's some way to make this into a holiday project in our Geometry classes.) The other picture is of a different fractal. The caption is:
"The first three stages of the Peano curve. The Peano curve was made in the 1890's by repeatedly applying successive generation to a segment."
And let's go right back to Wolfram for the picture:
We'll be in this fractal section for the next several days, and so I'll continue to link to Wolfram and other sites for pictures of the fractals being described.
This is finals week, and as is my tradition, I post the final on the first day of finals week. Because it's a test day, it's also time for another "traditionalists" post.
Actually, Barry Garelick has been mostly quiet lately, but he did make one post a week ago today:
This post has drawn only one comment -- of course, the lone commenter is "You Know Who":
So here SteveH makes a sharp distinction between the fields of "math" and "math education." He makes it clear that he has much more respect for the former than the latter. The implication SteveH makes here, of course, is that if professors of "math" were in charge of developing curriculum instead of professors of "math education," then the resulting curriculum would be much more traditionalist.
Where’s the data that shows that what NCTM does works and what Taiwan does doesn’t work? What help have all of our successful (true) STEM students gotten from us parents and tutors over the years? Why is the successful path in high school math the traditional AP Calculus track?
Now we already know what SteveH means by the "traditional AP Calculus track":
8th grade: Algebra I
9th grade: Geometry
10th grade: Algebra II
11th grade: Precalculus
12th grade: AP Calculus
In particular, Integrated Math doesn't appear on SteveH's "traditional AP Calculus track" at all. I've noticed recently that some middle schools are offering Integrated Math I to honors eighth graders. In past posts, I pointed out that there's not much difference between Common Core 8 and Integrated Math I, especially on the algebra side. (There's a bit more geometry in Integrated Math I.) And so there's an almost seamless transition from Common Core 7 to Integrated Math I in eighth grade and then Integrated II as freshmen, onward up to senior year Calculus. Nonetheless, SteveH doesn't count any Integrated Math class as leading to Calculus.
What math classes, then, are students taking in Taiwan? Let's find out:
Under Year 8, it doesn't just say "Algebra." Instead, we see:
Decimals, Fractions, Percentages, Algebraic Expressions, Algebraic Equations, Area, Geometry -- angles, Geometry -- triangles, Pythagoras, Statistics, Statistic -- probability
And under Year 9, it doesn't just say "Geometry." Instead, we see:
Algebraic equations, Rules for indices/exponents, Scientific Notation, Area, Surface Area, Statistics, Statistic -- probability, Geometry -- congruence, Pythagoras, Trigonometry -- ratios
The following link is more direct and to the point:
"For example, math is taught as math, not algebra, geometry, and calculus."
This link couldn't have made it any plainer -- in Taiwan, students take Integrated Math. So if SteveH really wants the American education system to be more like Taiwan's, he should be in favor of more, not less, Integrated Math. Again, as far as I'm aware, the only Asian education system that uses SteveH's so-called "traditional Calculus track" is Vietnam.
Unfortunately, it’s much harder to recover from incompetent and slow math education in K-8 than it is for English. For many kids, it’s all over in math by 7th grade because of their low slope and lack of INSURING mastery of content and skills – two things specifically cherished by the ninth grade Pre-AP classes pushed by the College Board – that also pushes low slope CCSS! Pre-AP is a study in trying to reconnect low and bad CCSS math in K-8 with proper math in high school that leads to AP Calculus and a STEM career. Hello reality!
SteveH mentioned "Pre-AP" in an earlier comment. Again, we see here that to SteveH, if curriculum developers really wanted to prepare students for the AP, they would encourage traditionalist math in elementary and middle school. Then designating high school classes as "Pre-AP" is unnecessary.
Never mind that in the 50’s and 60’s I got to Calculus in high school with absolutely no help from my parents, but had to recently help my math brain son [emphasis mine -- dw] survive a K-6 brew of MathLand and Everyday Math [U of Chicago elementary math texts -- dw] to get to the high school AP Calculus track where I didn’t have to do a thing.
OK then, but why should students who aren't our "math brain" sons be forced to take classes that prepare them for STEM careers that they don't even want? Actually, SteveH gives a partial answer:
This is not only an attack of their ideas of a mixed-ability Project-Based Learning process, but an attack of their ideas and philosophy on content and skills (beyond their turf) with NO opt-out choice.
Here SteveH implies that he wouldn't mind Project-Based Learning provided that the students and parents are given the "opt-out choice" of taking traditionalist math instead. To be sure, he and his math brain son would have easily chosen the traditionalist track.
Last year, I taught at a charter middle school with a strong Project-Based Learning program, the Illinois State text. (There's no need to repeat what's in the Illinois State text -- it suffices to know that there's nothing in the Illinois State text that SteveH would approve of.) Of course, my school didn't have an opt-out program for several reasons -- not the least of which was that our school was too small to allow students any choices at all.
There were at most fourteen eighth graders at any one time last year. Suppose we wanted to offer the eighth graders a choice -- Illinois State Math 8 or traditional Algebra I, and let's say that seven of them chose each text. Then we would have been forced to hire another teacher to teach Algebra I while the rest of the eighth graders were learning Illinois State Math 8 -- and this at a school that couldn't even hire a science teacher!
On the other hand, suppose our school didn't use the Illinois State text, and instead required all eighth graders to take traditional Algebra I. I'm sure that SteveH would be defending the students who don't desire a STEM career and wish they were given an "opt-out choice" out of Algebra I. (Yeah, right!)
Since Barry Garelick hasn't been writing much lately, let's go back to the Joanne Jacobs website, where I found the following:
Stop teaching calculus to high school students unless they’ve mastered foundational concepts and are ready to pass a college-level class, writes Jeffrey Forrester. As a Dickinson College math professor, he sees students who passed calculus in high school, then fail the college’s pre-calculus placement exam.
Well, it's traditionalists like Garelick and SteveH who keep wanting to push students into Calculus in high school. So these problems with high school Calculus are on them, not me.
I used to go to the Jacobs website to read the traditionalist "Bill." I notice that there's a "Bill Parker" in the comments, but I don't know whether it's the same Bill:
Calculus is nothing more than advanced algebra and trig, and often deals with variable rates of change. That being said, if a student is taking calc in high school and can’t pass the pre-calc exam in college, sounds like the student didn’t master algebra and/or trig.
Here is basic calculus (a limit):
Lim (x -> 5) (x^2 – 25) / (x – 5)
If a student who has taken and passed calc in high school cannot solve this, they don’t know calculus, much less algebra.
Another case where grades don’t match test scores…
Of course, we already know that if Bill had his way, the F grade would be much more common. For example, the chart at the link above shows 30.9% of students earning a 1 on the AP, so if Bill had his way, 30.9% of Calc students would have the letter F on their report cards.
Another commenter in this thread writes:
I believe that students should be encouraged to take Calculus in high school.
So far, Arvanites sounds like a traditionalist, but I don't want to pass judgment so soon. Let's look at what else he writes:
These students are still being exposed to the ways of thinking that calculus requires, its interpretation of rates of change, and its emphasis on showing that the infinite and the infinitesimal are one and the same. Even some students taking calculus at the college level have not mastered foundational concepts. In fact, most college students who have difficulty with calculus have weak algebra and trig skills. The concepts of calculus are not extremely difficult to explain (such as why the average rate of change must equal the instantaneous rate of change at some point in the Mean Value Theorem); however, some students’ weak algebra and trig skills prevent them from being able to do the manipulation required to apply the concepts to particular functions.
But if the students struggle with Algebra, then why shouldn't we place them in an Algebra class rather than a Calculus class?
Finally, here's one more link at the Jacobs website:
Before the Common Core, did teachers ask students to memorize facts without understanding? That’s a myth, argues Sandra Stotsky, who helped develop Massachusetts’ pre-Core standards, in the New Boston Post. The best pre-Core standards expected students to understand ideas.
I only point out this post because Sandra Stotsky -- a known traditionalist and opponent of the Common Core -- quotes Barry Garelick later on:
Barry Garelick blew up that false charge earlier this year by pointing out what had been in his old arithmetic book (Arithmetic We Need by William A. Brownell) from the 1950s, as well as in others: “With respect to the math books of earlier eras, they started with teaching of the standard algorithm first. Alternatives to the standards using drawings or other techniques were given afterwards to provide further information on how and why the algorithm worked.”
The comments mention other old texts favored by traditionalists, including Saxon math -- with the emphasis on the older (and more integrated) versions of the text. (Once again, for opponents of Integrated Math, let's just call it "Saxon Math" and use the old integrated texts -- would traditionalists like Integrated Math more then?)
But now I must post the first semester final exam. Here are the answers to the questions on the final:
BBBCA AADAB BBADA CADBC ABBAA BDDAC DDBCC ADBBA BDABC ADDCA