"Applications for fractals range from acid rain to zeolites, from astronomy to medicine, from cinematography to cartography to economics, and on and on."
This is the third page of the section "Fractal Worlds." So let's continue to use the label "Benoit B Mandelbrot" as we read more about fractals.
Here are some more excerpts from this page:
"Mathematically speaking, a fractal is a form which begins with an object -- such as a segment, a point, a triangle -- that is constantly being altered by reapplying a rule ad infinitum. The rule can be described by a mathematical formula or by words.
"One can think of a fractal as an ever growing curve. It is constantly developing. It was equally fortunate that Benoit Mandelbrot, in the spirit of the early mathematicians, studied and expanded the ideas and applications of fractals almost singlehandedly from 1951-75. How astonished the adventurous mathematicians of the 19th century, who first dared to look at these ideas most considered monstrous and psychopathic, would be to see the wondrous geometry of fractals in motion. In essence it is this idea of growth or change that links fractals dramatically to nature."
There is one picture on this page, of a particular fractal. Its caption reads:
"The first four stages of the Sierpinski triangle. Begin with an equilateral triangle. Divide it into four congruent triangles as shown and remove the middle one. Repeat this process to the smaller triangles formed ad infinitum. The resulting fractal has infinite perimeter and zero area!"
Once again, let's ask Wolfram to show us the Sierpinski triangle:
http://mathworld.wolfram.com/SierpinskiSieve.html
Today is the second day of finals week. It has been my tradition on the blog on the second finals day to explore the Math Twitter Blogosphere, or MTBoS.
But I must admit that I feel somewhat detached from the MTBoS. Since I left my old school, I am no longer a math teacher, and so I can't truly be a member of MTBoS. If I'm ever hired again to be a math teacher, I'll rejoin the MTBoS once again.
Moreover, last year during various MTBoS challenges, I discovered the blogs of several other middle school math teachers. But now I'm no longer a middle school teacher, so those blogs are no longer as relevant to me. Of course, I did link to the blog of Fawn Nguyen -- a middle school teacher and the new Queen of the MTBoS -- on Friday, so I guess I started thinking about the MTBoS a little early.
In the past, the main blogger I'd link to on MTBoS Day was Dan Meyer. I used to call him the King of the MTBoS as many other bloggers use his lessons -- notice how many of Meyer's 3-Act lessons appear on Stauver's pacing plan. But I no longer consider Meyer the King of the MTBoS because he has rejected the MTBoS label. Instead, Fawn Nguyen is now the Queen.
But last month, Dan Meyer wrote a thought-provoking post that has drawn dozens of comments. Let's look at this post in the link below:
http://blog.mrmeyer.com/2017/dismantling-the-privilege-of-the-mathematical-1/
"Dismantling the Privilege of the Mathematical 1%."
Meyer writes:
The economic 1% gets a lot of grief lately and whether we know it or not, whether we like it or not, we are all also in the 1% – the mathematical 1%.
In 2014, 2.8 million degrees were awarded in US universities – bachelors, masters, and doctorates – and 1.1% of them were in mathematics. If you change the denominator to reflect not advanced degree holders but anyone with a high school diploma our elitism becomes even more apparent.
Again, Meyer wrote this post in November. The economic 1% has been mentioned in the news even more since he posted this, considering tonight's Congress vote on tax reform. But let's avoid politics and stick to the mathematical 1%.
Math teachers know that only a minority of students would consider themselves good at math. My estimate might have been closer to 10%. Here Meyer counts 1% by considering only math majors, but surely physics majors, for example, would be good at math. For the sake of argument, let's stick to math majors as a 1% class in order to discuss Meyer's post. This means that yes, even I myself am member of his 1% class, even though I'm not currently a math teacher.
Meyer continues:
When someone says, “I’m not a math person,” what do you say back? Barring certain disabilities or exceptionalities, everyone starts life a math person. Infants can recognize changing quantity. Brazilian street vendors develop sophisticated arithmetic algorithms before they set foot in school.
It is our action and inaction that teach people they are not mathematical. So please consider taking two actions to extend your privilege to the other 99% of humanity.
First, change the definition of mathematics that people experience.
http://www.101qs.com/2859-circlesquare
to a textbook version of the same problem:
Mark an arbitrary point P on a line segment
The former king considers this to be just verbiage designed to shut out the 99%:
You might think, “Well that’s what math is,” but the definition of math isn’t a physical constant in the universe. It’s defined by people, just as people define the ways that wealth and power accrue in the world. That definition is then underlined, reflected, and enforced in public policy, curriculum, and syllabi.
So, second, let’s change the definition of mathematics in public policy, curriculum, and syllabi.
Of course, this goes right back to the traditionalist debate from yesterday's post. Traditionalists would consider the text question to be real math, and Meyer's version to be a non-math activity designed to lock students out of STEM careers.
We know that traditionalists like SteveH want to get as many students as possible into Calculus. On the other hand, let's see what Meyer writes about higher math:
To begin with, let’s eliminate policies that require intermediate algebra for college study.
The facts as I understand them are that:
- College completion is increasingly essential to even partial economic participation.
- College study is generally predicated on a student’s ability to pass a mathematics entrance exam. In the California State University system, that exam is heavily weighted towards intermediate algebra, problems like these, the majority of which depend on the recollection of an obscure and abstract procedure
- Students fail these exams in staggering numbers (68% nationally) placing into “developmental math” courses, courses which cost time and money and don’t offer credit towards graduation.
So SteveH wants students to take Calculus, while Meyer thinks that even Algebra II is too high. The former king makes a suggestion as to another class students can take instead of Algebra II:
Those goals are served poorly by intermediate algebra. And better alternatives to intermediate algebra exist to serve the CSU’s desire to “assess mathematical skills needed in CSU General Education (GE) programs in quantitative reasoning.”
Specifically, statistics.
Oops -- we already know what traditionalists like Bill from yesterday's post say about Statistics. To Bill, it's impossible to understand Stats without knowing Algebra II. A true Stats class would have Algebra II as a prerequisite, and to Bill, a Stats course that doesn't require Algebra II isn't a real Stats course --and probably not a real mathematics course.
Speaking of Stats, Meyer provides some:
When 907 CUNY students were assigned either to remedial algebra, remedial algebra and supplementary workshops, or college-level statistics and workshops, that latter group a) passed their course in greater numbers (earning credit!) and also b) accumulated more credits in later courses.
Of course, I can already see a traditionalist being skeptical here. What "later courses" are these? Are these courses like Calculus, Differential Equations, or Linear Algebra -- those that lead to and are required for STEM degrees? (Later in the post, social and natural science courses are mentioned, and of course natural science is required for a STEM degree.)
I agree with Meyer that Stats is a useful class for students to know. But suppose that Bill is also correct, that without Algebra II there can be no Statistics?
- All students need to learn Stats, a class with real-world applications. (Meyer)
- To learn Stats, students need to learn Algebra II. (Bill)
- Therefore all students need to learn Algebra II.
And of course, that's news to kids sitting in their Algebra II classes asking "When will we ever have to use Algebra II?"
In the end, I must disagree with Bill and side with the former king here. This is a Geometry blog, and so my bias lies with Geometry. If someone were to propose eliminating Geometry, I'd be strongly opposed to it., But I know that there's a huge jump in difficulty between Geometry and Algebra II, and so I sympathize with those who want to eliminate Algebra II. Still, I wonder whether there can be a compromise -- if Bill is right and Algebra II really is needed for Stats, then I wonder whether we can a class with just enough Algebra II to prepare students for Stats. Then all other Algebra II material could be pushed into Precalculus.
Meyer writes:
So we should be excited to see the California State University drop its intermediate algebra requirement for graduation. We should be excited to see a proposal from NCTM that reserves intermediate algebra concepts for elective courses in high school. But we should regard both proposals as tenuous, and understand that as people of privilege, our support should be vocal and persistent.
I do have one concern here -- how does this affect Integrated Math? Notice that the three courses Integrated Math I, II, and III are equivalent to Algebra I, Geometry, and Algebra II. But how would just two of these courses, Integrated I and II, count for? Two years ago (on Black Friday), I wrote about how as a young student, I attended the rare high school to offer Integrated Science, and that day, I wrote that while all three years are equivalent to three years of traditionalist science, two years aren't worth squat. So a student would have to take Integrated III just for I and II to count, and by taking Integrated III they would have completed the equivalent of Algebra II anyway.
The NCTM link Meyer provides above mentions new courses called "Essential Concepts." These are integrated courses that fit the plan above, and the Integrated III problem is avoided. Five semesters of Essential Concepts will be required for all students. And the "B Pathway" listed there explicitly leads to Calculus, even starting from Common Core 8, so traditionalists theoretically shouldn't have a problem there (but of course, to SteveH and other traditionalists, anything other than the so-called "AP Calculus track," with eighth grade Algebra I and so on, is undesirable).
Meyer concludes with a quote from NCTM:
Hi. We’re NCTM. We want to restore purpose, joy, and wonder to your high school math classrooms. We know that goal sounds ambitious, and maybe even impossible, but we have a lot of experience, a lot of ideas, a lot of resources, and a lot of ways to help you grow into it. We’re here for you, and we also can’t do any of this without you. Let’s do this!
Let me include a few comments from Dan Meyer's blog:
Chris:
Still chewing on this…
“You might think, “Well that’s what math is,” but the definition of math isn’t a physical constant in the universe. It’s defined by people, just as people define the ways that wealth and power accrue in the world. That definition is then underlined, reflected, and enforced in public policy, curriculum, and syllabi.”
I think it’s hard for me because one thing you always hear about math(ematics) is how it IS a physical constant in the universe. Seems like we’ve let that belief spill over into our understanding of how math is done.
I agree with Chris that math is a constant. On the other hand, math class doesn't have to be a constant (after all, even Garelick yesterday distinguished "math" from "math education"). For example, both the Quadratic and Cubic Formulas are part of math, but only the Quadratic Formula is considered a part of high school math class.
John Chase:
Love the post, Dan. As others have said, this is very timely and important!
I have to react to the comments about Statistics, though, so please indulge me for a minute.
I think pushing Statistics instead of other math classes runs the risk of missing the transcendence that math offers. Statistics rarely offers the kind of sense-making that is the hallmark of mathematics. Non-calculus-based Statistics, as it is traditionally taught in High School, has many appeals to authority and lots of blind trust in formulas. This is similar, to some extent, to the sciences, where we find math in the service of other disciplines. Statistics, although under the umbrella of mathematics, is not often taught with sense-making as the driving paradigm. The hallmark of what mathematics and mathematics education is *sense-making*.
I’d love to see the Desmos “take” on Statistics beyond scatter plots and estimation. I’m sure this is a direction you guys are already planning to go, so I’m anxious to see what you come up with!
I don’t have a perfect solution, but I don’t think putting Statistics at the center is the answer. Statistics is vital and absolutely needs to be part of our curriculum, but I just don’t think it deserves center stage.
I also don’t think putting algebraic manipulation at the center is the answer either!
Hmmm, that's interesting -- I suppose as a Geometry blog writer, I can appreciate the need to justify statements in other branches of math just as we do in Geometry. (Note: I've mentioned John Chase back in my October 30th post as an advocate of the inclusive definition of "trapezoid" in Geometry.)
Sarah Ream:
Sorry I disagree with most of this article, especially the privilege. I believe if you work hard, you will reap the benefits of that hard work. There are now students in 3rd world countries out performing us. Have we American math teachers become so defeated that we refuse to challenge our students to take intermediate Algebra. The reason we have this problem is because on paper we are increasing in rigor but in reality these students are passed along, never really mastering math. The students then go to college with unrealistic expectations based on their high school experience only to be prescribed a course load of remedial classes. Most of the students in college should not be in college. So are we to appeal to the SAT and ACT and remove intermediate math for the test to make it more accessible for students to attend college?
Woo-hoo!!! Excellent!!!! Graduation rates have increased and test scores have decreased. This has lead to students not being prepared for college.
“To begin with, let’s eliminate policies that require intermediate algebra for college study.”
“Students fail these exams in staggering numbers (68% nationally) placing into “developmental math” courses, courses which cost time and money and don’t offer credit towards graduation.
Those courses are disproportionally composed of African American and Latinx students.
Only 32% of students in developmental math ever take a math course required for graduation.”
“Students fail these exams in staggering numbers (68% nationally) placing into “developmental math” courses, courses which cost time and money and don’t offer credit towards graduation.
Those courses are disproportionally composed of African American and Latinx students.
Only 32% of students in developmental math ever take a math course required for graduation.”
= Less qualified people in society.
So our answer to this dilemma is to lower our standards. I am a minority and I believe that if we sell minority students on studying math they will have access to high demand fields where its easier for them to enter the workforce and have a lucrative career especially since the don’t have “privilege.” (Which I think is a made up concept and sounds elitist in itself). I teach my students that there is no limit to what they can accomplish. What happened to “Stand and deliver?” Minorities are capable of learning math if they receive an adequate education and are not passed along. Minorities from other countries including 3rd world countries are surpassing Americans. Let’s learn from Ben Carson, Mae Carol Jemison, Clarence Thomas etc. Lets challenge all of our students to master math. This will build perseverance and character. Let’s learn from my family that immigrated from the dirt and poverty of Haiti to become teachers, doctors, lawyers, engineers etc. in one generation. Stop selling minorities short. I didn’t need standards lowered for me. I needed teachers who believed in me and cared about me my future. When my counselor tried to steer me to a community college I was able to get admitted into the top 3 colleges in Florida with AP Calculus college credit.
I know you are tired of the friction of teaching under prepared disgruntled students, hearing “Why do I have to study math?,” and feeling tired but you will produce a generation of hard workers.
By the way, for the person struggling with Latinx it’s the PC term for Latinos and Latinas.
So I assume that Sarah Ream would agree with the traditionalists here. Of course, she opens a can of worms by mentioning race in this post, but as usual, it's almost impossible to separate tracking from racial issues.
And in fact, this is a good point to end this post, since I wish to avoid controversial topics like race or politics in a school year post. I have one more thing to say about the MTBoS in tomorrow's post.
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