It also marks one full year since my very first post here on the blog. Yes, I've declared Pi Day of the Century -- March 14th, 2015, to be the ceremonial start date of this blog. But the chronological start date was July 22nd, 2014. I like celebrating both Pi Day and Pi Approximation Day -- including eating pie both days -- and so I celebrate both of them as launch dates for the blog.
Let me a little introspective here. Writing on this blog has taught me many things about myself that I have not realized.
For starters, I realize that I change my mind a lot. I say that I'm going to post something about a geometry topic, then I change my mind and post something else -- and then I say that I'll post a third thing when I revisit that topic next year. From the perspective of the blog readers -- you -- it must be infuriating to read something in the archives that I say that I'll post and then move forward in the archives only to find out that I never posted it.
Another thing I notice about myself is that too many posts contain too many topics. Let's just look at this very post, for starters. This post begins with this anniversary introspection, then it moves on to Eugenia Cheng's How to Bake Pi, a book about category theory, since I'm still in the middle of reading her book. Then this post is considered part of the "How to Fix Common Core" series -- and I wrote in the previous post of this series that I would discuss the Common Core tests. Finally, since it is after all Pi Approximation Day, I want to post some more links to videos about the number pi.
Of course, I want to say that this hodgepodge of topics is only because it's summer -- I could make several posts each with its own topic, but I don't want to make too many summer posts because this blog is all about the school year, not the summer. But too many of my posts during the school year contain several topics as well. For example, some of my posts which purport to be about the PARCC Practice Exam don't mention that test until deep within these posts.
I notice that some of the best blogs often use "tags" or "labels" -- these labels describe the content of each post. It's about time that I use these labels as well. If I used the labels correctly, today's post should have the following labels:
-- Eugenia Cheng
-- How to Fix Common Core
-- Pi Day
And so let me proceed with the FAQ, since today's anniversary is an excellent day for me to remind or inform the readers about my intentions with this blog.
1. Who am I? Am I a math teacher?
I am David Walker. I have recently earned my clear credential in Single Subject Math here in my home state of California, but I have not been hired as a full-time teacher. Instead, I have worked as a part-time math tutor since October 2012 and a substitute teacher in two districts since October 2014.
This already explains some of why I change my mind so frequently. I changed the blog calendar (where my posts are labeled Day 1, Day 2, etc.) last October so that it follows the actual school calendar in one of the districts where I sub. This year, I plan on using the calendar for the other district instead. Day 1, the first day of school, will be on August 26th in this district. If I am hired to become a full-time teacher in a district, then of course I will change the calendar again to reflect the school year in that district. But as it is already Pi Approximation Day and I haven't been hired anywhere yet, so I am expecting to return to substitute teaching on August 26th.
2. What is Common Core Geometry?
Common Core Geometry refers to the new geometry standards included in the Common Core. In particular, I refer to the Core's new focus on translations, rotations, and reflections -- these transformations were not emphasized before the Core. Transformations appear in the standards for eighth grade and above.
The Common Core is neutral in that it supports either a traditionalist pathway (Algebra I, Geometry, Algebra II) or an integrated pathway. On the traditionalist pathway, of course it's mainly the Geometry class where the new geometric transformations appear. On an integrated pathway, the transformations appear in both Math I and Math II.
One of the districts where I work uses the traditionalist pathway, while the other is making the transition to the integrated pathway.
3. Am I for or against the Common Core Standards?
There are some things that I don't mind about the Common Core and some things I'd change. I know that some people oppose the Core for political reasons. I try to avoid politics here on the blog, but unfortunately the Core is inherently political. In particular, opponents of the political party that was in power when the Core was first adopted tend to dislike the Core as well.
Some people believe that the Tenth Amendment to the Constitution forbids Common Core -- only fifty separate sets of standards, one for each state, would be constitutional. I prefer national standards because I find that the top math nations -- the ones that the United States seeks to emulate -- have strong national standards. Some of the top math nations do have standards at their equivalent level of the state, and so I don't mind having separate sets of state standards, even though 50 is a lot. On the other hand, I oppose having completely localized standards -- one set for each city, town, or district -- because the top math nations do not have localized standards.
Some people oppose Common Core math because they do not like integrated math. I strongly disagree with this because the top math nations all use integrated math -- the United States is an outlier with its traditionalist Algebra I, Geometry, and Algebra II. In particular, the highly respected Singapore Math is based on the integrated pathway. Nonetheless, this website is currently based on the traditionalist Geometry class since I wish to focus on Geometry.
I once heartily endorsed the Singapore Math standards as a great example of an integrated math program for the United States to adopt. But since then I fear that Singapore Math, especially its Secondary Year Two program, is too advanced for most American eighth graders and would only set them up for failure. So instead, I prefer the other highly respected integrated program, Saxon Math, since I feel that its Algebra 1/2 (half) text is more realistic for American eighth graders -- and this text covers much of the same material as the Common Core.
There are some things that I would change about the Common Core. The least popular aspect of the Common Core is, of course, the testing associated with it, especially PARCC and SBAC. In today's post, I want to discuss what I would change about the Common Core tests.
4. What is the "U of Chicago text"?
In 1991, the University of Chicago School Mathematics Project published a series of secondary mathematics texts. I was able to purchase both the Algebra I and Geometry texts for two dollars each at a local public library. One thing I discovered was that the U of Chicago Geometry text is based on the same transformations that appear in the Common Core Standards -- even though the U of Chicago published the text nearly two decades before the advent of the Core.
The common denominator is NCTM, the National Council of Teachers of Mathematics. This group was behind both the U of Chicago text and the Common Core Standards. Still, this means that one can satisfy most of the Common Core Standards by teaching out of the U of Chicago text.
There are newer versions of the Geometry text than the one that I own. I've read that while my old version introduces reflections in Chapter 4 and the other isometries in Chapter 6, in the new version all isometries appear in Chapter 4.
The U of Chicago text is not the only book that I mention here on the blog, not by a long shot. Many posts here refer to the Glencoe Geometry text. This is mainly because I am a math tutor, and I've been tutoring an eighth grader who is using the Glencoe text. His seventh grade teacher fell behind in Algebra I, so this year the eighth grade teacher finished the Algebra I text during the first quarter and started Geometry during the second quarter. He ended up reaching Chapter 9 of the Glencoe text by the end of the year -- which just happened to be the chapter on transformations.
Sometimes when I sub at various schools, I comment on the various texts that I see in the classrooms where I am working. I especially like listing the various Geometry texts as well as Integrated Math I and II texts so that I can point out the Geometry chapters.
Unfortunately, I've never seen a copy of the Saxon Algebra 1/2 text that I mentioned earlier. The only Saxon text that I own is the Saxon 65 text for fifth and sixth graders.
To learn more about some of these texts, here are some links to a webpage by Cathy Duffy. Duffy's goal is to help homeschooling parents choose a textbook for their children. Here is her review of the U of Chicago text:
and the Saxon series:
5. Who are David Joyce, Dr. M, and Hung-Hsi Wu?
These are mathematicians who have commented on how geometry should be taught. Dr. David Joyce actually discusses pre-Common Core Geometry standards. At the following website, he strongly criticizes a Prentice-Hall Geometry text:
Dr. Joyce strongly believes that theorems should be proved before they can be used -- that is the overarching theme of his criticism. I don't agree with everything he writes -- for example, he criticizes Chapter 3, the chapter on isometries. Pre-Common Core, his criticism would have made sense, but now with the advent of the Core, things have changed. We now actually use isometries to prove SSS, SAS, and ASA all as theorems. Still, I attempt to use the Joyce ideal of using nothing before it is proved -- especially considering what Joyce writes about coordinate geometry:
In a return to coordinate geometry it is implicitly assumed that a linear equation is the equation of a straight line. A proof would depend on the theory of similar triangles in chapter 10.
The Prentice-Hall text doesn't prove this, nor do most other texts (not even the U of Chicago text) -- but the proof is actually required in the Common Core eighth grade standards!
Oh, and as a bonus, Dr. Joyce links to a copy of Euclid's Elements -- the world's first geometry text:
Dr. Hung-Hsi Wu is a Berkeley mathematician. He has written extensively on how to teach Geometry according to the new Common Core Standards:
I have used some of Dr. Wu's proofs, including the use of rotations to prove some of the properties of parallel lines. I also once tried to use Dr. Wu's proof of the Fundamental Theorem of Similarity -- but this proof is extremely complex and inappropriate for high school students. After posting the long Dr. Wu proof, I changed my mind and suggested that a Dilation Postulate be used instead. This is what Dr. Joyce means when he writes:
Chapter 10 is on similarity and similar figures. (Appropriately for this level, the difficulties of proportions are buried in the implicit assumptions of real numbers.)
Dr. Wu calls this the "Fundamental Assumption of School Mathematics." It is Dr. Wu who defines pi as the area of the unit disk -- this definition favors pi above the numbers tau and lambda. (Yes, today is Pi Approximation Day, so here's a pi reference.) Meanwhile, Wu has also created an extensive document on how to teach fractions under the Common Core:
Dr. Franklin Mason is a Geometry teacher at an Indiana high school. He has created an online text:
He has changed his text several times since first posting it. I've surmised that most likely, Dr. M wrote the original version based on what the Common Core Standards say, but the new version is based on what actually appears on the PARCC and SBAC tests. The original text followed the David Joyce ideal of using nothing before it is proved, but the new text replaces theorems with postulates if the proofs are more difficult than anything that appears on the PARCC or SBAC.
Here's an example of the evolution of the Dr. M text: originally, Dr. M used several theorems, including SAS, to prove a Triangle Exterior Angle Inequality (TEAI), and then used the TEAI to prove his Parallel Line Tests. Later on, he dropped the proof of TEAI and made it a postulate. Still later on, he dropped the proof of the Corresponding Angles Test and made it a postulate.
But Dr. M still includes a proof that all circles are similar. Dr. Joyce writes:
So the content of the theorem is that all circles have the same ratio of circumference to diameter. This theorem is not proven. It would require the basic geometry that won't come for a couple of chapters yet, and it would require a definition of length of a curve and limiting processes.
Dr. M actually includes these "limiting processes" by introducing a least upper bound. (Yes, here's another pi reference for Pi Approximation Day.)
There are a few more resources for Common Core Geometry. One is the EngageNY website:
The State of New York has developed a strong Common Core curriculum. Much of this is based on the recommendations of Dr. Wu. For example, Lesson 18 uses 180-degree rotations to develop the Alternate Interior Angles Test. This idea goes back to Dr. Wu:
Here is one last resource that I wish to mention, the Geometry Common Core website:
The website creator, Mike Patterson, teaches Geometry at a Nevada high school. I sort of like his idea of using translations to prove the Corresponding Angles Test, rather than rotations to prove the Alternate Interior Angles Test, since I consider corresponding angles to be more basic:
But unfortunately, I'm not quite sure whether one can actually write this proof without introducing some circularity, as the properties of parallel lines are uses to prove the properties of translations (at least in the U of Chicago text).
6. Who are "the traditionalists"?
I use the phrase "traditionalists" to refer to those who oppose Common Core for mathematical -- not political -- reasons. Such mathematicians include:
Dr. Katharine Beals
Dr. Barry Garelick
Dr. James Milgram
Dr. Ze'ev Wurman
SteveH (often comments at the websites of the other traditionalists)
The traditionalists tend to believe that many of the topics in the Common Core Math Standards are taught too late. In particular, they believe that the standards in Grades 4-6 that require the use of the standard algorithm for arithmetic should move down at least one grade level.
I tend to agree with the traditionalists on this point here -- most of the reasons for opposing direct instruction, rote memorization, and drill don't apply to the youngest students. I remember once learning about two nonstandard algorithms for addition -- the Left-Right Method and the Plus-Minus Method -- on an old 15-minute math show for sixth graders, "Solve It," that used to air on our local PBS station back when I was that age. For mental math, these nonstandard algorithms are great, but for a young elementary student first learning to add two-digit numbers I prefer the standard algorithm.
The traditionalists also prefer that high school seniors take AP Calculus in order to open the door to college STEM majors. In particular, some STEM colleges here in Southern California, such as Cal Tech and Harvey Mudd, require calculus for admission. Even colleges that don't formally require calculus are likely to send students rejection letters if they apply without having calculus on the schedule, especially if they apply to STEM majors. Therefore they want the classes leading up to senior year to prepare students to take AP Calculus -- including Algebra I in the eighth grade. They oppose Common Core because its eighth grade math standards do not comprise a full Algebra I class.
I only partially agree because I don't want seniors who can't handle AP Calculus or eighth graders who can't handle Algebra I to be labeled failures. Just as AP Calculus can open the door to STEM, it can close the door to a non-STEM career if students are forced to spend too much effort trying to pass their high school math classes. Then again, my preferred eighth grade text, Saxon Algebra 1/2, does provide a path to a Calculus class: Saxon Algebra 1 (third edition) in freshman year, Saxon Algebra 2 (third edition) in sophomore year, Saxon Advanced Math in junior year, and then Calculus.
Most traditionalists dislike the U of Chicago math texts. However, when a traditionalist refers to U of Chicago math, he is nearly always referring to the elementary school texts -- he rarely, if ever, considers the U of Chicago texts for sixth grade and above. Most likely, traditionalists would dislike the high school texts as much as they do the elementary texts -- especially since they already oppose Common Core and I emphasize how well the U of Chicago texts are Core-aligned.
Here are links to some of the traditionalist websites:
(The last is an article on which SteveH has commented heavily.)
7. How would I fix the Common Core tests?
Well, I said that I would post this today, so let me do exactly that. I repeat that the testing is the least popular aspect by far about the Common Core. Myriads of parents around the country chose to opt their students out of the Common Core tests.
What I'd like to do in this section is design a test that parents would want their students to take, so that there wouldn't be so many opt-outs. Of course, PARCC and SBAC may have already poisoned the well so much that parents may opt their children out of any national standardized test, no matter how good the test may be. To such parents, nothing that I can write in this post would produce a test that they'd allow their children to take.
Throughout this "How to Fix Common Core" series, I would often go back to the traditionalists to see what their specific objects are and how I would address them. I link above to the comment section of an article where the traditionalist SteveH made several posts. Let's find out what SteveH had to say about the Common Core testing:
"The ACT and College Board are now in the process of developing and improving their own college readiness assessments, and the College Board, in particular, offers assessments geared towards AP classes and STEM careers. Some states, like Alabama, have dropped PARCC and are now going with ACT. We need more than CCSS, and my hope is that states will use tools from ACT and the College Board to define a high-level curriculum path back to the lowest grades. The words in the CCSS document are too vague and open to vast differences in interpretation and level of expectation. States need to pick the assessment tests and let those tests drive the curriculum."
OK, I agree with what SteveH writes here. One way to improve the testing is simply to reduce the number of tests that students must take. Since high school students already take the ACT, that can be already replace the PARCC and SBAC tests that high school students must take. Like SteveH, I also look forward to seeing what the ACT will do to design a test for the lower grades.
Unfortunately, I don't completely agree with what SteveH writes in the next paragraph:
"Life will go on for many students and parents who know better than to look to a one-size-fits-all standard for how their kids are progressing. Most middle schools will continue to offer proper pre-algebra and algebra courses as lead-ins to a rigorous geomptry, algebra II, pre-calc, and calc sequence in high school - NOT integrated math curricula.. These math classes will continue to be driven by teacher-centered direct instruction, and these courses will continue to produce the STEM-prepared students that colleges are looking for. These students have not, and will continue to not be created by standardized state tests. For the most part, these STEM kids will be created by parents who reteach at home and with private tutors. STEM students will continue to be NOT be created by hands-on, group work in class. They will be created by a steady diet of individual homework sets, and parents, tutors, and teachers who ensure that students master those skills. I find it ironic that K-8 educators talk about having little Johnnie or Suzie think like a real mathematician or engineer while those professionals are telling them that they got it completely wrong."
Here are the parts of this paragraph with which I disagree:
-- "NOT integrated math curricula." Many traditionalists are convinced that integrated math is inferior to the usual Algebra I, Geometry, and Algebra II. In reality, the majority of the world's students who succeed in Calculus learned Integrated Math to prepare them for Calculus. Ironically, many of the parents and tutors mentioned later in the paragraph probably used the Singapore and Saxon Integrated Math texts.
-- "driven by teacher-centered direct instruction." Like most traditionalists, SteveH strongly prefers the direct "sage on the stage" to the progressive "guide on the side." The only problem with this is that it presumes that students would willingly listen to the teacher and accept him or her as the "sage on the stage." This may be true for younger students -- and this is one reason that I agree with traditionalism in the younger grades -- but it isn't true for the teenagers who would be taking the classes discussed in this paragraph.
Teenagers are at an age where they begin to question authority. Therefore, in math classes, they ask questions like "Why do we have to learn this?" or "When will we use this?" I can only assume that if SteveH taught such a class, he'd answer something like "It will prepare you for Calculus, which is necessary for a STEM career." But such an answer might not necessarily convince teens to get up and do the work -- especially those who have no intention of going into a STEM career.
SteveH mentions "individual homework sets" later in the paragraph. He appears under the impression that all if he were the teacher, all he'd to do is give lectures everyday in class and long homework sets every night, and teens will just do it because he is the "sage on the stage." But merely assigning individual homework sets won't prepare students for Calculus if they don't do the assignments -- or if they are no longer individual because someone else is doing the homework.
I have a problem with forcing everyone onto the STEM path in high school, in particularly in the name of "keeping doors open." Actually, eighth grade Algebra I isn't as much of a problem because colleges mainly look at the high school grades, so a D or F in eighth grade Algebra I doesn't really close doors. (I do note that some middle schools don't allow eighth graders to participate in the promotion ceremony if they are failing English or math during the last semester/trimester. I'd hate to see an eighth grader miss promotion after passing every math class except the last term just because he can't factor a quadratic polynomial.)
The real problem with the SteveH plan is in high school -- especially sophomore Algebra II, as there is a huge leap in difficulty between Algebra II and any math class that precedes it. If a student can't participate in sports or get a part-time job -- both of which might lead to a great non-STEM career -- because he's too busy trying to finish SteveH's "individual homework sets" for an Algebra II class that's unnecessary for his career path, then he's likely to see Algebra II and Calculus as closing doors, not keeping them open.
SteveH writes that he doesn't want his standards to be "one-size-fits-all." He points out that while there are five scoring levels on the PARCC and four on the SBAC, there are actually only two levels that matter. I'll call these levels "proficient" and "deficient." Schools are judged by how many students score as "proficient." There is no incentive to raise students from "low deficient" to "high deficient," or, much to SteveH's dismay, from "low proficient" to "high proficient." Therefore students end up spending all of their resources trying to raise students from "deficient" to "proficient" and hardly any trying to raise them from "proficient" to "Calculus-ready." And so let's try to design a test that would break out of this "one-size-fits-all" mold.
Suggestion #5 (the first four suggestions appear in earlier June posts): Here is a detailed design for how I would create a new computer-based Common Core test.
We begin with the third grade test, as this is typically the lowest grade that the tests are given. The third grade test will begin with simple one-digit multiplication problems.
Now the most important part of the test will be the scoring system. I want the average third-grader at the end of the year to receive a score of 400. In fact, the scoring should go like this:
100: Average first-grader at start of year
200: Average second-grader at start of year
300: Average third-grader at start of year
400: Average fourth-grader at start of year
500: Average fifth-grader at start of year
600: Average sixth-grader at start of year
700: Average seventh-grader at start of year
800: Average eighth-grader at start of year
So our average third-grader, taking the test at the end of the year, should score 400, since he or she is getting ready to start the fourth grade.
The test begins with our third-grader having a baseline score of 300 points. The goal is for the student to reach 400 points in the 30 minutes allotted for the test -- since I've already stated that I want to reduce the test time to 30 minutes. We could set each one-digit multiplication problem to be worth one point each. So if the student gets one question correct every 18 seconds, then this will get them the 100 points that they need in 30 minutes.
Notice that this would obviously be a computer-adaptive test, just like the current SBAC. But I don't believe that the SBAC takes full advantage of its computer adaptivity. Our test here keeps a running score and asks questions based on the current student score. Each time a student answers a question correctly, he or she gains a point, and each time a student answers a question incorrectly, he or she loses a point.
Traditionalists say that the most important math for a third-grader to learn is multiplication -- but it need not be the only math that they learn. We could set it up so that once a student reaches a score of 350, the questions switch so cover the rest of the third grade Common Core Standards. This will mean that 50% of the third grade score is devoted to multiplication -- which is probably more than the current PARCC and SBAC third grade tests. Questions can be worth more than one point, especially if they take more time to answer, so as to maintain the average score of 400.
Notice this means that, at least during the 300-350 multiplication section, a student who gets only half of the questions right will gain and lose equal numbers of points, so the score remains at 300. This is consistent with the fact that 50% in class is a failing grade -- and we certainly want third graders to get more than half of the multiplication questions correct. I'd argue that for more complicated questions, students can gain several points for correct answers but lose only one point if the answers are incorrect. Otherwise, a student who gets all of the easy answers right and all of the hard answers wrong would have a low score (since they'd gain points one at a time and lose them two at a time).
If the score drops below 300, then the test would switch to second grade questions, with more of a focus on addition and subtraction. Once again, the types of questions a student receives is based on the running score.
If the score rises above 400, then the test would switch to fourth grade questions. Notice that an exceptionally bright third grader might make it all the way to a score of 500 before the 30 minutes are up -- if this happens, the test would switch to fifth grade questions. I don't want the running score to appear on the screen while the student is taking the test, but when the thirty minutes are up, the score is reported immediately, on both the screen and a computer printout. This means that every question must have either a multiple-choice or numeric answer. Performance tasks where the student has to answer with sentences will not appear on this test.
When a fourth-grader returns to take the test the following year, the questions with which the test begins are based on the score the student received the previous year. So a student with a score of 400 starts with fourth grade questions, a student with a score of 273 starts with second grade questions, a student with a score of 546 starts with fifth grade questions, and so on.
The maximum score of this test should be 800, just like the SAT. But this time, a score of 800 represents the beginning of eighth grade -- which should be Algebra I, according to SteveH. The problem is that we don't want the test to favor either the traditionalist or integrated pathway -- it is just as unfair for an Algebra I student to answer geometry questions as it is for an Integrated Math I student to answer polynomial questions. Recall that this was the problem when California tried to implement eighth grade Algebra I on top of the Common Core Standards -- the problem was that eighth-graders in Algebra I would be unable to answer geometry questions on the SBAC eighth grade math test.
So I want this test to end with seventh grade math. We can have separate end-of-course exams for Algebra I, Geometry, and Integrated Math I/II, or roll them into other tests for students. Otherwise, a student who reaches a score of 800 but doesn't take the Algebra I or Integrated Math I tests should not take any math test at all.
We notice that this scoring system fits very well with the path system described in other posts. Of course, no national test can force schools to use paths, but it can certainly encourage it by means of the scoring system described in this post. We can also subtly encourage eighth grade Algebra I via the scoring system, simply by reporting scores on the Algebra I test on a scale of 800 to 900.
Actually, I've changed my mind slightly about the path system. Recall that in my proposal, students are divided into paths based on their reading levels -- students reading at a first or second grade level would be on the Primary Path, those at a third or fourth grade level on the Transition Path, and so on.
This makes sense at elementary schools, since reading is such a critical skill. But an argument can be made that at the middle school level, the most critical skill is math -- this is, after all, the crux of what SteveH is saying. Back when California still reported state scores for ELA and math, I can just look at a school's score and determine whether it is elementary or secondary. If the math score is higher, then it's probably an elementary school (2+2 is easy, but reading is hard). If the ELA score is higher, then it's probably a secondary school (they know how to read, but now algebra is hard). This trend persisted regardless of the student demographics.
And so one might introduce a sort of path system where students are divided onto paths based on their reading levels in elementary school but their math scores in middle school. Of course, by the time the students reach middle school they are already having several teachers throughout the day, so a path system may be irrelevant at all except the smallest middle schools.
Let's get back to Eugenia Cheng's book How to Bake Pi. I've now reached Part II which is on Category Theory -- the main topic of the book. In particular, I've completed Chapter 13, "Sameness."
As I've said earlier, category theory is just as new for me as it is for most of you. Here is what Cheng says about category theory: "Category theory is the mathematics of mathematics. Category theory, then, is the process of working out exactly which parts of math is easy, and the process of making as many parts of math easy as possible.
In Chapter 11, Cheng draws several examples of categories -- many of these look just like the networks found in Section 1-4 of the U of Chicago text. Here the nodes are objects and the vertices are actually directed line segments, or arrows, that represent "morphisms," which are sort of like functions, but Cheng describes them better as relationships.
I'm still trying to make sense of what I'm reading in Cheng's book. But just as I'm struggling to learn category theory from her book, many students in our high school algebra and geometry classes are struggling to learn from us. Algebra is as difficult to them as category theory is to me. This is why I am not as eager to put students into higher math as SteveH is, but my hope is that with a test similar to the one that I propose here, we'll be able to get our best math students -- those who are able to make that great leap of abstraction -- into the top math classes necessarily for success in STEM.
I remember when I was learning about groups and rings at UCLA, the professor often drew what he called a "commutative diagram." Although he never used the term "category," it is now obvious to me after reading Cheng's book that my professor was using category theory. In the same way, sometimes it takes a higher class, like Calculus, to see why something in Algebra or Geometry makes sense -- and this goes right back to the SteveH door to Calculus. Here is a simple example of such a commutative diagram, except this time I used sets of geometry figures as the objects. Here the morphism is the inclusion map:
Parallelogram --> Rectangle
Rhombus --> Square
Cheng begins the last chapter that I've read so far, Chapter 13, with a recipe for "Raw Chocolate Cookies" -- another recipe that Cheng created herself. She points out that "raw chocolate cookies" may be so different from "cookies" that it's unclear whether they should still be called "cookies" -- just like her "olive oil plum cake" from earlier in the book. The chapter describes how it's not always clear what it means for two objects to be the same. She gives triangle similarity as an example of sameness (same shape). Of course, we think of congruence as being a stronger example of sameness, while topological equivalence is a weaker example of sameness.
But today is Pi Approximation Day, and I like to eat apple pie -- which really is the same as a pie -- since I associate apple pie with this time of year ("as American as apple pie," and this is the month that we celebrated Independence Day").
In honor of Pi Approximation Day, let me post some more videos about pi. The first few of these mention several ways to approximate pi.
The History of Pi. This video mentions that even Legendre -- the author of the geometry text we've been discussing in some of my recent posts -- contributed to the history of pi. In particular, Legendre proved that not only is pi irrational, but so is pi^2 (unlike sqrt(2), which is irrational yet has a rational square, namely 2).
This video also mentions a quote from the mathematician John von Neumann: “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” We've seen Eugenia Cheng, in her How to Bake Pi, say almost exactly the same thing.
The Infinite Life of Pi:
No Pi (Approximation) Day is complete without a Numberphile video. In this video, we learn that pi is not just irrational, but is also transcendental -- that is, there doesn't even exist a polynomial f, with integer or rational coefficients, such that f (pi) = 0 (unlike sqrt(2), which is algebraic). This is even stronger than Legendre's result that pi^2 is irrational.
Here is a longer video, but it contains some more series approximations of pi:
This one's in German, but it does contain some interesting information. The video credits the Greek mathematician Archimedes as the first to use today's approximation 22/7 for pi, then it moves on to the question of whether pi is a normal number -- i.e., with random digits. (Oh, and BTW, an episode of Futurama, in one of its background jokes, refers to 22/7 as "Fool's Pi.")
Here is a short Pi song:
No, I can't avoid another one of my favorite math YouTube posters, Vi Hart. Of course, in this video she argues that there's nothing special about pi at all:
Let's wrap up with one more Pi song, a longer one this time:
Finally, this link is not a video, but it's one of my favorite links for Pi Approximation Day. We know that 22/7 is approximately equal to pi. As it turns out, 22/7 is actually more than pi -- and we can find out why 22/7 > pi using calculus. It's possible that an AP Calculus student -- a senior on the SteveH plan -- might be able to calculate this integral:
A few other Pi Approximation Day (aka "Casual Pi Day") links:
And so I wish everyone a Happy Pi Approximation Day.