Monday, December 7, 2015

Lesson 6-6: Isometries (Day 66)

Today is a very significant day. First of all, today is my first post following the first Saturday in December -- also known as Putnam Saturday. This is what I wrote last year about the Putnam:

The first Saturday in December is the day of the annual William Lowell Putnam competition. It is a math contest for college students. Eleven years ago, a poster named John called it the "hardest math test in the world."

Every year, I like to inspire the students I tutor by showing them the first -- and usually easiest -- question, A-1, from this year's Putnam exam. I especially like to show this to my geometry students, since Putnam questions tend to be proofs. They may complain when they have to do proofs, but those proofs are nothing compared to Putnam proofs.

Just like last year, I will post the easiest Putnam question in my Tuesday post.

But there is also something else significant about today. I was born on December 7, 1980, which makes today my 35th birthday. As a birthday gift, I received a copy of Mind-Bending Math: Riddles and Paradoxes, a DVD lecture published by The Great Courses. The lecturer is David Kung, a professor at St. Mary's College of Maryland. Here is a link to Dr. Kung's website:

And you guessed it -- I'm going to discuss the lectures right here on the blog. Here are the 24 lectures included in this course:

1. Everything in This Lecture Is False
2. Elementary Math Isn't Elementary
3. Probability Paradoxes
4. Strangeness in Statistics
5. Zeno's Paradoxes of Motion
6. Infinity is Not a Number
7. More Than One Infinity
8. Cantor's Infinity of Infinities
9. Impossible Sets
10. Godel Proves the Unprovable
11. Voting Paradoxes
12. Why No Distribution Is Fully Fair
13. Games with Strange Loops
14. Losing to Win, Strategizing to Survive
15. Enigmas of Everyday Objects
16. Surprises of the Small and Speedy
17. Bending Space and Time
18. Filling the Gap between Dimensions
19. Crazy Knob of Connectedness
20. Twisted Topological Universes
21. More with Less, Something for Nothing
22. When Measurement is Impossible
23. Banach-Tarski's 1 = 1 + 1
24. The Paradox of Paradoxes

A paradox is a statement that seems to contradict itself, yet is true anyway. Indeed, there are many counter-intuitive statements in higher mathematics, and Dr. Kung explains some of them in this series of lectures.

Even before watching the lectures, I can already figure out what some of them are about. Lecture 16, "Surprises of the Small and Speedy," refers to Einstein's Theory of Relativity. Yes, a few weeks ago I briefly mentioned Einstein's theory to celebrate the 100th anniversary of its publishing. Well, when we get to that lecture, I'll discuss the theory in more detail. Actually, only the "speedy" part of the title refers to Einstein -- the "small" part refers to another theory, Quantum Mechanics. These two theories seem to conflict with each other, producing the Information Paradox, which is why it would be included in a series on paradoxes. (This is why, as I mentioned last year, Stephen Hawking was searching for a Theory of Everything that would incorporate both theories without paradox.) The following lecture, "Bending Space and Time," should also refer to Einstein's theory.

And I bet you readers of this blog can figure out what "Filling the Gap between Dimensions" is referring to. Yes, it's Benoit Mandelbrot and his theory of fractals. But one lecture that I can't figure out by the title is the last one -- "The Paradox of Paradoxes." I'm especially looking forward to it.

Due to the hectic holiday period, I've decided that I'll wait until January to watch the lectures and begin discussing them here on the blog.

Meanwhile, today I subbed in a math classroom -- the first time in over a month that I subbed for an actual math class as opposed to some other subject. The teacher of this class has three sections of Trigonometry/Pre-Calculus and two sections of Integrated Math I.

The Pre-Calculus students were in Chapter 6 of their text, "Additional Topics in Trigonometry," and this chapter includes the Laws of Sines and Cosines. Naturally, many students had trouble with the ambiguous case of the Law of Sines -- I would've been surprised if they didn't have trouble there -- so I had to show them. We've actually discussed this on the blog before -- remember the ambiguous SSA and determined SsA cases, from Chapter 7 of the U of Chicago Geometry text? And there were also a few word problems that weren't so easy to set up, such as:

From a certain distance, the angle of elevation to the top of a building is 17 degrees. At a point 50 meters closer to the building, the angle of elevation is 31 degrees. Approximate the height of the building.

We cover angle of elevation problems in Chapter 14 of the U of Chicago Geometry text, but not these problems where there are two angles of elevation and it's not apparent where the 50 meters goes. I told the students that the distances to the building are h / tan(17) and h / tan(31), but it appeared that students were still confused. (By the way, the 17 and 31 there should be degrees. At least one student worked on several problems before realizing that her calculator was set to radians!)

At that point, I told them that as hard as this problem may seem, it's nothing compared to the questions on the Putnam. This is the reason that I like to mention the Putnam in class, especially near the weekend the test is given -- hopefully it will inspire them to learn more when they see what truly difficult math looks like.

The Integrated Math I students were also in Module 6 of their respective text, "Forms of Linear Equations," and they were working on Lesson 6.3, "Forms of Linear Equations." In this class, I played the usual game where I start by asking the students, "What's my age?" And of course, I had fun telling them that I was 35 years old, because it's my birthday. Unfortunately, the students in sixth period must have heard from the ones in fifth period that it was my birthday, because many groups were trying to shout out 35 at the same time.

Some birthdays are more significant than others. Last year, I mentioned that I like to call every fourth birthday a "Julian birthday," referring to the fact that one cycle of the Julian calendar has been completed that day. This year is my 35th birthday, so it is not a Julian birthday.

Other birthdays are significant because of what you are old enough to do that day. At 16 you can drive, and at 18 you can vote for president, and at 35 -- you can be the president. So on this, my 35th birthday, my presidential birthday, I hereby declare my candidacy for President of the United States.

What will I do as your president? No, I won't discuss anything like foreign policy today, since that would be off-topic for the blog. Of course, I'm going to write about my education policy. I've devoted several posts to what I would do if I were in charge. I've changed my mind several times since I started this blog, based on new ideas I see as well as current test results and other outcomes. Of course, some of those ideas come from the traditionalists, and so this will be yet another post to receive the "traditionalists" label. But what I wish to post today is a brand new idea of mine and not merely a repetition of what I wrote in another post.

In the news lately is a new authorization of the No Child Left Behind Act. So the policies I propose in today's post could be imagined as a new version of NCLB. But also in the news are the Common Core Standards -- in particular, how many people and states are rejecting the standards.

Many people believe that there shouldn't be any national standards. They cite the Tenth Amendment to defend the idea that states should be in charge of education, not the federal government. Others believe that even the state level is too high, and that education should be determined at the local level. Still others reject even local school boards in charge of education, and that a free market approach should be used instead. I reject these last two because they deviate too far from the educational model of other nations -- especially the high-scoring nations we wish to emulate.

Some people may point out that one of the best educational systems in the world is in Shanghai, the largest city in China. Since Shanghai is a city, this would appear to belie my claim that no other country has locally-determined education.

But Shanghai is not comparable to most American cities. I've noticed that in many countries, there exist cities -- usually capitals or other large cities -- that aren't included in any state or province. In the mainland U.S., only our capital Washington D.C. is such a city. But in other countries, many cities are self-governed. China has four such cities -- Shanghai, the capital Beijing, and two others. And indeed, the municipality of Shanghai has a population of over 14 million, nearly double that of New York City. Outside of those four cities, education is determined at the provincial level. And so the education system in China is more comparable to state-level, not local-level, education.

And of course one of the top education systems in the world is in Singapore (which is why Singapore Math is so popular with traditionalists). But Singapore is considered to be a city-state or island country, so it's not directly comparable to national-, state-, or local-level education in the U.S. at all.

And so most countries have either national- or state-level educational systems. I am not completely opposed to education determined by the 50 states in this nation. One usual pro-Core argument in favor of national standards is portability. A pro-states response to this argument is that some states, such as my home state of California, are large enough that having pure California standards is justified.

But I'm not sure about the smaller states. Sometimes I wonder, if the (pre-Core) Massachusetts standards were so good, why couldn't they be adopted at least in the other New England states? Some opponents of national standards point out that children in Massachusetts are so different from those in, say, Mississippi, that making them conform to the same standards are unreasonable. Yet are Massachusetts children so different from those in the other New England states that they can't adhere to the same New England standards?

Those who oppose national standards on grounds of the Tenth Amendment would say that for even two states (say in New England) to have the same standards violates the Tenth Amendment, because it would be an illegal interstate compact. So nothing fewer than 50 separate standards, one for each state, would be constitutional in their eyes.

Here's one final argument -- pro-Core adherents say that national standards allows one to compare students in different states. Opponents of the Core counter, why is it so important for young first- and second-graders to be compared to their counterparts in other states? If we think about, the only students who really need to be compared are high school students, since they will often apply to colleges and jobs in other states.

And so, as your president, I will present the following compromise between state and federal control, which we may consider to be a replacement for No Child Left Behind:

-- Let high schools fall under federal control.
-- Let middle schools fall under state control.
-- Let elementary schools fall under local control.

I will call this my Presidential Birthday plan, since I'm proposing it on my presidential birthday. We see that this plan takes the pro-Core argument about comparing students across states a step further -- as mainly high school students need to be compared, only high schools will have national standards.

This means that the plan automatically repeals all Common Core Standards below high school. I already oppose many of the standards for the lowest grades, and this plan takes it a step further and eliminates Common Core completely for all elementary and middle school grades.

Here I define the high school grades as 9-12, which is by far the most common grade span. As for elementary and middle schools, as these are determined at the state level or below, it's actually up to the states to define the middle school grade span. Here in California, the most common middle school span is 6-8, so our state would control grades 6-8. In other states, 5-8 or 7-8 may be more common than 6-8, so those states would choose what makes the most sense for them.

And so, as your president, I will only have to determine the policy for grades 9-12. Of course, on this math blog, I'm only going to discuss the proposed math standards for grades 9-12.

Now in addition to the Presidential Birthday plan, recall that in previous posts, I also mentioned something I called Presidential Consistency Core. This addresses the anti-Core complaint that the president, the Secretary of Education, and many high-ranking officials promote the Common Core, yet send their own children to schools like Sidwell Friends which do not use the Core. They insinuate that the president and others are promoting a knowingly bad curriculum like the Common Core on other children because they want children to suffer, then protect their own children from it by enrolling them in private schools. The Presidential Consistency Core avoids this by automatically defining the Common Core to be the curriculum used at the schools where the president and other officials send their own children. Then the nation's children would, in principle, receive the same quality of education as the president's children do.

When I first mentioned the Consistency Core, I focused on Sidwell Friends, since that is the school that the First Daughters attend. We learned that Sidwell divided math students into three tracks, with the following courses offered on each track:

Low Track
9. Algebra I
10. Geometry: An Inductive Approach
11. Intermediate Algebra
12. Algebra III

Middle Track
9. Geometry
10. Algebra II
11. Pre-Calculus
12. (Non-AP) Calculus

High Track
9. Integrated Math I
10. Integrated Math II
11. Integrated Math III
12. AP Calculus

Students make take Statistics during their senior year on any of the three tracks -- AP Stats for the two higher tracks and Non-AP Stats for the low track.

Recall that the Math I, II, and III classes on the high track actually integrate Geometry, Algebra II, and Pre-Calculus, so these classes are at a higher level than the classes usually labeled Math I, II, and III at most schools.

But now we ask, what about eighth grade Algebra I? The middle track starts with Geometry, and the Math I course on the high track also starts with Geometry. So the intent is clearly for students on those two tracks to complete Algebra I in eighth grade. But my plan doesn't mandate Algebra I in eighth grade -- only because it doesn't control eighth grade at all, only Grades 9-12.

Of course, I could have made it so that the Consistency Core covers Grades 8-12 instead of 9-12, to make sure that eighth graders get Algebra I. But I want to avoid forcing middle schools to split between state accountability for Grades 6-7 and federal accountability for Grade 8. So it's just easier to put the state-federal divide at the middle-high school divide.

So states would make their own choice whether to allow any eighth graders to take Algebra I. The states that don't would find that all of their students are on the low track. I'm not sure my own state of California would do, but considering that the old pre-Core California State Standards allowed for Algebra I in eighth grade, that is what the state would do under the Presidential Birthday plan.

But now there's one very important part of the plan to discuss -- what about testing? One thing that I've always liked is that some states, rather than the PARCC or SBAC, use the ACT or SAT as its accountability test.

Under my Presidential Birthday plan, I'd make it so that the only accountability tests in high school are the SAT, ACT, and the tests that lead up to these (PSAT 8/9, PSAT 10, ACT Aspire). In fact, there will not be any additional End of Course/Year tests on top of the SAT/ACT (as many of those states have), but only the SAT/ACT itself. This also fits the Consistency Core, since even Sidwell students take the SAT/ACT, but not the PARCC/SBAC or any standardized End of Year exams.

Just as states can choose whether to use PARCC or SBAC as its test, states can select either the SAT or ACT as its Presidential Birthday accountability test. It's often said that the ACT is more popular in the Midwest while the SAT is preferred on the coasts. Therefore my home state of California would most likely choose the SAT.

I also believe that the accountability test should be given as close to the last day of school as possible, and not this 75% or 85% requirement as under the current NCLB. The SAT offers May and June exams, so I can't see why there can't be an SAT for the states that late in the year. Ideally, the standardized test would be given the penultimate week of the year, just before finals week.

Students would take the SAT at the end of their junior year, as is already common. There are already tests for younger high school students, the PSAT 8/9 for freshmen and PSAT 10 for sophomores. (Of course by definition, the PSAT 8/9 is for eighth graders as well, but once again, the Presidential Birthday plan only covers grades 9-12.)

Let's look at the new version of the SAT -- the math section, of course -- in more detail:

According to the College Board link, there are four domains on the new SAT -- Heart of Algebra, Problem Solving/Data Analysis, Passport to Advanced Math, and Additional Topics in Math. As it turns out, Heart of Algebra corresponds approximately to the first semester of Algebra I, while Passport to Advanced Math aligns with the second semester of Algebra I into Algebra II. Most of the Geometry content is squeezed into Additional Topics in Math.

The above link also tells us that PSAT 8/9 contains mostly Heart of Algebra and Problem Solving in equal percentages, while PSAT 10 contains these two domains plus Passport to Advanced Math. Only the SAT contains any significant Additional Topics -- and even there, Additional Topics constitute a smaller percentage of test questions than the other three domains.

As I mentioned over a week ago, one curriculum popular with traditionalists is Saxon Math. We notice that if we look at the classic third edition of Saxon, Algebra 1 contains much of the Heart of Algebra material, Algebra 2 has Passport to Advanced Math, and much Geometry material is delayed until Advanced Mathematics. So the three texts Algebra 1, Algebra 2, and Advanced Mathematics prepare students quite well for PSAT 8/9, PSAT 10, and SAT, respectively.

The SAT Suite of Assessments give Geometry the short shrift -- and I don't necessarily like this, since I am writing a Geometry blog, after all. Furthermore, using the three Saxon courses as the three high school courses from Grades 9-11 violates Consistency Core, as all three Sidwell tracks lead to Geometry courses (including Math I on the high track).

I've been thinking about how I'd like to design my own Geometry test. I believe that it's possible to design a single Geometry test that students on all three tracks can take.

To see how this can possible, let's review the differences among the three Geometry courses. On the low track, the class uses Michael Serra's Discovering Geometry, which, as we know, emphasizes the concepts of Geometry but contains few formal proofs until the final chapter. The middle track Geometry course has students work on more proofs. The high track Math I course contains the hardest proofs -- no, not Putnam-level proofs, but still are harder than those on the middle track. We see according to the Sidwell website:

Math I is an intensive and accelerated course in geometry recommended for very able and independent math students. The topics of Geometry are covered with greater attention to rigorous proof and the deduction of results from a small number of postulates. Additional topics include advanced constructions, loci, proof by contradiction, a more intensive study of trigonometry, and probability and data interpretation. This course is student-driven and inquiry based, and students must be prepared to take responsibility for their own progress.

So we see that Math I assumes fewer postulates than middle track Geometry -- that is, statements that are postulates in middle track Geometry are theorems in Math I. Note that transformational geometry actually satisfies this requirement -- statements such as the Corresponding Angles Test, SAS, SSS, and ASA are all postulates in most texts but are theorems in transformational geometry. But we don't know what Sidwell's Math I is like. It could be that Sidwell's Math I assumes SAS as a postulate, and uses it to prove ASA, as Euclid does. In that case Math I is still more rigorous than middle track Geometry, but doesn't use transformations.

For the sake of argument here on the blog, I will assume that the high track Geometry course uses transformations to prove the statements that are taken as postulates on the middle track. And the middle track in turn proves the statements that are taken as postulates (or conjectures) on the lowest track, following the Serra text.

And so here's how my single Geometry text would work -- the first few questions would be those that a student in the Serra class can answer, the next few questions would be those that a student on the middle track can answer, and the last few would be for students on the high track. To make it easy, let's say there are 30 questions -- Questions 1-10 are for Serra, 11-20 are for the middle track, and then 21-30 are for the high track.

Notice that the final Serra chapter is on proofs. A student in the Serra course should be able to answer Questions 1-10 easily, but might be able to sneak a few extra points in by using what they learned in the last chapter to answer, say Questions 11 or 12. Likewise, we can cover transformations in the last chapter on the middle track, so that students on that track may be able to answer the first few high track questions.

What would the texts for the three tracks look like? The only text we know is used at Sidwell is Serra's Discovering Geometry. It may sound appealing just to use the Serra text on the higher two tracks, but just prove more and more statements along the way rather than wait for the final chapter as Serra does. But this is problematic -- it's because Serra doesn't have to prove the conjectures that he can order them however he wishes. Let's recall the first conjecture in Serra's Chapter 3:

C-1. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints (Perpendicular Bisector Conjecture).

On the middle track, we'd probably use SAS to prove the Perpendicular Bisector Theorem. But SAS doesn't appear in Serra until later on (Chapter 5 old version, Chapter 4 new version). Serra can state Perpendicular Bisector before SAS because it's only a conjecture, so it doesn't really matter that SAS is needed to prove it. On the middle track, we'd have to state SAS first. Ironically, on the highest track, we can follow Serra's order, because the Perpendicular Bisector Theorem can be proved trivially using reflections (as the U of Chicago does) without SAS! But there are still later conjectures that need to be reordered, such as the concurrency theorems -- the Centroid Theorem appear in Serra's Lesson 3.8, but we're a long way from being able to prove it.

Other than that, there exist texts which save transformations for the last chapter, so these would serve the purpose that I stated above for the middle track -- use postulates to prove the main theorems and then introduce transformations at the end so that students can answer one or two more questions. The following is a link to a North Carolina high school that uses such a text -- unfortunately, the publisher of this text is not evident:

On the low and middle tracks, there wouldn't be lessons like today's on glide reflections anymore. We see that a glide reflection the composite of a reflection and translation. This is some of what I wrote last year about glide reflections:

What, exactly, is a glide reflection? Well, here's how the U of Chicago defines it:

Let r be the reflection in line m and T be any translation with nonzero magnitude and direction parallel to m. Then G, the composite of T and r, is a glide reflection.

Just as reflections, rotations, and translations have nicknames -- "flips," "slides," and "turns," respectively -- glide reflections have the nickname "walks." The U of Chicago gives the example of the isometry mapping the right footprint to the left footprint while walking as a glide reflection. Another name for glide reflection is "transflection," since it is the composite of a reflection and a translation.

I once tutored a geometry student who had a worksheet on glide reflections. The student had to use a coordinate plane to perform the glide reflections, which were given as the composite of a reflection and a translation. But the problem was that on the worksheet, the direction of the translation wasn't always parallel to the reflecting line! In fact, in one of the problems the translation was perpendicular to the reflecting line. That would mean that the resulting composite wasn't truly a glide reflection at all, but just a mere reflection!

At any rate, I could set up a Geometry test that assess students on all three tracks. But as soon as we introduce a Geometry test, we immediately violate Consistency Core, because the Geometry test would be one that Sidwell students don't have to take. That is, Sidwell students take a Geometry class, but not any Geometry standardized test. I'd have to decided whether it's preferable to allow students take my special Geometry test in lieu of the PSAT the year that they take Geometry, or else insist that they take only the PSAT and SAT as standardized tests.

My Presidential birthday plan involves tracking. I get worried whenever I divide students into tracks, because previous tracking systems ended up segregating students by demographic factors such as class, ethnicity, or gender. Of these, gender sounds like the easiest to focus on. My concern is that students will find themselves stuck on lower tracks leading to lower-paying jobs -- and this would contribute to the gender wage gap.

There are two ways to enforce equality -- either make sure that there are equal numbers of males and females on each track, or make sure that all tracks lead to well-paying jobs. Notice that inequality with respect to either of these (number on each track, pay to which each track leads) is acceptable as long as there is equality for the other.

Some people may point out that males and females may choose different jobs that they enjoy -- for example, females may prefer jobs with a lot of reading or writing, while males may prefer hands-on jobs and careers -- and the difference is okay because members of each gender choose the jobs that make them feel happy. The problem is that people may enjoy their jobs and find themselves happy 29 out of 30 days per month, yet complain on the last day, payday, when the gender wage gap appears. (I know, most jobs have paydays twice a month, but I always think in terms of being a teacher, where we get paid only once a month.) Only when the gender gap is resolved can we even think about other gaps, such as the ethnicity gap.

Interestingly enough, last night's Simpsons episode was about gender and academics -- in particular, Lisa Simpson discovers that a woman inventor who lived in Springfield in the 19th century wasn't taken seriously because of her gender. That's the sort of issue that I want to address when dividing students into tracks.

Here is also where Presidential Consistency comes into play -- some people accuse the current administration of being hypocritical when it comes to the gender wage gap. They say how the president claims that he is committed to closing the gap, yet they found that female White House staffers make less money than their male counterparts. To be consistent, I'll make sure that females on my own staff are paid the same as the males, in order to avoid that accusation.

There's one more important part of the plan to mention, and that's teacher accountability. We already know that many high school students don't take the PARCC/SBAC seriously -- they should take the SAT/ACT more seriously, but still, teachers worry whether they'll be judged by test scores when the students don't make any effort to do well on the tests that have no effect on their grade. Yet, as proponents of teacher accountability would point out, if teachers have little effect on student outcomes, then why pay teachers at all?

So here's my solution -- if (and that's a big "if") a percentage of teachers' evaluations are determined by test scores, then an equal percentage of the students' grade should be determined by the scores.

For example, in New Jersey, test scores count for 10% of teacher evaluations. Therefore, tests should count for 10% of a student's' grade. And New York is considering making test scores count for 50% of teacher evaluations, so in New York they should count for half of a student's grade. Of course, if tests count for 0% of a teacher's evaluation, then they don't have to count for students either.

Of course, we'd have to find a way to convert SAT/ACT scores into student grades. For taking a naive percentage out of 800 (with the verbal score contributing to English grades and the math score contributing to math grades) causes problems -- a blank test scores 200/800 or 25%, which is too high (though not as bad as the 0=50 rule), while the average test scores 500/800 or 62.5, a low D, which seems too low. (It sure beats trying to convert Putnam scores, where the average score is around 1 point out of 120.) So there would have to be some formula to convert scores on the 200-800 scale to those on the 0-100 scale.

Thus completes my definition of the Presidential Birthday plan. Of course, there are still problems I'd have if I were elected president. For one thing, presidents don't create legislation -- Congress, of course, does that. And the new NCLB authorization is being discussed in Congress right now, and it's likely to be completed and signed by the current president by the time I reach the White House. And of course, I don't have any idea as to what I'd do about foreign policy or any other issues that presidents have to deal with.

And so I hereby end my candidacy for President of the United States and drop out of the race. Well, it was fun while it lasted. I guess that I'll just have to enjoy the rest of my 35th birthday on my own.

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