Friday, March 2, 2018

Chapter 11 Test (Day 120)

Today is many things. According to the blog calendar, today is Day 120. Therefore, it is the end of the second trimester -- at least it would be if this calendar weren't based on a high school district. My new district does have K-8 students -- except in this district, today is only Day 112. Therefore it's actually the end of the fifth quaver at the high schools, not the end of the second trimester in K-8.

Today is Dr. Seuss Day. My new district has elementary schools, and so I assume that they celebrate the famous author today. I subbed today in a high school biology class, so of course Dr. Seuss Day isn't observed at this high school -- or is it? One student accidentally leaves his folder in my class, and I look at the papers inside to identify its author. The student's English assignment is to answer questions about a certain book -- The Lorax, by Dr. Seuss. So apparently, even high school English classes celebrate Dr. Seuss Day!

Today is the full moon (last night in some time zones). Several cultures with lunisolar calendars associate holidays with this particular full moon. In China, today is the Lantern Festival. In Israel, today is Shushan Purim. And in India, today is Holi. The last few years (including this year), there is a Google Doodle for Holi. It is said that Holi is a celebration of the coming of spring and the triumph of good over evil. In a way, this describes Purim and Lantern Festival as well -- the only difference is whether one is celebrating the triumph of Krishna, Esther, or Buddha. The three holidays are also associated with color, costume, and light. Typically Purim is one month after the Lantern Festival, with Holi sharing the same full moon with either of these. But 2018 is a special year in which all three holidays share the same full moon. This is the first of two full moons in March -- the next moon will be the second blue moon of the year.

And finally, today is the day of the Chapter 11 Test. All those other celebrations might be fun, but this is a Geometry blog, so I must post the test -- sorry.

But first, last night I watched the episode of NOVA: "Prediction by the Numbers." Here are some key ideas mentioned in the episode:
  • The Wisdom of the Crowd -- when estimating the number of jellybeans in a jar, individual guesses many vary wildly, but the mean of the guesses is often close to the actual number.
  • The Law of Large Numbers -- first described by Girolano Cardano (who also came up with the Cubic Formula and imaginary numbers), in the short run the ratio of heads to tails may vary wildly, but in the long run the ratio will approach unity. It's the Wisdom of the Crowd applied to probability. Casinos depend on it to make money.
  • Numerical Forecasting -- meterologists may use this to predict future weather. The first step is to use three-dimensional coordinates to label points in the atmosphere -- it sort of makes me regret posting a 2-D graph on Lesson 11-6 day! Measurements are taken at each point, and these are used to predict weather probability, as in the percent chance of rain.
  • Polling and Election Forecasting -- oops, I mentioned politics in this school-year post. The outcome of the 2016 presidential election led some people to distrust statistics, since the candidate overwhelmingly predicted by pollsters to win the election, well, didn't. Sampling and margin of error are described in fuller detail. It goes without saying that Nate Silver, the famous political statistician, makes an appearance here.
  • Sports Analytics -- baseball and other sports now use advanced statistics when evaluating the worth of their players. The most well-known example is Moneyball -- the Oakland A's of the late 1990's and early 2000's. General manager Billy Beane used analytics to construct the team roster, and it was successful in that the team made the playoffs in four consecutive years and once won 20 consecutive games.
  • Bayesian Inference -- conditional probability is used to make predictions. It has wide-ranging applications, from locating a victim in a search-and-rescue mission on the shore, all the way up to artificial intelligence.
The show ends with a short preview of an upcoming episode, "The Great Math Mystery."

And as a bonus, after NOVA is another math-related documentary, "Counting from Infinity: Yitang Zhang and the Twin Prime Conjecture," from a few years ago. I've mentioned twin primes only once on the blog in passing, back when discussing Stanley Ogilvy's book in my September 25th post.

Twin primes are pairs of primes whose difference is two (such as 3 and 5, or 11 and 13.) The Twin Prime Conjecture is the statement that there exist infinitely many such pairs. This conjecture hasn't been proved, but Yitang Zhang is a Chinese mathematician who made significant progress. Here is another link to a description of the film:

In April 2013, a lecturer at the University of New Hampshire submitted a paper to the Annals of Mathematics. Within weeks word spread-- a little-known mathematician, with no permanent job, working in complete isolation had made an important breakthrough towards solving the Twin Prime Conjecture. Yitang Zhang's techniques for bounding the gaps between primes soon led to rapid progress by the Polymath Group, and a further innovation by James Maynard. The film is a study of Zhang's rise from obscurity and a disadvantaged youth to mathematical celebrity. The story of quiet perseverance amidst adversity, and Zhang's preference for thinking and working in solitude, is interwoven with a history of the Twin Prime Conjecture as told by several mathematicians, many of whom have wrestled with this enormously challenging problem in Number Theory-- Daniel Goldston, Kannan Soundararajan, Andrew Granville, Peter Sarnak, Enrico Bombieri, James Maynard, Nicholas Katz, David Eisenbud, Ken Ribet, and Terry Tao.

The film explains how Zhang (or Tom as he's affectionately called) considers the primes that appear in arithmetic sequences -- for example 4n + 1 and/or 4n + 3. It turns out that if we go far enough in these sequences (There's that Law of Large Numbers again!) the number of primes in these sequences will be equal. Tom is able to show how this fact relates to the Twin Prime Conjecture:

Although mathematicians had tried for centuries to prove this simple statement, before Zhang's work they couldn't even show that there was any number N such that there were always exceptions where the gap is at most N.

Zhang's breakthrough was to show that there is such a bound N-that you can always find more pairs of primes with a gap at most 70,000,000. Though this number is a lot bigger than 2, it is really small compared to infinity!

Despite Tom's research, the Twin Prime Conjecture hasn't been completely proven yet:

Zhang's work unleashed a flood of new results. A remarkable collaborative effort chipped away at the 70,000,000, making many small breakthroughs. Then James Maynard of Oxford University, using a different approach from Zhang's, brought the number down even further. Today it is under 250. (The goal is still 2; and that still seems far away.) In addition to improving the bound, many other new results have become possible because of Zhang's work.

The film compares Twin Primes to a similar conjecture -- Goldbach. It's pointed out that the Goldbach Conjecture is all about the sum of primes, while Twin Primes is about their difference.

I enjoyed my two hours of math programming on PBS. But there's more I want to say about these two shows, particularly NOVA.

Today is a test day -- hence a "traditionalists" post. The NOVA episode emphasizes the significance of statistics and probability, and how stats and prob are used to understand predictions better. This is a strong argument in favor of including more stats and prob in the math curriculum -- and I mentioned earlier this week, the Common Core does exactly that. Again, here in California, statistics is included in Algebra I, and probability is included in Geometry. This is why Chapter 13 of the Glencoe Geometry text is all about probability.

But, as we know, some people are opposed to teaching statistics in high school -- and of course, I'm speaking of the traditionalists. We know that one traditionalist in particular, "Bill," argues that students can't truly understand stats effectively without first having taken Algebra II.

Let's assume for the sake of argument that both NOVA and Bill are correct:

  • In order to understand predictions better, we must understand statistics.
  • In order to understand statistics better, we must understand Algebra II.
  • Therefore, in order to understand predictions better, we must understand Algebra II.
If this conclusion holds, then this would make Algebra II a very useful subject to learn, with many real-world applications. Yet, of course, the students who are actually sitting in Algebra II classes tend to ask questions like "Why should be learn this?" and "When will we ever use this?"

And so I wonder whether there can be a middle ground. People are bombarded with statistics and probabilistic predictions in the media all the time. Is it possible for us to gain a better understanding of all these figures without having to sit through an Algebra II class?

The TV program NOVA, like many PBS shows, is funded by the National Science Foundation. The NSF is mentioned in the latest Barry Garelick post:

The Education and Human Resources Dept (EHR) of the National Science Foundation (NSF) has been granting millions of dollars to beef up math education in the US. In the early 90’s they gave grants to entities to create math programs/textbooks that parents have been protesting for years:  Everyday Math [...] other words, the U of Chicago elementary text. We can see why traditionalists aren't fond of the NSF -- it funds TV shows that push math classes they don't like (such as Statistics) as well as textbooks they don't like (such as the U of Chicago elementary text).

Garelick continues:

Their latest gift is $2.8 million to the University of Houston’s education school  The grant covers new courses in the ed school as part of a masters program whose title tells much of the story:  “Enhancing STEM Teacher Leadership Through Equity and Advocacy Development in Houston”

The courses (and of course coaching—what program would be complete without coaches that help teachers to stop teaching and start facilitating) include topics such as:

  • Culturally responsive teaching and addressing learning disparities in STEM education.
  • The roles of technology and inquiry-based instruction in STEM education
  • Engineering design
As long as we’re paused at the site of the bloody accident, let’s take a closer look. The first bullet doesn’t go into a lot of detail but I wondered if they were going to take the view that math is all about “white privilege” and should be taught differently.  Meaning, from what I’ve read of such arguments, that it be watered down or discarded entirely.
Oops -- I already brought up politics in this post, and now I mention race in this post. Fortunately, I'm protected by the "traditionalists" label, since after all I'm quoting Garelick who brings up race.

The relationship between traditionalism and race is a bit complex. We know, for example, that many traditionalists advocate tracking as an effective method of teaching math. But tracking is discouraged, because our past experience with tracking shows that students of certain races tend to be placed on certain tracks.

Here Garelick suspects that progressive pedagogists are trying to associate traditional math with "white privilege" in order to make it as toxic as tracking, and then propose constructivist math as a race-neutral alternative. But Garelick, most likely, doesn't want to think about race at all -- he just wants to teach traditional math. From his perspective, it's the constructivists who inserted race into the discussion of pedagogy. (Again, this is all Garelick's suspicion -- he admits he doesn't entirely know yet what this course is all about.)

As for Garelick's second bullet, we pick up the story from Garelick's most frequent commenter. (So SteveH has only one comment in this thread -- so far, that is.)

I saw something on the TV the other day that talked about “Girls in STEM” and showed a hands-on display where kids could make various sand mountains while a system would project line contours on the sand. The kids could see how the contours changed as they moved the sand. How about going to Google maps, turning on “terrain”, and zooming in on a ski area to see why the trails are shown in green, blue, and black? Have them count the contour lines for a trail length to estimate the slope needed for each color. These facilitators have absolutely NO CLUE even for engagement.

I'm not quite sure what SteveH is saying here. Apparently he's suggesting the Google maps activity as an alternative to the sand mountain project he saw on TV (another PBS show?). To SteveH, the progressive pedagogy doesn't even successfully reach its stated goal of engagement, much less full preparation for STEM.

It’s all really quite incredible. Meanwhile, traditional high school AP Calculus track math teachers keep quiet and create real STEM prepared students. For all other teachers trying to create STEM students, they are left struggling to just promote ‘T’, put all of the onus on the students, and are thrilled if they get to a vo’T’ech school. That’s what engagement-driven learning is all about – hacking your way to a vocational degree – never to a true STEM degree that requires at least differential equations.

As usual, we see here which classes SteveH prefers -- AP Calculus and Differential Equations. Of course, Statistics is definitely not on his "AP Calculus track." So to SteveH, progressive STEM only prepares students for one of the four initials in STEM.

This is what I wrote three years ago about today's test. I decided to preserve some of the traditionalist discussion that I wrote about in that post as well:

On the other hand, I read anecdotes from traditionalist homeschooling parents about how their second grader learned fifth grade math effectively via direct instruction. I don't consider this to be a valid argument that the fifth grade math standards should be written from a purely traditionalist view -- because fifth graders, more than second graders, will start questioning why they have to learn how to compute with fractions. [2018 update: This isn't discussed at the traditionalist link above, but notice that one commenter wrote "homeschool" as the only rational response to progressive pedagogy.]

But this does mean that the traditionalists' favored standard algorithms and memorization of basic math facts are to be taught as soon as possible, and not delayed a year as in Common Core. One common complaint among traditionalists is that students are never made to memorize basic multiplication facts. Questions such as six times nine or seven times eight should be considered very easy questions that take no more than a second to answer. But not only do many people consider such problems to be difficult, but it has become fashionable to consider those who have difficulties with such problems to be normal and those who find such problems easy to be outliers -- nerds.

It's often pointed out that people would feel deeply ashamed to admit that that can't read at a third grade level, yet are proud to admit that they can't do third grade math. Since I've stated that third grade math is something that students should have learned traditionally -- that is, have memorized -- I should do something about it in my classes.

The thought is that, rather than have those who find single-digit multiplication to be easy be outliers who get the label nerd, it's those who can't multiply by the time they reach middle and high school who should be considered outliers -- just as someone who can't read at a third grade level is taken to be an outlier. But of course, it's improper for me, a teacher, to start calling my students derogatory names such as idiot, no matter how low their understanding of math is.

So I need a word that criticizes the student, yet is proper for me to use in a classroom. Well, since I want my word to have the opposite effect of the word nerd, I briefly mentioned at the end of one of my posts a few months back that I made up my own word, by spelling the word nerd backwards, to obtain "dren."

My plan is to use my new word "dren" in such a way to make it sound as if a "dren" is not what a student wants to be. For example, when we reach the unit on area, students will need to multiply the length and width to find the area of a rectangle. So I might say something like, "A dren will have trouble multiplying six inches by nine inches. Luckily you guys are too smart to be drens, so you already know that the area is ...," and so on. Similarly, if a student, say, starts to reach for a calculator to perform the single-digit multiplication. I can say, "You're not a dren. You know how to multiply six times nine ...," and so on.

Notice that in these examples, I don't call anyone a dren directly. But every time I say the word "dren," I want to be annoying enough so that the students will want to do what it takes to avoid my having to say that word.

I coined the word "dren" to be the word nerd spelled backwards. But ironically -- according to my new Simpsons book -- the word nerd is already spelled backwards! Originally, the word was "knurd," which is drunk spelled backwards. The net result is that my word "dren" is basically just an abbreviation of drunk. Of course, the word drunk isn't a word that I should use in the classroom!

[2018 update: Earlier this week I wrote about I might strengthen my anti-dren crusade. I can tell math classes that there are three types of students -- "smart," "almost smart," and "dren." Now I don't teach drens -- I choose to work at your school because the students are all "smart" or "almost smart." Those who are struggling with a concept are "almost smart" until they master the concept. This extends the idea from three years ago that I don't call any student a "dren" directly. Every student who works hard is at least "almost smart" -- and I regret having never said this to the "special scholar" last year.]

Here are the answers to today's test.

1. Using the distance formula, two of the sides have the same length, namely sqrt(170). This is how we write the square root of 170 in ASCII. To the nearest hundredth, it is 13.04.

2. The slopes of the four sides are opposite reciprocals, 2 and -1/2. Yes, I included this question as it is specifically mentioned in the Common Core Standards!

3. Using the distance formula, all four sides have length sqrt(a^2 + b^2).

4. Using the distance formula, two of the medians have length sqrt(9a^2 + b^2).

5. 60.

6. From the Midpoint Connector Theorem, ZV | | YW. The result follows from the Corresponding Angles Parallel Consequence.

7. From the Midpoint Connector Theorem, BD | | EF. The result follows by definition of trapezoid.

8. 4.5.

9. (0.6, -0.6). Notice that four of the coordinates add up to zero, so only (3, -3) matters.

10. At its midpoint.

11. 49.5 cm. The new meter stick goes from 2 to 97 cm and we want the midpoint.

12. Using the distance formula, it is sqrt(4.5), or 2.12 km to the nearest hundredth.

13. sqrt(10), or 3.16 to the nearest hundredth.

14. 1 + sqrt(113) + sqrt (130), or 23.03 to the nearest hundredth.

15. sqrt(3925), or 62.65 to the nearest hundredth. (I said length, not slope!)

16. -1/2. (I said slope, not length!)

17. (2a, 2b), (-2a, 2b), (-2a, -2b), (2a, -2b). Hint: look at Question 5 from U of Chicago!

18. (0, 5).

Oh, and here's one last 2018 update -- if you want, you can add the following questions, as the equation of a circle is still missing:

In 19-20, determine a. the center, b. the radius, and c. one point on the circle with the given equation.

19. (x - 6)^2 + (y + 3)^2 = 169
20. x^2 + y^2 = 50

Here are the answers:

19. a. (6, -3) b. 13
Possible answers for c: (19, -3), (18, 2), (11, 9), (6, 10), and so on.

20. a. (0, 0) b. 5sqrt(2)
Possible answers for c: (5, 5), (-5, 5), and so on.

I hope you enjoyed your Holi/Lantern Festival/Purim/Dr. Seuss Day/end of trimester/full moon day/whatever you celebrated today!

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