Tuesday, January 19, 2016

Semester 1 Final Exam (Day 86)

Lecture 12 of David Kung's Mind-Bending Math is called "Why No Distribution Is Fully Fair." In this lecture, Dave Kung reveals another election-related paradox. Last time he gives Arrow's Theorem which shows that a perfectly fair voting system is impossible, and now he will show what's wrong with a perfectly fair apportionment system.

Kung is talking about representatives from states, such as in Congress. He begins by telling us that in an ideal representation system, the number of seats should be proportional to the population. That is, we simply divide the population by some constant -- which he calls the standard divisor -- and obtain the number of seats -- the standard quota. It sounds simple -- it's what we teach to our middle school students when they are learning about proportions, and of course Geometry students need to consider proportions when it's time to study similar triangles.

But here's the problem -- most of the time, the standard quota is not a whole number -- and of course states should get a whole number of seats. All of the paradoxes that Kung describes involves getting whole numbers out of the standard quotas.

At this point, you may be wondering, why not just round the standard quotas, just as we teach elementary school students to use 4/5 rounding? Kung gives a simplified example where the entire population is 60 and there are 13 seats. Let's make the example even easier to understand -- there are 100 citizens and 10 seats:

State A has a population of 53.
State B has a population of 33.
State C has a population of 14.

The standard divisor is obviously 10. Dividing each population by the standard divisor gives:

State A has a standard quota of 5.3, which rounds down to 5.
State B has a standard quota of 3.3, which rounds down to 3.
State C has a standard quota of 1.4, which rounds down to 1.

But now the three states have a total of nine seats, when there are supposed to be ten seats in our hypothetical legislature. It's obvious what the problem is -- all three states rounded down. If at least one of the three states had rounded up, it's possible that the total number of seats really could be 10.

At this point, it seems that the best thing to do is assign State C the extra seat, since the .4 fractional part of 1.4 is closer to being rounded up than the .3 fractional parts. This method of apportionment was first described by Alexander Hamilton, and so it's called Hamilton's method. So under Hamilton's method, State A gets five seats, State B three seats, and State C two. With Hamilton's method, every state gets either the floor or the ceiling of its standard quota. Any method for which this is true follows, according to Kung, the quota rule.

But Hamilton's method leads to problems. Kung mentions that in 1880, it was decided that the House of Representatives should gain an extra seat -- instead of 299 congressmen, there would be 300. So which state got the extra seat? As it turned out, two states would gain a seat, and one state, Alabama, would actually lose a seat. This is now known as the Alabama paradox.

Suppose the three states decided to increase the legislature, so there are now 11 seats. Let's see how many seats each state gets:

State A now has a standard quota of 5.83.
State B now has a standard quota of 3.63.
State C now has a standard quota of 1.54.

We can't round all three states up because that would give us 12 seats. So we must round one of them down -- and that must be the state with fractional part .54, as this is the smallest. So we see that State A gets six seats, State B four seats, and State C one. So notice -- poor State C! When there are 10 seats, State C got two of them, but when there are 11 seats, State C gets only one! This shows how Hamilton's method leads to the Alabama paradox.

Before watching this lecture, I had heard of the Alabama paradox, but I didn't know how it was eventually resolved. Kung tells us that nowadays, we use the Huntington-Hill method. The new formula is too complicated for him to describe, but he tells us that it has something to do with the geometric mean of two consecutive integers. (At this point Kung gives an aside -- why do we call the square root of the product of two numbers the "geometric mean"? He says that this mean is indirectly related to Geometry. The geometric mean of two numbers is the side length of a square whose area equals that of a rectangle whose side lengths are the two given numbers.)

Kung proceeds to tell us that even though Huntington-Hill avoids the Alabama paradox, it doesn't adhere to the quota rule. We could try to come up with another method that adheres to the quota rule, but this isn't be desirable. Two mathematicians, Balinski and Young, proved that any method that adheres to the quota rule leads to the population paradox -- it's possible for one state's population to increase and another's to decrease, yet the former state must give up a seat to the latter! In other words, just as with Arrow's Theorem, Balinski's Theorem proves that there is no completely fair apportionment method!

Kung shows us why, using more hypothetical states and populations. I notice that in his examples, when the quota rule is violated, it's just barely violated -- for example, a state with a standard quota of 4.99 may receive six seats, or a state with a standard quota of 4.01 may get three seats. It's not as if a state with a standard quota of 4.99 is getting ten seats while another with a standard quota of 4.01 is getting only two.

The Quick Conundrum for today concerns light filters. Kung sets up a screen with an image, and shows how placing a filter on top of the screen makes the image almost impossible to see. Yet when he places a second filter on top of the image, the image is easily visible! Kung tells us how, in some sense, the first filter bends light waves in a certain direction while the second filter bends them back.

There's one thing about apportionment that I'm surprised Kung didn't mention. I notice that in some of his examples, a state may get zero representatives. But I believe that in reality, every state must get at least one representative, no matter how small its population is. If I somehow became the only citizen of the 51st state, Davidia, then Davidia gets one congressman -- and of course, I would be that congressman by default. (But now I wonder who Davidia's two senators would be.) Because state gets at least one representative, larger states are now even more likely to get fewer representatives than the quota rule would require.

Today's lecture is the last one related to elections. And so it's a good idea for me to hold my weekly traditionalist post today. Since I've already fantasized about being president or governor, maybe I should pretend to be a congressman and start writing proposals to fix Common Core...

I do read that one New York congressman, Lee Zeldin was responsible for the Zeldin Amendment to the Every Student Succeeds Act. The Zeldin Amendment makes it easier for states to drop out of the Common Core. I'm actually curious whether any Congress members from my home state (California, not Davidia) contributed to any key parts of the bill.

By the way, I'm curious about how traditionalists are reacting to last month's passage of ESSA. Well, here is what the traditionalist Dr. Ze'ev Wurman has to say:


“ESSA still talks about ‘post-secondary readiness,’ albeit in sotto voce,” he [Wurman -- dw] explains. “Further, the Secretary of Education still has to approve state standards as a part of state plans, which allows the Secretary to push his way around.”
“What the ESSA does cement is the annual testing monstrosity,” Wurman continues. “A massive annual multiple-grades testing regime that I am unaware of any country on earth having anything like it.”
Like NCLB, the ESSA still requires annual testing in grades 3-8 and then again in high school.
Wurman explains:
It makes some sense for annual testing if one wants to create somewhat reliable trajectories of students to full proficiency, or if one wants to use the annual scores to create “growth” scores, so teachers’ added value can be measured. But ESSA gave up on getting all students to proficiency, and ESSA gave up on any kind of teacher accountability based on students’ scores. In that case, who needs annual testing with all the inevitable teaching-to-the test? That is, besides testing companies like Achieve, Pearson, and the like. What’s wrong with grade-span testing, say once in 3-5, and once in 6-8?
Also, if we don’t use test results for teacher evaluation, why the ESSA’s insistence that the test must be aligned with the curriculum? Now that teacher evaluation is dead, shouldn’t we at least get rid of this “alignment” monstrosity?
“ESSA makes annual testing of grades 3-8 a goal in itself, without any educational rhyme or reason,” Wurman adds. “In this sense ESSA is worse than NCLB.”

Notice that my Gubernatorial Plan actually does make use of the annual testing, as my proposed scoring system does "create growth scores." But since the actual ESSA doesn't do anything of the sort, Wurman prefers to reduce testing by having it occur less often than once a year.

It's also interesting that Wurman doesn't want the test to be aligned with the curriculum -- but then again, how can students pass the test if it isn't aligned with what they learned? Actually, I assume what he means is for testing to be aligned with the Common Core curriculum, or state standards that deviate little from Common Core (because they're still influenced by the Secretary of Education, as he writes above). If the tests were aligned with the curriculum that the traditionalists favor, I assume they'd have no problem with that alignment.

Recall that my Gubernatorial Plan is a work in progress. I will continue to seek out what the Common Core opponents would prefer instead of the Core, and try to fix my Gubernatorial Plan so that it satisfies their preferences.

Here's a link to another opponent of Common Core, Peter Wood:


I don't know enough about Wood to conclude that he's a traditionalist, but in the above link, he expresses several desires compatible with those of the traditionalists. He writes the following [with my comments included]:

Slowing Down Math Instruction
Common Core math hurts students in other ways. First, it slows the pace of instruction. Before Common Core was in place, some states reasonably expected students to master basic addition and subtraction by 3rd grade; Common Core decided 4th grade would do. A similar situation arose with the multiplication table. Long division was generally a 5th grade skill; Common Core defers it to 6th grade.
[Actually, the Core does teach the indicated operations in 3rd and 5th grade -- the problem is that it teaches alternate algorithms and waits an extra year to teach the standard algorithms. We already know what the traditionalists' solution is -- just teach the standard algorithms a year early and nix the nonstandard algorithms.]
These changes may seem small by themselves, but they are extremely important in cumulative effect. At a time when other developed nations are racing ahead in science, technology, engineering, and math education, the United States has decided not to accelerate but to move into the slow lane.
Because math builds on itself, a slow pace in early education means a more significant slowdown later on. For instance, algebra often gets pushed back to 9th grade, and then Common Core tapers off. 
[Wood doesn't actually say that Algebra I should be taught in 8th grade, as most traditionalists agree, but he strongly implies it by referring to teaching it in 9th as "pushed back."]
It has no room at all, for example, for pre-calculus instruction, which used to provide the bridge for students about to head off to college.
[We've discussed this before as well. If Algebra I is taught in freshman year, then Pre-Calculus doesn't get taught until senior year, yet the Common Core tests are given to juniors. Of course Wood and the other traditionalists want Algebra I to be taught in 8th grade, thereby placing Pre-Calculus in junior year -- meaning that it would be on the tests.]
Logarithms are barely mentioned. Parametric equations are absent. Arithmetic series are omitted. Polar forms of functions never come up.
[Here's where I disagree with Wood the most. I myself, who went to high school before Common Core, didn't learn about polar functions until Calculus BC and parametric equations not until my first year of college. If it weren't for my exploring the "Par" and "Pol" modes on my TI-83, I might not have heard much about of either of these at all. Meanwhile, arithmetic series absolutely appear in Common Core Math -- indeed, they were on the quiz I gave the 8th graders last Friday. Notice that Wood mentions all of these topics right after mentioning a Pre-Calculus course, implying that's when he believes all of these should be taught.]
Of course, most of us adults live without these pieces of mathematical knowledge. We studied them once and moved on to study other subjects that didn’t depend on “parametric equations.” 
[But not everyone is so lucky. Some people can't study them once and move on because they can't get a passing grade during the one time that they're studying them. That failing grade may prevent them from graduating high school or getting into college, therefore blocking them from moving on to study other subjects.]
What’s the harm of not teaching them in the first place? The harm is by not providing instruction to young people at the age in which they can absorb the knowledge, we preempt the whole possibility of their going further. We are effectively slamming the door shut for millions of children on possible careers in the sciences, engineering, and many technical fields, in which a solid foundation in math is crucial.
[But Wood's plan harms students who can't ever pass the parametric equations course. By forcing students to take it and fail, we preempt the whole possibility of their going further. We are effectively slamming the door shut for millions of children on possible non-STEM fields, in which a solid foundation in math is irrelevant. Also, it's ironic that Wood begins the article with a picture of Common Core forcing all pegs into a square hole, yet the traditionalists want to force all pegs into a round hole.]
Similarly, Common Core approaches geometry from 5th grade to 8th grade as though it is nothing more than measurement, and then it abruptly turns to an effort to derive the rest of geometry from the study of “rigid motions”—a way of teaching geometry tried once or twice before, notably in the Soviet Union in the 1980s, where the experiment was deemed a failure and discarded.
[The idea that transformational geometry dates back to the Soviet Union comes directly from Wurman, so again, Wood's opinion overlaps with those of the traditionalists. Meanwhile, he criticizes the middle school standards for treating geometry like "nothing more than measurement." But I wonder what his alternative would be -- what should middle scholars learn for geometry besides measurement, if not transformations?]
(Okay, I said that this will be the only traditionalist topic this week, but I lied. At the end of the week I will post my Semester 2 Preview, and the way that I'll approach a certain topic early in the second semester comes directly from one traditionalist's website.)

Speaking of traditionalists, since I've announced that I'm participating in the MTBoS four-week blogging challenge, it's interesting to look at this from the traditionalists' perspective. I've mentioned before that most members of the MTBoS aren't traditionalists. After all, we don't blog to come up with traditional teaching methods -- we blog to come up with new teaching methods. So don't expect traditionalists to like anything that I, or any other participant, posts during these four weeks.

In my last post, I described a day of subbing in a math class in more detail. But let's look at that day again from the traditionalists' perspective:

I called the fourth period class an "honors" eighth grade math class even though it wasn't officially an honors class. Traditionalists have their own version of "honors" eighth grade math -- and of course, it's called Algebra I. Certainly, they'd argue that these students in that fourth period class should be in Algebra I, not Common Core 8 and bound to take Integrated Math I next year (which they dislike).

Notice that sprinkled throughout that post, I found "one good thing" to say about several of the classes I had that day. But traditionalists argue that we, as teachers, focus so much more of our time and effort on the struggling students and not the ones who are excelling. I thought it was great when the students in the third period class -- many of whom were failing -- were working hard to understand the material before the quiz (if only I'd had more time to help them). I suspect that if a traditionalist were the one subbing that class, he or she would only be able to find "one good thing" to say about the fourth period "honors" class. Only the students in that class were getting most of the answers right and answering the more difficult questions with fractions and decimals. (And I bet traditionalists would throw out all the questions on that test that required students to give explanations.)

Tomorrow I'll make my MTBoS Week 2 post. But I like to give links to interesting math teacher blogs that I find (whether it's MTBoS month or not), so let me comment on the three blogs I mentioned at the end of my Week 1 post, now that I've had time to peruse them:


Math Milla (the username refers to her last name, Mrs. Miller) is an Ohio high school teacher -- she teaches Algebra I and Calculus at a Catholic school. Her "One Day in the Life" post mentions only three classes, because her school is on a block schedule. In my comment to her, I wrote about my own experience as a high school student with the block schedule:

Yes, I attended a high school with a block schedule as well. Our school didn't have late days, and this is California, so of course we didn't have snow days either.

When I was a student teacher, I taught Algebra I, and I used foldables similar to the one you posted. Using foldables definitely encourages freshmen who might otherwise tune out to take notes.

You say that this lesson went well. Assuming that by "the 3rd method of solving systems" you mean elimination (the 3rd method on the foldable), then yes, students do seem to perform better with this method than the other two.

And yes, even Calculus students have trouble setting up the equations for word problems. After that, it's all downhill -- just take the derivative and set it to zero!

Thanks for participating with the MTBoS!

One of the two districts where I sub has a block schedule at most of its high schools, while the other has a hybrid schedule (with two block days per week) at one of its high schools. Since I rarely sub in a math class on any of the block days, I don't discuss block schedules here on my blog.

The other two blogs where I commented chose the "One Good Thing" option:


Marisa Aoki is a Central California middle school math teacher. Actually, her "One Good Thing" is not about math, but about a project she had her eighth graders complete. Here's what I wrote to her:

Hello to a fellow California math blogger!

I've heard of this concept -- Preferred Activity Time. In this case, the get well soon cards represent the Preferred Activity -- one that they'll hurry through their academic work to get through. And so apparently even here in 8th grade, PAT works!

I hope your mom treasures the get well poster she received. Yesterday, my friend showed me that he still had the get well cards that some other students and I had given him back in 3rd grade -- 26 years ago!

It's sad that she needed to have surgery so close to the holidays, and I hope that she is recovering well.

Thanks for participating with the MTBoS!


Jen Gravel is a Canadian high school teacher. In my comment to her, I wrote (where the numbered items refer to her list of "Good Things"):

Regarding #4, I admit that sometimes I get a little upset when someone asks me a question that I can't answer right away -- but your post reminds me that it's a GOOD thing when they ask such questions, as it shows that they're thinking.

As for #5, I've heard of Desmos, but I've never had the opportunity to see it used in any class. I'm looking back at your September post, which I assume shows Desmos. I can see why students can learn more about the graphs than using, say, a graphing calculator.

As for #9, I assume that the students say "Math was fun" after you show them some sort of activity -- such as Desmos. I assume this is why you do Desmos in class. It's great when we have the technology available in the classroom and use it.

As an American, I enjoy reading about how math is taught in Canada and other countries.

Thanks for participating in the MTBoS.

Meanwhile, here's a link to the blog just above mine in the list -- the one to which I was supposed to respond, but didn't want to because it was on WordPress:


The author is a South Dakota high school teacher who goes by an anonymous username. Her "One Good Thing" was about a Geometry lesson on dilations! So even though I tried to avoid WordPress, I couldn't resist commenting on her blog:

Welcome to the MTBoS!
Let me reply to your two “good things” — regarding your first, sorry, but I live in California, so I make no comment about your South Dakota weather.
As for your second comment, I know how tricky it is to teach Geometry under the new Common Core Standards. Back when we were taking Geometry in high school, we’d never heard of “dilations,” and now suddenly we have to teach them!
Those are some amazing drawings there! I’m a huge Simpsons (and Family Guy) fan, so I especially like the Homer and Meg drawings. I can’t believe that these were drawn by students who said that they couldn’t draw!
Then again, this demonstrates the power of Geometry, and why we teach students about coordinates and dilations. This is exactly how computer animation works — each pixel is a pair of coordinates that tells you or the computer where to draw. If the students didn’t have a coordinate plane, their pictures would have been as bad as they feared.
As you can tell by the title of my own blog, I write about Common Core Geometry. Your project sounds so great that I’m considering writing all about it on my blog.
I wish you luck as you complete your first year of teaching. Hopefully teaching will get easier as you progress in your career.
Note: Dilations will be one of the first topics I will post in the second semester, so expect to see a lesson similar to this one here, on this blog, very soon!

But now I must post the first semester final exam. Here are the answers to the questions on the final:


As Kung would say, democracy is tough -- whether it's trying to come up with fair elections despite the results of Arrow, Balinksi, and Young, or trying to come up with an alternative to Common Core that helps out students, teachers, and parents. I hope that Kung's next lecture will be more fun...

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