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:
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]:
(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:
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:
BBBCA AADAB BBADA CADBC ABBAA BDDAC DDBCC ADBBA BDABC ADDCA
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...