Friday, January 15, 2016

Review for Final Exam, Continued (Day 85)

This is my 300th post. It's the first time that I've used the "milestone" label on my post -- so just in case you're curious, here's a link to my 100th and 200th posts:

100th post: Section 12-10: The Side-Splitting Theorem (Day 88)
200th post: How to Fix Common Core: Tau Day Edition

Today I subbed in an eighth grade math class. Actually, this is the second time that I've subbed in the room, the first being back in late September (my 256th post -- Day 22 on the blog), and I talked a little about the California Glencoe text that the eighth graders are using back in that old post.

The students were working on patterns and linear functions, including the recursive and explicit formulas for arithmetic sequences. I noticed that the lesson was numbered 3.4 -- but this 3.4 refers not to the California Glencoe text, but to the old Mathcounts packets from last year -- the teacher had decided just assign copies of them as worksheets this year. Actually, this lesson corresponds to Chapter 4 of the California Glencoe text. As this is the last chapter in Volume I, it does make sense that they would be studying this right at the end of the semester.

Middle school students don't take finals, but they did have to take a quiz today. I had to remind the students to work hard on them, as this is their last big grade of the semester.

Lecture 11 of David Kung's Mind-Bending Math is called "Voting Paradoxes." In this lecture, Dave Kung moves from something as extremely abstract as Godel's Incompleteness Theorem to something with which most of us are familiar -- voting in elections.

Kung begins by describing U.S. presidential elections -- and of course this is a very relevant topic here in 2016. He points out how complicated the Electoral College is. It's possible for a candidate to win the popular vote yet lose in the Electoral College -- this most recently happened to former Vice President Al Gore in the year 2000. Kung also points out how the presence of a third candidate also affects the outcome of an election, such as Ralph Nader, also in 2000.

Kung proceeds to describe various possible voting systems -- such as runoff elections and agenda voting -- and points out that under most of these voting systems, Jesse Ventura isn't even close to being elected Minnesota Governor in 1998. Only by plurality voting -- each voter chooses one candidate, and the one with the most votes wins -- could Ventura win the election. Of course, that was exactly the system the North Star State had for its actual gubernatorial election.

As it turns out, none of these voting systems are completely fair. In fact, an American economist, Kenneth Arrow, proved that there is no completely fair voting system. Actually, if there are only two candidates, majority rules is a fair system according to another theorem, May's Theorem. But as soon as there's a third candidate, Arrow's Theorem applies.

Of course, this may be especially relevant if there's a third-party run by any major candidate -- by which I mean Donald Trump (the name most associated with a third-party run in 2016). Again, we only have to look back to 2000 to see the impact of a potential third-party run.

The Quick Conundrum for today asks, when does the full moon rise? Kung provides a simple answer to this -- the full moon rises at sunset. He points out that if the moon is full, then the moon must be on the opposite side of the earth from the sun. Therefore, the full moon must rise at sunset (and of course set at sunrise).

Let's continue our review for the final exam. This is what I wrote last year about the second half of the review worksheet:

Most of the questions on this half, which cover Chapters 5 through 7 of the U of Chicago text, are mostly self-explanatory. Notice that in Question 47, we are given that ABCD is a trapezoid with one pair of opposite angles congruent and we are to prove that it is a parallelogram. In other words, we are using the inclusive definition where a parallelogram is a trapezoid. If teachers prefer the exclusive definition, they can change the Given section to: AB and CD are parallel and angles A and C are congruent, to prove that ABCD is a parallelogram.

This is a good time to have a topic about the traditionalists and the politics of Common Core. After all, I did begin today's post by discussing voting and elections.

The week began with the current president giving his final State of the Union debates, and last night, the Republicans held a presidential debate. In the State of the Union address, President Obama didsn't mention Common Core, but did briefly refer to education in math (along with computer science):

[Obama] then went on to say that in the years ahead “we should build on that progress, by providing pre-K for all, offering every student the hands-on computer science and math classes that make them job-ready on day one, and we should recruit and support more great teachers for our kids.”

Education came up in the GOP State of the Union response:

Following the address, South Carolina Governor Nikki Haley gave the response on behalf of the Republicans:  “We would reform education so it worked best for students, parents and teachers, not Washington bureaucrats and union bosses.”

Governor Haley proceeded to criticize former Florida Governor Jeb Bush, a presidential debate, over his support of Common Core.

Meanwhile, in last night's sixth Republican debate, Florida Senator Marco Rubio accused New Jersey Governor Chris Christie of supporting Common Core. Christie replied that Common Core had been eliminated in New Jersey. Notice that the situation in the Garden State is more complicated than that -- the Common Core Standards have been repealed, but the Common Core Test -- PARCC, remains. In many ways, it's more logical to keep the standards and repeal the test than vice versa -- but of course, doing so doesn't win you points in a GOP debate.

I've already mentioned what I'd do if I were president or governor, as part of my Presidential Birthday and Gubernatorial Christmas plans. I don't have much more to say about the Presidential Birthday plan, since it amounted to not much more than adoption of the curriculum at the First Daughters' school, Sidwell Friends.

But I have much more to say about the Gubernatorial Christmas plan, since there are more degrees of freedom there. Recall that the goal of that plan is to break the connection between grade levels and the tests by allowing students to test at a different grade level. The reported test score reveals what level the student is ready to enter, so a score of 400 indicates that a student is ready to begin fourth-grade math, a score of 500 indicates readiness for fifth-grade math, and so on.

Yet, as much as many traditionalists would like to see something like this, I can see them fear that too many schools would simply ignore the test scores and place the students by their grade level.

There are ways to avoid this problem. One would be to use something like the Path Plan that I've mentioned so many times on the blog. So we call Grades 1-2 the Primary Path, Grades 3-4 the Transition Path, and so on. Then the scores can report only paths, not grade levels.

I've noticed that in other nations such as Australia, the standards aren't organized by single grade levels, but in two-grade spans such as Grades 1-2, Grades 3-4, and so on. So in a way, this is similar to the Path Plan. But still, it's transparent which grades correspond to which paths, and so schools may still naively sort the students by grade level.

So here's something we can try -- we can divide the entire K-12 span into several levels -- but not into 13 levels, where there's a clear 1-1 correspondence between grades and levels, not into six or seven levels, where each level plainly corresponds to two grades. Instead, we can choose some number in between these.

For example, let's divide the K-12 span into ten levels. Ten makes sense -- after all, I pointed out how each level can be subdivided into ten units. Then each student can receive a three-digit score -- the first digit gives the level from Level 0 to Level 9. The second digit tells which unit a student is on, and the third digit tells where within each level a student is approximately.

So these Levels 0 through 9 don't readily correspond to grades. I have no problem assigning the top four levels, Levels 6 through 9, to the traditional high school disciplines of Algebra I, Geometry, Algebra II, and Pre-Calculus, especially since these themselves don't correspond exactly to high school years. I still like the idea of having this testing plan max out at Pre-Calc, since there's already a separate test for Calculus students to take -- the AP exam.

This means that Levels 0 through 5 will take us approximately from grades K-8. With six levels to take us through elementary and middle school math, it's not as obvious which level, say, a fourth grader should be placed -- and this is the intent.

Of course, when I am creating the standards, I need to know how to convert from grade to level -- even though the schools shouldn't know this. Let's figure it out -- we need six levels to take us through the nine years K-8, so each level is approximately three semesters:

Level 0: Kindergarten and first semester of 1st grade
Level 1: Second semester of 1st grade and all of 2nd grade
Level 2: 3rd grade and first semester of 4th grade
Level 3: Second semester of 4th grade and all of 5th grade
Level 4: 6th grade and first semester of 7th grade
Level 5: Second semester of 7th grade and all of 8th grade

This conversion guide helps me determine what standards to place at each level. It's obvious from this chart that Level 1 students should be working on addition and subtraction, while multiplication and division belong at Level 2.

Of course, school administrators might still figure out this chart for themselves and still blindly use it to place students. But notice that even if they do this, we still have grades 1, 4, and 7 split between two levels, so schools still have to put some thought into placing them. In some ways, grades 1, 4, and 7 are great years to focus on the placement of students. I don't really the idea of separating kindergartners into different math classes -- the lowest acceptable grade for such is first grade. Fourth grade marks the transition from the early years -- where traditionalism should rule -- and the later years, when students begin to question what they are learning. Seventh grade is another critical year of transition, since it determines whether students are ready for Algebra I. Indeed, there's no excuse not to place a student who completes Level 4 in sixth grade and Level 5 in seventh grade into Algebra I (that is, Level 6) in eighth.

Other than dividing K-12 into ten levels, we can keep the rest of the Gubernatorial Plan intact. So we can still provide Integrated Math classes to let students accelerate or decelerate in high school. The old plan had students begin taking computerized tests in third grade, with a base score of 300. We can do the same here, except that 200 would be the base score.

I was tempted to make the scale run from 200 to 800, just like the SAT. On the low end, this would mean that students can't score below 200 -- so if they can't get the first few Level 2 questions (which would be on the times tables) right, the test ends immediately. On the high end, making the test end at 800 -- which marks the completion of Geometry and readiness for Algebra II -- means that students don't have to take a computerized test during the years when they take Algebra II or Pre-Calc. This simulates the current idea that students go two years, grades 9-10, without needing to test.

Then again, perhaps making the score range identical to the SAT range is dangerous. A score of 800 indicates only that a student is ready to take Algebra II -- not that the student is likely to score 800 on the actual SAT! And besides, on the low end, if a student fails to answer any Level 2 questions, we can still ask that student Level 0 and 1 questions to find out what he or she does know.

But once again, this is all what I'd do if I were elected governor or president. I wonder whether I could ever win a fair election for such a high office -- until I remember Arrow's Theorem that there is no such thing as a fair election.

By the way, Kung wraps up his lecture by alluding to down-ticket offices, such as our representatives in Congress. In particular, he will discuss the division of states into districts and tell us whether this can ever be done fairly.

Here is the Review for Final, Part II. Don't forget that this upcoming Monday is Martin Luther King Day, so my next post will be on Tuesday, January 19th. On that day I will post the final itself.

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