Thursday, November 9, 2017

Chapter 5 Test (Day 60)

This is what Theoni Pappas writes on page 313 of her Magic of Mathematics:

"page 303. Taking a Checkerboard Apart. Solution: Step 1...."

Oh yeah, that's right -- this is in the middle of the solution section, so we started reading again from the beginning of the book. Let's try that again....

This is what Theoni Pappas writes on page 9 of her Magic of Mathematics:

"Every time you pick-up the telephone receiver to place a call, send a fax, or modem information -- you are entering a phenomenally complicated and enormous network."

This is the first page of the subsection "The Mathematics of a Telephone Call." Of course, Pappas writes this book in 1994 -- cell phones existed but weren't as common as they are today. Therefore, she focuses mainly on landlines in this subsection.

Here are some excerpts from this page:

"It is difficult to imagine how many calls are fielded and directed each day over this network. How does a single phone call find its way to someone in your city, state, or another country? In the early years of the telephone, one picked-up the receiver and cranked the phone to get an operator. A local operator came on the line from the local switch board and said 'number please,' and from there connected you with the party you were trying to reach."

The lone picture on this page is a map of the world, displayed with many phone connections from one nation to another. Binary digits (0's and 1's) represent the information being sent over the wires.

Unfortunately, this first page is the only page that we'll read in this subsection. Veteran's Day is coming up -- and remember that here in California, Vets Day is celebrated on November 11th no matter what day of the week it is. This year, the 11th falls on a Saturday -- a day when schools are already closed. According to the rules for the observance of federal holidays, when the holiday falls on a Saturday, the schools are closed on Friday. Therefore every public school in California -- including the district whose calendar we're following -- is closed tomorrow.

And so my next post after today won't be until Monday, when we read page 13. The rest of the phone subsection, on pages 10-12, are therefore blocked by the three-day weekend.

It's a shame, though, because the math of a phone call is an interesting subject. Most teachers are very familiar with one thing that students would rather do than learn math -- use a cell phone. Many students wish that math would disappear, so that the time during which they would have studied math would be spent playing on their phones instead. The problem with this idea is that if math didn't exist, neither would phones -- as Pappas explains in this subsection.

Last year, whenever I confiscated a cell phone, before returning the phone, I made the student repeat the phrase "Without math, cell phones wouldn't exist." This way, students would see a connection between something they don't like (math) and something they do (phones).

Recall that I didn't purchase the Pappas book until after I left the classroom (which explains why we're rereading page 9 right now). If I'd had this book while still a teacher, I could have told the students how phone technology uses Danzig's simplex method to solve a linear algebra problem (and that's mathematician George Danzig -- not Sheila Danzig and her plan to eliminate DST). Also, the Karmarkar algorithm is a shortcut that replaces algebra with geometry and graph theory.

It would be interesting if Pappas would write an updated version of this subsection that incorporates cell phone as well as landline technology. Then it would definitely be something to read to students who'd rather play with cell phones than learn math.

Anyway, tomorrow I would have read Pappas page 10 if it weren't a holiday. And I would have read page 314 had it been part of the main Pappas text and not part of an appendix, since tomorrow is the 314th day of the year....

That's right -- tomorrow is the third Pi Day. Last year, the 314th day of the year was November 9th because of February 29th, but this year it is the usual November 10th. This is what I wrote last year about how I (inadvertently) celebrated third Pi Day in my classroom:

Hold on a minute! You probably thought that Pi Day was on March 14th -- and the date on which this blog was launched was Pi Approximation Day, July 22nd. So how can November 10th be yet another Pi Day?

Well, November 10th is the 314th day of the year. And so some people have declared the day to be a third Pi Day:

I like the idea of a third Pi Day, based on the ordinal date (where January 1 = 1, November 10 = 314, and December 31 = 365). As the author at the above link pointed out, the three Pi Days are nearly equally spaced throughout the year. So I can celebrate Pi Day every fourth month.

I wouldn't mention the third Pi Day in a classroom, unless I was at a school that was on a 4x4 block schedule, where a student may take math first semester and then have absolutely no math class in the second semester (when the original Pi Day occurs). The only chance a student has to celebrate Pi Day would be the November Pi Day. (Likewise, the second Pi Day -- July 22nd -- may be convenient for a summer school class.)

Both November 10th [or 9th] and March 14th suffer from falling near the ends of trimesters or quarters (depending on whether the school started in August or after Labor Day). Classes may be too busy with trimester or quarter tests to have any sort of Pi Day party.

[And indeed, in 2017 today is Day 60 -- the end of the trimester, except that high schools usually don't have a trimester calendar. Also, as we see this year, another problem with third Pi Day is that it falls close to Veteran's Day, and so third Pi Day is also likely to be blocked by a long holiday weekend.]

At home, I like to celebrate and eat pie for all three Pi Days. The pie that I choose is the pie most associated with the season in which that Pi Day occurs. Today, I will eat either pumpkin or sweet potato pie due to its proximity to Thanksgiving. On Pi Approximation Day, I ate apple pie, since it occurs right after the Fourth of July, a date as American as apple pie. And for the original Pi Day in March, I eat cherry pie -- the National Cherry Blossom Festival usually occurs between a week and a month after Pi Day.

Last year, right in the middle of class, some eighth graders start singing the song "Thrift Shop" by Macklemore & Ryan Lewis for who knows what reason. So then I told them that there's a math parody of this song called "Pi Shop."

Unfortunately, I wasn't prepared to sing Kevin Lee's "Pi Shop" today -- and of course my mentioning of this song had nothing to do with Pumpkin Pi Day. Some of the students tried to guess the lyrics of "Pi Shop" -- for some reason, they thought it had something to do with buying fourteen pies!

Sadly, third Pi Day was the only Pi Day I actually spent in my class last year, since I ended up leaving just before the first (original) Pi Day. Perhaps I would have made a bigger deal about third Pi Day if I knew I'd be gone by first Pi Day -- but then again, back in November there was no reason for me to think that I wouldn't complete the school year.

For each of the three Pi Days I like to post some sort of video. Last year on Pumpkin Pi Day, I posted "Pi Shop" since we sang it in my class, but I don't wish to post it this year.

This year, I think I'll post a version of "Oh Number Pi," a parody of "O Christmas Tree":

Notice that of the three Pi Days, third Pi Day falls closest to Christmas, and so it's a good day to sing Christmas parodies of pi songs -- especially if students are starting to practice holiday songs in their music classes now. I also point out that KOST, a Southern California music radio station, will switch to Christmas music on third Pi Day for the second straight year.

Today is Day 60, the day of the Chapter 5 Test. This is what I wrote two years ago about the answers to today's test:

Now here are the answers to my test.

1-2. constructions (or drawings). Notice that a construction for #2 is halfway to constructing a square inscribed in a circle for Common Core.

3. 95 degrees.

4. 152.5 degrees.

5. 87 degrees.

6. 86 degrees.

7. x degrees. This is almost like part of Euclid's proof of the Isosceles Triangle Theorem (except I think that his proof focused on the linear pairs, not the vertical angles).

8. 27 degrees.

9a. x = 60

9b. 61, 62, 58 degrees.

10. 25, 69, 128, 138 degrees.

11. polygon, quadrilateral, parallelogram, rectangle, square.

12. kite.

13. rectangle.

14. false. A counterexample is found easily.

15. Yes, the perpendicular bisector of the bases.

16. 46. Although I mentioned it briefly this year during Chapter 2, perimeter is a concept that could be developed more in these early lessons. My question actually defines perimeter since my lessons haven't stressed the concept yet. This is the simplest possible perimeter problem that I could have covered, where only the definition of kite is needed to find the two missing lengths. I could have given an isosceles trapezoid instead, where the Isosceles Trapezoid Theorem is needed to find a missing side length. Or since I squeezed in the Properties of a Parallelogram Theorem in our Lesson 5-6 (as part of proving that every rhombus is a parallelogram), I could have even put a parallelogram here with only two consecutive side lengths given.

17. The conjecture is true, and is a key part of the proof of Centroid Concurrency Theorem.

18. Statements                     Reasons
      1. angle G = angle FHI 1. Given
      2. EG | | FH                   2. Corresponding Angles Test
      3. EFHG is a trapezoid 3. Definition of trapezoid (inclusive def. -- it could be a parallelogram)

19. Statements                   Reasons
      1. O and P are circles  1. Given
      2. OQ = ORPQ = PR 2. Definition of circle (meaning)
      3. OQPR is a kite         3. Definition of kite (inclusive def. -- it could be a rhombus)

20. Figure is at the top, then below it is quadrilateral. Branching out from it are kite, trapezoid. Then below trapezoid is parallelogram. Kite and parallelogram rejoin to have rhombus below. (Once again, these are inclusive definitions!).

As today is a test day, it ought to be time for another discussion about traditionalists. Let's see what our favorite traditionalist, Barry Garelick, is writing about now:

This article about a “math festival” held at an elementary school contains all the usual tropes about what math education is supposed to be about. I almost stopped reading here, but like being stuck in a traffic jam because of an accident, I found myself staring at the gory site at the side of the road.

Notice that the link above goes to Edsource. I've linked to Edsource directly on my own before, but lately I've only visited the site via a traditionalist's link.

In this post, Garelick tells us that progressives repeat the same disparaging comments on traditional math so often that they're like "tropes" -- as in the archetypes and cliches that appear on many TV shows and other popular media. (Recall that over a month ago, I mentioned the TV show Young Sheldon in a traditionalists post. Now we see that Garelick would probably just consider the jokes on that show to be yet another example of anti-math -- or anti-traditional math -- tropes.)

So far, Garelick's post has drawn seven comments. Of these, five of them were posted by -- you guessed it! -- SteveH, the co-author of Garelick's blog. Let's look at some of SteveH's comments:

Ashley Hopkinson, Edsource editor:
“Once parents understand what we’re doing and what is happening you’ll hear them say ‘why didn’t someone teach me this way ?’ or ‘why didn’t I know this before?’”
And the parents of their best students keep quiet (because they get trashed – been there) and do the opposite at home.
Here SteveH implies that parental comments such as "Why didn't someone teach me this way?" should not be considered evidence that progressive math is superior to traditional math. This is because only the ones who agree with the pedagogy speak out, while those who disagree with it, as SteveH points out, "keep quiet."

I've often compared the debate between traditionalists and progressives to my own struggles with classroom management. A student breaks a rule and I try to punish him -- but then his friends call out, "That's unfair!" But this doesn't mean that all the students believe that my rule is unfair -- some students secretly agree with traditional management and that the first student should be punished. But the best students keep quiet, and indeed for the same reason that SteveH mentions above -- "because they get trashed."

Yes, they [students taught under progressive pedagogy -- dw] love math, but what they love is NOT math that leads to a STEM degree or many other college programs.

But that's exactly the problem -- they don't love, or even like, the rigorous level of math that leads to a STEM degree. So merely assigning the students problem sets and other forms of traditional math won't magically turn haters of math into lovers of math!

Then how will they feel when they aren’t “math phobic”, but can’t pass college statistics – when they have to change their degree program and career path?  If skills don’t automatically follow from good “feelings”, then don’t look at them.

I agree with SteveH that good feelings aren't sufficient to the development of math skills. But I believe that good feelings are necessary to the development of math skills. It doesn't matter how rigorous and excellent a traditional p-set is -- a math student won't even attempt -- or even look at -- Problem #1 in the p-set. Eliminating math phobia and generating good feelings in class are needed just to get the students to look at Problem #1 that night.

How many more kids (with no skill help at home or with tutors) get a ‘B’ or higher in a proper Algebra I class in 8th grade?

As we already know, for SteveH, eighth grade Algebra I is the gold standard (along with senior-year AP Calculus, of course). To SteveH, we should test so see how many students taught math go on to do well in eighth grade Algebra I -- and if it isn't enough, then the curriculum should be replaced with one that does lead to success in eighth grade Algebra I (presumably a traditional one).

This one's tricky. Let's say we have a fourth grade student here. If we teach a traditional curriculum in class, then there will be less work needed at home (as in tutors, as mentioned by SteveH) to get this fourth grader to Algebra I in four years. It might be possible to sell this fourth grader on traditional math by telling him that he'll have less work -- there will be p-sets, of course, but he can work on them in class and just do the finishing touches on them at home, as opposed to hours with tutors.

But in reality, fourth graders aren't thinking ahead to eighth grade Algebra I. Our fourth grader would prefer having a progressive math class with little to no math homework. Any attempt to get him to eighth grade Algebra I -- whether via traditionalism in the class or via tutors -- is likely to induce the student's "math phobia."

I still believe that to get this student to Algebra I, we should tell him why it's important to learn math, again using examples such as "without math, cell phones wouldn't exist." Or even more to the point, "cell phone technology was developed by those who get good grades in hard math classes."

Or, better yet still, we can use SteveH's nursing example:

Good luck passing the college statistics course to even get a nursing degree at some colleges.

Imagine the following hypothetical conversation:

Teacher: What do you want to be when you grow up?
4th Grader: A nurse.
Teacher: To be a nurse, you have to get A's and B's in all your classes, especially science and math.
4th Grader: Boo! I hate math!
Teacher: You know the nurses you see at the hospital? All of them are great at math. They won't let you be a nurse unless you're good at math.
4th Grader: But math is so hard!
Teacher: Don't worry -- there's still time. For the next ten years, you'll have to work hard on these p-sets.
4th Grader: That's no fun!
Teacher: I know that they're boring, but it's the only way you'll ever be good at math. Work on these p-sets, and ten years from now you'll be able to say, "Yes, I'm good at math!" Then they'll let you become a nurse, or anything else you want to be.
4th Grader: Um, OK.

This is what traditionalists need to do more often. Maybe then they could achieve their goal -- more young students who actually attempt #1 on the p-sets.

Let's conclude this post by pointing out regarding "Faith" -- it's traditionalists who show faith. They have faith that simply by giving "individual practice with conceptually challenging math problems," the students will actually do the math problems. Again, students learn more by doing a progressive lesson than by throwing away a traditionalist problem set on the way out the door.

Happy Pumpkin Pi Day tomorrow -- and this time, I'll make sure that I eat some pumpkin pie tomorrow. And Happy Veteran's Day -- my next post will be on Monday.

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