"They [John Schwartz and Michael Green] had been working on it for over a decade in spite of little encouragement from colleagues, who found the 10-dimensional world hard to accept. Their published paper finally caused physicists to take their idea seriously."
This is the last page of the section on string theory, and the final page in the science chapter. Pappas concludes this chapter by writing:
"These are just the beginning discoveries and applications of an emerging mathematical field -- knot theory."
Today I'm posting the Benchmark Tests. Just as I've done in the past, I'll be using these test days for tying up loose ends, including the traditionalists label.
Actually, the traditionalist debate has been quiet lately. The most prominent traditionalist, Barry Garelick, hasn't posted in a week (probably because he's focused on his first day of school).
Here is the link to Garelick's most recent post:
https://traditionalmath.wordpress.com/2017/08/22/articles-i-didnt-finish-reading-dept/
This post contains a link to an article, which I'll link here as well:
http://www.businessnorth.com/businessnorth_exclusives/teachers-learn-to-make-math-fun-engaging/article_9e3a534a-8692-11e7-9460-33af357a8dfc.html
The article asks a typical question students ask in math class, "When will I ever use this?" The article suggests that engagement is the key to answering this question. Naturally, Garelick disagrees:
I usually find that students ask this when they are frustrated and/or having difficulty with a particular type of procedure or problem. When they are capable of doing the procedure, they tend to be just as engaged as they would with any activity. Not to mention that the prevalence of this question is helped along by TV sitcoms that may feature such a situation. The question is met with the predictable laugh track as the camera zooms in to a close up of teacher’s frustrated expression.
Well, I partly agree here. I've written before that students don't complain about tasks that are easy, fun, or high-status. So when they ask "When will I ever use this?" it really means that the math is hard, but as soon as they are "capable of doing the procedure" so that it's no longer hard, the complaint vanishes.
In my old class, a different sort of complaint appeared -- when I tried to get the students to be quiet using call-and-response and other ideas, they complained that this was "juvenile." But they had no problem with sing songs during music break, which is arguably even more juvenile. That was because singing songs was fun, but being quiet during class isn't.
Garelick mentions TV sitcoms here, and alluded to "tropes" earlier in his post. The famous website TV Tropes lists "Everybody hates mathematics" as the relevant trope here.
There is one comment on Garelick's post here -- and of course, it's SteveH, the blog co-author:
ALL STEM career paths require individual focus and success on p-sets [problem sets-dw]. That IS a one size fits all requirement. If it doesn’t fit the student, then engagement is no alternate approach. Engagement can only be built on top of high expectations and an emphasis on individual homework. These educators NEVER talk about the importance of individual homework – the missing component in K-6. They can’t justify full inclusion and low expectations by claiming that “engagement is key.”
Here's what I think is meant by "engagement is key" -- many students only do assignments if they find them enjoyable. If they don't want to do it, then they don't do it. Yes, students are wrong to believe this, but that doesn't affect the fact that they do believe it. Engagement is the attempt to make students want to do math so that they'll actually do it.
I've said it once and I'll say it again -- students may learn very little from doing projects and activities, but they learn absolutely nothing from p-sets and homework assignments that they leave blank. I've even seen students leave the worksheets I give them on their desks -- they don't even bother to put them in backpacks since they have no intention of doing it. "Very little" trumps "absolutely nothing."
SteveH mentions "full inclusion" here -- the opposite of "full inclusion" is "tracking." He proceeds to write that tracking still occurs today, except it's parents and tutors who track, not teachers or schools.
Now there is something that I do wish to discuss in this traditionalists-labeled post -- the debate over Common Core and testing. In the past, I've written that I was of two minds on this issue. I could see why many people would oppose testing, but I conceded that it has its uses.
But here's the thing -- I went through my first testing season as a teacher last year. And as I continue to reflect on what went wrong in my first year, I'm starting to believe that standardized testing is much to blame for my struggles last year. In fact, I might not have left my classroom had it not been for the state test.
Last year, I wrote that I couldn't survive the Big March -- the stretch of the year between President's Day and Easter when there are no off days from school. It's not the longest holiday-free stretch -- in fact, as Columbus Day isn't a holiday in California, many schools in our state have no days off between Labor Day and Veteran's Day. But the Big March feels much worse -- and part of the reason is that the Big March is right in the middle of test-prep season, whereas Labor Day to Vets Day isn't as pressure-filled.
Test prep at my school began at the start of second "semester" -- that is, the mathematical midpoint of the year. As I mentioned before, we already lack conference periods at our school due to the lack of staff -- instead, our only quasi-break had been during P.E. time. But P.E. time most days of the week was replaced with SBAC Prep for sixth and seventh grades, while eighth graders had their test prep on Wednesday mornings.
Moreover, consider the problems I had with classroom management. The students decided that they only had to obey my support staff member and didn't have to listen to me. For all three grades, SBAC Prep took place at times when my support staff member wasn't present -- Grades 6-7 had it after she left while eighth grade had it before she arrived. Therefore the students misbehaved the most in my class during SBAC Prep time.
Then, of course, there was the issue that many students didn't remember what they were supposed to have learned, so they were resistant to doing the reviews. SBAC Prep time turned into copying down the questions and answers that I wrote on the board, which of course isn't learning at all.
The whole point is that had there been no Common Core, SBAC, or standardized testing, students might have been better behaved the second half of the year. There would have been P.E. time for the students and planning time for me, instead of my having to figure out what to do for SBAC Prep with no planning time. And I might not have left my school.
Traditionalists are split when it comes to standardized testing. Some of them favor standardized testing in order to combat grade inflation and increase learning.
Here's what I mean -- suppose a student is failing a class. The parent then complains to the teacher -- and the gist of the complaint is, "please make the grading scale easier." The teacher acquiesces to the parent by lowering the standards -- the accumulation of such lowering leads to grade inflation.
Now suppose the parent decides to complain to a tutor instead of a teacher. Now the gist of the complaint can't be "make the grading scale easier" since tutors don't give grades. So instead they end up saying, "please teach my kids more math" (so they can pass the teacher's tests). Traditionalists like this, since it results in the students learning more math.
So standardized testing makes teachers into tutors -- there will be fewer requests to "make the grading scale easier" and more requests to "teach the kids more math," since teachers don't grade the standardized tests. In fact, some traditionalists would like to see standardized test grades override teacher grades. This is to reduce grade inflation and increase learning.
Of course, while may they favor standardized testing, they still oppose the Common Core tests. In the post above, SteveH writes again about how the highest score on the PARCC and SBAC corresponds to "can pass a college Algebra course" instead of "can pass a high school Calculus course." Standards that don't encourage AP Calculus are said to decrease learning rather than increase it.
Another issue I had in my class was with computerized testing. During SBAC Prep time, I was required to have the students take the practice tests from the CAASPP website. Some 8th graders -- specifically those who couldn't remember how to do the problems -- refused to do it. Meanwhile, in Grades 6-7, there weren't enough laptops. Some students who didn't want to take the test just passed the laptop to another student -- and there were no consequences for this, since there was nothing else for them to do except take the online test. Also, sometimes the CAASPP website didn't work -- and many students tried to take advantage of this.
For these reasons, traditionalists oppose computer tests like PARCC and SBAC. They believe that any test worth its salt must be taken with paper and pencil.
I like the idea of a computer-adaptive test like the SBAC. For those who complain that a standardized test is "one size fits all," the point of an adaptive test is to avoid being "one size fits all." But that doesn't erase the issues I had with computers this year.
I remember once when I was subbing at a school during SBAC time (not just prep for the SBAC, but the actual test). School was set up for two-hour blocks to accommodate the SBAC. I was in an English class on a day when math tests were being administered. The regular teacher mentioned in her notes that students are not allowed to go to the restroom, period. She made no exceptions for emergencies, probably because students would lie about having an emergency. (And she wrote the bell schedule in her notes, so it's not as if she was unaware of the block schedule.)
Sure enough, one girl claimed that she had to go to the restroom for an emergency. This led to a long, heated argument. Regardless of what you think I should have done in that situation, the truth is that the whole argument is much less likely had there not been two-hour blocks forced by the SBAC.
I stated before that the entire state math test should be only 30 minutes long -- especially in middle and high school where students attend class by periods. With only a half-hour test, plus leeway to set up and put away the laptops, the entire exam fits within a single period, so there would be no need for a block schedule. With only one period required for math, if the computers break down, the test can simply be given the next day. This isn't possible under a block schedule, where the test must be given on the day specified by the block schedule.
I'm not sure what to do about the other problem -- the need for test prep months before the test. In theory, there should be no need for test prep, as simply following the curriculum ought to prepare the students for the test. Perhaps there should be fewer confusing test questions, since much of test prep time is to teach students how to decipher them.
Oh, and if there's going to be a computerized test, then scores should be released instantly. That's one of the main advantages of taking tests on computers, but in reality, scoring still takes months. Part of this is the existence of "performance tasks" that can't be graded electronically.
The other issue currently being discussed in California is the need for remedial classes at our Cal States and community colleges. Of course, traditionalists oppose these -- they want to see more students taking Calculus in high school, not Algebra in college.
There are a few more things that I wish to discuss in this post. I mentioned earlier that I'm posting the Benchmark Tests, so let me include a full version of the Benchmark Test song:
Benchmark Tests -- by Mr. Walker
Verse 1:
Why do we take Benchmark Tests?
It's the start of the year so let's
See how much we know, know know!
It's much new stuff on Benchmark Tests.
If we don't know it, we take a guess.
We leave none blank, oh no, no, no!
The teacher sees our Benchmark Tests,
Knows what to teach more or less.
That's the way to go, go, go!
Verse 2:
Why do we take Benchmark Tests?
The first trimester is done so let's
See how much we know, know know!
It's some new stuff on Benchmark Tests.
If we don't know it, we take a guess.
'Cause there's still time to grow, grow, grow!
The teacher sees our Benchmark Tests,
Knows what to teach more or less.
That's the way to go, go, go!
Verse 3:
Why do we take Benchmark Tests?
The second trimester is done so let's
See how much we know, know know!
It's some new stuff on Benchmark Tests.
If we don't know it, we take a guess.
'Cause there's still time to grow, grow, grow!
The teacher sees our Benchmark Tests,
Knows what to teach more or less.
That's the way to go, go, go!
Verse 4:
Why do we take Benchmark Tests?
It's the end of the year so let's
See how much we know, know know!
It's all old stuff on Benchmark Tests.
There's no need to take a guess.
We leave none blank, oh no, no, no!
The teacher sees our Benchmark Tests,
Just like the SBAC more or less.
That's the way to go, go, go!
I'm still in the process of updating the Fraction Fever song, to include a new verse or so on addition, subtraction, multiplication, and division of fractions. In doing so, I found something that will help me greatly -- a YouTube video of the actual 1980's Fraction Fever game!
If you watch the actual video, you might notice that no addition of fractions is actually needed to play this game. I want to include arithmetic in my version of the game, since that's what middle school students are actually learning. On the other hand, I retain the basic rules -- right answers=elevators to go up a floor, wrong answers=holes to go down a floor, and so on. The player points out that the scoring system is confusing. In my version, the floor (or level) itself is the team score. Oh, and I post my answers on a wall, and students have to jump up to reach them. This corresponds to the fact that in this game, you must jump (using a pogo stick) on the right answers.
While I was searching for Fraction Fever, I stumbled on another old website:
http://www.haplessgenius.com/mocha/
This website actually emulates my old 1980's computer. Notice that several games are available (but unfortunately, not Fraction Fever).
Anyway, recall that earlier this summer when I was writing about my old computer (in the Pappas music posts), I mentioned that there was a SOUND command (in BASIC) that played musical notes (numbered 1-255). But I never tried to play music with it, especially as there was also a PLAY command that played notes lettered A-G (as in real music). Now with this emulator, I can finally figure out what the Notes 1-255 actually correspond to. If so, then this emulator can be used to play microtonal music (that is, other than 12EDO).
I had made two guesses regarding the conversion of SOUND numbers to musical notes:
-- The SOUND notes are an n-EDO system for some n. Then Notes n, 2n, 3n, 4n, and so on would sound an octave apart.
-- The SOUND notes correspond to frequencies (hertz). Then Notes n, 2n, 4n, 8n, and so on would sound an octave apart.
As it turns out, both guesses are wrong. On the emulator, let's click "Sound" under the screen on the left side of the page. Then we type in:
SOUND 1,10
This plays Note 1 for a length of 10 "SOUND-units." I believe that on my old computer, one SOUND unit is about 1/15 second. On this emulator, it doesn't play a continuous note but beeps instead -- but right now we're more concerned with the note. (Of course, if we're trying to play microtonal music, that beeping would be annoying!)
Now, let's try the highest note:
SOUND 255,10
This clearly sounds several octaves higher than Note 1. But let's try the middle note:
SOUND 1,10
SOUND 128,10
Now Notes 1 and 128 sound just about an octave apart. But Notes 128 and 255 still sound much more than an octave apart. Instead, let's try:
SOUND 128,10
SOUND 192,10
Note 192 (which is around 3/4 of the range) now sounds near an octave above Note 128. This should now remind us of a string -- if we play half of its length, it sounds an octave higher, and if we fret it at the 3/4 mark (so the vibrating string is 1/4 of the original string), it sounds two octaves higher.
And now, after 30 years, I finally know the secret of SOUND. The numbered notes correspond to EDL (equal divisions of length), as if we had a string and were dividing its length up into equal parts.
We can test this out by writing a short program to create a major scale. On page 176 of Pappas, she tells us how to create a musical scale:
"For example, starting with a string that produces the note C, then 16/15 of C's length gives B, 6/5 of C's length gives A, 4/3 of C's gives G, 3/2 of C's gives F, 8/5 of C's gives E, 16/9 of C's gives D, and 2/1 of C's length gives low C."
Notice that the common denominator of all the fractions mentioned here is 90. So let's try using an octave with 90 equal divisions of length (180EDL). Now since Note 255 is the highest note, let's think of 256 as the "end of the string" -- the bridge, to use a guitarist's term. (After all, 255/256 still plays a note, but 256/256 means that we have no string vibrating, hence no note.)
So 90 notes below the bridge is Note 166, and 90 notes below this is Note 76. These notes are an octave apart, and we can easily divide this into a major scale using the ratios above.
Note Ratio Out of 180 From Bridge 256
C 1/1 90 166
B 16/15 96 160
A 6/5 108 148
G 4/3 120 136
F 3/2 135 121
E 8/5 144 112
D 16/9 160 96
C 2/1 180 76
Finally, we write a program to play all of these notes:
10 SOUND 76,10
20 SOUND 96,10
30 SOUND 112,10
40 SOUND 121,10
50 SOUND 136,10
60 SOUND 148,10
70 SOUND 160,10
80 SOUND 166,10
At the end of the program, we type in RUN to run the program. This does sound like a major scale, but let's try adding another note an octave above the last note:
90 SOUND 211,10
This last note sounds a little bit narrower than a true octave. In fact, when we played Notes 128 and 192 together earlier, it also sounds slightly narrower than an octave.
This seems to imply that maybe 256 isn't the bridge after all. Unfortunately, we can't play a true octave unless we know what the exact bridge is. Notice that if 256 were the bridge, then Notes 254 and 255 would be a full octave apart, but they clearly aren't. Using trial and error, the true bridge seems to be around 261 or 260. Let's use 261 as our bridge:
Note Ratio Out of 180 From Bridge 261
C 1/1 90 171
B 16/15 96 165
A 6/5 108 153
G 4/3 120 141
F 3/2 135 126
E 8/5 144 117
D 16/9 160 101
C 2/1 180 81
In case you're curious how I determined the bridge to be 261, type NEW for a new program, then try the following:
10 B=261
20 FOR X=3 TO 7
30 SOUND B-2^X,10
40 NEXT X
(Here the ^ symbol is actually an up-arrow -- press the arrow on the keyboard.) The played notes now sound like descending octaves. If we change line 10 to B=260 or B=262, the octaves are slightly off.
Let's type in NEW for a new program, then we run the following:
10 SOUND 81,10
20 SOUND 101,10
30 SOUND 117,10
40 SOUND 126,10
50 SOUND 141,10
60 SOUND 153,10
70 SOUND 165,10
80 SOUND 171,10
90 SOUND 216,10
This time, the last two notes sound like a true octave. This means that the first eight notes do form a major scale -- a justly tuned major scale, rather than the 12EDO major scale.
Of course, we've been calling the top note a C, but is it actually a C? It appears that this note is closer to concert B than to C -- we can confirm this by using PLAY:
90 PLAY "B"
The last two notes sound alike, indicating that this is really a just major scale on B. But in practice, we can call this a "transposing instrument" and just pretend that the note is C, since we'd rather write a C major scale than a B major scale with all of its sharps.
At this point, you may wonder why we don't just use PLAY, where we can play a true C note? The whole point of using SOUND is that we can actually play notes in just intonation, whereas PLAY will just give you the 12EDO scale.
Before we leave, suppose we wanted to play a septimal interval, such as the harmonic seventh. We can't play this exactly, since 90 isn't divisible by seven. The desired ratio is 8/7, and if we multiply this by 90, we can round this to 103 (or Note 158 as measured from Bridge 261). This isn't exact (but it's at least more accurate than 12EDO).
To obtain a just harmonic seventh, we must use a division other than 90EDL. The closest multiple of seven is 91 -- this produces a harmonic seventh between Notes 79 and 157. But of course, with 91 we lose all of the 3-limit and 5-limit intervals. The correct octave to use depends on what other intervals we're using in the song -- for example, 126EDL gives us a no-fives tuning.
The following is a link to music played in EDL systems:
https://sites.google.com/site/240edo/equaldivisionsoflength(edl)
Naturally, no scale larger than 255EDL can be played on the emulator. Don't forget to subtract the notes from Bridge 261 -- so the first three notes of 120EDL -- given as 120, 114, 108 -- correspond to Notes 141, 147, and 153 on SOUND.
OK, let's finally post the Benchmark Tests. These are based on old finals posted to the blog. I admit that the tricky thing about Benchmark Tests in Geometry is that the students are coming off of a year of Algebra I, when they've thought little about Geometry at all. This is different from Benchmark Tests in middle school or Integrated Math, where there should be some continuity from year to year.
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