Both of the co-teaching classes are seventh grade. Many seventh grade English classes, both special ed and gen ed, are continuing to read Rudyard Kipling's "Rikki-Tikki-Tavi," part of his Jungle Book.
Meanwhile, of the three classes with aides, two are seventh grade and the other is eighth grade. To make it easier, all of the classes are learning how to punctuate quotes. They learn five rules, the first two of which involve capitalization: capitalize the first word of a quote, but don't capitalize the first word of the second part of a quote in the same sentence. (For example, "Despite becoming ever more fascinated by geometry," said Stewart, "mathematicians did not lose their interest in numbers.")
As far as behavior problems are concerned, notice that this is the same school that I subbed in last Friday -- including many of the same students I placed on the bad list. The regular teacher comes in before her meeting to warn some of those students during homeroom. As it turns out, the main culprit that day is out at the nurse due to an injury he sustains during P.E. class.
Still, the aide and I must place four seventh graders on the bad list (lunch detention). In one class, two of the students are talking loudly to each other. In the other, one guy fails to bring in a book during silent reading (with a lame excuse), while the other has his head down and does no work. As a result, the eighth grade class is the best class of the day.
Today is a test day, and so it's a traditionalists post. Once again, our main traditionalists remain inactive, so let's check out the Joanne Jacobs. Today, not only did I sub in an English class, but the class has an explicit lesson on punctuation. So let's start with the traditionalist Momof4, who often laments that writing and grammar aren't taught well anymore.
https://www.joannejacobs.com/2019/10/economists-we-need-english-majors-too/
Momof4:
Being able to write grammatically correct English is a valuable skill, but not necessarily a stand-alone one. Just recently, I was talking with a high-level guy in corporate finance, with a very strong banking background, who said that every eval he had had praised his very strong writing skills, as well as others – but the writing ability made him a standout, since most didn’t have it.
https://www.joannejacobs.com/2019/10/free-college-leads-to-low-value-degrees/
Momof4:
3 cheers and amen! My poor, small-town school – with about 10 college grads in town and many grandparents of my classmates without HS attendance – produced eighth-grade graduates with solid literacy, numeracy (phonics, grammar composition and traditional math) and general knowledge (science, geography, government and history – including art/architecture and music history/appreciation- every week) – likely the equivalent of many, if not most, HS grads and far too many college grads Of course, kids came from intact, married families (widowhood aside, thanks to WWII and Korean War), were appropriately socialized prior to school entry, and held to appropriate behavioral standards after.
If Momof4 could have seen today's punctuation class, she might give it three cheers and amen too.
As for Bill -- the other main traditionalist who posts at Joanne Jacobs -- we must go back more than a week to two threads about gifted education:
https://www.joannejacobs.com/2019/10/what-about-the-smart-kids-2/
Bill:
IMO, what will probably wind up happening is that if the GATE and other programs are eliminated, it’s quite possible that students enrolled in those programs will simply withdraw from public school and find alternatives to continue high academic achievement…
https://www.joannejacobs.com/2019/10/does-gifted-ed-work-not-very-well/
Bill:
In my day, we tracked/grouped students by ability, low, average, and high…everyone learned, but just not at the same pace as everyone else (after high school, many students are going to learn the hard and fast reality of life, when they actually have to compete in the real world)…
I couldn’t care less about how the demographics track (though I’m sure a lot of people do as to what it implies).
Notice that these comments involve tracking -- and it's inevitable that any mention of tracking leads to a discussion of "demographics" (read: race). And of course, the main articles are about New York, where the recent elimination of GATE is definitely for racial reasons.
Bill states that he's in favor of tracking regardless of "demographics" -- that is, he considers tracking to be good even if every single student of one race (blacks, say) lands on the lowest track. Of course, parents of that race won't like their children being placed on the lowest track.
In his first comment, he implies that if GATE is eliminated, then parents of the races that land on the highest track (which in NYC are white and Asian) will become disillusioned and take their kids out of public school to enroll in schools that will place them on the highest track. But notice that if a strong tracking system is implemented, then parents of the races that land on the lowest track (which in NYC are black and Hispanic) will become disillusioned and take their kids out of public school, to give them opportunities that don't land them on the lowest track.
Thus the tracking question becomes zero-sum. Eliminating tracking makes high-track races flee the public schools, while implementing tracking makes low-track races flee the public schools. Neither system results in keeping all races satisfied. Thus Bill's final comment is fitting:
Bill:
I wish there was a way to provide a high quality education to every kid, and that every kid came to school fully equipped to learn on a daily basis, but in reality, that is pretty much wishful thinking…
Chapter 7 of Ian Stewart's The Story of Mathematics is called "Patterns in Numbers: The Origins of Number Theory." Here's how it begins:
"Despite becoming ever more fascinated by geometry," said Stewart, "mathematicians did not lose their interest in numbers."
This chapter is all about number theory -- the basic properties of whole numbers. (A few years ago, our side-along reading book was Ogilvy's number theory book.)
One major part of number theory are prime numbers -- those that cannot be expressed as the product of two smaller numbers. But as Stewart points out:
"According to this definition, the number 1 should be considered prime, but for good reason it is placed in a special class of its own and called a unit."
One of the first mathematicians to write about primes was Euclid -- the same Euclid who also wrote axioms for Geometry. He first proved unique factorization -- that is, given any composite number, the primes that multiply to give that number are unique, such as 30 = 2 * 3 * 5 = 2 * 5 * 3:
"The same three primes occur, but multiplied in a different order -- which of course does not affect the result."
Euclid also proves that prime numbers are more than any assigned multitude of prime numbers:
"In modern terms, this means that the list of primes is infinite. The proof is given in a representative case: suppose that there are only three prime numbers, a, b, and c."
Of course, we've already seen this proof -- multiply all the primes and add one. Then this number must have a prime factor other than a, b, and c, a contradiction:
"Therefore no finite list of primes can ever be complete. We have mentioned Diophantus of Alexandria in connection with algebraic notation, but his greatest influence was in number theory."
Indeed, Diophantus eventually proved that there are infinitely many Pythagorean triples -- that is, the Diophantine equation a^2 + b^2 = c^2 has infinitely many natural number solutions.
Stewart now jumps forward over a thousand years, to Fermat. One of his results is to show which primes can be written as the sum of squares:
"Fermat observed that there are three basic types of prime: (i) The number 2, the only even prime. (ii) Primes that are 1 greater than a multiple of 4, such as 5, 13, 17 and so on -- these primes are all odd. (iii) Primes that are 1 less than a multiple of 4, such as 3, 7, 11 and so on -- these primes are also odd."
Then all primes in (i) and (ii) are indeed the sum of two squares, but those in (iii) aren't. Another result is Fermat's Little Theorem, which states that if p is any prime and a is any whole number, then a^p - a is a multiple of p:
"The corresponding property is usually false if p is composite, but not always. Fermat's most celebrated result took 350 years to prove."
Of course, we're talking about Fermat's Last Theorem, a^n + b^n = c^n is impossible if n > 3. This was proved by Andrew Wiles in 1994. Earlier mathematicians, such as Euler and Lagrange, also tried to finish Fermat's work:
"Most of the theorems that Fermat had stated but not proved were polished off during this period. The next big advance in number theory was made by Gauss, who published his masterpiece, the Disquisitiones Arithmeticae (Investigations in Arithmetic) in 1801."
Stewart tries to describe the modular arithmetic in which Gauss worked. It's similar to clock arithmetic -- the inspiration of the Square One TV song "Time Keeper." His signature result is the Law of Quadratic Reciprocity. It involves the perfect squares in a given modulus -- for example, 11:
"Then the possible perfect squares (of the numbers less than 11) are 0 1 4 9 16 25 36 49 64 81 100 which, when reduced (mod 11) yield 0 1 3 4 5 9 which each non-zero appearing twice."
These are called the quadratic residues, mod 11. Gauss asks, for what values of p and q are p a quadratic residue (mod q) and q a quadratic residue (mod p)? In this case, we call it "reciprocity" when both questions have the same answer:
"Gauss proved that this law of reciprocity holds for any pair of odd primes, except when both primes are of the form 4k - 1, in which case the two questions always have opposite answers."
He also worked on classical geometric constructions -- particularly of regular polygons. We know that the Common Core expects students to construct triangles, squares, and hexagons. Euclid showed how to construct regular pentagons and 15-gons and declared that no other regular n-gon (n odd) can be constructed:
"For some two thousand years, the mathematical world assumed that Euclid had said the last word, and no other regular polygons were constructable. Gauss proved them wrong."
Indeed, he showed how to construct a regular 17-gon. It turns out that a regular p-gon, for p prime, is constructible if p is a Fermat prime of the form 2^2^m + 1. For m up to 4, 2^2^m + 1 is prime:
"Euler discovered that when m = 5 there is a factor 641. It follows that there must also exist ruler-and-compass constructions for the regular 257-gon and 65537-gon."
Returning to Fermat, the author writes:
"His annoying tendency not to supply proofs was put right by Euler, Lagrange, and a few less prominent figures, with the sole exception of his Last Theorem, but number theory seemed to consist of isolated theorems -- often deep and difficult, but not closely connected to each other. All of that changed when Gauss got in on the act and devised general conceptual foundations for number theory, such as modular arithmetic."
Stewart concludes the chapter by mentioning the importance of number theory to modern computers:
"It often takes time for a good mathematical idea to acquire practical importance -- sometimes hundreds of years -- but eventually most topics that mathematicians find significant for their own sake turn out to be valuable in the real world too."
The sidebars in this chapter discuss unique factorization, the largest known prime (of course, seven more Mersenne primes have been found since this book was published), what we don't know about primes, three mathematicians (Fermat, Gauss, and Marie-Sophie Germain), what number theory did for them, and what number theory does for us.
This is what I wrote last year about the answers to today's test:
1. C' is the same as C, but D' goes up diagonally to the left. This is tricky because the line of reflection is not perfectly vertical.
2. I' goes up diagonally to the right.
3. There are two symmetry lines -- the segment joining the two points and its perpendicular bisector.
4. The angle measures 62 degrees.
5. The angle measures 2x degrees.
6. The reflection image over line AD of ray AB is ray AC. This is tricky because it's been a while since we've seen the Side-Switching Theorem.
7. This is officially the Figure Reflection Theorem -- just make the right vertices correspond.
8. Reflections preserve distance.
9. The orientation is clockwise.
10. The orientation is counterclockwise, because reflections switch orientation.
11. There are three pairs: angles B and C, angles BAD and CAD, angles ADB and CDB.
12. There is one line of symmetry -- the line containing the angle bisector. This follows from the Angle Symmetry Theorem.
13. F' = E follows from the Flip-Flop Theorem. FG = EH is because reflections preserve distance.
14. Proof:
Statements Reasons
1. MO = MN 1. Given
2. M' = N 2. Given
3. MO = NO 3. Reflections preserve distance
4. MNO is equil. 4. Definition of equilateral
It's possible to add more details, such as O' = O, Transitive Property, etc.
15. The rectangle has two lines of symmetry, one horizontal, one vertical.
16. The isosceles triangle has one line of symmetry, and it's horizontal.
17. The images of the vertices are (1,3), (7,1), and (6,-2).
18. The image is (c, -d).
19. The angle measures 140 degrees.
20. The shortest distance is the perpendicular distance.
1. C' is the same as C, but D' goes up diagonally to the left. This is tricky because the line of reflection is not perfectly vertical.
2. I' goes up diagonally to the right.
3. There are two symmetry lines -- the segment joining the two points and its perpendicular bisector.
4. The angle measures 62 degrees.
5. The angle measures 2x degrees.
6. The reflection image over line AD of ray AB is ray AC. This is tricky because it's been a while since we've seen the Side-Switching Theorem.
7. This is officially the Figure Reflection Theorem -- just make the right vertices correspond.
8. Reflections preserve distance.
9. The orientation is clockwise.
10. The orientation is counterclockwise, because reflections switch orientation.
11. There are three pairs: angles B and C, angles BAD and CAD, angles ADB and CDB.
12. There is one line of symmetry -- the line containing the angle bisector. This follows from the Angle Symmetry Theorem.
13. F' = E follows from the Flip-Flop Theorem. FG = EH is because reflections preserve distance.
14. Proof:
Statements Reasons
1. MO = MN 1. Given
2. M' = N 2. Given
3. MO = NO 3. Reflections preserve distance
4. MNO is equil. 4. Definition of equilateral
It's possible to add more details, such as O' = O, Transitive Property, etc.
15. The rectangle has two lines of symmetry, one horizontal, one vertical.
16. The isosceles triangle has one line of symmetry, and it's horizontal.
17. The images of the vertices are (1,3), (7,1), and (6,-2).
18. The image is (c, -d).
19. The angle measures 140 degrees.
20. The shortest distance is the perpendicular distance.
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