Today is Day 100. As I've explained in previous years, Day 100 is significant in many kindergarten and first grade classrooms. Indeed, three years ago our K-1 teacher (who eventually succeeded me as middle school math teacher at the old charter school) celebrated Day 100. But the district whose calendar is observed on the blog is a high school only district.
Believe it or not, I did see a special ed teacher have "Day 100" on her calendar when I subbed there last week (on Days 97-98). I have no idea whether she actually has a party with her students or if it's just something she writes on the calendar. It's possible that some special ed students in high school celebrate Day 100.
Meanwhile, I've also once seen a reference to a Day 1000 celebration. This refers to counting continuously from the first day of kindergarten. Since 1000 - 100 = 900 and 900 = 180 * 5, we find that Day 1000 of elementary school works out to be Day 100 of fifth grade. Thus it's possible for four grades in an elementary school (Grades K, 1, 2, 5) to have a big celebration on Day 100.
On the other hand, in my new district (where I subbed today), we have elementary students, but in that district today isn't Day 100. Instead, it's only Day 91 -- the first day of the new semester. But then again, elementary and middle schools use trimesters, not semesters.
This teacher has both science and history classes. In my October 11th post, I didn't do "A Day in the Life" because three of the five classes had aides. Today, aides were available for only two of the classes -- with a majority of solo classes, "A Day in the Life" is in order here.
8:15 -- This is the school that starts with homeroom and has all periods rotating.
8:20 -- Today, the rotation begins with sixth period -- which just happens to be conference period.
9:20 -- First period arrives. This is the only eighth grade class -- US History -- and the first of two classes with an aide.
10:10 -- First period leaves for snack.
10:20 -- Second period arrives. This is the first of two seventh grade science classes, and the first of three solo classes.
First of all, this is a science class, so I must compare this to the old charter school as usual. It turns out that this class is still learning about plants. On October 11th, they were studying photosynthesis, and today is all about classifying plants (angiosperms, gymnosperms, etc.).
Like most science lessons these days, the assignment was on Chromebooks. Once again, this reminds me that I should have had more online science lessons at the old school, especially for the seventh graders who didn't get much science. Even if all I did was spend three-plus months on plants/botany and nothing else, that's much better than what I actually did with the seventh graders that year.
As for classroom management, last Friday's problems are still fresh in my mind. I believe that many of the problems all go back to a flaw in my music incentive -- I sang at the beginning of class, then expected the class to be good for the entire block before I proceeded to the next verse. By the time the students misbehaved, they'd long forgotten about the song!
So here's how I changed it up today. First, what song did I choose? For life science classes, I usually select "Meet Me in Pomona," but this class already heard that song in October. Thus instead I choose one I haven't sung in a while -- Square One TV's "Count on It."
I decide to take attendance first. But then three girls refuse to sit in their assigned seats. Going back to last Friday again, two of them claim restroom "emergencies," but only one actually leaves. Upon her return, I kick all three out of the room -- sending them again to the room in the center of the building.
And then I sing the first verse of "Count on It." Even though I believe that I successfully avoid arguing that day, starting the class on a negative note often ruins the song. I don't put as much energy into my performance, and the students don't enjoy it as much.
This is the problem if I try to do anything before singing a verse of the song (even attendance). If I must start with something negative, then the students will resent me so much that they'll hate anything I say next, even a song. Thus if I want the students to enjoy the song, then I must start the class positively, with the song itself. Even if there's a seating chart and I can easily tell that the students aren't in the correct seats, I should perform the song first. Just a single verse is enough (since as the students remain in the wrong seats longer, they'll start to think, "Hey, this teacher doesn't mind if we move!" and then others will try to switch seats).
An aide from another class hears the three girls talking loudly in the center room, and so she escorts them back to my room. This time they go back to their correct seats. One of the trio starts to work, and so I avoid writing her name on the bad list, while the other two names must remain there. (This is something I should have done more at the old charter school -- if there's a group of misbehaving students, try to isolate them somehow. The hope is that at least one of them will then choose to show good behavior.)
Then I set the timer on Google -- whenever the class is good for ten straight minutes, then I sing another verse of the song. This is better than expecting them to be good the entire period (even when it's not a block schedule like last Friday). Today, the class earns the incentive for two ten-minute periods, enough for me to sing the remaining two verses of "Count on It."
11:15 -- Second period leaves and third period arrives. This is the other seventh grade science class.
This time, I adjust by singing the song before taking attendance. And indeed, this class enjoys my song much better than second period did. I must still place a few guys' names on the bad list for failing to do any work on the Chromebook.
12:05 -- Third period leaves for lunch.
12:50 -- Fourth period arrives. This is the first of two seventh grade World History classes, and the last of three classes without an aide. Many of the same students from science stay here for history.
As usual, the class after lunch begins with silent reading. Then the students move on to their assignments on Ancient China. (This lesson is timely since last weekend was Chinese New Year -- but of course, it's tricky to try timing a unit to a lunar holiday.) This lesson apparently contains both a written component and a Chromebook component.
I must write two guys on the bad list in this class. One of them moves to a different seat and then refuses to do any work, while the other keeps making loud noises on the Chromebooks.
Since this class already hears "Count on It" during their science classes, for history I end up reverting to what's become my old standby, "One Billion Is Big."
2:00 -- Fifth period arrives. This seventh grade World History class has an aide.
The aide tells me that she often likes to find a YouTube video on China to play for the class when it rotates into the last position, because these students sometimes act up when it's the last class of the day. (Recall that the fact that students act differently at 2:00 than they do at 10:30 is the reason for having period rotations in the first place!) Moreover, many of these are the same students from the troublesome class I mentioned three weeks ago, in my January 8th post. So I know how bad these students can become.
And sure enough, the students cause trouble. First, one guy insults the aide for no reason. He quickly apologizes, but she doesn't accept it. Instead, she makes him write a "positive paragraph." (This reminds me of the 150-word essays that the instructional aide at the old charter school suggested that I assign the students.) He's also been placed on the bad list for one of the science classes.
Then a group of four girls starts to misbehave. One of them complains about having to having to watch the boring China video -- and this is after she complains to me when she thinks I'm about to sing "Count on It" instead of play the video! (She's also in the second period class -- that's the class that hates "Count on It," if you recall.) The aide ends up sending all four girls to the office, where they receive a referral.
I also make some questionable timing decisions in this class. Since the aide has already informed me in the morning that fifth period will get a video (so that I'd have time to search for one on YouTube), I know that fourth period is the last class that needs the Chromebooks or history texts. But if I'd asked the students to put them away at the end of fourth period, they might have wondered why fifth period wouldn't be using them. (It would have worked better if the penultimate class had been, say, an eighth grade class that would be doing something different from the seventh graders anyway. Unfortunately, today the last two classes are both History 7.)
And so I end up asking fifth period to clean up during the video. I remember early in the period to have some volunteers put the Chromebooks back (and this is how I establish a good list for fifth period -- the clean-up volunteers make the list). But I don't remember to have students put away the printed textbooks until there are about ten minutes left -- and some students interpret this as time to put away the notebooks (where they're taking video notes) as well, thus leaving the class with nothing to do the last ten minutes. (This is also a problem in math classes if I pass out homework or give them time to do with homework with about ten minutes left -- they see this as "Class is over now!" So I must find a better time to pass out the HW.)
The regular teacher shows up between his all-day district meeting and his after school meeting with about five minutes to go. The aide and I describe what's happened good and bad today.
2:55 -- Fifth period leaves. But my day isn't over yet -- at this school, both regular teachers and subs often have supervision duty.
3:05 -- After-school supervision ends, thus finally completing my day.
Today is Fourday on the Eleven Calendar:
Decade Resolution #4: We need to inflate the wheels of our bike.
Here "the bike" refers to easily remembered math knowledge, and so it's not very relevant in a science or history class. Still, the "Count on It" song indirectly refers to the fact that the students can "count on" using math in real life. Thus they need to recall at least some math skills as easily as remembering how to ride a bike.
Lecture 16 of Michael Starbird's Change and Motion is called "Zeno's Arrow -> The Concept of Limit," and here is an outline of this lecture:
I. The limit plays an essential role in calculus.
II. Historically speaking, limits were hard to formalize.
III. The limit arises in the context of an infinite process. Limit refers to the idea that the totality of an infinite process or collection of numbers can have a single number as the answer.
IV. Zeno's arrow paradox leads to the idea that is central to the mathematical underpinning of calculus -- the idea of limit.
V. The formal definition of limit is challenging but illustrates an interesting approach to understanding.
VI. Let's think about what characteristics a limit has, then identify features that are so fundamental that the limit is precisely the number that satisfies those conditions.
VII. The derivative and integral entail taking limits.
VIII. The definition of limit was finally formalized in the middle of the nineteenth century.
Starbird's lecture today is all about limits in Calculus. He opens this lecture with a quote from Isaac Newton, who defined derivatives before limits were formalized. Instead, he referred to "nascent" and "effervescent" quantities, which Bishop Berkeley later called "the ghosts of departed quantities."
The professor refers to Zeno's Paradoxes, which we also saw in David Kung's lectures. First he mentions the Arrow Paradox -- an arrow must travel halfway to its destination, then half of the remaining distance, then half of the still remaining distance, and so on. The other paradox he mentions is Achilles and the Tortoise, which we also explored in Douglas Hofstadter's book. Zeno tells us that Achilles can never catch the Tortoise if the latter has a head start.
He shows how to resolve the Achilles/Tortoise paradox in a diagram. When Achilles is at point 1, the Tortoise is at position 1.1. When Achilles is at point 1.1, the Tortoise is at position 1.11. When Achilles is at point 1.11, the Tortoise is at position 1.111, and so on. Therefore the warrior finally reaches his reptile friend at position 1.111....
Starbird now gives his first example of a limit. Once again, we're familiar the notation, but this looks weird in ASCII:
lim _x->2 ((x^2 - 4)/(x - 2))
The professor now shows the following chart:
x (x^2 - 4)/(x - 2)
2.1 4.1
2.01 4.01
2.001 4.001
1.9 3.9
1.99 3.99
1.999 3.999
lim _x->2 ((x^2 - 4)/(x - 2)) = 4
Starbird reminds us that limits are needed to calculate derivatives. He shows us the graph to find the derivative of f (x) = x^2 at x = 1.
The professor finally defines the limit -- lim _x->c f(x) = l means:
For every epsilon > 0, there exists delta > 0 s.t. if |x - c| < delta, then |f(x) - l| < epsilon.
Recall that epsilon is a small number (hence Rebecca Rapoport's Your Daily Epsilon of Math -- that is, your daily "little bit" of math). Starbird compares this to a great tennis player:
For every tennis ball you serve me, there exists a return swing from me.
For every challenge (epsilon) you serve me, there exists a response (delta) from me such that if x is delta-close to c, then f(x) must be epsilon-close to l.
The professor uses this to define the derivative. We can finally prove the derivative of x^2 to be 2x.
In addition to Day 100, today is also test day. Here are the answers to the Chapter 9 Test:
1. Draw two triangles, one not directly above the other, with corresponding vertices joined.
2. Draw a picture identical to #3.
3. Draw and identify a circle and an ellipse.
4. Draw and identify two circles.
5. circle (Yes, I had to make an eclipse reference because of last week's event!)
6. a. Draw a circle. b. Draw a parallelogram (not a rectangle). c. Draw a rectangle.
7. c or d.
8. Draw #7c or d again. (A cube is a rectangular parallelepiped!) The faces don't need to be squares.
9. a. 144 square units b. 8 units
10. a. Draw #3 again, with both heights labeled. b. 25pi square units
11. a. 2 stories b. 3 sections c. back middle
12. a. tetrahedron, regular triangular pyramid b. 6 edges c. (
13. sphere
14. solid sphere (This is yet another eclipse reference!)
15. rectangular solid
16 a. yes b. 3 planes
17. lw(h + 2)
18. xy(x + 2 + y)
19. pi r^2 (4 + r)
20. planes
There's no Euclid on the test, but let's look at the next proposition in Euclid anyway:
Euclid's proof of this Uniqueness of Perpendiculars proposition is indirect.
Indirect Proof:
Assume that point A lies in plane P, and both
Today is a traditionalists' post -- as if this post wasn't already jam-packed. By the way, it's interesting to comment on traditionalists during our viewing of Michael Starbird's lectures on Calculus -- the class that the traditionalists want high school seniors to take. Then again, no traditional AP Calc class would teach it the way that Starbird does -- both derivatives and integrals coming before limits!
Barry Garelick has made several posts recently. But the two that have drawn the most comments (five each) were posted over MLK weekend -- that is, on non-posting days for me.
https://traditionalmath.wordpress.com/2020/01/18/some-thoughts-on-devlin-and-boaler/
For those who have read and heard Keith Devlin, he is pretty close with Jo Boaler who you may also have heard about. Keith Devlin, you will recall, writes a column called Devlin’s Angle in MAA and also is known as “that math guy” at NPR.
This is the first time that I've heard of Keith Devlin -- but of course, this is the second straight post in which he invokes the name Jo Boaler as his opponent.
He made a big name for himself some years ago when he claimed that multiplication “Ain’t no repeated addition”.
OK, so Devlin believes that multiplication is more than merely repeated addition, but Garelick clearly disagrees with him:
Devlin and others of like mind think that teaching multiplication as repeated addition results in confusion when we teach fractional multiplication. Actually it isn’t that confusing, and using an area model incorporates the ‘repeated addition’ form of multiplication to get the end result.
I thought that Garelick disliked the area model -- especially when students are forced to draw pictures of areas just to do a simple fraction multiplication. Let's see what he says about Boaler:
Boaler doesn’t go quite so far, but she is more for “fluently deriving” the math facts than straight memorization. So 9 x 8, should be looked at as 9×9 – 9. That way, kids who don’t know all their facts can derive them. And also satisfy her idea of what “deeper understanding” is. Really, folks, multiplication isn’t that hard. Nothing against the strategy she talks about, but the disdain for memorization (because it supposedly eclipses understanding) is just more money-making nonsense for Boaler and her ilk.
Another Boaler method is to look at 9 * 8 as 8 * 8 + 8. This truly goes back to the definition of multiplication as repeated addition (that Garelick embraces), since we're adding 8 eight times to get 64, then once more to get 72. Would that be more acceptable to Garelick?
Also, it's interesting that Garelick criticizes Boaler's method as "money-making nonsense," even as the traditionalists claim their preferred methods as money-makers for tutoring companies.
Some comments:
Chester Draws:
Do you know how Devlin would try to teach that a + a + a = 3 x a = 3a ?
I’d be interested to see how one would approach that without using multiplication can be thought of as repeated addition. He’s bound to be big on “understanding” so you wouldn’t be able to merely say “because I say so”.
Does he oppose teaching powers as repeated multiplication? Because, again, that would make understanding it very difficult.
(I’m not going to read him myself. It would just make me angry.)
Hmm, perhaps I need to read Devlin myself to learn more about his method.You know I'm going to get to SteveH, of course:
SteveH:
Full inclusion academic classes have increased the range of willingness and ability in K-6 and their assumption is that natural differentiated learning will handle it all. Expectations have been lowered to a statistical low CC slope to no remediation in a college algebra course. This starts in kindergarten. K-6 is now officially a NO-STEM zone, and our state CC provider has officially stated that. The College Board recognizes this so they instituted their Pre-AP algebra math class in 9th grade to FINALLY emphasize individual mastery of knowledge and skills and P-sets. It’s too little and too late, and they know it. They’re just providing cover for the low CC expectations of K-8 – that they helped define. Students expect that they can get to AP Calculus, but what does the College Board tell them when nobody can squeeze in 4 years of traditional math into 3 years? Oops? It’s your fault? You didn’t have enough passion or engagement? You didn’t learn how to struggle productively? It’s you, not them?
Education is not about statistics. It’s about individual educational opportunity and equality, and that gap is being increased by the work we parents now have to do at home. We help support their statistics and they really don’t want to add parental help in as part of their “big data” answer to all problems. Anecdotes like my son contain all of the information and understanding and different ones might point to a number of problems and different solutions. When you break anecdotes into “big data”, a lot of understanding is lost and cannot be reconstructed. The only understandings people find from big data are the ones they are looking for, and they usually look for just one explanation.
That last sentence applies to the traditionalists as well as the progressives. The only understandings the traditionalists find from big data are the ones they are looking for, and they usually look for just one explanation -- namely the one that supports traditionalism.
https://traditionalmath.wordpress.com/2020/01/19/much-ado-about-distribution-dept/
A common complaint–as in “see, math is being taught wrong”– is that students fail to see that equations like 3(x-5)=60 can be solved by dividing both sides by 3 first. Progressives seem to make a big deal about this to the tune of “If students are doing this, they lack ‘deeper understanding’ about equations.
Textbooks that claim alignment to the Common Core now make it a practice to show this. The problem is that if you have 7(x-5)=60, the process isn’t so neat.
I mentioned this recently on the blog, only because one teacher started using a mnemonic SADMEP (the reverse of PEMDAS). So according to SADMEP, we should divide 3(x - 5) = 60 by 3 first, since division (D) comes before parentheses (P). But to me, SADMEP is really there to remind students to subtract/add (SA) first before dividing/multiplying (DM). I have no idea whether this teacher really expects students to follow SADMEP all the way to the P.Chester Draws:
[Quoting Garelick: The problem is that if you have 7(x-5)=60, the process isn’t so neat.]
But at least possible.
When they get x(x – 5) = 50, then they have no option but to expand.
Then again, quadratics can't be solved using SADMEP, even after it's written as x^2 - 5x = 50. I suspect that SADMEP is something to teach seventh graders, not Algebra I students.SteveH:
I never really learned algebra until Algebra II. There were lots of problems where I could not “see” the fastest or alternative approaches. That comes with practice and lots of individual homework. Do they offer any specific curriculum and process (other than slow down and take more time) to do a better job? What is full understanding and how do they test for it? Do they have any proof that what they do works?
As problems become more advanced and tricky, like logarithms on the AMC/12 test, it’s very easy to mislead very good math students. The best way for my son to prepare for the AMC test was to do as many of their past test problems as possible. There might be a small amount of general problem solving transference, but in the vast majority of problems, the best help is to see something like that problem before.
As for 3(x-5) = 60, solving it any which way shows understanding. Solve enough problem variations correctly at that course level, then everything is fine.
Here SteveH refers to the AMC high school contest. I took its predecessor, the AHSME. I graduated from high school in 1999, just barely before the AMC started in 2000.
Recall that today's fourth resolution on inflating the bicycle was made with traditionalists in mind -- the only way to be good at math is for as much math as possible to be as easily remembered as riding a bike is.
To traditionalists like SteveH, the only way to inflate the bike is to do traditional p-sets. He mentions that his son was willing to work on p-sets -- but then again, students who take the AMC are hardly representative of most students actually sitting in math classes. We need a way for students who tend to leave traditional p-sets blank to inflate their bicycles.
Here is today's test:
No comments:
Post a Comment