8:00 -- Recall that about three weeks ago, I subbed at a middle school on another testing day -- not the SBAC, but the ELPAC for English learners. Well, this high school incorporates this test as part of SBAC week -- today, the students indeed take the ELPAC. This leads into snack break.
10:20 -- Back in my April Fool's Day post (from last year), I noted that somehow that day there were more tardies when school starts at 10:20 than when it starts at 8:00. It's not because the students arrived late to school -- instead, they just hung outside the classroom, and when the bell rang, they had no sense of urgency.
Well, the same thing happens today. On 8:00 days (such as last Tuesday -- the last time I saw this class of seniors), most of the tardies were at 8:01 or 8:02, but today, a whopping eight (out of a total of twelve) are late, with the arrival times being more like 10:25 or 10:30. I can only assume that the same thing happens today -- they simply linger in the hallways instead of enter the classroom.
I do try to warn the students about having excessive tardies, but almost by definition, such warning are futile. These are late students, so by the time they hear a warning it's already too late.
For today's song, I repeat yesterday's "One Billion Is Big" on the guitar, since these classes haven't heard it yet.
11:25 -- Second period leaves and fourth period arrives. This is another senior class -- and the last of three to meet on these two days. It's also the only class with an aide.
Afterward, I count the vocab words and find the average to determine the winning class. Much to my surprise, it's a virtual tie between this class and yesterday's fifth period, with 5.7 words each! If we take an extra decimal place, yesterday had 5 + 11/16 = 5.69 while today has 5 + 2/3 = 5.67. But it's often tricky to count denominators against them (for example, 5 + 2/3 is exactly achievable only when the number of students is a multiple of three). I certainly wasn't expecting the difference between the top scores to be basically a rounding error.
But there is something I can penalize this class on and award the victory to fifth period -- on the board, I wrote down that students who leave class early will be penalized. Around noon, two students ask the aide for passes to go the restroom and drop something off at the office. They never return. I originally wrote this on the board because when I was here last week, two student sneaked out of class early -- but that was a minute or two, not an entire half hour!
This is a tricky one. On one hand, they do ask the aide for passes, but on the other, they clearly want to leave the classroom just because they don't feel like being there. Of course, their zeroes have already factored into their average.
Here's my solution -- it all depends on how many pies I actually see at the store tomorrow. If there are many pies, then I declare it a tie and buy enough for both fourth and fifth periods. But if there are only a few pies, then I penalize fourth for the early exits and give fifth the victory outright.
12:35 -- Fifth period leaves for lunch.
1:20 -- Sixth period arrives. This is a junior class.
And as expected, this is the worst behaved class by far. I'm forced to call security because two guys keep confronting each other. Then when security arrives, one student claims that I'd made the whole story -- he's just asking the other guy for a pencil like a perfect angel. I write a referral and request that security remove only the liar from the room. Meanwhile, other students make the bad list for touching and pouring ice water on each other.
For the pizza incentive, this class has an average of 7.2 vocab words, ahead of the 4.7 that seventh period finishes during yesterday's half-block. That class is about to arrive to try to beat the sixth period score.
There is also a significant tardy in this class -- one guy arrives a half-hour late because he tries to eat off campus and is stuck in a long line at a particular eatery.
2:30 -- Sixth period leaves and seventh period arrives. This is the other junior class.
I urge this class to beat sixth period -- not just because it's a smaller class (my wallet will thank me, especially if I have to buy extra fruit pie for the tied senior classes), but also because there's no way that those misbehaving sixth period students truly deserve a pizza. I finally get them over the hump as they build upon yesterday's half-block with an average of 7.7 words.
3:05 -- Seventh period leaves, thus ending my day.
But just before the final bell rings, an announcement is made. Tomorrow, there is supposed to be an extended snack for a Pep Rally for spring sports. But due to the coronavirus, there will be no Pep Rally and no spring sports.
Instead, it will be a Late Start day, just like an ordinary Monday. There are no teachers' meetings scheduled during the SBAC testing week, but an emergency meeting will be called tomorrow, most likely to discuss what to do about the pandemic.
(Speaking of sports and the coronavirus -- in the real world, most major college and professional sports have now been canceled until further notice.)
Let's see what happens with second period tardies tomorrow with the Late Start day. Notice that the announcement was made late -- seniors scheduled Periods 2-6 only don't hear it, and so some of them might arrive early tomorrow. But then those students might just linger outside of the classroom until well after the tardy bell again, just like yesterday.
On my calendar, today is Friday, the first day of the week:
Resolution #1: We are good at math. We just need to improve at other things.
This shows up only in the seventh period class. As this is the only class that meets both yesterday and today (half-blocks), this class requests a different song besides "One Billion Is Big." They end up choosing "No Drens," which fits the first resolution. We are good at math -- we are no drens.
This is a traditionalists' post. Our main traditionalist, Barry Garelick, posted yesterday -- but that post hasn't drawn many comments yet.
But Garelick was also active around Leap Day/Super Tuesday -- I made a traditionalists' post at that time, but that was mainly to justify writing about politics, rather than the traditionalists. So let's go back to that post right now:
https://traditionalmath.wordpress.com/2020/02/29/help-wanted-dept/
This article in USA Today purports to explain why the US lags other countries in math. Here’s one of the reasons their supposedly-researched article provides:
“One likely reason: U.S. high schools teach math completely differently than other countries. Classes here often focus on formulas and procedures rather than teaching students to think creatively about solving complex problems involving all sorts of mathematics, experts say. That makes it harder for students to compete globally, be it on an international exam or in colleges and careers that value sophisticated thinking and data science.”
As in many traditionalists' debate posts, Garelick mentions politics and race:
3) Make math more practical and inclusive: This is the other side of their mouth speaking. Students should be given more of the everyday problems held to be un-exciting and which turn children off of math. As far as inclusivity, problems should embody aspects of different cultures rather than the white western culture which has prevailed and oppressed free thought for centuries.
In this post, Garelick addresses the Integrated Math debate -- why traditionalists opposed Integrated Math despite its use in high-scoring math nations:
As far as math being taught differently in high schools overseas, it is true that they use an integrated approach: i.e., a mixture of algebra, geometry and trigonometry throughout the year, with each year getting more advanced. Greg Ashman, who teaches in Australia and who writes a blog about education is embarking upon a PhD in education from John Sweller. Greg makes this comment about integrated math:
“In Australia, I teach integrated maths and I always have. However, I teach it directly and explicitly. Other schools may choose less effective approaches. It seems to me that the math reformers are trying to use integrated maths as a Trojan horse for discovery maths. That’s a hard tactic to counter because there is some logic to integrating maths – it should lead to more contact with the various concepts over time, more retrieval practice and better formed schemas.”
Let's see what the commenters have to say here. We begin with Jeff Ulrich:
Jeff Ulrich:
They went from all 8th graders must take Algebra I, even if not ready, to no 8th graders taking Algebra I. Of course the passing rate will increase no matter what methods were used. Why not have students take Algebra I when they are ready, whatever grade level that would happen to be?
This is a tricky one. I can't speak for all districts (especially San Francisco, the district Ulrich likely has in mind here). But I do know that at the old charter school, there were so few eighth graders that we couldn't afford to have separate Math 8 and Algebra I classes for them.The next commenter, Wayne Bishop, builds on the Integrated Math discussion:
Wayne Bishop:
It is very old “news” and dates from the earliest days of so-called “integrated math”. Moreover, it has always represented ignorance on the part of the writer. It is true that the years of secondary math are often named Maths 1, Maths 2, etc., (mathematics are plural, you know) but presentation of standard topics are presented in an entirely standard form from a traditional American perspective. In spite of the non-informing names, everybody in the system knows exactly what topic is presented when (e.g., 2nd semester of Maths 2 – sometimes almost to the same page on the same day as dictated by the Ministry of Education) and opening the maths book at random (my son’s Mathematicas from our year in Mexico, for example) looks like a real mathematics textbook and very different from our “integrated” books. One huge difference is that once a topic has been presented, it is assumed to be done and available for use in deeper settings without extensive review.
In other words, they assume that everyone remembers everything they've learned previously (the "bicycle") -- which, as we know, is a false assumption.
Wayne Bishop:
By contrast, our “integrated” books cover everything every year and involve a great deal of overlap because it cannot be assumed that the students remember – or ever previously studied – basic material needed in the “new” setting. The very embodiment of “mile-wide and inch deep”.
How would Bishop then guarantee that the students have previously studied the material so that it doesn't need to be repeated? The only suggestion he gives here is micromanaging so that everyone in the country is on the same page on the same day. They oppose Common Core for micromanaging the curriculum so much, and here Bishop wishes to micromanage even more?
Of course, we can't leave this comment thread without SteveH:
SteveH:
So where are the voices of traditional AP/IB Calculus track teachers to rebut these ideas? Why don’t they point out that all STEM degrees require that path? Boaler, etal know that they don’t produce these students. They could say that what they promote are solutions for the other students, but they actually claim better understanding and interest and excitement for all – with no proof.
I do see the proof that traditionalists' p-sets and assignments regularly fail -- students don't learn from them because they leave them blank.
Even though today's classes I subbed in are English, I can see how four or five students in each class simply refuses to do the traditional vocab assignments -- they simply spend 100% of the class time on phones and entertainment. Not even the potential reward of a Pi Day pizza is enough to convince these students to do anything.
What Jo Boaler and the other anti-traditionalists promote are assignments that these sort of students don't leave blank. They learn more from assignments they don't leave blank than the ones they do.
Lesson 12-10 of the U of Chicago text is called "The Side-Splitting Theorem." This is what I wrote last year about today's lesson:
The U of Chicago version of the theorem is:
Side-Splitting Theorem:
If a line is parallel to a side of a triangle and intersects the other two sides in distinct points, it "splits" these sides into proportional segments.
And here's Dr. Wu's version of the theorem:
Theorem 24. Let triangle OPQ be given, and let P' be a point on the ray OP not equal toO. Suppose a line parallel to PQ and passing through P' intersects OQ at Q'. ThenOP'/OP = OQ'/OQ = P'Q'/PQ.
Notice that while the U of Chicago theorem only states that the two sides are split proportionally, Wu's version states that all three corresponding sides of both sides are proportional.
Moreover, the two proofs are very different. The U of Chicago proof appears to be a straightforward application of the Corresponding Angles Parallel Consequence and AA Similarity. But on this blog, we have yet to give the AA Similarity Theorem. So how can Wu prove his theorem?
We've seen several examples during the first semester -- a theorem may be proved in a traditionalist text using SSS, SAS, or ASA Congruence, but these three in Common Core Geometry are theorems whose proofs go back to reflections, rotations, and translations. Instead, here we skip the middle man and prove the original high-level theorem directly from the transformations. We saw this both with the Isosceles Triangle Theorem (proved from reflections in the U of Chicago) and the Parallelogram Consequences (proved from rotations in Wu).
So we shouldn't be surprised that Wu proves his version of the Side-Splitting Theorem using transformations as well. Naturally, Wu uses dilations. In fact, the names that Wu gives the points gives the game away -- O will be the center of the dilation, and P and Q are the preimage points, while P' and Q' are the images.
Here is Wu's proof: He considers the case where point P' lies on
Given: P' on
Prove: OP'/OP = OQ'/OQ = P'Q'/PQ
Statements Reasons
1. P' on
2. OP' = r * OP 2. Multiplication Property of Equality
3. Exists Q0 such that OQ0 = r * OQ 3. Point-Line/Ruler Postulate
4. For D dilation with scale factor r, 4. Definition of dilation
D(Q) = Q0, D(P) = P'
5.
6. Lines P'Q0 and P'Q' are identical 6. Uniqueness of Parallels Theorem (Playfair)
7. Points Q0 and Q' are identical 7. Line Intersection Theorem
8. OQ' = r * OQ, OP' = r * OP, 8. Substitution (Q' for Q0)
P'Q' = r * PQ
9. OP'/OP = OQ'/OQ = P'Q'/PQ = r 9. Division Property of Equality
Now the U of Chicago text also provides a converse to its Side-Splitting Theorem:
Side-Splitting Converse:
If a line intersects rays OP and OQ in distinct points X and Y so that OX/XP = OY/YQ, then XY | | PQ.
The Side-Splitting Converse isn't used that often, but it can be used to prove yet another possible construction for parallel lines:
To draw a line through P parallel to line l:
1. Let X, Y be any two points on line l.
2. Draw line XP.
3. Use compass to locate O on line XP such that OX = XP.
4. Draw line OY.
5. Use compass to locate Q on line OY such that OY = YQ.
6. Draw line PQ, the line through P parallel to line l.
This works because OX = XP and OY = YQ, so OX/XP = OY/YQ = 1.
The U of Chicago uses SAS Similarity to prove the Side-Splitting Converse, but Wu doesn't prove any sort of converse to his Theorem 24 at all. Notice that many of our previous theorems for which we used transformations to skip the middle-man, yet the proofs of their converses revert to the traditionalist proof -- once again, the Parallelogram Consequences.
Another difference between U of Chicago and Wu is that the former focuses on the two segments into which the side of the larger triangle is split, while Wu looks at the entire sides of the larger and smaller triangles. This is often tricky for students solving similarity problems!
Now it's time to give the Chapter 12 Test. Again, it's awkward to combine Lesson 12-10 with the test, but that's the way it goes. I can argue that by doing so, we're actually following the modern Third Edition of the text. Yesterday's lesson on SAS~ and AA~ is the final lesson of Chapter 12 in the newer version (Lesson 12-7). In the Third Edition of the text, the Side-Splitting Theorem is the first lesson of the next chapter, Lesson 13-1.
Test Answers:
10. b.
11. Yes, by AA Similarity. (The angles of a triangle add up to 180 degrees.)
12. Yes, by SAS Similarity. (The two sides of length 4 don't correspond to each other.)
13. Hint: Use Corresponding Angles Consequence and AA Similarity.
14. Hint: Use Reflexive Angles Property and AA Similarity.
15. 3000 ft., if you choose to include this question. It's based on today's Lesson 12-10.
16. 9 in. (No, not 4 in. 6 in. is the shorter dimension, not the longer.)
17. 2.6 m, to the nearest tenth. (No, not 1.5 m. 2 m is the height, not the length.)
18. 10 m. (No, not 40 m. 20 m is the height, not the length.)
19. $3.60. (No, not $2.50. $3 is for five pounds, not six.)
20. 32 in. (No, not 24.5 in. 28 in. is the width, not the diagonal. I had to change this question because HD TV's didn't exist when the U of Chicago text was written. My own TV is a 32 in. model!)
By the way, I changed the first worksheet below in order to give the U of Chicago version of the Side-Splitting Theorem proof, rather than the Wu/EngageNY version. When I did so, I notice that the old worksheet had a nine-step proof. The U of Chicago proof also lists nine steps (actually eight since that text never provides the "Given" step).
The test, meanwhile, starts with Question #10. This numbering has nothing to do with nine-step proof, since I used to post Questions #1-9 on an old defunct version of my test. But it suddenly gave me the bright idea to treat this lesson as part of the test, since we're giving Lesson 12-10 on the same day as the Chapter 12 test. (But this might be awkward for the students, especially Steps #5-9 of the proof which is all algebraic manipulation. We wouldn't want the students to miss 5/20 or 25% of the test due to confusion with the algebra.)
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