For the sixth and seventh graders, I can show them ideas from the first six or seven chapters of the book, minus the trig parts. For the eighth graders, I can go a bit further in the book, since here in California eighth graders focus on the physical sciences. This may change under Next Generation Science standards, though, to a more integrated approach.
Today is a traditionalists post, but I think I'll do the PARCC question first. Question 28 of the PARCC Practice Exam is on trig:
28. The figure shows right triangle ABC.
[The triangle is labeled, with the right angle at A and a, b, and c opposite A, B, and C -- dw]
Which of the listed values are equal to the sine of B?
Select all that apply.
D. the cosine of B
E. the cosine of C
F. the cosine of (90 - B)
G. the sine of (90 - C)
And now we have a trig problem -- the other subject I taught during that full week I spent in a math classroom, this time the Algebra II class rather than Honors Integrated Math I. Again, all these great problems that match up with the worksheets are appearing one and two weeks after I subbed there.
The usual definition of sin B is choice (C). But we know the definition of cosine as complementary sine -- and this is a frequent PARCC question, as we've seen in the past. As it turns out, choices (E), (F), and (G) are all correct.
Frequent errors include marking only one answer instead of four, and getting confused with the relationship between sine and cosine. It may help to change 90 - B to C and 90 - C to B before attempting to answer -- then it becomes obvious that (E) and (F) both say cos C and (G) says sin B, which is the original question.
This is a great point to transition into another traditionalists post. I've said before that my classes will not be traditionalist, except when it comes to basic skills, where I agree with traditionalism. One traditionalist I haven't discussed on the blog yet, but whose ideas will be a strong influence in my own classes, goes only by the username "Bill." Bill often posts at the following link:
The creator of this website (Joanne Jacobs) regularly links to various education articles -- I often skip the middle woman and link directly to those articles, but if I do link to Jacobs herself, it's to highlight the comments of Bill and a few other traditionalists who post in the comments.
Bill strongly believes that students graduating these days lack basic skills. Here are a few of his most recent comments:
Your 2nd grade teacher sounds like an idiot, if he or she couldn’t spot the properly spelled word and the improperly spelled one, but then again, in many elementary schools these days in the U.S., a student who fails all of their coursework in every subject isn’t held back, either, so they don’t have an issue with their self-esteem.
What happens in 6-7 years when they reach middle school and cannot read, write, or add/subtract/multiply/divide…
I'm actually confused with the first part of Bill's comment, since the "2nd grade teacher" mentioned here was trying to catch cheating. The important part of the comment is the second part. Bill complains that grade retention isn't used often enough, because students are allowed to move on to the next grade without having basic skills. This is a frequent traditionalist complaint.
That’s because in the 70’s and 80’s, persons who excelled academically or showed a strong interest in math/science got labeled as ‘geeks’, and the kudos always went to the football/baseball/basketball teams (or insert your sport here)…This is the way it has always been in high school/college/professional sports.
Given that it’s darn near impossible to hold back a student who has failed all of their courses in elementary school, is it any wonder why students who put the time and excel academically are in those ethnic groups which make education a priority?
I was looking at a article on using CRISPR-cas9 to edit DNA, and this was being discussed in a high school biotechnology classroom, I’d say all of the students were either caucasian, asian, or asian-american (go figure)…
By the way, most of Bill's comments are not about racial politics, but unfortunately, his most recent comment as of today is about racial politics (and I'd planned on writing this traditionalist post today about Bill long before I read this comment). Then again, I am burying this in a post just before a three-day weekend, which I often do in order to minimize the controversial subject of race. (The original article also mentioned gender -- not enough girls are going into STEM, but Bill and the other commenters chose to ignore gender and focus only on in this thread.)
Of course, the first part of the comment has nothing to do with race. Words like "geek" and "nerd" aren't specific to any race -- Bill laments that sports are considered high status and math is considered a low-status activity. We've seen before that students are willing to work on athletic drills, which are neither easy nor fun, if the drills would make them a champion athlete. But the same students, if they are asked to perform mathematical drills, will insist that they instead do only activities that are easy and fun, and ask questions like "Why?" and "When will we ever use this?" whenever asked to perform a task in math class that is not easy or fun.
Now it's the second part of Bill's comment where race places a factor. He repeats his complaint that there isn't enough retention -- a crude form of tracking -- in elementary school. But here's the thing -- Bill claims that this failure to track students, far from reducing the achievement gap, is actually contributing to the gap! Bill's post is implying that if elementary schools were to bring back retention, perhaps after repeating a grade, students will suddenly see the need to work hard in their classes in order to avoid being held back again. Then students of all races would suddenly have an incentive to work hard, and with everyone working hard, the achievement gap would be reduced.
This is a typical traditionalist argument, and I don't fully agree with it at all. I suspect that if an aggressive retention plan were adopted, students wouldn't start studying harder. Instead, the natural human reaction is to blame the system, rather than themselves, for their failure. And if we add the highly volatile factor of race, people are even more likely to blame the system -- and this is exactly why tracking no longer exists! Once again, the traditionalist plan won't give the results that the traditionalists desire.
And here's another, non-racial factor that the traditionalists fail to consider. Some students will refuse to learn, and so there will be huge age gaps in the classroom. Bill mentions self-esteem and the broken feelings of the students who must repeat grades, but I'm worried about the broken bones of the smarter students who are beaten up by kids in the same class who may be four or more years older than they are.
Last night was the Scripps National Spelling Bee, and every year after the competition, I watch one of my favorite movies, Akeelah and the Bee. It is about a girl who is branded as a nerd for being such an excellent speller, but becomes a local heroine as she advances to the National Spelling Bee. (Oh, and race definitely plays a factor among the characters in the movie as well.)
Now I said that I want to apply Bill's thoughts to my own classroom. I'm a middle school teacher, so there's obviously nothing I can do about elementary school retention (regardless of whether I agree with Bill's idea or not). But I do want to turn around the idea that students who are successful at math are "nerds" who are worthy of scorn.
Indeed, I turn around the word "nerd" to form the word "dren" -- an anti-nerd who is incapable of performing elementary-level math. Even though Bill doesn't use the word "dren" (of course not, since I made it up), it encapsulates Bill's opinion in a single syllable -- it's not the students who are good at math who have the problem. The students who can't do math are the ones with the problem!
In some of Bill's posts, he writes about college and career readiness. I will continue to write Bill's comments throughout the summer, and I may even repeat some of these in the classroom. The point is that the so-called "nerds" are actually popular with the only people who matter -- college admissions officers and employers -- and it's the "drens" who are unpopular with the people who matter. I'll have more to say about this idea as we read more of Bill's comments together on the blog.
My next post will be on May 31st, which is after Memorial Day.
PARCC Practice EOY Question 28
U of Chicago Correspondence: Lesson 14-4, The Sine and Cosine Ratios
Key Postulates: Definition of Sine and Cosine
Common Core Standard:
Explain and use the relationship between the sine and cosine of complementary angles.
Commentary: The worksheet is based on the Pizzazz Worksheet from Algebra II. I changed it so that the hypotenuse is a, to match the PARCC problem on the previous page.