1. Pappas Question of the Day
2. An Interesting Thought Experiment
3. Optimal Living Conditions?
4. Summer School Grades
5. Conclusion
Pappas Question of the Day
Today on her Mathematics Calendar 2018, Theoni Pappas writes:
Find the slope of the perpendicular bisector of segment
[Here are the givens from the diagram: A(-5, 6), B(2, 5).]
To solve this problem, we first find the slope of
m = (6 - 5)/(-5 - 2) = 1/-7
Now the slope of any line perpendicular to
I wonder whether this should be considered a Geometry question at all. Yes, it does contain the words "perpendicular bisector." But notice that the slope of any line perpendicular to
An Interesting Thought Experiment
Today I wish to take a break from Van Brummelen and spherical geometry. I was going to return to the traditionalists' debate, but the traditionalists have been quiet lately.
Instead, I'd like to begin this post with an interesting question. I actually found this question over on the Mersenne Forum website -- the site where newly discovered Mersenne primes are first reported.
http://www.mersenneforum.org/showthread.php?t=23412
Suppose you took a hundred or so babies at random and gave them optimal living conditions. Among other things, they get a healthy diet, a safe environment, and private tutors. Their life goal is to get a passing grade in a university-level math class such as linear algebra, differential equations, or multivariable calculus. Assume that they are sufficiently motivated and that none of them cheat.
Given those conditions, would more than half of them eventually pass? I think they would, but a lot of people I've talked to disagree.
They are given unlimited attempts and do not have to pass the course by a certain age.
In many ways, this question is at the heart of the traditionalists' debate -- and indeed, much of our educational policy depends on the answer to this question.
Our usual traditionalists might latch on to that phrase "private tutors." The author of this post, MooMoo2, probably included "private tutors" for the benefit of those students who are slightly below grade level.
But to the traditionalists, any student who isn't in eighth grade Algebra I is "below grade level." And moreover, they believe that more students would be in eighth grade Algebra I if only they were taught traditional math in Grades K-7. Thus not only would the traditionalists answer a resounding "yes," but to them, the private tutors are unnecessary -- just replace them with traditionalist teachers.
MooMoo2 tells us that these hypothetical students "do not have to pass the course by a certain age," yet traditionalists insist that students take Algebra I and Calculus by a certain age. To them, students who aren't in AP Calculus as high school seniors won't ever pass lower division math -- not even if they're given years or decades to do so.
Of course, that's where the phrase "sufficiently motivated" comes in. According to traditionalists, most students aren't motivated to pass lower division math after the age of 19 -- which is why they need to be exposed to Algebra I and Calculus as soon as possible. It's only in this thought experiment where our students are still motivated to pass math at age 29, 49, or 59.
I'm the opposite -- I'm worried about whether students are still motivated to do math past age nine. In fact, I agree with the traditionalists that traditional math is the best and fastest way to teach the students math. The problem is that many students are unmotivated -- that is, we assign them a p-set and they won't even start on Question #1 (or #2). The sole purpose of project-based learning and other nontraditional forms of math is to get unmotivated students who won't do #1 on the p-set to do something, anything.
In this gedanken experiment, all of the students are motivated, so we don't have to worry about any of these students not answering #1 on the traditional p-set. And the private tutors are there to keep them motivated just in case. Again, it's easier for you to escape starting #1 on the p-set when you're one of 30 students in the class, and when you're expected to do the p-sets at home, which could be miles away from the teacher. It's harder to avoid #1 when the private tutor is mere inches away from you and demanding that you complete the p-sets.
Optimal Living Conditions?
Included in the constraints given by MooMoo2 are "optimal living conditions" -- specifically, "a healthy diet" and "a safe environment."
There are many people who believe that the answer to MooMoo2's question is "no" -- and therefore, money is being "wasted" providing "optimal living conditions" for students for whom the answer to the question is "no." For example, providing "a healthy diet" leads directly to highly-charged political debates about the free lunch program in schools and food stamps in general.
There is a Google Doodle today -- Helen Rodriguez Trias, a pediatrician. (On the blog, I only discuss STEM-related Google Doodles -- and it's debatable whether medicine counts as STEM.) In reading her biography, she attended schools in New York City and was placed into low-track classes despite being qualified for high-track classes -- likely because she was of Puerto Rican descent. In other words, here we go again with tracking and race.
Fast forward 80 years, and there's another Big Apple tracking debate. The mayor is to eliminate the SHSAT entrance exam at some of the special magnet schools. And the reason he gives is that not enough students of certain races are being admitted.
I'm not sure what the situation was back in Helen's day. The SHSAT didn't exist back then, and some of these elite high schools didn't admit girls at all. Thus I can't directly tie the young Helen's tracking problems to the current SHSAT debate. But it's all about the same issue -- access to higher classes.
Even though our main traditionalists have been silent lately, there are a few recent posts at the Joanne Jacobs website that are indirectly related to this issue:
http://www.joannejacobs.com/2018/07/a-shiny-new-school-fails/
San Francisco spent $54 million to build and equip a new STEM-focused middle school to serve low-income black students — and lure middle-class students from nearby neighborhoods, writes Daniel Duane in Wired. “Tech titans” contributed. It sounded great, he writes. “Then it opened.”
And sure enough, race -- as well as income -- is mentioned right in the first paragraph. Again, this underlies the inevitability that any discussion of tracking leads to race and income.
The commenter Momof4 has leaned traditionalist in earlier comments, so let's see what she says here:
Momof4:
The whole concept is seriously flawed- because you cannot start with MS for students who are so far behind grade level. Like the Core Knowledge schools I have encountered, and the way Singapore Primary Math was implemented at my grandkids’ school, you start the program with k-1 and add a grade a year, for the first cohort and those following – and those were schools filled with bright, motivated kids. It is like building a house; the foundation must come first.
This part isn't quite related, but it does go back to some criticisms of Common Core. Any new curriculum -- even traditionalist-approved curricula like Singapore Math (mentioned in this thread) -- should be introduced a year at a time. In other words, Common Core was doomed to fail unless those implementing it waited at least a decade before introducing it to high school students.
But this is the part I want to focus on:
Momof4:
Such a school also must be willing to remove kids who are uninterested in the programs, disruptive, uncooperative, have attendance problems, are unwilling to work hard, cannot keep up with classmates etc. As Robert Weissberg says, the best way to improve a school is to get better students – and vice versa. The district should have known that any attempt to attract middle-class-plus students was/is doomed to fail if the campus is unsafe, the classes chaotic or their kids’ classmates are significantly behind them in academics.
And this goes back to MooMoo2's "safe environment." Momof4, quoting Robert Weissberg, implies that the students are the environment. By this line of reasoning it's impossible to improve the environment of "bad" (whatever that means) students -- instead, moving "bad" students to another environment makes the environment "bad," as opposed to making the students "good."
There would be no problem except that "bad" ends up gaining the de facto definition "of the wrong race," and then the big tracking/race debate begins. And this is what happened to Helen, the subject of today's Google Doodle, 80 years ago -- she was left off the gifted track not because she wasn't smart, but she was considered to be a "bad environment" due to her ethnicity.
Here's a link to a similar recent article at the Joanne Jacobs website:
http://www.joannejacobs.com/2018/07/new-schools-end-up-like-old-schools/
High-tech schools of the future end up looking a lot like schools of the past, writes Larry Cuban in Regression to the Mean, Part 1 and Part 2.
His examples are a public middle school in New York City, called the Downtown School in a study, and School of the Future, a public high school in Philadelphia.
By the way, this "Downtown School" is described as a new school, so it obviously didn't exist back in Helen's day.Anyway, even though this post is ostensibly about technology in education (Edgenuity, anyone?) here's Momof4's comment, where she jumps right back into the tracking issue:
Momof4:
There is nothing preventing city schools from grouping kids by academic level. Since leftists are whining about “segregated” schools, where almost all kids are black and/or Hispanic, there are no pesky racial/ethnic issues to cause political problems. Given appropriate coursework and acceleration, many kids would be prepared for this kind of MS and/or HS. Their failure to do so suggests that those schools do not really care much about well-behaved, bright, motivated poor kids who are disproportionately easy – and cheaper- to teach and whose test scores would reflect that.
Here Momof4 implies that the racial part of tracking is a red herring. If the only reason that tracking is avoided is that it inevitably leads to racial segregation, then why don't we at least have tracking at schools where nearly everyone is the same race? Thus to her, race is just an excuse -- the powers that be simply don't want tracking, period.
Of course, I admit that there are non-racial reasons to avoid tracking. In previous posts, Momof4 promotes grouping kids by academic level "in each subject." This is impossible unless elementary students have more than one teacher during the day (at the very least in ELA and math). I've promoted ways to accomplish this in previous posts (the "Path Plan"), but I'm starting to get wary after reading the article that fifth graders do better with a single teacher the whole day -- that is, being taught by a single teacher who knows them outweighs the advantages of tracking.
By the way, here are a few more related articles:
https://www.newsobserver.com/opinion/article214394284.html
When [North Carolina] Gov. Roy Cooper signed House Bill 986, which requires students who earned the highest possible score on end-of-course math exams to be placed in advanced math classes the following school year, I thought about one particular student.
She barely said a word in my standard-level English class three years ago, didn’t always turn in her assignments and occasionally skipped class. By October she was at risk of failing, so I challenged her to do more. Then, a few weeks later, she turned in a writing assignment that blew my mind.
Naturally, the girl in this story reminds me of Helen.https://arbitrarilyclose.com/2018/07/06/blackbrilliance-blog-section-1-cultural-historical-perspectives/
Before I get started, I’d like to just talk briefly about my current feelings towards the tome. And it does feel like that a bit to me – even though it’s a collection of essays from a number of authors, knowing Dr. Danny Martin (I’m less familiar with Jacqueline Leonard, his co-editor) I guess I expected the book to not exactly be a “summer read”. Dr. Martin is a brilliant speaker and author, but his works are always challenging reads and demand full attention. That’s partially why this post is late – I’ve needed to find dedicated time to read any of these essays.
The "tome" that Minnesota math teacher Annie Perkins mentions here is called The Brilliance of Black Children in Mathematics. (Perkins actually posts this as an MTBoS challenge -- yet another MTBoS challenge in which I won't participate, since I don't have this book.)
Summer School Grades
Yesterday, the students in the Algebra I class I almost taught completed the district final. But unfortunately, I can't see what the final scores are when I log into Edgenuity.
I remember back during the Wednesday meetings leading up to summer school, we were informed that the students would take the district final, so we should delete the Edgenuity final. But I recall being confused at the time -- it's possible that we needed to leave an empty spot in Edgenuity for us to enter the final exam scores manually. Unless we took that step, once the class is over, it's impossible to go back and enter the results of the final exam. The two remaining teachers gave the students the final and know their scores, but the scores won't appear in Edgenuity.
And so I can only report the grades excluding the final exam. The top student in the class has a grade of 87%, even though his test scores are 92, 94, 90, 90. It appears what's holding his grade down to a B is assignments -- his averages for two categories of assignments are 69.2 and 83.3. He's likely to improve in the second session, since the Algebra 1B teachers won't count the assignments. Actually, a second student also has a grade of 87%. Her test scores are 84, 76, 92, 94.
On the low end, one student failed the class again with a grade of 57%. His test scores are 80, 40, 55, 50 -- graphing linear functions got the best of him. Another student has a grade of 59% -- his test scores are 62, 68, 50, 45. Recall that the second test is an introduction to functions, while linear functions appear in earnest on the third and fourth tests. These are the only two students who failed (barring a miracle on the final). There's also a student who earned 60 on every test -- so technically she's barely passing up to the final, but if she fails the final, she fails the course. Most likely, she failed at least one of the tests and then retook it until her score reached 60.
(By the way, in giving the lowest grades, I exclude two students with a grade of 0% -- most likely, these two signed up for summer school but never attended a single class.)
By the way, the Algebra II grades do include a final -- either because the Algebra II teacher (unlike us Algebra I folks) remembered to leave a placeholder in Edgenuity for manual entry, or the district doesn't require a written test in Algebra II (so they could take the Edgenuity final). Three students are tied for the highest grade in the class with 93% -- the final exams for these trio are 94, 94, 96. The lowest score is 65% (62 on the final) -- and she has the only D in the class, with no 0's or F's. A few other students earn D's on the final yet are able to maintain C's in the class.
What's going on here? Undoubtedly Algebra II is more difficult than Algebra I -- even though many difficult topics won't appear until the B session, polynomials are still taught in A session, Yet the grades are higher in the Algebra II class. Well, for one thing, the Algebra II students are more mature than the Algebra I kids, so the older students understand how important it is to pass their classes. The students who take Algebra I, fail it, take it in summer school, and then fail it again are unlikely ever to take an Algebra II class -- such students are weeded out, so they don't appear in Algebra II at all.
Conclusion
This post is much shorter than most of my other traditionalists' debate posts -- again, this reflects the fact that most traditionalists haven't been saying much lately.
So how would I answer the original question asked by MooMoo2? I admit that I trust the opinion of VBCurtis the most, as he/she is an experienced community college professor:
VBCurtis:
I've taught community-college and freshman-level university courses for 20 years. Under the circumstances you posit, I think this number is over 90%.
The community-college offers courses equivalent to 7th-grade math, up through the courses you mention. It's really quite rare to run into a student in one of the developmental courses who honestly lacks the talent to pass multivariable calc or linear algebra. Of those I would put into that category, at least some of them might not lack the talent if they'd grown up in a different setting.
Within the American course progression, I believe that nearly everyone who passes a college-level precalculus/trig course will pass each term of calculus within two tries. Most of the ones who could not meet your criteria would likely fail in the Intermediate Algebra or Trig levels, rather than making through single-variable calculus but then failing all 3 of your mentioned second-year university courses.
But the real debate is, how do we satisfy the constraints required to attain that 90%? Traditionalists tell us that traditional math can replace "private tutors," while some commenters tells us that tracking is necessary to maintain "a safe environment." And still others are concerned about how much money must be spent to provide "a healthy diet."
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