In my new district, where it is only Day 21 -- which would be near the end of the first quaver, except that today I was at a middle school where trimesters and hexters are actually relevant. So in other words, today is nothing -- except Back to School Night, that is. Thus it's a minimum day, with students released at 12:35.
I once thought about my old idea of handing out free pencils on special days. The first day of school is a special day, and so is Halloween (since there are holiday-themed pencils). But with schools starting earlier on the calendar than they did when I was a young student, Halloween is more than two months after the first day of school. That's a long time between pencil giveaways.
And so to my old idea, I add an additional pencil giveaway day about halfway between the first day of school and Halloween. Back to School Night (or during classes that morning) is just about the right time in many districts, including this new district. It also should have been a good time to give away pencils at my old char-... oops, ixnay on the arterchay! (I promised that I wouldn't mention that topic during the Eugenia Cheng reading!)
This is the first minimum day of the year, and since sixth grade is still elementary school, this is the first minimum day that these seventh graders have experienced at their new school. And so as I've done in seventh grade classes throughout September, I help guide the students as they figure out what classes to go to and when. There is no lunch after school, and snack is halfway through the day (right before fifth period). Fortunately, no one asks for a restroom pass in fifth period. One student is tardy to third period, but I cut him some slack -- especially since the minimum day bells ring manually and are slightly off (leaving the kids with only 2 1/2 minutes to make it to class).
In today's science classes, students are watching a video. A British woman named Jane visits the African wilderness to meet -- no, not Tarzan. Actually, it's Jane Goodall who visits Tanzania to live Among the Wild Chimpanzees. This was back in 1960. (As it turns out, Goodall is still alive -- she's currently 84 years old.)
The regular teacher has been out all week. Yesterday's sub finished the video, but today's assignment is for students to add details that they missed the first time to their video notes (in red ink).
Since most of today is just playing a video, I won't write out a full "Day in the Life." But I will write about the management issues that I'm hoping to address now. Again, note that this is a minimum day with the middle school rotation beginning with second period. Sixth period is conference, while all other classes are Science 7. (There are no honors classes, as opposed to last Thursday.)
When each class enters the room, I try to keep the students quiet right away. I tell the students to take out their journals and a red pen. Therefore students know what is expected right away, rather than assume from the start that I'm a pushover.
Also, I decide to reprimand students for talking. It would have been too easy just to create a good list and a bad list of students, with the criteria for making the lists "how much red ink I see." But as I figured out last week, this sends the wrong message. Hardworking students who simply lack a red pen would end up on the bad list, while talkative students who don't pay attention to the film and write random crap on their paper in red ink would end up on the good list.
And so I don't even consider red ink at all. I go around to each student and ask (not tell) them to take out a red pen, but more importantly, I tell (not ask) them to stop talking.
I name fourth period the best class of the day, followed by third period. But I wonder whether I'm able to keep it up in fifth period, which is slightly more talkative throughout the movie. (I give the students a break if they laugh at something funny the chimps do on the screen.)
Meanwhile, I worry about first period (the last class of the day) from the get-go, because yesterday's sub labeled this class merely "OK" rather than "good class." Like fifth period, first period seems to talk throughout the movie. In the end, I decide that first period made a "slight improvement" compared to the note from yesterday's sub -- and indeed, it means that I myself made a slight improvement compared to last week's management.
But still, I wonder whether I could have done better. At this school, homeroom consists of the first period students. It's possible that both yesterday's sub and I had trouble keeping these kids quiet during the morning homeroom announcements, and this extends into first period, regardless of when first period actually meets in the day. (Yesterday the rotation started with first period right after homeroom, and today it ends with first period.)
The fact that the students are watching an excellent video again makes me lament what could have been back in my own -- oops, ixnay on the arterchay iencescay! Let's just jump into Cheng's book.
Chapter 8 of Eugenia Cheng's The Art of Logic in an Illogical World is "Truth and Humans." It's the first chapter of Part II, "The Limits of Logic." Here's how it begins:
"We have seen the power of logic in producing rigorous unambiguous justifications. Now we are going to address the limits of that power."
Cheng warns us logic has its limits -- it works well in the mathematical world, but it doesn't always work in the real world. In some ways, logic is a social construct. She writes:
"This might seem to fly in the face of everything I've said about mathematics being completely rooted in logic, but the situation is more subtle than that."
The author compares this to a trial by jury. Sometimes we lack time to make a logical argument:
"But another way in which logic isn't powerful enough is when we need to convince someone else of our argument. Logic turns out to be a good way to verify truth, but this is not the same as convincing others of truth."
Cheng writes that one way to make an argument is divide it into parts, and then subdivide each part of your argument into parts. She compares this to something we've seen before -- fractals:
"A fractal is a mathematical object that resembles itself at all scales, so that if you zoom in on a small part of it, that small part looks like the whole thing."
Three years ago, our side-along reading book was Benoit Mandelbrot's book on fractals -- and of course, Mandelbrot was the creator of fractal theory. And another side-along reading author, Wickelgren, also compared problem-solving to a fractal-like tree. Cheng draws such a tree in her book -- back in April, I wrote about how to program a computer to draw the tree. She explains:
"This tree represents how finding a proof works, in my head. The base is the thing you're trying to show is true, a but like in our diagrams of causation in Chapter 5. The two branches going into it are the main factors that logically imply it."
Fractal trees are infinite, but proofs are finite. So when do proofs stop? Cheng answers:
"In arguments in real life, we should keep going until the other person is convinced, or until we realize that our fundamental starting points are so different that we will never be able to convince them unless we can change their fundamental beliefs."
According to Cheng, even mathematical proofs are subject to a trial by jury, except that this trial by jury is called "peer review." We've discussed peer review back during another side-along reading book (on Perelman's proof of the Poincare conjecture). And just as we found out, submitting a paper anonymously does little to avoid its acceptance being swayed by human emotions, because the world of mathematicians is a small world after all:
"It's a bit like marking exams anonymously if your class has only three students and you've been working with them closely all year -- whether or not they write their names on the exam paper, you will know exactly which student is which."
The author tells us that to appeal to the human emotional side, research papers include material that isn't strictly logical:
"The help comes in the form of analogies, ideas, informal explanations, pictures, background discussion, test-case examples, and more. None of this is part of the formal proof, but is part of the process of helping mathematicians get their intuition to match the logic of the proof."
Cheng now leaves the world of mathematical papers and enters the world of politics:
"For politicians, the 'peer review' is the election -- it doesn't matter if they're right or not, and it doesn't matter if their if their arguments are sound or not, it just matters if people vote for them or not. Voters do not have to justify their vote either."
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next two weeks and skip all posts that have the "Eugenia Cheng" label.
Politics is all about raising skepticism. She tells us that reasonable skepticism about a mathematical proof can arise in two ways:
- Someone might think there's a gap or error in your logic.
- Your conclusion might contradict someone's intuition.
The second type of objection happens all the time in politics. Cheng writes:
"It's why some people still believe that vaccinations cause autism although there is no scientific evidence for it. It's why some people believe the universe is only a few thousand years old or that the earth is flat or that human life did not originate in Africa or that Barack Obama was not born in Hawaii, despite evidence."
When writing a mathematical paper, Cheng tries to anticipate her skeptics in advance:
"We are allowed to imagine that they are intelligent as us, which is why it's call peer review and not idiot review, but I imagine they they are highly skeptical of everything I'm saying, or that they are actively trying to find a mistake in my proof, so that I can find any possible mistakes myself."
At this point, Cheng wishes to point out a major distinction. She writes:
"Whenever there is an article about dogs there is bound to be a thread of comments declaring, apropos of nothing in particular, that 'they eat dogs in China.' This brings us to the difference between truth and illumination."
The statement "they eat dogs in China" is true, but it isn't very illuminating, especially if the current conversation is all about, say, a dog walking app.
Cheng explains that the only equations that are true in first-order logic are the equalities x = x:
1 = 1
2 = 2
3 = 3
4 = 4
...
and so on. These are true, but they aren't illuminating. As for all other equations, Cheng flatly states:
All equations that are illuminating are lies.
For example, she writes:
10 + 1 = 1 + 10
As far as first-order logic is concerned, this statement is a "lie," in the sense that it's not necessarily true on its own. It's true only in a system that has the Commutative Property of Addition -- without that property, the statement could be false. But Cheng explains why this statement is illuminating:
"One children work this out, they can use it to help them add up by counting on, knowing that it will always be easier to start with the larger number in their head and count on by the smaller number."
Notice that this is going to be a traditionalists-labeled post. And here's Cheng writing about things that traditionalists don't like -- adding by "counting on," or any method other than the standard algorithm of addition.
Anyway, Cheng tells us that the equations that really have nothing different about the two sides are the ones of the form:
x = x
and these ones are never useful.
At this point, Cheng repeats what she says about research papers containing stuff other than logic:
"Often in a research paper a logical proof will be accompanied by a description of what 'the idea is,' which is something more informal, not rigorous, but invokes ideas and imagery that might help us to understand the logic."
This leads back to the traditionalists' debate. Cheng tells us that some people -- traditionalists -- believe that rote memorization is necessary:
"But other people, often professional research mathematicians themselves (including me), are convinced that they have never really memorized anything in math. In fact, one of the main reasons I always loved math was exactly the fact that it didn't require memorization, only understanding."
And in fact, Cheng continues:
"However, I am perfectly fine at basic arithmetic and certainly above average compared with the general population, and yet I have never memorized my times tables. I know my times tables by some other, more subtle route that does not involve memorizing."
According to traditionalists, for those who don't memorize their times tables, "it's all over" and "doors are closed to STEM careers." Yet Cheng is a living, breathing counterexample -- she has a PhD in math yet never memorized her times tables.
Cheng wraps up the chapter with some Internet memes. Just like equations, memes can often be illuminating without being true. For example, here is such a meme:
HOW IT SHOULD BE:
Scientists (experts): There's a problem.
Politicians (non-experts): Let's debate a solution.
HOW IT ACTUALLY IS:
Scientists (experts): There's a problem.
Politicians (non-experts): Let's debate if there's a problem.
Cheng herself wants to edit this meme, as follows:
HOW IT SHOULD BE:
Scientists (experts): We think there's a problem. Here's a solution.
Politicians (non-experts): Let's debate funding their solution.
HOW IT ACTUALLY IS:
Scientists (experts): We think there's a problem.
Politicians (non-experts): Let's debate if there's a problem.
Funny how no country has ever tried to repeal universal healthcare.
It's almost like it works.
But the author responds:
"However, I'm not sure if the meme is true -- arguably some people have been trying to destroy and privatize the National Health Service in the UK."
Cheng concludes her chapter with another warning:
"We should particularly not pit emotions against logic. They are not opposites, but can work together to make things that are both defensible and believable."
As I wrote earlier, this is my regularly scheduled traditionalists' post. As it turns out, the Barbara Oakley discussion has died down. Once again, I'm late to the party -- by the time I finally agreed to add extra traditionalists' posts to accommodate the growing discussion, everyone else has stopped talking about it.
Instead, here are more recent battles being fought in the traditionalists' debate:
- Rochelle Gutierrez is probably the most controversial anti-traditionalist. So far I've been avoiding articles from either Gutierrez or her opponents like the plague. But in our last traditionalists' post, one blogger linked Gutierrez to Eugenia Cheng. And since we're reading Cheng's book now anyway, I think it won't hurt to look at the Gutierrez debate for today's traditionalists' post only.
- The "0 -> 50" grading controversy has appeared again, this time in Florida.
Let's start with Gutierrez. She is mentioned in this article at the Joanne Jacobs website:
Joy Pullman rants well. In The Federalist, she takes on social justice math.
University of Illinois Professor Rochelle Gutierrez, who teaches future teachers, will “argue for a movement against objects, truths, and knowledge” in a keynote to the Mathematics Education and Society conference in January, writes Pullman.
Let's jump directly to Pullmann's article, linked to above. Here are some key excerpts:
A U.S. professor who teaches future public school teachers will “argue for a movement against objects, truths, and knowledge” in a keynote to the Mathematics Education and Society conference this coming January, says her talk description.
“The relationship between humans, mathematics, and the planet has been one steeped too long in domination and destruction,” the talk summary says. “What are appropriate responses to reverse such a relationship?” We can already guess University of Illinois at Urbana-Champaign professor Rochelle Gutierrez’s answer, from reviewing her published writings and comments. Her plans for “an insurgency by the people” to subvert public institutions and American self-rule through “ethnomathematics” will knock your eyebrows off your face. Let’s take a look.
OK, so far I'm not quite sure what this "ethnomathematics" is, but I hope it's supposed to be math taught in such a way so that students can be successful independently of their ethnicity. Not everyone agrees that this "ethnomathematics" is something worth teaching -- especially not Pullmann. But then the author proceeds:
Gutierrez is an education professor who also teaches in Urbana-Champaign’s “Latino studies” program, of course. Her CV says she helped write federally funded Common Core math tests and has been on a host of taxpayer-funded committees, including several of the National Science Foundation.
She’s also affiliated with the National Council of Teachers of Mathematics, which wrote notoriously terrible curriculum rules that destroyed math instruction in many states before helping form Common Core. She’s helped decide which education professors to grant tenure at more than a dozen public universities, and been given visiting lecture position at Vanderbilt University, which is reputed to have one of the most pre-eminent teaching degree programs.
And here lies the problem. Gutierrez, the "ethnomathematician," is a co-author of Common Core. So opponents of ethnomathematics, such as Pullman, now reasonably concludes that Common Core has something to do with this ethnomathematics. Thus Pullman uses Gutierrez and her ethnomathematics as a reason to oppose the Common Core.
(Moreover, Pullman also criticizes the NCTM standards here. The U of Chicago text is based on these NCTM standards, which explains the resemblance of our text to Common Core Geometry.)
And not only does Gutierrez bring up politics and race, but economics as well. Pullman writes:
Gutierrez’s December article is, perhaps not surprisingly, more of a blog post that quotes sources such as Medium.com and activist websites. Rather than being scholarship worthy of the name, it is essentially a political game plan for using math classes and teachers to follow the Marxist political playbook in “[d]ismantling White supremacist capitalist patriarchy.”
So Gutierrez lists capitalism as something that needs to be dismantled, and Pullman responds by calling her a Marxist. And in fact, Pullman continues:
If we don’t preference competence over political correctness, kids lose big. An understanding of basic mathematics is crucial to competence in many lucrative jobs, plus an introduction to one of the great mysteries of the universe, as well as centuries of human inquiry. These are kids’ lives and minds we’re talking about here, which don’t deserve to be pawns in somebody’s ideological war for social engineering. But far too often, that’s what they are, and it’s American education’s many interlocking monopolies and cartels that are chiefly to blame, because cartels inherently prioritize tribalism over excellence.
So Pullmann responds to Gutierrez and her discussion of economics by providing an economic reason for why students are failing (the "monopolies" and "cartels").
I said that I'd compare Gutierrez to Eugenia Cheng, so let me do so. Earlier this week, we saw how Cheng draws a cuboid to represent privilege, with "rich white male" at the top. When Gutierrez refers to "patriarchy," it means "male privilege" in Cheng's terms, and "White supremacist" is a stronger way of saying "white privilege." This leaves "capitalist" to refer to Cheng's third privilege -- in other words, "rich privilege."
But my main takeaway here is that because of Gutierrez is a Common Core author and writes about ethnomathematics and opposing capitalism, her opponents like Pullmann are going to associate the Core with ethnomathematics and capitalism. I believe that the Common Core is race-agnostic and has nothing to do with Marxism, capitalism, or social justice.
Let's return to Jacobs. She continues by writing about another anti-traditionalist, Cacey Wells:
Tracking math students is rooted in “capitalist exploitations and settler colonialism” and leads to academic apartheid, argues a University of Oklahoma math education professor. Cacey Wells is the author of recently published study, writes Toni Airaksinen on PJ Media.
And here's yet another mention of capitalism. Once again, I'm not sure whether capitalism per se is the problem but rather the "rich privilege" of Cheng.
But here again is the problem -- Airaksinen continues:
He suggests placing all students in the same math class, regardless of ability, and replacing “Algebra 1” and “Geometry” with “Math 1” or “Math 2” to reduce the stigma.
The problem is that "Math 1" and "Math 2" sound like Integrated Math 1 and 2. So now just as Gutierrez causes Pullman to associate Common Core with anti-capitalism and social justice, now suddenly Casey causes Airaksinen to associate integrated math with anti-capitalism and social justice.
I strongly disagree that integrated math has anything to do with white privilege. Integrated math is how secondary math is taught in nearly every country in the world -- including those with and without white majorities.
Most of the commenters at the Jacobs website are actually responding to Wells, not Gutierrez. For example, our main traditionalist at the Jacobs website is Bill:
Bill:
Wells sounds like an idiot… Math doesn’t care how someone feels about it and yes we had tracking when I was in middle and high school.
Trying to label Algebra 1 as math 1 so that every one doesn’t have a self esteem issue will only result in the student who doesn’t have the skills to handle the coursework to simply fail.
A student who doesn’t have a solid grasp of addition, subtraction, multiplication, division, percentages and fractions will simply flunk Algebra (or any other higher level math course)…
Moronic psycho babble.
Regarding the great tracking debate, it reminds me of the debate regarding ID's and voting. Both major political parties agree with the following statements:
- Stronger ID laws lead to lower black turnout at the polls.
- Weaker ID laws lead to greater black turnout at the polls.
But there isn't a consensus as to why these statements are true. Both sides get into heated arguments, and questions such as "Do blacks know how to obtain an ID?" get thrown around. All that matters here is that these statements are true.
Each party accuses the other party of using politics to determine whether to strengthen ID laws:
- Members of a party that expects high black support will favor weaker ID laws.
- Members of a party that expects low black support will favor stronger ID laws.
Chances are that both parties are at least partly correct here. Notice that Cheng's "false positives" and "false negatives" are at play here. Here a "false negative" is someone who deserves to vote but can't (a citizen without ID) and a "false positive" is someone who doesn't deserve to vote but tries anyway (a non-citizen).
Ideally, we want to avoid both false negatives and false positives. Ideally, there should be strong ID laws that don't reduce black turnout. But no one from either party knows how to achieve this.
Another commenter, CT, writes:
CT:
“Apartheid” means “separation.” Is Wells seriously arguing that merely separating students into groups in order to teach to their level is somehow equivalent to an official, brutally-enforced national policy of racial segregation? What a joke. Wells’ ideas, carried out, inevitably lead to lower math performance by nearly everyone in public schools because people cannot learn when a discussion is too far out of their zone of proximal development. Has Wells ever actually taught a classroom of children?
No, tracking isn't equivalent to brutal racial segregation. But, just as with voter ID laws, every time a school adopts tracking, most blacks wind up on the lowest track. If it wasn't for this fact, more people would probably be in favor of tracking.
Suppose we divided everyone into two groups -- say those with an odd number of letters in their full name and those with an even number of letters. We'd be surprised if most of the even-numbered people found themselves on the lowest track. I want to live in a world where we'd be surprised if most members of a track were of one race, or one gender, or one income bracket.
But we can't even come up with a strong ID law that is race-independent (that is, without diminishing black turnout). An education system where every student is learning within his/her ZPD, yet remains race-independent, is even more challenging to create.
Repeating Pullmann's words:
If we don’t preference competence over political correctness, kids lose big. An understanding of basic mathematics is crucial to competence in many lucrative jobs
I do know that students stuck on the lowest track aren't being well-prepared for the lucrative jobs mentioned by Pullmann here. That's all the more reason to ensure that "low track" doesn't correlate strongly with a particular race.
Before we leave Jacobs, let's look at one final comment:
Jean @ Howling Frog:
Good old ‘story problems relevant to real life.’ Always the solution, until you get them and find out that real life is BORING. Give me Saxon’s fanciful story problems any day.
This comment refers to word problems in Algebra I/II classes, as well as the common student question "When will we ever use this in real life?" I can't tell what Jean is saying about word problems in most common texts, but I do know about the Saxon series of texts -- text that are commonly recommended by traditionalists. My question is, does the question "When will we ever use this in real life?" frequently arise when working on Saxon word problems. (I mean with the majority of students sitting in math classes, not the traditionalists and their math-bright children.)
Before we leave Joanne Jacobs, there's another issue to which I want to apply Cheng's logic. A frequent commenter at the Jacobs website is lgm -- and lgm often criticizes "full inclusion," especially in elementary schools. For example, we check out the following thread:
https://www.joannejacobs.com/2018/09/when-high-hopes-meet-low-expectations/
lgm:
The big issue is the dumbing down of each elementary school grade level so all are ‘included’….the result is capable students who need remediation because large portions of the grade level material simply were not done. A test prep book isn’t enough to catch them up.
lgm:
Before we leave Joanne Jacobs, there's another issue to which I want to apply Cheng's logic. A frequent commenter at the Jacobs website is lgm -- and lgm often criticizes "full inclusion," especially in elementary schools. For example, we check out the following thread:
https://www.joannejacobs.com/2018/09/when-high-hopes-meet-low-expectations/
lgm:
The big issue is the dumbing down of each elementary school grade level so all are ‘included’….the result is capable students who need remediation because large portions of the grade level material simply were not done. A test prep book isn’t enough to catch them up.
lgm:
The elementary reading here has changed due to full inclusion — there is just not enough funding for enough adult aides to supervise those who cannot work independently while the teacher is giving small group lesson.
So lgm is clearly opposed to this "full inclusion," so maybe lgm supports its negation. Let's use Cheng's quantifiers here to find the negation. So "full inclusion" means something like:
For all elementary students, that student is included (in the gen ed classroom).
The negation of a "for all" statement is a "there exists" statement:
There exists an elementary student who is excluded (from the gen ed classroom).
The negation of "all students are included" is "some students are excluded." So if lgm is opposed to full inclusion, maybe lgm supports the negation, "partial exclusion."
And of course, it's easy to name someone who might be opposed to "partial exclusion" -- the parents of the students who would be excluded. This explains why schools use full inclusion -- otherwise parents of the ones to be excluded might riot, or sue.
OK, let's move on to the "0 -> 50" grading controversy. Apparently, a teacher in Florida was just fired because she disagrees with the "0 -> 50" grading system:
https://www.wftv.com/news/local/teacher-fired-after-refusing-to-abide-by-no-zero-policy-when-students-didnt-hand-in-work/840999722
I first mentioned "0 -> 50" on the blog three years ago, then I alluded to it when I needed to do science grades quickly when -- oops, ixnay on the arterchay iencesay (for the third time today).
So let's focus on what I wrote in 2015. I have much to say about 0 -> 50, but unfortunately this post is already jam-packed with other parts of the traditionalists' debate. (And that was after I pointed out that the Barbara Oakley material has dried up!)
So lgm is clearly opposed to this "full inclusion," so maybe lgm supports its negation. Let's use Cheng's quantifiers here to find the negation. So "full inclusion" means something like:
For all elementary students, that student is included (in the gen ed classroom).
The negation of a "for all" statement is a "there exists" statement:
There exists an elementary student who is excluded (from the gen ed classroom).
The negation of "all students are included" is "some students are excluded." So if lgm is opposed to full inclusion, maybe lgm supports the negation, "partial exclusion."
And of course, it's easy to name someone who might be opposed to "partial exclusion" -- the parents of the students who would be excluded. This explains why schools use full inclusion -- otherwise parents of the ones to be excluded might riot, or sue.
OK, let's move on to the "0 -> 50" grading controversy. Apparently, a teacher in Florida was just fired because she disagrees with the "0 -> 50" grading system:
https://www.wftv.com/news/local/teacher-fired-after-refusing-to-abide-by-no-zero-policy-when-students-didnt-hand-in-work/840999722
I first mentioned "0 -> 50" on the blog three years ago, then I alluded to it when I needed to do science grades quickly when -- oops, ixnay on the arterchay iencesay (for the third time today).
So let's focus on what I wrote in 2015. I have much to say about 0 -> 50, but unfortunately this post is already jam-packed with other parts of the traditionalists' debate. (And that was after I pointed out that the Barbara Oakley material has dried up!)
So let me just quote a few parts of that old 2015 post about 0 -> 50. I'll insert a few comments relevant to the Florida situation:
A grade represents a percentage, a scale from 0 to 100. One of the most commonly used grading scales in schools is something like this:
90-100 = A
80-90 = B
70-80 = C
60-70 = D
0-60 = F
Now let's return to school grades. For batting averages .300 is excellent, but in school 30% is surely a failing grade. For winning percentages .600 is one of the best teams in the majors, but in school 60% is the lowest passing grade. All of the grade boundaries -- from B to A, C to B, and so on -- occur in the upper half of the scale. The lower half of the scale -- from 0 to 50 percent -- is irrelevant as far as determining letter grades is concerned.
To the extent that C is average, the average student is earning between 70% and 80%. So we expect the average score in the class to be around the midpoint of this range -- 75%. So the average batter gets one hit every four at-bats, the average team wins two out of every four games, and the average student gets three out of four questions right.
And so, unlike the batter who gets a hit then makes an out and sees that overall his average has risen, the student who gets 100% on one test and 0% the next can never have a higher letter grade -- only a lower letter grade is possible. Therefore, a grade is much more likely to drop very rapidly than it is to rise very rapidly. Since the average grade is around 75%, a student would have to receive 150% on a test in order for the grade to rise as rapidly as a 0% drops it. (And all of this is assuming that the student isn't at one of those schools that has abolished D grades and makes 70% the lowest passing score -- that 10% difference means that a student would have to earn 160% on a test to have the same impact on the grade as a 0%!)
I've seen students come up to me and ask me why their grade has dropped so quickly -- especially when their grades have dropped from B to D seemingly overnight, while no one's grade is rising from D to B as quickly. They think that I'm a mean teacher and a harsh grader -- when the true reason is the nature of the grading scale, where the F grade takes up three-fifths of the scale.
A grade represents a percentage, a scale from 0 to 100. One of the most commonly used grading scales in schools is something like this:
90-100 = A
80-90 = B
70-80 = C
60-70 = D
0-60 = F
Now let's return to school grades. For batting averages .300 is excellent, but in school 30% is surely a failing grade. For winning percentages .600 is one of the best teams in the majors, but in school 60% is the lowest passing grade. All of the grade boundaries -- from B to A, C to B, and so on -- occur in the upper half of the scale. The lower half of the scale -- from 0 to 50 percent -- is irrelevant as far as determining letter grades is concerned.
To the extent that C is average, the average student is earning between 70% and 80%. So we expect the average score in the class to be around the midpoint of this range -- 75%. So the average batter gets one hit every four at-bats, the average team wins two out of every four games, and the average student gets three out of four questions right.
And so, unlike the batter who gets a hit then makes an out and sees that overall his average has risen, the student who gets 100% on one test and 0% the next can never have a higher letter grade -- only a lower letter grade is possible. Therefore, a grade is much more likely to drop very rapidly than it is to rise very rapidly. Since the average grade is around 75%, a student would have to receive 150% on a test in order for the grade to rise as rapidly as a 0% drops it. (And all of this is assuming that the student isn't at one of those schools that has abolished D grades and makes 70% the lowest passing score -- that 10% difference means that a student would have to earn 160% on a test to have the same impact on the grade as a 0%!)
I've seen students come up to me and ask me why their grade has dropped so quickly -- especially when their grades have dropped from B to D seemingly overnight, while no one's grade is rising from D to B as quickly. They think that I'm a mean teacher and a harsh grader -- when the true reason is the nature of the grading scale, where the F grade takes up three-fifths of the scale.
Now it's the end of the quarter, and grades are about to come out. Let's assume for simplicity that there are equal numbers of points possible in the first and second quarters (in reality, the second quarter may have more points because that quarter has a final exam while the first quarter doesn't).
[2018 update: In this particular Florida school district, each quarter is actually worth 40%, and the final is worth 20%.]
Let's say a student is earning 10% at the quarter. Then this student will surely fail the class -- even if 100% is earned the second quarter, the average grade is only 55%, an F. This is akin to [the 2018 Baltimore Orioles, who were mathematically eliminated on August 19th].
A student earning 20% at the quarter can still pass the class -- but only if 100% is earned the second quarter, which isn't very likely. And a student who gets 30% the first quarter would still need to earn 90% (an A) the second quarter to pass the class. The type of student who would earn only 30% in a quarter is not the type of student who would earn an A the second quarter -- which typically covers more difficult material than the first quarter. This is akin to a baseball team who is nine games back with only 10 to play -- although the team hasn't been mathematically eliminated, they arerealistically eliminated from the division. It would take a miracle for the team to come back and claim the division, and a similar miracle is needed for the student to get any semester grade other than F.
In fact, in some ways, any quarter grade below 50% would realistically eliminate a student from receiving a grade higher than F at the semester. A 50% quarter grade would require a second quarter of 70%, a C, to pass the class -- and this might be doable, if the student works a little harder during the second quarter.
(And that, of course, assumes that 60% is the lowest passing grade. If the 50% student is at a no-D school, then that student will need 90%, an A, in the second semester to get any letter other than F on the semester report card -- once again, realistic elimination.)
When I am teaching class, what I want is for as few students as possible to be mathematically or realistically eliminated from getting any semester grade other than F, when there are still plenty of weeks separating them from the end of the semester. I have no sympathy for a student who is realistically eliminated when there are only nine or ten days left in the semester, but nine or ten weeks, that's another matter.
[2018 update: In this particular Florida school district, each quarter is actually worth 40%, and the final is worth 20%.]
Let's say a student is earning 10% at the quarter. Then this student will surely fail the class -- even if 100% is earned the second quarter, the average grade is only 55%, an F. This is akin to [the 2018 Baltimore Orioles, who were mathematically eliminated on August 19th].
A student earning 20% at the quarter can still pass the class -- but only if 100% is earned the second quarter, which isn't very likely. And a student who gets 30% the first quarter would still need to earn 90% (an A) the second quarter to pass the class. The type of student who would earn only 30% in a quarter is not the type of student who would earn an A the second quarter -- which typically covers more difficult material than the first quarter. This is akin to a baseball team who is nine games back with only 10 to play -- although the team hasn't been mathematically eliminated, they arerealistically eliminated from the division. It would take a miracle for the team to come back and claim the division, and a similar miracle is needed for the student to get any semester grade other than F.
In fact, in some ways, any quarter grade below 50% would realistically eliminate a student from receiving a grade higher than F at the semester. A 50% quarter grade would require a second quarter of 70%, a C, to pass the class -- and this might be doable, if the student works a little harder during the second quarter.
(And that, of course, assumes that 60% is the lowest passing grade. If the 50% student is at a no-D school, then that student will need 90%, an A, in the second semester to get any letter other than F on the semester report card -- once again, realistic elimination.)
When I am teaching class, what I want is for as few students as possible to be mathematically or realistically eliminated from getting any semester grade other than F, when there are still plenty of weeks separating them from the end of the semester. I have no sympathy for a student who is realistically eliminated when there are only nine or ten days left in the semester, but nine or ten weeks, that's another matter.
A player on a baseball team that is realistically eliminated from the playoffs may be traded to a team that has a shot of winning [like some Orioles]. And even if the player isn't traded, he will still play out the rest of the season, no matter how little his heart is in it, because he's under contract to do so. Even a college student realistically eliminated from passing a class can drop out and take a W instead. But for high school students, there is no choice but for the student to take the F.
I mentioned last year that one way to avoid realistic elimination would be to allow students to retake the tests on which they received a low score. But in this article, there is mention of two separate -- and I repeat, separate -- grading scales that help students avoid realistic elimination.
[2018 update: only one of these grading scales is relevant to Florida.]
The first is to record scores of zero as equaling 50%. After all, it's the devastating effect of 0% on the grades that eliminate students from passing the course. The average of 0% and 100% is 50%, an F, but the average of 50% and 100% is 75%, a C.
Back when I was an Algebra I student, my teacher did something similar. When she recorded test scores, she only recorded a letter grade, so if one received an F, one can't tell whether the F was really a score of 1% or 59%. Then to determine the average at the quarter or semester, she converted the grades back into percentages by letting each letter represent the midpoint of the interval. So an A was 95%, a B was 85%, a C was 75%, and a D was 65%. But an F was converted into 55%. So this represents a similar scheme to the 0 -> 50 system, since a score of 5% instantly became 55%.
And at this point, I kept writing about the alternative to 0 -> 50. This has nothing to do with Florida, and so I don't repeat it.
Since this post is already so long, let me end it with some illuminating insight from Cheng. Those who support 0 -> 50 (like the Florida district) are afraid of "false negatives," while those who oppose 0 -> 50 (like the fired Florida teacher) are afraid of "false positives."
"False negative" students:
Student misses a few early assignments and gets a zero.
Student studies three hours per night to make up the work.
Student gets 100% on every remaining assignment.
Student deeply cares about learning and getting high grades.
Teacher calculates semester average as 59.99% or less, gives student an F.
"False positive" students:
Student misses a few early assignments and gets 0 -> 50.
Student studies very little each night.
Student determines which assignments to do and which assignments he/she can afford 0 -> 50.
Student doesn't care about learning, only about placating parents with a D.
With 0 -> 50, teacher calculates semester average as 60.00% or more, gives student a D.
The whole 0 -> 50 boils down to the fact that a single digit -- the digit in the tens place of the percentage -- determines the student's letter grade. If the student works hard but that digit is a 5, the teacher must give the student an F. If the student slacks off but that digit is a 6, the teacher must give the student an D. The teacher's hands are tied -- especially since it's probably a computer that's calculating the grades. The teacher must assign the grade that appears on the computer. Basing grades on a single digit is almost like determining whether to scold students based on what's written in red ink in a journal -- which takes us back full circle to my day of subbing.
Yes, writing a name on the bad list for not writing in red ink (when the class is so loud that the student never hears the instruction "write it in red ink") is a "false negative." And writing a name on the good list for copying someone else's paper in red ink (while talking all throughout the lesson) is a "false positive." I avoid both the false negative and the false positive by not using the red ink to determine what names to write on each list.
Likewise, perhaps if we know that a student is making a deep and honest effort to make up the work yet can't mathematically reach a semester 60%, then we show that student mercy with 0 -> 50. On the other hand, if we know the student isn't trying to make up the work at all, then we keep 0 as 0. This might be common sense but would be difficult to justify to parents. (Why does another child get the benefit of 0 -> 50 but not my child?)
Here is the Chapter 2 Test:
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