Thanksgiving Eve Post
Table of Contents
Introduction
Today is Wednesday, the day before Thanksgiving. The day has a few alcohol-related nicknames, none of which I care to post on an education blog.
This is the first of two special posts I plan on writing during the school break. I wish to catch up on a few topics that I didn't have time to blog about during the school year. In particular, last week's visit to the Calculus teacher at the main high school is still fresh on my mind. I want to discuss how I'll implement some of the ideas I saw into my own classroom, as well as why incorporating other ideas into my class will be more challenging.
I'll definitely be comparing my current classes to previous classes I taught, at the old charter school as well as my long-term subbing last year, both at the middle-school level
The Problems With Weighting Grades
Most teachers, in their syllabi at the start of the year, declare that the student grades will be comprised of certain categories, with a certain set percentage set to each category. For example, in my classroom, I stated that 40% of the grade will be tests, 30% will be quizzes, 20% homework, and 10% classwork.
Now there are two ways to achieve these percentages. One way is to enter the grades into the computer and declare a category for each assignment, and the computer will automatically "weight" each grade so that the percentages work out to be correct. For example, suppose at the end of the semester, the total points in each category just happens to be 200 -- that is, 200 test points, 200 quiz points, 200 homework points, and 200 classwork points. The the computer will weight these points so that 40% of the final grade comes from tests, 30% from quizzes, and so on. In this example, since homework is 20% while classwork is 10%, each homework point is worth twice as much as a classwork point. And since 40% of the grade comes from tests, each test point is worth twice as much as a homework point -- and therefore four times as much as a classwork point.
The other way is simply to give assignments and then make sure that 40% of the points are devoted to tests, 30% to quizzes, and so on, without any grade weighting. For example, if the entire class is worth 1000 points, then the tests can add up to 400 points, the quizzes to 300 points, and so on.
Many of the classes I've covered recently used the first method -- grade weighting. Back when I was working on BTSA, the district had already pre-programmed weighting into the computer. If I recall, there were three categories devoted to assessments -- regular assessments given by the teacher, chapter tests that were common to the district, and the final exam. The old charter middle school where I worked also had grade weighting in PowerSchool -- 40% of the grade was for "formal assessment and projects," 20% for "quizzes," then there were a few more categories, including 10% for classwork.
The long-term school from last year also had grade weighting in Canvas. Here I assume that the regular teacher set up the weighting, since only the Math 8 classes were weighted -- and for some reason, the weighting was different in the two sections of Math 8. There was a category for "assignments" and one for "imported assignments" -- the former was worth 40% of the grade (in one of the Math 8 classes -- 30% in the other) while the latter was worth 0%. And when the regular teacher set it up, he placed only one assignment in the "assignments" category -- a certain Edpuzzle. Thus I had to explain to the students why a single Edpuzzle was worth 40% of their grade.
I don't like grade weighting. Indeed, even without this absurd 40% Edpuzzle, I don't like it when students ask "How far away am I from getting a C?" and having to answer "ten points -- if it's a test, or 40 points if it's classwork." I'd much rather let points just be points. And so when I'm in control of the assignments, I try to avoid grade weighting.
Back when I was at the old charter school, I actually tried to circumvent the grade weighting in PowerSchool simply by counting the number of points each assignment is worth, and then making sure that 40% of the total points are for tests and projects, 20% for quizzes, and so on. Then since the percentages were already correct, the weighting by PowerSchool did nothing. This worked out for the first trimester, but then it fell apart in second trimester when I suddenly had to assign a science grade.
This year, it's up to me to set up the gradebook in Aeries and decide whether to weight the grades. So of course, I choose not to. Instead, I decided in advance that the entire semester would be 1000 points, and so I need 400 points of tests, 300 points of quizzes, 200 points of homework, and 100 points of classwork to satisfy the percentages I declared in the syllabus.
The Problems With NOT Weighting Grades: Quizzes
Let's see what this looks like in my Calculus class. The 400 points of tests are easy to achieve -- each test is worth 100 points, so I need to give four tests in the semester. I've already given three tests, and the fourth test will be the final exam.
The 100 points of classwork are also easy to attain. I don't give much "classwork" at all -- I only consider the Vanderwerf name tents at the start of the year and the weekly Warm-Up sheets to be in the classwork category. The name tents, given the first two weeks of school, were worth ten points, and the other 90 points consist of 15 Warm-Ups, worth six points each, given during Weeks 3-17. Week 18 will be finals week, and so I won't do Warm-Ups that week. I stamp each sheet six times -- three for the Warm-Ups (Monday, block Tuesday, block Thursday) and three for the Exit Passes, and so by counting each stamp as a point, these are the six points.
But the other two categories are very problematic. Let's start with the quizzes category -- I need this to add up to 300 points. There are a few ways to reach 300 points -- have five quizzes worth 60 points each, for example, or six quizzes worth 50 points each. But now consider how often I give these assessments -- I have a summative assessment (either a quiz or a test) about once every two weeks, and I didn't give the first assessment until Week 4. An assessment every other week from Week 4-18 (counting the final) works out to be eight assessments -- four of which are tests. Thus there must be four quizzes as well -- and so each quiz must be worth 75 points, in order for the four quizzes to total 300.
So suddenly, this means that each "quiz" is worth almost as much as a test. And this also affects how long each quiz is. A 100-point test can have 20 questions each worth five points, but a 75-point quiz would need to have 15 questions if each is to be worth five points. The first quiz I gave my Calculus class was indeed 15 questions, but it was a bit long, and students struggled to finish. For subsequent quizzes I doubled the point value of many questions to ten points, but even then, the shortest such a quiz could be is eight questions. And even this was a bit long when I gave the third and most recent quiz on derivatives, with students still being unable to finish it.
There's another thing about the quizzes and tests here. My original plan was for the each test to represent a chapter test. Thus Test #1 was the Chapter 1 Test, Test #2 was the Chapter 2 Test, and Test #3 was the Chapter 3 Test. Then Test #4, instead of a Chapter 4 Test, would be the final exam covering all of Chapters 1-4. The second semester would begin with Test #5, the Chapter 5 Test. This would fit the original plan suggested by the presenter of the online AP boot camp I attended -- begin the second semester with integral calculus, which in our text is Chapter 5. Meanwhile, the quizzes sort of follow this same pattern. Actually, I didn't give a quiz in Chapter 1, so Quiz #1 was in Chapter 2. But Chapter 3 had both Quizzes #2-3, so the quiz numbers would finally line up with chapters starting in Chapter 4.
Of course, keep this in mind as I compare this to the Calc class at the flagship high school. That teacher isn't trying to finish differential calculus this first semester -- instead, his plan is to move more slowly and finish only Chapter 3 first semester, then go faster in the second semester. If I were to follow this pattern, then I can get away with just calling the final exam "the final" rather than Test #4, and save Test #4 for the second semester, where it can serve as the Chapter 4 Test. But this doesn't work for the quizzes -- Quiz #4 is first semester while Chapter 4 is second semester, so I wouldn't be able to give Quiz #4 during Chapter 4.
Then again, my primary goal needs to be student understanding. Giving Quiz #4 first semester just so that the quiz points add up to 300 shouldn't the primary goal -- and rushing Chapter 4 in December just to make the quiz and chapter numbers line up is a terrible idea.
Besides, the Calc teacher at the flagship school appears to give shorter quizzes more often -- as in once a week. The quiz I saw him give the AB class I observed had six questions and was worth 21 points. I didn't ask him about grades or percentages though, so I don't know what "21 points" means in the context of his class. In this class, the correspondence between quiz numbers and chapter numbers is completely nonexistent. (Interestingly enough, he makes the quiz number into a short Calculus problem that he prints on top of the test -- a bit like my Exit Passes with the date.)
I could try to switch to something like this for the last few weeks of the semester -- instead of the fourth 75-point quiz, give two shorter quizzes and make sure that they add up to 75 points. But once again, I'm not sure whether I want to change horses midstream and suddenly throw an extra quiz at the students, even if it is shorter. If I switch to weekly quizzes, it will be second semester. Instead, I'll just keep Quiz #4 as originally planned -- but instead of Chapter 4, I might give some extra Chapter 3 questions that are relevant to the AP exam.
The Problems With NOT Weighting Grades: Homework
The other problem with avoiding grade weighting is with the homework. I need the HW assignments to add up to 200 points. Now, the way I assign homework is based on the way my own Calculus teacher did when I was a young high school student -- each Monday, I give one assignment, and on each block day I give two, so that there are a total of five assignments per week -- and the total number of assignments is approximately equal to the number of school days. The first assignment was given just before Day 10, and the last assignments will be given on Day 80, so there are just over 70 possible days of assignments. It will be less, since I don't always assign HW on the day of the quiz or test -- the number of assignments will be around 60.
Now let's try to make these 60 assignments add up to 200 points. We can't make each HW be worth four points, since that will be too many points. If I instead make each HW be worth three points, I can give as many as 67 assignments -- and this is what I currently do. If an assignment is late, then I give the student two out of three points.
But as it turns out, 67 assignments is still difficult to reach -- if I don't give HW on the day of a quiz/test and it's a block day, it means that I skip the two assignments I would have given that day. It would be easier to reach 67 if I skip only one assignment on such days and still give one assignment. But what sort of assignment would I have on a quiz day?
The answer is obvious -- some sort of review assignment. On the day of a quiz or test, I pass out the markers and have the students answer a few questions on their desk whiteboards. When I did this last week, I even had them rotate the whiteboard into a VNPS. I count this as one of the HW assignments, even though they clearly do it at school, not at home (but the 100 classwork points have already been accounted for).
Now notice what ends up happening here. I wrote earlier that the students sometimes run out of time to finish their quizzes -- but I don't give them the entire block to take the quizzes, because I start with the whiteboard assignment. If I skipped the whiteboards, then they'd have more time for the quiz. But both the whiteboard assignment and the lengthy quizzes are driven by the points system and the need to make the points add up.
Let's compare this to the Calc teacher at the flagship school. I know that he doesn't have give two assignments per block day. In fact, he doesn't even give one assignment per block, much less two. His method is to give fewer, longer HW assignments -- perhaps as many as 50 questions, but only once a week, or however long it takes for him to go over some of them.
This would be tricky to implement under my points system -- how can I make all the homework add up to 200 points if I don't know how in advance many HW assignments there will be? Once again, I didn't ask the veteran teacher about his grading system, but most likely he'd say that he weights the grades, so that he doesn't need to know in advance how many assignments there will be. Aeries automatically weights his grades so that they match the correct percentage.
By the way, I keep writing about Calculus. In Stats, I don't wish to give two assignments in a day. So instead, I make some of the assignments be worth six points -- in particular, each day I assign two three-pointers in Calculus, I assign a six-pointer in Stats. Then the points are equal in both classes, so I know that the percentages are correct as well. (This also safeguards me against errors -- for example, I accidentally entered a three-point HW as a six-pointer in Aeries, and this slightly hurt the students' grades because they all had 3/6 on a HW that they completed. Once I saw that the points were different in Calculus and Stats, I realized my error.)
My Plans for December
OK, so the purpose of my visit to the flagship high school last week was to observe practices from a veteran teacher that I can incorporate into my own class. But, as I just pointed out, some of the practices that I observed are so radically different from my own that I don't want to make so many changes -- especially not so close to the end of the semester.
So here are some plans for what I really wish to do once the students return after Thanksgiving. The first day is a regular Monday, so as usual, I will play a Michael Starbird video. But I also want to reintroduce them to AP Classroom.
That's right -- this week the College Board finally validated my course, and I can use AP Classroom. I was expecting the veteran teacher to use AP Classroom during my visit, but he didn't -- nor did he use any other form of technology (such as DeltaMath or Edpuzzle). He did refer to a Desmos lesson that he'd given the previous week, but that was it.
So I haven't completely decided what role DeltaMath, Edpuzzle, or Desmos will have in my class moving forward, but I at least want to show the students AP Classroom. I have to look more into what sort of activities are available on AP Classroom -- there might be one connecting Chapter 3 material and the content of the particular Starbird video (and the site, I notice, has videos of its own).
Then the rest of that week, I'll give more complex problems from Chapter 3. That's right -- I've decided not to start Chapter 4 until second semester, so that I'm following the veteran teacher at the charter school rather than the presenter of my online workshop. (It just doesn't seem right to speed up and start Chapter 10 in Stats based on my observations from last week without slowing down and stay in Chapter 3 in Calc due to those same observations.)
The problems I will focus on are some of the more advanced problems in each chapter that might appear on the AP exam. When I first covered Chapter 3 in October and November, my assignments were something like #1-20 (odd or even) and #21-40 (odd or even) -- now I wish to give those questions in the 40's and/or 50's that require higher thinking, just as the veteran teacher did. And Quiz #4 will be on these questions. I wouldn't mind giving this quiz on DeltaMath, but I suspect that DeltaMath won't have these higher thinking problems, so it might have to be on paper.
I'll continue to give two assignments on block days. The last assignment I gave before Thanksgiving was Assignment #57 (which was Test #3 corrections). If I give eight more assignments, then that takes me to Assignment #65. Then Assignments #66-67 can be given during finals week as review for those final exams (with Assignment #67 worth just two points, since 67 * 3 = 201, not 200). This means that I can afford not to give any assignment on the day of Quiz #4 (since I need only eight assignments over the next two weeks, not nine).
And so I want at least the day of Quiz #4 to be just like the day I observed at the flagship school. I'll start with the quiz (rather than give any VNPS filler assignment), and then follow this up with questions from the previous day's homework -- which then won't be collected until the next day. This will be followed by a main lesson and then an activity -- maybe even the group activity I witnessed at the flagship school.
The next time I see them, as a Warm-Up question, I'll ask them whether the previous lesson helped them learn the material better, since it's based on my observations at the other school. Their responses will help me decide how to teach the class in the second semester -- whether I'll continue to follow the veteran teacher, or keep teaching the way I've been doing it so far. Some students might like the veteran teacher's methods, but others might at least wish to continue using DeltaMath or other software.
No matter what happens, I want to continue some habits into second semester. I want opportunities for the students to ask questions from the HW, and for them to collaborate with each other to show understanding of what they've learned. Even if they're learning something brand-new, I can let them discuss the steps with each other and try to figure out why I'm doing what I'm doing.
Still, I'm not sure whether I'll ever implement the most radical changes, such as giving them longer assignments with more time to do them. They're already accustomed to how I do HW, and so I'm not sure whether I want to change the HW itself -- at least not in Calculus class, that is.
My Trigonometry Class
Recall that my general Stats class is only one semester. It will be replaced with another class for the second semester, namely Trig. And since that's a different class, it's easy for me to justify implementing some of the ideas I observed from the veteran teacher, but in Trig class instead of Calc.
Indeed, that text is more conducive to the veteran teacher's methods anyway. There are only seven chapters, and each chapter contains only five sections. I won't finish the book, and so I don't need to cover an entire section in a block period. This means that I have time to slow down and have the students discuss the math with each other and ask questions from the homework.
Let's imagine what the class might look like. After covering Section 1.1, I give the assignment that day, but it's not due yet. On the second day, I answer questions from 1.1 and start 1.2. On the third day, I finish 1.2 and the questions from 1.1 -- meaning that the HW is due the next (fourth) day, when I start teaching Section 1.3. Then on the fifth day, I finish 1.3, and assign new HW -- from 1.3, that is.
I never assign work from 1.2 -- which also matches what the veteran teacher did on my visit. He was teaching Section 3.4 of the Calc text and answering questions from the 3.2 homework. The next day, he assigned HW from 3.4, never assigning 3.3 work at all. Then again, some of the questions in Section 3.4 on the Chain Rule involve the trig functions taught in 3.3, so the students show understanding of 3.3 by getting the 3.4 questions correct.
By spending two days (which may be Mondays or block days) on each section, I hope to reach Section 3.2, on radian measure, by Pi Day. I know -- we just barely had third Pi Day (the 314th day of the year, on November 10th), and I'm already thinking about the first Pi Day on March 14th. But that's how I think -- I'm always looking forward to the next Pi Day and trying to figure out what I'm doing. And now I know what I'm teaching on Pi Day -- a trig class, with a timely lesson on radian measure.
In this class, I can give weekly quizzes, just like the veteran Calc teacher. It might make sense to give these quizzes on Tuesdays, so that I can go over them on Thursdays.
But the tests can be once per chapter, following the practice I established with Calc. While it's not necessary for the Calc and Trig tests to line up, it's possible -- I give the Calc Chapter 4 and Trig Chapter 1 Tests at the same time, and then Calc Chapter 2 and Trig Chapter 5 on the same day. These means that by mid-March, I'll have started Chapter 6 of Calc -- which makes sense, as I want to reach Chapter 7 by the day of the AP exam.
What should I do about grading and percentages in the Trig class? One thing I could do is make the HW assignments be worth as much as the sum of all HW assignments given in Calc in the meantime. For quizzes, I do the opposite -- every few weeks I give a 75-point quiz in Calc, so make all the quizzes I give in Trig during those weeks add up to 75 points. This way, all the points -- and more importantly, all the percentages -- work out in both classes.
Of course, there's another possibility -- I can just give in and weight the Trig grades. Then I don't have to worry about making the points add up at all -- the computer will weight the grades accordingly.
By the way, I still plan on giving Warm-Ups and Exit Passes in my Trig classes, even though the veteran teacher I'm emulating doesn't give them. My classes are 100 minutes each while his are close to 90 minutes, so this leaves about 5-6 minutes near the start and end of each period for Warm-Up and Exit Pass, while still leaving enough time to cover all parts of the veteran teacher's lesson plans.
Tutoring
OK, that's enough about next semester -- let's get back to this semester. With finals looming, some students can benefit from some extra tutoring leading up to the big test. Well, our district has just approved funding for tutoring. All teachers are allowed to stay after school for two hours per week to help students out with their assignments.
Last week, I singled out two students who could benefit from tutoring -- the guy who earned the lowest score on the Calculus test, and the girl who scored the lowest on the Stats test. I asked them when they would be the most available for tutoring. The Calculus guy replied that he would be willing to see me on Tuesdays, and the Stats girl chose Thursday. This suggests that I should plan on having tutoring for one hour each on Tuesdays and Thursdays.
Then again, I wonder whether those days are the most convenient for me. Looking at the block schedule, notice that fifth period Stats meets on Tuesdays and Thursdays. We teachers are no longer allowed to buy lunch from the school, so I must rush out and get a salad those days -- and sometimes I don't have time to finish my lunch.
Normally, I'd wait until after school to eat -- but now I'm on the verge of scheduling tutoring those days, forcing me to wait even longer to eat. On the other hand, if I schedule tutoring Mondays and Wednesdays, then I can eat during sixth period, so I can be nourished rather than hungry heading into the tutoring session.
Still, I'm leaning towards Tuesdays and Thursdays anyway. First of all, I'm supposed to be focused on what's convenient for my students, not what's convenient for me. Moreover, it's a bad look if they chose Tuesday/Thursday and I overrule them by making it Monday/Wednesday instead -- if I really wanted to tutor Monday/Wednesday, then I should just have selected those days without consulting them, instead of leading them on.
And finally, who says that I'd get to eat lunch before tutoring on Monday/Wednesday either? More often than not, I must cover a class during sixth period, and then I still wouldn't get to eat. Thus I should schedule my tutoring around the students, not my stomach.
By tutoring on Tuesdays and Thursdays right after Stats, I can remind the Stats girl right there during class that she needs to stay for tutoring -- it's so easy to forget when the tutoring is on another day. And while tutoring won't be right after Calculus, at least it's the same day, so I'll be able to remind the Calc guy to attend tutoring as well.
Meanwhile, I'm also looking ahead into second semester and Trig. If the lone girl in Stats is struggling in this class, imagine how she'll fare in Trig -- a much tougher class. There's a good chance that she'll need tutoring for Trig, and at least now the tutoring is already set up.
I'm also thinking about the special ed guy in Trig. Since he has his one-on-one aide, he won't need me to tutor him after school. Instead, I'm hoping that using the veteran Calc teacher's methods in Trig will help him out as well. In particular, I'm hoping that he'll ask questions from the homework assignments and collaborate with the other students.
That's all the more reason to adopt the veteran teacher's methods in Trig. I'm assuming that all four Stats students will be back for Trig, and I want to prepare all four of them for success in that class.
By the way, the other math teacher at our school (my partner teacher) will be too busy to provide tutoring now. I asked her whether it would benefit her students if I tutored them as well, and her response was a resounding yes. So I must prepare to tutor her Integrated Math III and Pre-Calculus kids as well.
Retakes and Extra Credit
As we approach the end of the semester, one question that will inevitably come up is extra credit. Many teachers lament how students who do hardly work during the semester will ask for extra credit right at the end. In other words, what they're really asking is, how can I pass this math class without actually knowing any math?
During my many years as a sub, I was thinking about what I should do about extra credit. I think back to my master teacher during student teaching, and my BTSA mentor. Those teachers would sometimes give extra credit assignments during Thanksgiving or spring break. But in reality, this doesn't work -- the students simply ignore the assignment during the break and then ask for extra credit more than a week after returning from that break.
Notice that if they really wanted the extra credit, they would have done the work over the break. But they didn't really want the extra credit -- what they wanted was to get a good grade in math class without knowing any math. This is why I didn't even bother with extra credit during the current Thanksgiving break.
And so here's my extra credit plan -- if students ask for it, I remind those students that the reason for the low grade is that they showed that they hadn't learned the math yet. "So if you can prove to me that you have learned the math, then I'll raise the grade." The students can do whatever they want to do to prove to me that yes, they really did learn the math. It's up to the students to decide the proving behavior, but of course, it starts with their going back and figuring out what the math was about on the assignment or assessment where they received the low score.
Now that I'm establishing tutoring sessions, I can tie the extra credit to tutoring. Any students wanting extra credit must come in during tutoring to demonstrate that they've learned the math.
Of course, retakes (and test corrections) go hand-in-hand with extra credit. Some teachers (including my partner teacher) allow retakes, but others (including the veteran Calc teacher) don't. I entered this year without a properly defined retake plan -- I would allow the retakes only if I was setting up a makeup test for another absent student, then the makeups and retakes can be given simultaneously. But as you'll recall, this led to a big argument -- one Calculus girl was absent, and I let another girl retake during the absent student's makeup. But then the absent girl wanted a retake herself.
On one hand, I'm leaning towards following the veteran teacher's class and not allowing retakes. On the other hand, my partner teacher pointed out that those monthly minimum days are often convenient for test corrections, since there's not much time to do anything else.
Hmm -- perhaps that should be my plan. If the Calculus quiz falls before a minimum day, then I allow test corrections to take place on that minimum day. But if there's no minimum day, then there will be no test corrections allowed.
In particular, no monthly minimum day yet has been announced for December. This means that Quiz #4 will most likely not have corrections. (There might not be a monthly minimum day for December, since there are short days for finals anyway.)
By the way, as for the final itself, the veteran teacher has announced that his test will have 30 questions (multiple-choice, of course). But finals blocks at his school are relatively short -- as opposed to my school, with the traditional two-hour exam blocks. Perhaps I could give a longer test -- maybe even 50 questions, which is proportionate to the lengths of the finals blocks at our respective schools, and is also convenient with a 100-point test, with each question worth two points. (That's right -- here we go with counting points again!)
But after having so many problems with assessment lengths this semester, I'm considering sticking to the 30 questions that the veteran teacher is giving. So my students would have two full hours to answer the 30 questions.
I am aware that the actual AP exam has 45 multiple-choice questions, so I'd be justified in making my test just as long. The AP has both calculator and non-calculator sections, and I'll do the same, regardless of whether I'm giving 30 or 45 questions.
Cheng's Art of Logic in an Illogical World, Chapter 1
So far, I haven't written anything about my Ethnostats class in this post. There's actually an author whose books are highly relevant to some of the topics of Ethnostats -- Eugenia Cheng.
I read her books and discussed about them on the old blog. She's written five books, and her third (on logic and race) and fourth (on gender) are most relevant to Ethnostats. Her most recent book is a children's book, and I blogged about it over the summer. I returned that book to the library just 24 hours before interviewing and being hired for this position, and one week before finding out that I'd be teaching Ethnostats.
Anyway, I think I'll go back and repost what I wrote about her third book on this new blog. I might as well, especially since with my general Stats class becoming Trig, soon Ethnostats will be the only Stats class that I teach (and hence the only class that justifies the name of this blog, Common Core Stats). In fact, I might even go out and purchase either this or her fourth book and use them in my class.
Chapter 1 of Eugenia Cheng's The Art of Logic in an Illogical World is called "Why Logic?" It's the first chapter of Part I, "The Power of Logic." Here's how it begins:
"The world is a vast and complicated place. If we want to understand it we need to simplify it. There are two ways to make something simpler -- we can forget parts of it, or we can become cleverer so that things that used to seem incomprehensible become clear to us."
Cheng tells us that logic involves both aspects of making something simpler. She continues:
"Anyone can make claims about what they think is true, but unless they back up their claims in some way, maybe no one will believe them, and rightly so. So different subjects have different ways of accessing truth."
But she points out that many people believe their arguments are logical when they really aren't:
"Instead of simply lamenting the misunderstanding of logic and mathematics, I choose to address it, in the hope that their power might actually be used to good purpose. That's why I've written this book."
The author explains that logic establishes clear rules so that the same conclusions can be arrived at by different people:
"This is wonderful in theory, and perhaps here 'in theory' means in the abstract world of mathematics. Mathematics has a remarkable ability to make progress."
She tells us that its logical grounding is the reason that, while science is always being updated, math remains a constant. The Pythagorean Theorem is just as valid now as it was when the ancient Greek mathematician first devised it:
"We are going to show that accessing the logical abstract world enables us to get further in the real world, just as flying through the sky enables us to travel further and faster in real life. In essence, this is the whole point of mathematics."
First, Cheng seeks to define what math is -- and isn't. Her ideas here echo the big traditionalists' debate that has heated up in recent weeks. This example hits home, since it's a Geometry example:
"Show that angle A is half of angle B."
The diagram shows a bunch of lines drawn at random, with two angles marked A and B. The author warns -- or reassures -- us that this example is a spoof and can't be solved. She tells us that math is:
"Not a series of hoops to jump through, not an attempt to get the 'right answer,' but a world to explore, discover and understand: the logical world. At this point some people realize that the thing they liked about 'math' up until then was jumping through hoops and getting the right answer."
And those people are called "traditionalists." Anyway, here is Cheng's definition of mathematics:
"Mathematics is the logical study of how logical things work."
Anyway, logic has rules, just as sports have rules:
"One problem with logic, as with sport, is that the rules can be baffling if you're not very used to them. I am pretty baffled by the rules of American football."
And Cheng points out that her being British has nothing to do with it -- she's confused by the rules of soccer as well.
Similarly, many people are confused by the rules of logic. Part of the confusion has to do with the meaning of the word "theory":
"In science, a 'theory' is an explanation that is rigorously tested according to a clear framework, and deemed to be statistically highly likely to be correct. In mathematics, though, a 'theory' is a set of results that has been proved to be true according to logic."
While math is all about certainty, the real world is uncertain:
"Some of the disagreement around arguments in real life is unavoidable, as it stems from genuine uncertainty about the world. But some of the disagreement is avoidable, and we can avoid it by using logic."
And indeed, the real world isn't completely logical. Cheng gives a real-world example:
"If you give a child a cookie and another cookie, how many cookies will they have? Possibly none, as they will have eaten them."
(Yes, it's only Chapter 1 and Cheng has already mentioned her favorite example -- baked foods.) Now she compares this to the world of the abstract:
"If I add one and one under exactly the same conditions in the abstract world repeatedly, I will always get 2. (I can change the conditions and get the answer as something else instead, but then I'll always get the same answer with those new conditions too.)
The author tells us that math is all about searching for the similarities between different things:
"When we look for similarities between things we often have to discard more and more layers of outer details, until we get to the deep structures that are holding things together. This is just like the fact that we humans don't look extremely alike on the surface, but if we strip ourselves all the way down to our skeletons we are all pretty much the same."
In her next example, Cheng writes about the newly painted green lines near subway tracks in London:
"The aim was to try and improve the flow of people and reduce the terrible congestion, especially during the rush hour. This sounds like a good idea to me, but it was met with outcry from some regular commuters."
And this is because they liked the "competitive edge" they'd gained from knowing exactly where to stand so that the doors opened right in front of them. Now those green lines gave away their secrets!
Here's the first of many references to politics and race in Cheng's book:
"This complaint was met with ridicule in return, but I thought it gave an interesting insight into one of the thorny aspects of affirmative action: if we give particular help to some previously disadvantaged people, then some of the people who don't get help are likely to feel hard done by."
Cheng tells us that logic is all about the inherent, not the coincidental:
"The inherentness means that we should not have to rely on context to understand something. We will see that our normal use of language depends on context all the time, as the same words can mean different things in different contexts, just as 'quite' can mean 'very' or 'not much.'"
Because of this, if we start with ambiguous concepts, we end up with ambiguous results:
"We can use extremely secure building techniques, but if we use bricks made of polystyrene we'll never get a very strong building. However, understanding mathematical logic helps us understand ambiguity and disagreement."
At this point Cheng summarizes the contents of her third book:
"In the first part of this book we are examining what logic is as a discipline for building arguments, and as a piece of mathematics. In the second part we'll see what the limitations of logic are. And in the third part we'll see how important it is, given those limitations, to take our emotions seriously."
But the English language contains many ambiguities -- after all, it's being spoken by humans, not the super-logical Vulcans of Star Trek:
"As adults we develop the ability to become more relaxed about figurative language, and more relaxed about how precise we need things to be in order to get on with our lives. This is a bit like how accurately you need to measure things."
(And here comes Cheng's second baking example already!) The author tells us how she can be a little off in measuring sugar to bake a cake, but when she's making macaroons she must be more precise.
In conclusion, Cheng tells us that the purpose of logic is to shine a light on something:
"In all cases the aim should be illumination of some kind. First we need some light, and then we can decide where, and how, to shine it."
Conclusion
Well, in this post I explored all of my classes, and I started making plans for the rest of this semester and the new semester, based on my observation of the veteran Calculus teacher. I hope that my students will be more successful when I teach them more effectively.
I wish everyone a happy Thanksgiving -- the 400th anniversary of the first feast in Plymouth with the Pilgrims and the Wampanoag Tribe.
(Note: Speaking of Trig class and Native Americans, I heard that recently, a California Trig teacher wore a Native American headdress to teach her students the mnemonic "SOH-CAH-TOA." This lesson was considered offensive to the Native population. I haven't decided yet whether I will use this mnemonic in my own Trig class, but as an Ethnostats teacher, I will be sensitive to Native cultures and avoid any stereotypes if I do use the mnemonic.)
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