Fruitcake Toss Day Post (Yule Blog Challenge #9)

Table of Contents

1. Introduction

2. Yule Blog Prompt #13: Intervention Strategies for Helping the Struggling Student

3. Cheng's Art of Logic in an Illogical World, Chapter 11

4. Conclusion

Introduction

Today is Fruitcake Toss Day -- a week after Fruitcake Day itself. Presumably the original Fruitcake Day is a few days after Christmas, when many people eat fruitcake. Then a week later, we toss the old uneaten fruitcake out, hence Fruitcake Toss Day.

Actually, when I researched these food names for the days after Christmas, I found an interesting one -- Candy Cane Day, on the day after Christmas (the same as Boxing Day). According to some sources, its origin goes back to the idea that the day before Christmas Eve is Christmas Adam. Then the sequence of days goes Christmas Adam, Christmas Eve, Christmas, Christmas Cain (the son of Adam/Eve). But Christmas Cain sounds like Christmas Cane -- as in candy cane. So December 26th is Candy Cane Day.

But Christmas is over -- let's get back to today. Some teachers and students give today another name -- the dreaded day when winter break and school reopens. Yet I'm not going to call it that -- because my district is one of those with a three-week winter break. (LAUSD is one such district, but I'm not at LAUSD). And since it's still winter break for me, this counts as a Yule Blog post:

Yule Blog Prompt #13: Intervention Strategies for Helping the Struggling Student

How did I help my struggling students? I think back to the last week of October -- the week of Parent Conferences, when I first identified some of my struggling students. In particular, I had to call the parents of one Calculus student, one Ethnostats student, and one Advisory student. For this post, I wish to focus on the Calc kid -- after all, he's the one who will be taking an AP exam in May. If he continues to struggle, he'll end up with a low grade on the AP.

After the parent conference, the first thing he did was submit many of his late assignments. But he continues to do poorly on his quizzes and exams. Indeed, it's easy to see why -- some of my homework assignments are odd problems and some are even problems. He made up his late odd assignments rather quickly -- most likely, he just looked up the answers in the back of the book. Of course he had to make a little more effort to make up his even assignments, but this doesn't help him if he's not doing the work until well after the quiz/test is over.

In December, my district started a tutoring program, and I had him attend two sessions. I also referred to the tutor.com website that our district is promoting to all students. He told me that he also received help from his older brothers (in college). Indeed, they were the ones who told him about l'Hospital's rule, which aided him with the limit problems on the final exam. (L'Hospital's rule doesn't appear in our text until Chapter 4, while the final was only on Chapters 1-3.)

His final grade was 20/30, which is a D. But this was, ironically, one more correct question than the most common score in the class. Thus in a way, all of my students are struggling, in that students who get D's on the final are unlikely to pass the AP exam in May without extra help.

I already mentioned, in an earlier Yule Blog post, the girl who was absent for most of December. She earned the highest score in the class, a 24/30 or the lowest B, but only by taking the final after school and asking for extra help along the way. She won't be able to do that on the actual AP exam.

During this third week of winter break, I wish to preview the upcoming second semester. I want to make plans for this Calculus class to maximize my students' chances of being successful.

There are 16 weeks in the second semester before the AP Calculus exam -- that is, the test will be on Tuesday of Week 17. And there are four chapters of the text (Chapters 4-7) to cover. Thus a rough pacing guide is to assign four weeks to each chapter:

Weeks 1-4: Chapter 4 (Applications of Differentiation)

Weeks 5-8: Chapter 5 (Integrals)

Weeks 9-12: Chapter 6 (Applications of Integration)

Weeks 13-16: Chapter 7 (Differential Equations)

Week 17: AP Exam

Notice that I did originally spend four weeks on each of Chapters 1-2, before spending essentially two months on Chapter 3. Also, note that not all of Chapter 7 will appear on the AB exam (since some of these topics are BC), so that leaves some time during Weeks 13-16 to give a final review before the test (including giving several practice exams).

Let's look at Chapter 4 in more detail, since this is the next chapter for me to cover:

Section 4.1: Related Rates

Section 4.2: Maximum and Minimum Values

Section 4.3: Derivatives and the Shapes of Curves

Section 4.4: Graphing with Calculus and Calculators

Section 4.5: Indeterminate Forms and l'Hospital's Rule

Section 4.6: Optimization Problems

Section 4.7: Newton's Method

Section 4.8: Antiderivatives

Thus the two block periods next week will be for Sections 4.1 and 4.2, while the block periods the following week will be for Sections 4.3 and 4.4. Section 4.4 specifically mentions calculators -- and recall that my students struggle with the TI, so that will be a day to give them much needed practice on the calculators.

Recall that the plan is to give a quiz on the Thursday before the monthly Monday minimum day, so that the minimum day itself can be used for quiz corrections. But unfortunately, my school doesn't list its minimum days in advance. I don't believe that January 10th will be the short day (although I know that meetings to discuss the future of our magnet are imminent -- there's a slight chance the 10th becomes a minimum day in order to launch those meetings), and the 17th is MLK Day. This leaves the 24th and the 31st as the most likely minimum days.

Section 4.5 is l'Hospital's Rule. Since I already briefly discussed l'Hospital before the final (when my struggling student brought it up), I can cover it on an all-classes Monday, thus saving the block days for more complex topics. (Once again, the exact Monday depends on the timing of the minimum day). This is followed by Section 4.6 is on optimization. The last Starbird video in the differentiation section is on optimization, so I can play that video on the day I teach Section 4.6

On the other hand, I doubt that Newton's Method appears on the AP, and the antiderivatives in Section 4.8 are really just a preview of Chapter 5. So the final sections of Chapter 4 can be replaced with review for the Chapter 4 Test, to be given on Thursday, February 3rd.

During Chapter 4, I'll incorporate the text, AP Classroom, and DeltaMath. I do believe that DeltaMath has some questions on related rates, maxima/minima, and other Chapter 4 topics. Still, I need to ask more AP-like questions that might appear either in the text or AP Classroom. The quiz can be given on DeltaMath as well, but the test might need to be on paper if DeltaMath doesn't provide enough strong AP-like questions for the students.

I will keep in mind what Cheryl Leung wrote about test anxiety. I'll start out with shorter quizzes for the students, but they will get longer as we approach the AP, as the students really do need to be able to answer questions at the pace required for the AP. An easy way for me to plan quizzes is to ask as many questions as the chapter number -- so the Chapter 4 Quiz will have four questions, and the Chapter 7 Quiz will have seven questions.

And once again, I plan on giving the students extra time to discuss homework questions. Most practice will be done using markers and whiteboards. Partner activities will mainly appear at the beginnings of chapters and after quizzes, when there is no homework to discuss. (Again, the derivative activity I observed at the main high school will be given on the day of Section 4.1, with another partner activity to be given after Quiz #4.)

I hope that these lesson plans will prepare my students adequately for the AP exam. As a teacher, I owe it to the students to give them every opportunity to learn what they need to pass the big test in May. As I pointed out before, I have six students in Calculus, with a possible seventh student coming. (I was told that this student was in independent study and that she might join my class in the second semester -- but that was before omicron, which might discourage her from attending class in person next semester.)

Cheng's Art of Logic in an Illogical World, Chapter 11

Chapter 11 of Eugenia Cheng's The Art of Logic in an Illogical World is called "Axioms." It's the first chapter of Part III, "Beyond Logic." Here's how it begins:

"Logic itself has no starting point. It consists of a way of making deductions from things we already know."

This chapter is all about axioms, which are also known as "postulates." Here's how Cheng describes axioms in her book:

"In mathematics, the things we decide to start with are called axioms, and in life these are our core beliefs. Axioms are the basic rules in the system. We do not try to prove axioms, we just accept or choose them as basic truths that generate other truths."

Cheng tells us that we can imagine a world with different axioms -- a thought experiment about how the world would be different, such as a world without a gender pay gap:

"Or a fantasy world in which perpetrators of sexual harassment are not tolerated, particularly not in positions of power or influence. It is informative to imagine a world in which everyone reporting sexual harassment were automatically believed."

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 week and skip all posts that have the "Eugenia Cheng" label.

Returning to math, Cheng writes:

"One famous axiomatization is Euclid's axiomatization of geometry, where he came up with five rules from which all other rules of geometry should be deducible."

I've devoted many posts on my old Geometry blog to discussing Euclid's postulates. Let's continue with what the author has to say about Geometry:

"For some years mathematicians suspected that Euclid's fifth axiom (about parallel lines) was redundant, and tried to prove it from the other four. But they turned out to be wrong -- dropping the fifth axioms leaves you with a perfectly good mathematical system, just a somewhat different type of geometry."

Cheng explains that axioms in math are analogous to our personal core beliefs. She writes:

"Anything is valid, really, as long as it doesn't cause a logical contradiction, in which case your system of beliefs will collapse. Of course, this is only the case if you are trying to be a logical person."

And indeed, the author explains the three main groups into which her personal axioms fall:

1. Kindness: I believe in being kind to people. From this I deduce other beliefs about helping others, contributing to society, education, equality, and fairness.
2. Knowledge: I believe in the frameworks that we have set up to access knowledge in different disciplines. So I believe in scientific research and historical research, for example, within the confidence levels that those disciplines have established.
3. Existence: I believe we exist, mainly as a pragmatic approach to getting on with life. I'm not so sure about this one and I suspect that if I believed the opposite it wouldn't make much difference, so I've chosen to include it as it seems more helpful than the opposite.

Now Cheng asks the question, where did we get our axioms? She answers:

"Some things are instilled in us by our parents, but most of us don't have exactly the same opinions as our parents, which means something else must influence us. Education can expand people's world view and lead them to see things differently from their parents."

In fact, let's continue her line of thinking about education:

'

"I believe that education is the most important way in which I can contribute to the world, and again, I simply feel this very strongly. But if I examine the source of that feeling, it is a combination of values instilled in me by my parents and piano teacher, together with evidence that I am not really cut out to be a doctor (too much memorizing in medical school) [...]."

And while she concedes that memorization is necessary to become a doctor, it's not necessary to study math. In fact, she makes it quite clear -- she chose to become a mathematician rather than a doctor precisely because she doesn't want to memorize a lot. In fact, she admits earlier in her book that she's never even memorized her multiplication tables!

Cheng also writes about another source of axioms/fundamental beliefs -- religion:

"Understanding what people's fundamental beliefs are helps us find the root of disagreements, and understanding where they get their beliefs can help us understand how we might be able to address those beliefs."

The author tells us that some people operate as if all of their beliefs are axioms:

"For example, a person might say, 'I oppose same-sex marriage because I believe marriage should be between a man and a woman.' This might sound like a justification as it does have the word 'because' in it, but it's really a restatement of the initial belief."

Cheng believes that everyone's situation is a combination of their own input and circumstances. No man is an island:

"This in turn comes down to my more fundamental belief that we should understand all things in terms of systems rather than individuals, as in Chapter 5, whether this is people, factors causing a situation, or mathematical objects -- the latter being why I do research in category theory, a discipline that focuses on relationships between things, and the systems those form."

Cheng concludes this chapter with a summary of this third final part of the book, including chapters on gray areas, analogies, and equivalence -- particularly false equivalence:

"In Chapter 15 we'll look at how we can use all these techniques to engage our and other people's emotions, to try and better see eye to eye with other humans, and finally in the last chapter we'll paint a portrait of a good rational human -- not a perfect computer -- and what good rational arguments should look like."

Conclusion

Since New Year's Day is over, so are my annual Calendar Reform posts. But there's one more calendar that's briefly worth mentioning, the Yerm Lunar Calendar:

https://calendars.fandom.com/wiki/Yerm_Lunar_Calendar

This is a pure lunar calendar in that a "yerm" has nothing to do with a solar year. Because of this, I don't wish to spend too much time on figuring out what a school "yerm" would look like. Even if we assume a standard five-day school week and two-day weekend, the school year would have to fit the "yerm" -- so in particular, there can be no "summer break" (or any break based on a season). The link states that three yerms are approximately equal to four years, so we can have 240 school days per yerm, and then Yerms 1-3 might correspond to Grades 1-4, Yerms 4-6 to Grades 5-8, and Yerms 7-9 to high school.

Yerms begin on the night of a new moon -- and according to the link, today just happens to be the first day of a new yerm. (It's the closest that any New Yerm's Day gets to any New Year's Day, at least among the yerm dates listed at the link above.) So since I already wished everyone a Happy New Year, let me now wish everyone a Happy New Yerm!