1. Introduction
2. Rapoport Question of the Day
3. New Decade's Resolutions: Attitude
4. New Decade's Resolutions: Bicycle
5. New Decade's Resolutions: Heroes
6. New Decade's Resolutions: Music
7. New Decade's Resolutions: Time
8. Traditionalists, Calculus, and Michael Starbird
9. Loose Ends with Music: The Whodunnit Song
10. Conclusion
Introduction
This is my third and final winter break post. It is also my first post of the decade of the 2020's. Now normally, my first post of the year contains some New Year's Resolutions. But since it's a new decade, this post will contain some New Decade's Resolutions.
As usual, the New Decade's Resolutions will contain ideas on how I can become a better math teacher someday. My biggest failure of the old decade was, of course, at the old charter school. So my hope is to be hired again as a math teacher at some point during the new decade. The New Decade's Resolutions will provide me with guidance, both as a sub attempting to get hired as a teacher, and once I become that math teacher trying to keep that new job.
Last month, I yelled at a student for trying to leave art class early -- even though at least one student was doing the same. The reason that I singled her out over the other guy is shameful. It's because she looks a little like -- and thus reminded me a little of -- the special scholar at the old charter school.
I was wrong to yell at the special scholar, and I was wrong to yell at this girl. I apologized to this girl the next day, and she accepted my apology. But unfortunately, I can never truly apologize to the special scholar, who will never forgive me for the mistakes I made with her that year.
The last couple of years, my first post of the year was all about the special scholar. This year, I won't repeat that story again. Instead, my focus is on the New Decade's Resolutions, which will guide me in avoiding arguments and yelling as we move forward. I never want to treat another student the way I once treated the special scholar.
Rapoport Question of the Day
Theoni Pappas -- she's so last decade. It's time for us to begin my new calendar, Mathematics: Your Daily Epsilon of Math 2020 by Rebecca Rapoport.
Notice that "epsilon" is the fifth letter of the Greek alphabet. A few years ago, I explained "epsilon" during a side-along reading of Paul Erdos. Anyway, "epsilon" represents a small quantity. It's used to make Calculus more rigorous, so that the math is based on actual small real numbers that approach zero in the limit, rather than "ghosts of departed quantities" (as seen by the originators of Calculus, Isaac Newton and Gottfried Leibniz).
Then Erdos used the term "epsilon" to refer not to small numbers but small people -- in other words, young children. Now Rapoport is using "epsilon" to refer to something else that's small -- a small problem to be solved each day. Anyway, let's begin without any further ado.
Today on her Daily Epsilon of Math 2020, Rebecca Rapoport writes:
log_10(x) = 2 - log_10(25)
I know that this isn't a Geometry problem, but since it's my first post of the year, I wanted to start out with a Rapoport problem anyway. This problem is on common logarithms base 10. Rapoport includes the subscript 10 to emphasize the base, but these don't necessarily look good in ASCII. But since most Algebra II texts omit writing the base of a log when it's 10 (including Lesson 9-4 of the U of Chicago Advanced Algebra text), we can rewrite this problem as:
log(x) = 2 - log(25)
Since the equation has log(x) and we wish to find x, let's raise 10 to the power of both sides. In doing so, we use the fact that subtraction of logs really indicates division:
x = 10^2/25
x = 100/25
x = 4
Therefore, the desired value of x is 4 -- and of course, today's date is the fourth.
Yes, the answer to every Rapoport problem -- just like every Pappas problem -- is simply the date. I reckon that Rapoport got the idea of making the answer the date from the earlier calendar author. And I'm glad that this familiar aspect of Pappas problems will be maintained by the new author.
On her calendar, Rapoport explains that there will be three exceptions to "the answer is the date" rule, all of which are on special days. On those three days, she has a math joke instead of a problem. She also tells us that some of her questions will require research to answer. Indeed, the very first problem on her calendar is:
Let P(n) be the prime-counting function. Find lim n->oo P(n)ln(n)/n.
We must be familiar with the Prime Number Theorem (first conjectured by Gauss, proved by subsequent mathematicians) in order to obtain the answer as 1 (on New Year's Day, of course).
Meanwhile, Rapoport's first Geometry question isn't until next week.
New Decade's Resolutions: Attitude
Let's get ready for our discussion of my New Decade's Resolutions. These don't exactly replace the previous resolutions posted on the blog. Instead, the new resolutions will show me how exactly to achieve my old ones.
My two most important resolutions previously posted on the blog are to improve my classroom management and to reduce/eliminate arguing and yelling. The New Decade's Resolutions are written to tell me what exactly I need to say to students to improve management and reduce arguments.
The first two resolutions address student attitudes towards learning math. Many students enter our classrooms wondering why they need to learn math. They ask questions such as the dreaded "When will we use this in real life?" Their underlying assumption is no, they don't need to learn math -- that they'd be better off if they didn't learn math at all.
Closely related is the statement "I'm not good at math." Many students assume that they aren't good at math, and thus there's no point in even trying. What often happens is they work on a problem and get the wrong answer due to incomplete knowledge of a prior topic (for example, they can't solve an equation because they struggle with fractions or negative numbers). They don't understand why their answer is wrong, so they just give up and say, "I'm not good at math."
This extends to calculator use. I've blogged so much about "drens" who need calculators to solve simple problems like 3 * 4. It's not exactly that they don't know what 3 * 4 is -- instead, they think to themselves, "3 * 4 is math and math is hard, therefore 3 * 4 is hard." They don't allow themselves to distinguish between the math they know (without a calculator) and the math they don't know -- instead, they just use a calculator for all math.
All of these ideas are related to student attitudes. Thus my first New Decade's Resolution shows me how to deal with student attitudes in the classroom:
Decade Resolution #1: We are good at math. We just need to improve at other things.
I've decided to word all of these new resolutions in the first person plural. The reasons for doing so here are twofold. First, mathematicians often use the royal "we" when writing complex proofs. The U of Chicago text doesn't often do this (especially not in two-column proofs), but the pronoun "we" does appear in the proofs of the SSS and SAS Congruence Theorems in Lesson 7-2:
SSS Congruence Theorem:
The proof uses the idea of transitivity of congruence. That is, to prove Triangle ABC = DEF, we prove that each triangle is congruent to a third triangle
SAS Congruence Theorem:
Since
But the more important reason I use "we" here is to avoid certain student responses. If I were to say "I am good at math," a student response might be, "That's for you. That's not for me." This regularly occurred at the old charter school when I told the students that I (back when I was their age) had no problem following a certain rule. And as I found out, replacing "I" with "students in the other class" leads only to "That's for them. That's not for me."
And so the new resolutions use "we" to emphasize that the students in front of me are included as part of "we." We are good at math -- I don't mean just myself or the kids in the honors class, but all of us in this room right now are good at math.
Many Calculus teachers often tell their students, "You are good at Calculus -- you just need to improve at algebra." And the algebra students in turn say, "You are good at algebra -- you just need to improve at fractions." This pattern may continue -- some students may know the rules for fraction arithmetic yet struggle at basic arithmetic. The next level below basic arithmetic is logic.
But I personally believe that everyone is good at logic, to some extent. (We might not be Vulcans who always act logically, but we at least think logically.) Therefore, I believe that we all have the potential to be good at math.
The first resolution of the new decade is intended to get students to separate what they're good at from what they're not, and to improve at what they're not good at. But it's impossible to do so if what they say they're not good at is "math."
In implementing the first resolution, I begin by saying, "We are good at math. We just need to improve at...," followed by the specific skill at which they need to improve. For example, if a student solves a fraction addition problem by adding the denominator, I can say, "We are good at math. We just need to improve at remembering when we need to find a common denominator."
Of course, working on that specific skill isn't easy. This takes us to our second resolution:
Decade Resolution #2: We make sacrifices in order to be successful in math.
Many traditionalists point out that students happily make sacrifices in order to be successful in sports or music, but not in math. The students' thoughts here are, "I'm so desperate to be good at sports or music that I'll make sacrifices," but "I don't mind being bad in math, so if I have to make a sacrifice to improve, I simply won't." The goal of the second resolution is to encourage students to be just as desperate to be good at math as they are at sports or music.
The second resolution also deals with another issue that regular math teachers often see -- students who ask for extra credit right at the end of the semester. What they're really asking here is, "How can I get a good grade this semester without making any sacrifices whatsoever?" Thus the second resolution is here to remind them that they need to make sacrifices to be truly successful at math.
I'm thinking back to the Washington Post article I mentioned in my last traditionalists' post. If I recall correctly, one commenter pointed out that all math taught in school can be divided into three levels:
- basic arithmetic that most adults use regularly
- medium-level math whose usefulness isn't obvious ("When will we use this in real life?")
- higher-level math that only specialists learn and is used for science, technology, engineering
The boundaries among these three levels are difficult to place. Clearly most secondary math (middle and high school) fits in the middle level. Indeed, if we assume that the middle level begins as soon as the question "When will we use this in real life?" is asked, then according to another commenter in that thread, this could be as early as the grade she teaches -- fifth grade.
I actually once suggested that fourth grade is the year that "When will we use this?" begins. Even though she doesn't state this, as a multiple-subjects teacher, she strongly implies that "When will we use this in real life?" is asked only during math and not during her other subjects. And we can also guess during which unit the question is asked the most -- the fractions unit.
Hey -- this post is all about New Decade's Resolutions. Anyway, I just noticed that the three levels correspond roughly to decades. The basic level corresponds to the arithmetic that is taught during the first decade of life, when students have single-digit ages. The middle level is taught in the second decade, when students have "1" in the tens digit of their ages -- this begins roughly during fourth or fifth grade and lasts through the first two years of college (gen ed requirements). Then the last level is the third decade, when students have "2" in the tens digit of their ages. This is upper-division and graduate-level courses, taken only by math (or other STEM) majors.
The commenter who mentioned the three levels writes that medium-level (second decade) math is taught mainly to prepare students for higher-level (third decade) math -- admittedly, this math isn't used much in the real world on its own. That's why "When will we use this?" is asked.
But, if we allow students to opt out of taking second decade math, then many will -- and then there won't be enough students to get to the very useful third decade math. And without third decade math, much of our current technology wouldn't exist.
That's the whole problem in a nutshell. As math teachers, we need a way just to get students through this second decade of math -- for them to make sacrifices to be successful even though its usefulness won't be apparent until the third decade. This is what being a math teacher is all about.
(If you don't like that phrase "second decade math," it can be replaced with "secondary math" without changing its meaning much. But once again, the New Decade's Resolutions are all about decades.)
New Decade's Resolutions: Bicycle
Recently, I've been reading articles about something called "winter melt." It refers to the idea that, just as with "summer melt," students often forget things over winter break. It's often pointed out that much of the achievement gap can be attributed to summer, and possibly winter, melt. (Yes, we can't get through a holiday break without some mention of politics and race.)
Of course, ideally there should be zero melt -- students should remember what they learn. Yet we all look forward to summer and winter breaks. We want there to be time off when we don't have to work, but is it possible to have a break from working without forgetting what was taught?
(I think back to the Eleven Calendar. With five days off in every 11-day week, this reduces the length of summer so that there are just three annual breaks of 15 days each. This would seemingly reduce summer melt, but even that wouldn't eliminate it, considering that summer break would be about the length of winter break and winter melt exists.)
So what is my solution to the summer/winter break melt problem? Well, it's often said that once we learn how to ride a bicycle, we never forget it. If we ride our bike everyday to school, then not ride it at all during winter or even summer break, when school starts up again we'll still remember how to ride our bike. It's only other subjects, in particular math, that we forget after long breaks.
This leads to my third New Decade's Resolution:
Decade Resolution #3: We remember math like riding a bicycle.
Traditionalists often lament that students aren't being taught basic math anymore. In particular, they refer to students taking high school math classes but don't know the math they should have learned in previous years.
I believe that the students are still being taught basic math -- it's just that they've forgotten it by the time they reached the higher class. Students recall material long enough to pass the test, but then forget it over summer or even winter break. In other words, they are definitely not remembering math the same way they remember how to ride a bike.
Thus this is the goal of my third resolution. I want my students to remember as much math as possible like riding a bike -- to remember what they've learned forever, not just until the next test or next school break.
Of course, I must start with yours truly. Can I honestly claim that someone today can hand me the final exam from any math class I took at UCLA during my third decade (that is, after my 20th birthday) and that I'd be able to pass that final cold? The answer is, of course not. (And of course, some finals, such as that infamous graduate analysis class where I earned a C+, I had trouble passing at the time I originally took it!) Thus third decade math isn't part of my "bicycle."
But I at least like to believe that I can pass my second decade finals today. (For me, this starts with the Advanced Pre-Algebra text that I studied independently from starting around my tenth birthday, up to the Infinite Series class I took at UCLA the summer when I was 19.) Some of the finals I took at age 19 might be a little tricky, but I hope I still know enough just to pass them. As for the material I learned in APA at age ten, I shouldn't merely pass the tests today, but ace them. Thus second decade math is part of my "bicycle."
My bicycle is thus larger than those of many of my students. We can remind the students over and over not to add the denominators, and they'll remember it for the test. We can even have the students take notes on adding fractions in an interactive notebook. But then in a subsequent unit on something else (say simple equations), the students forget how to add fractions. They don't even check the notes, but simply add the denominators, since they so certain that they're doing it correctly. Thus adding fractions isn't part of our students' bicycles -- soon after they've learned it, they forget it.
How much math should be part of a student's bicycle? Certainly first decade math should be a part of everyone's bicycle. Some second decade math probably needs to be included to -- especially math that will come up again in subsequent math classes that the students will take. This certainly includes fraction arithmetic as well as integer arithmetic.
But as we've seen before, sometimes students have trouble even with first decade math. With examples such as 3 * 4, chances are that the students know the answer, but they just enter it on a calculator anyway instead of asking themselves whether it's part of their bicycle or not. For other problems like 7 * 8, this one truly isn't part of many students' bicycles.
Another problem that's come up lately is telling time on an analog clock. Traditionalists complain that students aren't being taught how to tell time anymore. In reality, they still are taught telling time, but they don't include it as part of their bicycle. They pass the test on telling time, then forget it as soon as they've learned it.
And so what I wish to do is make student's bicycles bigger -- increase the amount of math that they can still remember to do months, years, and even decades after they've learned it. Hmm, let's try to maintain the bicycle analogy. We don't really enlarge bikes, but we do inflate their tires:
Decade Resolution #4: We need to inflate the wheels of our bike.
Traditionalists believe that students would learn and remember basic facts more effectively if only they were taught traditionally. I mentioned recently their complaints about 6 + 8 (that is, students are being taught to "make tens" rather than just memorize what 6 + 8 is). I disagreed in part here and suspect that both traditional and modern methods can aid in inflating the bike wheels. (Thus making tens out of 6 + 8 can help students learn 6 + 8 -- and then once they've learned it, they should try to memorize it and add it to their bike wheels.)
Once again, I wish to incorporate both traditional and modern methods into math classes. My stance on this blog has always been for first decade math to be taught more traditionally, and for second decade math to contain a mix of traditionalism and modernism.
For example, I've noticed that some teacher bloggers refer to "killing kittens" whenever their students make a common error. Let me find such a website here:
http://eatplaymath.blogspot.com/2015/09/dont-kill-kitten-please.html?m=1
The author of this website, Lisa Winer, tells gives her students a list of common errors that count as "killing a kitten" in her Algebra II class. The first she lists here is:
(x + 2)^2 = x^2 + 4
That is, students believe that they can distribute an exponent over a sum. It's not that the students were never taught how to expand (x + 2)^2 in Algebra I -- it's just that they learned and remembered it long enough to pass the Algebra I final, then forget it by the time they reach Algebra II. And it's likely that Winer might have her students do this problem in their notes -- and then when it's time for the homework, they just write x^2 + 4 without even looking at the notes. Oh, and if Winer were to tell the students that x^2 + 4 is wrong, they just start arguing that they're right.
The whole idea behind the New Decade's Resolutions is to avoid arguments. Thus Winer came up with something simple to say whenever a student makes this error. The statement she chose to say is "Don't kill a kitten!" This avoids arguments -- the students hear several times during the class that distributing an exponent over a sum is feline murder, and so they know quickly to fix the error.
Notice that two of the other dead cats on her list are just generalizations of this first one:
sqrt(x^2 + y^2) = x + y
(sqrt(z) + sqrt(x))^2 = z + x
Thus I should strongly consider saying this in class when I see students make this type of error. Of course, if I sub for an Algebra II class and just start saying "Don't kill a kitten!" the students will have no idea what I'm talking about (unless I just happen to be subbing for Winer's class). But this is something I should come up with if I teach my own Algebra II class at some point this decade.
Oh, and I might not wish to mix metaphors here. The kitten metaphor doesn't necessarily fit with my bicycle metaphor here.
Hmm, or maybe it does. I can say that writing (x + 2)^2 = x^2 + 4 is like running over a kitten with our bicycle. Then fixing the error, (x + 2)^2 = x^2 + 4x + 4, is like avoiding the kitten by steering around it. After all, part of riding a bike is knowing how to steer and avoid running over obstacles, like kittens playing on the road.
I suspect that at some point, Winer's students get tired of hearing about dead cats. But then again, that's the whole point -- the way to get the teacher to stop talking about kittens is just to remember the laws of algebra and avoid the errors in the first place.
New Decade's Resolutions: Heroes
I've written much about getting rid of the concept that someone good at math is a "nerd." I replace this with two new ideas -- one is that someone who can't do basic (first decade) math is a "dren" (or "nerd" spelled backwards), and that someone who excels at math is a "hero."
Last month, I subbed in a seventh grade math class and read them a graphic novel about the NASA scientists who contributed to the Apollo moon landings. Clearly, my point there was that we should treat mathematicians and scientists like heroes, not nerds -- and that students should try to emulate the heroes in math class.
I wish to take this a step further. I wish to treat the heroes almost like celebrities, so that students would want to emulate them the way that many kids want to imitate their entertainment heroes. In particular, I wish to highlight three particular individuals:
New Decade's Resolutions: Bicycle
Recently, I've been reading articles about something called "winter melt." It refers to the idea that, just as with "summer melt," students often forget things over winter break. It's often pointed out that much of the achievement gap can be attributed to summer, and possibly winter, melt. (Yes, we can't get through a holiday break without some mention of politics and race.)
Of course, ideally there should be zero melt -- students should remember what they learn. Yet we all look forward to summer and winter breaks. We want there to be time off when we don't have to work, but is it possible to have a break from working without forgetting what was taught?
(I think back to the Eleven Calendar. With five days off in every 11-day week, this reduces the length of summer so that there are just three annual breaks of 15 days each. This would seemingly reduce summer melt, but even that wouldn't eliminate it, considering that summer break would be about the length of winter break and winter melt exists.)
So what is my solution to the summer/winter break melt problem? Well, it's often said that once we learn how to ride a bicycle, we never forget it. If we ride our bike everyday to school, then not ride it at all during winter or even summer break, when school starts up again we'll still remember how to ride our bike. It's only other subjects, in particular math, that we forget after long breaks.
This leads to my third New Decade's Resolution:
Decade Resolution #3: We remember math like riding a bicycle.
Traditionalists often lament that students aren't being taught basic math anymore. In particular, they refer to students taking high school math classes but don't know the math they should have learned in previous years.
I believe that the students are still being taught basic math -- it's just that they've forgotten it by the time they reached the higher class. Students recall material long enough to pass the test, but then forget it over summer or even winter break. In other words, they are definitely not remembering math the same way they remember how to ride a bike.
Thus this is the goal of my third resolution. I want my students to remember as much math as possible like riding a bike -- to remember what they've learned forever, not just until the next test or next school break.
Of course, I must start with yours truly. Can I honestly claim that someone today can hand me the final exam from any math class I took at UCLA during my third decade (that is, after my 20th birthday) and that I'd be able to pass that final cold? The answer is, of course not. (And of course, some finals, such as that infamous graduate analysis class where I earned a C+, I had trouble passing at the time I originally took it!) Thus third decade math isn't part of my "bicycle."
But I at least like to believe that I can pass my second decade finals today. (For me, this starts with the Advanced Pre-Algebra text that I studied independently from starting around my tenth birthday, up to the Infinite Series class I took at UCLA the summer when I was 19.) Some of the finals I took at age 19 might be a little tricky, but I hope I still know enough just to pass them. As for the material I learned in APA at age ten, I shouldn't merely pass the tests today, but ace them. Thus second decade math is part of my "bicycle."
My bicycle is thus larger than those of many of my students. We can remind the students over and over not to add the denominators, and they'll remember it for the test. We can even have the students take notes on adding fractions in an interactive notebook. But then in a subsequent unit on something else (say simple equations), the students forget how to add fractions. They don't even check the notes, but simply add the denominators, since they so certain that they're doing it correctly. Thus adding fractions isn't part of our students' bicycles -- soon after they've learned it, they forget it.
How much math should be part of a student's bicycle? Certainly first decade math should be a part of everyone's bicycle. Some second decade math probably needs to be included to -- especially math that will come up again in subsequent math classes that the students will take. This certainly includes fraction arithmetic as well as integer arithmetic.
But as we've seen before, sometimes students have trouble even with first decade math. With examples such as 3 * 4, chances are that the students know the answer, but they just enter it on a calculator anyway instead of asking themselves whether it's part of their bicycle or not. For other problems like 7 * 8, this one truly isn't part of many students' bicycles.
Another problem that's come up lately is telling time on an analog clock. Traditionalists complain that students aren't being taught how to tell time anymore. In reality, they still are taught telling time, but they don't include it as part of their bicycle. They pass the test on telling time, then forget it as soon as they've learned it.
And so what I wish to do is make student's bicycles bigger -- increase the amount of math that they can still remember to do months, years, and even decades after they've learned it. Hmm, let's try to maintain the bicycle analogy. We don't really enlarge bikes, but we do inflate their tires:
Decade Resolution #4: We need to inflate the wheels of our bike.
Traditionalists believe that students would learn and remember basic facts more effectively if only they were taught traditionally. I mentioned recently their complaints about 6 + 8 (that is, students are being taught to "make tens" rather than just memorize what 6 + 8 is). I disagreed in part here and suspect that both traditional and modern methods can aid in inflating the bike wheels. (Thus making tens out of 6 + 8 can help students learn 6 + 8 -- and then once they've learned it, they should try to memorize it and add it to their bike wheels.)
Once again, I wish to incorporate both traditional and modern methods into math classes. My stance on this blog has always been for first decade math to be taught more traditionally, and for second decade math to contain a mix of traditionalism and modernism.
For example, I've noticed that some teacher bloggers refer to "killing kittens" whenever their students make a common error. Let me find such a website here:
http://eatplaymath.blogspot.com/2015/09/dont-kill-kitten-please.html?m=1
The author of this website, Lisa Winer, tells gives her students a list of common errors that count as "killing a kitten" in her Algebra II class. The first she lists here is:
(x + 2)^2 = x^2 + 4
That is, students believe that they can distribute an exponent over a sum. It's not that the students were never taught how to expand (x + 2)^2 in Algebra I -- it's just that they learned and remembered it long enough to pass the Algebra I final, then forget it by the time they reach Algebra II. And it's likely that Winer might have her students do this problem in their notes -- and then when it's time for the homework, they just write x^2 + 4 without even looking at the notes. Oh, and if Winer were to tell the students that x^2 + 4 is wrong, they just start arguing that they're right.
The whole idea behind the New Decade's Resolutions is to avoid arguments. Thus Winer came up with something simple to say whenever a student makes this error. The statement she chose to say is "Don't kill a kitten!" This avoids arguments -- the students hear several times during the class that distributing an exponent over a sum is feline murder, and so they know quickly to fix the error.
Notice that two of the other dead cats on her list are just generalizations of this first one:
sqrt(x^2 + y^2) = x + y
(sqrt(z) + sqrt(x))^2 = z + x
Thus I should strongly consider saying this in class when I see students make this type of error. Of course, if I sub for an Algebra II class and just start saying "Don't kill a kitten!" the students will have no idea what I'm talking about (unless I just happen to be subbing for Winer's class). But this is something I should come up with if I teach my own Algebra II class at some point this decade.
Oh, and I might not wish to mix metaphors here. The kitten metaphor doesn't necessarily fit with my bicycle metaphor here.
Hmm, or maybe it does. I can say that writing (x + 2)^2 = x^2 + 4 is like running over a kitten with our bicycle. Then fixing the error, (x + 2)^2 = x^2 + 4x + 4, is like avoiding the kitten by steering around it. After all, part of riding a bike is knowing how to steer and avoid running over obstacles, like kittens playing on the road.
I suspect that at some point, Winer's students get tired of hearing about dead cats. But then again, that's the whole point -- the way to get the teacher to stop talking about kittens is just to remember the laws of algebra and avoid the errors in the first place.
New Decade's Resolutions: Heroes
I've written much about getting rid of the concept that someone good at math is a "nerd." I replace this with two new ideas -- one is that someone who can't do basic (first decade) math is a "dren" (or "nerd" spelled backwards), and that someone who excels at math is a "hero."
Last month, I subbed in a seventh grade math class and read them a graphic novel about the NASA scientists who contributed to the Apollo moon landings. Clearly, my point there was that we should treat mathematicians and scientists like heroes, not nerds -- and that students should try to emulate the heroes in math class.
I wish to take this a step further. I wish to treat the heroes almost like celebrities, so that students would want to emulate them the way that many kids want to imitate their entertainment heroes. In particular, I wish to highlight three particular individuals:
- Bill Gates
- Steve Jobs
- Tim Berners-Lee
All three of these individuals were born in the same year -- 1955. And all three of them created the technology that we use everyday -- Gates founded Microsoft, Jobs founded Apple, and Berners-Lee created the Internet.
Any teacher today knows that cell phone use is prevalent among young people -- to the point that they are often too distracted during class. In other words, young people enjoy the technology that Gates, Jobs, and Berners-Lee created. Yet they often hate math and don't wish to learn it -- even though Gates, Jobs, and Berners-Lee used math in their inventions.
One website seeks to delay cell phone use until at least eighth grade:
I partly agree with the goal of this website. I'd prefer to say that smartphone use should be delayed until the student reaches eighth grade math (that is, has completed Math 7 with a passing grade). This is to emphasize the link between the phones they enjoy and the math used to develop the apps they use on those phones.
One idea mentioned on this website is "We follow the research." This statement refers to the research that demonstrates how students learn more when they are bored, not distracted. For example, it's used to counter students' arguments that using a phone (say to listen to music) while they study helps them learn better. On the contrary, "the research" shows that this is false -- even those who claim that they are good at "multitasking" are even more successful when they "uni-task."
Even though "we follow the research" is written in the first person plural, I didn't include it as one of my New Decade's Resolutions. Still, I respect the idea.
One idea mentioned on this website is "We follow the research." This statement refers to the research that demonstrates how students learn more when they are bored, not distracted. For example, it's used to counter students' arguments that using a phone (say to listen to music) while they study helps them learn better. On the contrary, "the research" shows that this is false -- even those who claim that they are good at "multitasking" are even more successful when they "uni-task."
Even though "we follow the research" is written in the first person plural, I didn't include it as one of my New Decade's Resolutions. Still, I respect the idea.
If I had it my way, strong math students will get phones sooner (seventh, sixth grade or earlier), while weaker math students will have to wait later (high school, etc.) to get them. Younger siblings will see their older siblings get phones -- and those who are the oldest in their families will see their friends with older siblings get phones -- and they'll be motivated to study math and pass it so that they can receive their own phones.
Of course, as a teacher I have no way of making that idea a reality. Instead, I should focus on what I can control about phone use in my classroom. And it begins with seeing the founders of that technology -- Gates, Jobs, and Berners-Lee -- as heroes:
Decade Resolution #5: We treat the ones born in 1955 like heroes.
Some people may question whether Bill Gates, for example, is truly a hero. Many computer users criticize Windows or Office software and wish that there were more alternatives. I myself compare Microsoft to McDonald's here -- we often complain about how greasy McDonald's food is or how it tastes like cardboard. Yet the drive-through is always crowded at lunchtime -- and many of us would not wish to live in a world where McDonald's doesn't exist. The same is true of Microsoft. And in my case, I wish to emphasize how hard-working Gates was in high school (he earned a near-perfect SAT score), and how he was able to apply the math he learned to developing Microsoft.
(Note: Some anti-Common Core traditionalists dislike Gates because he was influential in the spread of the Common Core Standards here. Once again, my emphasis is on the math he learned and how he applied it, not what he did with his money after he earned it.)
It's also often pointed out that the technology kings born in 1955 were college dropouts. (Actually, Berners-Lee completed a physics degree -- only Gates and Jobs were dropouts.) That being said, they must have excelled in high school in order to be admitted to the colleges they dropped out from.
We can't simply say, "Gates dropped out from college because education was holding him back, as he proved by creating Windows 95. Thus if he'd dropped out of kindergarten, he would have created Windows 85 instead of 95." (Actually, there really was a Windows 85, but we called it "Windows 1" or just "Windows" instead of "Windows 85." Say "Windows 75" instead.) Instead, we conclude that he applied the math he learned in his second decade, before dropping out, to his business. And it's because math classes existed that the three kings was able to do their magic. Therefore we owe the technology we have today to math classes.
Note: It's also pointed out sometimes that the fifty-fivers don't allow their own children/grandchildren to use the technology that their companies created. The idea is to show that the fifty-fivers are at best hypocrites and at worst evil -- they sell products that they know are harmful, because they care more about profits than actually benefiting the public. This is a tricky one -- the students believe that phones and other forms of technology are good. My goal is to take advantage of this belief in order to get them to realize that learning math is good, so that they'll want to work hard in my classes.
Note: It's also pointed out sometimes that the fifty-fivers don't allow their own children/grandchildren to use the technology that their companies created. The idea is to show that the fifty-fivers are at best hypocrites and at worst evil -- they sell products that they know are harmful, because they care more about profits than actually benefiting the public. This is a tricky one -- the students believe that phones and other forms of technology are good. My goal is to take advantage of this belief in order to get them to realize that learning math is good, so that they'll want to work hard in my classes.
And so it's settled. Gates, Jobs, and Berners-Lee -- the generation born in 1955 -- are the heroes I will celebrate in my math classes.
Now what does this mean as far as learning math is concerned? Well, let's compare the generation born in 1955 to the students who sit in our classes today.
Many of our students today are distracted playing games and watching videos on phones. Of course, the fifty-fivers didn't play on their smartphones in class -- since they, the fifty-fivers, hadn't invented that technology yet! Instead, the fifty-fivers were focused in their high school classes -- and that's why they learned enough to make their inventions.
The fifty-fivers didn't need to listen to music while they studied. The fifty-fivers were able to concentrate on their work in silence.
As a young person, I just barely missed the cellphone generation. The closest I ever got to being distracted by technology was as a library clerk, when I might visit websites on the computers. The supervisor told me to avoid browsing -- and when I did, I found that I was suddenly more attentive to patron needs, because I was bored. This is first-hand research showing that bored students and workers are better students and workers.
The fifty-fivers didn't need to listen to music while they studied. The fifty-fivers were able to concentrate on their work in silence.
As a young person, I just barely missed the cellphone generation. The closest I ever got to being distracted by technology was as a library clerk, when I might visit websites on the computers. The supervisor told me to avoid browsing -- and when I did, I found that I was suddenly more attentive to patron needs, because I was bored. This is first-hand research showing that bored students and workers are better students and workers.
Meanwhile, many students want to use their calculators to solve basic math problems. The fifty-fivers, of course, didn't do so, because pocket calculators hadn't been invented yet. Thus the fifty-fivers actually had to learn and memorize their basic math facts.
I can't really enforce the "no smartphones until eighth grade" rule suggested by the link above, since I, as a teacher, am not the one purchasing the phones for the students. Instead, I can limit how much technology is used in my classes, so that my students can learn as much math as the heroes who were born in 1955.
One area where I've seen technology use gone wild is during high school "tutorial." Many high schools in my newer district have tutorial scheduled in the morning, when students are supposed to study or make-up work. But it's too easy for students to claim "I've finished all my work for all my classes," take out their phones, and then use them for non-academic entertainment during the entire tutorial period.
Some teachers have ways of making sure that the students are actually working. But it's almost impossible for me as a sub to do so. By now I've basically given up -- if a group of students enter my tutorial and use 100% of that time for entertainment, there's nothing I can do to stop it.
But I have come up with a plan if someday, I'm hired to be a regular teacher at a school that has a tutorial period. And it invokes the idea that the 1955 generation is the one to emulate.
By the way, this all reminds me of a YouTube video I once saw about a math song. It is called "Gettin' Triggy wit It," a parody of Will Smith's "Gettin' Jiggy wit It":
My point here isn't about the song itself, but the first few seconds of the video. It begins as a teacher announces that she's about to start the trig unit. Then one student complains, "Is there any way you can make this more fun or something?" and several students nod in agreement.
All of this is just to introduce the "Triggy" parody, but that's the problem right there -- many students believe that they shouldn't have to learn anything or do anything unless it's fun. In stark contrast, the fifty-fivers were willing to learn things even when they weren't fun -- and that's why they succeeded in high school and were able to apply what they learned. That something be "fun" is an unnatural constraint, and learning will skyrocket for students who remove that constraint.
By the way, this all reminds me of a YouTube video I once saw about a math song. It is called "Gettin' Triggy wit It," a parody of Will Smith's "Gettin' Jiggy wit It":
My point here isn't about the song itself, but the first few seconds of the video. It begins as a teacher announces that she's about to start the trig unit. Then one student complains, "Is there any way you can make this more fun or something?" and several students nod in agreement.
All of this is just to introduce the "Triggy" parody, but that's the problem right there -- many students believe that they shouldn't have to learn anything or do anything unless it's fun. In stark contrast, the fifty-fivers were willing to learn things even when they weren't fun -- and that's why they succeeded in high school and were able to apply what they learned. That something be "fun" is an unnatural constraint, and learning will skyrocket for students who remove that constraint.
By the way, sometimes I wonder whether "fifty-fivers" is a good name for this generation. (Actually, that generation already has a name -- 1955 is considered part of the Baby Boomers.) I myself was born in 1980 and graduated high school in 1999. And indeed I think of myself as a "ninety-niner," not an "eightier." As high school teachers, we want students to look ahead to their graduations and how they'll get there, not back at their births.
Thus the three technology kings should really be called "seventy-three'ers." (Actually, Jobs skipped fifth grade and so he graduated in 1972. Both Gates and Berners-Lee indeed completed secondary school in 1973.) At any rate, both 1955 and 1973 will be mentioned in the tutorial plan that I will describe here:
Decade Resolution #6: We ask, what would our heroes do?
And this sixth resolution demands that we ask ourselves, what did the generation born in 1955 and graduated in 1973 do when they needed to study?
Here's how my plan works. On Mondays in my district, there is no tutorial (since school starts later on Mondays). On Tuesdays, I declare it to be "Tutorial 1955." This means that students are not to use any technology that was created after 1955. Students can't claim, for example, that they need the Chromebook or cell phone to do work for another class (and then use it for entertainment as soon as I'm not looking). If they know in advance that they need technology to complete an assignment, then they should attend a different tutorial on Tuesdays. And they can't even use pocket calculators on Tuesdays, since they didn't exist in 1955. This forces the students to practice their math skills without using a calculator.
Wednesdays would be "Tutorial 1973." This is around the time that simple and scientific calculators would begin to be sold. Hence these calculators would be allowed on Wednesdays, but laptops and cell phones are still forbidden.
Thursdays would be "Tutorial 1991," to continue the pattern. Graphing calculators were first sold around this time. The forerunners of cell phones also appeared -- you know, those huge things that don't fit in our pockets. If a student actually has a 1991-style phone, he/she can use it on Thursdays.
And Fridays would be "Tutorial 2020" (or whatever the current year is). In this case, a student can use a cell phone, even if it's for entertainment -- I don't mind giving the students one day of entertainment if they've been working the rest of the week. The only thing I check during the week is what level of technology is being used, not what it's being used for.
This plan mainly describes tutorial, but it's possible to extend it to the main math class. It's also possible to write the lessons and assignments so that they're geared towards the technology that's permissible each day.
For example, suppose the Geometry lesson for one week is on the Triangle-Sum Theorem. On 1955 day, all the angles in the triangles should be multiples of ten. There's no excuse for students to be unable to add and subtract multiples of ten up to 180 without a calculator. On 1973 and 1991 days, I can include problems with angles that aren't multiples of ten (including decimals). Of course, technically students should be able to add non-multiples of ten and decimals without a calculator, but this is strongly the case for tens, so I enforce it then.
Quizzes and tests can also be labeled 1955, 1973, 1991, etc., to indicate how much technology can be used on the assessment. The one thing that throws a wrench into this is if I'm in a classroom that depends on Chromebooks. For Chromebooks didn't appear until about the first iPhone, and so there's no year I can name that allows students to use Chromebooks but not cell phones in class. Well, I'll cross that bridge when I come to it.
Once again, this is something for me to do as a regular teacher, not as a sub. If I sub next week at a school with tutorial, no one will know what I mean if I say "It's Tutorial 1955 today!"
But I did briefly mention this idea in class a few times. One day I subbed at a tutorial school, saw many students on phones, and told the students that they should work like the generation of 1951. On another day, a middle school special ed English class was watching a video on Anne Frank, and the aide and I suddenly came up with telling the students that they didn't have cell phones in 1942 (the year the diary was written). Some students asked whether phones exists in 1942 -- they did, but of course they were landlines only back then.
Anyway, the sixth resolution demands that I use 1955 as the year of heroes, not 1951 or 1942. And this will indeed be the focus year if I get my own classroom during the 2020's.
New Decade's Resolutions: Music
Ever since the earliest years of the old decade, I envisioned a classroom where I would regularly sing songs to help the students learn. It was in 2012 when I discovered that many old songs from my favorite TV show Square One TV were posted to YouTube, and so it's likely that around then, I imagined what it would be like to sing some of those old songs in a math class at school.
But except for a few instances of "Quadratic Weasel" in Algebra I classes, I didn't really make music a regular feature of my classes until I was hired at the old charter school. I had the first five songs ready to start the year at my new job. "The Dren Song" was based on an old tune that I had created but never added lyrics until then. "Count on It" was the first Square One TV show that I performed in class, due to its message on the importance of learning math. "The Benchmark Test Song" is a simple tune that I randomly generated using a pentatonic scale. "Fraction Fever" was based on a tune from an old computer game. (It's the same system as Mocha, but it's not playable on that emulator.) And "Need for Speed" (the mousetrap car song) was also based on a random song in a minor key.
One of my most successful uses of singing in class was "Measures of Center." This song, set to the tune of "Row, Row, Row Your Boat," was actually suggested by a new sixth grader who had heard it from the previous teacher at her old school. During a seventh grade computer lesson, the coding teacher asked the students to calculate mean, median, and mode on the computer. Many students couldn't remember what any of those were -- until I started singing the song again. And it definitely helped jog their memories.
Decade Resolution #7: We sing to help us learn math.
Some readers might point out that this seventh resolution apparently contradicts earlier resolutions about doing what the heroes of 1955 did. After all, singing is entertainment, and thus it reinforces the idea that class time should be for entertainment. I seriously doubt that Gates, Jobs, or Berners-Lee -- or anyone else in their generation -- had math teachers who needed to sing for them. Didn't I just complain about the "Triggy" video and the need to have songs in class? (Once again, the 1955 generation had much more tolerance for boredom than the current generation does!)
But in this case, I believe that the seventh resolution reinforces the fourth resolution on enlarging the size of the bicycle. In many cases, it's easier to remember a song than spoken prose. Thus my songs help students remember mathematical laws and possibly reduce common errors. (And these songs are more relevant than the ones the students would try to listen to on their phones.)
If it increases student learning, then I might even sing "Triggy" in class someday, perhaps for a Geometry class when we reach the trig chapter (equivalent to Chapter 14 of the U of Chicago text) and it's time for the students to learn SOH-CAH-TOA. But I keep it balanced and remind the students that not everything about math needs to be "fun."
(Note: I mentioned "Triggy" on the blog when I first heard the song -- it was for a class I subbed for about three months before I started at the old charter school.)
My only lament about singing at the old charter school was that I didn't do it enough. The day I sang in seventh grade coding was one of the few times that I'd sung on a Monday, and I also seldom sang on Wednesdays either. My thought was that on days when they really didn't have "math class" (or "STEM class"), they didn't need any music.
This might have made sense on middle school Wednesdays in my new district -- the entire day is shorter as is each period. But at the charter school, while school was also out early on Wednesdays, I actually saw my eighth graders for more minutes that day (and seventh grade not at all). Sometimes the eighth graders would grow restless, and songs might have broken up the monotony.
Also, many of my songs were enjoyable for the students. My eighth graders seemed to like "Count on It," while "Fraction Fever" was a seventh grade favorite. But I never had time to memorize all the lyrics -- I would write them on a poster, and then put the poster away when it was time for me to sing a different song. The students might have enjoyed it -- and perhaps behaved a little better -- if I could have sung their favorites again on demand. After I left the old charter school, I started writing the lyrics in a small notebook, but it was too late to help my charter students.
When I returned to subbing, my original plan wasn't to keep singing in class. But this suddenly changed within the last ten months of the decade -- and now, suddenly I'm expecting myself to sing almost everyday that I sub.
It all started back on Pi Day. I found myself to my delight in an eighth grade math class -- one I'd spent several days in earlier that year. As it was one of the few times that I've ever found myself in a math class on Pi Day, I couldn't resist singing songs for them. I began with "Oh Number Pi" and then added some more festive songs from YouTube.
From that day forward, singing quickly spread. I would return to that same school, and a student would ask whether I had a song to sing or not. Then at other schools, I found I could help those kids out by singing some of those same songs, and so on. One day, I subbed in an English class that had a parody of Meghan Trainor's "All About That Bass" as part of the lesson. So of course I couldn't resist making a math parody of that song, "All About That Base and Height." And so it continued.
As a sub, I've found that singing can help the students to remember math. In addition to "All About That Base and Height," one particularly nice song to sing is "Solving Equations." Even though I'd originally written it for eighth grade, it also fits seventh grade and Algebra I.
When I sub for non-math classes, I've found that singing song can be a powerful incentive for students to control their behavior. This leads to the next resolution:
Decade Resolution #8: We sing to help us remember procedures.
In many cases, the song serves only as an incentive to work. I've lately used Square One TV's "One Billion Is Big" for this reason. But other songs help students remember classroom procedures. The main song of this type is "The Packet Rap" -- which was originally "The Packet Song" at the old charter school, but I've since then decided that it sounds better as a rap.
I've found that singing songs definitely reduces arguments. When I get the urge to yell at a student, the best thing for me to do is start singing a song instead.
And so here are some of my plans for songs moving forward in the new decade. Each morning, I decide whether I'm going to sing or not, and which song. If it's a math class, then I might choose a song related to the lesson. Lately, I've saved a few songs from the old charter for science -- "Meet Me in Pomona" for life science, "Earth, Moon, and Sun" for physical science. Any of my other songs can work in a non-STEM class.
If the class isn't math -- or if it's math but the song isn't directly related to the lesson (for example, if it's an Algebra I lesson on factoring, I have no factoring song) -- then the song is an incentive. I enter the class, take attendance, and then sing the first verse of the song. Then I tell the students that they need to earn the rest of song, and what they need to accomplish to earn it.
If it's a math song directly related to the lesson, then I do examples on the board and sing the song for each example I do on the board. This also serves as a mini-incentive -- if the students wish to hear more of the song, then they should be quiet so we can get through more examples.
My rule of thumb is that students need a music incentive if they're too immature to work hard for subs without it. I like singing for middle school and younger high school students (say up to sophomore year, or Geometry if it's math) and expect older students to work hard on their own without a song.
Special ed classes are tricky. If there is an aide, or if it's co-teaching, I expect the other adult to run the class, and I dare not interrupt that other teacher with a song. Sometimes, there may be one or two periods that day when I'm teaching solo, and thus the song returns for that period only. It's also likely that I'll need to sing even for older grades in a special class, since they're often not motivated to work.
The eighth resolution tells us that singing can help with procedures. As a sub, a common procedure the students need to follow is putting away Chromebooks. Often, only the last period needs to put the laptops away. Thus I might sing an incentive song for that last period only -- or if it's a middle school class where I'm already singing all day, change it the last period so that the incentive is for putting Chromebooks away rather than for completing their work.
Classes that aren't representative of what I'd like to teach someday don't need songs. This includes P.E. and self-contained special ed classes. And of course, if the class is only watching a video or taking a test, then a song might take time away from it, so I don't sing.
It often happens that I enter a classroom deciding not to sing a song, perhaps because I fear that the song might take time away from the lesson. In general, it's best to sing anyway. I'd rather set up an incentive and get the students to work to earn that incentive, even if it takes time away from that work, then give them the whole period and end up with no one doing the work at all (with talking or playing with technology, etc., instead).
I also mentioned in the past that repeating the importance of earning A's over and over was a huge turn-off at the old charter school. Instead of talking about the importance of earning A's, it's possible for me to sing about it instead. In fact, two of the parody songs that I include in my songbook mention grades -- "All About That Base and Height" and "x's and y's." (In both cases, someone else first came up with the parodies and posted these lines on YouTube -- I modified some of the lyrics to create my own parodies but retain the mention of grades.)
Some of the songs I wrote at the old charter school -- "Ratios" and "Another Ratio Song," for example -- are recorded in my songbook (and on the blog), but I either don't like or don't even remember the tunes I made for them. This is where Mocha comes in -- I can use Mocha to add tunes to these songs and then reintroduce them while subbing.
It's also possible to create songs to remind the students about rules and procedures, especially if they get tired of hearing them over and over. My song "Show Me the Numbers" at the old charter school was originally such a song, but then it fell apart as it referred to an old participation points system ("demerits") that I stopped using soon after I wrote it.
Many teachers have a strong teacher tone, and they manage effectively because of it. I don't have a strong teacher tone -- instead I have a strong musical tone. I need to take advantage of what I have.
New Decade's Resolutions: Time
One thing that I've found to annoy me more than many other things in the classroom is when students find excuses not to be in the classroom. This includes students who regularly arrive late, sneak out of class early, or ask for restroom passes all the time.
Sometimes a student might have tried avoiding missing class time and failed. But a student who asks within seconds of the tardy bell ringing after snack or lunch clearly didn't try at all. And on some days that I've subbed, it appears that more students ask to go during the periods right after snack and lunch than the other periods (when a restroom visit might be more justifiable).
When I was a young student, I went three whole years (Grades 10-12) without asking a teacher for a restroom pass even once. Yet I believe that I actually visited the restroom much more than the average student did. I often went twice during lunch, and twice during P.E. (in the locker room, once when getting dressed and once when returning to street close), and yet never needed a restroom pass at all.
But of course, I've seen that telling stories about myself doesn't convince anyone at all. (That is, responding to the student complaint "It's impossible to go the whole day without a restroom pass!" with "I myself used to go years without needing a pass!" doesn't change any student's minds.) So instead, I have the following New Decade's Resolution:
Decade Resolution #9: We attend every single second of class.
I believe that almost any student can go days without needing a restroom pass if they really want to -- the problem is that many students don't. To them, class time is unpleasant, and so they want to find any excuse to get out of class. Far from striving to use the restroom twice during lunch in order to avoid missing any class time, they wish to go during class even when they don't want to in order to shorten class time a little.
This resolution is a bit difficult to enforce as a sub, since I enter many classrooms not knowing what the specific restroom policy is. But I include this in my decade resolutions because I hope to return as a regular teacher at some point during the decade. And so here are a few ideas I have for trying to enforce this resolution.
First, I can rework that failed participation points system so that it's based more on attendance. A student simply earns points for attending every single second of class. If a student misses even one second of class (arriving late, leaving before the bell, or with a restroom pass), then the point isn't earned. Of course, there might be exceptions for situations that are beyond a student's control (such as being summoned to the office).
I can also come up with a stricter punishment for the most annoying habit of asking for passes right after break. Suppose a student has a half-hour lunch at noon, then returns to class at 12:30 asking for a pass. If the student leaves, then upon returning, that student must write standards:
At 12:00, I could have used the restroom, but I didn't.
At 12:01, I could have used the restroom, but I didn't.
At 12:02, I could have used the restroom, but I didn't.
...
At 12:29, I could have used the restroom, but I didn't.
In some situations, I might give the student a warning by asking, "What could you have done today in order to miss zero seconds of class?" Then it's standards the next time a pass is needed.
There's one excuse I've heard often for needing passes right after lunch -- "I had detention for all of lunch, so I couldn't go." There's no way to verify if it's true -- and even so, it's not a valid excuse. If you need to use the restroom at lunch, then behave well in all your classes, and never let any of your grades drop below a C. Then you'll never get lunch detention! It's not my job as a teacher to accommodate your detention -- it's your job to become a good student who never gets detention.
Of course, this is another rule that I can't really enforce as a sub. It's expected that students will take advantage of subs by shortening class a little.
And there's one more time-related resolution that I wish to introduce here. Unfortunately, it's another resolution that a sub can't enforce, but it's an important one. It refers to those students who rush through their assignments just so they can say "I'm done with everything. Can I use the extra time to play on my Chromebook or phone?"
For these students, I have the following resolution:
Decade Resolution #10: We are not truly done until we have achieved excellence.
Some students enter class expecting at least a third of the period to be "extra" time for playing on computers or phones. Of course, members of the 1955 generation didn't expect so much time for entertainment in every single class. And it's really bad when they rush through the assignments to get that free time that they believe they're entitled to.
As far as I'm concerned, a student isn't truly done until his or her grade in the class is an "A." If the student is getting anything less than an "A," then the extra time should be reviewing the material or studying in order to achieve the "A" on the next assessment. Students who never reach an "A" in math class should never have free time for entertainment during class time.
So the way to enforce this resolution is to avoid letting students use phones at the end of class unless their grades are high. I might consider having "Technology 2020" time very seldom -- such as right after they finish a test or quiz. (They want to do well on the quiz/test, so most won't just leave the test blank just to get free time, especially when compared to classwork/homework that many of them don't care about.)
If I ever have "Dren Quizzes" today, then such quizzes count for "Technology 2020" time, provided that the grade earned on the quiz is an "A." I wouldn't want to grade regular quizzes and tests quickly for the Technology 2020 incentive, but Dren Quizzes should be easy to grade.
I want the current generation to know that they aren't done until they excel. I want the current generation to produce the next great cohort of inventors, so that technology in 2055 and beyond can surpass anything (or anywhere, as in outer space) that we can imagine today.
With the Eleven Calendar still fresh on my mind, note there are two ways that I'll actually use my new calendar. One is for food -- lately I've been buying one of my favorite foods, buffalo wings, on Fridays in the Eleven Calendar. Usually these are cheap wings at 7-Eleven, but when the weekends (Friday-Sunday) line up exactly in both calendars, I purchase a different brand. That includes this weekend -- January 3rd-5th Gregorian, December 12th-14th Eleven Calendar. I observed this special day today (Saturday in both calendars), when I ate wings at Fuddrucker's instead.
The other use that I'll start this year is, when I sub at a school, my focus resolution for the day, the one I work on the most that day, will be determined by the Eleven days of the week. So on Fourdays I look at the fourth resolution, the seventh on Sevendays, the tenth on Tendays. Friday-Sunday will be for the first three resolutions respectively. On Elevendays and blank days, I can focus on all of the New Decade's Resolutions as a whole.
Loose Ends with Traditionalists
My New Decade's Resolutions are intended to take a balanced approach to teaching -- balanced between the traditionalist and modern approaches, that is. As I mentioned above, the idea of singing in class is quite modern, as is of course any form of technology or projects (including mention of these in my old New Year's Resolutions).
But this is also balanced out with some traditionalist-leaning resolutions. The two resolutions that mention 1955 are clearly traditionalist. After all, the generation born in 1955 and graduated in 1973 grew up during the Golden Age of textbooks, when Dolciani and similar authors prevailed. Thus on occasions when I focus on these resolutions, my lessons will lean towards traditionalism.
Our main traditionalists have indeed been active during the past week. It's interesting to compare and contrast what the traditionalists wrote this week about the best teaching methods to my goals as stated in my resolutions.
The head traditionalist, Barry Garelick, posted three times this week. One of his posts has drawn ten comments so we'll focus on that one, but let's glance at the other two posts first. Back on Monday (New Year's Adam), Garelick wrote the following:
https://traditionalmath.wordpress.com/2019/12/30/bad-pd-517/
At a six-hour PD I had the misfortune of having to attend, the moderator put this slide on the screen in a defense against the call for evidence that certain teaching practices are effective. It was a slide from a presentation by David Theriault, who teaches English and has a blog
For this post, I wish to emphasize only the comment by a major traditionalist, Ze'ev Wurman:
Ze'ev Wurman:
Actually, bloodletting worked for me! I don’t need any research to show me right!
Wurman is clearly being sarcastic here. He is also trying to make an analogy here -- "bloodletting" corresponds to reformist, non-traditionalist math, while "modern medicine" corresponds to traditionalist math. To Wurman, only traditional math -- like modern medicine -- has the research to back it up.
Research is important to me. I did mention "we follow the research" earlier in this post -- and not surprisingly, I mentioned it in connection with the 1955 (that is, the traditionalist) resolutions.
I agree with the research showing that traditionalist math instruction is the most effective -- provided, of course, that the students actually do the traditionalist assignments. No worksheet, no matter how traditionalist it is, can teach anything to a student who leaves it blank. And I do have the research to show this -- namely the number of worksheets that students leave blank in my classes, including ones I see tossed on the floor after school.
Therefore my resolutions provide for songs and activities that the students will actually do. As I wrote earlier, the resolutions provide for some days to be "Technology 1955" days when traditional lessons are taught, and others to be "Technology 2020" days when modern lessons are taught. On the other hand, the traditionalists rarely compromise -- to them, only traditional lessons are effective, and just ignore the existence of students who leave those traditional worksheets blank.
Here's the other post to glance at. Garelick just barely posted this one today, and thus there aren't many commenters yet. Some readers might comment here in the next few days:
https://traditionalmath.wordpress.com/2020/01/04/pbl-a-guide-to-the-hype/
Edutopia has advice for those math teachers who believe that Problem (Project) Based Learning (PBL) actually has something of value to offer.
Project Based Learning was a cornerstone of the Illinois State curriculum that we used back at the old charter school. Of course, I already knew that there's not much in that curriculum that traditionalists would like.
OK, let's get to our main Garelick post now. This he posted the day before yesterday:
https://traditionalmath.wordpress.com/2020/01/02/the-prevailing-caricature-of-traditional-math-dept/
In a recent article about math education that ballyhoos the “latest approach” in how to teach math, this statement was made
“If the teachers are telling students how to solve a problem, and then that problem isn’t exactly what’s on the test, it creates this disequilibrium for a student,” said Beverly Velloff, the math and science curriculum coordinator for the University City School District.
Let's look at the comments here, starting with pkadams:
pkadams:
Your articles provide more and more evidence that , for whatever reason, some people do NOT want children to learn, math or other subjects. “New Math’ has been a failure since it came out! Just like teaching reading without phonics is a failure. And not teaching actual history is producing little Socialist/Fascists. Call it conspiracy theory or stupidity, but it is the only logical conclusion.
In her comment, pkadams refers to two other subjects -- ELA and history. (Here I use feminine pronouns to refer to pkadams because the selfie avatar next to the username appears female.) I'll omit a discussion of history and politics here (even though traditionalist discussions regularly lead to political arguments).
But her mention of phonics is quite interesting, since it often comes up in math discussions. I've reached a conclusion about phonics and traditional math, but this conclusion is likely not the one the traditionalists want me to reach.
Phonics, after all, is all about breaking up words into their individual sounds. Pro-phonics advocates believe that this is method of learning to read is superior -- when a student sees an unfamiliar word, it's better for that student to "sound out" the word by breaking it down, as opposed to the "whole language" approach of learning whole words at a time.
Anyway, to me, learning how to add 6 + 8 via making tens is akin to learning to read via phonics. In fact, we see how both involve breaking things down -- "cat" into c, a, t, and 6 + 8 into 6 + 4 + 4. On the other hand, foregoing breaking sums down and simply learning the whole table at once is just like whole language.
Thus if a traditionalist is pro-phonics, as pkadams appears to be, then she should also be in favor of making tens and other methods of breaking down sums and products. But somehow, I doubt that she or many other traditionalists will agree with this.
Chester Draws:
The reporter is an idiot, since memorising “equations” would be worthless. I think he means either formulas or algorithms. That he cannot get even that right makes me suspect that the reporter has mischaracterised the whole approach. I wouldn’t trust him to report anything — the Maths314 might be entirely traditional for all he knows.
It helps that in my jurisdiction the students are provided formula sheets. All the US has to do is provide a formula sheet at the start of each test if they want to limit memorisation.
Ironically, the U of Chicago Geometry text (which wouldn't typically be considered traditional, though this is mostly due to the U of Chicago elementary texts) outright encourages students to memorize formulas. Lesson 10-6 (in the chapter on surface area and volume) is simply titled, "Remembering Formulas."
This is a tricky one, since some teachers allow students to use notes on quizzes or tests, or provide formula sheets similar to the ones that Draws provides to his students.
If we compare this to my resolutions, we clearly need to look at the "bicycle" resolutions. Should formulas be included as part of the "bicycle" of math to memorize, or not? To me, let's just look at a test that some traditionalists still hold in high regard -- the SAT:
https://blog.prepscholar.com/critical-sat-math-formulas-you-must-know
According to this list, the SAT provides most formulas needed for Geometry. Thus if I were to become a Geometry teacher, I can safely take the Draws approach of providing the formulas.
On the other hand, many formulas of Algebra I and II, such as Slope Formula, Quadratic Formula, and definitions of trig functions, are listed under "need to know." Thus these formulas should be included as part of the "bicycle." We can use traditional methods to get students to memorize these, or we can use non-traditional methods such as songs (including Slope Dude, Quadratic Weasel, and yes, even Triggy).
Stephanie Sawyer:
In this comment as well as her response to one of the other Garelick posts, Sawyer tells us that even her Catholic school is using non-traditional methods (even though many private schools are not subject to the Common Core Standards). She writes about how the school she attends as a young girl taught the standard algorithm for addition ("carrying") as well as adding money in first grade, since it helped prepare her for subsequent grades. The Common Core Standards, of course, teaches this in later grades.
Barry Garelick responds:
Absolutely right. When I teach Math 7, I’m getting them prepared for algebra: either Math 8 algebra, or Algebra 1 in 8th grade. For Math 8, I teach more algebra than is currently in the CC curriculum so that when they take Algebra 1 in 9th grade, they are familiar with things like factoring, algebraic fractions and various types of word problems.
This sounds a bit advanced for Common Core Math 8. Indeed, algebraic fractions hardly even appear in Algebra I any more (saving it for Algebra II) -- and even when I student taught Algebra I using a "proper Glencoe" text, the chapter on algebraic fractions was skipped. Meanwhile, factoring still appears in Algebra I, but do we really need to prepare students for it the year before?
Many of my eighth graders struggled in math the year I taught at the old charter school. Imagine if I'd tried to teach them Garelick's Math 8 curriculum. (Indeed, many of these topics are likely to lead to eighth graders asking, "Why do need to learn this?" Garelick explicitly tells us his answer -- to prepare them for freshman Algebra I. I wonder whether students will accept this answer.)
Of course, I save our main commenter for last:
SteveH:
I no longer have any hope that these educators will understand the realities of mathematical education when they see right before their eyes the success of AP/IB math curricula in their own high schools. They don’t even seem to want to look at individual success cases to see how they can duplicate them. They see that success in a traditional algebra course in 8th grade is a key to success, but have no interest in digging deeper to see how that happens. It’s all guess (what they hope for) and check (badly and with low expectation statistics).
We already know what SteveH would prefer the levels "proficiency" or "distinguished" to correspond to -- something like a 99% likelihood of passing College Algebra, and/or a 75% chance of passing a Calculus course (preferably AP/IB in high school, since he mentioned these again).
SteveH:
Colleges and admissions competition drive reality down through high schools via AP and IB curricula. That’s when students and parents really begin to pay attention and little of their pedagogical silliness sticks, or more likely, is ignored. Seventh and eighth are the transition battleground grades where reality begins to sink in for many students and parents. Some recover, but many don’t. Clearly, the low CC slope to no remediation for college algebra has to change to a proper STEM AP/IB slope, and that has to start in 7th grade. Everyone knows this, but many CC lovers never seem to want to make that transition clear. They desperately try to make lower expectations sound like better math understanding and results.
The only reason our K-8 schools got rid of CMP and replaced it with proper Glencoe textbooks was because students and parents complained that CMP did not do a proper curriculum job of preparing students for Geometry as a freshman. However, that just pushed the fairlyland/reality wall back to the end of 6th grade. Now, everything is based on the CC, but the school website does talk about an accelerated CC 7th and 8th grade coverage in 7th grade to prepare for Algebra I in 8th grade. However, the site says nothing about what textbooks, if any, they use or how formal the algebra class is.
Of course, what did I just say about "proper Glencoe" texts earlier? Some chapters -- specifically the one on algebraic fractions in Algebra I -- are usually not reached by most teachers.
SteveH also mentions an accelerated path for Grades 7-8. Many districts have similar paths, including the one where I work and receive the most calls. I have no problem with offering such a path, since after all, AP Calc exists and some students might be able to take it.
Speaking of Calculus, today I begin watching the Great Courses video that I mentioned back in my October 15th post. It is called Change and Motion: Calculus Made Clear. Our lecturer is Professor Michael Starbird, from the U of Texas at Austin. I will create a new label, "Michael Starbird," as we begin to discuss these lectures. Part I consists of Lectures 1-12.
Lecture 1 of Michael Starbird's Change and Motion is called "Two Ideas, Vast Implications." Here is an outline of this lecture:
I. Calculus is all around us.
II. Calculus is an idea of enormous importance and historical impact.
III. Calculus comes from everyday observations.
IV. Calculus deals with change and motion and allows us to view our world as dynamic rather than just static. Calculus provides a tool for measuring change whether it is change in position, change in temperature, or change in demand.
V. Zeno's paradoxes of motion confront us with questions about change. Calculus helps to resolve these ancient conundrums.
VI. The history of calculus spans two and a half millennia.
VII. Mathematical relationships between dependent quantities are described by functions.
VIII. The power of calculus lies in its abstraction, that is, that the insights and methods that arise from investigating physical problems also apply to many other settings.
IX. Calculus is the study of two fundamental ideas (called the derivative and the integral), which we will describe and define simply and understandably in Lectures Two and Three.
X. Here is an overview of the lectures [omitted here -- dw].
Starbird, like Fawn Nguyen and many other teachers, mentions how his fellow travelers cowered in fear upon learning that he is a math teacher. He jokes that many people find Calculus hard because the word comes from a Greek word meaning "stone" -- as in the pebbles used in an abacus.
The professor quotes Winston Churchill, who mastered advanced mathematics in just six months. The English statesman compared differential calculus to a fiery dragon. But he also reminds us that we learn by teaching -- students don't truly understand Calculus until they have to teach it.
I can also tie Starbird's lecture back to my New Decade's Resolutions. The professor mentions that nearly all technological developments of the last 300 years can be traced back to Calculus. Thus if math classes were to disappear, we wouldn't merely go back to 1955 technology -- we'd revert all the back to 1700.
Still, the resolutions mention the 1955 generation as an answer to "When will we use this in real life?" as the heroes used Calculus to build the technology that students use and love. Calculus is considered the tail end of second decade math -- the main purpose of most of it is to prepare students for future math courses (culminating in the third decade).
Loose Ends with Music: The Whodunnit Song
The seventh and eighth resolutions demand that I include music more often in class. My newest song is the "Whodunnit" song, which I introduced last month in a multi-day subbing assignment. In this post, I will describe this song in more detail, including how to play it in Mocha.
(By the way, I was reading some old blog posts and realized that last month was not the first time I saw the Whodunnit activity in a class I subbed for. I'd actually seen the Whodunnit clues posted on the wall about one year earlier, but we didn't play it on the day I subbed there.)
Let's me start with how I generated the song. I decided to use the 12EDL Mocha generator (that I posted here back on Halloween 2018). We know that 12EDL is very minor sounding, which fits into the murder mystery theme.
At first, I tried the generator that adds quick sixteenth notes. This is great when I already had lyrics with many syllables and need a fast song to fit the lyrics too. But since I was creating the lyrics from scratch as well, I feared that I couldn't invent enough lyrics to fit such a fast song.
As it turned out, the second line generated by Mocha didn't include any sixteenth notes. Instead, it alternated eighth and dotted quarter notes. This produces a simple rhythm, to which I easily came up with lyrics overnight.
The song I created repeats this line three times. The fourth line produced by the generator is different, but it also includes eighth and dotted quarter notes, so it fits the rest of the song.
Let me remind you of the 12EDL scale:
Degree Note
12 white A
11 lavender B
10 green C
9 white D
8 white E
7 red F#
6 white A
Recall that the "lavender B" is about halfway between white Bb and white B. In the 12EDL scale I usually call it "lu B," but sometimes I call it "lu Bb" when I feel this name is more convenient.
Here are the notes for the song:
First Three Lines:
E-C-E-B-E-E-F#-B-A12
Last Line:
D-E-A-D-A12
Originally, Mocha chose Degree 6 (high A) for the last note of the first three lines. But I replaced it with Degree 12 (low A) because it sounds better after Degree 11 (lu B). Remember the first rule of music is that it should sound right. I have the right to change the random notes generated by Mocha in order to make the song sound better.
The fourth note in the last line was originally Degree 7 (ru F#). But I couldn't resist changing it to Degree 9 (wa D) in order to make the riff D-E-A-D appear in this murder mystery song. Of course, we could keep D-E-A-F# if the culprit had merely taken the victim's hearing. (This isn't the first time that I intentionally included notes in a song that spelled something -- at the old charter school, I wrote a GCF song for the sixth grades that included the notes G-C-F.) This line sounds better if we use Degree 6 for the first A and Degree 12 for the second.
Here are the lyrics for two verses that I wrote. (They come from the first two clues last month.)
First Verse:
Was Mrs. fixing snacks in kitchen?
Was Dr. hearing songs in atrium?
Professor's iPad by the pool now?
Did Miss kill the victim?
Second Verse:
A cougar's mauling was the method?
Or had a green snake's bit begun it?
Or object's falling was the weapon?
Just tell me whodunnit!
To make the rhythm fit the notes, we start with an eighth note and then switch to a dotted quarter note per syllable. So the first line is (bold = longer note): Was Miss - us fix - ing snacks in kitch -en? In the second line: Was Doc - tor hear - ing songs in a - tr'um? (Say "atrium" with only two syllables in order to make it fit.)
But then each line adds up to 8 1/2 beats, or two bars in 4/4 plus an extra eighth note. Originally, the randomizer made one of the dotted quarters an ordinary quarter so that it adds to to eight beats, but it was always easier for me to keep the rhythm consistent with dotted quarters throughout.
Thus I decided to make the the first eighth note a pickup note -- an opening note that occurs before the first bar. Then the other eight beats form two full bars. But then the second line of lyrics also starts with this same eighth note pickup. In class last month, I decided to put a 3 1/2-beat rest between the first two lines. I like this -- the silence makes the song eerier (as befitting a murder mystery song).
The third line is just like the first two. The fourth verse starts with a dotted quarter on Did, and so there is no 3 1/2-beat rest before it. The last note is a double whole note (length 32 in Mocha) -- we can devote a whole note to each syllable vic- and -tim. Then there's one more 3 1/2-beat rest before the second verse begins.
Here's what it looks like in Mocha:
https://www.haplessgenius.com/mocha/
10 N=16
20 FOR V=1 TO 2
30 FOR X=1 TO 3
40 IF X=3 THEN Y=14 ELSE Y=9
50 FOR Z=1 TO Y
60 READ A,T
70 SOUND 261-N*A,T
80 NEXT Z
90 FOR R=1 TO 1400
100 NEXT R
110 RESTORE
120 NEXT X,V
130 DATA 8,2,10,6,8,2,11,6,8,2
140 DATA 8,6,7,2,11,6,12,2
150 DATA 9,6,8,2,6,6,9,2,6,32
As usual, don't forget to click on the Sound box before RUN-ning the program.
Last month, I also mentioned attempting to add an accompaniment (or harmony) to this song by opening a second window and running Mocha in it. This is tricky -- we'll have to time it perfectly so that the two parts line up exactly, or they won't sound good at all. We'll have to know when exactly to type in RUN in each of the two window.
"Whodunnit?" is a tricky song for our first attempt at Mocha harmony. That pickup eighth note is likely to throw our timing off. (The rests might be problematic at well. It might be nice to have the accompaniment playing notes while the main voice is resting, but this is tricky. We estimated the rest of 3 1/2 beats with an empty FOR loop of length 1400 in Line 90. This is a good estimate, but we need to be exact if we want to have an accompaniment. Of course, this can be avoided if we simply have the accompaniment rest with a 1400 FOR-loop as well.)
Let's try an easier song for our first crack at accompaniment -- "Row, Row, Row Your Boat," which, as I mentioned above, can serve as "Measures of Center" (as well as "Same Sign, Add and Keep" for integer addition) in a math class.
This song is written in 6/8 time, with the eighth note as length 2 and a set of three eighth notes, a dotted quarter note, as length 6. Since the original song in in a major key, we'll use 18EDL:
Degree Note
18 white D
16 white E
14 red F#
13 thu G
12 white A
9 white D
NEW
10 N=4
20 FOR V=1 TO 2
30 FOR Z=1 TO 27
40 READ A,T
50 SOUND 261-N*A,T
60 NEXT Z
70 RESTORE
80 NEXT V
90 END
100 DATA 18,6,18,6,18,4,16,2,14,6
110 DATA 14,4,16,2,14,4,13,2,12,12
120 DATA 9,2,9,2,9,2,12,2,12,2,12,2
130 DATA 14,2,14,2,14,2,18,2,18,2,18,2
140 DATA 12,4,13,2,14,4,16,2,18,12
Now let's try to give this song an accompaniment. A simple one that fits this song is a running bass line that alternates between white D and white A, each a dotted quarter note in length. Indeed, this is so easy that we won't even use DATA lines for this song. Since these will be bass notes, we drop an octave to Degree 36 (for D) and Degree 24 (for A) in our accompaniment.
We start by opening up a second window and launching Mocha. A second tab isn't sufficient, since the browser (at least if it's Microsoft Edge) will block the song from the other tab. In this second tab, we type in the following code:
10 N=4
20 FOR X=1 TO 18
30 SOUND 261-N*36,6
40 SOUND 261-N*24,6
50 NEXT X
Make sure that Sound is clicked on in both windows. Now for the tricky part -- the timing. We need to enter RUN in both windows at the right time. Indeed, we should have the three letters RUN typed already in both windows, so all we'll need to do is press a single key (ENTER) at the proper time.
The song contains eight bars, and since it plays twice, we need 16 bars in the bass. But the FOR loop in Line 20 goes up to 18. That's because the two extra bars are there for timing. We start by hitting ENTER key (after the word RUN) in the window with the accompaniment, and then while the first two bars are playing, we switch to the window with the melody. Just as the third bar is about to begin, we hit ENTER in the second window and then RUN it. (And if we miss our cue, it will be harmless if we wait until the fourth or fifth bar, since all bars are identical in this simple walking bass.)
I tried it a few times, and the timing was tricky! It's worth trying this a few more times before attempting a more complex accompaniment.
By the way, in both lines, N must be the same number. Here N must be in the 1-7 range -- sadly, we can't quite reach the low D needed for N=8. I'm more likely to sing along with N=5 (key of green Bb) or N=7 (key of red E).
Conclusion
Recently I found some old papers -- something that I'd written twenty years ago. It was, in fact, a list of New Millennium Resolutions that I had written on January 1st-4th, 2000.
As a young boy, I'd always fantasized about building a time machine and traveling to the future, the new century. Because I didn't know at the time about the nonexistence of the year zero, 2000 was always my target year. Back then, I'd believed that robots would take over in the year 2000 -- and that these robots would try to destroy all humans.
And so on January 1st, 2000, I couldn't help but want to retell the story of my old game. But in addition, I also added New Millennium's Resolutions to the document. Let me reproduce them here:
A. Each day when I am in contact with fellow students, I will try to begin a conversation. In order to fulfill this resolution, I will wait for someone to ask me a question, such as "What are your classes like?" Then I will "perpetuate" the conversation by attempting to give a full, detailed answer.
B. If the first resolution does not lead to a conversation, then I will "initiate" a conversation. The best way to do so is to repeat a commonly asked question, such as "What are your classes like?" to another student distant from the one who originally posed the question.
C. I will maintain these secret files each day by writing a long story of my game. By doing so, I finally reveal to myself and others the reasons for my silent personality.
You might ask, did I ever keep any of these resolutions? Well, I didn't literally keep Resolution C, since I only found four days' worth of "secret files" (and I doubt that I wrote on January 5th). On the other hand, I do remember creating blogs -- and indeed, we can even count this very blog as an extension of Resolution C.
On the other hand, I must count Resolutions A-B as abject failures. In the year 2000 I was in my first year as UCLA, which is why I kept mentioning "What are your classes like?" as an example of small talk (similar to "How's the weather?"). At the time, I wasn't quite sure whether I would become a teacher, where "What are your classes like?" suddenly has a different meaning. (Now it would be something to say in the teacher's lounge, asking another teacher what his/her classes are like. This sort of small talk occurs all the time in lounges!)
Back at the old charter school, I had trouble coming up with science lessons. I once confessed to my eighth graders that I didn't know what to teach them, and one girl simply replied, "Why don't you ask any of your teacher friends?"
Notice that if I had kept Resolutions A-B, I would have found a teacher to discuss science classes with, and thus I would have had a better science class. And as I mentioned on the blog, I failed even to communicate with other teachers online by creating a Twitter account, which I should have done.
On those first four days of the 2000's, I also wrote some basic assumptions about myself. And as I wrote back in my September 6th post (the day we covered Lesson 1-7), even back then I referred to those assumptions as "postulates":
Postulate 1. Organization is the key to success.
Postulate 2. When possible, one must travel with another.
Postulate 3. Persons of wisdom deserve obedience.
Postulate 4. Not all guns are fake.
The fourth postulate referred to an incident from my story -- the original game of time machines, which I played in the first grade, had evolved into a game where kids pretended to have guns. The fourth postulate takes into account the Columbine incident (which had occurred about 8 1/2 months before I wrote it) -- and indeed, it's exactly because of Columbine and subsequent school violence that I don't discuss it fully on the blog.
The important thing was that I was suspended from school -- and I told my eighth graders at my old charter school a version of this incident on the anniversary of the day it happened. (So once again, I kept Resolution C in spirit by telling my students this story.)
Their response to this story was that since I was once suspended from school, I wasn't really the "perfect student" I'd tried to present myself as. This refers to when a student claims that it's "impossible" to follow rules such as avoid restroom trips or eating food during class, and I would say "I never went to the restroom/ate during class." All they had to counter was "Yeah, but you were once suspended from school." And in fact, one girl told me that she respected her father -- who admitted that he'd made mistakes as a young student and that she should avoid them -- much more than she respected me, the one who acted as if I was perfect but really wasn't.
That's what the New Decade's Resolutions are all about. The Resolutions #9-10 on time are all about why the students should avoid going to the restroom during class, instead of claiming that I'd been a perfect student who never broke rules. I should avoid comparing students to myself (or those in other classes) unless I'm making a favorable comparison. I should admit that I was once suspended from school -- and guide my students to avoid suspensions. I should admit that I received C's in many classes, starting with C's in first grade handwriting, sixth grade health, and seventh grade art (just because I was bored) all the way up to that C+ in graduate analysis at UCLA -- and guide my students to make more sacrifices to earn A's and B's.
And most importantly, the New Decade's Resolutions are meant to guide me as much, if not more, than my students. If I want my students to value every second of class (Resolution #9), then I should provide them a valuable learning experience during the entire period. If I want my students to make sacrifices to become a more effective student (Resolution #2), then I should make sacrifices to become a better teacher. If I want my students to sing songs to help them remember math (Resolution #5), then I should write enjoyable songs for them to sing.
Officially, my New Millennium's Resolutions are still in effect -- after all, it's still the millennium of the 2000's, isn't it? These apply to both my students and to myself. If I want the students to be organized (Postulate 1), then so should I as a teacher. If I want the students to "travel with another" and communicate with each other to be successful in this world (Postulate 2, Resolutions A-B), then so should I travel with my fellow teachers. If I want the students to obey me as a wise teacher (Postulate 3), then so should I obey the wise principal (which incorporates last year's resolutions -- adhering to the principal's wise ideas involving proper use of the curriculum/technology).
And finally, if the students aren't truly done until they've achieved excellence (New Decade's Resolution #10), then I'm not truly done until I've achieved excellence. I'll keep revisiting these resolutions over and over -- and discussing my progress on the blog (New Millennium's Resolution C), until I've achieved excellence as a teacher.
And I'll continue to do so until I'm indeed an excellent teacher -- even if it takes me all decade.
New Decade's Resolutions: Music
Ever since the earliest years of the old decade, I envisioned a classroom where I would regularly sing songs to help the students learn. It was in 2012 when I discovered that many old songs from my favorite TV show Square One TV were posted to YouTube, and so it's likely that around then, I imagined what it would be like to sing some of those old songs in a math class at school.
But except for a few instances of "Quadratic Weasel" in Algebra I classes, I didn't really make music a regular feature of my classes until I was hired at the old charter school. I had the first five songs ready to start the year at my new job. "The Dren Song" was based on an old tune that I had created but never added lyrics until then. "Count on It" was the first Square One TV show that I performed in class, due to its message on the importance of learning math. "The Benchmark Test Song" is a simple tune that I randomly generated using a pentatonic scale. "Fraction Fever" was based on a tune from an old computer game. (It's the same system as Mocha, but it's not playable on that emulator.) And "Need for Speed" (the mousetrap car song) was also based on a random song in a minor key.
One of my most successful uses of singing in class was "Measures of Center." This song, set to the tune of "Row, Row, Row Your Boat," was actually suggested by a new sixth grader who had heard it from the previous teacher at her old school. During a seventh grade computer lesson, the coding teacher asked the students to calculate mean, median, and mode on the computer. Many students couldn't remember what any of those were -- until I started singing the song again. And it definitely helped jog their memories.
Decade Resolution #7: We sing to help us learn math.
Some readers might point out that this seventh resolution apparently contradicts earlier resolutions about doing what the heroes of 1955 did. After all, singing is entertainment, and thus it reinforces the idea that class time should be for entertainment. I seriously doubt that Gates, Jobs, or Berners-Lee -- or anyone else in their generation -- had math teachers who needed to sing for them. Didn't I just complain about the "Triggy" video and the need to have songs in class? (Once again, the 1955 generation had much more tolerance for boredom than the current generation does!)
But in this case, I believe that the seventh resolution reinforces the fourth resolution on enlarging the size of the bicycle. In many cases, it's easier to remember a song than spoken prose. Thus my songs help students remember mathematical laws and possibly reduce common errors. (And these songs are more relevant than the ones the students would try to listen to on their phones.)
If it increases student learning, then I might even sing "Triggy" in class someday, perhaps for a Geometry class when we reach the trig chapter (equivalent to Chapter 14 of the U of Chicago text) and it's time for the students to learn SOH-CAH-TOA. But I keep it balanced and remind the students that not everything about math needs to be "fun."
(Note: I mentioned "Triggy" on the blog when I first heard the song -- it was for a class I subbed for about three months before I started at the old charter school.)
My only lament about singing at the old charter school was that I didn't do it enough. The day I sang in seventh grade coding was one of the few times that I'd sung on a Monday, and I also seldom sang on Wednesdays either. My thought was that on days when they really didn't have "math class" (or "STEM class"), they didn't need any music.
This might have made sense on middle school Wednesdays in my new district -- the entire day is shorter as is each period. But at the charter school, while school was also out early on Wednesdays, I actually saw my eighth graders for more minutes that day (and seventh grade not at all). Sometimes the eighth graders would grow restless, and songs might have broken up the monotony.
Also, many of my songs were enjoyable for the students. My eighth graders seemed to like "Count on It," while "Fraction Fever" was a seventh grade favorite. But I never had time to memorize all the lyrics -- I would write them on a poster, and then put the poster away when it was time for me to sing a different song. The students might have enjoyed it -- and perhaps behaved a little better -- if I could have sung their favorites again on demand. After I left the old charter school, I started writing the lyrics in a small notebook, but it was too late to help my charter students.
When I returned to subbing, my original plan wasn't to keep singing in class. But this suddenly changed within the last ten months of the decade -- and now, suddenly I'm expecting myself to sing almost everyday that I sub.
It all started back on Pi Day. I found myself to my delight in an eighth grade math class -- one I'd spent several days in earlier that year. As it was one of the few times that I've ever found myself in a math class on Pi Day, I couldn't resist singing songs for them. I began with "Oh Number Pi" and then added some more festive songs from YouTube.
From that day forward, singing quickly spread. I would return to that same school, and a student would ask whether I had a song to sing or not. Then at other schools, I found I could help those kids out by singing some of those same songs, and so on. One day, I subbed in an English class that had a parody of Meghan Trainor's "All About That Bass" as part of the lesson. So of course I couldn't resist making a math parody of that song, "All About That Base and Height." And so it continued.
As a sub, I've found that singing can help the students to remember math. In addition to "All About That Base and Height," one particularly nice song to sing is "Solving Equations." Even though I'd originally written it for eighth grade, it also fits seventh grade and Algebra I.
When I sub for non-math classes, I've found that singing song can be a powerful incentive for students to control their behavior. This leads to the next resolution:
Decade Resolution #8: We sing to help us remember procedures.
In many cases, the song serves only as an incentive to work. I've lately used Square One TV's "One Billion Is Big" for this reason. But other songs help students remember classroom procedures. The main song of this type is "The Packet Rap" -- which was originally "The Packet Song" at the old charter school, but I've since then decided that it sounds better as a rap.
I've found that singing songs definitely reduces arguments. When I get the urge to yell at a student, the best thing for me to do is start singing a song instead.
And so here are some of my plans for songs moving forward in the new decade. Each morning, I decide whether I'm going to sing or not, and which song. If it's a math class, then I might choose a song related to the lesson. Lately, I've saved a few songs from the old charter for science -- "Meet Me in Pomona" for life science, "Earth, Moon, and Sun" for physical science. Any of my other songs can work in a non-STEM class.
If the class isn't math -- or if it's math but the song isn't directly related to the lesson (for example, if it's an Algebra I lesson on factoring, I have no factoring song) -- then the song is an incentive. I enter the class, take attendance, and then sing the first verse of the song. Then I tell the students that they need to earn the rest of song, and what they need to accomplish to earn it.
If it's a math song directly related to the lesson, then I do examples on the board and sing the song for each example I do on the board. This also serves as a mini-incentive -- if the students wish to hear more of the song, then they should be quiet so we can get through more examples.
My rule of thumb is that students need a music incentive if they're too immature to work hard for subs without it. I like singing for middle school and younger high school students (say up to sophomore year, or Geometry if it's math) and expect older students to work hard on their own without a song.
Special ed classes are tricky. If there is an aide, or if it's co-teaching, I expect the other adult to run the class, and I dare not interrupt that other teacher with a song. Sometimes, there may be one or two periods that day when I'm teaching solo, and thus the song returns for that period only. It's also likely that I'll need to sing even for older grades in a special class, since they're often not motivated to work.
The eighth resolution tells us that singing can help with procedures. As a sub, a common procedure the students need to follow is putting away Chromebooks. Often, only the last period needs to put the laptops away. Thus I might sing an incentive song for that last period only -- or if it's a middle school class where I'm already singing all day, change it the last period so that the incentive is for putting Chromebooks away rather than for completing their work.
Classes that aren't representative of what I'd like to teach someday don't need songs. This includes P.E. and self-contained special ed classes. And of course, if the class is only watching a video or taking a test, then a song might take time away from it, so I don't sing.
It often happens that I enter a classroom deciding not to sing a song, perhaps because I fear that the song might take time away from the lesson. In general, it's best to sing anyway. I'd rather set up an incentive and get the students to work to earn that incentive, even if it takes time away from that work, then give them the whole period and end up with no one doing the work at all (with talking or playing with technology, etc., instead).
I also mentioned in the past that repeating the importance of earning A's over and over was a huge turn-off at the old charter school. Instead of talking about the importance of earning A's, it's possible for me to sing about it instead. In fact, two of the parody songs that I include in my songbook mention grades -- "All About That Base and Height" and "x's and y's." (In both cases, someone else first came up with the parodies and posted these lines on YouTube -- I modified some of the lyrics to create my own parodies but retain the mention of grades.)
Some of the songs I wrote at the old charter school -- "Ratios" and "Another Ratio Song," for example -- are recorded in my songbook (and on the blog), but I either don't like or don't even remember the tunes I made for them. This is where Mocha comes in -- I can use Mocha to add tunes to these songs and then reintroduce them while subbing.
It's also possible to create songs to remind the students about rules and procedures, especially if they get tired of hearing them over and over. My song "Show Me the Numbers" at the old charter school was originally such a song, but then it fell apart as it referred to an old participation points system ("demerits") that I stopped using soon after I wrote it.
Many teachers have a strong teacher tone, and they manage effectively because of it. I don't have a strong teacher tone -- instead I have a strong musical tone. I need to take advantage of what I have.
New Decade's Resolutions: Time
One thing that I've found to annoy me more than many other things in the classroom is when students find excuses not to be in the classroom. This includes students who regularly arrive late, sneak out of class early, or ask for restroom passes all the time.
Sometimes a student might have tried avoiding missing class time and failed. But a student who asks within seconds of the tardy bell ringing after snack or lunch clearly didn't try at all. And on some days that I've subbed, it appears that more students ask to go during the periods right after snack and lunch than the other periods (when a restroom visit might be more justifiable).
When I was a young student, I went three whole years (Grades 10-12) without asking a teacher for a restroom pass even once. Yet I believe that I actually visited the restroom much more than the average student did. I often went twice during lunch, and twice during P.E. (in the locker room, once when getting dressed and once when returning to street close), and yet never needed a restroom pass at all.
But of course, I've seen that telling stories about myself doesn't convince anyone at all. (That is, responding to the student complaint "It's impossible to go the whole day without a restroom pass!" with "I myself used to go years without needing a pass!" doesn't change any student's minds.) So instead, I have the following New Decade's Resolution:
Decade Resolution #9: We attend every single second of class.
I believe that almost any student can go days without needing a restroom pass if they really want to -- the problem is that many students don't. To them, class time is unpleasant, and so they want to find any excuse to get out of class. Far from striving to use the restroom twice during lunch in order to avoid missing any class time, they wish to go during class even when they don't want to in order to shorten class time a little.
This resolution is a bit difficult to enforce as a sub, since I enter many classrooms not knowing what the specific restroom policy is. But I include this in my decade resolutions because I hope to return as a regular teacher at some point during the decade. And so here are a few ideas I have for trying to enforce this resolution.
First, I can rework that failed participation points system so that it's based more on attendance. A student simply earns points for attending every single second of class. If a student misses even one second of class (arriving late, leaving before the bell, or with a restroom pass), then the point isn't earned. Of course, there might be exceptions for situations that are beyond a student's control (such as being summoned to the office).
I can also come up with a stricter punishment for the most annoying habit of asking for passes right after break. Suppose a student has a half-hour lunch at noon, then returns to class at 12:30 asking for a pass. If the student leaves, then upon returning, that student must write standards:
At 12:00, I could have used the restroom, but I didn't.
At 12:01, I could have used the restroom, but I didn't.
At 12:02, I could have used the restroom, but I didn't.
...
At 12:29, I could have used the restroom, but I didn't.
In some situations, I might give the student a warning by asking, "What could you have done today in order to miss zero seconds of class?" Then it's standards the next time a pass is needed.
There's one excuse I've heard often for needing passes right after lunch -- "I had detention for all of lunch, so I couldn't go." There's no way to verify if it's true -- and even so, it's not a valid excuse. If you need to use the restroom at lunch, then behave well in all your classes, and never let any of your grades drop below a C. Then you'll never get lunch detention! It's not my job as a teacher to accommodate your detention -- it's your job to become a good student who never gets detention.
Of course, this is another rule that I can't really enforce as a sub. It's expected that students will take advantage of subs by shortening class a little.
And there's one more time-related resolution that I wish to introduce here. Unfortunately, it's another resolution that a sub can't enforce, but it's an important one. It refers to those students who rush through their assignments just so they can say "I'm done with everything. Can I use the extra time to play on my Chromebook or phone?"
For these students, I have the following resolution:
Decade Resolution #10: We are not truly done until we have achieved excellence.
Some students enter class expecting at least a third of the period to be "extra" time for playing on computers or phones. Of course, members of the 1955 generation didn't expect so much time for entertainment in every single class. And it's really bad when they rush through the assignments to get that free time that they believe they're entitled to.
As far as I'm concerned, a student isn't truly done until his or her grade in the class is an "A." If the student is getting anything less than an "A," then the extra time should be reviewing the material or studying in order to achieve the "A" on the next assessment. Students who never reach an "A" in math class should never have free time for entertainment during class time.
So the way to enforce this resolution is to avoid letting students use phones at the end of class unless their grades are high. I might consider having "Technology 2020" time very seldom -- such as right after they finish a test or quiz. (They want to do well on the quiz/test, so most won't just leave the test blank just to get free time, especially when compared to classwork/homework that many of them don't care about.)
If I ever have "Dren Quizzes" today, then such quizzes count for "Technology 2020" time, provided that the grade earned on the quiz is an "A." I wouldn't want to grade regular quizzes and tests quickly for the Technology 2020 incentive, but Dren Quizzes should be easy to grade.
I want the current generation to know that they aren't done until they excel. I want the current generation to produce the next great cohort of inventors, so that technology in 2055 and beyond can surpass anything (or anywhere, as in outer space) that we can imagine today.
With the Eleven Calendar still fresh on my mind, note there are two ways that I'll actually use my new calendar. One is for food -- lately I've been buying one of my favorite foods, buffalo wings, on Fridays in the Eleven Calendar. Usually these are cheap wings at 7-Eleven, but when the weekends (Friday-Sunday) line up exactly in both calendars, I purchase a different brand. That includes this weekend -- January 3rd-5th Gregorian, December 12th-14th Eleven Calendar. I observed this special day today (Saturday in both calendars), when I ate wings at Fuddrucker's instead.
The other use that I'll start this year is, when I sub at a school, my focus resolution for the day, the one I work on the most that day, will be determined by the Eleven days of the week. So on Fourdays I look at the fourth resolution, the seventh on Sevendays, the tenth on Tendays. Friday-Sunday will be for the first three resolutions respectively. On Elevendays and blank days, I can focus on all of the New Decade's Resolutions as a whole.
Loose Ends with Traditionalists
My New Decade's Resolutions are intended to take a balanced approach to teaching -- balanced between the traditionalist and modern approaches, that is. As I mentioned above, the idea of singing in class is quite modern, as is of course any form of technology or projects (including mention of these in my old New Year's Resolutions).
But this is also balanced out with some traditionalist-leaning resolutions. The two resolutions that mention 1955 are clearly traditionalist. After all, the generation born in 1955 and graduated in 1973 grew up during the Golden Age of textbooks, when Dolciani and similar authors prevailed. Thus on occasions when I focus on these resolutions, my lessons will lean towards traditionalism.
Our main traditionalists have indeed been active during the past week. It's interesting to compare and contrast what the traditionalists wrote this week about the best teaching methods to my goals as stated in my resolutions.
The head traditionalist, Barry Garelick, posted three times this week. One of his posts has drawn ten comments so we'll focus on that one, but let's glance at the other two posts first. Back on Monday (New Year's Adam), Garelick wrote the following:
https://traditionalmath.wordpress.com/2019/12/30/bad-pd-517/
At a six-hour PD I had the misfortune of having to attend, the moderator put this slide on the screen in a defense against the call for evidence that certain teaching practices are effective. It was a slide from a presentation by David Theriault, who teaches English and has a blog
For this post, I wish to emphasize only the comment by a major traditionalist, Ze'ev Wurman:
Ze'ev Wurman:
Actually, bloodletting worked for me! I don’t need any research to show me right!
Wurman is clearly being sarcastic here. He is also trying to make an analogy here -- "bloodletting" corresponds to reformist, non-traditionalist math, while "modern medicine" corresponds to traditionalist math. To Wurman, only traditional math -- like modern medicine -- has the research to back it up.
Research is important to me. I did mention "we follow the research" earlier in this post -- and not surprisingly, I mentioned it in connection with the 1955 (that is, the traditionalist) resolutions.
I agree with the research showing that traditionalist math instruction is the most effective -- provided, of course, that the students actually do the traditionalist assignments. No worksheet, no matter how traditionalist it is, can teach anything to a student who leaves it blank. And I do have the research to show this -- namely the number of worksheets that students leave blank in my classes, including ones I see tossed on the floor after school.
Therefore my resolutions provide for songs and activities that the students will actually do. As I wrote earlier, the resolutions provide for some days to be "Technology 1955" days when traditional lessons are taught, and others to be "Technology 2020" days when modern lessons are taught. On the other hand, the traditionalists rarely compromise -- to them, only traditional lessons are effective, and just ignore the existence of students who leave those traditional worksheets blank.
Here's the other post to glance at. Garelick just barely posted this one today, and thus there aren't many commenters yet. Some readers might comment here in the next few days:
https://traditionalmath.wordpress.com/2020/01/04/pbl-a-guide-to-the-hype/
Edutopia has advice for those math teachers who believe that Problem (Project) Based Learning (PBL) actually has something of value to offer.
Project Based Learning was a cornerstone of the Illinois State curriculum that we used back at the old charter school. Of course, I already knew that there's not much in that curriculum that traditionalists would like.
OK, let's get to our main Garelick post now. This he posted the day before yesterday:
https://traditionalmath.wordpress.com/2020/01/02/the-prevailing-caricature-of-traditional-math-dept/
In a recent article about math education that ballyhoos the “latest approach” in how to teach math, this statement was made
“If the teachers are telling students how to solve a problem, and then that problem isn’t exactly what’s on the test, it creates this disequilibrium for a student,” said Beverly Velloff, the math and science curriculum coordinator for the University City School District.
Let's look at the comments here, starting with pkadams:
pkadams:
Your articles provide more and more evidence that , for whatever reason, some people do NOT want children to learn, math or other subjects. “New Math’ has been a failure since it came out! Just like teaching reading without phonics is a failure. And not teaching actual history is producing little Socialist/Fascists. Call it conspiracy theory or stupidity, but it is the only logical conclusion.
In her comment, pkadams refers to two other subjects -- ELA and history. (Here I use feminine pronouns to refer to pkadams because the selfie avatar next to the username appears female.) I'll omit a discussion of history and politics here (even though traditionalist discussions regularly lead to political arguments).
But her mention of phonics is quite interesting, since it often comes up in math discussions. I've reached a conclusion about phonics and traditional math, but this conclusion is likely not the one the traditionalists want me to reach.
Phonics, after all, is all about breaking up words into their individual sounds. Pro-phonics advocates believe that this is method of learning to read is superior -- when a student sees an unfamiliar word, it's better for that student to "sound out" the word by breaking it down, as opposed to the "whole language" approach of learning whole words at a time.
Anyway, to me, learning how to add 6 + 8 via making tens is akin to learning to read via phonics. In fact, we see how both involve breaking things down -- "cat" into c, a, t, and 6 + 8 into 6 + 4 + 4. On the other hand, foregoing breaking sums down and simply learning the whole table at once is just like whole language.
Thus if a traditionalist is pro-phonics, as pkadams appears to be, then she should also be in favor of making tens and other methods of breaking down sums and products. But somehow, I doubt that she or many other traditionalists will agree with this.
Chester Draws:
The reporter is an idiot, since memorising “equations” would be worthless. I think he means either formulas or algorithms. That he cannot get even that right makes me suspect that the reporter has mischaracterised the whole approach. I wouldn’t trust him to report anything — the Maths314 might be entirely traditional for all he knows.
It helps that in my jurisdiction the students are provided formula sheets. All the US has to do is provide a formula sheet at the start of each test if they want to limit memorisation.
Ironically, the U of Chicago Geometry text (which wouldn't typically be considered traditional, though this is mostly due to the U of Chicago elementary texts) outright encourages students to memorize formulas. Lesson 10-6 (in the chapter on surface area and volume) is simply titled, "Remembering Formulas."
This is a tricky one, since some teachers allow students to use notes on quizzes or tests, or provide formula sheets similar to the ones that Draws provides to his students.
If we compare this to my resolutions, we clearly need to look at the "bicycle" resolutions. Should formulas be included as part of the "bicycle" of math to memorize, or not? To me, let's just look at a test that some traditionalists still hold in high regard -- the SAT:
https://blog.prepscholar.com/critical-sat-math-formulas-you-must-know
According to this list, the SAT provides most formulas needed for Geometry. Thus if I were to become a Geometry teacher, I can safely take the Draws approach of providing the formulas.
On the other hand, many formulas of Algebra I and II, such as Slope Formula, Quadratic Formula, and definitions of trig functions, are listed under "need to know." Thus these formulas should be included as part of the "bicycle." We can use traditional methods to get students to memorize these, or we can use non-traditional methods such as songs (including Slope Dude, Quadratic Weasel, and yes, even Triggy).
Stephanie Sawyer:
Agree with all comments made so far, but I think we forget the obvious: good “traditional” math teachers know what has passed and what is coming. We know why we teach the curriculum we do, because we know what’s coming in the next year or 2 or 3 years. We also know why we should be able to teach our level because of what has supposedly come before.
I think I am probably the end of an era, where my Catholic school teachers all lived together and could literally vertical team over the dinner table every night. I can just hear it.
Sawyer mentions the Catholic religion because she learned at a Catholic school, and she is currently a Catholic school teacher. It's safe for me to mention religion in this post since I gave it the "Calendar" label (due mainly to my brief mention of my Eleven Calendar earlier).In this comment as well as her response to one of the other Garelick posts, Sawyer tells us that even her Catholic school is using non-traditional methods (even though many private schools are not subject to the Common Core Standards). She writes about how the school she attends as a young girl taught the standard algorithm for addition ("carrying") as well as adding money in first grade, since it helped prepare her for subsequent grades. The Common Core Standards, of course, teaches this in later grades.
Barry Garelick responds:
Absolutely right. When I teach Math 7, I’m getting them prepared for algebra: either Math 8 algebra, or Algebra 1 in 8th grade. For Math 8, I teach more algebra than is currently in the CC curriculum so that when they take Algebra 1 in 9th grade, they are familiar with things like factoring, algebraic fractions and various types of word problems.
This sounds a bit advanced for Common Core Math 8. Indeed, algebraic fractions hardly even appear in Algebra I any more (saving it for Algebra II) -- and even when I student taught Algebra I using a "proper Glencoe" text, the chapter on algebraic fractions was skipped. Meanwhile, factoring still appears in Algebra I, but do we really need to prepare students for it the year before?
Many of my eighth graders struggled in math the year I taught at the old charter school. Imagine if I'd tried to teach them Garelick's Math 8 curriculum. (Indeed, many of these topics are likely to lead to eighth graders asking, "Why do need to learn this?" Garelick explicitly tells us his answer -- to prepare them for freshman Algebra I. I wonder whether students will accept this answer.)
Of course, I save our main commenter for last:
SteveH:
This is one of their goals:
“OUTCOMES ORIENTED We care about results. Your student results become our student outcomes. We hold ourselves accountable to measuring the impact of our work together.”
Statistics. So much for individuals or increasing STEM degree readiness, since “proficiency” on CC tests only means the probability of needing no remediation if you end up taking a college algebra course. “Distinguished” only means that you have a 75% likelihood of passing the course. I really should have saved that link I read years ago. It was deep inside a PDF of our state CC test provider.I no longer have any hope that these educators will understand the realities of mathematical education when they see right before their eyes the success of AP/IB math curricula in their own high schools. They don’t even seem to want to look at individual success cases to see how they can duplicate them. They see that success in a traditional algebra course in 8th grade is a key to success, but have no interest in digging deeper to see how that happens. It’s all guess (what they hope for) and check (badly and with low expectation statistics).
We already know what SteveH would prefer the levels "proficiency" or "distinguished" to correspond to -- something like a 99% likelihood of passing College Algebra, and/or a 75% chance of passing a Calculus course (preferably AP/IB in high school, since he mentioned these again).
SteveH:
Colleges and admissions competition drive reality down through high schools via AP and IB curricula. That’s when students and parents really begin to pay attention and little of their pedagogical silliness sticks, or more likely, is ignored. Seventh and eighth are the transition battleground grades where reality begins to sink in for many students and parents. Some recover, but many don’t. Clearly, the low CC slope to no remediation for college algebra has to change to a proper STEM AP/IB slope, and that has to start in 7th grade. Everyone knows this, but many CC lovers never seem to want to make that transition clear. They desperately try to make lower expectations sound like better math understanding and results.
The only reason our K-8 schools got rid of CMP and replaced it with proper Glencoe textbooks was because students and parents complained that CMP did not do a proper curriculum job of preparing students for Geometry as a freshman. However, that just pushed the fairlyland/reality wall back to the end of 6th grade. Now, everything is based on the CC, but the school website does talk about an accelerated CC 7th and 8th grade coverage in 7th grade to prepare for Algebra I in 8th grade. However, the site says nothing about what textbooks, if any, they use or how formal the algebra class is.
Of course, what did I just say about "proper Glencoe" texts earlier? Some chapters -- specifically the one on algebraic fractions in Algebra I -- are usually not reached by most teachers.
SteveH also mentions an accelerated path for Grades 7-8. Many districts have similar paths, including the one where I work and receive the most calls. I have no problem with offering such a path, since after all, AP Calc exists and some students might be able to take it.
Speaking of Calculus, today I begin watching the Great Courses video that I mentioned back in my October 15th post. It is called Change and Motion: Calculus Made Clear. Our lecturer is Professor Michael Starbird, from the U of Texas at Austin. I will create a new label, "Michael Starbird," as we begin to discuss these lectures. Part I consists of Lectures 1-12.
Lecture 1 of Michael Starbird's Change and Motion is called "Two Ideas, Vast Implications." Here is an outline of this lecture:
I. Calculus is all around us.
II. Calculus is an idea of enormous importance and historical impact.
III. Calculus comes from everyday observations.
IV. Calculus deals with change and motion and allows us to view our world as dynamic rather than just static. Calculus provides a tool for measuring change whether it is change in position, change in temperature, or change in demand.
V. Zeno's paradoxes of motion confront us with questions about change. Calculus helps to resolve these ancient conundrums.
VI. The history of calculus spans two and a half millennia.
VII. Mathematical relationships between dependent quantities are described by functions.
VIII. The power of calculus lies in its abstraction, that is, that the insights and methods that arise from investigating physical problems also apply to many other settings.
IX. Calculus is the study of two fundamental ideas (called the derivative and the integral), which we will describe and define simply and understandably in Lectures Two and Three.
X. Here is an overview of the lectures [omitted here -- dw].
Starbird, like Fawn Nguyen and many other teachers, mentions how his fellow travelers cowered in fear upon learning that he is a math teacher. He jokes that many people find Calculus hard because the word comes from a Greek word meaning "stone" -- as in the pebbles used in an abacus.
The professor quotes Winston Churchill, who mastered advanced mathematics in just six months. The English statesman compared differential calculus to a fiery dragon. But he also reminds us that we learn by teaching -- students don't truly understand Calculus until they have to teach it.
I can also tie Starbird's lecture back to my New Decade's Resolutions. The professor mentions that nearly all technological developments of the last 300 years can be traced back to Calculus. Thus if math classes were to disappear, we wouldn't merely go back to 1955 technology -- we'd revert all the back to 1700.
Still, the resolutions mention the 1955 generation as an answer to "When will we use this in real life?" as the heroes used Calculus to build the technology that students use and love. Calculus is considered the tail end of second decade math -- the main purpose of most of it is to prepare students for future math courses (culminating in the third decade).
Loose Ends with Music: The Whodunnit Song
The seventh and eighth resolutions demand that I include music more often in class. My newest song is the "Whodunnit" song, which I introduced last month in a multi-day subbing assignment. In this post, I will describe this song in more detail, including how to play it in Mocha.
(By the way, I was reading some old blog posts and realized that last month was not the first time I saw the Whodunnit activity in a class I subbed for. I'd actually seen the Whodunnit clues posted on the wall about one year earlier, but we didn't play it on the day I subbed there.)
Let's me start with how I generated the song. I decided to use the 12EDL Mocha generator (that I posted here back on Halloween 2018). We know that 12EDL is very minor sounding, which fits into the murder mystery theme.
At first, I tried the generator that adds quick sixteenth notes. This is great when I already had lyrics with many syllables and need a fast song to fit the lyrics too. But since I was creating the lyrics from scratch as well, I feared that I couldn't invent enough lyrics to fit such a fast song.
As it turned out, the second line generated by Mocha didn't include any sixteenth notes. Instead, it alternated eighth and dotted quarter notes. This produces a simple rhythm, to which I easily came up with lyrics overnight.
The song I created repeats this line three times. The fourth line produced by the generator is different, but it also includes eighth and dotted quarter notes, so it fits the rest of the song.
Let me remind you of the 12EDL scale:
Degree Note
12 white A
11 lavender B
10 green C
9 white D
8 white E
7 red F#
6 white A
Recall that the "lavender B" is about halfway between white Bb and white B. In the 12EDL scale I usually call it "lu B," but sometimes I call it "lu Bb" when I feel this name is more convenient.
Here are the notes for the song:
First Three Lines:
E-C-E-B-E-E-F#-B-A12
Last Line:
D-E-A-D-A12
Originally, Mocha chose Degree 6 (high A) for the last note of the first three lines. But I replaced it with Degree 12 (low A) because it sounds better after Degree 11 (lu B). Remember the first rule of music is that it should sound right. I have the right to change the random notes generated by Mocha in order to make the song sound better.
The fourth note in the last line was originally Degree 7 (ru F#). But I couldn't resist changing it to Degree 9 (wa D) in order to make the riff D-E-A-D appear in this murder mystery song. Of course, we could keep D-E-A-F# if the culprit had merely taken the victim's hearing. (This isn't the first time that I intentionally included notes in a song that spelled something -- at the old charter school, I wrote a GCF song for the sixth grades that included the notes G-C-F.) This line sounds better if we use Degree 6 for the first A and Degree 12 for the second.
Here are the lyrics for two verses that I wrote. (They come from the first two clues last month.)
First Verse:
Was Mrs. fixing snacks in kitchen?
Was Dr. hearing songs in atrium?
Professor's iPad by the pool now?
Did Miss kill the victim?
Second Verse:
A cougar's mauling was the method?
Or had a green snake's bit begun it?
Or object's falling was the weapon?
Just tell me whodunnit!
To make the rhythm fit the notes, we start with an eighth note and then switch to a dotted quarter note per syllable. So the first line is (bold = longer note): Was Miss - us fix - ing snacks in kitch -en? In the second line: Was Doc - tor hear - ing songs in a - tr'um? (Say "atrium" with only two syllables in order to make it fit.)
But then each line adds up to 8 1/2 beats, or two bars in 4/4 plus an extra eighth note. Originally, the randomizer made one of the dotted quarters an ordinary quarter so that it adds to to eight beats, but it was always easier for me to keep the rhythm consistent with dotted quarters throughout.
Thus I decided to make the the first eighth note a pickup note -- an opening note that occurs before the first bar. Then the other eight beats form two full bars. But then the second line of lyrics also starts with this same eighth note pickup. In class last month, I decided to put a 3 1/2-beat rest between the first two lines. I like this -- the silence makes the song eerier (as befitting a murder mystery song).
The third line is just like the first two. The fourth verse starts with a dotted quarter on Did, and so there is no 3 1/2-beat rest before it. The last note is a double whole note (length 32 in Mocha) -- we can devote a whole note to each syllable vic- and -tim. Then there's one more 3 1/2-beat rest before the second verse begins.
Here's what it looks like in Mocha:
https://www.haplessgenius.com/mocha/
10 N=16
20 FOR V=1 TO 2
30 FOR X=1 TO 3
40 IF X=3 THEN Y=14 ELSE Y=9
50 FOR Z=1 TO Y
60 READ A,T
70 SOUND 261-N*A,T
80 NEXT Z
90 FOR R=1 TO 1400
100 NEXT R
110 RESTORE
120 NEXT X,V
130 DATA 8,2,10,6,8,2,11,6,8,2
140 DATA 8,6,7,2,11,6,12,2
150 DATA 9,6,8,2,6,6,9,2,6,32
As usual, don't forget to click on the Sound box before RUN-ning the program.
Last month, I also mentioned attempting to add an accompaniment (or harmony) to this song by opening a second window and running Mocha in it. This is tricky -- we'll have to time it perfectly so that the two parts line up exactly, or they won't sound good at all. We'll have to know when exactly to type in RUN in each of the two window.
"Whodunnit?" is a tricky song for our first attempt at Mocha harmony. That pickup eighth note is likely to throw our timing off. (The rests might be problematic at well. It might be nice to have the accompaniment playing notes while the main voice is resting, but this is tricky. We estimated the rest of 3 1/2 beats with an empty FOR loop of length 1400 in Line 90. This is a good estimate, but we need to be exact if we want to have an accompaniment. Of course, this can be avoided if we simply have the accompaniment rest with a 1400 FOR-loop as well.)
Let's try an easier song for our first crack at accompaniment -- "Row, Row, Row Your Boat," which, as I mentioned above, can serve as "Measures of Center" (as well as "Same Sign, Add and Keep" for integer addition) in a math class.
This song is written in 6/8 time, with the eighth note as length 2 and a set of three eighth notes, a dotted quarter note, as length 6. Since the original song in in a major key, we'll use 18EDL:
Degree Note
18 white D
16 white E
14 red F#
13 thu G
12 white A
9 white D
NEW
10 N=4
20 FOR V=1 TO 2
30 FOR Z=1 TO 27
40 READ A,T
50 SOUND 261-N*A,T
60 NEXT Z
70 RESTORE
80 NEXT V
90 END
100 DATA 18,6,18,6,18,4,16,2,14,6
110 DATA 14,4,16,2,14,4,13,2,12,12
120 DATA 9,2,9,2,9,2,12,2,12,2,12,2
130 DATA 14,2,14,2,14,2,18,2,18,2,18,2
140 DATA 12,4,13,2,14,4,16,2,18,12
Now let's try to give this song an accompaniment. A simple one that fits this song is a running bass line that alternates between white D and white A, each a dotted quarter note in length. Indeed, this is so easy that we won't even use DATA lines for this song. Since these will be bass notes, we drop an octave to Degree 36 (for D) and Degree 24 (for A) in our accompaniment.
We start by opening up a second window and launching Mocha. A second tab isn't sufficient, since the browser (at least if it's Microsoft Edge) will block the song from the other tab. In this second tab, we type in the following code:
10 N=4
20 FOR X=1 TO 18
30 SOUND 261-N*36,6
40 SOUND 261-N*24,6
50 NEXT X
Make sure that Sound is clicked on in both windows. Now for the tricky part -- the timing. We need to enter RUN in both windows at the right time. Indeed, we should have the three letters RUN typed already in both windows, so all we'll need to do is press a single key (ENTER) at the proper time.
The song contains eight bars, and since it plays twice, we need 16 bars in the bass. But the FOR loop in Line 20 goes up to 18. That's because the two extra bars are there for timing. We start by hitting ENTER key (after the word RUN) in the window with the accompaniment, and then while the first two bars are playing, we switch to the window with the melody. Just as the third bar is about to begin, we hit ENTER in the second window and then RUN it. (And if we miss our cue, it will be harmless if we wait until the fourth or fifth bar, since all bars are identical in this simple walking bass.)
I tried it a few times, and the timing was tricky! It's worth trying this a few more times before attempting a more complex accompaniment.
By the way, in both lines, N must be the same number. Here N must be in the 1-7 range -- sadly, we can't quite reach the low D needed for N=8. I'm more likely to sing along with N=5 (key of green Bb) or N=7 (key of red E).
Conclusion
Recently I found some old papers -- something that I'd written twenty years ago. It was, in fact, a list of New Millennium Resolutions that I had written on January 1st-4th, 2000.
As a young boy, I'd always fantasized about building a time machine and traveling to the future, the new century. Because I didn't know at the time about the nonexistence of the year zero, 2000 was always my target year. Back then, I'd believed that robots would take over in the year 2000 -- and that these robots would try to destroy all humans.
And so on January 1st, 2000, I couldn't help but want to retell the story of my old game. But in addition, I also added New Millennium's Resolutions to the document. Let me reproduce them here:
A. Each day when I am in contact with fellow students, I will try to begin a conversation. In order to fulfill this resolution, I will wait for someone to ask me a question, such as "What are your classes like?" Then I will "perpetuate" the conversation by attempting to give a full, detailed answer.
B. If the first resolution does not lead to a conversation, then I will "initiate" a conversation. The best way to do so is to repeat a commonly asked question, such as "What are your classes like?" to another student distant from the one who originally posed the question.
C. I will maintain these secret files each day by writing a long story of my game. By doing so, I finally reveal to myself and others the reasons for my silent personality.
You might ask, did I ever keep any of these resolutions? Well, I didn't literally keep Resolution C, since I only found four days' worth of "secret files" (and I doubt that I wrote on January 5th). On the other hand, I do remember creating blogs -- and indeed, we can even count this very blog as an extension of Resolution C.
On the other hand, I must count Resolutions A-B as abject failures. In the year 2000 I was in my first year as UCLA, which is why I kept mentioning "What are your classes like?" as an example of small talk (similar to "How's the weather?"). At the time, I wasn't quite sure whether I would become a teacher, where "What are your classes like?" suddenly has a different meaning. (Now it would be something to say in the teacher's lounge, asking another teacher what his/her classes are like. This sort of small talk occurs all the time in lounges!)
Back at the old charter school, I had trouble coming up with science lessons. I once confessed to my eighth graders that I didn't know what to teach them, and one girl simply replied, "Why don't you ask any of your teacher friends?"
Notice that if I had kept Resolutions A-B, I would have found a teacher to discuss science classes with, and thus I would have had a better science class. And as I mentioned on the blog, I failed even to communicate with other teachers online by creating a Twitter account, which I should have done.
On those first four days of the 2000's, I also wrote some basic assumptions about myself. And as I wrote back in my September 6th post (the day we covered Lesson 1-7), even back then I referred to those assumptions as "postulates":
Postulate 1. Organization is the key to success.
Postulate 2. When possible, one must travel with another.
Postulate 3. Persons of wisdom deserve obedience.
Postulate 4. Not all guns are fake.
The fourth postulate referred to an incident from my story -- the original game of time machines, which I played in the first grade, had evolved into a game where kids pretended to have guns. The fourth postulate takes into account the Columbine incident (which had occurred about 8 1/2 months before I wrote it) -- and indeed, it's exactly because of Columbine and subsequent school violence that I don't discuss it fully on the blog.
The important thing was that I was suspended from school -- and I told my eighth graders at my old charter school a version of this incident on the anniversary of the day it happened. (So once again, I kept Resolution C in spirit by telling my students this story.)
Their response to this story was that since I was once suspended from school, I wasn't really the "perfect student" I'd tried to present myself as. This refers to when a student claims that it's "impossible" to follow rules such as avoid restroom trips or eating food during class, and I would say "I never went to the restroom/ate during class." All they had to counter was "Yeah, but you were once suspended from school." And in fact, one girl told me that she respected her father -- who admitted that he'd made mistakes as a young student and that she should avoid them -- much more than she respected me, the one who acted as if I was perfect but really wasn't.
That's what the New Decade's Resolutions are all about. The Resolutions #9-10 on time are all about why the students should avoid going to the restroom during class, instead of claiming that I'd been a perfect student who never broke rules. I should avoid comparing students to myself (or those in other classes) unless I'm making a favorable comparison. I should admit that I was once suspended from school -- and guide my students to avoid suspensions. I should admit that I received C's in many classes, starting with C's in first grade handwriting, sixth grade health, and seventh grade art (just because I was bored) all the way up to that C+ in graduate analysis at UCLA -- and guide my students to make more sacrifices to earn A's and B's.
And most importantly, the New Decade's Resolutions are meant to guide me as much, if not more, than my students. If I want my students to value every second of class (Resolution #9), then I should provide them a valuable learning experience during the entire period. If I want my students to make sacrifices to become a more effective student (Resolution #2), then I should make sacrifices to become a better teacher. If I want my students to sing songs to help them remember math (Resolution #5), then I should write enjoyable songs for them to sing.
Officially, my New Millennium's Resolutions are still in effect -- after all, it's still the millennium of the 2000's, isn't it? These apply to both my students and to myself. If I want the students to be organized (Postulate 1), then so should I as a teacher. If I want the students to "travel with another" and communicate with each other to be successful in this world (Postulate 2, Resolutions A-B), then so should I travel with my fellow teachers. If I want the students to obey me as a wise teacher (Postulate 3), then so should I obey the wise principal (which incorporates last year's resolutions -- adhering to the principal's wise ideas involving proper use of the curriculum/technology).
And finally, if the students aren't truly done until they've achieved excellence (New Decade's Resolution #10), then I'm not truly done until I've achieved excellence. I'll keep revisiting these resolutions over and over -- and discussing my progress on the blog (New Millennium's Resolution C), until I've achieved excellence as a teacher.
And I'll continue to do so until I'm indeed an excellent teacher -- even if it takes me all decade.
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