"Richard Buckminster Fuller was an inventor, a designer, an engineer, an author. He was an architect of ideas, a man whose visions were often ahead of their time."
This is the first page of the first section, "Buckminster Fuller, Geodesic Domes & Buckyballs." Here Pappas gives us an excellent description of Buckminster Fuller. Affectionately known as "Bucky," he was one of the most famous American architects.
Pappas includes a drawing of Bucky's most famous invention on this page, but she doesn't describe it until page 251, so I'll have to wait until tomorrow to blog about it.
Instead, I write about one of his other inventions from page 250, the Dymaxion House (since we are after all in the architecture chapter). Pappas tells us that Dymaxion Houses "were designed to be completely transportable living units." Of course, this description is just begging for a corresponding photo, so here's a link to pictures of both it and the Wichita House:
https://www.inverse.com/article/13715-forget-geodesic-domes-buckminster-fuller-s-dymaxion-house-was-his-masterpiece
By the way, the word "Dymaxion" can be defined -- it means "dynamic maximum tension." But now let's move on to some words which can't be defined.
Lesson 1-6 of the U of Chicago text is called "The Need for Undefined Terms." (It appears as Lesson 1-4 in the modern edition of the text.) Earlier this week we revisited Lessons 1-4 and 1-5 as they were my opening week activities last year. But those lessons are exceptions -- most of my Geometry lessons have nothing to do with middle school. For most lessons, we'll have to go back two years to find my previous notes on the lesson:
This is what I wrote two years ago about today's lesson. I have updated the post to reflect the number of results of a certain Google search.
Lesson 1-6 of the U of Chicago text is where the study of geometry formally begins. This section states that three important words in geometry -- point, line, and plane -- are undefined. This may seem strange, for mathematics is all about definitions, yet these three important concepts are undefined.
In college-level math, one learns that these undefined terms are called primitives, or primitive notions. Just over a hundred years ago, the German mathematician David Hilbert declared that there are in fact six primitive notions in geometry: point, line, plane, betweenness, lies on, and congruence. But most textbooks list only the first three as undefined terms. This is because texts actually define the last three using concepts from other branches of mathematics. "Lies on" or "containment" -- that is, what it means for, say, a line to contain a point -- is defined using set theory (which is why the very first sentence of this section states that a set is a collection of objects called elements). "Betweenness" of points -- that is, what it means for a point to be between two other points -- is defined later in this chapter in terms of betweenness for real numbers (their coordinates of course). And the definition of "congruence" is the cornerstone of Common Core Geometry -- we use reflections, rotations, and translations to define "congruence." So we're left with only three primitive notions -- points, lines, and planes.
Lesson 1-6 is a fairly light lesson. So point, line, and plane are undefined -- big deal! Of course, we can do things with points, lines, and planes, but that's not until 1-7. So instead, I use this as an opportunity to remind the students the reasons for taking a geometry course.
The students in a geometry course are around the age where thoughts such as "I hate math" become more and more common. This is the age where they wonder whether they'll ever have any use for the math that they're learning. They begin to wonder whether they'll ever use any math beyond what they learned in elementary school and wish that math classes were no longer required beyond elementary school, for can't they live very successful lives not knowing anything higher than fifth grade math?
As of today, a Google search for "I hate math" returns 602,000 results. And we can easily predict the most common reason for hating math -- of course it's because it's hard. We don't hate things that are easy -- we hate things that are hard. And the class that turns so many off from math is algebra. Indeed, if you choose some school and tell me only its standardized test scores in ELA and math, I can very reliably tell you whether it's an elementary or a secondary school. If the math score is higher, it's probably an elementary school -- if the ELA is higher, it's likely a secondary school. And so now we, as geometry teachers, have the students for the math course right after the one that caused them to hate math in the first place.
The number of search results for "I hate math" has increased almost 50% over two years ago. But I notice that much of the increase is generated by a number of tutors that have taken the name "I hate math" -- that is, they tutor for students who hate math, as opposed to hating math themselves.
But we do see some results that are obviously from genuine math haters. One girl has posted a YouTube video of about 6 1/2 minutes on why she hates math. The girl in the video is an eighth grader who is struggling with the Quadratic Formula in her Algebra I class. She says that she hates math because without the class, she'd have a 4.0 GPA, but with math she struggles just to get a D+. I don't link to the video, but anyone can find it via a Google/YouTube search.
There are also images that say "I'm still waiting for the day that I will actually use xy + (420) > x - 5y[2 + 9 = 7] in real life." Well, of course we will probably never use a non-linear (because of the xy term) inequality such as that one in the real world. This image would have been much funnier if, instead of that inequality, the image contained the type of equation that traditionalists lament don't appear in Common Core texts, such as "I'm still waiting for the day that I will actually use (a quadratic-in-form equation with radicals) in real life."
So why do we require students to take so much of a class they hate in order to graduate high school? As it turns out, we can answer this question from one of the sections that we've skipped, Lesson 1-1:
"A point is a dot."
And this section gives many examples of dots -- the pixels on a computer screen. The shapes that appear on our screens consists of dots, which can be modeled in geometry by points. We look at images on our TV screens all the time. And one of the most geometry-intensive computer programs that we have are video games -- we must create images consisting of dots that move rapidly.
The point of all this is that we can surely have math without entertainment, but we can't have entertainment -- at least not most modern forms of entertainment -- without math. We can only imagine how much technology would disappear if math were to disappear.
Elementary school math -- at least early elementary arithmetic (before the dreaded fractions) -- is easy. And college majors majoring in STEM know the importance of learning math. The problem is those in-between years in middle and high school. If math were merely an elective in secondary school, many students would avoid it and choose easier classes. Then there wouldn't be enough STEM majors in college because they wouldn't have had the necessary algebra background. The only way to bridge the gap between "math is easy" (early elementary) and "math is important" (college STEM majors) is to require the subject during the intervening middle and high school years. Otherwise we'd have no modern technology or entertainment.
When I give notes in class, I prefer the use of guided notes. This is not just because I think the students always need the extra guidance, but that I, the teacher, need the guidance. In the middle of a lesson, I often forget what to teach, or forget how to explain it, unless I have guided notes in front of me.
And so today's images consist of guided notes. I begin with Lesson 1-6 and its definitions. Here I emphasize the fact that point, line, and plane are undefined by leaving spaces for the students to write in their definitions -- which they are to leave blank (or just write "undefined")! Notice that Lesson 1-6 distinguishes between plane geometry and solid geometry -- a crucial distinction in Common Core Geometry because the reflections, rotations, etc., that we discuss are transformations of the plane.
Then I move on to Lesson 1-1. This is based on an online discussion I had a few years ago on why students should learn math. I also include it as guided notes so that the students are listening when the teacher gives the reasons that they are taking this course. (The answers to the blanks beginning with the conversation are MBA, polynomial, investing, data, supermarket, and -- the object Americans use that has more computing power than the A-bomb -- cell phone!)
In the year since I first posted this lesson, I've been thinking about how to rewrite the lesson so that students are more responsive to it. In particular, I was thinking about last week's bridge puzzle, on which I wrote, "Back then, people spent their Sundays taking walks over bridges." Think about that statement for a moment -- entertainment back then was limited to Sundays. Back then, six days a week were workdays, on which no one expected to be entertained. Even on Sundays, the morning were devoted to church, so only the afternoons were amusing. And when we finally get to Sunday afternoon, all people did was cross bridges -- something that we wouldn't find entertaining today.
What has changed since the 18th century? The answer is technology -- that is, mathematics. Just as I mentioned in the worksheet, one especially widespread form of entertainment is the cell phone. We don't have to wait until Sunday afternoon for entertainment -- with our modern phones, we can be entertained at almost any time. Games and videos can be played anywhere, and if our friends live across the bridge, we don't need to cross it, since we can call or text them. All of this technology is available now because of mathematics.
Yet the greatest paradox is that, while math makes all of this technology possible, students use this technology to justify avoiding the study of mathematics. Traditionalists don't like the fact that students don't study as much now as they did in the past. Nowadays, the idea that one should study for two hours at once -- that is, go two hours without cell phones, TV, or other entertainment -- is unthinkable for many students, yet before modern technology, the idea of being entertained as often as once every two hours was equally unthinkable. The girl in the YouTube video says that she must study two hours per day just to pass her Algebra I class -- and that she's lucky if she can finish her weekend homework by Sunday afternoon. If the hypothetical "Math God" that she mentions in the video could make math disappear, she'd have a 4.0 GPA, and much less time needed to study -- but then the technology that makes YouTube possible would no longer exist, and she'd be spending Sundays crossing bridges to entertain herself.
We don't need to go back to Euler's day, 300 years ago, to find generations of students who were willing to work hard and forego entertainment. But some traditionalists go back to 100-year-old texts because they feel that newer texts have too many pictures. Technology progressed so much that photography, even in the mid-20th century, was inexpensive (going back to "a point is dot,") but that photo technology made texts even as early as then too entertaining, and therefore, not educational enough for the traditionalists.
The phrase Millennial Generation refers of course to the millennium. Strictly speaking a millennial is one who was born in the old millennium and graduated from high school in the new millennium. By this definition, I am not a millennial, since I was born in December 1980 and graduated high school in June 1999 -- still the old millennium. But some authors, such as Mark Bauerlein, consider the Dumbest Generation to be anyone under 30 at the time of its publication (2008). By this definition, I am a member of the "dumbest generation."
Naturally, most traditionalists and members of older generations who criticize millennials blame the problems of our generation on technology. This is why, when I teach this lesson, I want to point out that using technology to justify being a "dren" who can't count change makes us -- including myself as a member of the generation -- look bad. Of course, in a few years, I can't credibly claim to be in the same generation as my students -- some incoming students starting high school this year are already born in the new millennium (and so are no longer "millennials"). The important thing is that all of us, my age and younger, need to avoid being the "dren" who can't solve simple math problems and instead work on becoming the hero whose knowledge of math saves the day. This is what I want my students -- including those like the girl from the video, if she ever scrapes by Algebra I and is placed in a Geometry class like mine -- to realize.
OK, let's return to 2017. Just as many students may hate math, I bet many of my readers hate it when I "cry over spilled milk" and write about my class from last year. But I'd doing so yet again in today's post, because that comment about Mark Bauerlein's book reminds me of something from last year.
Almost exactly one year ago today, I told my eighth graders about The Dumbest Generation, and I mentioned it here on the blog. But what I never blogged was what happened when I tried to mention the book to my seventh graders the following week.
Once again, this was just before the seventh graders took their first Dren Quiz. I wanted to explain to the students why they had to take a quiz on multiplying by ten, and so I tried to tell them about Bauerlein's book.
But the seventh graders were talking loudly. And this was right at the point when I realized that the students were quiet for my support staff member, but not for me. I knew that my experience as a teacher would be miserable unless I could get the students to be quiet. So I knew I had to do something at that point.
So I told the seventh graders to line up and stand outside until they were quiet. But they never did become quiet, and I knew that if I didn't proceed with the lesson, the students might even consider standing outside as a reward. Thus I continued with the lesson -- which, as you recall, was about Bauerlein's book and the Dren Quiz.
But because the class was still loud, not everyone heard all of what I was saying. One girl had missed everything I spoke except the word "dumbest." And when her mother came to pick her up, she told her what I'd said, and the mom angrily ran up to me and asked, "How dare you insult a 12-year-old girl like that!"
Eventually the mom and I met in the dean's office, where I calmly explained that I was referring to our generation, not to her daughter. The mom apologized, and she scolded her daughter for not telling her the full story. But I don't believe the girl ever forgave me for she thought I said. She was one of the better behaved students and I think she could have enjoyed my class, but instead she always spoke coldly to me and even refused to give me a high-five when I greeted the class in the mornings.
There are several issues here. First of all, the whole incident occurred because -- as was so often true in my classes -- the students were talking loudly and couldn't hear me. I knew that I had to do something about the fact that they obeyed my support aide and not me -- but clearly, making them stand outside changed nothing.
I've said before that I should have made it clear from the first day of school what my expectations were -- namely that they remain silent for most of the period. In particular, I should have ignored all complaints that making them be quiet is "unfair."
But suppose I'd made it to that point (almost a month into the year) and it became obvious that the students only listened to others and not to me. Then this is what I should have done at that point -- first, when my support aide is in the room, I say, "I like how quiet the class is now. This is how quiet I expect the class to be all the time, including when the aide isn't present."
Then I get ready to send my aide out of the room for any reason -- it could be as simple as making copies of anything (such as the Warm-Up sheets). At this point, I anticipate that students will start talking the instant she leaves the room. I make sure that all students have pencil and paper before she leaves, so that they can't claim "I was trying to borrow a pencil!" as a reason to talk.
At this point, my aide leaves the room. I expect that the first student will talk possibly as soon as she opens the door, or at the least before she closes it as she leaves. When this is happening, I scan the room like a hawk, looking for the first student to speak. Then I call out that student. If the student claims "That's unfair!" or "I wasn't talking!" then I give that student the teacher look, and proceed as I mentioned in previous classroom management posts. By following this, the students get the message that I mean it when I tell them not to talk.
Some may question the wisdom of mentioning Bauerlein's book in justifying the Dren Quiz. On one hand, if the class is quiet, then I can tell them about Bauerlein without fear of anyone misconstruing it as a personal insult. On the other hand, my counterpart at our sister school had the students had the students fill in times tables almost everyday -- and I doubt she mentioned Bauerlein's book. So I could have just assigned the Dren Quiz and gave the reason as "Because I said so!"
Finally, I know that several students regularly told me that I was "unfair" -- and that includes the day that I made the students stand outside. But notice that the girl who thought I'd insulted her (that is, the girl who genuinely thought that I was being unfair) said nothing to me at all. Indeed, this is how students whom I genuinely treated unfair would act -- by being cold with me, not by calling out "Unfair!" over and over. Therefore, I should assume that anyone calling me "Unfair!" is just trying to get out of following the rules cheaply. Such students deserve only a teacher look -- not a full-blown argument over why my rules aren't "unfair."
I know that I keep using other blogs, such as Fawn Nguyen and Julie Reulbach, as excuses to keep going back to classroom management. Now it's Sarah Carter's turn:
http://mathequalslove.blogspot.com/2017/09/seating-preference-form.html
Carter writes about a "Seating Preference Form" where students can choose their own seats. I've written about some of the issues I had with seats in both the seventh and eighth grade classes. I wonder whether using this would have solved some of those issues.
For example, recall that there were four students per group, while the clique of five girls in eighth grade wanted to sit together. Maybe by knowing this, they might have decided for themselves which girl would have to sit elsewhere, rather than have me choose.
Meanwhile, in seventh grade, the four troublemaker boys needed to be separated. Most likely they would have requested to sit together -- but then I could tell them that in trying to seat everyone else according to their own preferences, the only remaining seats for them were separated!
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