Monday, August 31, 2015

Lesson 1-6: The Need for Undefined Terms (Day 4)

This is what I wrote last year 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 -- pointline, 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 pointline, 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 406,000 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 10% over last year. 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 pointline, 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 MBApolynomialinvestingdatasupermarket, 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.

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