"Here mathematical units and symbols were used to get the point across about recycling paper!"
This is the only page of the subsection "Recycling the Numbers." On this page, Pappas provides a whole list of statistics related to recycling. Here are some of them:
- A ton of virgin paper =/= [ASCII for "does not equal"] a ton of recycled paper
- A ton of recycled paper uses 4102 kwh less energy.
- A ton of recycled paper uses 7000 gallons less water to produce.
- A ton of recycled paper produces 3 cubic yards less solid waste.
- A ton of recycled paper uses 17 fewer logged trees.
-- the numbers behind recycling and landfill --
- Only 29% of all newspapers produced are recycled by the consumer.
- 97% of the virgin forests of the continental USA have been cut down in the past 200 years.
It's only fitting that I write about recycling and the environment today. That's because one year ago today, I received the email suggesting me to have my class participate in the Green Team. This is what I blogged about Green Team on today's date last year:
1:00 -- During lunch, I decide to check my e-mail. Today I receive a very interesting message -- it is from a senior specialist at a local non-profit organization. She proposes to have a special meeting with me to discuss a special middle school science program, called Green Team. In short, her organization would join up with the LA Department of Water and Power to teach students about energy and water -- specifically how to conserve these resources.
As I wrote earlier in this post, I welcome any and every opportunity to sneak in extra science lessons into my science-starved class. So I reply that I'm willing to participate in Green Team. Next week I'm scheduled to meet with her to discuss the program in more detail and how to implement it.
As I wrote earlier in this post, I welcome any and every opportunity to sneak in extra science lessons into my science-starved class. So I reply that I'm willing to participate in Green Team. Next week I'm scheduled to meet with her to discuss the program in more detail and how to implement it.
And as Pappas points out, one way to save energy and water is to recycle paper -- it indirectly saves thousands of kilowatt hours of energy and thousands of gallons of water.
Of course, every time I read Pappas discussing something to do with science, I think back to how I failed to teach science properly in my class. In the end, I have no idea how successful Green Team was at my school, since the program didn't really heat up until right around the time I left.
The real problem is when I wrote the above in last year's post, "sneak extra science lessons into my science-starved class." That is, my science lessons were completely dependent on Green Team and, if you recall, the Bruin Corps students from UCLA who visited my class (and whom I also mention in that same post from last year). This was a huge mistake -- I should have started with a science curriculum from the get go and use Bruin Corps and Green Team to supplement my curriculum.
One reason I was so hesitant to teach science was that I was afraid of teaching the wrong thing. I knew that there was a California Science Test for eighth graders coming up, but I had no idea what would be on the test -- and indeed, there was a standoff between the feds and the state regarding whether the old California Standards or new NGSS would be tested.
At my tiny charter school, no one informed me of which science standards to teach. But now -- one year too late -- I finally know what I should have done. I should have consulted the nearest large school district, LAUSD, and find out what they were teaching, and how they were making the transition from California Standards to NGSS.
Here's a matrix I found on the LAUSD website:
And so last year -- the 2016-2017 school year -- I should have followed the first row of boxes and followed the recommendations for each grade level. In other words, I use this matrix to justify -- to both myself and my class -- what science lessons I'm teaching.
For example, how would I deal with the unknown California Science Test? That's easy -- I just teach whatever it says in the eighth grade box on the chart. If this matches the content of the state test, then my students would be well-prepared for it. And if this doesn't match the content of the test, then at least my students would be no more unprepared for it than the students of LAUSD.
Last year, I wrote about an argument I had with some students regarding my science lessons:
-- On Thursday, the eighth graders were doing a science lesson with my Bruin Corps member (again, see yesterday's "Day in the Life" post). This was a life science lesson, but it appeared in the eighth grade standards according to our online software. One girl (yes, the same girl from Monday) correctly noted that she learned this lesson in seventh grade (under the old pre-NGSS standards) and therefore she'd refuse to work on the lesson. I was about to login to our website and show her the standards, but she continued to complain before I could get to the site, so I yelled at her again.
I could have avoided this type of argument by returning to the chart again. I ask the complaining student three questions:
- What school year is this?
- What grade are you in?
- What does it say in the box under this school year and your grade?
There's no need to attempt to explain what NGSS is or what's going to appear on the test in May -- I just show the students the chart. If a student says, "You don't know what you're talking about," my response can be, "You're right -- I don't know what I'm talking about, but this chart was created by people who do know what they're talking about. We're following the chart, and that's final." And if the student refuses to learn the content mentioned on the chart, then I can just whip out the chart at parent conferences and point to the proper box. "This is what your child was supposed to learn in science this year, and your student failed to learn it."
I'm upset at myself that I didn't go online and look for this chart the instant I found out that I would be teaching science last year. And I should have started teaching science -- for all three grades -- in September, not November. (Allow some leeway in August for Benchmark Tests, but in September there was no excuse.) You may notice that the chart is dated December 2016, but I suspect that the chart was last tweaked in December (and those tweaks might have affected only the 2017-2018 or 2018-2019 school years). I seriously doubt that there was no chart online explaining what to teach in 2016-2017 until December of that year.
Let's look at the chart in more detail. We see that all three grades have a mix of the old California standards ("98 Standards," as in 1998, the date the old standards were formulated) and the new science standards. Note that the transition is from the old standards (6th Grade Earth Science, 7th Grade Life Science, and 8th Grade Physical Science) to the new standards (following the preferred Integrated Model).
In both sixth and eighth grade, there are two "98 Standards" and a choice (indicated by "Or") of NGSS strands to teach. It's easy just to tie each strand to a trimester, so sixth graders get "Plate Tectonics and Heat Transfer" in the fall, "Ecosystems" in the winter, and one of the NGSS strands in the spring.
In seventh grade, there are three "98 Standards," but it might be possible to squeeze them into two trimesters anyway, saving NGSS for the third trimester. Notice that Green Team can be used to make a choice between "Or" strands -- "Natural Processes and Human Activities Shape Earth's Resources and Ecosystems" fits Green Team much better. (Yes, I know that the rock cycle appears on the state test, but this cohort also saw the rock cycle in sixth grade, which probably explains why it's given in seventh grade as an "Or" standard.) That's the extent of Green Team in determining curriculum -- it's used merely to choose between "Or" standards, not as the entire curriculum.
Eighth grade is an interesting year. The first strand listed there is "Forces and Interactions," and so this ought to be taught during the first trimester. But in my Mole Day post, I wrote about how I could have used that week to engage the kids in chemistry, so maybe "Chemical Reactions" should be the first trimester strand. Then again, I used the holidays of Rosh Hashanah and Yom Kippur to engage the students in astronomy (as in the earth, moon, and sun cycles that determine the date of those holidays), and so maybe "Astronomy" should be the first trimester strand.
Here's how we could accommodate all of those dates. I begin the year with "Astronomy," from the start of the year until Yom Kippur. Then I switch to "Chemical Reactions" -- notice that there were only 11 days last year from Yom Kippur to Mole Day -- and this lasts until the end of the trimester. In second trimester, I teach "Forces and Interactions." And then in third trimester, I teach "Sustaining Local and Global Biodiversity" (which lines up with Green Team, as one way to sustain biodiversity is to take care of the environment) until the day of the state test. This means that the eighth graders would get both sides of the "Or" choice.
In all three grades, the first two trimesters follow the old earth, life, physical sciences pattern. This means that I can use the corresponding Illinois State science text for each grade -- including the accompanying projects. In the third trimester I might jump between texts in each grade.
In all three grades, the first two trimesters follow the old earth, life, physical sciences pattern. This means that I can use the corresponding Illinois State science text for each grade -- including the accompanying projects. In the third trimester I might jump between texts in each grade.
So this plan incorporates the LAUSD chart, Green Team, Illinois State, and the special days to teach certain eighth grade lessons. The only thing I left out was Bruin Corps. Well, the Bruin Corps member who came on Thursdays was a molecular bio major who could help out the seventh graders during the first unit, "Cell Biology." The other Bruin Corps member who came in three days a week was an atmospheric and oceanic sciences major. She might have been able to assist the sixth graders during their second unit, "Ecosystems," since the atmosphere and ocean are indeed ecosystems. (I guess the atmosphere is technically part of an ecosystem.)
It's so sad that I wasn't able to teach this last year. Again, I'm crying over spilled milk -- but I can't help myself today, when the Pappas page reminds me of last year's Green Team.
Chapter 5 of George Szpiro's Poincare's Prize is called "Geometry Without Euclid." Here's how the chapter begins:
"Three mathematicians are shown a cube and asked to describe what they see. The first, a geometer, says, 'I see a cube.' The second is a graph theorist. She ventures, 'I see eight points, connected by twelve edges.' The third, a topologist, declares, 'I see a sphere.'"
In this chapter, Szpiro will introduce us to topology -- the subject of the Poincare Conjecture. We've already seen some topology in Pappas (see, for example, my May 10th and August 23rd posts) as well as in Kung's lectures. Along the way, he presents us with the biographies of the mathematicians who pioneered the field. I won't be able to give all the biographies here -- again, you can read the book yourself if you want to learn more.
Spziro begins by telling us about the famous Bernouilli brothers of 18th-century Switzerland:
"The Bernoulli family produced a few additional remarkable men of science [besides the famous Johann and his son Daniel -- dw]. In fact, the position in St. Petersburg had become available because the previous incumbent, Daniel's brother Nikolaus Bernouilli, had just dies."
But this is actually the biography of the man who fills that faculty position -- Leonhard Euler. As the author tells us:
"While at the Academy of Sciences in St. Petersburg, Euler received an intriguing letter from the picturesque town in Konigsberg in Prussia (Kaliningrad in today's Russia)."
Yes, you already know what's coming -- the Konigsberg bridge problem yet again. But arguably, this problem is the world's first topology problem -- and is, indeed, the only topology problem highlighted in the U of Chicago text. By the way, when I was searching for the Numberphile video on the French grading scale, I stumbled upon a Numberphile video of the Konigsberg bridge problem. It was first posted around this time last year, so it was after my first day of school project at my old school. But still, I wish I'd found this video before we reached Lesson 1-4 in the U of Chicago text this year:
Szpiro gives us more details about how Euler comes to solve this problem. He is first notified of the problem by Carl Leonhard Ehler, the mayor of the nearby town of Danzig. (I assume they observe Year-Round Standard Time in Danzig, as in Sheila Danzig.):
"In a somewhat indignant letter he answered Ehler that the problem 'bears little relationship to mathematics and I do not understand why [solutions] are expected from mathematicians rather than from anybody else, since they are based solely on common sense."
But of course, the mayor ultimately convinces Euler to work on the problem anyway:
"The thought that Leibniz's geometry of position was the appropriate tool had not occurred to Euler, but rather to Ehler, who had mentioned in his letter that the Konigsberg problem 'would prove to be an outstanding example of the calculus of position, worthy of your great genius.'"
Of course, we already know the solution to the problem (if you don't, watch the video above). The important thing is that measurement is irrelevant:
"Here the lengths of the paths were of no interest, and neither were the exact shapes of the town's four quarters. All that mattered was how the different parts of the town were connected by bridges."
OK, we're done with Konigsberg, but not with Euler. As Szpiro tells us:
"Fourteen years later, in 1750, Euler struck again. Of course, Euler did many things in between, but here we are only interested in work that relates to topology."
Now here we are referring to Euler's magic formula: F + V - E = 2, where V is the number of vertices, E the number of edges, and F the number of faces of a polyhedron. Pappas refers to this formula in Chapter 6, and I listed the formula in giving the table of contents, but I never reached the page. She lists the formula on page 171, and the 171st day of the year was June 20th, a summer day when I simply didn't post. Also, the formula is mentioned in an exercise in Lesson 9-7 of the U of Chicago text (as one of the formulas in a multiple choice problem), but we've never fully covered Chapter 9 on the blog at all.
But it was discovered a few years after Euler's death that the formula is only valid if the polyhedron happens to be convex. This term is defined in the U of Chicago text, and Szpiro explains why a cube is convex:
"The line connecting the corners lies wholly inside the cube; hence the cube is convex. Now drill a hole through the cube."
And once we do so, the polyhedron is no longer convex, and Euler's formula no longer applies. The author writes about one of the mathematicians who fills in the details for nonconvex polyhedra, Swiss mathematician Simon Antoine l'Huilier:
"[Tutoring the prince's son] must have been a cushy job because his teaching duties left him enough spare time to do research and to write mathematical papers."
His discovery is that F + V - E isn't just 2, but two plus the number of holes it contains. Thus a polyhedron with two holes has an Euler characteristic of 6:
"Again it does not matter whether the body is a cube or an icosahedron and the cavities are two octahedra, or an octahedron and a pyramid, or a cube and a tetrahedron, or anything else."
The connection to topology is that just as the Euler characteristic categorizes polyhedra by the number of holes they contain, topology also classifies figures by the number of holes.
Now the author turns to the mathematician who gives this fledgling field the name "topology" -- the 19th century German mathematician Johann Benedict Listing:
"From the age of thirteen, he helped his parents augment their meager income by doing drawings and calligraphy for the public."
Szpiro continues with many details about Listing's life -- and here our biographies must end, since I want to get on to Geometry. So let's just skip to the end, where he writes about Heinz Hopf, a topologist who ultimately escapes from the Holocaust:
"When he was informed in the late 1930's in a letter from an ostensible friend, the head of Hopf's student fraternity, that 'with regret' he had to be kicked out of the association of former members because his father was Jewish and his wife's brother an anti-Nazi, Hopf drew the consequences: He applied for, and received Swiss citizenship."
And so Hopf become a citizen of the nation that's produced the first topologist, Euler.
Lesson 6-5 of the U of Chicago text is called "Congruent Figures." In the modern Third Edition, we must backtrack to Lesson 5-1 to learn about congruent figures.
Here's a link to a more recent comment from Ze'ev Wurman:
https://edsource.org/2017/understanding-the-common-core-state-standards-in-california-a-quick-guide
Let's skip to what Wurman writes about Geometry:
Today is the last day before the week-long Thanksgiving break. (It's also approximately the end of the third quaver.) It's also an activity day -- after all, it would be awkward for yesterday's lesson to be basically an activity (and even Wednesday's lesson contained a semi-activity) and then not give an activity today. And so I'm only repeating one of the worksheets from last year and then adding an activity page.
Last year at my old school, I gave the students an activity on graphing turkeys. The idea is that I want to give an activity that is somewhat fun on the last day before the holiday -- and perhaps when they are done graphing, they can color the turkeys. The problem is that the turkeys take much too long to graph, and the activity ends up not being much fun at all.
This is what I wrote about the turkey activity -- again, it was one year ago today:
10:15 -- My eighth graders leave and my sixth graders arrive. Just like the seventh graders, this class works on the turkey graphing activity. During this time, I hand out four licorice sticks to all students who earned an A on yesterday's quiz.
11:05 -- My sixth graders leave for nutrition, except for one girl who has incurred a short detention for going to the restroom yesterday when our school has a no-pass policy.
11:25 -- My sixth grade class returns from nutrition. Ordinarily we don't have the same class before and after the break, but our schedule is awkward this week due to Parent Conferences. The three middle school teachers decided that we would see all three grades before the break, and then rotate so that we see one grade each day after the break. Today happens to be sixth grade for me.
Ordinarily when I see sixth grade for the second time in a day, it's considered Math Intervention and the students use our other software program, IXL. (Again I don't give the link here.) I wrote about Math Intervention in my "Day in the Life" post for October.
The sixth graders are often very loud during IXL time, and today is no exception. As I implied back in October, some students still have trouble with their passwords. Since then, I came up with the idea of handing out computers to only some of the students and putting the rest on a waiting list. All students who fail to login to IXL within five minutes must forfeit their computers and give them to someone on the waiting list.
I intend for the students who lose their computers to go back to the turkey activity. Instead, these students interpret this time to be free time -- they talk very loud and run around the classroom, and even some of the students who have laptops join them. In short, IXL time for sixth graders has become a big mess! Today, it might have been better just to have all of the students continue to graph the turkey, since none of them have actually finished it. I should have either foregone IXL entirely or reserved it for those few students who actually complete the turkey.
And so now in 2017, I want to post an activity that the students will enjoy more. We see how the sixth graders started playing around because they got tired of graphing.
Today, I discovered what activity the middle school students at my old school did this week. They were to draw pictures (not necessarily turkeys) on graph paper by coloring in the boxes (rather than graphing coordinates).
This activity reminds me of Lesson 1-1 of the U of Chicago text. The first description of a point is "A point is a dot," and the squares on the graph paper represent dots that form the image. And the very first word defined in our Geometry text is pixels -- and we notice that this is exactly what the students created, pixelated art.
I've spent so much of this post writing about science, but art is the other part of the curriculum that I failed to teach last year. Art is the "A" of STEAM, and the Illinois State text encourages that art be taught in conjunction with math and science.
OK, I said I was done with the biographies in Szpiro, yet I'm returning to his book yet again. Let's continue to biography of Johann Listing, the namer of topology, that we started earlier:
"From the age of thirteen, he helped his parents augment their meager income by doing drawings and calligraphy for the public. Despite his modest upbringing he attended good schools and received a sound education in modern and classical languages, mathematics, and the sciences. Such were his achievements at school that he was awarded a scholarship that enabled him to continue his studies at university. Since the stipend was granted only for the study of arts, not science or mathematics, Listing chose architecture as a compromise."
Thus we see how Listing's art skills allowed him to learn math as well -- the power of STEAM. In the Illinois State weekly plan I posted earlier, science would have been taught on Thursdays, while art is part of the Learning Centers that would take place on Wednesdays.
Anyway, I notice that the middle school students worked on this project the entire week before Thanksgiving break -- not just one day. This allows the students to take pride in their work -- and if they don't finish the first day, they can continue the next day.
Indeed, perhaps I could have kept the turkey coordinate graphing project intact last year by simply spreading it out over the entire week. Monday of course was for coding, but then I could have let them graph on Tuesday, Wednesday, and Thursday. Then perhaps the sixth grade scene I posted earlier doesn't occur -- students would be eager to finish the graph, since there would be plenty of time left to color the turkeys.
I also mentioned that there was some leeway to take a week off from Common Core Math, especially during short three-days weeks (due to PD days). The week before Thanksgiving was a four-day week since there was only one PD day, but the weeks leading up to winter break, Presidents Day, and Chavez Day were all three-day weeks, hence suitable for similar arts projects.
This is one arts project that doesn't require the die cut machine, which I didn't know how to use. In the end, whether I would have given last year's turkey graph or this year's pixel graph depends on Illinois State. It's possible that my successor teacher found this assignment in the Illinois State text (even though it was difficult to find arts projects above Grade 5) -- and of course, priority always goes to Illinois State projects. (Ironically, the only middle school arts projects that I could easily find in the Illinois State text were in conjunction with science, not math.)
But for this year, I will post the pixel version of this assignment -- the one that the middle school students enjoyed this week. (Yes, I know I'm posting it as a one-day activity since I've never posted multi-day projects on the blog.)
You may ask, what does this have to do with today's Lesson 6-5? Well, suppose the students choose to draw a human face, or even a turkey viewed head-on. Humans, turkeys, and other animals exhibit bilateral symmetry. So the two eyes will be congruent.
And so let me announce my holiday posting schedule. Last year I made only one post during Thanksgiving break -- on Monday, the day I met the Green Team coordinator in person. Two years ago, I established the tradition of posting twice during the break -- on Turkey Day itself and the next day, Black Friday. This year I return to my 2015 posting schedule, with posts Thursday and Friday.
In addition to Pappas and Poincare, the holiday posts will contain some music. I want to have some actual songs to be played on the computer emulator, using the new 7-limit scales that we discovered.
Enjoy your Thanksgiving break. The district whose calendar I'm observing will resume school on Monday, November 27th.
"Three mathematicians are shown a cube and asked to describe what they see. The first, a geometer, says, 'I see a cube.' The second is a graph theorist. She ventures, 'I see eight points, connected by twelve edges.' The third, a topologist, declares, 'I see a sphere.'"
In this chapter, Szpiro will introduce us to topology -- the subject of the Poincare Conjecture. We've already seen some topology in Pappas (see, for example, my May 10th and August 23rd posts) as well as in Kung's lectures. Along the way, he presents us with the biographies of the mathematicians who pioneered the field. I won't be able to give all the biographies here -- again, you can read the book yourself if you want to learn more.
Spziro begins by telling us about the famous Bernouilli brothers of 18th-century Switzerland:
"The Bernoulli family produced a few additional remarkable men of science [besides the famous Johann and his son Daniel -- dw]. In fact, the position in St. Petersburg had become available because the previous incumbent, Daniel's brother Nikolaus Bernouilli, had just dies."
But this is actually the biography of the man who fills that faculty position -- Leonhard Euler. As the author tells us:
"While at the Academy of Sciences in St. Petersburg, Euler received an intriguing letter from the picturesque town in Konigsberg in Prussia (Kaliningrad in today's Russia)."
Yes, you already know what's coming -- the Konigsberg bridge problem yet again. But arguably, this problem is the world's first topology problem -- and is, indeed, the only topology problem highlighted in the U of Chicago text. By the way, when I was searching for the Numberphile video on the French grading scale, I stumbled upon a Numberphile video of the Konigsberg bridge problem. It was first posted around this time last year, so it was after my first day of school project at my old school. But still, I wish I'd found this video before we reached Lesson 1-4 in the U of Chicago text this year:
Szpiro gives us more details about how Euler comes to solve this problem. He is first notified of the problem by Carl Leonhard Ehler, the mayor of the nearby town of Danzig. (I assume they observe Year-Round Standard Time in Danzig, as in Sheila Danzig.):
"In a somewhat indignant letter he answered Ehler that the problem 'bears little relationship to mathematics and I do not understand why [solutions] are expected from mathematicians rather than from anybody else, since they are based solely on common sense."
But of course, the mayor ultimately convinces Euler to work on the problem anyway:
"The thought that Leibniz's geometry of position was the appropriate tool had not occurred to Euler, but rather to Ehler, who had mentioned in his letter that the Konigsberg problem 'would prove to be an outstanding example of the calculus of position, worthy of your great genius.'"
Of course, we already know the solution to the problem (if you don't, watch the video above). The important thing is that measurement is irrelevant:
"Here the lengths of the paths were of no interest, and neither were the exact shapes of the town's four quarters. All that mattered was how the different parts of the town were connected by bridges."
OK, we're done with Konigsberg, but not with Euler. As Szpiro tells us:
"Fourteen years later, in 1750, Euler struck again. Of course, Euler did many things in between, but here we are only interested in work that relates to topology."
Now here we are referring to Euler's magic formula: F + V - E = 2, where V is the number of vertices, E the number of edges, and F the number of faces of a polyhedron. Pappas refers to this formula in Chapter 6, and I listed the formula in giving the table of contents, but I never reached the page. She lists the formula on page 171, and the 171st day of the year was June 20th, a summer day when I simply didn't post. Also, the formula is mentioned in an exercise in Lesson 9-7 of the U of Chicago text (as one of the formulas in a multiple choice problem), but we've never fully covered Chapter 9 on the blog at all.
But it was discovered a few years after Euler's death that the formula is only valid if the polyhedron happens to be convex. This term is defined in the U of Chicago text, and Szpiro explains why a cube is convex:
"The line connecting the corners lies wholly inside the cube; hence the cube is convex. Now drill a hole through the cube."
And once we do so, the polyhedron is no longer convex, and Euler's formula no longer applies. The author writes about one of the mathematicians who fills in the details for nonconvex polyhedra, Swiss mathematician Simon Antoine l'Huilier:
"[Tutoring the prince's son] must have been a cushy job because his teaching duties left him enough spare time to do research and to write mathematical papers."
His discovery is that F + V - E isn't just 2, but two plus the number of holes it contains. Thus a polyhedron with two holes has an Euler characteristic of 6:
"Again it does not matter whether the body is a cube or an icosahedron and the cavities are two octahedra, or an octahedron and a pyramid, or a cube and a tetrahedron, or anything else."
The connection to topology is that just as the Euler characteristic categorizes polyhedra by the number of holes they contain, topology also classifies figures by the number of holes.
Now the author turns to the mathematician who gives this fledgling field the name "topology" -- the 19th century German mathematician Johann Benedict Listing:
"From the age of thirteen, he helped his parents augment their meager income by doing drawings and calligraphy for the public."
Szpiro continues with many details about Listing's life -- and here our biographies must end, since I want to get on to Geometry. So let's just skip to the end, where he writes about Heinz Hopf, a topologist who ultimately escapes from the Holocaust:
"When he was informed in the late 1930's in a letter from an ostensible friend, the head of Hopf's student fraternity, that 'with regret' he had to be kicked out of the association of former members because his father was Jewish and his wife's brother an anti-Nazi, Hopf drew the consequences: He applied for, and received Swiss citizenship."
And so Hopf become a citizen of the nation that's produced the first topologist, Euler.
Lesson 6-5 of the U of Chicago text is called "Congruent Figures." In the modern Third Edition, we must backtrack to Lesson 5-1 to learn about congruent figures.
This is what I wrote two years ago about today's lesson:
Lesson 6-5 of the U of Chicago text is on congruent figures. Congruence is one of the most important concepts in all of geometry, especially Common Core Geometry.
As I mentioned many times on this blog, the word congruent is defined very differently in Common Core Geometry than under previous standards. We all know what it means for two segments to be congruent -- that is, that they have equal length -- or for two angles to be congruent -- that is, that they have equal measure. The new definition ofcongruent appears to be original to the Common Core, and yet, it isn't. Years before the Core, the U of Chicago text used the following definition of congruent -- indeed, it is mainly because of this definition that I chose the U of Chicago as the textbook on which this blog is based:
Definition:
Two figures, F and G, are congruent figures [...] if and only if G is the image of F under a translation, a reflection, a rotation, or any composite of these.
And there we have it -- this definition of congruent predates Common Core. But many opponents of Common Core do not like this new definition. One such opponent is Ze'ev Wurman, a member of the commission here in California that reviewed the Common Core standards. Although his views are posted at several websites, one of the best Wurman articles I found is at this link:
http://www.libertylawsite.org/2014/03/27/the-common-cores-pedagogical-tomfoolery/
Skipping down to the discussion of the math standards -- since, as Wurman himself points out, math is his area of expertise -- the author begins with some elementary school standards. For example, Wurman gives this standard:
1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Wurman states that this standard should have stopped after the first sentence. Instead, it goes on to prescribe some nonstandard algorithms for addition. I already discussed much of this in many previous posts. Since much of what Wurman writes about grades 1-3 echo what I wrote about the lower grades, I am in full agreement with Wurman for the lower grades.
But then Wurman moves on to Common Core Geometry. Here's what he writes about it:
A true content standard would simply say “Students prove triangle congruence” or, perhaps, “Students understand triangle congruence,” leaving the method of instruction to the teacher. Instead, Common Core not only dictates how to teach congruence, it insists on a specific experimental method of instruction that has an established a track record of failure where it was invented [...]
[emphasis Wurman's]
He then gives a link to a PDF file about the Russian mathematician and scientist A.N. Kolmogorov, whom the PDF credits as the creator of geometry based on transformations. Wurman implies that this geometry was tried out in Russia (i.e., the Soviet Union) and was a big failure.
As I mentioned many times on this blog, the word congruent is defined very differently in Common Core Geometry than under previous standards. We all know what it means for two segments to be congruent -- that is, that they have equal length -- or for two angles to be congruent -- that is, that they have equal measure. The new definition ofcongruent appears to be original to the Common Core, and yet, it isn't. Years before the Core, the U of Chicago text used the following definition of congruent -- indeed, it is mainly because of this definition that I chose the U of Chicago as the textbook on which this blog is based:
Definition:
Two figures, F and G, are congruent figures [...] if and only if G is the image of F under a translation, a reflection, a rotation, or any composite of these.
And there we have it -- this definition of congruent predates Common Core. But many opponents of Common Core do not like this new definition. One such opponent is Ze'ev Wurman, a member of the commission here in California that reviewed the Common Core standards. Although his views are posted at several websites, one of the best Wurman articles I found is at this link:
http://www.libertylawsite.org/2014/03/27/the-common-cores-pedagogical-tomfoolery/
Skipping down to the discussion of the math standards -- since, as Wurman himself points out, math is his area of expertise -- the author begins with some elementary school standards. For example, Wurman gives this standard:
1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Wurman states that this standard should have stopped after the first sentence. Instead, it goes on to prescribe some nonstandard algorithms for addition. I already discussed much of this in many previous posts. Since much of what Wurman writes about grades 1-3 echo what I wrote about the lower grades, I am in full agreement with Wurman for the lower grades.
But then Wurman moves on to Common Core Geometry. Here's what he writes about it:
A true content standard would simply say “Students prove triangle congruence” or, perhaps, “Students understand triangle congruence,” leaving the method of instruction to the teacher. Instead, Common Core not only dictates how to teach congruence, it insists on a specific experimental method of instruction that has an established a track record of failure where it was invented [...]
[emphasis Wurman's]
He then gives a link to a PDF file about the Russian mathematician and scientist A.N. Kolmogorov, whom the PDF credits as the creator of geometry based on transformations. Wurman implies that this geometry was tried out in Russia (i.e., the Soviet Union) and was a big failure.
[2017 update: Kolmogorov is one of the mathematicians mentioned in Chapter 4 of Szpiro's book, and indeed I mentioned him in yesterday's post. Yes, I said I wasn't going to write any more biographies in this post, but now I wish to reread what Szpiro writes about him and compare it to the things that Wurman says about him.]
Szpiro writes:
"In 1954, the Russian Andrei Nikolaevich Kolmogorov gave a plenary lecture about the n-body problem at the congress of the International Mathematical Union in Amsterdam. The subject was the thorny question of what happens to periodic orbits of bodies when small perturbations disturb their course.
"At the Courant Instititute in New York, the German Fulbright scholar Jurgen Moser was asked by an editor of Mathematical Reviews to write a report on Kolmogorov's paper. As Moser studied it more closely, it seemed to him that Kolmogorov's central thesis was not completely proven. For six long years Moser worked on the problem before he could definitely close the missing gap in Kolmogorov's proof. Some mathematicians maintain that Kolmogorov's proof was not lacking to start with.
"At the same time that Moser was looking for the missing piece of the puzzle, one of Kolmogorov's students, Vladimir Igorevich Arnold, attacked the n-body problem from a different angle. In honor of the three mathematicians -- Kolmogorov, Arnold, and Moser -- the new theory was named after their initiials: KAM theory."
Of course, Szpiro doesn't tell us whether transformational geometry appears in KAM theory. The link between Kolmogorov's axioms and Common Core, to me, is uncertain -- but it is possible that both U of Chicago and Common Core derived their ideas from Kolmogorov.
Why should we use the transformation approach? In pre-Core Geometry, we must define the word congruent three times -- first for segments, then for angles, and finally for figures. But in some ways, this is an ad hoc approach. In the Common Core, we only definecongruent once, and it applies to segments, angles, and figures all at once.
In the Common Core, congruent means "identical up to isometry" (and later on, we see that similar is defined as "identical up to a similarity transformation"). There are many concepts in college-level mathematics that are defined similarly -- such as topologically equivalent ("identical up to homeomorphism"), equinumerous ("identical up to bijection"), and so on. Furthermore, the Lebesgue measure of a set is defined so that two sets such that there is an isometry mapping one to the other have the same measure.
So in some ways, the Common Core definition is more rigorous than the pre-Core definitions. Also, in some ways, the Common Core definition predates Kolmogorov by a wide margin -- Euclid himself used it as the Principle of Superposition in his proof of SAS (Proposition I.4).
In Hilbert's formulation of Euclid's axioms, congruence is a primitive notion -- that is, it is undefined just as point, line, and plane are. Actually, it's two undefined terms, since Hilbert considers segment and angle congruence separately. As I mentioned before, we can't define an undefined term, but instead we know what it means through the use of axioms or postulates. Hilbert provides six axioms of congruence -- these cover the Equivalence Properties and Segment and Angle Congruence Theorems as given in this section, some of the Point-Line-Plane and Angle Measure Postulates, and SAS. We notice that Hilbert's congruence is completely nonmetric -- there is no notion of distance or angle measure anywhere.
So which formulation should we use? This is a Common Core site and so I use the Common Core definition of congruence, but in the long run, which is best for the students? The usual guiding principles is that if a concept is easy for the students to understand and leads to a higher concept, then the students should learn how to prove it. But if the lower concept is difficult for the students, it should be made into a postulate and not proved in class.
So we can see a full continuum, from more proofs to more postulates:
Szpiro writes:
"In 1954, the Russian Andrei Nikolaevich Kolmogorov gave a plenary lecture about the n-body problem at the congress of the International Mathematical Union in Amsterdam. The subject was the thorny question of what happens to periodic orbits of bodies when small perturbations disturb their course.
"At the Courant Instititute in New York, the German Fulbright scholar Jurgen Moser was asked by an editor of Mathematical Reviews to write a report on Kolmogorov's paper. As Moser studied it more closely, it seemed to him that Kolmogorov's central thesis was not completely proven. For six long years Moser worked on the problem before he could definitely close the missing gap in Kolmogorov's proof. Some mathematicians maintain that Kolmogorov's proof was not lacking to start with.
"At the same time that Moser was looking for the missing piece of the puzzle, one of Kolmogorov's students, Vladimir Igorevich Arnold, attacked the n-body problem from a different angle. In honor of the three mathematicians -- Kolmogorov, Arnold, and Moser -- the new theory was named after their initiials: KAM theory."
Of course, Szpiro doesn't tell us whether transformational geometry appears in KAM theory. The link between Kolmogorov's axioms and Common Core, to me, is uncertain -- but it is possible that both U of Chicago and Common Core derived their ideas from Kolmogorov.
Why should we use the transformation approach? In pre-Core Geometry, we must define the word congruent three times -- first for segments, then for angles, and finally for figures. But in some ways, this is an ad hoc approach. In the Common Core, we only definecongruent once, and it applies to segments, angles, and figures all at once.
In the Common Core, congruent means "identical up to isometry" (and later on, we see that similar is defined as "identical up to a similarity transformation"). There are many concepts in college-level mathematics that are defined similarly -- such as topologically equivalent ("identical up to homeomorphism"), equinumerous ("identical up to bijection"), and so on. Furthermore, the Lebesgue measure of a set is defined so that two sets such that there is an isometry mapping one to the other have the same measure.
So in some ways, the Common Core definition is more rigorous than the pre-Core definitions. Also, in some ways, the Common Core definition predates Kolmogorov by a wide margin -- Euclid himself used it as the Principle of Superposition in his proof of SAS (Proposition I.4).
In Hilbert's formulation of Euclid's axioms, congruence is a primitive notion -- that is, it is undefined just as point, line, and plane are. Actually, it's two undefined terms, since Hilbert considers segment and angle congruence separately. As I mentioned before, we can't define an undefined term, but instead we know what it means through the use of axioms or postulates. Hilbert provides six axioms of congruence -- these cover the Equivalence Properties and Segment and Angle Congruence Theorems as given in this section, some of the Point-Line-Plane and Angle Measure Postulates, and SAS. We notice that Hilbert's congruence is completely nonmetric -- there is no notion of distance or angle measure anywhere.
So which formulation should we use? This is a Common Core site and so I use the Common Core definition of congruence, but in the long run, which is best for the students? The usual guiding principles is that if a concept is easy for the students to understand and leads to a higher concept, then the students should learn how to prove it. But if the lower concept is difficult for the students, it should be made into a postulate and not proved in class.
So we can see a full continuum, from more proofs to more postulates:
- Common Core: SAS, ASA, SSS all proved (using transformations)
- Hilbert (supported by David Joyce): SAS postulated, ASA, SSS proved (using SAS)
- Status quo: SAS, ASA, SSS all postulated (most texts)
- Minimalist: SAS, ASA, SSS not mentioned (isosceles/parallelogram properties postulated)
The argument from Wurman and other Common Core opponents is that proving SAS, ASA, SSS from transformations only confused students (which would be the reason why this would have been a big failure in the Soviet Union) and that they should be assumed as postulates. But then, we wonder, why not go one step further and state that all proofs confuse students, so that all proofs involving SAS, ASA, SSS should be dropped, and the properties of triangles and parallelograms assumed? Why is the status quo, where SAS, ASA, SSS are assumed and used in proofs, exactly the right level of complexity for the students?
Well, this is what I hope to find out through this blog. It could be that these Core opponents are correct, and that the status quo level of complexity is exactly appropriate for high school students taking geometry. To me, this is not as clear-cut as elementary math, where the standard algorithms for addition and subtraction are clearly superior to the nonstandard algorithms. This is the reason that I agree with the traditionalists for K-3 math, but not high school math yet.
As for the other theorems proved in this chapter, the Equivalence Properties of Congruence is proved in a way that is standard for many types of transformations -- by using the identity, inverse, and composite functions. The Segment and Angle Congruence Theorems are proved using reflections only, since the text states (in the "Shorter Form" of the definition of congruence) that only reflections, or a composite thereof, are needed to establish congruence. But sometimes it's easier for students to visualize other transformations -- for example, in the Segment Congruence Theorem, one can simply translate X to Z, so that X' and Z coincide. Then one can rotate W' to Y, so that W" andY coincide. In the text, both of these are reflections instead.
Notice that this lesson, 6-5, is the first lesson in which the word congruent appears. The U of Chicago text is careful to use phrases such as "of equal length (measure)" instead of congruent.
I've mentioned before that many people -- both teachers and subjects -- use the words equal and congruent interchangeably. There are two distinctions to make -- one is that numbers (including lengths and angle measures) are equal, while segments and angles are congruent. The other is that we don't know that any figures are congruent until we know of an isometry mapping one to the other, which the Segment and Angle Congruence Theorems provide.
In this course, the latter distinction has priority. I admit that I myself have called angles "equal" (when it's their measures that are equal) on this blog -- because I don't want to call them "congruent" until reaching the Segment and Angle Congruence Theorems. I am especially guilty of this when I write phrases such as "Angle A = Angle B" because it's so much easier than trying to write an angle symbol in ASCII. Occasionally, I would underline a slash: m / A = m / B is the best I can do. Of course, I can't really draw a congruent sign at all, unless I write ~= and you just imagine that the tilde is directly above the equal sign.
I've mentioned before that many people -- both teachers and subjects -- use the words equal and congruent interchangeably. There are two distinctions to make -- one is that numbers (including lengths and angle measures) are equal, while segments and angles are congruent. The other is that we don't know that any figures are congruent until we know of an isometry mapping one to the other, which the Segment and Angle Congruence Theorems provide.
In this course, the latter distinction has priority. I admit that I myself have called angles "equal" (when it's their measures that are equal) on this blog -- because I don't want to call them "congruent" until reaching the Segment and Angle Congruence Theorems. I am especially guilty of this when I write phrases such as "Angle A = Angle B" because it's so much easier than trying to write an angle symbol in ASCII. Occasionally, I would underline a slash: m / A = m / B is the best I can do. Of course, I can't really draw a congruent sign at all, unless I write ~= and you just imagine that the tilde is directly above the equal sign.
Now all of this was three years ago. So what does Wurman have to say this year? [This post is now a 2017 update from this point forward.]
https://edsource.org/2017/understanding-the-common-core-state-standards-in-california-a-quick-guide
Let's skip to what Wurman writes about Geometry:
Wurman:
And here is a key geometry standard:
Common Core: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Old: Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.
Anyone with a bit of sense can see how the Common Core specifies not only the WHAT but also the HOW. In contrast, our previous standards defined only the WHAT.
Another commenter, Brandon Michaels, asks Wurman for more clarification:
Michaels:
“For Geometry, which parts of triangles would you propose should be used if not angles and sides? And how is that in any way dictating a curriculum?”
Wurman:
In my original comment I observed that anyone “with a bit of sense” can see “how the Common Core specifies not only the WHAT but also the HOW.” Yet Mr. Micheals seems confused, as he speaks of the WHAT, the corresponding triangle parts that are in the definition of congruency, and that are present in BOTH the old and the new standards, while completely ignoring the HOW – that Common Core demands the use of rigid motions, and only rigid motion – to prove the congruency. Our old standards properly left the question of how to prove open ended.
Some refresher of geometry might be in order.
Today is the last day before the week-long Thanksgiving break. (It's also approximately the end of the third quaver.) It's also an activity day -- after all, it would be awkward for yesterday's lesson to be basically an activity (and even Wednesday's lesson contained a semi-activity) and then not give an activity today. And so I'm only repeating one of the worksheets from last year and then adding an activity page.
Last year at my old school, I gave the students an activity on graphing turkeys. The idea is that I want to give an activity that is somewhat fun on the last day before the holiday -- and perhaps when they are done graphing, they can color the turkeys. The problem is that the turkeys take much too long to graph, and the activity ends up not being much fun at all.
This is what I wrote about the turkey activity -- again, it was one year ago today:
10:15 -- My eighth graders leave and my sixth graders arrive. Just like the seventh graders, this class works on the turkey graphing activity. During this time, I hand out four licorice sticks to all students who earned an A on yesterday's quiz.
11:05 -- My sixth graders leave for nutrition, except for one girl who has incurred a short detention for going to the restroom yesterday when our school has a no-pass policy.
11:25 -- My sixth grade class returns from nutrition. Ordinarily we don't have the same class before and after the break, but our schedule is awkward this week due to Parent Conferences. The three middle school teachers decided that we would see all three grades before the break, and then rotate so that we see one grade each day after the break. Today happens to be sixth grade for me.
Ordinarily when I see sixth grade for the second time in a day, it's considered Math Intervention and the students use our other software program, IXL. (Again I don't give the link here.) I wrote about Math Intervention in my "Day in the Life" post for October.
The sixth graders are often very loud during IXL time, and today is no exception. As I implied back in October, some students still have trouble with their passwords. Since then, I came up with the idea of handing out computers to only some of the students and putting the rest on a waiting list. All students who fail to login to IXL within five minutes must forfeit their computers and give them to someone on the waiting list.
I intend for the students who lose their computers to go back to the turkey activity. Instead, these students interpret this time to be free time -- they talk very loud and run around the classroom, and even some of the students who have laptops join them. In short, IXL time for sixth graders has become a big mess! Today, it might have been better just to have all of the students continue to graph the turkey, since none of them have actually finished it. I should have either foregone IXL entirely or reserved it for those few students who actually complete the turkey.
And so now in 2017, I want to post an activity that the students will enjoy more. We see how the sixth graders started playing around because they got tired of graphing.
Today, I discovered what activity the middle school students at my old school did this week. They were to draw pictures (not necessarily turkeys) on graph paper by coloring in the boxes (rather than graphing coordinates).
This activity reminds me of Lesson 1-1 of the U of Chicago text. The first description of a point is "A point is a dot," and the squares on the graph paper represent dots that form the image. And the very first word defined in our Geometry text is pixels -- and we notice that this is exactly what the students created, pixelated art.
I've spent so much of this post writing about science, but art is the other part of the curriculum that I failed to teach last year. Art is the "A" of STEAM, and the Illinois State text encourages that art be taught in conjunction with math and science.
OK, I said I was done with the biographies in Szpiro, yet I'm returning to his book yet again. Let's continue to biography of Johann Listing, the namer of topology, that we started earlier:
"From the age of thirteen, he helped his parents augment their meager income by doing drawings and calligraphy for the public. Despite his modest upbringing he attended good schools and received a sound education in modern and classical languages, mathematics, and the sciences. Such were his achievements at school that he was awarded a scholarship that enabled him to continue his studies at university. Since the stipend was granted only for the study of arts, not science or mathematics, Listing chose architecture as a compromise."
Thus we see how Listing's art skills allowed him to learn math as well -- the power of STEAM. In the Illinois State weekly plan I posted earlier, science would have been taught on Thursdays, while art is part of the Learning Centers that would take place on Wednesdays.
Anyway, I notice that the middle school students worked on this project the entire week before Thanksgiving break -- not just one day. This allows the students to take pride in their work -- and if they don't finish the first day, they can continue the next day.
Indeed, perhaps I could have kept the turkey coordinate graphing project intact last year by simply spreading it out over the entire week. Monday of course was for coding, but then I could have let them graph on Tuesday, Wednesday, and Thursday. Then perhaps the sixth grade scene I posted earlier doesn't occur -- students would be eager to finish the graph, since there would be plenty of time left to color the turkeys.
I also mentioned that there was some leeway to take a week off from Common Core Math, especially during short three-days weeks (due to PD days). The week before Thanksgiving was a four-day week since there was only one PD day, but the weeks leading up to winter break, Presidents Day, and Chavez Day were all three-day weeks, hence suitable for similar arts projects.
This is one arts project that doesn't require the die cut machine, which I didn't know how to use. In the end, whether I would have given last year's turkey graph or this year's pixel graph depends on Illinois State. It's possible that my successor teacher found this assignment in the Illinois State text (even though it was difficult to find arts projects above Grade 5) -- and of course, priority always goes to Illinois State projects. (Ironically, the only middle school arts projects that I could easily find in the Illinois State text were in conjunction with science, not math.)
But for this year, I will post the pixel version of this assignment -- the one that the middle school students enjoyed this week. (Yes, I know I'm posting it as a one-day activity since I've never posted multi-day projects on the blog.)
You may ask, what does this have to do with today's Lesson 6-5? Well, suppose the students choose to draw a human face, or even a turkey viewed head-on. Humans, turkeys, and other animals exhibit bilateral symmetry. So the two eyes will be congruent.
And so let me announce my holiday posting schedule. Last year I made only one post during Thanksgiving break -- on Monday, the day I met the Green Team coordinator in person. Two years ago, I established the tradition of posting twice during the break -- on Turkey Day itself and the next day, Black Friday. This year I return to my 2015 posting schedule, with posts Thursday and Friday.
In addition to Pappas and Poincare, the holiday posts will contain some music. I want to have some actual songs to be played on the computer emulator, using the new 7-limit scales that we discovered.
Enjoy your Thanksgiving break. The district whose calendar I'm observing will resume school on Monday, November 27th.
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