Meanwhile, today was also the "super blue blood moon" I mentioned in my New Year's Eve post. So here in California, the lunar eclipse occurred at around 5:30 this morning. Even though my bedroom window faces north, I was actually able to see the eclipse through the window.

How was this possible? Yes, the sun and moon are usually in the southern sky. But sunrise and sunset are only due east and west at the equinoxes. In spring and summer, sunrise and sunset are slightly north of due east/west, while in fall and winter, they're slightly south of due. Back in my August 21st post (the day of the solar eclipse), I mentioned an annular eclipse from a few years earlier. That eclipse occurred in May just before sunset. Thus the sun was north of due west, and so it was actually visible from my bedroom. (But I didn't have eclipse glasses that day, and so the visibility of the eclipse from my bedroom was irrelevant.)

Now that's the sun, but what about the moon? Well, the full moon and sun are approximately in opposition -- and they're exactly in opposition during an eclipse. (After all, that's why it's an eclipse in the first place!) Thus the full moon and sun follow the opposite rules -- moonrise is at sunset, and moonset is at sunrise. In winter, sunrise is south of due east -- thus moonset must be slightly north of due west. And so once again, the eclipse was visible from my bedroom. (And since it's a lunar eclipse, I didn't have to worry about eclipse glasses!)

When I woke up, most of the moon was reddish. A few minutes later it changed to its usual color, but only a small crescent was visible. The moon sank below the garage of house next door before more than a crescent could appear -- and besides, by then it was approaching sunrise anyway.

Of course, we could have figured this out if we had finished our Spherical Trigonometry book. Recall that we began Glen Van Brummelen's book this summer (but then I abruptly stopped as uncertainty regarding my employment situation grew). Actually, Van Brummelen writes only a little about lunar eclipses (and nothing about solar eclipses) in his book. First, he writes:

"Centuries before that, Aristotle had given several arguments for the sphericity of the Earth, including the observation that the shadow cast by the Earth on the Moon during a lunar eclipse is always a circle."

This, by the way, is related to the plane sections of Chapter 9. Since any plane section of a sphere is a circle, the shadow cast by a sphere is likewise always round. On the other hand, were we living on a disk, the shadow could be rectangular, since cylinders also both rectangular and round plane sections.

Here is Van Brummelen's other reference to eclipses:

"One way around this is to observe during a lunar eclipse, which takes place simultaneously for all Earthly observers."

The fact that lunar eclipses take place everywhere at the same time means that it is visible in any time zone where it is nighttime. The eclipse was visible from the west coast of North America as well as Asia and Australia. But in Europe, Africa, and the east coast of the Americas, it was daytime, hence the eclipse was not visible there.

If I were still teaching in the classroom today, it's possible that I might have taught a lesson on lunar eclipses as a follow-up to the solar eclipse activity of August 21st (our first day of school). Now Wednesdays would be Learning Center days, so I'd have to make it fit. Again, die cuts could be used to cut out moon shapes. It's also possible for me to award extra credit to any student who takes a photo of the eclipse. This would be the rare day when cell phones are allowed, as they would be part of the lesson.

In addition to eclipse day, today is also test day. Here are the answers to the Chapter 9 Test:

1. Draw two triangles, one not directly above the other, with corresponding vertices joined.

2. Draw a picture identical to #3.

3. Draw and identify a circle and an ellipse.

4. Draw and identify two circles.

5. circle (Yes, I had to make an eclipse reference!)

6. a. Draw a circle. b. Draw a parallelogram (not a rectangle). c. Draw a rectangle.

7. c or d.

8. Draw #7c or d again. (A cube is a rectangular parallelepiped!) The faces don't need to be squares.

9. a. 144 square units b. 8 units

10. a. Draw #3 again, with both heights labeled. b. 25pi square units

11. a. 2 stories b. 3 sections c. back middle

12. a. tetrahedron, regular triangular pyramid b. 6 edges c. (

13. sphere

14. solid sphere (This is yet another eclipse reference!)

15. rectangular solid

16 a. yes b. 3 planes

17.

*lw*(

*h*+ 2)

18.

*xy*(

*x*+ 2 +

*y*)

19. pi

*r*^2 (4 +

*r*)

20. planes

There's no Euclid on the test, but let's look at the next proposition in Euclid anyway:

**Proposition 13.**

Euclid's proof of this Uniqueness of Perpendiculars proposition is indirect.

Indirect Proof:

Assume that point

*A*lies in plane

*P*, and both

*P*. Since three noncollinear points determine a plane,

*A*,

*B*, and

*C*lie in some plane

*Q*. Since two planes intersect in a line, planes

*P*and

*Q*intersect in some line containing

*A*-- call it line

*DE*. By definition of line perpendicular to a plane,

*Q*, there are two lines through

*A*perpendicular to

Today is a traditionalists post. But today, I want to get as far away from traditionalism as possible. I was recently reading about perhaps the most progressive school possible:

http://www.sudval.org/

This is the link for Sudbury Valley School, a Massachusetts private school. Sudbury Valley School is indeed

*completely*different from what the traditionalists want. It's better for me to link directly to the Sudbury Valley website than to explain what it is:

http://www.sudburyvalley.org/01_abou_01.html

*Sudbury Valley School, since its founding in 1968, has been a place where children can enjoy life, liberty and the pursuit of happiness as they grow up in the newly emerging world. From the beginning of their enrollment, no matter what their age, students are given the freedom to use their time as they wish, and the responsibility for designing their path to adulthood.*

*In our environment,students are able to develop traits that are key to achieving success: They are comfortable learning new things; confident enough to rely on their own judgment; and capable of pursuing their passions to a high level of competence. Children at Sudbury Valley are adaptable to rapid change, open to innovation and creative in solving new problems. Beyond that, they grow to be trustworthy and responsible individuals, and function as contributing members of a free society.*

*At Sudbury Valley, students from pre-school through high school age explore the world freely at their own pace and in their own unique ways. They develop the ability to direct their own lives, be accountable for their actions, set priorities, allocate resources, deal with complex ethical issues, and work with others in a vibrant community.*

So this is the Golden 50th Anniversary of Sudbury Valley. Now let's see in more detail how Sudbury Valley differs from traditionalism.

Traditionalists believe that teachers are "the sage on the stage," while progressives believe that teachers are "the guide on the side." Well, Sudbury Valley doesn't deign to call its adult employees "teachers" at all!. Instead, they are called "staff members." And these staff members not only avoid giving homework, they don't even give tests either.

The traditionalist wonders, how are students' grades determined without tests? The Sudbury answer is, students don't receive grades at all. So A-F, 1-4, and other grades are distinctions that Sudbury doesn't make.

The traditionalist wonders, by what criteria are students promoted to the next grade level without letter grades? The Sudbury answer is, students aren't divided into grade levels. Kindergarten, first grade, and other grade levels are distinctions that Sudbury doesn't make.

The traditionalist wonders, by what criteria are students promoted from elementary to middle school, or relatedly from middle school to high school, without grade levels? The Sudbury answer is, students of all ages, from four to the teens, attend the same school. Elementary, middle, and high school are distinctions that Sudbury doesn't make.

The only distinction that Sudbury makes is graduation. The following link is supposedly about how students can graduate, but it doesn't explain much at all:

http://www.sudburyvalley.org/01_abou_07.html

Apparently, prospective graduates are to write some sort of thesis statement, and the oral defense of this thesis is what determines who gets to graduate.

The following link is to a FAQ page:

http://www.sudburyvalley.org/01_abou_09.html

Let's skip to what matters the most to the traditionalists -- college admissions:

**Daria:**Let's go back to the learning strategy of the school. We presuppose that all children learn, they learn at their own pace, and they learn because they're passionate about learning. How does this prepare a child, for example, to go on to college when they must have SAT scores and must function in a very different setting in a college situation?

But within six weeks, he had learned everything I had to teach him." That speaks to motivation. This child was not interested in learning math until he needed it for the SAT's and when he needed it for the SAT's, he learned it quickly. Colleges are not as different from Sudbury Valley, I think, as high schools are, because you're expected to have a lot more autonomy and a lot more responsibility for doing what you need to do in college than you have in most high schools.

**Mimsy:**Kids who leave here are usually extremely well-prepared to go to college. First of all, they're quite knowledgeable, and they're very articulate. If you want to go to a college for which you need SAT scores (which certainly is not every college at all) then that's one of the things you're motivated to do, and you apply yourself to learning how to do well on the SAT's. This is not a strange idea. Recently, a guest came to the school, and he said, "Oh, I knew someone from this school once," – this person was a teacher – "I tutored him in math. He had graduated from your school and yet he knew very little math and he wanted to take the SAT's.But within six weeks, he had learned everything I had to teach him." That speaks to motivation. This child was not interested in learning math until he needed it for the SAT's and when he needed it for the SAT's, he learned it quickly. Colleges are not as different from Sudbury Valley, I think, as high schools are, because you're expected to have a lot more autonomy and a lot more responsibility for doing what you need to do in college than you have in most high schools.

Really -- students can study for the SAT math section in six weeks? We know that traditionalist SteveH complains that three years (of "Pre-AP" math) is too little time to prepare for AP Calculus, and now Sudbury claims to get a student from "knowing very little math" to success in SAT math in 1/20 of that time!

It's interesting that Sudbury Valley is in Framingham, Massachusetts -- just 20 miles away from a little town called Cambridge. And that town is home to two universities that you might have heard of, namely Harvard and MIT. I'm sure that traditionalists are very interested whether Sudbury students are getting into colleges like Harvard and MIT and not just Framingham Community College. So far, I don't see much evidence of Sudbury grads entering the Ivy League.

The Sudbury website contains a blog. Notice that traditionalist schools don't require blogs to justify their existence the way that Sudbury schools do. And yes, I'm aware that I myself am a blogger criticizing another blog. The difference is that I'm just trying to explain Common Core Geometry, while the Sudbury bloggers are trying to explain how Sudbury Valley qualifies as a "school." And I know that I often criticize traditionalists. But compared to Sudbury, I and nearly everyone reading this blog are traditionalists. I can't see imagine that many Sudbury students are studying high school Geometry, not even the SAT crammer (as there's little Geometry on the new SAT).

Here is a link to the most recent blog post:

http://blog.sudburyvalley.org/2018/01/whole-language-phonics-and-two-boys-learning-to-read/

The author writes that one Sudbury student is learning to read via whole language while the other is reading via phonics:

*One thing that I love about Sudbury schools is that children not only have the freedom to chose when and what to learn, but also how to learn. While the experts debate, our children are lucky to be at a school that respects children’s innate wisdom and intelligence and supports them on the unique path it takes them on.*

Traditionalists, by the way, prefer phonics as the only valid method of teaching reading.

The second most recent blog post merely announces the 50th anniversary of Sudbury:

http://blog.sudburyvalley.org/2018/01/fifty-years-front-and-center/

*Throughout the 50 years, we have expected excellence. The expectation has guided us unfailingly, and is, I believe, what has brought us to this point. It, more than anything, has fostered a continuously supportive community, inspired by our vision and our efforts to make that vision real.*

Here's an article from October in which math is mentioned:

http://blog.sudburyvalley.org/2017/10/lost-over-the-summer/

*Naturally, I find myself wondering what lessons learned by my SVS children have been “lost” over the summer? Maybe they have forgotten the exact list of ingredients for gingerbread. Perhaps they forgot what time JC meets? Perhaps they have forgotten the price of their favorite snack at concession? But these are all simple facts that can be easily re-accessed when the need for them arises. In the research on Math skill loss there is discussion about how the kids lose knowledge on how to carry out mathematical procedures. This I can imagine. Once in my adult life, I was surprised to find myself needing to solve a quadratic equation. The only part I could remember was something about “4ac.” This was before the Internet, or at least before I knew how to use the Internet, so I found an old notebook that had common math equations on the inside front and back pages, and there it was! It took me much longer to solve my one problem as an adult than I could have done it in high school. But does that really represent an important “loss?” If I forced myself to 15 or 20 of them, I bet I could rapidly regain my high school proficiency level, but for what purpose?*

*I experienced a similar problem more recently, when my daughter asked for help doing a long division problem. Much to my surprise, I had actually forgotten the mechanics of doing long division by hand. Quite embarrassed, I spent a few minutes on the Internet looking at examples, before I could help my daughter.*

We forget. All the time. Especially when we don’t understand why the algorithm works or if it has been a long time since we last used it. I can guarantee if I took a survey at Starbucks right now, and asked people how to perform long division, convert a mixed number to a fraction, recite the quadratic formula, factor a binomial, and complete the square—most would fail. Not because math is too hard or people are bad at it, but because memorization and algorithms are not the best ways to retain information. We remember through context,understandingandapplication.

*Garelick:*

*We also remember through continued application. For example, if I haven’t worked with percent calculations for a while, I have to brush up on it. Same with finding derivatives of certain functions. The survey of people at Starbucks might be different if the majority of customers were practicing engineers. That people forget how how to do something if they haven’t worked with it for years is not evidence that the traditional method of math teaching is ineffective. And like many authors of similar rants and polemics, she also does not provide evidence that her methods are superior to those that she feels do not work.*

http://www.openschooloc.com/wp/

Like most Sudbury schools, the Open School of Orange County has a blog to justify its existence. I link to the most recent blog entry below:

http://www.openschooloc.com/wp/2018/01/21/why-do-we-have-academic-categories/

The author of this article, by the way, is an alumna of the original Sudbury Valley School, and now she's a teacher -- um, "staff member" -- at the Open School. She writes:

*The artificial separation of categories — say, separating “mathematics” from the entire rest of the world — creates a series of illusions that educational institutions then spend an overwhelming amount of time and energy working against. Once you accept that “mathematics” is a separate category of activity from knitting, Pokemon, music, cooking, graphic design, physics, writing sonnets, hiking, sales, or getting a weekly allowance, you have undermined the mathematics intrinsic to all of those common life activities.*

*Then you’re stuck with dissociated word problems like “Sue has eleven apples. If she gives five to Peter, how many will she have?” In the meantime, all the activities I listed require exactly that kind of problem, while also yielding some additional product, such as a scarf, a sonnet, social time with your friends, exercise, a poster, or a song.*

I can already imagine the traditionalist rebuttal to this -- students actually have to know math before they can apply it to cooking, physics, and so on -- and they won't ever know any actual math unless they are taught it traditionally. Otherwise students are being asked to "put the cart before the horse."

It goes without saying that Sudbury schools lack bell schedules. They do have school calendars, and students are required to attend school for a certain number of hourse (such as five) each day.

Meanwhile, Garelick has posted more recently than the MLK Day post I linked to above, but we might as well discuss the old post since I already linked to it. Let me link to the original article that Garelick is discussing here:

https://medium.com/i-math/common-core-math-is-not-the-enemy-c05b68f46b3e

As usual, SteveH comments on it:

*“Why is Common Core Math Hated by Parents?*

*Because the Common Core Math standards are trying to teach number sense and mental math techniques through various forms of diagrams and step-wise procedures that are new and look confusing.”*

*SteveH:*

*Rote strawman 2 – Really. Stop reading! I guess I’m not paying attention.*

*Forget the fact that CCSS is non-STEM by definition, starts in Kindergarten, and that the highest level expectation is no remediation for community college math.*

SteveH:

*Forget the fact that the College Board knows that this is a problem and has defined Pre-AP in ninth grade as an attempt at crossing the chasm of content and skills between fuzzy K-6 CCSS math and AP Calculus.*

Well, at least three years of Pre-AP math are better than six weeks of SAT Prep math.

*“Respect the Teachers*

They are experts on child education. They have the best intentions for the students in mind. This teacher made a decision [5X3 = 5+5+5 marked wrong] based off a lot more information about the student and class setting than we can tell from a photo. We don’t have to agree with it, but we can respect it. If you are confused, ask them why they did something before you discredit a teacher on the Internet.”

They are experts on child education. They have the best intentions for the students in mind. This teacher made a decision [5X3 = 5+5+5 marked wrong] based off a lot more information about the student and class setting than we can tell from a photo. We don’t have to agree with it, but we can respect it. If you are confused, ask them why they did something before you discredit a teacher on the Internet.”

*SteveH:*

*I had to put my hip boots on for that one. They are the experts of their own opinions. They are especially not experts in math. They might not be held to peer review, but mathemeticians are, and the mathemetician of this opinion piece is getting some bad peer reviews.*

Any traditionalists reading this post might think I'm playing with them -- I'm defending mainstream progressive math by comparing it favorably to Sudbury. It's just like a thief's justification that at least he's not a killer.

Actually, it's possible to go in a different direction. Perhaps if we support traditional schools, then we should support them completely with a traditional pedagogy, and if we oppose traditionalism, then we should oppose it completely and support Sudbury. Maybe it's the middle ground -- mainstream progressive pedagogy -- that needs to be questioned.

Well, as of now I'm still not sold on Sudbury as a viable school model. I'll continue to post about Common Core Geometry, including worksheets based on it. And I'll continue to use the traditionalists label to compare Common Core to what they prefer.

Let's end today's post with a link to the second commenter from the Garelick post:

BL:

*I never did memorize the quadratic formula. But my math teacher showed us how it’s derived, and I quickly did that before each test in the margin of the test paper.*

*Problem solved. (And, as a result, I never forgot about “completing the square”!)*

If I were teaching the Quadratic Formula, I'd forget about both BL and Sudbury and just teach it to the students using the tune of "Pop Goes the Weasel." As I've mentioned in previous posts, I like using music to help my students learn and remember math.

Here is today's test: