*approximate*coordinates, but when we take these to be

*exact*coordinates.

Points on the globe whose coordinates are both integers are called

*degree confluences*. Here is a link to a website that describes these confluences:

http://confluence.org/

http://confluence.org/faq.php

I first discovered this website over 10 years ago -- I forgot what I was looking up when I stumbled upon the website. But now that I've been making several posts about spherical geometry, I decided to go back to revisit the website. The author of this webpage, Alex Jarrett, is a New England pianist. His goal is to take a picture of every confluence in the world.

Los Angeles County in California is the most populous county in the nation, so it figures that the confluence that lies in this county, 34N, 118W, is one of the most visited. Here is a link to the 14th official visit to this confluence:

http://confluence.org/confluence.php?id=162

The most recent visit to the confluence is numbered 26th. Notice that despite its location in the most populous county, the confluence itself is located in the steep hills northeast of Whittier. Visitors to the confluence report thick vegetation and slippery slopes leading up to the confluence.

I decided that today, since I was driving near this area anyway, I'd drive near the confluence. No, I didn't really make an official visit to the confluence, since I'm not up to hiking on these steep hills, nor do I even have a GPS, to show a visit to the exact location.

But I decided to drive on the street mentioned at the link, Turnbull Canyon Road, from which the confluence should be visible. Turnbull Canyon is one of the most difficult streets I've ever driven -- it's a winding road through the hills of Hacienda Heights. I noticed that there are mile markers off the side of Turnbull Canyon -- but unfortunately none of the visits mentioned on the link tell me what mile marker to look for, oh well!

Notice that the webpage gives a link to the antipode of this point, which is 34S, 62E. But as I've mentioned before, the antipodes of nearly the entire USA lie in the Indian Ocean. The rules state that oceanic confluences don't count unless they are within site of land -- and according to the link, land is over 900 miles away from the confluence. Here is a link to some antipodal pairs where both have been officially visited:

http://confluence.org/antipodes.php

The first pair listed on that page has one confluence in Hawaii (22N, 160W) and its antipodes in Botswana (22S, 20E):

http://confluence.org/confluence.php?visitid=11367

http://confluence.org/confluence.php?visitid=6014

Here are visits to a very special pair of antipodal points -- the North and South Poles:

http://confluence.org/confluence.php?visitid=16370

http://confluence.org/confluence.php?lat=-90&lon=-180

As it turns out, the same person visited both Poles during the centennials of their respective discoveries -- and he pointed out that his first ever visit to a pair of antipodal points were the Poles.

Finally, here's a link to the very first confluence that the author Alex Jarrett visited, 43N, 72W:

http://www.confluence.org/confluence.php?visitid=1

Let's move on to the book that I mentioned in my last post -- Eugenia Cheng's

*How to Bake Pi*. The author is a British mathematician who current works at the U of Chicago -- yes, I talk about the U of Chicago math texts on this blog all the time, but the book I'm describing today is a popular book about math, not a textbook. I actually stumbled on this book when I was in the Los Angeles County Public Library -- it was on hold for another patron whose last name also started with W. So I couldn't resist requesting my own copy of the book from the library.

Cheng's book is about a branch of mathematics known as

*category theory*. This theory is quite complex -- even more so than group theory or set theory. But I'm hoping that Cheng will be able to introduce category theory in an understandable manner. So far, I've read the first two chapters of Cheng's book -- and as usual, I'll report anything that may be relevant to geometry on the blog.

Chapter 1 is titled, "What Is Math?" In this chapter, Cheng gives her own definition of "math":

"Mathematics is the study of anything that obeys the rules of logic, using the rules of logic."

And of course, the underlying logical structure of mathematics is apparent in Geometry class. The U of Chicago text (by which I mean the Geometry text, not Cheng's book) devotes parts of Chapters 1, 2, and 13 to logic. The theorems of geometry follow logically from definitions and postulates, ever since the days of Euclid.

Chapter 2 is titled, "Abstraction." Abstraction is something that we do in all levels of math, from preschool to grad school. Cheng's first example of an abstraction is group theory -- and she gives the symmetry group of an equilateral triangle as a example of a group -- this group is isomorphic to the permutation group of a three-element set.

Actually, this is her

*second*example of an abstraction. The first example is a recipe -- the title of the book does mention baking, and Cheng frequently compares recipes to math throughout the book. She points out that hollandaise sauce is just like mayonnaise, except one replaces olive oil with butter.

Here's my own example of such an abstraction: On Sunday, the American women scored early and often en route to a dominating victory over Japan to secure the championship in the World Cup ... of

*Softball,*that is. (Did you think I was going to say "soccer"?)

Cheng points out that students must make several leaps of abstraction as they proceed through their study of mathematics. They make the first jump fairly early in their academic career, as they abstract from

*objects*to

*numbers*. But the next leap is one at which many students struggle -- the abstraction from

*numbers*to

*variables*. This marks, of course, the study of algebra. Subsequent leaps occur from algebra to AP Calculus, then from AP Calculus to college-level calculus. She writes that some people reach their abstraction limit -- which is where I myself did, so I ended up settling for a Masters Degree in math. One of the highest levels of mathematical abstraction is category theory -- and one for which Cheng has proudly cleared the bar.

Notice that this post is part of my "How to Fix Common Core" series -- which means that I'm going to find a way to tie this back to Common Core. One frequent complaint about the Common Core Math Standards is that they expect students to

*abstract*at too young an age. This complaint mainly refers to the elementary standards.

But returning to the high school standards, we notice that many students reach their upper limit of abstraction at Algebra I, with a few moving on to Calculus. The problem that occurs is that in order to get seniors into Calculus classes, they need to take Algebra I in eighth grade, so that they can get into those very abstract math classes, graduate, and move on to do great things. But if we make eighth graders take Algebra I, the students who can't clear that level of abstraction end up getting F grades in every math class from eighth grade on. They fail to graduate, drop out of school, get a low-paying job, and now they can't even

*buy*great things. It's a dilemma that anyone who tries to write math standards ends up facing.

On this blog, I proposed a solution by introducing the concept of the

*Dickens age*. The Dickens age represents the age at which a student is considered mature enough to work -- anything below that would be considered Dickensian (hence the name "Dickens age"). The idea is that students only have to attend school up to the Dickens age, then they can leave to find work. This way, students who can handle algebra or calculus can take those classes. Those who can't would be protected from having their inability to abstract at the level needed for algebra block them from attaining a comfortable, middle-class lifestyle.

In various posts, I've waffled between setting 15 (end of 9th grade) and 16 (end of 10th grade) as the Dickens age. My original choice was 15 -- I switched to 16 because I wanted to endorse the Singapore Math standards, which are set up for the common leaving age in that country, 16. But now I no longer endorse Singapore Math -- instead I endorse Saxon Math all the way through high school.

So therefore I set the Dickens age back to 15, or the end of 9th grade. This will allow students to complete Integrated Math I before leaving (but recall that those who are headed to Calculus would still skip directly into Integrated Math II as freshmen). In this Saxon series, this corresponds roughly to the Algebra 1/2 text. For traditionalists, this is also half of Algebra I -- the first semester. Finally, it is roughly the level of math required to pass the CAHSEE, California High School Exit Exam. This test could be given a year earlier, to 9th graders, to accommodate the Dickens age.

Now let's look at the younger grades again. I say that a criticism of Common Core Math is that it expects students to abstract when they are so young. But Eugenia Cheng writes that nearly

*every*child abstracts at a young age, since

*numbers*are themselves an abstraction. So we must see what these critics mean when they say that Common Core requires developmentally inappropriate abstraction -- I link to a letter which makes this argument:

http://deyproject.org/2014/06/28/common-core-pushes-abstract-topics-too-early/

Seeing that children will need “high order” conceptual thinking in college and the workplace, the Common Core introduces such thinking early on. For example, it introduces mathematical place value, an abstract topic, in kindergarten and the first grade. But before the age of 7 or so, children’s minds aren’t inclined toward such conceptual matters. Young children are more naturally motivated to develop their powers through the arts, play, and the exploration of nature. They are enthusiastic about these activities, which enable them to develop their imaginations and sense of wonder.

So the letter writer -- Dr. William Crain, a New York professor of psychology -- declares that "place value" is too abstract for first graders. Let's look at the specific first grade standards that refer to place value:

#### Understand place value.

CCSS.MATH.CONTENT.1.NBT.B.2

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

CCSS.MATH.CONTENT.1.NBT.B.2.A

10 can be thought of as a bundle of ten ones — called a "ten."

10 can be thought of as a bundle of ten ones — called a "ten."

CCSS.MATH.CONTENT.1.NBT.B.2.B

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

CCSS.MATH.CONTENT.1.NBT.B.2.C

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

CCSS.MATH.CONTENT.1.NBT.B.3

Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

So Dr. Crain argues that these standards are too abstract for first graders. But without place value, the largest number that students can count to is nine. Does Crain really believe that students shouldn't know how to count to any number greater than nine until the second grade?Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

Of course, it's possible that Crain is actually referring to the specific place value questions that appear in first graders' homework -- that is, the addition problems that are based on place value "strategies" rather than the standard algorithm. It's possible that, if first graders were simply given two two-digit numbers and asked to add them using only the standard algorithm, Dr. Crain might

*not*consider that to be a developmentally inappropriate abstraction of place value.

Notice that Crain proposes a more play-based kindergarten and first grade. Recall that this was also the philosophy of Dr. Rudolf Steiner and his Waldorf schools. The problem is that those who say that kindergarten and first-grade should be play-based are usually not the same critics as the ones who say that first-graders should learn the standard algorithms so that they'll be ready for an eighth-grade Algebra I class. Is it possible for one plan to satisfy

*both*types of Common Core critics -- those who want the lowest grades to be play-based

*and*those who want the middle grades to be algebra-based?

Recall that on this blog, I also once proposed a sort of tracking plan, called the

*path plan*, which was based on a scheme that my own elementary school implemented for a few years. The plan involved placing students on

*paths*based on their

*reading levels*(with the paths corresponding to the nominal grades of kindergarten, Grades 1-2, Grades 3-4, and Grades 5-6). These reading classes represent the students' homerooms. Then students move around to various teachers for different subjects during the day, so that they could learn other subjects above or below their reading level. My proposal involved having the following subjects be taught by a teacher other than the homeroom teacher:

Kindergarten: None (homeroom all day)

Grades 1-2: Math

Grades 3-4: Math, Elective

Grades 5-6: Math, Elective, Science

Grades 7-8: Math, Elective, Science, PE

From time to time, people post proposals to have students grouped "homogeneously by subject" in elementary school on various education articles and blogs. Oftentimes, a comparison is made to homeschooling parents -- if their child is reading at a first-grade level yet doing math at a fourth-grade level, the parent can teach the child first-grade reading and fourth-grade math. If the student masters first-grade reading, the parent can move to second-grade reading that very day -- not wait until the end of the year, trimester, or week.

The problem is that this plan, while perfect for homeschooling, doesn't scale up. I don't like the idea of a five- or six-year-old having five different teachers in the day just because he or she is at five different grade levels in five different subjects -- whereas a homeschooling six-year-old can be taught those five different subjects by just one teacher (the parent). Also, the idea that a student can move to a different class on any of the 180 days of the school year just because he or she has mastered a subject obviously isn't convenient for either the student or the schoolteacher.

And so my plan gradually increases the number of teachers a student may see in a day. Notice that in this plan, the only class that a first- or second-grader has outside the homeroom is math. But keep in mind that as originally envisioned at my elementary school, students on the Primary Path (the path corresponding to Grades 1-2) attended something called "Developmental Center." I can't be exactly sure what the DC was like, since I was already past the Primary Path when the path system was first implemented at my school. But based on what little I saw of it, classes were divided into two groups, with each group spending one hour reading and the other hour in the DC, which apparently provided the sort of play-based education that Crain desires.

Here is an example of a bell schedule for grades 1-6 under the path system. Recall that the names of the paths are Primary (normative grades 1-2), Transition (grades 3-4), and Preparatory (grades 5-6) -- and it's a student's

*reading*level that determines which path he or she is on.

8:30 First Bell, Pledge of Allegiance

8:45 Reading/Language Arts (all paths in homeroom)

10:15 Paths take turns going to AM Recess (paths not at recess remain in homeroom)

10:50 First Block (all paths)

11:30 Lunch (Primary), Second Block (other paths)

12:10 Lunch (Transition), Third Block (Preparatory)

12:50 Lunch (Preparatory), Homeroom (other paths)

1:30 Paths take turns going to PM Recess (paths not at recess remain in homeroom)

2:10 Homeroom (all paths)

2:55 Dismissal

So we see that all paths switch teachers between AM recess and lunch, and the older paths have later lunches, so they end up with more teachers, as desired. Each block ends up being 40 minutes -- including the lunch block. Students return to homeroom to pick up their sack lunches, money, or free/reduced lunch cards for the first five minutes, and then eat lunch the remaining 35 minutes.

Now this bell schedule has Primary Path students in math from 10:50-11:30. It's based on the concern of many proponents of this scheme that if students are to be grouped homogeneously in any subject, that subject should be math. But this schedule doesn't incorporate the play-based Developmental Center that we wish to implement following Crain's recommendation.

Here's a Primary Path schedule that includes the Developmental Center:

8:30 First Bell, Pledge of Allegiance

8:45 First Hour of DC (half in DC, half reading in homeroom)

9:45 Second Hour of DC

10:45 All students return to homeroom

10:50 AM Recess

11:05 Homeroom

12:25 Lunch

1:00 Homeroom

1:45 PM Recess

1:55 Homeroom

2:55 Dismissal

On this schedule, the lunches overlap slightly -- which is desirable since the lunch block is divided into eating time and play time. Here, I would probably move the start of the Transition lunch period up a few minutes, to 12 noon.

In reality, the path system only lasted a few years. It began when I was already on the Preparatory Path and ended a few years after I left the school -- when the California State Standards were implemented with tests based on

*grade levels*, not paths. For example, the state science test was to be given to all fifth graders, not whatever path.

Of course, the choice to implement a path system is a local, not a national, decision. But the Common Core testing can be done so that it is compatible with the path system. This deserves its own post, so I will save it for the next "How to Fix Common Core" post.

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