Spherical Geometry (Legendre 496-501)

Today is a spherical geometry day. There are six theorems to cover today -- three "scholia" and three main theorems, including the big deal theorem that gives the ultimate link between angle measure and area.

Let's begin with Proposition 496, which is considered by Legendre to be a mere "scholium":

496. Scholium. The spherical wedge comprehended by the planes AMB, ANB is to the entire sphere as the angle A is to four right angles. For, the lunary surfaces being equal, the spherical wedges will also be equal; therefore two spherical wedges are to each other as the angles formed by the planes which comprehend them.

The final Legendre theorem we discussed in our last spherical geometry post compared the area of a lune to the surface area of the entire sphere. This scholium follows up with a comparison of the volume of a spherical wedge to that of the entire sphere. A spherical "wedge" is exactly what you think it is -- it's just like the wedge of an orange. A wedge is closely related to a lune -- if we start on the surface of a lune, any point we can reach by digging directly down from the lune towards the center of the sphere is in the wedge.

Legendre tells us that the volume of a wedge is exactly what you expect it to be -- take the volume of the sphere and multiply it by the measure of its angle divided by "four right angles" -- that is, 360 degrees or 2pi (sorry, make that tau, or 4lambda) radians.

We proceed with Proposition 497, our first main theorem:

497. Two symmetrical spherical triangles are equal in surface.

Now Legendre uses the word "symmetrical" in different ways throughout his text. Sometimes he uses it to mean merely "congruent," but at other times he uses it to denote the existence of a certain reflection mapping one to the other. For example, earlier in the text (before the unit on spherical geometry), Legendre calls two pyramids S-ABC and T-ABC "symmetrical." In this case the two pyramids have the same base ABC, and a reflection in the plane ABC maps S to T. In the first paragraph of his proof, Legendre explains exactly what he means by "symmetrical" here:

Demonstration. Let ABC, DEF, be two symmetrical triangles, that is, two triangles which have their sides equal, namely, AB = DE, AC = DF, CB = EF, and which at the same time do not admit of being applied one to the other; we say that the surface ABC is equal to the surface DEF.

Clearly Legendre is using "symmetrical" here to mean "congruent." After all, he lists three pairs of congruent sides, and he's already proved the spherical equivalent of SSS, so we know that the two triangles are congruent. Since he writes that the two triangles "do not admit of being applied one to the other," we assume that there is not necessarily any reflection (or any other isometry) mapping ABC to DEF (though there could be).

Legendre says that he is to prove that ABC and DEF are equal in "surface" -- that is to say, that they are equal in area. This reminds us of the Area Postulate found in Lesson 8-3 of the U of Chicago text:

Area Postulate:
c. Congruence Property: Congruent figures have the same area.

Given this postulate, it appears to be obvious that ABC and DEF have the same area -- we just said that the triangles are congruent, and the postulate tells us that congruent figures have the same area. so therefore the triangles have the same area, right? So what is there for Legendre to prove?

Here's the problem -- the postulate tells us that "congruent" figures have the same area, but what does it mean for two figures to be "congruent"? The U of Chicago text uses the Common Core definition of congruence -- that is, the existence of an isometry mapping one to the other. But Legendre expressly mentioned that we don't have such an isometry available! He tells us that the two triangles are "congruent" by SSS -- but his proof of SSS only tells us that two triangles with equal sides have equal angles, not that any isometry exists between them! In other words, the two triangles are congruent by the old pre-Core definition, but not necessarily by the Common Core definition.

Maybe we should be consistent about our own use of terminology. Let's call two triangles ABC and DEF "symmetrical," just as Legendre does, and reserve the word "congruent" only when there's a proved isometry between the two triangles. As it turns out, in this proof Legendre will ultimately divide the triangles ABC and DEF into smaller triangles, and then he does prove the existence of isometries between pairs of these smaller triangles. This isn't enough to prove that there's an isometry between the entire triangles ABC and DEF, but it is sufficient to prove (via the Additive Property of Area) that the triangles have the same area.

Okay, so now let's proceed with Legendre's proof.

"Let P be the pole of the small circle which passes through the three points A, B, C; from this point draw equal arcs PA, PB, PC (464); at the point F make the angle DFQ = ACP, the arc FQ = CP, and join DQF = APC."

Let's see what's going on so far here in Legendre. We know that a circle passes through A, B, and C (since a circle passes through any three points) -- the only fear is that the circle might be a great circle instead of a small circle. Legendre tells us in a footnote that if ABC can't be a great circle -- basically because otherwise ABC couldn't be a triangle.

Now any circle, small or great, has a pole (two poles, in fact). The pole of any circle is essentially the center of that circle. On the earth, the poles of the Equator, or any parallel of latitude, are the actual North and South Poles. The poles of a circle are equidistant from all points on the circle. (Note that Proposition 464 mentioned here details how to find the pole of a circle -- the poles are the endpoints perpendicular to the plane containing the circle.) The rest of this part of the proof is the construction of a certain triangle on side DF, the new triangle DFQ.

"The sides DF, FQ, are equal to the sides AC, CP; the angle DFQ = ACP; consequently the two triangles are equal in all their parts (480); therefore the side DQ = AP, and the angle DQF = APC.

"In the proposed triangles DFE, ABC, the angles DFE, ACB, opposite to the equal sides DE, AB, being equal (481), if we subtract from them the angles DFQ, ACP equal, by construction, there will remain the angle QFE equal to PCB. Moreover, the sides QF, FE are equal to the sides PC, CB; consequently the two triangles FQE, CPB, are equal in all their parts; therefore the side QE = PB, and the angle FQE = CPB.

"If we observe, now, that the triangles DFQ, ACP, which have the sides equal each to each, are at the same time isosceles, we shall perceive that they may be applied one to the other; for having placed PA upon its equal QF, the side PC will fall upon its equal QD, and thus the two triangles will coincide; consequently they are equal, and the surface DQF = APC. For a similar reason the surface FQE = CPB, and the surface DQE = APB; we have, accordingly,
DQF + FQE - DQE = APC + CPB - APB, or DEF = ABC; therefore the two symmetrical triangles ABC, DEF, are equal in surface. QED"

Let's move on to Proposition 498, which is considered by Legendre to be a mere "scholium":

498. Scholium. The poles P and Q may be situated within the triangles ABC, DEF; then it would be necessary to add the three triangles DQF, FQE, DQE, in order to obtain the triangle DEF, and also the three triangles APC, CPB, APB, in order to obtain the triangle ABC. In other respects the demonstration would always be the same and the conclusion the same.

Notice that Legendre's P is the pole (center) of the circle containing the points A, B, and C -- that is, P is the circumcenter of triangle ABC. In Euclidean geometry the circumcenter of ABC lies inside the triangle if it's acute, and outside the triangle if it's obtuse. If ABC is a right triangle, then the circumcenter lies on the triangle -- indeed, it's the midpoint of the hypotenuse.

This case is quite suited for a two-column proof. Here is the beginning of such a proof (which I admit will be long):

Given: AB = DE, AC = DF, CB = EF, P pole of ABC lies inside triangle ABC
Prove: Area(ABC) = Area(DEF)

Proof:
Statements                                               Reasons
1. AB = DE, AC = DF, CB = EF, P pole 1. Given
2. PA = PB = PC                                      2. Definition of pole of a circle
3. Q s.t. DFQ = ACP, FQ = CP               3. Ruler/Protractor Postulates
4. Triangle DFQ = ACP                          4. SAS
5. DQ = AP, DQF = APC                        5. CPCTC

and so on.

We proceed with Proposition 499, our second main theorem:

499. If two great circles AOB, COD, cut each other in any manner in the surface of a hemisphere AOCBD, the sum of the opposite triangles AOC, BOD, will equal to the lunary surface of which the angle is BOD.

OK, so we're getting closer to the connection between angle measure and area. Notice that Legendre clearly intends A and B to be antipodal points since he calls AOB a "semicircumference" later on, and likewise C and D are antipodal points.

Let's look at Legendre's proof now:

Demonstration. By producing the arcs OB, OD, into the surface of the hemisphere until they meet in N [that is, O and N are antipodal points -- dw], OBN will be a semicircumference as well as AOB; taking from each OB, we shall have BN = AO. For a similar reason DN = CO, and BD = AC; consequently the two triangles AOC, BDN [I prefer writing the second triangle as BND -- dw], have the three sides of the one respectively equal to the three sides of the other; moreover, their position is such that they are symmetrical; therefore they are equal in surface (496) [Legendre clearly means 497 here -- dw] and the sum of the triangles AOC, BOD is equivalent to the lunary surface OBNDO, of which the angle is BOD. QED

As usual, this is easier to visualize if we let N and O denote the North and South Poles. Notice that if A and C are in the Southern Hemisphere, their antipodes B and D are in the Northern Hemisphere (so the triangle BOD is a rather large triangle). The Southern Hemisphere triangle AOC and the Northern Hemisphere triangle BND are congr -- er, symmet -- er, congruent. (Notice that the antipodal map -- that is, the function mapping every point to its antipodes -- is in fact an isometry!) And so we fit BND and BOD together to form a lune.

Let's move on to Proposition 500, which is considered by Legendre to be a mere "scholium":

500. Scholium. It is evident, also, that the two spherical pyramids, which have for their bases the triangles AOC, BOD, taken together, are the spherical wedge of which the angle is BOD.

Recall that a spherical pyramid is formed by taking a triangle and digging towards the center of the sphere, just as a spherical wedge is formed by taking a lune and digging down. Therefore Proposition 500 follows from 499 exactly as Proposition 496 follows from 495.

We proceed with Proposition 501, our third main theorem:

501. The surface of a spherical triangle has for its measure the excess of the sum of the three angles over two right angles.

And this is the amazing link between angle measure and area that we've been preparing for! We already know from earlier that the sum of the angles of a triangle in spherical geometry is always greater than 180 degrees. This theorem tells us that the triangle sum is greater for larger triangles than it is for smaller triangles -- in fact, the area of the triangle is exactly proportional to the number of degrees past 180 that the sum of the angles is. And if we choose the correct units -- as usual, radian measure with the radius of the sphere being unity -- the area of the triangle is exactly equal to the sum of the angles minus tau/2.

Legendre uses the word "excess" to denote the triangle sum minus tau/2. In Euclidean geometry the excess of any triangle is zero, but in spherical geometry the excess of any triangle is its area. In fact, if we wanted to, we can use the words "excess" and "area" interchangeably in spherical geometry, as they are always equal.

Let's see how Legendre proves how excess equals area:

Demonstration. Let ABC be the triangle proposed; produce the sides until they meet the great circle DEFG drawn at pleasure without the triangle. By the preceding theorem the two triangles ADE, AGH , taken together, are equal to the lunary surface of which the angle is A, and which has for its measure 2A (495); thus we shall have ADE + AGH = 2A; for a similar reason BGF + BID = 2B, [and] CIH + CFE = 2C.

Oops -- Legendre calls the great circle DEFG, yet he throws in H and I in as well. To make the proof work, let's assume that our triangle lies in the Northern Hemisphere and the Equator is the great circle "drawn at pleasure." Then AB is extended to intersect the Equator at D and G (which are antipodal of course), AC intersects the Equator at E and H, and BC intersects the Equator at F and I.

"But the sum of these six triangles exceeds the surface of a hemisphere by twice the triangle ABC; moreover, the surface of a hemisphere is represented by 4 [right angles -- that is, 4lambda -- dw], consequently, the double of the triangle ABC is equal to 2A + 2B + 2C - 4[lambda], and consequently ABC = A + B + C - 2[lambda]; therefore every spherical triangle has for its measure the sum of its angles minus two right angles. QED"

(Notice that the six triangles in question cover ABC thrice and the rest of the Northern Hemisphere once, which is why Legendre writes that the six triangles cover the hemisphere plus twice ABC.)

We'll get to more consequences of the fact that excess equals area in my next spherical post. Next week, we'll continue with the next rule of my new middle school classroom.

Rule #2: Respect Your Honesty

Last week, I wrote that the first rule in my classroom is to "Respect your grade." I devoted this post to a number of ideas to get the students to strive for an A in every class, and how I would give copious rewards to the "heroes" who earn the top grades in my classes.

But by placing such an emphasis on high grades, I'm tempting the students to cheat. If Rule #1 is all about earning good grades, then Rule #2 needs to address cheating somehow. Just as in my last post, I will draw from the ideas of the traditionalists -- at least when I agree with what they're saying.

Contents:
1. Have I Ever Cheated Before?
2. The Traditionalists on Cheating
3. How to Avoid Thinking
4. Other Comments From Traditionalists
5. Avoid Trying to Get an A Without Learning Anything
6. A Warm-Up Equation
7. What It Means to Show Work
8. Foldable Notes
9. Grading Group Work
10. Rule #2: Respect your honesty.


Have I Ever Cheated Before?

I'd love to be able to say that I've never cheated before in my life. But I am not perfect. My eyes have wandered to other students' work before, especially around the time I got that C+ in my Graduate Analysis class. But I always felt guilty whenever I did that -- and so I always wrote only some of what I saw on the other paper, never all of it. (I even remember once when I wrote down everything I saw on a separate sheet of paper, then copied only some of it onto the paper I turned in!) Because of this, I truly earned all of the A (or A-) grades that I ever received in any math class. The only grades that could have resulted by my eye-wandering were B- grades (and probably that C+ as well), since I only copied a little bit of what I saw.

But that's the thing -- since I succumbed to the temptation to cheat in Graduate Analysis, how much more will my students be tempted to cheat in my middle school classes? And this particularly applies to the students who find middle school math as challenging as I found graduate school math.

The Traditionalists on Cheating

Two years ago, there was an article about an app that solves math problems. (I don't link to it here, since students might see this and try to use it to cheat.) Now Bill (the traditionalist whose writing I plan on quoting the most) replied:

I’d agree, but many students go through school unable to actually do math, and they fail when they get to college and find out that the calculator can’t do it all. Perhaps teachers should give the grade to the calculator or app?

This is the biggest concern of the traditionalists -- too many students are able to get good grades in math class without actually knowing any math.

Now here's a link to a more recent article about this concern. As usual, I will link first to the article, and then to the Joanne Jacobs website where most of the traditionalists are responding:

https://www.washingtonpost.com/local/education/is-it-becoming-too-hard-to-fail-schools-are-shifting-toward-no-zero-grading-policies/2016/07/05/3c464f5e-3cb0-11e6-80bc-d06711fd2125_story.html

http://www.joannejacobs.com/2016/07/too-hard-to-fail/

The article is actually about the controversial 0=50 grading system. I will not repeat here what I've written before about 0=50, as that has nothing to do with my current grading system.

I'm surprised that so far, Bill hasn't commented in this thread (but he might soon). Instead, let's look at what some of the other commenters have to say about 0=50:

lee says:

That’s not the worst of it. I teach at the local CC, and a number of the schools here have a “50% minimum” policy. As might be anticipated, when these snowflakes eventually enroll in community college, they expect the same mollycoddling they’ve been accustomed to. They don’t do homework unless points are attached to it, they bomb exams and then pester you for opportunities for extra credit. Of course, you are the big, bad meanie who stands in the way of their future when you refuse to cater to the nonsense.

Enough. I’m looking to make a lateral move to some other profession ~

Here "lee" doesn't write what subject he teaches at the community college. But I must assume that some students do see math teachers as "big, bad meanies" who block their futures -- especially if they need to earn a degree to work at a job where they don't need to know math beyond arithmetic. The students enter the class thinking "How can I get a good grade without knowing any math?" instead of "How can I know lots and lots of math?"

The next poster has a homework policy that could be similar to my plans for my upcoming classes:

I have no problem with not basing grades on HW; tests, quizzes, and projects that indicate mastery of the lack thereof is all that is needed. But these administrators should be careful what they wish for. Many marginal students who diligently do their HW keep themselves afloat that way (i.e., failing but near-passing test scores and excellent HW records).

I still like giving HW, though, as it gives me an idea what the students understood…and what I need to re-teach.

By the way, the Los Angeles charter school mentioned the article is not -- I repeat, not -- the charter school where I'm going to teach in the fall. (The article mentions a high school, but I'll teach at a middle school in the fall.) The student interviewed for the article said that she actually opposes the school's former retake policy, since colleges won't let students retake tests.

Recall that in my classes, the only assessments students can retake are the Dren Quizzes -- and this is only because 90% correct is required to pass them.

How to Avoid Thinking

I recall some traditionalists who said that the real problem is that students don't want to think long and hard about math problems. The avoidance of thinking leads to several problems cited by traditionalists, including:

-- Students enter a number incorrectly into the calculator, obtain a result more than an order of magnitude away from the correct answer, and barely flinch.
-- Students groan when asked to solve a multi-step problem or one that takes more than a minute.
-- (Here's one I've seen myself, rather than mentioned by any traditionalists.) Students cheer when an equation has no solution.

Notice that this last one runs counter to the history of mathematics. Mathematicians hate -- I repeat, hate -- it when an equation has no solution. Indeed, they hate it so much that they invented new numbers for the sole purpose of preventing an equation from having no solution -- beginning with fractions, irrational numbers, negative numbers, and finally imaginary numbers.

But this is part of the problem -- students don't want to think. An equation without a solution means nothing to think about. Equations with solutions means something to think about -- and students especially find those invented numbers mentioned above difficult to think about.

I point out that if someone views math as a barrier, as many students do, then it's reasonable to want as many math problems as possible to have either no solution or a quickly found solution. That way the students can pass the barrier as soon as possible and they can engage in non-math activities that may be more relevant for their futures.

But if math is actually a door, as many traditionalists do, then it's not reasonable to hope that problems have no solution. If someone really wants to save time, then look for faster and better solutions rather than hope that a problem has no solution.

Other Comments from Traditionalists

Before I get to my own Rule #2 to stop cheating in the classroom, let me comment on a few other posts I've seen around the web. Unlike most Jacobs posts, the following links to a video rather than an article:

http://www.joannejacobs.com/2016/07/doing-science-beyond-dinosaurs/

This video discusses the Next Generation Science Standards (i.e. "Common Core Science"). As I will be teaching in an integrated STEM program, the science standards will be relevant.

The poster Dennis Ashendorf comments on the Next Gen Science Standards:

Dennis Ashendorf says:

BACKGROUND
Frankly, the curriculum people face unfortunate choices in high school:
1. NGSS is a four-year program. California has a two-year requirement. UC wants three lab years
2. Making NGSS compatible with AP science takes consideration. NGSS intentionally ignored AP in its design.

This reminds us of the traditionalists' complaints regarding Common Core Math and AP Calculus. So just as Common Core Math doesn't fit with AP Calculus, NGSS doesn't fit with AP science.

Let's skip down to the end of the Ashendorf's comment:

Solution

1. Earth Science, a course that provides no benefit to getting into college, becomes a senior elective. This decision places pressures on Earth Science teachers to get more official credentials, but they and we have known this for three years.

2. The easiest standard path that is most compatible with AP would be either:
(a) 9th-Chemistry, 10th-Biology, 11th-Physics or (b) 9th-Physics, 10th-Biology, 11th Chemistry
The “best” sequence (see Leon Lederman) that sadly conflicts with the two-year CA requirement would be: 9th-Physics, 10th-Chemistry, 11th-Biology

3. Simplest solution would be Integrated Physics/Bio/Chem for 9, 10, 11, where Biology needs completion by the end of the second year.
Students could opt out after two years and take any AP Science course or stop taking science, and three years completes the UC requirements, which is useful for CSU schools!

Notice that what Ashendorf calls the "best" sequence is often known as Physics First. I've mentioned Physics First a few times earlier on the blog.

Another poster in this thread is Mike:

Mike says:

Ocean waves ARE very similar to radio waves. And the sine waves generated by spinning the radius in a unit circle demonstrates how we can model them mathematically. Teaching wave phases teaches how waves can add and cancel, something that can be seen and demonstrated in a tank or a resonance tube.

I’m a big fan of hands on science and hands on math. A lot of phenomena ARE related within and across disciplines. Similar principles, different expressions of energy. Hands on science also teaches students they can reason from doing. It teaches them they can teach themselves.

The Illinois State text from which I'll be teaching takes this "hands on math" approach. And so I lean more towards Mike and away from the traditionalists in this regard.

Last week I mentioned most of my usual traditionalists. But in that post, I didn't mention Katharine Beals, who was inactive that week. I don't want Beals to feel left out of my traditionalist party, so here's a link to one of her posts from this week:

http://oilf.blogspot.com/2016/07/who-are-real-educational-colonialists.html

Many of us education bloggers have complained about people who advocate policies for other people's children that they would never inflict on their own. My own litany includes these types:

I've addressed this issue in the past when discussing "Presidential Consistency," but here Beals is referring to various education experts. Let's just skip down to the types that are the most relevant to what I will be teaching in the fall:

5. Parents who say charters and vouchers are destroying public education, but opt their own kids out of public education.

Well, technically I will be teaching at a charter school in the fall so this is relevant, but the list item I wish to discuss is:

7. People (typically education experts or education software developers) who would like to see the latest education fads--heterogenous group work, child-centered discovery, Everyday/Investigations Math, online, project-based learning--applied to children in general, but send their kids to more traditional schools that evade these fads.
[emphasis mine -- dw]

And here's the problem -- the Illinois State text from which I'm teaching is project-based learning. Of course we already know that Beals, like most traditionalists, dislikes project-based learning. But here she's mentioning PBL as an example of inconsistency -- those who advocate PBL know that it's "inferior" to traditionalism, so much that they prefer traditionalism to PBL for their own children.

Here's my response -- what about those kids who are actually sitting in those traditional schools, especially the middle and high schools, particularly in the math classes? Do those students actually enjoy their traditional Algebra classes? Or do they hope that the equations all have no solution so they can be done with the homework quickly -- deep down, do they wish they were in the PBL and other nontraditional classes for which their parents advocate?

The latter are the students I'm preparing myself to teach in my classroom. My goal is to get them to learn something rather than try to get a good grade without learning anything. So I need to have a rule that goes something like:

Avoid trying to get an A without learning anything.

And so here I will discuss my plans for how I will encourage students to learn actual math, rather than attempt to get a good grade without the math.

As is common in many math classes today, I'll begin each day with a warm-up. This will consist of a single question whose answer is the date. If this sounds familiar, it should -- the idea comes directly from Theoni Pappas and her Mathematical Calendar 2016.

At this point, you may be asking, doesn't this make it easier to cheat, not harder? After all, if the answer is just the date, any student who knows the date can just write down the answer without doing any math. And any student who doesn't know the date can just ask another, "What's today's date?" So far, this appears to accomplish the opposite of what we want.

But here's the thing -- since the students already know that the answer is the date, they'll have to show more work in order to to receive credit. So the typical student complaint, "How come I can't get credit just for writing the answer?" has a ready-made response: "You don't get credit just for knowing what today's date is!"

Today is July 9th, Let's look at the question Pappas wrote for today on her calendar:

To write 39065.21 * 10^3 in scientific notation, the decimal point must be moved in front of the digit ____.

This very well could be a warm-up question that I'll ask my eighth graders. After all, scientific notation appears in Learning Cycle 5 of the Illinois State eighth grade text: "H20 + ?: Measuring Using Parts Per Million (ppm)." This is the first cycle in Unit 1 of the text. (Recall that Unit 0 is Tools for Learning -- the first four modules are identical for all three grades.)

Since today's the ninth, we already know that the correct answer to the question is 9. So when I check the students' warm-up papers, I obviously need to see more than just the number 9 written down. The most obvious thing to have the students write is the actual scientific notation of the number, which in this case is 3.906521 * 10^7. Here knowing that the date is the 9th serves to provide the students with a hint that the decimal place should be moved. When I check the papers -- and I'll need to check 30 papers within a few minutes -- I won't look at the decimal points (since they already know that it belongs in front of the 9) and instead look for the exponent, which is 7 (not 9). Thus giving students the date hint helps the students out and allows me to check more papers in less time.

The problem of converting 39065.21 * 10^3 to scientific notation is tricky (bordering on deceptive), since the given form is not the standard form either (that would be 39,065,210). In the actual classroom, I'm likely to ask the students just to convert 39065.21 to scientific notation -- then the right answer is 3.906521 * 10^4. I'm not bound to use the exact same question as Pappas -- in fact, I'll almost never do so as it's rare that the question from the Pappas calendar is on the exact same topic that I'm teaching in any of the three classes. (For example, tomorrow's question involves finding the major axis of an ellipse. This is definitely not a middle school question -- it belongs in Algebra II, if not Precalculus.)

Here's another good scientific notation question that I could ask on the 9th of a month:

-- (2 * 10^2)(5 * 10^6) = 1 * 10^____

Here the students must write something to show why the correct exponent is nine. In this case, I'll be scanning the papers for an intermediate step such as 10 * 10^8.

A Warm-Up Equation

Of all the questions that appeared on the actual Pappas calendar this week, the only one that might be appropriate in a middle school classroom was the one from the 6th:

3/5 - 1/5 (14 + 9x) = -13

Let's look at all the steps written out:

3/5 - 14/5 - 9x/5 = -13
3 - 14 - 9x = -65
-9x - 11 = -65
9x + 11 = 65
9x = 54
x = 6

For this problem, I'll glance at the papers for key numbers, such as -65 all by itself on the right hand side of the equation. If I don't see -65, I'll know that the student didn't really do the work.

Then again, I'm not sure whether I'd ever assign this exact equation to eighth graders. If the lesson is all about clearing fractions in equations, then it's not quite obvious how to clear the fractions in:

3/5 - 1/5 (14 + 9x) = -13

We need to multiply every term in the equation by five, but it's not evident why -1/5 (14 + 9x) counts as only one term:

3/5 - 1/5 (14 + 9x) = -13
5(3/5) - 5(1/5)(14 + 9x) = -13
3 - (14 + 9x) = -65

Here I think it's better to distribute the -1/5 before clearing fractions. But then again, we tell the students to clear fractions in order to avoid calculating with them, yet we must calculate with them in order to perform the distribution! This is why I don't necessarily like to teach equations with both fractions and distribution required.

Moreover, the question contains many negative signs -- in fact, every term turned out to be negative, so I multiplied every term by -1 in order to make the terms positive:

-9x - 11 = -65
9x + 11 = 65

So this question really assesses three different topics: clearing fractions, distribution, and negatives. I don't believe that warm-up questions should assess three difficult topics -- especially when this increases the danger of "two wrongs make a right," such as:

3 - (14 + 9x) = -65
3 - 14 + 9x = -65
-11 + 9x = -65
9x = 54
x = 6

which they'll know to be correct if the question is asked on the sixth of the month. I can even imagine a situation where students make several mistakes along the way:

3/5 - 1/5 (14 + 9x) = -13
5(3/5) - 5(1/5)5(14) + 9x = -13 (multiplies wrong values by 5)
3 - 70 + 9x = -13 (fails to distribute anything to 9x)
-67 + 9x = -13
9x = 54
x = 6 (Hey look, today's the 6th!)

So I will avoid this giving sort of problem on the warm-up. My warm-up questions will focus on a single concept, which students must understand in order to get the right answer. So to assess clearing of fractions, I might give a question such as:

9x/5 + 11/5 = 13

where issues with distribution and sign have been eliminated.

Of course, notice that it's still possible for students to copy all the work from each other. But this takes more effort than just copying a single-number answer. And besides, a student who copies all the work could actually learn more than a student who just copies a number. The emphasis now is on the process, not just the correct answer.

It's consistent with this idea that I want students to focus on the process and not just the answer for me to assign odd-numbered exercises from a traditional text for homework. This refers to the idea that many texts contain the answers in the back of the book. So the students know the all the answers and so they realize that they need to write down the process in order to get credit. But this idea doesn't apply to the Illinois State text, which doesn't provide any answers at all.

What It Means to Show Work

Let's go back to the traditionalist Beals now. Lately, her "Math problems of the week" series has been focusing on scored free-response questions from various Common Core tests. The idea behind these recent posts is, students can know lots of math, yet receive low scores for these questions because the Common Core graders don't like their explanations.

When I say that I want the students to show work, I mean that I want them to show me just enough to indicate that the understand the math rather than copied a cheap answer. I assume that traditionalists like Beals don't like it when students just copy answers rather than learn the math (though sometimes they forget that with their emphasis on individual problem sets for homework, many students will just copy instead of do the work).

Indeed, when I'm checking warm-ups quickly, I don't want to see Common Core-type explanations. I want to see something that shows that the student really knows the math. In my examples above, I'm looking for a specific number or value that the student can't reach without understanding the material, such as -65 in the equation example or 10 in the scientific notation example (first 10 in 10 * 10^8).

Foldable Notes

When I was a student teacher, my school used the Glencoe text for Algebra I. This text encourages teachers to use Foldable notes in class. Students would take several sheets of construction paper and fold them together to make a miniature "notebook," or "foldable," on which to take notes.

Several members of the MTBoS use foldables, and I've linked to some of their blogs before. Back in March, the blogger "Math Easy As Pi" used a foldable to teach surface area. In January, the blogger "Math Milla" (Mrs. Miller) used a foldable to teach systems of equations. And last year, the blogger Sarah Hagan used a foldable to teach HOY-VUX, a mnemonic to help students remember the properties of horizontal and vertical lines.

(Oops! Of course by "Sarah Hagan" I mean Sarah Carter, the newlywed and one of the best-known members of the MTBoS. Recently she wrote a post to make a big deal about how much trouble she went through with her marriage and name change -- especially as her new husband is Australian. So the least we can do is call her by the correct last name.)

Meanwhile, I can't see the traditionalists liking foldables one bit. In March, there was a post on the Jacobs website about a Day 100 project at an elementary school (as usual for Jacobs links, I give both the article and the comment thread):

http://nypost.com/2016/02/28/im-always-stuck-doing-my-kids-homework-and-its-not-their-fault/
http://www.joannejacobs.com/2016/03/im-tired-of-doing-my-kids-homework/

I won't quote the comments here, but looking at them, we can see what the complaints are -- the project (in which students had to create a poster with 100 of something) was just artsy busywork with no educational value. And so if traditionalists don't even like this sort of assignment for young kindergartners and first graders, how much less then will they want to see middle schools creating foldables in math class? They'd deride foldables as pointless artsy busywork as well.

But here's the problem -- any traditionalist will say that many students don't take notes properly. I have the students create foldables -- sure, the time the students spend making the foldable is better spent writing notes in a college-ruled notebook. Yet -- this is par for the course for traditionalists -- they assume that the students will just diligently take notes in the notebook. I know firsthand that some students are willing to take notes on the foldable but won't take them in a notebook -- and in some cases, won't bother even to purchase a notebook to class. As usual, I'd rather a student spend a few minutes making a foldable and taking notes in it than spend the entire period staring blankly at a traditional notebook.

I mentioned in my Father's Day post that I might consider giving my students open-note tests. When I was a student teacher, my master teacher suggested that we let the students use the foldables on some (but not all) of the tests. And so I may do the same in my upcoming class this fall.

Grading Group Work

Of course, no discussion of trying to get a good grade without knowing the math is complete without addressing grading during group projects. With group projects, the traditionalists' biggest fear is that some students don't even attempt to contribute to the group.

I'm thinking back to my own days as a student. I didn't have group assignments that often in my math classes, but I remember one in particular that I had back in my Geometry class. As it turns out, the blogger Sarah Carter (yes, I just mentioned her earlier in this post) recently wrote about a group assignment that she plans on giving the first day of school -- and this assignment is very similar to the one I had in Geometry twenty years ago:

http://mathequalslove.blogspot.com/2016/07/broken-circles-planning-for-day-1.html

Students are placed in groups of 3-6.  Each student is given an envelope that contains 2-3 puzzle pieces.  The objective of the activity is for students to put their pieces together in such a way that each student has a complete circle.  

There are some very specific rules that must be followed, though.  

1.  No talking.  I think this will be the hardest rule for my students to follow. 

2.  No point or hand signals may be used at any time.  This will also be very tricky for my students.  As soon as they realize they can't talk, this will be the next thing they want to do. 

3.  Each player must put together his or her own circle.  No one may show another player how to put together his or her circle or do it for him or her.  

4.  Students may not take pieces from another student.  However, they may give one or more of their pieces to another student.  They may not put the piece in another person's puzzle.  Instead, they must hand it to the other person or lay it down on their desk.  

And I remember not faring particularly well on this assignment. I was placed in a group where all of the other students were older girls (recall that I was the only eighth grader in the class). I was looking around to see whether I could give my pieces to anyone, but the others just assumed that I wasn't going to participate fully. So they just took my pieces away -- and they claimed that they weren't breaking Rule #4 because one girl was taking my pieces to give to one of the other players. I wanted to tell them that I really was looking to give them pieces, but I couldn't say anything, because of Rule #1 of course. (Note: Unlike Carter, my teacher didn't give this on the first day of school.)

Throughout my years, I've had a mixed record with group assignments. I recall my fourth grade teacher saying that I had trouble with "cooperative learning groups," and there were some group assignments in high school history classes that I completely bombed. On the other hand, I usually did well on group lab assignments for science.

At this point, the traditionalists will point out that just as I didn't always do well on my group assignments, many of my students won't succeed either. So they conclude that I shouldn't torture my students with group projects.

In fact, I now think about a group assignment I once gave while student teaching. The rules for this assignment came from the district, and it was given near the first week of school (like Carter's). The students were divided into groups of four. The first student was supposed to build an equation by starting with a simple equation and performing the four operations on each side of the equation:

x = 1
2x = 2
2x + 4 = 6
(2x + 4)/3 = 2
(2x + 4)/3 - 1 = 1

Then the first person would give the second person in each group only that final equation, which now must be solved:

(2x + 4)/3 - 1 = 1
(2x + 4)/3 = 2
2x + 4 = 6
2x = 2
x = 1

This way, the students would see that to solve an equation, they must perform the inverse of each of the operations that they see in the equation, in reverse order. But here was the problem -- the first person in each group didn't build the equation correctly! This ended up frustrating the second group members, since their success was contingent on the first students' work. And I believe that many of these students disliked my teaching method the entire time I was there. When they had trouble with a later lesson, some didn't want to ask me for help because they thought I wouldn't really help them -- not after I'd refused to help them on this equation builder activity.

So these are the things I must consider when assigning group assignments from the STEM book. I will state the second rule of my classroom as:

Rule #2: Respect your honesty.

I must always be on the lookout for cheaters. The warm-ups and foldable notes should be set up to reward learners and punish cheaters.

And how should I handle group projects? I should monitor them to make sure that the students are actually learning something from the assignment -- and they aren't, find out why? If the activity is too complex (as that equation builder activity was), then I should simply it -- even if my simpler activity deviates from the instructions given by the district or the textbook. (So for the activity above, I should have had the students solve first one-, then two-step equations, even though the district insisted that the students build and solve four-step equations.)

And if the reason for the lack of success is a particular group member, then I should address it. I can break up groups to divide workers from slackers, to allay the traditionalists' fear that slackers can get a good project grade without learning any math.

My next post will be in about a week, when we'll return to spherical geometry.

Spherical Geometry (Legendre 490-495)

Today is a spherical geometry day. In this post we will discuss the next six propositions in Legendre's text, namely 490 to 495.

But first, I do want to say something about female mathematicians and scientists from my last post. In particular, yesterday the Google Doodle featured the American scientist Nettie Stevens. At the turn of the century, Stevens helped to discover that she, as a woman, had XX chromosomes, while males have XY chromosomes, but unfortunately, sexism stripped her of recognition. She performed her research right here in California (at Stanford). As a math teacher, I want to emphasize female mathematicians and physical scientists (as physics is more closely allied with math) rather than biologists like Stevens. But still, I wrote in my last post that I want to encourage girls to pursue STEM careers -- to create a world in which females are recognized for their contributions to STEM as much as males -- and biology definitely falls under the STEM umbrella. And so I celebrate the scientist Nettie Stevens here on the blog.

Now here's a female mathematician -- Theoni Pappas, author of the Mathematical Calendar 2016. As it turns out, her feature for July is "Geometric Worlds: From Euclidean to digital geometry." She writes about several different types of geometry -- including spherical geometry. And so let me post what Pappas has to say about this particular brand of non-Euclidean geometry:

"In elliptic geometry, such as on the surface of a sphere, its lines are also not straight, but are defined as great circles. In this world two lines (i.e. two great circles) always intersect in two points -- no parallel lines exist here. Unlike Euclidean geometry, the sides of a triangle are curved and its three angles always total more than 180 degrees."

The question for May 2nd was, "In spherical geometry, all pairs of lines intersect in ____ points." As Pappas writes above, the correct answer is two -- and of course, this question appeared on the second day of a month.

Pappas tells us that another way to distinguish among Euclidean, hyperbolic, and elliptic (spherical) geometry is curvature:

"The world of a plane in Euclidean geometry is flat, and therefore is said to have zero curvature. A hyperbolic plane, on the other hand, curves inward and therefore has negative curvature through it. In an elliptic geometric world such as a sphere, it continually curves outward which means it has positive curvature."

There are a few more interesting geometries mentioned in the Pappas article. She continues:

"In 1736, Leonhard Euler solved the famous Konigsberg Bridge problem in an innovative way using networks, and his work launched the geometry of topology."

This was, of course, our first day of school activity posted here on the blog! Indeed, I'm considering using the Konigsberg bridge problem as a first day of school activity in my actual classroom this upcoming school year!

Notice that the title of this article mentions "digital geometry." Pappas explains:

"Today a new geometry -- digital geometry -- is evolving in the world of the computer monitor."

I've made a big deal about this in the past as well. In Lesson 1-1 of the U of Chicago text, we see the following profound statement:

A point is a dot.

Whenever students wonder why they have to learn Geometry, we can remind them that they most likely enjoy the products of digital geometry everyday! (But let's not get carried away here -- recall that I'll be teaching middle school math in the fall, not Geometry.)

With that, let's get back to Legendre. But as Pappas points out, one of the most important theorems of spherical geometry is Triangle-Sum -- whereas the sum of the angle measures of a triangle is always exactly 180 degrees in Euclidean geometry, it's always more than 180 in spherical geometry. Using radian measure, this angle sum always exceeds pi -- or should I say, tau/2. (Just like last summer, I plan on using the new circle constant tau as much as possible from Tau Day until Pi Approximation Day, when I'll return to using pi.)

Now Triangle-Sum was Legendre's Proposition 489. As it turns out, most of today's theorems are basically corollaries of Triangle-Sum. Here's Legendre's Proposition 490:

490. Corollary I. The sum of the angles of a spherical triangle is not constant like that of a plane triangle; it varies from two right angles to six, without the possibility of being equal to either limit. Thus, two angles being given, we cannot thence determine the third.

Proposition 490 follows trivially from 489. Legendre -- like Euclid two millennia earlier -- uses right angles, not degrees, as units. In my last post, I mentioned that we can use lambda as a constant to denote the right angle, so we conclude that the sum can vary from 2lambda to 6lambda. In terms of tau, the sum varies from tau/2 to 3tau/2.

Legendre tells us that we can't determine the third angle of a triangle in spherical geometry given two angles as we can in Euclidean geometry. Thus there is no equivalent of what some Geometry texts call the "Third Angle Theorem."

Let's move on to Legendre's Proposition 491:

491. Corollary II. A spherical triangle may have two or three right angles, also two or three obtuse angles.

This is in stark contrast to a Euclidean triangle in which at least two angles must be acute. It's easy to see why this is the case -- if a Euclidean triangle had two right angles, then the sum of these angles would be 2lambda, which is the triangle sum for all three angles. This would leave us a measure of zero for the third angle. A triangle with two obtuse angles would be even worse -- the third angle would have to have a negative measure! On the other hand, in spherical geometry the sum of the angles is always more than 2lambda, so we can comfortably fit two right, or even two obtuse, angles and still have some measure left for the third angle.

When thinking about a triangle with three right angles, I can't help but think about the following brainteaser, which I've posted several times here on the blog:

http://www.murderousmaths.co.uk/books/bearpuz.htm

• A bear hunter sets out from camp and walks one mile south.
• He sees a bear and is about to shoot it.
• The bear grabs his gun and eats it.
• The hunter runs away one mile east.
• He then walks one mile north and gets back to his camp and changes his underwear.
• What colour was the bear?

The answer is that the "colour" (sorry -- this is obviously from a British website) of the bear is white, since the puzzle describes a polar bear at the North Pole. Technically, this is not a spherical triangle, since the "one mile east" is along a parallel of latitude, not a great circle. It's not even close to being a great circle -- if the hunter ran approximately six miles east he would have walked in a complete circle around the pole.

Indeed, Legendre tells us what a true spherical triangle with two or three right angles would look like:

"If the triangle ABC has two right angles B and C, the vertex A will be the pole of the base BC (467); and the sides AB, AC will be quadrants.

"If the angle A also is a right angle, the triangle ABC will have all of its angles right angles, and all of its sides quadrants. The triangle having three right angles is contained eight times in the surface of the sphere."

Here Legendre tells us that if ABC has right angles B and C, we can still let A be at the North Pole just as in the original "What color was the bear?" puzzle. But then points B and C would end up lying on the Equator! Proposition 467 cited by Legendre here is one that we discussed last year -- it refers to the relationship between a line (great circle) and its poles. Every line has two poles -- the name "pole" is justified by the fact that the poles of the Equator are the actual North and South Poles. (Last year I came up the habit of using lowercase "pole" to refer to the poles of any line, but the capitalized "Poles" are the North and South Poles only.) The two sides opposite the right angles (the two "hypotenuses"?) are each a quadrant in length -- a quadrant being the distance from the North Pole to the Equator.

Then Legendre's triangle with three right angles has all three sides equal to a quadrant (equiangular triangles are always equilateral in both Euclidean and spherical geometry). He writes that the entire globe can be divided into eight such triangles. It's easy to see how to divide the Northern Hemisphere into four such triangles -- the sides of these triangles can be the Prime Meridian, the 90th Meridians (both east and west), and the 180th Meridian (the International Date Line). We can divide the Southern Hemisphere into four triangles in exactly the same manner, so that there are eight total triangles with three right angles each. (This also tells us how to find the poles of any meridian -- the poles of a meridian lie on the Equator, 90 degrees east and west of the original meridian.)

It's trickier to find a triangle with two obtuse angles. If we again take the North Pole to be the vertex angle A of an isosceles triangle, where must B and C be placed so that both B and C are obtuse? The answer is that they must lie in the Southern Hemisphere. This is difficult to visualize -- B and C might be on the same parallel of latitude and any parallel forms a right angle with any meridian, but the problem is that parallels of latitude (other than the Equator) are not great circles. The symbol BC can refer to the actual great circle joining B and C, but then it's not obvious why BC must form obtuse angles with both AB and AC. Indeed, the only great circles that are easy to visualize are the Equator and all meridians, and BC is neither. A full proof would be quite long, and would only distract us from the theorems that actually appear in Legendre. This is probably why Legendre doesn't give us a triangle with two obtuse angles in the first place.

Here is Legendre's Proposition 492:

492. Scholium. We have supposed in all that precedes, conformably to the definition, art. 442, that spherical triangles always have their sides less each than a semicircumference; then it follows that the angles are always less than two right angles. For the side AB is less than a semicircumference, as also AC; these arcs must both be produced in order to meet in D. Now the two angles ABC, CBD taken together, are equal to two right angles; therefore angle ABC is by itself less than two right angles.

Here Legendre hints at the idea that a triangle may have an angle greater than 180 degrees. (Some texts give the name "reflex angle" to an angle that is larger than a straight angle.) In Euclidean geometry, the angles of a triangle add up to 180, so no angle can be greater than 180 without one of the other angles being negative. But the sum of the angles of a Euclidean quadrilateral is 360, and so there can -- and do -- exist quadrilaterals with an angle greater than 180. Polygons containing reflex angles are called "concave" (or "nonconvex").

So Legendre is telling us that in spherical geometry, there might exist concave triangles. But, as he writes, all triangles with sides less than a "semicircumference" are convex. Most of the time, we expect sides of a triangle to be less than a semicircumference indeed -- after all, to get from A to B, we usually want to go the short way (which is a minor arc less than a semicircumference) rather than the long way (a major arc greater than a circumference). Legendre proves that in this case, the angles must all be less than 180. This is because Legendre extends AB and AC to meet at D -- recall that all great circles passing through A meet at the antipodal point of A, which is D. Now D is exactly a semicircumference away, so we truly are extending AB and AC to meet at D. Then the two angles ABC and CBD form a linear pair which adds up to 180. So each angle must be less than 180.

Legendre continues:

"We will remark, however, that there are spherical triangles of which certain sides are greater than a semicircumference, and certain angles greater than two right angles. For if we produce the side AC till it becomes an entire circumference ACE, what remains, after taking from the surface of the hemisphere the triangle ABC, is a new triangle, which may also be designated by ABC, and the sides of which are AB, BC, AEDC. We see, then, that the side AEDC is greater than the semicircumference AED, but, at the same time, the opposite angle B exceeds two right angles by the quantity CBD."

The easiest way to visualize what Legendre is writing here is to place B at the North Pole (rather than A) and all the other named points on the Equator. Since he calls AED a "semicircumference," his intent is for D to be the antipodal point of A. He doesn't state where E is, but we might as well let E be the antipodal point of C. Then there are two triangles with vertices ABC -- one where we go the short way AC, and the other where we go the long way AEDC. Now it becomes obvious why the angle B opposite AEDC must be more than 180 -- the large angle ABC must equal ABD plus CBD, but ABD is exactly 180 as A and D are antipodal. That is, the great circle through AB (a meridian, as B is the North Pole) must pass through D since all great circles through A pass through D. So ABD is a straight line (great circle), and all straight angles measure 180. Then the excess CBD makes the full angle ABC a reflex angle.(Adding the smaller angle ABC to this gives 360 degrees or tau.)

Let's proceed with Legendre's Proposition 493, our only major theorem for today:

493. The lunary surface AMBNA is to the surface of the sphere as the angle MAN of this surface is to four right angles, or as the arc MN, which measures this angle, is to the circumference.

What is this theorem stating? First of all, let's recall what a "lunary surface" is -- often abbreviated to lune, a lunary surface is the region bounded by two intersecting line segments (great circle arcs). As all lines intersect at antipodal points, the two arcs must be semicircles. A lune is essentially a polygon with two sides -- which doesn't exist in Euclidean geometry.

The name lune or lunary surface reminds us of the word lunar, which means "of the moon." And indeed, every time we look up at the moon, we see a lune -- the portion of the moon illuminated at any time (regardless of the phase of the moon) is a lune. Another way to imagine a lune is to let points A and B be two antipodal points -- say the North and South Poles. The points M and N can be placed anywhere on the Equator -- this allows us to measure either the angle MAN or the arc MN, just as Legendre does in the statement of the theorem.

This is the first theorem in Legendre that hints at the concept of area, but he doesn't use "square units" when considering area. Instead, he uses ratios -- just as the use of a right angle as a unit of angle measure, the use of ratios to determine area is a tradition going back to Euclid. But notice that there's an obvious natural unit of area to consider -- the surface area of the sphere itself. Let's assume that our sphere is the unit sphere -- that is, it has radius 1 -- and we will use radians throughout.

Legendre mentions the ratio of the angle MAN to four right angles -- that is, the ratio of the angle MAN to tau. Notice that the ratio of the arc MN to the circumference is this exact same ratio, because we chose radians and the unit sphere.

Now the theorem tells us that this equals the ratio of the area of the lune AMBNA to the surface area of the entire sphere. The surface area of a sphere is 4pi r^2 and this is a unit sphere, so the area of the sphere is 4pi -- oops, make that 2tau. So the theorem is essentially telling us:

Area(AMBNA)/(2tau) = Angle MAN/tau

Let's see how Legendre proves this theorem:

Demonstration. Let us suppose, in the first place, that the arc MN is to the circumference MNPQ in the ratio of two entire numbers, as 5 to 48, for example. [That is, suppose that the measure of the arc MN is 5tau/48 -- dw] The circumference MNPQ can be divided into 48 equal parts, of which MN will contain 5; then joining the pole A and the points of division by as many quadrants, we shall have 48 triangles in the surface of the hemisphere AMNPQ, which will be equal among themselves, since they have all of their parts equal. The entire sphere will therefore contain 96 of these partial triangles, and the lunary surface AMBNA will contain 10 of them; therefore the lunary surface is to the sphere as 10 is to 96, or as 5 is to 48, that is, as the arc is to the circumference.

So we can easily see what Legendre is doing here -- he is essentially dividing both the lune and the entire sphere into many small triangles. In his example, the lunes has angle measure 5tau/48, so he divides each half of the sphere into 48 triangles, with the lune taking up 5 of these triangles. We must consider both halves of the lune or sphere, which is why the entire lune takes up 10 out of the 96 triangles into which the entire sphere is divided. Using algebra, we can generalize this by saying that if the measure of the angle is (p/q)tau, we can divide each hemisphere into q triangles of which the lune takes up p. So the entire lune contains 2p out of 2q triangles into which the entire sphere is divided, and 2p/2q reduces to p/q, so the lune takes up p/q of the surface area of the sphere -- which, if you remember, is 2tau. So the area of the lune is (p/q)2tau.

The proof seems to be complete, yet Legendre adds the following cryptic lines:

"If the arc MN is not commensurable with the circumference, it may be shown by a course of reasoning, of which we have already had many examples, that the lunary surface is always to that of the sphere as the arc MN is to the circumference. QED"

What is Legendre saying here? Here's the problem -- what if the angle measure equals k tau for some irrational value k? ("Commensurable" essentially means "rational" -- a tradition going all the way back to Pythagoras and his sqrt(2) irrationality proof.) Then we can't just blindly write k = p/q and divide the hemisphere into q triangles. Instead, Legendre resorts to a "course of reasoning" that allows us to conclude that the theorem holds for both rational and irrational k. Dr. Hung-Hsi Wu gives this "course of reasoning" a name -- the Fundamental Assumption of School Mathematics.

By the way, notice that the equation:

Area(AMBNA)/(2tau) = Angle MAN/tau

can be rewritten as:

Area(AMBNA) = 2Angle MAN

And notice that the lune contains another angle at the South Pole, MBN, that is congruent to MAN (as both angles have the same measure as arc MN). So we can also write this as:

Area(AMBNA) = Angle MAN + Angle MBN

This begins to hint at a relationship between area and angle measure that exists in spherical, but not Euclidean, geometry.

The last two propositions are corollaries of this theorem:

494. Corollary I. Two lunary surfaces are to each other as their respective angles.

This is too trivial to prove as it follows directly from that equation we just wrote:

Area(AMBNA) = 2Angle MAN

Let's look at the final proposition for today:

495. Corollary II. We have already seen the entire surface of the sphere is equal to eight triangles having each three right angles (491); consequently, if the area of one of these triangles be taken for unity, the surface of the sphere will be represented by eight. This being supposed, the lunary surface, of which the angle is A, will be expressed by 2A [Yes, this is the formula we just wrote again -- dw], the angle A being estimated by taking the right angle for unity; for we have 2A : 8 :: A : 4. Here there are two kinds of units; one for angles, this is the right angle; the other for surfaces, this is the spherical triangle, of which all the angles are right angles, and the sides quadrants.

We see that Legendre's equilateral right triangle has area one-eighth that of the entire sphere -- since the sphere has area 2tau, the triangle has area 2tau/8 = tau/4 = lambda. So his two units -- the right angle for angle measure and the equilateral right triangle for area measure -- are both equal to lambda (again, assuming the unit sphere and radian measure).

In fact, we see that this is true for any triangle with two right angles -- if the third angle has measure A, then the triangle is half of a lune with angle A and measure 2A, so its measure is also A:

Area(AMN) = Angle A

And notice that the sum of the angles of Triangle AMN is exactly two right angles (tau/2) plus the measure of this Angle A. Legendre mentions the special case where Angle A = lambda. Again we hint at a connection between angle measure and area.

This is a good place for us to end this post. My next post will be tomorrow, when I'll continue with the rules of my new classroom.

Rule #1: The Teacher Respects You

This is the first of several posts I'm making this summer in order to prepare for my first year of teaching at a charter middle school.

I've decided to write this series of posts in terms of the classroom rules that I plan on having. As a sub, I've seldom had to come up with classroom rules -- my job was to enforce the rules established by the regular teacher. In the rare situations where I needed my own rules, my first rule would be something like, "Follow all adult directions." The emphasis here was that if the students wouldn't respect me as a teacher (the attitude of many students toward subs), perhaps they would at least respect me as an adult, hence the rule "Follow all adult directions."

But now I'm going to be a regular teacher myself, so I need my first rule to have a higher purpose than simply to treat me as an adult. The thing is, before the students can respect me, I must learn to respect them.

This post is quite long, so I've divided it into sections.

1. Suggestions from Traditionalists: Math Curmudgeon
2. Strive to earn an A in this class and every class!
3. Suggestions from Traditionalists: Bill
4. Suggestions from Traditionalists: Wurman and Garelick
5. Grade School to Grad School (or From C to Shining C)
6. The Consequences of my C's
7. My Departure from the Traditionalists: Real World Math
8. Let's Break Down My Grades
9. Don't Be a Dren!
10. Heroes and Fraction Fever
11. On Gender
12. Rule #1: The Teacher Respects You

Suggestions from Traditionalists: Math Curmudgeon

Recall that I'm using traditionalist links to guide me through my first year of teaching. The first link I'll give today is a post mentioned by the traditionalist Math Curmudgeon, whose blog I mentioned back in my Father's Day post:

http://mathcurmudgeon.blogspot.com/2016/05/ego-of-ignorant.html

In this post from two months ago, Curmudgeon quotes another person's tweet:

"I just took the 2016 SATs test. I failed. 25% in Maths, 40% in English. Kids, you don't need to know what a modal verb is or a subordinating conjunctive is to get where you want to go in life. You need ideas & passion -- so go on adventures, dream BIG and don't worry about your SATs scores."

Curmudgeon then laments that the original tweet drew comments like "You're my hero!"

There are several things going on here. This is a math blog, so we won't worry about modal verbs and go directly to the math score. At first Curmudgeon assumes that the tweeter is referring to the SAT test, but it's awkward to give SAT scores as a percentage, as this tweeter does. And if the tweeter intends to say 25% of the top score of 800, then this is 200 -- the lowest possible score.

Then a commenter on Curmudgeon's post points out that the tweeter is British (the spelling "Maths" should give it away), and she's actually referring to SATs that are taken by students completing a U.K. elementary school. Then again, this makes the tweet even worse. The tweeter is bragging that she only knows 25% of elementary school arithmetic -- in other words, she's a "dren"!

But again, Curmudgeon's biggest complaint is about the "You're my hero" responses to the tweet. We can easily see why the blog author is upset -- suppose the original tweet said, "I got 85% in Maths and 100% in English," and guess how many "You're my hero" responses this tweet now receives. A good estimate to this question is zero. Instead of "You're my hero," we expect "You're a nerd" responses to be more common.

Many traditionalists like to compare those who excel in academics to those who excel in sports. The NBA season recently concluded. Throughout the regular season, there had been much excitement about Steph Curry, the first unanimous MVP. Then in the playoffs, the accolades were now directed towards LeBron James, the Finals MVP, as he won his third championship. Traditionalists lament that Curry and James are treated as heroes, yet the mathematical equivalents of Curry and James are treated as nerds.

We know that math is near easy nor fun, but neither is training to be a world-class athlete on the level of Curry or James. But I've said many times on the blog that hard work is respected in "high status" fields, which includes basketball, but not mathematics. Yet this leads to yet another question -- why exactly is basketball considered higher status than mathematics?

After all, it is possible to have mathematics without entertainment (which is why many students don't like math classes), but it's impossible to have entertainment without mathematics (especially not modern forms of entertainment). We enjoy watching sports on a variety of media thanks to the inventions of people who earned A's in their math and science classes -- without them, we wouldn't be able to watch the games unless we bought tickets. In fact, what would students rather be doing instead of studying math and science? The answer is almost certainly to use something that was invented by people who earned A's in their math and science classes -- without them, the inventions that they enjoy wouldn't exist. Therefore the real heroes are the students who get lots of A's -- especially in math and science classes. These students should be treated as heroes, not nerds.

Here's another way to think about this problem. Someone who calls Curry and James heroes is thinking, "Curry and James are people just like me, except better at basketball." But if a student excels in math, the thought is, "That student isn't just like me -- he/she is a nerd." It doesn't matter that students are more likely to ace math than they are to beat Curry or James one-on-one -- more people identify with basketball stars than mathematicians.

So my goal as a math teacher is to get my students thinking, "That great math person is just like me, except better at math," and eventually, "That great math person is my hero." Keeping this in mind, I was considering the following as the first rule in my classroom:

Strive to earn an A in this class and in every class!

It's not enough merely to try to earn an A in my class (Yoda: "There is no try.") -- instead, students must work hard and strive to earn an A. This rule is an umbrella rule -- it covers other common classroom rules such as "Bring all books and materials to class." Students who don't bring materials to class aren't really striving to earn an A.

When I was a student, it never even occurred to me not to strive for an A in every class. Of course I was a good math student, but I wasn't great at other subjects, such as history. I will go as far to say that I hated history and didn't find the subject relevant to my future. Yet I never earned any grade less than a B in any history class. It's possible to hate a subject yet still score above 80% on nearly every test in that subject.

I don't expect every single student to earn an A in my class, but I want every student to strive to earn the top grade. After all, every single player on the court strives to win the championship, even though only one team can actually win the championship. Likewise, every single student at a school should be striving to be the valedictorian, even though not every student can actually be the valedictorian.

At this point, you may ask, what about tanking? Aren't there players and teams who aren't actually striving to win a ring? But think about it: teams tank because they want to draft great players -- players who will eventually win them a ring. So in reality, every team is trying to win, either in the current year or in a near future year. On the other hand, students who "tank" in math class aren't trying to pass math ever, either now or in the future.

In theory, every single student should be striving for an A, but in reality, some students aren't trying to pass at all. I'm getting ready to teach at a middle school -- and often it's in middle school when students decide to stop working hard in their classes.

Just before Memorial Day, I blogged about the movie Akeelah and the Bee, where the title character is a girl who's trying to win the National Spelling Bee. At the beginning of the movie, Akeelah is encountered by two girls who criticize her for earning so many A's, then beat her up. Of course, these girls are wrong to hit Akeelah, but that doesn't change the fact that they hit her. This is something that I must watch out for as I teach -- even though Akeelah's school in the movie is a fictional middle school in Los Angeles, the charter school where I'll be teaching is, frankly, located not that far from where the movie is set.

The tweet mentioned in Curmudgeon's post earlier is a symptom of a larger problem -- we simply don't want to hear that the most successful people in life are those who earned the top grades. Yes, it's possible to be successful without earning A's, but such people are exceptions to the rule. There have even been books written about how A students aren't the most successful people in life -- just look at the title of Robert Kiyosaki's 2012 bestseller: Why A Students Work for C Students. Meanwhile, a book with a more truthful title such as Why A Students Run the World wouldn't have been a bestseller.

In my class, an A isn't the only acceptable grade, but it is the only acceptable goal. It's okay to earn a B if you were striving to get an A. It's okay to earn a C if you were striving to get an A. It's even okay to earn a D -- but by that point, I'm skeptical that you were really striving to get an A.

I want to tell the story about my own grades, from grade school to grad school. But first, I have several more traditionalists to discuss, as they were particularly active in posting this week.

Suggestions from Traditionalists: Bill

As usual, I'm getting these comments from the Joanne Jacobs site, which in turn links to articles from other websites, but the traditionalists comment only on the Jacobs site. The first link involves my home state of California, except it's Northern California:

http://www.joannejacobs.com/2016/06/sf-no-child-gets-ahead-in-math/

The link describes how in San Francisco, all eighth graders take Common Core Math 8 rather than the class that traditionalists prefer them to take, Algebra I. I've discussed this topic so many times on the blog, so let's just skip to the traditionalists' comments.

I was expecting the traditionalist Bill to comment on the Jacobs site, and he didn't disappoint:

Bill says:
It looks like high schools are more interested in diversity rather than actually learning math, the foundations of which start in Elementary School…if they don’t have the basics down by the time they head to 6th grade, they’re gonna struggle in math the rest of their lives…UGH

Another poster, a community college professor, wrote about a student in his class who didn't know how to calculate the average or arithmetic mean. Here is Bill's response:

Bill says:
The student should have never been allowed to be enrolled in this class without a math placement examination (IMO), but I’ve seen students like that myself who couldn’t handle basic stats/probability…it’s painfully evident how math challenged our society has become…

Suggestions from Traditionalists: Wurman and Garelick

But there are other traditionalists in the comment thread as well. I wasn't expecting the traditionalist Ze'ev Wurman to post here, but he does. He basically echoes Bill's comments:

Ze'ev Wurman says:
That’s what happens when your goal is to assure equal outcomes rather than pursuit of excellence.

Another surprise traditionalist posting a comment is Barry Garelick. Jacobs herself wrote that in order to get to senior year AP Calculus in districts that don't offer eighth grade Algebra I, some districts offer a choice between taking a single course that combines Algebra II and Pre-Calculus, and simply doubling up in math one year. Here is Garelick's response:

Barry Garelick says:
That isn’t the case here in San Luis Coast USD. Students have to double up courses one year.

As it turns out, Garelick has a comment on the original article as well:

My grandmother didn't take Algebra 1, but I took it in the 60's and I suppose it's people like me that Ryan [STEM director at the SFUSD -- dw] is referring to. I have a bunch of textbooks from that era. I'll be teaching 8th grade algebra at a school in California, in which the school district doesn't sit on the high horse that SFUSD likes to occupy. In looking through the Common Core-aligned algebra book I'm forced to teach from, I'm aghast at the dearth of good solid word problems, the short shrift given to exponentials, to rational expressions, not to mention the omission of solving quadratic equations by factoring--I guess the quadratic formula saves a lot of time and there's no value in teaching that approach. There is a chapter on statistics (as if that's needed in an algebra class), and a superficial look at exponential functions, which I suppose allows people like Ryan to say "Look how deep this course is. Not your grandmother's algebra 1".

Furthermore, the algebra Ryan feels is taught in regular 8th grade math, isn't that much different than what used to be offered in 7th grade pre-algebra classes. The exception is that they teach simultaneous linear equations--and spend an inordinate amount of time on that, as well as developing a "deep understanding" of slope. I observed an 8th grade class going through this supposed "deep understanding"--spending five weeks on slope and functions which could have been taught fairly well in 2 weeks.

I will be supplementing the algebra book heavily and giving lots of word problems, as well as problems with exponentials, powers, and rational expressions. That aside, the policy that 8th graders shouldn't be taking algebra 1 is an ill-thought one. The school district in which I reside (but do not teach in, and refuse to do so because of a constructivist-oriented superintendent and a very student-centered approach to education in general) has implemented a similar policy. Algebra 1 for those middle schoolers who are "truly gifted"--a term left undefined, but tracked by a very poor readiness exam put together by Silicon Valley Math Initiative (SVMI). SVMI is made up of constructivist group-thinkers who 1) haven't a clue what works, nor 2) do they realize that the "grandmothers" who took algebra 1 learned a hell of a lot more than today's youth.

I've already mentioned the traditionalists' solution to the Algebra I problem. They don't make students take two classes in one year to reach Calculus -- instead, they compress nine years of math (that is, Common Core K-8) into eight years (K-7). To accomplish this, they cut out all Common Core K-8 Standards that they don't like (especially those involving nonstandard algorithms) and drop down a grade those that they do like (especially those involving the standard algorithm). Garelick hints above at what the resulting seventh grade class would look like -- it is basically Common Core 8 without the unit on stats (and probably the unit on transformations as well) and cutting three weeks out of the unit on slope and functions.

Our next thread is a little closer to home -- it involves high schools in LAUSD. Remember that I will be teaching at a middle school (not a high school) and it's a charter school (not LAUSD proper). But it's possible that many of my students will be moving on to LAUSD high schools if they fail to be admitted to a charter high school. The topic is credit recovery.

http://www.joannejacobs.com/2016/06/fudging-grad-rates-via-credit-recovery/

Bill writes:
Credit Recovery programs are a scam designed to boost graduation rates, period…you cannot learn a semester’s worth of information in less than usually 80-100 hours of instruction time (give 5 hours a week for 13-15 weeks), and the issue of taking a 10 question multiple choice exam and getting a pass for 60% is a joke, since the students failed the class in the first place (the cut score should be at least 75% using 20-30 questions of multiple choice, fill in the blank, and true/false)…

Later on Bill responds to a teacher describing a similar situation in South Carolina. I'll leave that part out and stick to California in my post.

Based on these traditionalists' posts, my goal is to make sure that the students see the value in actually learning the material. Even if there are high school "credit recovery" classes where students can graduate without actually knowing any math, it's far. far better to know math anyway. As a student, it never even occurred to me to try to graduate without learning anything. I'd like to say that I earned all A's and B's in my classes, but unfortunately, I did get a few C's along the way.

Grade School to Grad School (or, From C to Shining C)

The first C grade that I ever received was in first grade, when I earned a C in handwriting. But when I was in middle school, I earned a few more C grades. Even at the time, I considered each C grade to be a mark of deep shame, and I still am ashamed of my C's to this today. Of course I earned A's in all of my math classes, but in several other classes I earned C's.

Here are all the C's that I earned in middle school. My school divided the year into four quarters, and so I will give each class plus the quarter in which I earned the C:

-- 6th Grade Health/Self Esteem, third quarter
-- 7th Grade Art, first quarter
-- 7th Grade Science, fourth quarter
-- 8th Grade Science, first quarter
-- 8th Grade Science, second quarter
-- 8th Grade English, second quarter
-- 8th Grade English, third quarter
-- 8th Grade Library Aide, fourth quarter

I'm the most ashamed of my eighth grade science C's. This is because 8th grade science has always been a physical science class (as opposed to 6th grade earth science, which wasn't established in California until after I passed the 6th grade). Of all the sciences, physical science is the most allied with mathematics. No student with A's in math -- especially as the only 8th grader in Geometry -- has any business earning C's in physical science, yet that's exactly what happened to me.

Throughout high school, I strove to make sure that I earn all A's and B's in order to erase the shame of my "seven C's" of middle school. Notice that I used the pun "seven seas" = "seven C's" of middle school, yet in reality I earned eight C's in Grades 6-8. This is because at the time, sixth grade was considered elementary, not middle school, so I didn't count my sixth grade C with the seven C's of middle school.

In high school, I sometimes had low grades after the first quaver (half of a quarter) of a class. This often occurred in my English classes, where there were very few graded assignments the first quaver (when the focus was on just reading the material rather than turning in work for a grade). Sometimes I received a low grade on the first major assignment -- and since there was hardly any other assignment to balance out my grade, I'd receive a D grade on the first quaver progress report.

I even once had an F grade as my first quarter grade in my junior-year English class. I don't wish to make excuses here, but that year, my teacher injured herself the second week of school, and so there were a series of subs for over six weeks. One of the subs assigned a major assignment -- I think we had to write a poem. But I forgot about it because none of the other subs had given us longer assignments, and so I'd become accustomed to doing homework for other classes, not for English. At the end of the quarter (by when the regular teacher had returned), this poem made up the largest part of our grade, and so I ended up with something like 55% for the quarter.

Ironically, this was right around the time when I was being considered for the magnet program (as I explained in a previous blog post the second week of June). I was transferred anyway despite the F -- as it turned out, the grade appeared as D on the quarter progress report. This might have been because I was being transferred from Honors English 11 in the old program to English 10 in the magnet (as the magnet is a year ahead of the regular program) -- as honors classes didn't begin until 11th grade, I was graded on the non-honors grading scale, where 55% was a D! (Notice that if I hadn't switched districts and applied to the magnet at the end of 8th grade as was standard, I probably would have been rejected for earning too many C's in the 8th grade!)

My new English teacher told me that she had to include the quarter grade as part of the semester grade, but she would weigh the second quarter more heavily. In the end, I earned a B that semester -- and indeed, I ended up with all A's and B's every semester in high school.

My college career, however, was a different story. The first C grade I earned at UCLA was in a biology class. But I'm most ashamed of my lowest grade I ever earned in any class -- a C- in the third quarter of first-year Physics. I'd passed AP Physics C Electricity and Magnetism with a grade of 4 (and unlike Calculus, Physics AP's never receive equivalence at UCLA), and so I'd probably assumed that I could get a good grade in this Physics class without much effort. Obviously, I was wrong.

Even though my Physics C- was my lowest grade ever, my most destructive grade was actually the C+ that I earned in a Graduate Analysis class. The problem is that this was the first quarter in the grad program at UCLA, and grad students are really supposed to earn only A's and B's. Actually, what matters the most is the overall GPA, which should be at least 3.0. That quarter, my grades in the other two classes were B and B+, which made my overall GPA 2.87 (as plus-grades are worth an additional 0.3 point). Just as with my missing poem in 11th grade English, the problem was that there were no other quarters to balance out my grade, as undergrad quarters aren't included in calculating the grad GPA at all. My 2.87 grad GPA meant that I was officially on academic probation until I raised the GPA to 3.0 or better.

Again, I don't want to make excuses, but I heard that the Analysis prof was a tough grader. Indeed, so many students were failing the class that the prof was encouraging students to drop the class and move down a level to Honors Upper Division Analysis -- again, this was a class that I had already taken and passed with a B. So once again, I'd become jaded and assumed that just because I'd passed the previous class, I could pass the later class without much effort.

I still remember one test question that I had particular trouble with -- it was about determining whether a certain function was Lebesgue measurable. (I actually mentioned Lebesgue measure and integration earlier on the blog, in the process of discussing David Kung's DVD course.) I remember the prof saying several times that the open intervals generate all the measurable sets, yet I didn't use this when working on the test. And so I ended up failing the test and getting only a C+ in the course.

Why am I posting all of this on the blog? It's because I want to inspire my students to earn as many A's in their classes as possible -- and I want to show that I wasn't perfect myself, so the students should try to avoid my mistakes. I plan on telling my students about my own grades. This also allows me to empathize with my students -- for many of them, working with fractions is as difficult as Grad Analysis was for me.

I never earned a grade as low as C+ again -- even though I did get a few B- grades. I raised my GPA to above 3.0 in my second grad quarter and ultimately earned my Masters degree with a final GPA of around 3.3 or 3.4. But as I'd struggled so much with my Masters, I decided not to seek a Ph.D degree.

The Consequences of My C's

After leaving UCLA, I wasn't sure what I wanted to do with my life. Yes, I did say in earlier posts that as a young kid, I'd wanted to become a math teacher. But by middle school I wasn't so sure -- I'd seen the way that students treated subs, and I knew that I had to become a sub before I could become a regular teacher.

Of course I wanted to apply my STEM degree, so I applied to some local engineering companies. I remember one interview when I was asked about my GPA. My interviewer had noticed that my undergrad GPA was 3.6, but my grad GPA was only 3.3-3.4. I told him that grad classes are more difficult than undergrad classes, and his reply was that yes, but I should have been a stronger, more mature student by then as well. In the end, I was denied the job -- and based on the sequence of questions, I concluded that it was because of my low grad GPA. In other words, I was denied the job because I hadn't earned enough A's in my classes.

Back in the seventh grade, I'd learned that the largest public employer of mathematicians is in fact the Department of Defense -- in particular, the NSA. After grad school, I ended up applying to the NSA, and I was flown across the country to its Maryland headquarters for an interview. One of the questions I was asked was, "Have you ever been on academic probation?" And I was forced to answer "yes" because of my bad first quarter as a grad student. In the end, I was denied the job -- and based on the sequence of questions, I concluded that it was because of my low grad GPA. In other words, I was denied the job because I hadn't earned enough A's in my classes.

Eventually, I gave up on a STEM career and sought out a teaching credential instead. I worked hard to earn my credential -- with A's or B's in all my classes, of course -- and I'll realize the culmination of that work this fall when I begin my first teaching assignment. In other words, I was offered the job because I'd earned enough A's in my classes.

Students who earn A's in their classes are setting themselves up for a bright future -- students who don't earn A's in their classes are setting themselves up for a dismal future. The only ages that truly matter in a person's life are your 20's, 30's, 40's, 50's, and maybe 60's -- the years when you earn money -- and the only people who truly matter in a person's life are your employers -- the ones who give you money. (Well, of course your family matters -- but you can't start a family unless you earn enough money.) The only people who truly matter want to see as many A's as possible, and their opinion trumps anyone who says that A students are "nerds" or worse.

In particular, a student's peers don't matter, since they aren't employers. Not even I, their teacher, matter, since I'm not an employer. The traditionalist Bill isn't an employer either -- but he often writes about what employers are looking for. This is why I plan on reading some of Bill's posts in class.

I don't really want to tell my students my NSA story -- as interesting as it sounds, I don't want to give the impression that teaching them is only my "second choice." Instead, I want to tell the students about the grades I earned when I was their age -- in Grades 6-8, with emphasis on the C's. In a way, those C's, despite being middle school grades, almost cost me admission to UCLA. This is because ordinarily those C's would have cost me admission to the magnet program -- and many Honors and AP courses, the ones that look attractive to UCLA, were offered only to magnet students. It was only because of a loophole (that is, changing districts during freshman year) that I was allowed to enter the magnet program. I'll tell my students that they shouldn't count on loopholes like changing districts (or "credit recovery" classes) and that it's much better just to learn the material, so that they'll actually know the things that employers want them to know.

My Departure From the Traditionalists: Real World Math

Here is the third post from the Jacobs site that I want to mention:

http://www.joannejacobs.com/2016/06/pure-math-real-world-math/

This post is especially relevant to my upcoming class. It cites a study which purportedly shows that it's better to teach pure (i.e., traditionalist) math than applied math. Recall that my classes will be using the Illinois State text, which is heavy on STEM and applications. This is so important that I will link to the original article as well:

http://washingtonmonthly.com/2016/06/27/is-it-better-to-teach-pure-math-instead-of-applied-math/

Strangely enough, Bill hasn't posted in this thread yet (as of the time I posted this current entry). But Wurman does have something to say here (after the thread went on a tangent with phonics).

Ze'ev Wurman writes:
All the discussion about phonics aside, the report is about math. And there is little argument that Common Core overall is heavily “problem solving” oriented, problem-solving being a stand in for “real-life problems.”

Another poster, Ray (who isn't a traditionalist), responds to Wurman:

Ray writes:
There is something about education issues that can get people so upset that they can start to think that facts don’t matter. Sometimes people get so upset that they start to make things up. Mr. Wurman, you lost a lot of credibility when you wrote that the Common Core math standards defer fluency in division until sixth grade. In fact, Common Core standards require students to “Fluently multiply and divide within 100,” in third grade.

In Wurman's defense, I assume in sixth grade he's referring to long (multi-digit) division, for which the standard algorithm doesn't appear until Common Core Math 6 (but of course nonstandard algorithms appear in Grade 5). On the other hand, Ray cites the third grade standard which is mostly about single-digit division (or two-digits, as in 81 divided by 9 is 9).

But in the end, I can't agree with Wurman or the other traditionalists here. As usual, traditionalists forget that many students sitting in math classes often ask questions like "Why do we have to learn this?" or "When will we ever use this in real life?" The applications are provided in order to provide answers to those common questions -- for without such answers, the students will refuse to work hard enough to learn the material.

From the same thread, here is another poster named Michael Hiteshew:

Michael D. Hiteshew writes:
I think the reason that people who are taught pure math do better all around is that abstract math teaches you (forces you) to think logically and to reason from a known set of information to a solution. It also teaches the use of tools (techniques) that you keep in mental toolbox, and teaches you to ask yourself “What tools do I need to solve this problem?”

So after solving equations, they ask questions like, "What tools do I need to solve this problem?" But before solving equations, they ask questions like, "Why do I need to solve equations?" and "When will I need to solve equations in real life?"

Michael D. Hiteshew writes:
In addition, anyone who was taught Euclidean geometry by proofs forever after asks themselves ‘What do I actually know to be true?’ and keep that separate from what you surmise may be true.

So after doing proofs they ask questions like, "What do I actually know to be true?" But before doing proofs, they ask questions like, "Why do I need to do proofs?" and "When will I need to do proofs in real life?"

As usual, traditionalists like Hiteshew just assume that the students will work on the abstract pure math simply because they are told to work on them, even though they won't see the benefits of doing so until afterward. But the students will never reach a point where they're asking themselves those great questions if they're throwing the worksheets full of equations or proofs in the trash -- because they see no relevance of those equations or proofs to real life.

The goal of the Illinois State text is to show students how math is relevant by way of various math and science projects. Even if students working in the Illinois State text don't learn as much math as students working on worksheets full of equations, they definitely learn more math than students throwing away worksheets full of equations. My hope is that I can open a unit with a project, the students will see how math will help them with their project, and then they'll be motivated to learn the math and start earning those A's.

Of course, suppose the students do finally see the importance of earning A's in math class. But some students will feel frustrated as they feel that they were never good in math. Even in kindergarten, they were struggling to learn the concepts, and not even in kindergarten did they get most of the questions right on a worksheet or hear their teacher say "Great job!" after a math worksheet is completed. So by the time they get to my class, they'd been failing math for six, seven, or eight years, and so they certainly won't start to try hard in math now.

But to me, that's a lousy excuse not to strive for an A in my class. Recall that the Cleveland Cavaliers had never won the championship before this year, and teams from the Forest City hadn't won rings in any sport in 52 years. So Clevelanders had been failing in sports seven times as long as my students have been failing in math. Yet LeBron James and the rest of the Cavs squad didn't use that as an excuse not to strive for a championship.

And of course, the ultimate example of failure in sports is in baseball -- the Chicago Cubs haven't won a championship in 108 years. So the Northsiders have been failing thirteen to eighteen times as long as my 6th-8th graders have been failing in math. Yet Jake Arrieta and the rest of the Cubs squad aren't using that as an excuse not to strive for a championship -- and indeed, right now they have the best record in all of baseball.

Let's Break Down My Grades

This post is all about grades. One consideration I must make is the grading breakdown, including what percentage of the grade is devoted to tests, classwork, and so on. I think back to my student teaching days, and that district had the following policy:

40% -- Tests
30% -- Quizzes
20% -- Classwork
10% -- Homework

These percentages are approximate -- for example, I think there was a separate components of the grade for teacher tests and district tests, as well as the final. Of course, finals won't be a consideration this year as I'm teaching at a middle school, not a high school.

All grades were entered into a computer, and then the grades were weighted so that the percentages for each component (tests, quizzes, etc.) are correct. For example, let's say there are ten 5-point homework assignments followed by a 100-point test. So there are 100 test points and 50 homework points, but tests are supposed to be worth four times as much as the homework. Therefore a single test point is worth twice as much as a single homework point. I disagree with this grading method, as it's deceptive -- a 5-point question on a test is worth more than turning in a 5-point HW paper. I know why the computer is programmed this way -- it helps non-math teachers get the correct percentages without having to perform calculations.

But as a math teacher, I hold myself to a higher standard. I'd much rather do the calculations myself so that a point is a point no matter what. To make the calculations easier, I can choose a number such as 1000 points for the whole trimester. Then there will be 400 points for the tests, 300 points for the quizzes, 200 points for classwork, and 100 points for homework.

For the tests, I can hold four tests each trimester, each worth 100 points. The tests can be staggered so that I'm not testing all my students at the same time. The first test can be given the second week of school to my 8th graders, then to my 7th graders the third week, then to my 6th graders the fourth week, and back to the 8th graders again the fifth week.

On these tests, a grade of A isn't the only acceptable grade -- it's OK to get a grade just below A provided that every effort is made to strive for the A. That is, A is the only acceptable goal on most tests and quizzes.  But there will be one instance in my class where A really is the only acceptable grade -- the Dren Quizzes.

Don't Be a Dren!

Back in the district where I student taught, there would often be basic skills tests given. Often these would be on integer operations. Students would be given 100 questions, and they had to get 90 of them right -- an A -- in order to pass. In fact, if the student receives any grade other than A, the test doesn't count and students receive a score of 0 (or possibly 1%, so that parents don't wonder why there is a 0% test on their student's grade report). The test must then be repeated until the student gets at least 90 correct.

In my class, I will strongly state that a "dren" is a reverse-nerd -- someone who isn't proficient at basic math (third grade and below). And so I will give out Dren Quizzes, where students are asked to solve 50 basic multiplication problems. Just as in my former district, there are only two possible grades -- A and Dren. Any grade other than an A is a Dren grade, and students get only 1/50 (or 2%) until the quiz is retaken and 90% (or 45/50) is earned.

I'd like it so that the second Dren Quiz is on the 2's times tables, the third Dren Quiz is on the 3's times tables, and so on. The first Dren Quiz, meanwhile, won't be on 1's but on the 10's instead. In theory, every single student should get 90% -- forget that, every student ought to get 100% on a quiz on the 10's times tables. Yet the traditionalist Bill has often lamented that there really are some students who would fail a 10's times tables quiz -- for example, here's a Bill comment from about five years ago:

http://www.joannejacobs.com/2011/07/chalkboards-pencils-e-readers/

Bill says:

Overuse of technology has left most young persons in society unable to handle many tasks considered common knowledge 25-30 years ago. All a calculator, computer, or a e-reader are tools, but if persons do not understand the basics of reading, writing, and math, all the technology in the classroom will do NOTHING to help them later on.

In another topic, a teacher in chemistry class cringes when his students have to rely on a calculator to handle basic basic math or to multiply a number by 10 (add 1 zero to the right hand side of a whole number to multiply by 10), these are facts that students should have MASTERED in elementary school (grades 1 through 5).

The fact that people are experts in email or text messaging isn’t going to help them when they cannot write a business letter, or a resume, or fill out a job application properly. Ever watch a person who is engrossed in using their phone in public, they almost block out the entire world around them (which in some cases can have deadly consequences, or as I call it, the darwin effect).

Lets get back to using what worked more than a quarter century ago, and quit buying into the latest fad craze, generally foisted upon us by individuals who have spent their entire careers in academia, with no knowledge of the real world, per se.

We may ask, why is a question like "What is 2 times 10?" so difficult for students? Here is what too many students are thinking:

-- 2 times 10 is math, and math is hard, therefore 2 times 10 is hard.

And so when they are asked to find 2 times 10, they don't even think about whether this is an easier or a harder question (like 7 times 8) -- they just react to any math problem with "This is hard!" and either ask for a calculator or answer "I don't know!" And of course, it goes without saying that students will not be allowed calculators on Dren Quizzes.

So just before giving my students the Dren Quiz, I'll read this last five-year-old comment from Bill, so that the students understand why I'm giving them a Dren Quiz. The idea is that students will see that a "dren" is something that they don't want to be, so that they'll be motivated to do well on the Dren Quizzes.

My plan is to rotate so that students take a Dren Quiz the first week after a test (when the students haven't learned much new material yet), then an ordinary quiz the second week (which is about halfway through the new material), and finally the test the third week. All quizzes, ordinary and Dren, can be given on Wednesdays when the classes are shorter (as 50 minutes should be more than enough time for these quizzes).

So the plan for the first few weeks of eighth grade will look like this:

First Test: Friday, August 26th
First Dren Quiz (10's): Wednesday, August 31st
First Ordinary Quiz: Wednesday, September 7th
Second Test: Friday, September 16th

Then on Wednesday, September 21st, the eighth graders take their second Dren Quiz. It will be on the 2's times tables except for those who didn't pass their 10's (which hopefully will be no one except for students absent on the 31st). On the 2's Dren Quiz, I'll sneak in a few 10's. After all, the idea isn't just to learn the 10's for the first Dren Quiz and forget them, but to remember them forever. That way, when the science teacher (as in the Bill quote above) asks for 2 times 10, the reaction isn't:

-- 2 times 10 is math, and math is hard, therefore 2 times 10 is hard.

but for "20" to pop into their head even before they get to "2 times 10 is math."

I'll repeat the idea that students are to avoid being drens over and over again. For example, I might tell the students "dren jokes." Dren jokes are basically blonde jokes, except that I change the word "blonde" to "dren" (and make them gender-neutral, of course). The original version of the following blonde joke is inappropriate for the classroom, but as a dren joke it is very appropriate:

Q: How is a dren like a solar-powered calculator?
A: Neither works in the dark.

I must be careful, though. When telling dren jokes, I will make sure that I'm not directly calling any student in the class a "dren." The only time I'll ever call a student a "dren" directly is in a situation like the following:

Me: What is 2 times 10?
(Student reaches for a calculator.)
Me: Don't be such a dren! Only drens need a calculator to multiply 2 times 10.

Will all of my Dren Quizzes be on multiplication? Perhaps if after the 10's test students make it all the way from 2's to 9's, the next Dren Quiz will be subtraction of decimals, as another traditionalist complaint about youngsters is that they can't make change correctly.

I want my students to view being a dren as a mark of shame, not a mark of pride. If a student is a dren, the first thing is for the student not to admit it. If a student is a dren, the best thing is to hide this fact as much as possible, not brag about it on her Twitter page or laugh at how she was able to make it to middle school without knowing math (as one girl did during my last week of subbing). In other words, fake it until you make it.

So far, so much of this post is about shaming the drens. But what should I do with the students on the other side -- the ones who are actually learning math and doing it well?

Heroes and Fraction Fever

As I wrote earlier, students who do great in math aren't nerds -- they're heroes, and I want them to know that they are heroes. The first step is for me to admit that they are heroes, and treat those students as heroes. It's far too easy for me to focus only on the drens -- and this is terrible.

Let's think back to Akeelah and the Bee. We know that the other students make fun of the title character because she is so smart. Now suppose that I have a mathematical Akeelah in my class -- a student who is doing great. I can't really stop the other students from making fun of her. But it's one thing to be ridiculed by the students outside of class -- it's another to be ridiculed by the students outside of class and ignored by the teacher inside the classroom. I admit that when I was a student teacher, I sometimes ignored the good students -- and this I must change.

In fact, I want to go out of my way to make the top students feel like heroes even if it means embarrassing myself, for the priority is to make the top students feel good, not myself. For example, if I make a silly error and a student corrects me, I want to make a show of it. I plan on pounding the table in "anger" and yelling at myself for making the mistake. This will cause the other students to laugh at me -- which is exactly what I want. I want the students to congratulate their classmate who "annoyed" the teacher so much -- because that's the student who really understands the math enough to spot my error. Finally, I may even award a bonus point to the student who catches the error. If this is successful, I might even make some errors intentionally in hopes that the students will catch it!

It would be one thing if the other students feel that they simply don't want to be friends with the smart kids and just ignore them. But it's another for them to physically beat up the smart kids, just as the other girls did to Akeelah. And that's a huge problem -- it's hard to get even the smart kids to know that they're preparing for the only years that matter in their lives (their twenties and beyond) if they can't survive their preteen and teenage years without being beat up.

The best I can do is offer up my classroom as a "safe space" on certain days at lunch, so that students who fear being attacked due to their success in math have a place to stay. Perhaps on an especially rainy day (when students would want to eat in the classroom anyway), I can invite the students with the highest grades into my classroom and inform them that they can eat there on certain dry days as well if they are having problems with the other students.

Again, the idea is to reward students who earn A's as much as possible. I may have two types of rewards -- an individual award to each student who earns an A on a test, and a class reward if sufficiently many students in the class earn A's. Perhaps the existence of a class reward will lead to the kids viewing the top students as the heroes they are.

And the idea that I should celebrate bright students extends beyond the classroom. I don't follow anyone like that British dren on Twitter, but I do have online contact with several of my former classmates, who often write about their children's academic accomplishments. Sometimes I congratulate the children for doing well in school, but I'm not consistent at it. I want to put my mouth where my money is, and make sure that I tell my friend's children that they are heroes the next time that their parents brag about them online.

It's often said that you never forget how to ride a bike. Unfortunately, learning math (or science) is very unlike riding a bike -- people forget it all the time (otherwise, I wouldn't have ended up with C's in my college science classes). Let's get back to our first traditionalist link, Math Curmudgeon, who comments on how often certain topics in math are forgotten by the time students take the SAT (the American SAT, not the British SATs):

http://mathcurmudgeon.blogspot.com/2016/06/sat-prep-course-set-up.html

Even the best ones have forgotten the most basic ideas. √300 = 10√3 - what magic is this? Proportions and fractions - who knew?

Now sqrt(300) = 10sqrt(3) isn't relevant to the classes that I'll be teaching -- but fractions definitely are (after all, middle school math is A Story of Ratios). For some reason, adding fractions is nothing like riding a bike. According to Curmudgeon, even students who get A's in all their math classes can't add 1/2 + 1/3 = 5/6 instantly. I myself, of course, could have taken a fractions test the very first day of school as a junior year and get a perfect score without studying at all, but this is rare among most students -- even most A students. People ought to remember how to add fractions forever, but they simply don't.

When I was a young child, I remember one of my first computer games. (This was one of those old computers from the 1980's.) One of my favorite games was Fraction Fever. Even though this is an old game, I was able to find a link describing the game:

http://www.giantbomb.com/fraction-fever/3030-22016/

An educational game dealing with fractions. The player must navigate a bridge on a pogo stick with the objective of landing on the space that answers the fraction shown on screen. Wrong answers remove a space. If too many spaces are removed the player falls to a lower level.

Now it might be possible for me to implement a version of this game in the classroom. No, the students wouldn't jump around on pogo sticks, but I would post answers to fractions near the ceiling so that students would still have to jump (in the spirit of the game) to reach the right answer.

Such a game could fit at the beginning of the school year. The Illinois State texts for all three grades begin with "Tools for Learning," which is essentially a Unit 0:

Tools for Learning:
1. The Need for Speed
2. Show Me the Numbers
3. What's the Best Advantage?
4. Learning to Communicate

I'm not sure how long I will spend in this Unit 0, since it all depends on how long it takes to complete all the projects (which begin with constructing a model car and measuring its speed). But in between these projects, the students will need fraction practice, so I can whip out my Fraction Fever game, which will have higher levels ("floors") where students must add, subtract, multiply, and divide fractions -- not just identify them as in the original computer game. The hope is that students will be prepared for the more difficult projects which require the students to know fractions.

But I suspect that even playing a game like Fraction Fever won't be enough. Curmudgeon's complaint was that students don't remember fractions from year to year, so that by the time they take the SAT they will have forgotten everything. Since it appears that I will be teaching the same students for three years, this puts me in a unique position to encourage the students to remember fractions as each of those three years passes.

Here's my plan: on the first day of school of my second year (that is, in 2017), I will give the seventh and eighth graders a fractions test. Students won't be penalized for failing it, but I want to reward them handsomely for passing the test, with an even better reward for those who get A's on it. I will inform the students at the end of the upcoming school year of this Fractions Challenge.

I know -- I'm supposed to be preparing for my first year of teaching, and here I am already writing about my second year! Still, this gives me an entire year to think up a suitable reward for those students -- those heroes -- who can remember fractions.

Now Bill and the other traditionalists once again mention race/ethnicity in their comments. This is because the idea of placing San Francisco eighth graders in Algebra I is sort of like tracking, and any discussion about tracking ultimately leads to race. But here I wish to discuss the other major demographic -- gender. This is especially important at a school that emphasizes STEM, since females tend to be underrepresented in the STEM fields.

On Gender

I remember back when I was student teaching, and I noticed that many of the girls were struggling in my Algebra I class. I fear that the reason for this is that I, as a male math teacher, was displaying an unconscious bias towards my male students when teaching. Here is a link from last September to an article that discusses the problems that male STEM teachers have with female students:

http://www.npr.org/2015/09/01/436525758/how-teachers-unconscious-bias-play-into-the-hands-ofgender-disparity

Here on the blog I've devoted posts to several prominent female mathematicians, including:

-- Hertha Ayrton
-- Eugenia Cheng
-- Dido
-- Vi Hart
-- Danica McKellar
-- Emmy Noether
-- Theoni Pappas

And speaking of Theoni Pappas, on her Mathematical Calendar 2016 she featured another female mathematician in June. Iranian-American mathematician Maryam Mirzakhani is the first female Fields Medalist (which is very prestigious considering that there is no Nobel Prize in math). She is currently a professor at Stanford, and her work is on topology -- remember how a doughnut is topologically equivalent to a coffee cup?

But again, I must make sure that I treat the girls in my classes who are doing well in math as the heroines they are. And I want to make sure that I encourage them to work hard on the STEM projects so that they'll be successful in my classes.

One of the women mathematicians I listed above is Danica McKellar. Last year, I wrote how I had bought two of the books in her Girls Get Curves series. I, a male, am not in the target demographic of her books -- but then again, I already know math. The girls in my classes are the ones in the target demo of these books. In fact, yesterday I finally purchased the other two books in that series (Kiss My Math and Hot x: Algebra Exposed! which are geared towards Pre-Algebra and Algebra I students, respectively), now that I'm teaching at a middle school. If at any point it appears that my girls are disengaged, I will bring out the McKellar books and read these books to them in hopes that the girls will get back on track.

Indeed -- and this goes for both males and females -- yet another trick to get the top students seen as heroes is for me to start talking about famous mathematicians and scientists as superheroes. I recently read a commenter (not a traditionalist -- indeed this wasn't even at an education website) who was upset that movies promote the idea of superheroes as people with special powers (not just Batman or Superman, but also wizards like Harry Potter) when the real heroes are scientists. Of course, I wrote last year about two scientist movies (featuring Stephen Hawking and Alan Turing), but those movies were not blockbusters (although Eddie Redmayne won an Oscar as Hawking).

And so whenever I have the chance, I can start talking about McKellar, Hawking, Turing, Euclid, and Dido as if they were celebrities. Vi Hart (another female mathematician from the list!) created a video about the life of Pythagoras, and his strange aversion to beans. I admit that I've never thought of scientists or mathematicians as heroes --one day, one of my elementary school teachers asked me to name my hero, expecting me to name someone like Einstein. I don't remember my response, but I was told later on that my reply was a certain game show host. (Perhaps I'd said Bob Barker -- but then again, The Price Is Right is, in fact, a very mathematical game show!)

My first rule will be:

Rule #1: The Teacher Respects You

I can't help but think back to my student teaching. In my Algebra I class, there was a girl who was a sophomore -- so she must have taken the class as a freshman and failed it. Still, I could tell that she was motivated to learn -- probably because she knew she didn't want to fail the class again, and moreover, she still remembered a little bit of Algebra I from the previous year.

Early in the year, this girl often volunteered to answer my questions -- and I often rewarded her with participation points when she answered correctly. She began the year with two participation points and soon ended up with ten a few days before the test. At that time, I told her that she had earned the maximum number of points I'd allow. Whenever she raised her hand, I would ignore her and use my random name generator (explained in a previous post) to choose a student instead.

The girl passed her test with a C, but I could tell she was upset that I wasn't calling on her. Even though the participation points started over again at the next unit, she no longer volunteered to answer any more questions. From that point on, she was frequently absent from class, and she ended up earning a D in class for the semester.

Two years later -- long after my student teaching was completed -- I happened upon the graduation program for the school. I still remember the names of many of the seniors graduating that day but -- and by now you've figured out where this is going -- the girl's name was not listed. Of course, it's possible that she switched schools (just as I once did my freshman year). But my fear is that girl joined the anti-school subculture -- and it's all because she didn't like how I ran her math class. My fear is that the student failed to graduate all because of me.

So what will I do in my new class to make sure that nothing like this ever happens again? My idea is that early in the year, there should be no limit on the number of participation points a student can earn. Recall that in my most successful student teaching class (sixth period Algebra II), the students were highly motivated to volunteer and learn. Perhaps if the other students saw this one girl always volunteering and receiving praise, they'd want to volunteer as well -- and then in subsequent units, there'd be no need to limit the number of points because there would always be several students volunteering on every question! Finally, I should mainly use the random name generator when there is no one volunteering on a question.

In short, I will respect the students in the way that I failed to respect this girl.