Sunday, June 18, 2017

Still Not My Father's Math Class

This is what Theoni Pappas writes on page 169 of her Magic of Mathematics -- a short timeline:

1637 -- Analytic Geometry
1639 -- Projective Geometry
1736 -- Topology
1829 -- Hyperbolic Geometry
1854 -- Elliptic Geometry
1860's -- Fractal Geometry
1951 -- My Father's Birth -- dw

This post fulfills my monthly posting requirement for Tina Cardone's "Day in the Life" project. My monthly posting date is the 18th. Today is also Father's Day, and so much of this is a repost from what I wrote last year on the holiday.

I rarely mention my family here on the blog. Except for right after my grandmother's passing, I chose not to write about family here on the blog. But today on Father's Day, I must mention my father, because he is a retired teacher. A list of influences on me as I embark on full-time first teaching job would be incomplete if I left my father off.

He was a fifth grade teacher for a few years (including the year that I was a fifth grader), until he switched to a sixth grade classroom in LAUSD. The district often had multiple-subject teachers, including my father, teach sixth grade math and science.

The summer after I graduated from high school, I was a volunteer in my father's classroom. At the time, LAUSD had year-round schools. Students were divided into three tracks, A, B, and C, and the students and the teachers attended school during different times of the year. My father was a C track teacher, meaning that he taught from July to October, and then again from January to April. This meant that I could volunteer in his classroom the entire summer, from July all the way up to my first day as a UCLA student in late September, as UCLA has a quarter college calendar.

I remember my first day in my father's classroom. I was so surprised to see how tiny the sixth graders were -- amazing considering that 13 years earlier, I'd thought that sixth graders were grownups!

Most of my volunteer work in that class was limited to helping my father grade papers. Still, I recall a few things about the way he taught his class:

-- Every third test, he would drop the lowest test grade. Therefore, he tried to give six tests per semester each in math and science, so that he could drop the two lowest test grades.
-- All tests were open-book. Despite these concessions, many students were earning D's and F's, especially in the math class.
-- The four remaining tests were the entirety of the students' grades. In particular, any homework assignments were extra credit.
-- I remember once when a student forgot to bring his pencil to class. My father told the student that he was just like a baseball player who went up to home plate without a baseball bat! Just as it would look silly for a player making millions of dollars to forget to bring a bat to the plate, that's how silly it is for a student to forget his or her pencil.
-- My father regularly gave out candy as a reward/incentive.
-- My father often said that the motto of the class was, "If you don't know the answer, at least know where to find it." In many ways this statement applies more to science than to math -- in math one must calculate the answer, but in science all the answers should be right there in the text.

This, in fact, reminds me of a day a few years back, when I was subbing. It was a seventh grade lesson on the human reproductive system. Due to the explicit nature of the material, students required parental permission in order to attend the lessons. Most likely, the teacher would send students without a permission slip out of the classroom, but she obviously didn't want a sub to deal with all of that. So instead, she provided an alternate packet for the students who didn't have parental permission to study the main lesson. Her expectations were that all students would work independently on their respective packets.

Now in two of the classes, the teacher's lesson plan worked. Again, these classes weren't labeled as "honors" classes, but it was obvious that the teacher expected more from those students -- their packets included an extra vocabulary page.

But in the non-honors classes, the students refused to work on their assignments. Their excuse was, "Normally, the teacher explains it to us." Thus many of the students left more than half of their worksheets blank -- and those who had been absent at least once that week left even more blanks.

In those classes I had more pressing issues to deal with -- including the fact that an explosive device was found in the classroom! But if I could have focused only on the academics, I would have told the students my father's motto: "If you don't know the answer, at least know where to find it."

Notice it's likely that what the students said was true -- usually they wait for the teacher to explain things on their worksheets, so when there is no one to explain anything, they do no work. And after an absence, the students wait until the teacher has a spare moment to tell them what they missed -- until then, they do no work. The teacher probably would have told me to help the students out with the worksheet -- except that she couldn't because of the parental permission problem. She didn't want me to deal with students would had to leave the room and she didn't want me to start discussing the material out loud in front of the students who didn't have permission. I might even have played the "Who Am I?" game to motivate the students to work, except that I couldn't speak out loud.

There were two issues in that class -- the sub issue and the parental permission issue. Had there been only one issue or the other, there'd have been no problem. It was the combination that doomed the class -- the teacher just hoped that the students would work independently, but they didn't. That's why I wished I could have given them my father's advice -- "If you don't know the answer, at least know where to find it." Maybe then, the students would have realized that they were to search for the answers in their science text.

Now that I have discussed my father's class, I ask myself, can I apply any of my father's teaching methods to my own classroom? Well, my new classroom is making some changes that will make my class very different from the one he taught.

First of all, the school has changed its bell schedule, to a block schedule with longer 80-minute blocks. Now here's the biggest change -- the math curriculum. Previously, our school used the McDougal Littell text. But a few schools in the LAUSD were selected to use a new STEM-based curriculum -- and my charter school chose to go along with it. This curriculum is called IMaST, and it was developed at Illinois State University.

Since the focus of this blog is eighth grade, let's look at the Illinois State text for eighth grade in much more detail:

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

Unit 1: Mathematics in Settlements
5. H20 + ? Measuring Using Parts Per Million (ppm)
6. The Capacity of Water-Carrying Structures
7. Shapes, Angles and Structures: How Strong Is It?
8. Tessellate a Structural Design: A Study of Polygons
9. Similarity

Unit 2: Mathematics in Systems
10. Input, Process, Output...Does It Work?
11. Bouncing Balls
12. Say it with Words, Pictures, Tables, and Symbols
13. Looking at Relationships
14. Where shall we Meet?
15. That Model Looks Good!
16. Atmospheric Layers

Unit 3: Mathematics in Animal Habitats
17. Give Me Space
18. Patterns in Data
19. Chances Are

Unit 4: Mathematics in Communication
20. Sound Waves
21. Codes I
22. Codes II: Tracking Data
23. Codes III
24. Matrices I
25. Matrices II

This text is full of projects, right from the very beginning. In fact, the first project, "Off to the Races: Design and Make a Car" (made out of cardboard) starts on page 3.

Oh, and it's finally time for me to address the elephant in the room -- homework. In fact, let's see what the Illinois State text itself says about homework, in its preface to teachers:

"Generally, the Creative Core Curriculum...does not rely on homework to reinforce what is done in the classroom. There are, however, instances where work must be completed at home. For example, in 'The Right Kind of Fuel' learning cycle, students must determine the nutritional content of various foods. The home refrigerator may be an excellent source of data. The data is used the following day in a class activity.

Let's compare this to what MTBoS blogger Elissa Miller writes about homework:

I have major issues with homework:
  • you don't know who actually did it
  • you don't know whether someone copied
  • you don't know whether you should 'check' or 'grade' it
  • you don't know if students understand enough to do the work alone
  • you don't know the best way to go over it without wasting class time
  • you don't know if it is effective
  • you don't know what students have to do/deal with at home
  • you don't know what other commitments/priorities students have
My normal teaching method starts with notes as a class in the INB for day one, then day two is some kind of practice activity/game, and the start of day 3 is a short quiz, then the notes for the next skill.

(And recall that the other MTBoS blogger, Fawn Nguyen, resolves to give less homework or none at all as #46 of her 51-item list.) Of course, my class will focus even more on activities and less on taking notes than Miller's and Nguyen's classes. In any case, there probably won't be much homework in my class (except for the aforementioned extra credit).

By the way, I've said that I would have to meet with the previous math teacher in order to maintain continuity for the students. Let me correct that -- I'll have to meet with the science teacher to maintain continuity, as the projects in math and science are supposed to link up. Therefore I can't really set up a pacing plan right now, as everything depends on the projects in both classes.

So I believe that I'll have much to contribute to the MTBoS, as I am perhaps the only member of the MTBoS who is teaching Illinois State math.

There's one thing I know as I prepare for the school year -- this is certainly not my father's math class.

OK, let's return to 2017. Last year, I'd written about what I was expecting my class to be like -- and some of my expectations turned out to be wrong.

First of all, it turned out that there really was homework to assign my class. But the homework from Illinois State was to be assigned online -- so it's even less like my father's class, since the technology for online HW didn't exist in his day. And I couldn't just make the homework extra credit, since PowerSchool required us to count homework as 15% of the grade.

More importantly, last year I was looking forward to meeting the science teacher to discuss some of the projects. As it turned out, I was the science teacher. My father had taught both math and science to sixth graders, but he had a multiple subject credential. I'd been expecting to teach only math.

Last year I wrote about my father's class motto, "If you don't know the answer, at least know where to find it." I remember telling my class this motto at the start of the year (see my First Day of School "Day in the Life" post). But then I don't recall saying it much ever again. I never really did embrace my father's motto the way I'd planned.

So in many ways I feel I've let my father down. When I told him that I'd been replaced at my old school, of course he was disappointed. But he saw a silver lining, in that I might do better teaching high school students at my new school than middle school students -- the toughest grade span.

In this post, I quoted the MTBoS blogger Elissa Miller. (Actually, Miller is also a fellow participant in Cardone's "Day of the Life" challenge -- her monthly posting date is the 10th.) I was quoting Miller's homework philosophy, but now I notice something else that I unwittingly quoted:

My normal teaching method starts with notes as a class in the INB for day one, then day two is some kind of practice activity/game, and the start of day 3 is a short quiz, then the notes for the next skill.

As it turns out, this was the best method for teaching from the Illinois State text as well. I could have expanded Miller's three-day plan to a full five-day weekly plan. One extra day was for coding Mondays (since the coding teacher came in every week), and the other was for Learning Centers. Of course, I had to squeeze in science somewhere -- either the activity could be a science lab, or else science could be included as one of the stations for the Learning Centers.

But unfortunately, I never truly embraced this weekly plan. If only I had taken Miller's advice to heart, I might have had a more successful year -- and perhaps never had to leave my old school. So instead, I used a ten-day cycle where the project and traditional lessons are extended. This fit well with our requirement from the administrators to submit some project photos to Illinois State every two weeks.

So now you might ask, why didn't I embrace Miller's plan? Well, I must admit that some of those projects intimidated me. Many of them required materials that I wasn't sure I had -- there were so many boxes to search each time I needed to start a project. And the fact that I was teaching three grades meant that I'd have to set up three different projects most of the time. So in the end, I wanted to slow down the pace to give me more time to prepare for the projects -- and this meant that I wouldn't be able to finish all the projects by the end of the year.

In hindsight, I realize that I shouldn't had let the projects dictate how I ran the class. Instead, I should have followed Miller's plan. If I reached a project that I didn't know how to implement, I could have set up an alternate activity, or even a science project if that was easier. I could have even given an extra day to the traditional lesson and just skipped the project. As long as I had one project in one grade every two weeks, I could have met the photo requirement.

The first four projects of the year ("Tools for Learning" above) were the same for all grades. And that car project I mentioned was to be created using mousetraps (which were provided). I didn't confident in building the mousetrap car myself. So of course, I asked my father to build the mousetrap car the night before I set up the project for the classes.

Of course, it's too late for me to do anything about this past year. I can now only worry about the upcoming school year and how I will make it a successful one. I don't know anything about the curriculum, so I don't know whether it's project-based. If it is, then I know that I should follow some sort of daily cycle similar to Miller's method above. At least I don't have to teach science, which will make it easier for me.

No matter what I'll teach, classroom management is a must. I'll continue to reflect all summer on why my management style failed, and consider what I'll do to improve my management for this year. I'll make sure that I actually say my father's -- make that my -- class motto throughout the year.

There's one thing I know as I prepare for the school year -- this is still not my father's math class.

Saturday, June 17, 2017

Van Brummelen Chapter 2: Exploring the Sphere

Table of Contents:

1. Pappas Page of the Day
2. Van Brummelen Chapter 2: Exploring the Sphere
3. Exercise 2.7: What Color Is the Bear?
4. Exercise 2.12: Spherical Polygon-Sum Theorem
5. Exercise 2.14: Polar Triangle Inequality
6. A Possible High School Geometry Course
7. Hilbert's Axioms for Spherical Geometry?
8. Plans for Tomorrow's Post

Pappas Page of the Day

This is what Theoni Pappas writes on page 168 of her Magic of Mathematics:

600 BC -- Thales introduces deductive geometry. It was developed over the years by such mathematicians and philosophers as Pythagoras, the Pythagoreans, Plato, Aristotle.

On this page, Pappas is providing us with a timeline of Euclidean and non-Euclidean geometry. Let me post some of the highlights from this timeline:

300 BC -- Euclid compiles, organizes and systematizes geometric ideas, which had been discovered and proven, into thirteen books, called The Elements.

140 BC -- Posedonius restates Euclid's 5th postulate.

1637 -- Rene Descartes formulates analytic geometry.

1795 -- Gaspard Monge (1746-1818) describes structures by plane projections.

1854 -- G.F. Bernhard Riemann (1826-1866) presents elliptical geometry.

1888 -- Giuseppe Peano (1858-1932) creates Peano space-filling curve (fractal).

1904 -- Helge von Koch (1870-1924) creates Koch snowflake curve (fractal).

Notice that Pappas credits Riemann with presenting spherical (elliptic) geometry. Even though Legendre had written about spherical geometry earlier, it was Riemann who recognized spherical geometry as a separate, non-Euclidean geometry.

Van Brummelen Chapter 2: Exploring the Sphere

Chapter 2 of Glen Van Brummelen's Heavenly Mathematics is "Exploring the Sphere." In Chapter 1 he focuses on the "trig" part of spherical trig, and here he writes about the "spherical" part, before he puts it all together in the next chapter. He begins:

"At first glance there's not much to see on a sphere; every point on its surface looks the same as any other. But give it some physical meaning -- call it the celestial sphere, or the Earth's surface -- place some identifying marks on it, and set it in motion, and visualizing what's happening can become rather complicated."

Van Brummelen assumes that we're already familiar with earth's surface, and so he devotes part of this chapter to describing the celestial sphere. He writes:

"We've already seen the most obvious feature of the celestial sphere, namely its daily rotation around us. Given the sphere's unfathomably large size, rendering the Earth as an infinitesimal pin prick at its center, one can only imagine how quickly it is actually moving."

The author defines several key terms here:

-- The celestial equator rises from the east point of the horizon and sets in the west.
-- The ecliptic is the path the sun takes as it makes a complete circuit around the celestial sphere.
-- The obliquity of the ecliptic, symbolized by the Greek letter "epsilon," is the tilt between the celestial equator and the ecliptic. Its current value is 23.44 degrees.
-- The equinoxes are the two points where the celestial equator and the ecliptic intersect.
-- The summer solstice is the most northerly point on the ecliptic, halfway between the equinoxes.

Notice that the summer solstice is coming up next week. I could have waited until the actual summer solstice to make today's post, but I've already manipulatedthis week's posting schedule too much -- and besides, the solstice is just after 9PM Pacific Time on June 20th (and therefore after midnight Eastern Time on June 21st), and so I'd be forced to choose between the 20th and the 21st to post. So instead I'm posting three times this weekend, and then taking several days off from posting, including both the 20th and the 21st.

Afterwards, Van Brummelen defines three coordinate systems, similar to longitude and latitude for earth's surface. One is called right ascension and declination, symbolized (alpha, delta). The second is called ecliptic longitude and latitude, symbolized (lambda, beta). The third is called azimuth and altitude -- this determines where a star is at a given time at night. Zero longitude for both the equatorial and ecliptic coordinates is set to the spring equinox. The symbol for this equinox can't be rendered in ASCII, but it's actually the symbol for "Aries."

Now Van Brummelen writes some basic theorems in spherical geometry. We've already proved some of these theorems last year with Legendre, so today's listing of them can serve as a review.

Theorem: Every cross-section of the sphere by a plane is a circle.

Given: Figure 2.6. O is the center of the sphere.

Consider the cross-section in figure 2.6. Let D be any point on the cross-section, and let OC be the perpendicular line dropped from O onto the intersecting plane. Then Angle OCD is right, and the Pythagorean Theorem applies: OD^2 = OC^2 + CD^2. But OD is constant regardless of D's position on the cross-section, since it is the radius of the sphere; and clearly OC doesn't depend on where D is either. Therefore CD^2 cannot change as we move D around the cross-section, and so neither does CD. QED

Lemma (Triangle Inequality):
The third side of any spherical triangle cannot exceed the sum of the other two.

Note: Van Brummelen defines the length of a segment AB on the sphere to be the measure, in degrees, of the central Angle AOB. Last year, I wrote that if we were to use radians instead of degrees on the unit sphere, then the measure of the central angle really is the length of the segment.

Given: Figure 2.8. O is the center of the sphere, and ABC is a spherical triangle.

Examine the angles at O corresponding to the sides in figure 2.8. Imagine allowing segment OA to fall onto the plane OBC, leaving O in place but bringing A downward. Then two of the angles would fit perfectly within the third. But if we lift A back into its original position, the two angles at O that rise with it become larger. So their sum must be greater than BOC on the plane. QED

The sum of sides in a spherical triangle cannot exceed 360 degrees.

In figure 2.8, join A, B, and C with straight lines, forming a tetrahedron with O. The nine angles in the tetrahedron, excluding the angles in face ABC, must add up to 3 * 180 = 540 degrees, since they form three triangles. Now, the sum of the two of those nine angles that are located at A exceeds Angle A in the plane triangle ABC (by the same argument that led to the lemma a few moments ago), and likewise for the pairs of angles at B and C. Therefore,

Sum of sides = Angles at O
                      = 540 - (angles at A + angles at B + angles at C)
                      < 540 - (Angle A + Angle B + Angle C)
                      = 540 - 180 = 360 degrees


The polar triangle of a polar triangle is the original triangle.

Given: Figure 2.10. The triangle ABC is shown with its polar triangle A'B'C'.

Proof: In figure 2.10, extend the arcs of the original triangle to intersect the sides of the polar triangle. Since C' is a pole of AB and A' is a pole of BC, both C' and A' are 90 degrees removed from B. So B must be a pole of A'C'. Likewise for the other arcs. QED

Polar Duality Theorem:
The sides of a polar triangle are the supplements of the angles of the original triangle, and the angles of a polar triangle are the supplements of the sides of the original.

In figure 2.10 both D and E (extensions of the sides of the original triangle to the sides of the polar triangle) are 90 degrees removed from A; therefore Angle A = DE. Now since C' is a pole of ABD and B' is a pole of ACE, both C'D and B'E are 90 degrees. Therefore

B'C' = B'E + C'D - DE = 180 - DE = 180 - Angle A.

Similarly for the other sides of the polar triangle; we have now dispatched the first half of the theorem. The second half follows immediately from the duality relation; simply apply the result we have just established to the polar triangle and its polar (i.e., the original triangle), rather than the original and the polar triangle. QED

The angle sum of a triangle must exceed 180 degrees.

We know that the sum of the sides of a polar triangle must be < 360 degrees. Since the sides of the polar triangle are the supplements of the angles of the original,

(180 - A) + (180 - B) + (180 - C) < 360,

so A + B + C > 180 degrees. QED

Van Brummelen writes, "And now, finally, we have enough spherical geometry under our belts to tackle some spherical trigonometry."

Exercise 2.7: What Color Is the Bear?

7. (a) A bear hunter walks one km south, then one km east, then one km north, and ends up back where he started. What color is the bear?

This question is a classic! I wrote about it here on the blog in November 2015. Instead of searching for that old post, let me just post the link I gave that day:

(b) The puzzle in (a) has a particular location in mind, but there are actually many locations where this journey is possible. Identify the others and say what animal must replace the bear in the story.

Again, let me answer that question with a link:

Exercise 2.12: Spherical Polygon-Sum Theorem

12. Show that a spherical polygon with n sides (each less than 180 degrees) has a sum of interior angles greater than 180n - 360 degrees. [paraphrased from Cresswell 1816, p.54]

As it turns out, we prove this theorem exactly like the Euclidean Polygon-Sum Theorem from Lesson 5-7 of the U of Chicago text.

Choose a vertex on the polygon. Call it A. Draw the diagonals from A. For the n-gon, the diagonals form n - 2 triangles. Since the angle sum of each triangle is greater than 180 degrees, the total for the entire n-gon is more than (n - 2)180 = 180n - 360 degrees. QED

Exercise 2.14: Polar Triangle Inequality

14. Show that in any spherical triangle, the difference between any angle and the sum of the other two is less than 180 degrees. [paraphrased from Moritz 1913, p. 12]. (Hint: use the polar triangle.)

Well, let's follow Van Brummelen's hint and consider the polar triangle. Since the theorem asks us to prove something about the angles of a triangle, we look at the sides of the polar triangle. In particular, to find a relationship between one angle and the sum of the other two angles, we look in the polar triangle for a relationship between one side and the sum of the other two sides. The obvious relationship to consider is the Triangle Inequality.

Polar Triangle: A'B' < A'C' + B'C'
Original Triangle: 180 - C < (180 - A) + (180 - B)
                              180 - C < 360 - (A + B)
              (A + B) + 180 - C < 360
                        (A + B) - C < 180


A Possible High School Geometry Course

Van Brummelen suggests that his spherical trig book can be used as a course text. I wouldn't teach spherical trig to high school students, but I can see a situation where we might introduce spherical geometry as the last unit of a high school Geometry class. Yesterday, I mentioned that the Honors Geometry course at Barron Trump's new school does exactly that.

I like the idea of such a course. Here is a unit plan for my ideal Geometry course, which revolves around the Common Core transformations and ends up with spherical geometry:

1. Reflections
2. Rotations
3. Translations
4. Glide Reflections
(first semester finals)
5. Dilations
6. Two Dimensional Measurement
7. Three Dimensional Measurement
(state testing)
8. Spherical Geometry
(second semester finals)

This was actually what I had in mind during the first two years of this blog. The first unit, on Reflections, actually introduces the basics of Geometry (points, lines, angles, and so on). Congruence of triangles is lumped in with Rotations, parallel lines are part of Translations, and Glide Reflections includes other miscellaneous topics (most notably parallelograms). Then similarity is obviously included in the Dilations unit.

I actually like the idea of all eight units being named after a transformation. The 2D measurement unit can be named Transvections, since these preserve area, and a transvection maps a parallelogram to a rectangle of the same area (hence the parallelogram area formula). The 3D measurement unit can be named Screws, as these are 3D isometries that don't exist in 2D. (Transvections also exist in 3D and are closely related to Cavalieri's Theorem.) This doesn't imply that students actually focus on transvections or screws, but they are just made aware of their existence.

The final section on spherical geometry can't really be named for a transformation, though. Had this been hyperbolic geometry, a good name would be Horolations. But unlike hyperbolic geometry with its extra isometry, spherical geometry has one fewer isometry. This is because a translation is the composite of two reflections in parallel lines -- but in spherical geometry there are no parallel lines and thus no translations.

My idea is that parallel lines and their properties are introduced in the translations unit. Thus every theorem mentioned in the first two units is valid in both Euclidean and spherical geometry -- and this fact can be brought up in the final unit on spherical geometry. The eight units of the year readily correspond to the eight quavers of the year.

During the first two years of this blog, I wanted to organize my course with this in mind. But there were two problems. The first is that the U of Chicago text wasn't organized this way, so I had to jump around the text to fit this scheme. Eventually I decided that it's better to just to follow the text in order rather than jump around.

The other reason has to do with a difficulty in spherical geometry that doesn't exist in either Euclidean or hyperbolic geometry. To see what this is, we go back to Van Brummelen and look at his commentary following the proof of the Spherical Triangle Inequality:

"If you are not happy with the informality of this argument, I can bring none other than Euclid to my defense. The planar equivalent of this statement, that the third side of any plane triangle cannot exceed the sum of the other two sides, is Proposition 20 in the first book of the Elements. Curiously, Euclid's proof works for spherical triangles just as well...

"Euclid avoided [5th postulate] as long as he possibly could, until finally he was forced to use it in Proposition 29. Now, it turns out that spherical geometry is one of the non-Euclidean geometries that is consistent with Euclid's other axioms, but not with the parallel postulate. Since Proposition 20 comes before 29, Euclid's proof works on the sphere as well as it does on the plane."

The problem is that according to David Joyce, Euclid's Proposition 16 fails in spherical geometry, despite coming before 29:

This theorem is known as the Triangle Exterior Angle Inequality, or TEAI. Proposition 20 is proved using 19, which is proved used 18, which in turn is proved using 16 or TEAI. Therefore, the proof of Proposition 20 is not valid in spherical geometry (even though the theorem itself is)!

One misconception about spherical geometry is that it, like hyperbolic geometry, is one of the three geometries that satisfies Euclid's first four postulates. It is elegant to assume that all we have to do is consider the parallel postulate:

-- Through a point not on a line, there is exactly one line parallel to the given line.

and replace "exactly one line" with "more than one line" for hyperbolic geometry and "no line" for spherical geometry, and be done. But as it turns out, this works for hyperbolic geometry, but not for spherical geometry. This is because spherical geometry does not satisfy the first four postulates!

The first postulate of Euclid is:

-- Through any two points, there is exactly one line.

And this fails in spherical geometry, since through the poles there are infinitely many lines. In fact, spherical geometry arguably satisfies the fifth postulate, since the postulate and many equivalent statements are written in the form "if bla bla bla, then the lines intersect" (which is trivially true as all lines intersect) or "if two lines are parallel, then bla bla bla" (which is vacuously true). And Playfair's axiom, the simplest equivalent of Euclid's fifth postulate, is correctly written as:

-- Through a point not on a line, there is at most one line parallel to the given line.

"At most one line" clearly includes the possibility that there are zero parallel lines -- which is exactly the case for spherical geometry.

Some people (like David Joyce) point out that spherical geometry fails the first postulate. Almost no one points out that spherical geometry actually satisfies the fifth postulate. And so technically, if we want to see how spherical geometry is different from Euclidean geometry, we shouldn't be looking at the fifth postulate at all, but rather the first postulate.

Hilbert's Axioms for Spherical Geometry?

Another way to look at this is to consider not Euclid's axioms, but Hilbert's axioms. Recall that Hilbert's axioms are more rigorous than Euclid's. Just as with Euclid's axioms, only changing the parallel postulate makes Euclidean geometry into hyperbolic geometry, but we must do more to make spherical geometry. Here is a link to a webpage that considers which of Hilbert's axioms are valid in spherical geometry:

According to this link, only three of Hilbert's axioms fail in spherical geometry:

(I1) Any two distinct points lie on a unique line. (Euclid's 1st postulate)
(B2) For any two distinct points A and B, there exists a point C such that A * B * C (that is, B is between A and C).
(B3) Given three distinct points on a line, exactly one lies between the other two.

Notice that a parallel postulate isn't listed among these axioms.

Arguably, Euclid's second postulate may fail in spherical geometry. This postulate states that any line segment can be extended, and (B2) involves extending a segment AB to a new segment AC. For simplicity, I interpret Euclid's second postulate as stating that a segment can be extended so that it divides the plane (or sphere) in half. (Joyce alludes to this at the link above.) This is so that denying a single postulate of Euclid's, the first, can give us spherical geometry (just as denying a single postulate of Euclid's, the fifth, can give us hyperbolic geometry). But in truth, we know that we must modify three of Hilbert's postulates to produce spherical geometry.

I often wonder how to rewrite these three postulates for spherical geometry. We might rewrite (I1):

(I1-sphere?) Any two distinct points lie on at least one line.

that is, we assert existence but not uniqueness of a line. Notice that we can even include uniqueness by adding a third point:

(I1-sphere?) Any two distinct points line on at least one line, and any three distinct points lie on at most one line.

This version of (I1-sphere?) is just a weaker version of (I1), and as such it still holds in Euclidean as well as spherical geometry.

Of course, the exceptions to both (I1) and (B2) for spherical geometry are antipodal points. Through antipodal points there are infinitely many lines, and indeed every other point lies between a pair of antipodal points. Somehow, a spherical version of either (I1) or (B2) should mention antipodal points.

As for (B3), this may be the hardest to rewrite spherically. Not all counterexamples to (B3) are antipodal -- for example, consider three points spaced 120 degrees apart on the Equator.

None of this matters for a high school Geometry course, where we don't use Hilbert's axioms. So instead, the idea is that in the first two units (Reflections and Rotations), we only prove theorems that are provable in both Euclidean and spherical geometry. There's already a term, neutral geometry, for geometry in which the first four of Euclid's postulates hold (Euclidean and hyperbolic). I propose referring to Euclidean and spherical geometry as natural geometry. (In previous posts, I used the term "normal geometry," but I like "natural" better.) In natural geometry we don't have Euclid's first postulate, but we do have his second though fifth postulates (under our interpretation of them).

The most obvious results of natural geometry are SSS, SAS, and ASA, since these hold in both Euclidean and spherical geometry. And so our first two units can focus on proofs that require only these three congruence theorems. On the other hand, AAS and HL aren't natural, since they hold in Euclidean and not spherical geometry. Also, AAA isn't natural, since it holds in spherical but not Euclidean geometry.

But there are a few problems here. We've seen that the Triangle Inequality holds in both Euclidean and spherical geometry, but we've yet to see a single rigorous proof that works for both. So the Euclidean proof that Van Brummelen mentions fails in spherical geometry due to the dependence of the proof on TEAI.

There are a number of inequalities that we prove in Geometry -- Triangle Inequality, SAS Inequality, SSS Inequality, Unequal Sides, and Unequal Angles. Euclid proves all of these using TEAI. But we wish to avoid TEAI in our proofs as it isn't part of natural geometry. The U of Chicago text assumes the Triangle Inequality as a postulate. But it seems as if we should be able to do better -- there ought to be a proof of one of the Inequality Theorems that doesn't ultimately depend on TEAI.

The concurrence theorems are also tricky. The angle bisectors of a triangle are concurrent in both Euclidean and spherical geometry, but no proof I can find is valid in both. The usual Euclidean proof uses AAS ot HL, while the following link gives a spherical proof using AAA!

Even the Converse Isosceles Triangle Theorem causes problems. It is true in both Euclidean and spherical geometry, but the usual Euclidean proof uses AAS, which isn't spherically valid. One way to unify the proofs (that is, give a single proof that works for both geometries) is the "quick and dirty" proof that a triangle with equal base angles is congruent to itself via ASA.

Notice that I'm trying to keep the theorems as natural as possible by avoiding theorems like AAS and HL as long as possible, and instead using only results valid in both geometries. This doesn't mean that we actually introduce spherical geometry (until the final unit). The following are to be avoided in a high school Geometry course:

-- terms like "Lambert quadrilateral" (or "Saccheri quadrilateral" for that matter). In both Euclidean and spherical geometries, a Lambert quadrilateral has three right angles. The difference is that in Euclidean geometry, the fourth angle is also right, but in hyperbolic geometry, it is obtuse.
-- theorems like "the sum of the angles of a triangle is at least 180 degrees." This is true for both Euclidean and spherical geometry (=180 for Euclidean, >180 for spherical), but the proofs are completely different. Even if we could unify the proofs, we should still avoid it in high school.

Again, during the first two years of this blog, I tried to rearrange the U of Chicago text so that it follows this pattern. It's tricky because SSS, SAS, and ASA appear so late (Chapter 7), while the first postulate that isn't natural appears in Lesson 3-4:

Corresponding Angles Postulate (Euclid's Proposition 28):
If corresponding angles have the same measure, then the lines are parallel.

Parallel Lines Postulate (Euclid's Proposition 29):
If two lines are parallel, corresponding angles have the same measure.

The first postulate holds in Euclidean and hyperbolic geometry, but fails in spherical geometry. The second postulate holds in Euclidean geometry and vacuously holds in spherical geometry, but fails in hyperbolic geometry. The second is equivalent to Euclid's fifth postulate, which we know fails in hyperbolic geometry. The first doesn't use Euclid's fifth at all, so it holds in hyperbolic geometry. The idea is that spherical geometry is the opposite of hyperbolic geometry -- the first somehow uses a postulate like Euclid's first which fails in spherical geometry, while the other avoids Euclid's first and so is valid in spherical geometry (even if vacuously so). In Lesson 3-5, the Two Perpendiculars Theorem follows from the Corresponding Angles Postulate and hence is spherically invalid.

Believe it or not, the next spherically invalid result isn't until Lesson 5-6. Notice along the way, in Lesson 5-5, we have several theorems involving isosceles trapezoids and rectangles, but these hold vacuously in spherical geometry. In Lesson 5-4, the theorems involve kites and rhombi. Kites aren't problematic -- they exist in spherical geometry, and all the theorems about them are true. The definition of "rhombus" is a quadrilateral with four sides equal in length -- and such equilateral quadrilaterals exist in spherical geometry. Nowhere in the definition of "rhombus" does it state that a rhombus is a parallelogram. Rhombi exist in spherical geometry, but not parallelograms.

In Lesson 5-6, we reach our next main non-natural (unnatural?) result -- the Alternate Interior Angles Test (abbreviated as AIA => || Lines Theorem). It's used to prove that a rhombus is a parallelogram, which we already know is spherically invalid since rhombi exist, but not parallelograms. (Also, notice that equiangular quadrilaterals exist as well. These quadrilaterals aren't rectangles -- instead all four angles are obtuse. Yet they have the same symmetry lines -- these symmetry lines divide the equiangular quadrilateral into two Saccheri or four Lambert quadrilaterals.)

Once we reach Lesson 5-6, we are fully Euclidean and no longer spherical. Still, it's bad style to claim that almost all theorems up to Lesson 5-6 are spherically valid by counting vacuous theorems as valid. I'd much rather go to non-vacuous results like SSS, SAS, and ASA before all of these vacuous quadrilateral theorems.

Before we leave my ideal geometry course, let's look at the Pearson Integrated Math I course, which was written with Common Core in mind. Geometry begins in Lesson 8-1 with translations on the coordinate plane, so we're already spherically invalid -- and this is followed in Chapter 9 with area formulas which are spherically invalid. But there are a few interesting things to note here.

In Lesson 11-2, there is an unusual postulate:

Same-Side Interior Angles Postulate:
If a transversal intersects two parallel lines, then same-side interior angles are supplementary.

This statement is equivalent to Euclid's fifth postulate, and so it fails in hyperbolic and holds in Euclidean and spherical geometry (vacuously). In fact, as every statement is equivalent to its contrapositive, we can rewrite this as:

If same-side interior angles are not supplementary, then the lines aren't parallel.

This is actually closer to Euclid's original fifth postulate than any parallel postulate that occurs in any modern text. The only difference is that Euclid's fifth tells us on which side of the transversal the two lines intersect. This statement holds in spherical geometry because lines are never parallel -- and they always intersect on both sides of the transversal.

I could go on with this all day -- but notice that none of this has anything to do with any actual courses that I might teach in the fall. I don't know whether I'll even teach Geometry at all -- and if I do, I won't rearrange the chapters to fit this ideal schedule. We've already seen how well rearranging chapters turned out the last three years (including the year I actually taught). Instead, I might sneak in a few of these ideas or present proofs of the main theorems slightly differently. Oh, and if there's any time left at the end of the year, I might introduce a little bit of spherical geometry. (But notice that most Geometry students will be freshmen or sophomores, and those two grades don't take the SBAC in California.)

Plans for Tomorrow's Post

Tomorrow is the last of these three consecutive posting days -- set up so that I can post the last page of the Pappas section on non-Euclidean Geometry. It's also Father's Day, and so as I promised, I'll reblog last year's holiday post and submit it to Tina Cardone's "Day in the Life." I'll add a little bit to that post, including thoughts about this post in the year since I blogged it.

So in short, tomorrow's post will be about Pappas and papas.

Friday, June 16, 2017

Van Brummelen Chapter 1: Heavenly Mathematics

Table of Contents

1. Pappas Page of the Day
2. Introduction to Van Brummelen's Book
3. Chapter 1: Heavenly Mathematics
4. Exercise 1.1: sin 3
5. Exercise 1.2: Subtraction Formula for Sine
6. Exercise 1.3: Concavity of Sine
7. Exercise 1.11: Velocity Due to Rotation
8. Glen Van Brummelen? What About Eugenia Cheng?

Pappas Page of the Day

This is what Theoni Pappas writes on page 167 of her Magic of Mathematics:

"I have discovered such wonderful things that I was amazed -- out of nothing I have created a strange new universe." -- Janos Bolyai from a letter to his father, 1823.

This is the first page of a new section, "Geometries -- Old & New." By "old geometry," Pappas is referring to Euclidean geometry, and so the "new geometries" are non-Euclidean.

As the readers of this blog already know, non-Euclidean geometry is one of my favorite topics. I've devoted the past two summers to writing about non-Euclidean geometry. Notice that Pappas quotes Bolyai, one of the creators of one non-Euclidean geometry -- hyperbolic geometry. But the geometry I wrote about the past two years is the other type -- spherical geometry.

Usually, I write a few paragraphs summarizing the Pappas page of the day, as well as add my own knowledge of the topic. But I've already lined up another side-along reading book -- Glen Van Brummelen's Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. This book extends spherical geometry by defining trigonometry on the sphere.

And so this will count as my explanation of the Pappas page -- a description and summary of Van Brummelen's book. Notice that last summer, I tried to alternate between my school posts and my spherical geometry posts. But this summer, I've changing it up so that my spherical geometry posts line up with Pappas pages. This is why my first two summer posts (June 1st and 9th) were both school posts, and now I'm following that up with multiple spherical trig posts.

In fact, I usually don't post three days in a row during the summer (and in fact, I rarely posted three days in a row last fall or winter either). But I want to post three straight days this weekend, in order to cover the entire "Geometries -- Old & New" section in Pappas -- pages 167, 168, and 169. I'll then follow up those Pappas pages with Van Brummelen chapters. This increases the likelihood that I'll actually finish Van Brummelen's book before my upcoming first day of school.

Introduction to Van Brummelen's Book

Van Brummelen begins his book with a preface:

"Mathematical subjects come and go. If you glance at a textbook from a century ago you may recognize some of the contents, but some will be unfamiliar or even baffling. A high school text in analytic geometry, for instance, once contained topics like involutes of circles, hypocycloids, and auxiliary circles of ellipses: topics that most college students today will never see. But spherical trigonometry may be the most spectacular example of changing fashions in the 20th century."

These opening sentences sound a lot like the complaints of our traditionalists (most notably Katharine Beals) -- so many topics from 100 years ago are missing under the Common Core Standards. But according to Van Brummelen, the loss of spherical trig dates back to the 1950's -- well before the Common Core era.

According to the author, the loss of spherical trig is a shame. He writes:

"This paucity comes at a time when new applications of spherical trigonometry are being found. GPS devices have some of its formulas built in. It's amusing to see bibliographies of research papers in computer graphics and animation (for use in movies like those made by Pixar) referring to nothing older than last week, except for that stodgy old spherical trig text."

Of course, today Pixar releases its latest movie, Cars III. Who'd have thought that spherical trig could have been used to create the animation in that movie?

Van Brummelen writes about how to use his book:

"Mathematics teachers may wish to use some of the material in their classes. The core of this book is chapters 1, 2, 5, and 6, although chapter 1 can stand on its own. Chapters 3 and 4 provide an interesting historical contrast to the modern theory, but may be skipped if the instructor wishes a briefer journey; my own course covers chapters 1 through 6. The remaining chapters evolved from student projects. I can vouch personally that the first six chapters work well in a class setting with an enthusiastic group."

Notice that Van Brummelen is a Canadian professor -- so clearly he's referring to college students, not high school students. I have no intention of teaching spherical trig to any of the high school students at my new school.

On the other hand, I mentioned in my May 15th post that Barron Trump, the president's son, attends a school that ends the Honors Geometry course with spherical geometry (not spherical trig). I've been thinking a lot about what such a course might look like. Indeed, I'll write more about some of my ideas in tomorrow's post.

Also, notice that the first two chapters of Van Brummelen aren't about spherical trig yet. Instead, they're more like review chapters -- Chapter 1 reviews the "trig" part, while Chapter 2 reviews the "spherical" part. Therefore in Chapter 2 tomorrow, we'll review much of the spherical geometry that we learned last year from Legendre's old book.

Chapter 1: Heavenly Mathematics

Chapter 1 of Glen Van Brummelen has the same title as the book, "Heavenly Mathematics." As I wrote above, this chapter actually reviews Euclidean trig, but the author does apply standard trig to answer a celestial question -- how far is it to the moon?

Van Brummelen begins:

"We're not ancient anymore. The birth and development of modern science have brought us to a point where we know much more about how the universe works. Not only do we know more; we also have reasons to believe what we know."

Here the author describes how much of science is based on evidence or proof -- and of course, Geometry class is all about proof. Then again, he laments that most trig students don't learn how to derive the sine addition formula.

"The goal of this chapter is twofold. Firstly, we will revisit topics in plane trigonometry in order to prepare for our passage to the sphere. But our second purpose takes precedence: to explore and learn without taking anything on faith that we cannot ascertain with our own eyes and minds. This is how mathematics works, and by necessity it was how ancient scientists worked. They had no one to build on. Our mission is as follows: Accepting nothing but the evidence of our senses and simple measurements we can take ourselves, determine the distance to the Moon."

I admit that sometimes it's difficult to appreciate how difficult it is to find this distance. We can type in "distance to the moon" into Google and obtain a value within seconds, but that doesn't show us where that number came from. It's not as if anyone took a yardstick or tape and actually measured this distance!

Van Brummelen begins with an easier task -- to find the size of the earth. He suggests a method discovered by the medieval scientist al-Biruni, who was actually trying to determine how to find the direction to Mecca, to aid Muslim prayer. (Yes, we are currently in the holy month of Ramadan, and so the qibla is definitely relevant right as we speak.)

The method involves climbing to the top of a mountain and constructing similar triangles. The author provides us with an equation:

cos theta = OT/OE = r / (r + 305.1 m)

where r = OT is the radius of the earth, OE is the radius plus the height of the mountain, and theta is the angle of depression to the horizon as viewed from the mountaintop, which al-Biruni found to be about 34 minutes of arc (as in 34/60 degrees). The author solves the equation to obtain r = 6238, but van Brummelen writes that there's a problem:

"Our goal was to work without relying on anyone or anything, and at the end we likely relied on Texas Instruments to tell us the value of cos 0.56667 degrees."

And so Van Brummelen proceeds to calculate exact values to create a trig table, just as the ancient Greek mathematician Claudius Ptolemy would have done. Some are easy to find -- Lesson 14-4 of the U of Chicago text gives us the sines of 30, 45, and 60 degrees. The author tells us that the next simplest value to find is sin 36, since this is related to the regular pentagon (though we don't show this in high school Geometry). Then, the sine addition formula is used to find more values -- he states and proves this formula:

Theorem: If alpha, beta < 90, then sin (alpha + beta) = sin alpha cos beta  + cos alpha sin beta.

Given: Van Brummelen's proof refers to a "Figure 1.9," so let me describe it here. There is a right triangle OCE, and its hypotenuse OC is the leg of another right triangle OCD, whose hypotenuse OD is equal to 1. Angle COE has measure alpha, and angle DOC has measure beta. We drop a perpendicular from D to OE and label the point of intersection G. Last, we drop a perpendicular from C to GD and label the point of intersection F.

Proof: In figure 1.9, since OD = 1, the quantity we're after is GD = sin (alpha + beta). It is conveniently broken into two parts, GF and FD. Now from Triangle OCD we know that OC = cos beta and CD = sin beta. So in Triangle OCE we know now the hypotenuse. Thus sin alpha = EC/cos beta, so EC = sin alpha cos beta. Since EC = FG, we're halfway there: we've found one of the two line segments comprising GD.

We can find FD by noticing first that Triangle OCE is similar to Triangle DCF. This statement is true because Angle FCO = alpha, so Angle FCD = 90 - alpha, and the two triangles share two angles, so they must share the third. So Angle FDC = alpha...and we already know the hypotenuse CD of Triangle DCF. So cos alpha = FD/sin beta, which gives FD = cos alpha sin beta, and finally we have sin (alpha + beta) = sin alpha cos beta + cos alpha sin beta. QED

Van Brummelen tells us that perhaps for the first time in our mathematics education, we have a reason to believe the sine addition law. He tells us that using this law, we can use the sines of 30 and 45 degrees to find the sin 75, and sin 36 twice to find sin 72. A similar-looking formula for subtraction then gives us sin 3 from the sines of 75 and 72 degrees -- and from there, we can find the sine of any angle that's a multiple of three degrees.

But unfortunately, much to Ptolemy's dismay, we can't find the sine of 1 degree this way. The best he could do is use the following theorem, which Van Brummelen states but doesn't prove:

Theorem: If beta < alpha < 90, then alpha/beta > sin alpha/sin beta.

According to the author, Ptolemy would use a half-angle formula with sin 3 to find the sines of 3/2 and 3/4 of a degree, and then interpolate these using the inequality above to find sin 1. The upper and lower bounds produced by this formula give sin 1 to within five decimal digits.

By the way, once a table of trig values has been produced, we can use it not only to find the size of the earth, but also the distance to the moon. Claudius Ptolemy may have used parallax -- the idea that two people in different locations would see the moon in two different spots in the sky. The value he came up with was 395,167 km or 245,545 miles. This is not that far off from the distance as given by Google -- 238,900 miles. (Note: Some of my posts dated April 2016 discuss these calculations in more detail, except here we focus more on calculating the actual trig values.)

Exercise 1.1: sin 3

Believe it or not, Van Brummelen's book contains exercises. Well, in the preface, he does admit that he uses this as a textbook in his college classes! Let's try some of these exercises here on the blog.

1. Using only a basic pocket calculator (no scientific calculators, although you may take square roots), determine the value of sin 3 in the most efficient way that you can. Include in your work the computation of any sine values you need along the way.

Let's use the method Van Brummelen suggests in the text above -- find the sines of 75 and 72 degrees first, then use the subtraction formula. And to find the sine of 75 degrees, we use the addition formula with the sines of 30 and 45 degrees:

sin 75 = sin 45 cos 30 + cos 45 sin 30
           = (sqrt(2)/2) (sqrt(3)/2) + (sqrt(2)/2) (1/2)
           = (sqrt(6) + sqrt(2))/4
           = 0.96593

I've seen some calculators display exact values not just for fractions, but square roots as well. (I've mentioned such calculators in my November 29th post.) With these calculators, if we enter sin 75, the calculator actually displays (sqrt(6) + sqrt(2))/4. Of course, I didn't ask my middle school students to find sin 75, but it did come up back when I was tutoring high school students a few years ago -- and of course they were confused by the appearance of that value.

Before we leave 75 degrees, notice that we must find cos 75 as well, since both sin 75 and cos 75 appear in the subtraction formula for 3 degrees. Since cos 75 = sin 15, it's easiest just to use the subtraction formula with 45 and 30 degrees:

sin 15 = sin 45 cos 30 - cos 45 sin 30
           = (sqrt(2)/2) (sqrt(3)/2) - (sqrt(2)/2) (1/2)
           = (sqrt(6) - sqrt(2))/4
           = 0.25882

Now let's try 72 degrees. Van Brummelen already shows us how to find sin 36 degrees -- he uses a regular pentagon with each side equal to 1. Here are the labels -- A, B, and C are consecutive vertices of the pentagon, D and E are where AC intersects the diagonals drawn from B (in the order A-D-E-C), and F is the midpoint of DE (and of AC).

The author finds BD by noting that triangles ABC and ADB are similar. He obtains 0.61803, which he notes is the golden ratio. This is often denoted by the symbol "phi" -- and it's exact value is easily found to be (sqrt(5) - 1)/2. (This is not to be confused with (sqrt(5) + 1)/2 = 1/phi = 1 + phi. This ratio is often written as a capital "Phi.")

He then finds the other two values DF and BF using decimals. But let's try to keep the exact values as long as possible. So we'll perform his calculations in terms of phi:

DF = AC/2 - AD
       = (1 + phi)/2 - phi
       = (1 - phi)/2

BF^2 + DF^2 = BD^2
BF^2 + (1 - phi)^2/4 = phi^2
BF^2 = phi^2 - (1 - phi)^2/4
BF^2 = (-1 + 2phi + 3phi^2)/4
BF^2 = (2 - phi)/4                     (using the known fact that phi^2 = 1 - phi)
BF = sqrt(2 - phi)/2

If we write this in terms of sqrt(5) instead of phi, we obtain BF = sqrt((5 - sqrt(5))/2)/2. At this point, we want to avoid the nested radicals and so we write it as a decimal, 0.58779.

At this point, Van Brummelen suggests that we use addition formulas to find sin 72. But notice that this isn't necessarily -- earlier, he points out that Angle BDF = 72. So Triangle BDF is a right triangle with a 72 degree angle, and so we can just find sin 72 directly using the values we already found!

sin 72 = BF/BD
           = (sqrt(2 - phi)/2)/phi
           = 0.58779/0.61803
           = 0.95107

cos 72 = DF/BD
           = ((1 - phi)/2)/phi
           = 0.19098/0.61803
           = 0.30901

We see that sin 72 requires nested parentheses, but cos 72 does not. Let's try to write cos 72 exactly in terms of sqrt(5):

cos 72 = (1 - phi)/(2phi)
           = (phi^2)/(2phi)    (taking full advantage of the known fact that phi^2 = 1 - phi)
           = phi/2
           = (sqrt(5) - 1)/4

This value is simple enough to appear on symbolic calculators along with sin 75 and cos 75. But unfortunately, sin 72 requires nested radicals and so it doesn't appear on our calculator.

Now we can finally use the subtraction formula with the trig values for 75 and 72 to find sin 3:

sin 3 = sin 75 cos 72 - cos 75 sin 72
         = (0.96593)(0.30901) - (0.25882)(0.95107)
         = 0.05233

There is a little rounding error in our calculation, of course -- the true value is closer to 0.05234. We can find an exact value avoiding nested radicals by using phi, sqrt(2), and sqrt(6):

sin 3 = sin 75 cos 72 - cos 75 sin 72
         = ((sqrt(6) + sqrt(2))/4)(phi/2) - ((sqrt(6) - sqrt(2))/4)((sqrt(2 - phi)/2)

There's no way to make this simpler, except maybe to write it all with a denominator of 8.

Exercise 1.2: Subtraction Formula for Sine

2. The sine subtraction law is sin(alpha - beta) = sin alpha cos beta - cos alpha sin beta
(a) Derive this result by replacing beta with -beta in the addition law.

Okay, this is easy, as sin (-beta) = -sin beta and cos (-beta) = cos beta. So we have:

sin(alpha - beta) = sin alpha cos (-beta ) + cos alpha sin (-beta)
                           = sin alpha cos beta - cos alpha sin beta

(b) Now attempt the more interesting task: prove it geometrically using figure E-1.2.

Let me describe the figure. We begin with right triangle OCA, with hypotenuse OC and Angle COA, with measure alpha. We choose point B on side CA and drop a perpendicular to side OC, and we label the intersection as D. We label as OBD is a right triangle with hypotenuse OB = 1 and Angle BOD, with measure beta. Then OBA is a right triangle with hypotenuse OB = 1 and Angle BOA, with measure alpha - beta (so AB is the side we want to find). Finally, we drop a perpendicular from D to side CA, and we label the intersection as E.

At first, it appeared that to find AB, we must determine AC and BC and then subtract them. But both of these are tricky -- and it turns out that this is wrong. Let's determine some easier values first:

BD = sin beta
OD = cos beta

Now we notice that Triangles OCA and BCD are similar. This is because both are right triangles with Angle C in common, hence they are similar by AA~. It also follows that the measure of the third angle, CBD, must be alpha. Now this leads to another triangle similar to OCA, namely BDE. This is because both are right triangles with an angle alpha, hence they are similar by AA~. Since Angle CBD has measure alpha, we can find its cosine:

cos CBD = BE/BD
cos alpha = BE/sin beta
BE = cos alpha sin beta

Notice that cos alpha sin beta looks like part of the formula -- the part that we subtract. This suggests that our goal is not to subtract BC from AC, but rather BE from AE!

But how do we find AE? This is the trickiest part of all. We know that sin alpha = AC/OC, but we can use the Side-Splitting Theorem to conclude that sin alpha = AE/OD, since OA and DE are parallel (as both are perpendicular to AC). We know that the Side-Splitting Theorem involves ratios and trig involves ratios, yet we rarely see them use together this way!

So we write:

sin alpha = AE/OD
                = AE/cos beta
AE = sin alpha cos beta

AB = AE - BE
      = sin alpha cos beta - cos alpha sin beta

which is exactly what we wanted to prove. QED

Exercise 1.3: Concavity of Sine

3. (a) Show by construction that 2 sin A > sin 2A.
[Wentworth 1894, p,8]

Hmmm, Wentworth -- where have we heard that name before? That's right -- traditionalist Katharine Beals likes to mention Wentworth as an example of how rigorous math texts were in the late 19th century as opposed to today:

In this link above, Beals compares Wentworth to the Singapore math texts. Most traditionalists are fond of the Singapore curriculum, and Beals is -- up to a point, at least. She likes Singapore math only for sixth grade and below. At that point, she wants to switch to full-blown Algebra I, and the only text she finds rigorous enough for her daughter is Wentworth.

Beals likes to point out how quickly Wentworth gets into difficult problems. In Chapter 5, Wentworth is already on polynomial long division -- and some of these questions are very abstract, with multiple variables (including in the exponents).

The Wentworth text that Van Brummelen mentions here probably isn't the same text -- according to Beals, her text is called New School Algebra and dated 1898, four years after Van Brummelen's. Most likely the 1894 text is trig, not algebra. But notice that this difficult-looking trig question appears on page eight of the old text!

I know, this isn't supposed to be yet another traditionalist post. But I can't help it -- Van Brummelen wrote about how old texts used to teach spherical trig and then I saw the name Wentworth, and I just had to bring up Beals. (By the way, her daughter is about to start her junior year. I wonder what level of math Beals is teaching her daughter now -- spherical trig, perhaps?)

OK, that's enough about Beals -- let's just try to solve the Wentworth problem. As it turns out, this is related to the sine addition formula, so we can use the same diagram that we used for that problem (the proof I gave before all of the exercises). Instead of calling COE "alpha" and DOC "beta," we're going to call both of those angles alpha (or just A, since Wentworth doesn't use Greek letters).

Now sin A appears twice in this diagram -- it is both CE/OC and CD/OD (= CD, since OD = 1). We use both of these in our problem:

2 sin A = sin A + sin A
            = CE/OC + CD

Now CD is clearly greater than FD, since CD is the hypotenuse of right Triangle CDF and FD is merely a leg. So we write:

            = CE/OC + CD
            > CE/OC + FD

So we see the trick here -- we want to make the RHS smaller so that we can use the > sign. We can make the term CE/OC smaller either by decreasing the numerator or increasing the denominator. We will do the latter, by replacing OC with OD, which is greater since OD is the hypotenuse of right Triangle OCD and OC is merely a leg:

           > CE/OC + FD
           > CE/OD + FD
           = CE/1 + FD
           = CE + FD

Notice that CE and GF are equal as these are opposite sides of a rectangle, so we obtain:

          = CE + FD
          = GF + FD
          = GD
          = sin 2A

So we have 2 sin A > sin 2A, which is what we wanted to prove. QED

(b) Given two angles A and B (A + B being less than 90), show that sin (A + B) < sin A + sin B.

This is basically the same problem as part (a), except that we generalize to beta (or B) as the measure of Angle DOC. Oh, and this time we start with sin (A + B) and reverse the steps:

sin (A + B) = GD
                   = GF + FD
                   = CE + FD
                   = CE/OD + FD
                   < CE/OC + FD
                   < CE/OC + CD
                   = sin A + sin B

Take that, Beals -- I just solved a Wentworth problem! Of course, I assume Beals won't be impressed unless I tried teaching this to high school trig students -- and close to the first day of school, since this is supposed to be a Page 8 problem. Never mind!

Exercise 1.11: Velocity Due to Rotation

Let's try one more of Van Brummelen's exercises. I don't want simply to do the problems in order, so we'll skip to Exercise 11:

11. The latitude of New York City is 40.75 degrees. Find the velocity of New York in space due to the rotation of the Earth on its axis.
[Welchons/Krickenberger 1954, p. 43]

I assume that we're allowed to use either the value of the Earth's radius that we already measured, 6238 km, or a more accurate value, (as opposed to having to use trig to find it again).

Notice that the radius of the circle of latitude 40.75 is in fact cos 40.75 times as long as the radius of the sphere (or a great circle such as the Equator). To see why it's cos 40.75, we consider a circle as a cross section of the sphere, and we see that the radius is just the x-coordinate of a point on the circle, and we know that x is cosine (while y is sine).

Also, I assume that we can use a calculator to find cos 40.75. Actually, 40.75 isn't one of the values for which a cosine can be determined exactly. (It is possible to find cos 40.5, however -- we can use addition with 45 and 36 to find cos 81 and then a half-angle formula.)

Notice that we actually need the circumference, not the radius, of the small circle. And so we'll multiply cos 40.75 by the approximate circumference given in the text, 40,000 km (which is closely related to how the meter was originally defined):

40,000(cos 40.75) = 30,303 km

The length of time it takes for the earth to rotate on its axis is about one day (which is closely related to how the day was originally defined, of course). So the velocity is 30,303 km/day. We can convert this easily to km/h to obtain about 1262 km/h, or about 785 mph.

I don't live in New York, of course. Since I live closer to the equator, I'm actually moving faster. In previous posts, I've given my latitude as 34 degrees N. My actual latitude is a bit closer to 33.75 degrees -- which incidentally is a value for which a cosine can determined.

The text gives a half-angle formula for sine -- sin(alpha/2) = sqrt((1 - cos alpha)/2). As it turns out, the formula for cosine replaces the - sign with +, cos(alpha/2) = sqrt((1 + cos alpha)/2). We will apply this formula twice, beginning with 135 degrees:

cos 135 = -sqrt(2)/2
cos 67.5 = sqrt((1 - sqrt(2)/2)/2)
              = 0.38268
cos 33.75 = sqrt((1 + sqrt((1 - sqrt(2)/2)/2))/2)
                = 0.83147

40,000(cos 33.75) = 40,000(0.83147)
                              = 33259

And 33,259 km/day is about 1386 km/h, or about 861 mph. I'm moving mighty fast now!

Glen Van Brummelen? What About Eugenia Cheng?

Van Brummelen isn't the only author I'm interested in this year. Back in 2015, I devoted several posts to the author Eugenia Cheng, who wrote the book How to Bake Pi. Well, Cheng published her second book this year: Beyond Infinity. And this book is all about one of my other favorite mathematical topics -- infinity.

I wouldn't mind reading and writing about Cheng's second book. But this summer is jam-packed with so many other blog topics that I don't know whether I'll have time to write about it this summer -- and I certainly won't have time when school starts up again.

I've been trying to line up my side-along reading books with Pappas pages. Notice that we actually reached the Pappas pages on infinity back in April, right around spring break in the district where I was substituting. I could have acquired Cheng's book, read it, and blogged about it then -- but I didn't want to spoil the surprise that I had left my old school and was subbing until I was hired for the new teaching job at the end of May. (Reading side-along books during the "wrong week" for spring break would have been a huge giveaway!)

I haven't decided what I'll do yet. I'm strongly considering reading Cheng's book, then going back and editing my old April posts in order to write about the book. I know -- it's a bad habit to edit posts months after they occurred. But there's no need to hide the surprise about my new job any longer. I'd rather write about infinity in the old posts devoted to infinity rather than tie up new posts on spherical geometry and actual school topics with infinity. And besides, spring break is right around the time that Cheng actually released her book.

I'll let the readers know what's happening, since I won't just edit old posts without warning.

Friday, June 9, 2017


Table of Contents

1. Pappas Page of the Day
2. The Best Laid Plans
3. Revisiting My "Rules"
4. Why? Because I said so!
5. Teacher Tone
6. Teacher Look
7. Traditionalists
8. Valedictorians
9. Participation Points
10. Plans for a Future Post

Pappas Page of the Day

This is what Theoni Pappas writes on page 160 of her Magic of Mathematics:

"Mathematics certainly offers an abundance of problems. In fact, mathematics and problems are inseparable."

Here Pappas is about to discuss problems that haven't been solved yet. This is the start of the new section "Unsolved Mathematical Mysteries."

On this page, she mentions three specific problems:

-- Is there a formula or a test to determine whether or not a given number is prime?
-- Is there an infinite number of prime pairs? ("Prime pairs" are commonly called "twin primes.")
-- Is there an odd perfect number?

Pappas defines a perfect number as one which is equal to the sum of its proper divisors. There are 49 known perfect numbers, and all of them are even: 6, 28, 496, 8128, and so on. No one knows whether an odd perfect number can exist.

The Best Laid Plans

Yesterday I visited my old school, since I needed the director (principal) to fill out a form confirming employment for my new school. It was after school, but there are always a few students who stay in the after school program. The first student who saw me was -- you guessed it! -- that same seventh grader who's always spotting my car from the street.

He and a sixth grader wanted me to sing to them, so of course I obliged them. The sixth grader wanted me to sing the Ratio Song, and the most popular song among seventh graders was always Fraction Fever.

While waiting for the director to fill out my paperwork, I spoke to the other administrators who were in the office. They told me about last Friday's eighth grade graduation -- all of the students were dressed up for the occasion. My predecessor teacher (who is now a tutor) had come for a visit -- he was present for the eighth grade ceremony. I was also told about the valedictorian -- he had given a wonderful speech that day.

I noticed a sign near the director's desk that listed the SBAC testing schedule. As it turned out, the tests were given later than expected. In fact, the day I submitted my May "Day in the Life" post (and called it the "last day of testing") was actually the first day of testing -- yes, the science tests I'd fretted about for so long were administered that day. The final day of testing was actually the day I found out I was hired -- the day before the eighth grade graduation!

I've always stated in Common Core debate blog posts that this is how it ought to be -- why should there be wasted time after the SBAC with nothing to do? The state tests should be given just before the last day of school -- and indeed they were at my old school.

But this does contradict something I wrote last summer. I said that the last thing I would do with my eighth graders, between the SBAC and graduation, is give them a "final quiz," a comprehensive exam to prepare them for the real finals that they would take in high school. We now see that even if I hadn't left, I wouldn't have given them a final quiz because there was no extra time between the end of the SBAC and graduation.

It just goes to show me that I shouldn't have spent so many blog posts last summer about academic plans for the upcoming year, since they were based on schedules that could change. Instead, I should have focused (and blogged) more on classroom management, as I'd need to have good management skills no matter what.

The director marked down that I had taught for 117 days at my old school. As I left the school, the seventh grader told me a little about my successor teacher -- recall that she had been teaching kindergarten and first grade at the time of my departure. My former student tells me that she was a little mean, but nonetheless she taught well.

This, of course, is to be expected -- she obviously has better management skills than I, which is why she was chosen to replace me. The students who had neutered all of my rules find out that they can't get away with neutering the rules with my successor. This is something I must keep in mind as I move forward.

Meanwhile, this was the last week before summer break at my old school, It is a week of service -- instead of lessons, the students spend time beautifying the classroom and campus. I actually did know (and write) about the week of service on my blog last year.

Revisiting My "Rules"

In this post, I'll continue to write about why I was unsuccessful my first year in the classroom. (So again, experienced teachers who already know all of this may want to skip this entire post.) In my last post, I listed the "rules" that I came up with last summer:

1. The Teacher Respects You
2. Respect Your Honesty
3. Respect Yourself and Others
4. Respect Your Class Equipment

In fact, my original plans were to have two of the rules be the following:

1. Strive to earn an "A" in every class.
2. Strive to earn an "E" in every class.

In my July 22nd post, I wrote that the form of this the rule as "Respect Yourself and Others" came originally from the MTBoS blogger Fawn Nguyen. But still, the idea behind earning A's and E's (excellent conduct grades) underlay this rule, as I mentioned in that post:

Still, I know that some members of our generation can't go seven waking minutes without using a cell phone, and yet teachers expect them to go seven hours without using one -- that is, 60 times as long as they naturally would. Indeed, I know that for some students, the only effective incentive that will motivate them to work is a few minutes' free time on their cell phone. Again, the idea is for me, after telling the students my library story, to remind them that the people who matter (employers) will criticize -- that is, make fun of -- those who can't go a few minutes without reaching for a phone, and those who do have self-control will get more A's and E's, and ultimately jobs and promotions.
Notice that this would be in response to a student question, "Why can't we use cell phones?" And so this would be the answer -- the students who avoid cell phones are more likely to earn A's and E's.

But the problem was that every time the students asked why they have to do something or are not allowed to do something, I mentioned A's and E's as the answer:

-- Why can't I talk?
-- Why must I do these problems in the book?
-- Why do I have to sit in my assigned seat?
-- Why can't I sit in the teacher's chair?

First of all, mentioning E's was a bad idea, because our report cards didn't actually have conduct grades (unlike the LAUSD report cards). So mentioning E's was meaningless to the students.

Of course, our reports do have A's and other academic grades. But as it turned out, this didn't help convince the students to change their behavior. One girl told me that she was tired of my mention of A's and grades over and over again.

To me, A's for a student are like W's (wins) for an athlete. Just as every single athlete wants to earn as many W's as possible, every single student should want as many A's as possible. Just as every single athlete wants to be a champion, every single student should want to be a valedictorian. Just as a coach can't stress the importance of getting W's too much, I as a teacher can't stress the importance of getting A's too much. And in case you think that there are some players or teams who aren't trying to win ("tankers"), notice that even tankers are trying to win championships -- in this case, they're trying to earn future rings by drafting top players, rather than present rings.

But a sports analogy didn't work in my classroom -- especially in my eighth grade class, where the majority of students were girls who didn't care about sports either. When I was a teenager, I'd much rather be bored in class with an A than entertained in class with an F, or even a C -- indeed, it wasn't even close. I highly valued the opinion of the people who mattered -- and those people wanted me to earn as many A's as possible. Praise from them was worth gold to me, and I wanted to get that gold by getting lots and lots of A's. On the other hand, the people who would make fun of me for being smart didn't matter to me at all. Their opinion was worth mud to me.

On the other hand, I see that to these eighth grade girls -- and several other students of all grades and genders -- they'd rather be entertained with a low grade than bored with a high grade. And so answering their "Why?" questions with "So you can get an A!" didn't change their behavior.

I remember on my final day in the classroom, the day began with eighth grade SBAC Prep. The administration had been getting on my case for not allowing the students to take the SBAC Practice Test online. And so I distributed the laptops and directed the students to take the test.

The test frustrated many students. The first question was on the laws of exponents -- which I'd taught back in October, but by now many students have forgotten them. (Think back to those few SBAC Prep worksheets that I posted last week.) The second question was on slope -- which I was scheduled to reach later in March or April. And so naturally, many students would ask, "Why? Why do we have to take this test?" And when I wouldn't help them on the test, another "Why?" question came up -- "Why can't you help us on the test?"

I replied, "So you can do better on the SBAC in May." But this didn't placate the students. One girl said, "Just help me. No one will ever know," and another added, "I promise I'll learn all of this by May, but help me now." In other words, I told them that the purpose of the test was to prepare them for May, and their response was that it wasn't May yet.

Why? Because I said so!

When I was a young student growing up, I was generally a compliant, obedient student -- especially as I completed eighth grade and headed for high school. Yet I always wanted to know the reasons that teachers were telling me to do something. I hated the phrase, "Because I said so!"

And so when I began teaching, I would do whatever it took to avoid saying "Because I said so!" So students would ask "Why?" and I wanted to give them a good reason -- a stupendous reason that will make them see "Oh, that's why!" and get them back to work. In other words, I wanted to convince them to follow the rules using logic. But unfortunately, this almost never happened. Students would continue to ask "Why" and insist that the reasons I was giving them were invalid.

The problem is that a teacher can never convince a student to obey using logic. After all, I've tried telling students to stop talking, and they claim "I wasn't talking!" though they knew they were. If a student knows that he's talking, then no amount of logic can ever convince him that he was -- since he already knew he was! And yet I always tried in vain to "prove" that he was talking -- which would degenerate into "Yes you were!" and "No I wasn't!"

Likewise, no amount of logic can convince students to take the SBAC Prep seriously. After all, if they were interested in doing well on the test, they'd already be working hard on the practice -- the fact that two girls weren't meant that they didn't care about the test, no matter what I said. (By the way, I knew better than to answer "We're taking the practice because the administrators are on my case to make sure that you take it!" -- after all, why would they care about that?)

The ideal classroom manager already knows how to handle the "Why?" questions. The correct answer to a "Why?" question is that phrase I wanted to avoid so much -- "Because I said so." I've learned that students who ask "Why?" are really looking for arguments, not reasons. By trying to give them reasons I'm legitimizing their argumentation, but by answering "Because I said so," the ideal manager is telling them that it's not okay to argue. And once the ideal manager gains a reputation for answering "Because I said so," the students might even stop asking "Why?" That's the only true way of getting rid of the "Why?" questions, as opposed to trying to answer them.

Yes, students may hate it when I reply "Because I said so!" -- but it's possible that they might hate my alternatives even more! I've mentioned that those eighth grade girls didn't like it when I answered "So you can get an A" and lecture them on the importance of earning high grades. They much would have preferred "Because I said so," if only since this phrase contains only four words.

I remember one day back when I was student teaching. My master teacher told me that a certain special ed student needed to be seated closer to me -- and she wanted me to be the one to tell the girl that she'd have a new seat. Of course, the girl wanted to know "Why?" she had to move -- and so she refused to budge. I started telling the other students affected by the move to pick up their belongings so that I could sit a special ed student near the front of the room. At this, the girl agreed to move -- but she was very offended that I'd told everyone she was a special ed student. (Naturally, I don't post any more details here on the blog -- if she didn't want me telling the whole class about her, of course she wouldn't want me telling the whole world about her. If I write about special ed students on the blog, I keep silent about the details, and of course the students remain anonymous.)

In this case, we see how the girl would have strongly preferred hearing "Because I said so" to hearing the actual reasons for the move. If I'd simply said "Because I said so," she might have changed the seat without being offended. And if I'd had a reputation for answering "Because I said so" -- say that I'd used that phrase to answer the previous ten "Why?" questions asked of me -- most likely she would have moved within ten seconds of my first telling her to move without even asking "Why?" In fact, this is most likely what she would have done if my master teacher had been the one asking her to change seats.

Another situation when "Because I said so" might be useful is when a student implies that a rule is difficult or impossible to follow. This most often occurs with the "no restroom" rule. I wrote back in my July 29th post about the tomb guards who can go up to 24 hours without going to the restroom, and my students complain about having to go 80 minutes without using the restroom. Only a few times did I tell my students about the tomb guards. Instead, I give the story about my own years as a young student, which I wrote on the blog last year:

I'm proud to say that I completed three full years of school -- Grades 10-12 -- without asking for a restroom pass even once. How did I do it? Each day, I would use the restroom:

-- Before school
-- During nutrition
-- During lunch
-- After changing from street clothes to P.E. clothes (as I was on the track team)
-- After changing from P.E. clothes back to street clothes

That's five times per day that I went to the restroom. Now 5 * 180 = 900, so that's nine hundred times I used the restroom per year, and that was for three years (Grades 10-12) . So that's a grand total of 2,700 times that I used the restroom -- without missing even one second of class time! Again, I might have been bored and didn't enjoy certain classes (like history), yet I didn't ask for restroom passes. I would much, much rather preserve a reputation as one of the brightest and best-behaved students in the class and earn as many A's, B's, and E's as I could.

During snack break, I'd drink no liquids at all. It's possible that I wouldn't have taken any liquids at lunch either, save for the fact that I ran cross country and for runners, it's good to hydrate. But if I drank at lunch, I'd make it all the way to practice time before having to use the restroom.

I'd tell my students this story whenever they asked for a restroom pass. I also told them that on the weekend, they'd probably gone three or more hours between restroom visits. They only go during class because they'd rather miss class time than friend time. I wasn't like that -- to me, I'd rather spend all of my break time in the restroom than miss even one second of class. To me, class time was more precious than not friend time. Even if I hated the class or found it boring, I didn't want to miss a second, because I wanted to maintain a reputation of being smart, compliant, and obedient.

Of course, the point is that this long story hardly convinced anyone to change restroom habits. And again, I point out that I as a young student hadn't avoided the restroom because some teacher told me a long story about tomb soldiers or anything like that. I avoided it because nearly every teacher gave some penalty to those who went during class, and I didn't want that penalty. In other words, it was because the teachers said so.

By the way, my original intention wasn't to have a "no restroom" rule. In July, I wrote about how I would wanted to give restroom passes to students who earned A's on quizzes. I warned myself that this might not work -- and it didn't. The restroom pass system was neutered simply because many students lost their passes (and often it was either a student I knew had earned an A, or it was right after a Dren Quiz when nearly everyone earned an A).

Teacher Tone

Notice that my new school is Grades 6-12 -- that is, students attend it for seven years, and they start at age 11. In the world of J.K. Rowling's Harry Potter, Hogwarts is also a school that students attend for seven years starting at age 11. Of course, my new school isn't Hogwarts and I'll be teaching the students math, not magic. Still, I often like to compare myself as a teacher to the professors of magic who appear in the series.

One teacher from the series is very laid back -- Professor Flitwick, Charms teacher. Another teacher is more strict -- Professor Snape, Potions teacher. A third teacher is firm yet fair -- Professor McGonagall, Transfiguration teacher. It's obvious which chararacter Rowling considers to be the ideal teacher -- McGonagall, whom the author admits is based on her favorite English teacher.

When I left my old school, the instructional aide suggested that I read Lee Canter. As it turns out, Canter also divides teachers into three categories -- non-assertive, assertive, and hostile. These three categories correspond directly to Flitwick, Snape, and McGonagall. Sometimes I also think of the first three teachers I had in the eighth grade (science, P.E., and history) as corresponding to the three types of teacher.

The ideal teacher is the assertive teacher -- the McGonagall category. I, unfortunately, leaned too much towards the other two categories. According to Canter, teachers who are not assertive may find themselves acting like Flitwick one day and Snape the next.

The three teacher categories also correspond to tones of voice. My normal voice is weak and fits in the Flitwick category. When I speak in a normal voice, students often respond by mocking it. Even the eighth grade guy who became the valedictorian mocked my voice once or twice -- and the girls mocked it all the time. When I raise my voice, it becomes a yell -- the Snape category. When I yelled, students often asked, "Why are you yelling?"

The McGonagall category corresponds to what I call a teacher tone. All ideal classroom managers use the teacher tone to show the students that they mean business. But unfortunately, I simply do not have a teacher tone. I remember back when I was student teaching, sometimes one student wouldn't pay attention to me. Another student suggested that I "yell" at the first student to get his attention -- but here "yell" really means "teacher tone." A true yell (Snape) would only anger the student rather than gain compliance.

Sometimes I hoped that what I lacked in teacher tone, I could make up in musical tone. During the music break I would sing songs to grab their attention. But I couldn't sing all the time -- and besides, when I yelled (Snape) too much, it would hurt my voice and I couldn't sing as well afterward. In the end, I really need a teacher tone.

In my last post, I wrote about the discipline hierarchy that the instructional aide suggested for me:

1st Infraction: Warning
2nd Infraction: Write a 150 word essay telling what did and how you will improve your behavior.
3rd Infraction: Call parents
4th Infraction: Send to Principal

I've said before that the ideal classroom manager only needs the first step in the hierarchy to get the students to comply. This is because the ideal classroom manager uses a teacher tone to issue the warning for the first infraction. In my class, my warnings were ineffective because I lacked the teacher tone to show the students that I meant business. (Actually, the instructional aide told me that it's okay to reach the second level -- it's the third and fourth levels, where other adults are involved, that the ideal manager seeks to avoid.)

So my warnings didn't work, and combined with the second level (already neutered when students didn't put much effort into their 150 words), I found myself at the third and fourth levels too often.

Teacher Look

No more pencils!
No more books!
No more teachers' dirty looks!

Tonight marks the start of summer vacation in the LAUSD and at my old school, and so this rhyme is appropriate today. It refers to another tool at the ideal classroom manager's disposal -- teacher look.

I already know that I don't have a strong teacher tone. But I don't even know whether I have a strong teacher look or not, because I've never tried to give one! As far as I know, I could have an excellent teacher look that will be very successful in getting the students to comply.

I've stated above that with a strong teacher tone, it's not necessary to go beyond the warning step of the discipline hierarchy. In fact, with a strong teacher look, even a warning it's necessary. Students know from the way the teacher is looking at them that the time for playing around is over.

And so when I tell a student to stop talking and the student replies, "I wasn't talking," this is a great time for me to try out the teacher look. I just stare coldly at the student and make sure that I'm not smiling or laughing. This might be enough to get the student to stop talking -- certainly more than trying to "prove" that the student was talking.

I feel bad for wanting to use the teacher look in class. After all, students don't like it -- which is why they would sing songs about "no more teachers' dirty looks," so I'd love to be the teacher who can be effective without teacher looks. But then again, students don't necessarily like pencils or books either, and I wouldn't consider trying to teach without those! The song is intended to be a celebration of summer, not a suggestion to eliminate pencils, books, or teacher looks in school.

And so now I see that if I want to be a better teacher for my new job in the fall, I need to develop my teacher tone -- or if that's impossible, my teacher look. The only real ways to win arguments with students are teacher tone and teacher look, not logic.


Let's check out the latest from our main traditionalist, Barry Garelick:

This article makes the point that the emphasis on getting students to “understand” by using alternatives to standard algorithms is a subterfuge. The purpose, the article contends, is to make students look smarter than they are.  They reason as follows:

At this point Garelick quotes the article to which he links above, so I'll do the same:

The problem is this: “number bonds” is a counterfeit of the way kids who are genuinely good at math act by the time they get into elementary. While the other kids are counting on their fingers, kids who’ve been playing with numbers in their heads since they were two or three have figured out all the relationships and will take numbers apart to make it easier to solve. Not something stupid like seven plus seven, of course. More something like 115 + 115.
Having figured out that number-gifted children will do this as 100+100=200, 15+15=30, so 115+115= 230. This is quite nifty for a first-grader, but the left thinks it can skip all the work getting there. If they just teach perfectly normal, average children to think in terms of taking numbers apart, voila! Everyone will be a math genius!
Oh yeah, I should've warned you that there would be politics ("the left") in this article. Actually, Garelick disagrees that it's "the left" who thinks this way. But Sarah Hoyt, the author of the article, associates Common Core math with leftism -- the first thing we see after clicking on the article is the Soviet Union flag. And many of the commenters on her own webpage agree with her.

Notice that this complaint is about first grade math -- and I've said before that I'm sympathetic to traditionalism in the early grades. Instead, Garelick and Hoyt are criticizing the non-traditional method of "number bonds."

Hoyt mentions 7 + 7 as an example of a question which is to be solved using "number bonds." And without even clicking on the article, we can guess what's go on here. The "number bond" method probably asks students to reduce 7 + 7 to an easier problem, such as 7 + 3, whose sum is 10. And so instead of 7 + 7, students must solve 7 + 3 + 4 = 10 + 4 = 14. I click on the article and, sure enough, that's exactly what "number bonds" means.

We already know how traditionalists like Garelick and Hoyt want students to solve 7 + 7 -- they want them to give the answer 14 in one second or less. The sum 7 + 7 shouldn't reduced to an easier sum -- instead 7 + 7 itself should be a basic sum to which other sums are reduced (and of course, this is via the standard algorithm, as in 77 + 77).

The addition table from 0 + 0 to 9 + 9 contains 100 different sums. To traditionalists, a student should be able to answer any of those 100 sums in one second or less. Remember that I agree with the traditionalists regarding early arithmetic, and so I do like the idea that students should be able to add any two one-digit numbers within one second.

But no one is going to memorize 100 different things at once. Some of the entries in the table are easier to memorize than others, so at some point a young student will know some, but not all, of the entries in the table. To tackle 100 table entries, it's good to look for patterns in the addition table.

An advocate of number bonding would argue that the tens -- 9 + 1, 8 + 2, 7 + 3, and so on -- are easier to add than the others. I have no problem with highlighting the sums that are easier en route to memorizing the harder ones. So a student who doesn't know 7 + 7 yet can add four more to 7 + 3, and then after doing this enough times, the student will memorize 7 + 7 as well.

Number bonders don't only emphasize tens. They also like to stress the doubles -- 1 + 1, 2 + 2, 3 + 3, 4 + 4, and so on. On Hoyt's page, the commenter "cirby" mentions doubles:

Instead of "4 + 5 = 9" you get "4 - 1 = 3, and 5 - 2 = 3, and 1 + 2 = 3, so you add 3 + 3 = 6, and 6 + 3 = 9. See how smart I am? Screw the kids, they're all stupid, and I'm going to make sure they stay that way."

Here cirby reduces 4 + 5 to the double 3 + 3. Actually, cirby is exaggerating a little -- in reality, a number bonder would reduce 4 + 5 to 4 + 4 (that is, by writing 4 + 5 = 4 + 4 + 1), not 3 + 3 as cirby does here. The point is that a child who knows 4 + 4 but hasn't memorized 4 + 5 yet (that's the key word -- yet) can reduce what's unknown to what's known, en route to memorizing the larger sum. Oh, and as a bonus, learning doubles is the key to learning the two's times tables.

But unfortunately, the last part of cirby's comment is not an exaggeration and represents the traditionalists' biggest fears. "I'm going to make sure that they stay that way" -- in this sentence "I" refers to a hypothetical progressive pedagogue. The fear is that the students "will stay that way" -- that is, they'll always add 4 + 5 as 4 + 4 + 1 and never learns 4 + 5 as its own fact.

Correct use of number bonding:

-- No student memorizes all 100 one-digit sums at once.
-- Student so far has memorized 4 + 4 = 8, but not 4 + 5 yet.
-- Student uses number bonds to write 4 + 5 as 4 + 4 + 1 = 8 + 1 = 9.
-- A day or two later, student uses this fact to memorize 4 + 5 = 9.
-- Student never needs number bonds to add 4 + 5 again.

Incorrect use of number bonding:

-- No student memorizes all 100 one-digit sums at once.
-- Student so far has memorized 4 + 4 = 8, but not 4 + 5 yet.
-- Student uses number bonds to write 4 + 5 as 4 + 4 + 1 = 8 + 1 = 9.
-- Student is never made to memorize 4 + 5 = 9
-- Student always needs number bonds to add 4 + 5 from that point on.

The traditionalists argue that too much number bonding is of the incorrect use above. Regarding this, I agree with the traditionalists.

Number bonding also appears in the multiplication table. No student memorizes all 100 products of one-digit numbers at once. Number bonding allows students to use the products they've memorized so far to find the products they haven't.

Tens don't appear as often in multiplication as in addition. In theory, students can use 2 * 5 = 10 to find the products 4 * 5, 6 * 5, and 8 * 5. But doubles -- or their multiplicative equivalent, squares -- can be used, and indeed they are.

I remember once reading an old math text, "Quick Arithmetic" (which I mentioned in an old blog post from October 2014). This text lists the "sexy six" -- the six most difficult products to learn -- these were 6 * 7, 6 * 8, 6 * 9, 7 * 8, 7 * 9, and 8 * 9. Notice that these include all of the products of two numbers from six to nine -- except the squares 6 * 6, 7 * 7, 8 * 8, and 9 * 9. The implication is that it's easier to find squares like 7 * 7 and 8 * 8 than 7 * 8,

I began this post with Pappas and primes. Her first unsolved mystery was a primality test -- a method of determining whether a number is prime. It's often said that the smallest composite number that "appears" to be prime is 91. This is because it's clearly not a multiple of two or five. Neither is 91 a multiple of three, due to the "omega" rule -- 9 + 1 = 10, which isn't a multiple of three. The omega rule explains why 91 is the smallest "apparent" prime rather than 87 -- since 8 + 7 is 15, a multiple of three, we conclude that 87 is itself a multiple of three.

Indeed, we see that 91 = 7 * 13, and both 7 and 13 are "opaque" primes -- that is, there is no simple divisibility rule for 7 or 13. But 91 isn't the smallest product of two opaque primes -- that honor goes to 49, which is 7 * 7. So why is 91 the smallest "apparent" prime rather than 49? It's because squares are easy -- 49 is obviously 7 * 7, but 91's factors aren't as obvious.

The whole point of this is that squares are easy -- and so we can number bonding can be used to convert non-squares into squares. A number bonder can multiply 7 * 8 by notice that it is seven more than 7 * 7 = 49, so the product is 49 + 7 = 56. Of course, this is good only if the student proceeds to memorize 7 * 8 itself within the next few days.

Before I leave this post, let me quote the co-author SteveH:

K-6 pedagogues claim that they love the balance of understanding and skills, but CCSS does not effectively isolate and test the skills portion. K-6 math classes are set up to trust the process and assume that, like thematic learning, facts and skills will be learned automatically, but they only check skills in the context of fuzzy problems or at very low NON-STEM levels. I don’t have any hope that we can find some way to make them see the understanding embedded in mastery of basic skills. I learned all about splitting and combining numbers in different ways in my traditional math classes.


In this post, I congratulated my old school's eighth grade valedictorian. Later on, I mentioned that just as every athlete's goal is to be a champion, every student's goal is to be a valedictorian.

But not everyone agrees with this. From time to time, people write books about how it's not good to be a valedictorian.

I used to link to the Joanne Jacobs site in order to quote the traditionalist Bill. Actually, Bill doesn't comment in the following thread, but I do link to this site to mention an anti-valedictorian book:

As usual, Jacobs herself provides another link:

According to the author, Eric Barker, a student is more likely to be a millionaire with a 2.9 GPA than a GPA of 3.6. This isn't the first anti-valedictorian book -- the most well-known is probably Robert Kiyosaki's Why "A" Students Work for "C" Students.

I believe that it's good to earn as many A's as possible, just as it's good for an athlete to earn as many W's as possible. But if I were to write a book about how "A" students rule the world, I'd expect the book to sell zero copies.

People don't like to read about how "A" students are successful -- they'd rather read about the "C" students who dropped out of school and started their own companies. I believe that these are exceptions to the general rule that "A" students are the most successful.

One commenter, Walter Underwood, writes:

That article points out that the key skill for getting straight As is to excel in things that do not interest you. That same skill may be useful in practicing law or medicine, but probably not in extending the frontiers of anything.

I think back to my own high school days. Of course, I earned straight A's in math all the way through high school, but not every student is interested in math. This is why when I see such a student, I try to compare that student's struggles in math to my own struggles in other subjects. For example, I didn't care for history, just as my students don't care for math.

I remember when I was a junior, I was enrolled in AP US History. But before we took the AP exam, we were to take the "Golden State Exams." The "Golden State Exams" were not state tests like the current PARCC or SBAC. In fact, to this day I don't know the purpose of the Golden States. The Golden States disappeared a few years after the old state tests, the CST's, started appearing (which was also my junior year).

The AP US History exam covers all of American history, but the Golden State History exam covered only the 11th grade curriculum -- basically the 20th century. (All earlier US History was considered to be an eighth grade subject.) In our class we had to rush through more recent history (to the extent that the following year, the teacher started giving summer homework). So I knew that I'd struggle on the essay section of the Golden States, which was on the Vietnam War.

When I received my Golden State score, I found out that I was in the Honors level -- reserved for the top 14% of juniors in the state. (In another subject, Economics, I was in the top 7%, High Honors.) I realized that somehow, in a subject I didn't care for and didn't think I was good at, I had scored in the top 14% range! At the time, I didn't know this, but hearing now about juniors who tank the SBAC, I wouldn't be surprised if 86% of my cohort just wrote a random essay or left it blank on a test that didn't truly count for anything. So simply taking the test seriously is enough for a top 14% score.

I also remember another Golden State Exam in my top subject, math. I was a seventh grade student taking eighth grade Algebra I -- the only such student at my school. As it turns out, the seventh grade Renaissance Fair for history was at the same time as the eighth grade Golden State Exam -- again, I was the only student affected by that scheduling conflict. And I chose to take the math test -- even though this was one time when the history class was actually fun!

At the end of my freshman year, I read about CSF, the California Scholarship Federation. To join this academic club, students must earn points based on grades -- three points for an A, one point for a B (unless it's an honors B for two points), and none for a C.  On my report card, I had two A's and three B's, with no C's or lower. This worked out to be nine points (since none of the classes were honors) -- but 10 points are needed to join! So I couldn't join CSF my sophomore year. My grades rose enough for me to join CSF as a junior, but by then I'd lost interest.

And so it surprised me when one day I subbed at a high school on report card day, and when I passed out the report cards, not one student had only A's and B's. Every single student had at least one grade of C or lower. Here I was, thinking myself a failure for not having enough points for CSF, and yet the entire class that day had lower grades. (And of course becoming valedictorian of my high school was out of the question -- our school used unweighted grades, and so straight A's were necessary for valedictorian status.)

The whole point of this story is to show that most of my students aren't as motivated to earn top grades as I was. And so any classroom management style based on motivating students to earn top grades is doomed to failure.

Participation Points

Back in July, I explained how I would use participation points as part of my management. But by January, the participation point system failed. This is what I wrote in my January 3rd post:

Moreover, in my current class, I started an individual participation points system where students can gain or lose points -- where most points were gained for "participating" (i.e., giving right answers) and points were lost for "not participating" (i.e. off-task behavior). I would begin giving out consequences when a student has lost all his or her points.

And there is the problem -- those smart yet loud students would rack up points for answering my questions, and when I took them away for talking, they'd never lose enough points for me to start giving consequences. So again, the students would talk and talk without any punishment (until I started yelling, of course) -- all because of the doublethink of conflating academics with behavior.

Eventually, I did change my participation points system. Students can now gain points whenever they participate, but now consequences are given separately from the point system. The problem is that it's far too late to introduce a change -- the students have already seen the loudest kids get away with being loud for too long, and so they're never quiet.

I still remember the first time my participation point system failed. It was on the third day of school -- and I mentioned it in my monthly "Day in the Life" post for August:

8:25 -- My first class, a seventh grade class, arrives. Today there is a confrontation with one of the seventh graders. She refuses to do her work, then argues with my student support aide, who asks her to leave the room. I am the teacher, so I should have tried to intervene sooner, though it still might not have made much difference. It is only Day 3, but I already know there's one girl I'll need to watch out for this year.

Well, as I reflect back on this third day of school, the reason I didn't intervene was my flawed participation points system. The girl had gained a few cheap points for turning in her emergency information on time and answering a few questions. So when she confronted my support staff member, I just kept deducting points instead of doing something more substantive.

Ironically, the phrase "I'm taking away a participation point" doesn't scream "I mean business" the same way that a silent teacher look does -- especially if I don't utter that phrase in teacher tone. By this time, the girl had already confronted my assistant, and so it was probably too late for teacher tone or teacher look to make a difference. The best thing for me to have done in that situation was back my assistant out and send the student out of the room, participation points be damned.

As I wrote in January, I'll continue to use participation points in my classes, but I'll only give students points, not deduct them. Participation points can't replace a real classroom management plan, which begins with teacher look and teacher tone, not fooling around with points.

Plans for a Future Post

In a future post, I plan to write about my interactions with one student in particular -- one of my eighth grade girls. I believe that the way I interacted with her is representative of the reasons that I was not successful in my old class, and so I'll write about them in more detail. I haven't decided what day I'll make this post.