Lesson 10-2: Surface Areas of Pyramids and Cones (Day 102)

Today I subbed in a high school English class. It is similar to the class I subbed in on December 6th in that it's at the same school and the teacher has only three periods. It's also similar to the class I subbed in on Tuesday, with both a double-blocked class for English learners (Periods 3-4) and a gen ed junior English class (fifth period).

As usual, the English learner class has an aide who speaks Spanish, and so there's definitely no need for "A Day in the Life" today. These students watch a movie that's definitely familiar to both me and the readers of this blog -- McFarland USA. I described this movie on the blog in several posts back during the last week of November. The class that watched it after Thanksgiving (around the time of the State Cross Country Meet) was a Spanish class, so we watched it in Spanish with subtitles in the same language.

But today is an English class, so we watch it in English with subtitles in the same language. We begin the movie just as we see the Diaz brothers work in the field for the first time. Since it's a double-block we essentially finish the movie -- the bell rings right after McFarland is announced as the winner of the State Meet.

In between the two periods is tutorial. I'm able to help one of the students -- an English learner who remains between his two classes -- on his Algebra I assignment, inequalities in two variables. It's tricky due to the language barrier, so I hope that my explanation helps him a little bit. Fortunately, there's another guy in the same class, so we look at his graphs for comparison.

In fifth period, the juniors must do a pre-write for their upcoming essay on what exactly it means to be American. Before, there's a Warm-Up on racism and prejudice. (Today is the 100th birthday of Jackie Robinson, a baseball player who faced more than his share!) It's a district assessment -- and yes, that was one of the issues during the LAUSD teacher strike. In the LAUSD, district assessments are supposed to be reduced 50% as part of the settlement -- but once again, the district where I'm working isn't LAUSD.

Lesson 10-2 of the U of Chicago text is called "Surface Areas of Pyramids and Cones." In the modern Third Edition of the text, surface areas of pyramids and cones appear in Lesson 9-10.

This is what I wrote last year ago about today's lesson. Notice that I spent much of that post comparing the U of Chicago text to three other math texts (and I decided to preserve this discussion):

One of the texts was published by Merrill, the others by McDougal Littell. I ended up purchasing the latter, which is dated 2001. I actually recognize this text from when I spent one month in an advanced seventh grade math classroom back in 2012. Geometry is covered in Chapters 8 through 10. Chapter 8 covers points, lines, polygons, transformations, and similarity. The transformation section covers reflections and translations (but not dilations in the similarity section), but of course, this is an old pre-Common Core text, so transformations aren't used to define congruence. Chapter 9 is officially called "Real Numbers and Solving Inequalities," but the real numbers portion of the chapter segues from square roots to the Pythagorean Theorem and to the Distance Formula.

That takes us to Chapter 10. As it turns out, much of Chapter 10 of this seventh-grade text matches up with the same numbered chapter of the U of Chicago geometry text. Here are the sections:

Section 10.1: Circumference and Area of a Circle

Section 10.2: Three-Dimensional Figures

Section 10.3: Surface Areas of Prisms and Cylinders

Section 10.4: Volume of a Prism

Section 10.5: Volume of a Cylinder

Section 10.6: Volumes of Pyramids and Cones

Section 10.7: Volume of a Sphere

Section 10.8: Similar Solids

It's often interesting to see how much surface area and volume appears in pre-algebra texts. Wee see that this text gives all of the volume formulas, while only the cylindric solids have their surface areas included in the text. But let's keep in mind that this text was specifically written for the old California state standards that we had before the Common Core.

The final chapter, Chapter 12, of this text is on polynomials. This chapter actually goes a bit beyond the seventh grade standards -- most notably, Section 12.5 is "Multiplying Polynomials" and actually teaches the FOIL method of multiplying two binomials. I was only in the classroom that taught using this text for a month, but I was told that the honors class would cover Chapter 12 around the start of the second semester, with the rest of the chapters taught in numerical order. (Non-honors classes would not cover Chapter 12 at all.) The next section, Section 12.6, may also seem a bit advanced for a pre-algebra class -- "Graphing y = ax^2 and y = ax^3" -- but it appears in the 7th grade standards. 

If we compare this to the Common Core Standards, we see that much of Chapter 10 of the McDougal Littell text corresponds to an eighth grade standard in Common Core:

CCSS.MATH.CONTENT.8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

This is to be expected. The Common Core Standards are based on Algebra I in ninth grade, while the California Standards were based on Algebra I in eighth grade. So many eighth grade Common Core Standards must have been seventh grade standards in California.

Before we leave the McDougal Littell text, let me note that Section 4.3 is on "Solving Equations Involving Negative Coefficients." For comparison purposes, let's look at the McDougal Littell Algebra Readiness text in more detail:

1. Expressions, Unit Analysis, and Problem Solving

2. Fractions

3. Decimals and Percents

4. Integers

5. Rational Numbers and Their Properties

6. Exponents

7. Square Roots and the Pythagorean Theorem

8. Equations in One Variable

9. Inequalities in One Variable

10. Linear Equations in Two Variables

The purpose of Algebra Readiness was to prepare students for Algebra I. Therefore, as we can see, there is very little geometry in this text compared to the McDougal Littell Math 7 text. The only geometry that appears is in Chapter 7, with even less geometry content than Chapter 9 of the Math 7 text (as the Distance Formula doesn't appear in the Algebra Readiness text). Area and volume are nowhere to be seen in the Algebra Readiness text (except the appendix, "Skills Review Handbook").

In many ways, Algebra Readiness was more like a Common Core 7 text than Common Core 8, as Common Core 8 contains more geometry (and even a little more algebra) than the Readiness text.

Now leave MacDougal Littell and continue with the Merrill text:

I didn't purchase the Merrill Pre-Algebra text, so I don't recall how old the text is. But I glanced at it and noticed that all of the equations that appear in Chapter 10 of the U of Chicago Geometry text also appear in this text, with the exception of the equations involving a sphere. That is, the surface area formulas of all cylindric and conic solids appear in this text. This is unusual since, as we've seen, neither the CAHSEE nor the Common Core Standard expect students to learn the more complex surface area formulas before high school Geometry. Since today's lesson is Lesson 10-2 of the U of Chicago text, which is on surface areas of pyramids and cones, I want to discuss what I remember about the Merrill lesson on these surface areas.

Both Merrill and the U of Chicago give the lateral area of the pyramid as the sum of the areas of its triangle lateral faces. But only the U of Chicago gives the formula for a regular pyramid, which it defines in Lesson 9-3 as a pyramid whose base is a regular polygon and the segment connecting the vertex to the center of this polygon is perpendicular to the plane of the base. The formula for the lateral area of a regular pyramid is LA = 1/2 * l * p.

But now we must consider the surface area of a cone. The Merrill text does something interesting here, as it considers the area of the net of the cone. We cut out the circular base and a slit in the lateral region, and then flatten this lateral region. What remains is a sector of a circle. Then the Merrill text simply gives the area of this sector as pi * r *s (where s, rather than l, is the slant height) without any further explanation.

The U of Chicago text, meanwhile, gives a limiting argument for the surface area of the cone, as its circular base is the limit of regular polygons as the number of sides approaches infinity. But there is Exploration Question 25, where the Merrill demonstration is done in reverse -- we begin with a sector of a disk and fold it into a cone.

But neither tells us why the area of the sector (and thus the lateral area of the cone) is pi *r * l. Let me give a demonstration of why the area of the sector is pi * r * l.

We begin with the area of a circle, pi * R^2. The reason why I used a capital R is to emphasize that the radius of the circle that appears in Question 25 is not the radius r of the base -- indeed, it's easy to see that the radius of the circle becomes the slant height l. So the area of the circle is pi * l^2 -- that is, before we cut out the sector. We want to fit the area after we cut it.

Let's recall another formula for the area of a circle given by Dr. Hung-Hsi Wu: A = 1/2 * C *R -- and once again, R = l, so we have A = 1/2 * C * l. But neither one of these gives us the circumference or area of a sector. If we let theta be the central angle of a sector, we obtain:

x = theta / 360 * C

L.A. = theta / 360 * A

        = theta / 360 * 1/2 * C * l

For lack of a better variable, I just let x be the arclength of our sector. But here I let L.A. be the area of the sector, since these equals the lateral area of the cone we seek. The big problem, of course, is that we don't know what angle theta is for the cone to have a particular shape. But we notice that we can simply substitute the first equation into the second:

L.A. = 1/2 * theta / 360 * C * l

        = 1/2 * x * l

And what exactly is the arclength x of our sector? Notice that once we fold the sector into a cone, the arclength of the sector becomes the circumference of the circular base of the cone! And this we know exactly what it is -- since the radius of the base is r, its circumference must be 2 * pi * r:

L.A. = 1/2 * (2 * pi * r) * l

        = pi * r * l

as desired. QED

I incorporate this demonstration into my lesson.

2019 Update: Today is no longer an activity day. The old worksheet mentions a quick activity based on the Exploration question that I just mentioned above. This can now be considered a Bonus question, with this year's new activity to appear tomorrow.

Let's continue with the next proposition in Euclid:

Proposition 15.

If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then the planes through them are parallel.

This proof should be easy to modernize:

Given: AB, BC distinct lines in Plane P, DE, EF distinct lines in Plane Q, AB | | DE, BC | | EF
Prove: Plane P | | Plane Q

Proof:
Statements                                Reasons
1. bla, bla, bla                          1. Given
2. Point G in plane Q so that   2. Proposition 11 from Monday (construction)
    BG perp. plane Q
3. H in Q so that GH | | ED,     3. Existence of Parallels
    K in Q so that GK | | EF         (sometimes called "Playfair," but uniqueness is not needed)
4. BG perp. GH, BG perp. GK 4. Definition of line perpendicular to a plane
5. BG perp. AB, BG perp. BC  5. Perpendicular to Parallels (planar)
6. BG perp. plane P                 6. Proposition 4 from two weeks ago
7. Plane P | | Plane Q               7. Proposition 14 from yesterday (a form of Two Perpendiculars)

Euclid's original proof should be simple enough for high school students to understand without the need to convert it to two columns. All that's need is to replace phrases such as "therefore the sum of the two angles...is two right angles" with "Perpendicular to Parallels," for example.

Lesson 10-1: Surface Area of Prisms and Cylinders (Day 101)

Chapter 10 of the U of Chicago text is on surface areas and volumes. Measurement is usually the focus of the three-dimensional chapters in a Geometry text, not Euclid's propositions that we've been discussing the past two weeks.

Lesson 10-1 of the U of Chicago text is called "Surface Areas of Prisms and Cylinders." In the modern Third Edition of the text, surface areas of prisms and cylinders appear in Lesson 9-9.

Three years ago, I actually borrowed a lesson from the (former) King of the MTBoS, Dan Meyer, and I repeated this post last year.

This means that if I were to repeat that activity this year, this would be yet another three-activity week, with the 2016 activity today, the 2018 activity tomorrow, and then this year's brand new activity on Friday. This year I'm taking charge of how many activities I'm including this year, and so the old activities from 2016 and 2018 will be dropped.

Once I drop the old activity, I don't have much to say about today's worksheet. I will say that I include the Exploration Questions as a bonus. One of them is open-ended -- don't let traditionalists see that problem, as they'll complain it's ill-posed. Everything else I have to say about lateral and surface area I mentioned in last week's parody song "All About That Base (and Height)."

Even though we're in Chapter 10 now, we might as well continue with Euclid. After all, David Joyce implies that he wouldn't mind teaching only "the basics of solid geometry" and throwing out surface area and volume altogether.

Proposition 14.

Planes to which the same straight line is at right angles are parallel.

This is another version of the Two Perpendiculars Theorem. Earlier, in Proposition 9, we had two line perpendicular to one plane, and now we have one line perpendicular to two planes. In all three theorems, two objects perpendicular to the same object are parallel.

Euclid's proof, once again, is indirect.

Indirect proof:
Point A lies in plane P and point B lies in plane Q, with AB perpendicular to both. Assume that planes P and Q intersect in point K. By the definition of a line perpendicular to a plane, AB is perp. to AK, and for the same reason, AB is perp. to BK. Then Triangle ABK would have two right angles, which is a contradiction since a triangle can have at most one right angle. (This is essentially Triangle Sum -- the two right angles add up to 180, so all three angles would be more than 180.) Therefore the planes P and Q can't intersect -- that is, they are parallel. QED

Euclid mentions a line GH where P and Q intersect, but Joyce tells us it's not necessary. Joyce also adds that Euclid forgot to mention the case where K, a point common to both planes, is actually on the original line that's perpendicular to them both. He doesn't tell us how to prove this case, but here's what I'm thinking -- consider a plane R that contains the original line. Planes P and R intersect in some line through K, and planes Q and R intersect in another such line, since the intersection of two planes is a line. By definition of a line perpendicular to a plane, these two new lines are both perp. to the original line. So in a single plane R, we have two lines perpendicular to a line through the same point K on the line, a contradiction.

Before I end this post, no, I'm not going to label three traditionalists' posts in a row. But as so often happens, with so much going on at Barry Garelick's blog, sometimes I forget to check the other websites where traditionalists often post:

https://www.joannejacobs.com/2019/01/wealthy-district-cuts-grade-level-math/

Apparently, a Northern California school district is on the verge of eliminating Common Core Math 8 for all eighth graders, thereby forcing all eighth graders to take Algebra I.

Many math teachers in the district are opposed to the change, since they know full well that many of their middle school students won't be able to handle Algebra I. The only people in favor of the change are -- you guessed it -- traditionalists like Ze'ev Wurman, who posted in the comments. Here's an excerpt of what he writes:

When California abandoned its 1997 standards in 2013, 2/3 of its 8th grade cohort took Algebra 1, with high and ever-increasing success rates. But if the goal is to have everyone with identical outcomes, the majority of students must be dragged down to the level of the worst 5-10%, which is what California has been busily doing.

Hmm, if 2/3 of the students are taking Algebra I, it follows that 1/3 is taking Math 8 -- and that's a lot more than the 5-10% that Wurman quotes. He then mentions a district that's doing the opposite of what the traditionalists want -- San Francisco, where all eighth graders take Math 8.

I have nothing more to say on this. We already know what the traditionalists want, and we've already had too many traditionalists' posts this week. Still, I wanted to link to this nearly week-old post before I forget about it.

Here is the worksheet for today's lesson:

Chapter 9 Test (Day 100)

Today is Day 100. As I've explained in previous years, Day 100 is significant in many kindergarten and first grade classrooms. Indeed, two years ago our K-1 teacher (who eventually succeeded me as middle school math teacher at the old charter school) celebrated Day 100. But the district whose calendar is observed on the blog is a high school only district. Thus there will be no celebration at any school following this calendar,

On the other hand, in my new district (where I subbed today), we have elementary students, but in that district today isn't Day 100. Instead, it's only Day 91 -- the first day of the new semester. But then again, elementary and middle schools use trimesters, not semesters.

Today I subbed at such a middle school. It's the same special ed English class that I subbed in about three weeks ago, on January 11th. Thus I wrote more about this class in my January 11th post.

Three weeks ago, I did "A Day in the Life" and chose my first New Year's Resolution as the focus:

1. Implement a classroom management system based on how students actually think.

But as it turns out, like many special ed classes, the lowest class has an aide. She apparently was sick back on January 11th, and so I never saw her that day. But today she was present, and so she handled classroom management for that class. She speaks Spanish and so is able to take charge of the Spanish speakers more effectively than I ever could.

And so I won't do a full "Day in the Life" today, since the class has an aide. But I will say a little more about how things went in that class. First of all, I see the regular teacher in the morning as she's about to head to her all-day meeting. I remind her of the January 11th incident, and she informs me that both the boy and the girl had been guilty that day. While the boy was calling the girl certain Spanish words starting with "G" and "F," she was insulting him back with a Spanish word starting with "A" (and starts with the same letter in English). I knew of the "G" and "F" words but never recognized the "A" word.

As you may recall from my January 11th post, the lowest class is double-blocked periods 2-3, but this is the middle school with a period rotation. On Tuesdays, the rotation is 1-3-4-5-6-2, and so this class is split between the morning in afternoon. Yes -- on January 11th, I wrote that the two double-blocked classes are split on Tuesdays and Fridays, and as luck would have it, my two visits to this class are on Friday and Tuesday.

(Note: Last Friday, I wrote how holidays like MLK Day affect day-of-the-week schedules. But that's more of a problem with block schedules, not period rotations. Yet there was no school in this district yesterday, so why didn't that affect today's period rotation? Well, with a period rotation, every class meets every day just in a different order, but with blocks, only certain classes meet. Thus missing a day is more detrimental to a block schedule than to a period rotation -- and so only block schedules need to make up the missing day.)

Recall that in the other two classes (first period gen ed and period 5-6 advanced English learners) were learning about writing claims (movie violence and droughts respectively). Well, now both of those classes are writing essays on those respective topics.

But as for the period 2-3 class, third period has a special assignment. You see, our school has declared this week to be Kindness Week. And with the "G"/"F" boy and the "A" girl, this class could definitely use a lesson on kindness!

As luck would have it, I'd stumbled onto a New York Times article on kindness:

https://www.nytimes.com/2019/01/28/opinion/kindness-politics.html

I show my students this article and some of the author's ideas on how to be kind:

The best icebreaker. To start such a gathering, have all participants go around the room and describe how they got their names. That gets them talking about their family, puts them in a long-term frame of mind and illustrates that most people share the same essential values.

Tough conversations are usually about tribal identity. Most disagreements are not about the subject purportedly at hand. They are over issues that make people feel their sense of self is disrespected and under threat. So when you’re debating some random topic, you are mostly either inflaming or pacifying the other person’s feeling of tribal identity. You rigidify tribal identity every time you make a request that contains a hint of blame. You make that identity less inflamed every time you lead with weakness: “I know I’m a piece of work, but I’m trying to do better, and I hope you can help me out.” When tribal differences are intractable, the best solution is to create a third tribe that encompasses both of the warring two.

Presume the good. Any disagreement will go better if you assume the other person has good intentions and if you demonstrate how much you over all admire him or her. Fake this, in all but extreme cases.

Our class actually begins with such an icebreaker where we go around the class, and each student must introduce his or her partner by stating the partner's name (though not its origin), age, grade, country, and what he or she did over the three-day weekend. In this class, eight different countries are represented among the 14 students -- China, Japan, Philippines, Thailand, Colombia, Guatemala, Mexico, and the United States.

The third period assignment is based on Kindness Week. The students log into Chromebooks and complete an activity on kindness. They must state at least two ways to be kind, and this may include some of the ones I write on the board (from the NY Times) or other students' ideas.

Second period at the end of the day consists essentially of Learning Centers. This works better than January 11th since there's now a second adult in the room to lead one of the three groups. She takes the lowest group, and I take the highest group (while the middle group works independently). My group is reading Louis Sachar's There's a Boy in the Girls' Bathroom, Chapters 6-9. I tell them that I've read and enjoyed this book as a young student. I don't recall how old I was at the time, but I was probably around the fifth grade, the same age as the story's protagonist (Bradley Chalkers).

In the end, I declare first period gen ed to be the best class of the day, but with the aide's help, the Period 2-3 class is now the second best class. Period 5-6 is a bit of a struggle, since some of the students play around instead of writing their essays (which are due today in this class). I think I do a good job keeping the students in check for most of fifth period, but as fifth leads into lunch and sixth, the playing and talking begin.

In addition to Day 100 (old district) and Kindness Week (new), today is also test day. Here are the answers to the Chapter 9 Test:

1. Draw two triangles, one not directly above the other, with corresponding vertices joined.
2. Draw a picture identical to #3.
3. Draw and identify a circle and an ellipse.
4. Draw and identify two circles.
5. circle (Yes, I had to make an eclipse reference because of last week's event!)
6. a. Draw a circle. b. Draw a parallelogram (not a rectangle). c. Draw a rectangle.
7. c or d.
8. Draw #7c or d again. (A cube is a rectangular parallelepiped!) The faces don't need to be squares.
9. a. 144 square units b. 8 units
10. a. Draw #3 again, with both heights labeled. b. 25pi square units
11. a. 2 stories b. 3 sections c. back middle
12. a. tetrahedron, regular triangular pyramid b. 6 edges c. (AB, CD) or (AC, BD) or (AD, BC)
13. sphere
14. solid sphere (This is yet another eclipse reference!)
15. rectangular solid
16 a. yes b. 3 planes
17. lw(h + 2)
18. xy(x + 2 + y)
19. pi r^2 (4 + r)
20. planes

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

Proposition 13.

From the same point two straight lines cannot be set up at right angles to the same plane on the same side.

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

Indirect Proof:
Assume that point A lies in plane P, and both AB and AC are perpendicular to plane P. Since three noncollinear points determine a plane, A, B, and C lie in some plane Q. Since two planes intersect in a line, planes P and Q intersect in some line containing A -- call it line DE. By definition of line perpendicular to a plane, CA perp. DE. And for the same reason, BA perp. DE. Then in plane Q, there are two lines through A perpendicular to DE, which contradicts planar Uniqueness of Perpendiculars (implied by Angle Measure/Protractor Postulate). Therefore through a point on a plane, there can be only one line perpendicular to the plane. QED

Today is a traditionalists' post.  As I wrote yesterday, now we're going right back to Barry Garelick's Sunday post:

https://traditionalmath.wordpress.com/2019/01/27/effective-math-instruction-hiding-in-plain-sight/

He begins by quoting an old article from 2015:

In a well-publicized paper that addressed why some students were not learning to read, Reid Lyon (2001) concluded that children from disadvantaged backgrounds where early childhood education was not available failed to read because they did not receive effective instruction in the early grades. Many of these children then required special education services to make up for this early failure in reading instruction, which were by and large instruction in phonics as the means of decoding. Some of these students had no specific learning disability other than lack of access to effective instruction.

And it goes without saying that by "effective instruction," Garelick means "traditional instruction."

I mentioned yesterday that only traditionalists have left comments on this post. Two of the comments are from You-Know-Who:

SteveH:

“She [Jo Boaler] is the author of the first MOOC on mathematics teaching and learning.”

If she really wanted to prove her ideas, she would create a MOOC (stupid name!) on actually learning mathematics – not mathematics teaching and learning. She needs to put her ideas to a reality test compared with other techniques in an open world, not an educator and school monopoly world.

Ah yes, we know that Jo Boaler is very unpopular with the traditionalists. (Oh, and by the way, SteveH's "stupid name" MOOC refers to a massive open online course.) An earlier commenter, Wayne Bishop, links to the YouCubed website. YouCubed is, I presume, a website similar to YouTube, but focuses more on math videos. I'm not completely sure how traditionalists feel about math videos such as Khan Academy, but we know they don't like Boaler's YouCubed videos.

SteveH suggests that Boaler should put her ideas to a "reality test," and this is the original topic of Garelick's post. Indeed, SteveH continues:

SteveH:

# 6 especially, since it would be a simple research project. All of my son’s STEM-prepared friends received help at home and with tutors – not so much for specific high school material, but for recovering from K-8 damage. Once my son got to the high school’s traditional AP track after K-8 help from me, I didn’t have to do a thing. Unfortunately, many parents (and educators) now feel that parents should be surrogate teachers at home who need to understand their fuzzy math techniques and/or to help them “flip” the classroom. Do they ever stop to think what expected parental help implies?

Everyone knows that traditional AP/IB math in high school won the battle, so what’s so special about K-8 that it requires something else? Nobody has EVER addressed this split. At best, we hear about some sort of Zombie Effect with no proof and no examples of ANY other method that works better. All colleges ask for GPA for AP/IB classes, SAT I and II in math, AP test scores, and any AMC test scores. Where is their better model?

By "#6," SteveH is referring to an item from Garelick's list:

6) The extent to which students who are doing well in a reform-based classroom are receiving outside help via parents, tutors or learning centers.

Of course, SteveH believes that he already knows the results of such a "reality test." According to his "reality test," most of the students who do well in math had learned traditional math from tutoring centers, while those who didn't go to tutoring don't do well in math. And his conclusion is that if all schools would only teach traditional math, test scores would skyrocket.

As usual, SteveH is probably correct that the tutored students will have higher scores, but his conclusion that all students should be taught traditionally doesn't follow. Once again, SteveH makes the unstated assumption that just because students are assigned traditional p-sets that the students will actually complete them.

Look, students are great at avoiding things they find boring. Traditional p-sets are boring, and so the students will try to avoid doing it. When the teacher and student are physically a few inches apart -- as they typically are at a Kumon center -- then it's difficult for the student to escape the p-set. But in a public school classroom with 39 students, it's much easier for the student to escape. And if the problems are on a worksheet, the students often just leave them on their desk or the floor without even bothering to put it in their backpacks.

Boaler's YouCubed videos are less boring than traditional p-sets. Therefore at least some students who are unwilling to do a traditional p-set might be willing to watch a Boaler video. So the true hierarchy of test scores is as follows:

traditional p-sets that are completed > Boaler's YouCubed videos > traditional p-sets left blank

Unfortunately, I missed an opportunity to perform my own "reality check." Last week, students in an Algebra I class were assigned twenty IXL problems on computer, while students in two Geometry classes were assigned a traditional p-set of 25 problems from a textbook. But I neglected to count what percentage of students in each class completed the assignments. I'd love to say "Of course a greater percentage completed IXL than the written p-set," but I didn't count it, so I can't claim it. It's unfortunate that regarding evidence, right now I don't have squat.

SteveH:
My own view is that their ideas are all part of their philosophy of full inclusion and natural learning – deliberately ignoring what many of us parents do at home. They can get away with it in the lower grades because reality is not pushing back, and when kids get to high school, there are enough parent and tutor-helped students who do well to provide them with enough cover. Perhaps they are still using my son as cover for their stinking bad choices of MathLand and then Everyday Math [U of Chicago elementary math text -- dw].

I've seen that phrase "full inclusion" appear in many traditionalists' posts, and it's always mentioned as something to which they are opposed.

Hmm, let's apply some of Eugenia Cheng's logic to this phrase. The phrase "full inclusion" would seem to mean "Every student is included." Since traditionalists oppose this, we should determine the negation of this phrase. As Cheng taught us, the negation of a "for all" or "for every" statement is a "there exists" or "for some" statement -- "Some students are excluded."

It's obvious to see why a "some students are excluded" would be objectionable to the parents of the students who are excluded! But SteveH doesn't mind, because he's sure that his son (precisely because he was tutored) would be one of the "included" students:

SteveH:
But then they would have to confront the issue that all of the kids in the accelerated groups are from families that give help at home.

Meanwhile, SteveH moves on to the issue of charter schools:

SteveH:

It could be that the only option is to fight for more charter schools, but that battle is entwined with political demagoguery about profit, not whether or not parents who opt-in know what they are getting. My view is that it’s easy for parents to compare school ‘A’ with school ‘B’. They can always switch back. Many charter schools set higher expectations and push, but unfortunately, they often still use fuzzy math.

Many educators complain about the 19th century school model of education, but what about their 19th century monopoly? I really don’t like the idea of charter schools, but that’s because I want public schools to get it right in the first place. 

Recall that my old charter school from two years ago used the Illinois State text. This curriculum was project-based, which a traditionalist like SteveH would definitely classify as "fuzzy math."

In most of my traditionalists' posts this month, I've been writing about the LAUSD teacher strike. As it turned out, one charter network was also on strike (not my old charter, of course), and that strike wasn't resolved until yesterday. It appears that this strike was a victory for teachers and their union, with job security being a major issue. I should know firsthand about how often teachers are replaced at charter schools, as in the two years since I've left my old charter, there have been three successor teachers in my old position!

Another issue with the main LAUSD strike is the growth of charters in the district. Part of the deal was for the Board of Education to vote on a moratorium or cap on new charters. The vote took place earlier today and it passed. It's now up to the state of California to limit the number of new charters.

There's a new commenter responding to Garelick here, P.K. Adams:

pkadams:
I mostly read your blog as confirmation that I did the right thing homeschooling my children. Once again, you reassure me. I thank God and Texas freedom that I taught my children to read using phonics and math using traditional methods! Homeschoolers have known for years that those methods are effective. Look at Classical homeschooling. It’s all about using the natural learning and developmental stages that all children go through to make learning facts and concepts EASIER! This includes rote memorization, copy work, and recitation at the younger ages.

The problem is that children don't think of rote memorization and copy work as making learning easier -- they just think of it as boring. Students who find work boring tend to leave it blank, unless the teacher is sitting inches away from them and it's impossible to escape. Notice that in the case of P.K. Adams, her children are indeed sitting right next to their teacher, namely Adams herself, since she's homeschooling! (Here I use feminine pronouns to refer to Adams, since there's a woman pictured next to the name.) Thus Adams is just like SteveH's tutors -- the student has no choice except to do the work and learn.

pkadams:
Of course the public schools really do not have the goal of education anyway. The goal is babysitting [...]

Here Adams criticizes public schools for "babysitting" too much. But during the recent strike, some parents criticized schools for not babysitting enough. This is understandable, as working parents are concerned that there is no one to supervise their students from 3:00 to 5:00 when they get off work, or during the summer at all. Working parents sometimes wish that public school were eight hours or more long, so that their children never go unsupervised.

But Adams, who has time to homeschool her children, probably isn't a working parent. She's either a homemaker or works at most part-time. Homeschoolers often brag that their school days often last four hours or less, since they don't need time for passing periods, attendance taking, and so on. The implication is that a homeschooling parent might reconsider public schools if only they could "trim the fat" and likewise make the school day four hours or less, while a working parent wishes that the school day could be eight hours or more. No public school schedule can satisfy them both.

pkadams:
[...] and keeping children out of the work force for as long as possible and preparing them to be compliant worker bees.

This sounds as if Adams is contradicting herself. First, she criticizes the public schools for being too anti-work ("keeping children out of the workforce"), and then she suddenly attacks them for being too pro-work ("compliant worker bees")!

But this is Kindness Week after all. In the NY Times article linked to above, David Brooks suggests that we "presume the good," the best of our opponents. My opponent, P.K. Adams, probably has a valid point somewhere. So instead of calling her out for contradicting herself, I should reread what she's saying to figure out what she really means.

pkadams:
[...] and keeping children out of the work force for as long as possible[...]

It's been almost exactly four years since I've written about the concept of "Dickens age." The Dickens age is the age at which it's acceptable for a child to leave school and go to work. Before this age, requiring a child to work is cruel or Dickensian. We think about the protagonist of Oliver Twist, who is ultimately based on author Charles Dickens himself. A child in a Dickensian situation is forced to work long hours for very low pay and no chance of advancement.

Here in the United States, the Dickens age is implicitly set to 18 or the completion of 12th grade. A child can't work earlier than age 18 without a work permit, entailing the permission of both the parent and the school.

Many other countries around the world have younger Dickens ages. The Dickens age I suggested on the blog four years ago was 15, the approximate average Dickens age around the world.

So apparently, Adams is criticizing public schools for setting the Dickens age to 18. If public schools would set the Dickens age younger, to say 15, and let students graduate at that age and get full-time jobs, then she'd reconsider sending her children to public schools.

But there still seems to be a contradiction. Adams attacks public schools for ignoring developmental stages and pushing difficult work too young, but then she pushes for younger employment. And we still can't ignore her "worker bee" statement. The problem with Dickens was that he was pushed to work too young. In other words, he was treated like a "worker bee."

To what developmental stages does Adams mean? She mentions the Classical Curriculum, and I've brought it up in previous traditionalists' posts. The Classical Curriculum consists of four stages -- grammar (Grades 1-4), logic (Grades 5-8), and rhetoric (Grades 9-12). Our Dickens age thus fits squarely in the rhetoric stage.

We can use the Classical stages to summarize why Adams dislikes public schools. In short, the schools treat the entire childhood as the logic stage. They eliminate the grammar stage by emphasizing understanding over rote memorization in the youngest grades. And they eliminate the rhetoric stage by not allowing the students to have full-time jobs as teenagers.

pkadams:
The strong ones who emerge prepared for life do it despite the system or sometimes thanks to a teacher who does not stick to the curriculum.

Recall that a "worker bee" is someone who works long hours for very low pay and without a chance of advancement. But the 15-year-olds under the Adams plan aren't worker bees. Her teens are skilled because of what they previously learned under the grammar and logic stages. And so they're able to qualify for higher-paying jobs and promotions, and if their employer doesn't raise their pay or grant them a promotion, they can quit and find another employer who will pay them more.

I don't fully agree with the Adams plan. But I've been fascinated by the four-year cycles that exist for English, history, and science under the Classical Curriculum. For example, in history, students learn Ancient, Medieval, Early Modern, and Late Modern History during the four-year cycle. So they have one year of each during the grammar, logic, and rhetoric stages.

As a math teacher, I've thought about how to extend these four-year cycles to math. The three stages "grammar," "logic," and "rhetoric" become "A Story of Units," "A Story of Ratios," and "A Story of Functions" (as some math textbooks have labeled their texts). Within each stage, students learn how to add, subtract, multiply, and divide.

For "A Story of Units," this corresponds roughly to the usual curriculum (for example, multiplying "units" or whole numbers in third grade). But it extends the arithmetic of  "ratios" (fractions, and possibly decimals) a bit beyond the usual curriculum (multiplication of fractions in Grade 7, division of fractions in Grade 8). In past posts, I've argued that this is beneficial. When I taught at the charter middle school, my sixth graders still did math without a calculator, while my eighth graders wanted to do simple arithmetic on a calculator. I suspect this is because of the year in between, as seventh graders feel that they are "done" learning arithmetic, so there's no need for them to do any arithmetic by hand anymore. Extending arithmetic of ratios through eighth grade thus forces the students to avoid a calculator a little longer, which is what traditionalists want.

The four operations for "A Story of Functions" is a little trickier. I've stated in previous posts how exactly this works. (Grade 9 = "adding polynomials," the linear function of Algebra I, Grade 10 = "subtracting polynomials," the distance function of Geometry. Grade 11 = "multiplying polynomials," the quadratic function/complex numbers of Algebra II. Grade 12 = "dividing polynomials," the rational function of Pre-Calculus.)

I agree with extending traditional math into the lower grades, though I usually don't use the phrase "grammar stage" of Classical Curriculum. But SteveH and Adams disagree with the "rhetoric stage," as SteveH wants his seniors to take AP Calculus, but Adams wants her 17-year-olds in the workplace (where they won't be learning AP Calc).

There's one more thing I want to say about traditionalists before I end this post. When I read There's a Boy in the Girls' Bathroom (Chapters 6-9) with my students day, I was reminded of one of my favorite lines from the entire book:

"A teacher can often learn a lot more from a student than a student can learn from a teacher."

The speaker here is Carla Davis, a counselor newly assigned to the protagonist's school. When she first meets Bradley, she says this line to him.

In a way, this line sums up the reformer/progressive (anti-traditionalist) view of education. To the traditionalists, of course this statement is false. The teacher is the sage on the stage and has nothing to learn from the student.

And of course Bradley, the class bully, goes on to prove the traditionalists right. He quickly replies that he's "taught" his teacher geography -- by cutting up his U.S. map and gluing the states in a completely different order. (Bradley's map isn't in Lesson 9-8 on the Four Color Theorem.) And indeed Carla goes out of her way to show kindness (the week's theme!) to Bradley, but he's determined not to show her even the slightest hint of a smile.

Yet I learned much from my students today. They ask me what my favorite foods are, and they tell me theirs in return. (It's actually related to their vocab worksheet -- the word "delicious.") Since they come from so many different cultures, I've never heard of many of these foods before. One of them tells me about Peck Peck, a Korean chicken dish.

Indeed, I believe if I'd taken the time to learn more from my students at the old charter school -- even if it's just to learn more about their names, as David Brooks suggests -- I might still be working over there today. I would have shown the students more kindness and respect, and this would have led to less arguing. I believe that this is more effective than "Listen to me because I'm the sage on the stage who knows everything and you know nothing!" This is something that I hope the traditionalists will think about even if just a little.

Here is today's test: