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, three years ago our K-1 teacher (who eventually succeeded me as middle school math teacher at the old charter school) celebrated Day 100.

Of course Day 100 is a bit awkward this year, even for elementary students. Since elementary kids do attend school everyday, and their hybrid is AM/PM rather than only on certain days, yes, today really is their 100th day of school. But since schools were online in August and most of September, not all 100 of those days are in person. I have no idea how elementary schools celebrate Day 100 this year.

Meanwhile, I've also once seen a reference to a Day 1000 celebration. This refers to  counting continuously from the first day of kindergarten. Since 1000 - 100 = 900 and 900 = 180 * 5, we find that Day 1000 of elementary school works out to be Day 100 of fifth grade. Thus it's possible for four grades in an elementary school (Grades K, 1, 2, 5) to have a big celebration on Day 100.

In addition to Day 100, 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
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

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. Our main traditionalists are still inactive, but I do have something else to say about the math debates.

We just finished Arthur Benjamin's course on the math of games and puzzles. But Benjamin also has other courses on mental math, as well as a book about this subject. I glanced at a copy of his book, and he does discuss some issues that are ultimately related to the traditionalist' debates.

In his book, Benjamin teaches his readers to multiply two-digit, three-digit, and ultimately five-digit numbers mentally. In each case, he begins with the case of squaring -- where both numbers to be multiplied are equal. Then he moves on to the general case.

He explains that he prefers thinking from left to right for addition, subtraction, and multiplication, as opposed to the standard algorithms that force users to think from right to left. By thinking from left to right, it's possible to start saying the first few digits in the answer while still working on the remaining few digits.

But before he begins multiplying two-digit numbers, he stresses the importance of memorizing the products of all one-digit numbers, since his multi-digit methods assume knowing them. Therefore while Benjamin is not a traditionalist with regards to the standard algorithms, he is definitely a traditionalist with regards to knowing the times tables.

In some ways, I agree with Benjamin. I like to take the middle position between traditionalism and reformism -- perhaps using traditionalist methods to teach certain topics and and reformist methods when teaching others. And so it makes sense to be a traditionalist when teaching the tables but use methods other than the standard algorithm to teach multi-digit multiplication.

Suppose we wanted to teach a math class using Benjamin's methods. The Common Core suggests that students should memorize the tables in third grade:

CCSS.MATH.CONTENT.3.OA.C.7
By the end of Grade 3, know from memory all products of two one-digit numbers.

and multiply multi-digit numbers using nonstandard algorithms in fourth grade:

CCSS.MATH.CONTENT.4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.

OK, so imagine that we're teaching a fourth grade class, and we wish to use Benjamin's method to multiply two-digit numbers -- especially since fourth graders are often confused when we try to teach them the standard algorithm. We tell them to think from left to right, as Benjamin suggests.

We can identify several groups of fourth graders based on their reaction to these lessons:

1. Students who already know the standard algorithm (perhaps from a traditionalist parent or a tutor) and find Benjamin's method a waste of time.

2. Students who enjoy Benjamin's method and are glad they don't have to learn the standard algorithm.

3. Students who are still confused with multiplication even using Benjamin's method.

Benjamin's method does emphasize speed, and so some Group 3 students may be upset that we're trying to equate "good at math" with "fast at math." But if we appeased them by not teaching Benjamin's method, the Group 2 students would lose a chance to learn.

And moreover, Group 3 students would feel under pressure to learn math quickly in order to avoid failing and possibly having to repeat the fourth grade. But a traditionalist would counter that if we don't make them learn a method of multiplying (either Benjamin's or standard algorithm), they might make it all the way to sixth grade, seventh grade, and beyond without learning how to multiply.

This problem, as usual, has no simple solution.

Even though our main traditionalists have been silent lately, I do see a few recent comments made by traditionalists at the Joanne Jacobs website (a common posting place for such posters). Yesterday, traditionalist Ze'ev Wurman responded to the following article:

https://www.joannejacobs.com/2021/01/pluribus-vs-unum/

A commenter named Jean wrote the following:

Jean @ Howling Frog:

“You’re not to think you are anything special.” This perfectly accurate translation utterly fails to communicate the crushing snottiness of the original “Du skal ikke tro, at du *er* noget.”

And Wurman responds:

Ze'ev Wurman:

Snotty or not, how does it compare with the idiocy of “you are special,” “you can be anything you want,” and “you know what’s best for you”?

If we return to the original article, we read:

This contradiction also helps explain our interminable curriculum battles. American hyper-individualism conflicts with any notion of education as a structured transmission of knowledge from one generation to the next.

The problem is that viewing education as only "a structured transmission of knowledge from one generation to the next" inevitably leads to the questions "Why do we have to learn this?" and "When will we use this knowledge in real life?" And such students are likely to leave assignments blank.

Ten days ago, I subbed in a special ed English class. One guy there refused to work and continued to complain about having to learn until he was kicked out of the classroom. When students feel that there's no reason to learn something, they leave their work blank. When they feel that learning has a purpose, they're less likely to leave their work blank.

Telling a student that he/she is special leads to fewer blank papers. Telling a student that he/she is nothing special leads to more blank papers.

The other article features a comment by the traditionalist Bruce William Smith:

https://www.joannejacobs.com/2021/01/ethnic-studies-mandate-is-revived/

Bruce William Smith:

In One World Education Centre, we won’t offer Ethnic Studies, and won’t recognize any credits (or grades) achieved in it, since it does not resemble any curriculum we offer, or think worth teaching; and all private college admission offices should do the same, to discourage this kind of planned social divisiveness.

Unfortunately, ethnicity and race often come up in traditionalists' posts. And I guess it's only fitting that we're discussing this on the first day of Black History Month -- the original Ethnic Studies month.

But the issue he describes here actually extends beyond race. In the article, we see:

A study of low-performing ninth graders who took ethnic studies in San Francisco found “highly encouraging” results, reported Thomas Dee, a Stanford education professor. D students became C+ students.

We can generalize here -- a study of low-performing freshmen are made to feel good. But feelings matter -- by taking the students' feelings into account, they're less likely to leave assignments blank and thus earn a grade of C+ instead of D.

There are nonracial ways of taking students' feelings into account, such as telling students that they are special -- but the traditionalists' disapprove just as much as they do with racial methods. But race here adds what they fear is a zero-sum element -- to them, making students of one race feel good necessarily makes students of another race feel bad. So this is what Bruce William Smith means here when he says "social divisiveness."

And so I ask the traditionalists, is there a non-zero sum way to teach Ethnic Studies, to make build up students of one group without diminishing another? And more importantly, since the traditionalists fear that they're too much emphasis on feelings, do they have a way to get students to do the work and not leave assignments blank without focusing on feelings? (Again, race matters only because feelings matter -- make it so that feelings no longer matter, and race becomes irrelevant.)

Here is today's test: