Let me include the answers as well as the rationale for including some of the questions that I did.
1. 4.
2. 24 square units.
3. 9pi cubic units.
4. sqrt(82) * pi square units, 3pi cubic units.
5. 48,000 cubic units.
6. 98 square units.
7. 28,224pi square units, 790,272 cubic units.
8. 5.5. Section 10-3 of the U of Chicago text asks the students to estimate cube roots. If one prefers to make it a volume question, simply change it to: The volume of a cube is 165 cubic units. What is the length of its sides to the nearest tenth?
9. Its volume is multiplied by 343. This is a big PARCC question!
10. Its volume is multiplied by 25 -- not 125 because only two of the dimensions are being multiplied by 5, not the thickness.
11. The volume of Neptune is 64 times that of Earth.
12. A ring -- specifically the area between the the circular cross section of the cylinder and the circular cross section of the cone. This is Cavalieri's Principle -- recall the comments I made about Dr. Beals?
Today on her Mathematics Calendar 2018, Theoni Pappas writes:
Find the height of this kite to the nearest meter.
(The length of the string is 58 m and the angle of elevation is 15 degrees.)
This is a simple trig problem that can be solved using the sine ratio from Lesson 14-4 of the U of Chicago text:
sin 15 = h/58
h = 58 sin 15
h = 15.01 meters
We are asked to round this height to the nearest meter. So the answer is 15 meters -- and of course, today's date is the 15th.
Our students won't be able to solve this since we haven't reached Chapter 14 yet. We definitely wouldn't want to include this question on today's Chapter 10 Test. If you included yesterday's Pappas question as part of the review, you might also wish to include a similar question on today's test. So here's a possible Question #13 for the test:
13. Find the prism's volume, if the volume of the pyramid is 5 cm^3.
Again, the prism and pyramid are drawn with the same height and base -- with the height to be given unnecessarily (as any random value). The correct answer, of course, is 15 cm^3.
It's also possible to give versions of this question so that the numbers work out friendly -- for example, make it a square pyramid with volume 6 cm^3 and height 2 cm:
V = (1/3)Bh
6 = (1/3)B(2)
B = 9 cm^2
Since the base is a square with area 9 cm^2, its side length must be 3 cm. Then we can find the volume of the prism -- which happens to be a box:
V = lwh
V = (3)(3)(2)
V = 18 cm^3
This doesn't change the fact that all we had to do is triple the pyramid's volume to find the prism's volume -- but this problem can reveal the student's thought patterns to the teacher.
If we really wanted to be devious, we could ask for surface area of the box rather than volume. Then students actually have to find the side length of the square base. The surface area is:
SA = 2lw + 2lh + 2wh
SA = 2(3)(3) + 2(3)(2) + 2(3)(2)
SA = 42 cm^2
Out of curiosity, what's the surface area of the pyramid? To find the lateral area, we need to find the slant height first. The slant height is the hypotenuse of a right triangle whose legs are the height and the apothem of the base (which is half the side length, since the base is a square). So the legs are 2 cm and 1.5 cm (half of 3 cm):
a^2 + b^2 = c^2
1.5^2 + 2^2 = c^2
c^2 = 6.25
c = 2.5 cm
Then we find the lateral area:
LA = (1/2)lp
LA = (1/2)(2.5)(12)
LA = 15 cm^2
\
To this we must add the area of the base to obtain the surface area:
SA = LA + B
SA = 15 + 9
SA = 24 cm^2
Thus the surface area of the box isn't a simple multiple of the surface area of the pyramid, in the same way that the volume of the box is triple the area of the pyramid. This is to be expected, since we needed to find different lengths (height vs. slant height) to find the surface area. Therefore, I wouldn't consider this to be a fair question to ask the students after yesterday's review question.
Today I subbed in a middle school classroom, on Day 103 in this district. (Day 100 in this district was back on Monday -- which was Lincoln's Birthday in my old district, so I didn't post that day. But my new district is a K-12 district, unlike my old high school district. Thus the kindergartners and first graders in my new district probably did celebrate Day 100 on Monday.)
It's at the same school as last Thursday -- and indeed, many of the same special ed students are in this English and history class as well. I wrote earlier that starting this week, I'm only posting under the "subbing" label for math classes. And so I won't write a "Day in the Life" for today.
Instead, today is my "traditionalists" post. Actually, I already mentioned a traditionalist in this post -- Dr. Katharine Beals. Actually, Beals is no longer an active blogger -- instead, she's focusing on writing her own book about traditionalism and language.
But three years ago, Beals wrote a post attacking Cavalieri's Principle as fluff and a waste of time in high school Geometry classes. Since I reblog my old posts from two and three years ago every time I write about Cavalieri, I keep dragging up her name and the old debates.
And so as usual, we return to our main active traditionalist, Barry Garelick. (Recall that Beals and Garelick co-wrote an explosive article defending traditionalism a few years ago. Also, Garelick is the mathematician Beals mentions in her posts -- the teacher who wants his daughter to date someone who can derive the Quadratic Formula.)
https://traditionalmath.wordpress.com/2018/02/14/weve-always-been-at-war-with-eurasia-dept/
What with Robert Pondiscio’s welcome and well-written article extolling the benefits of Direct Instruction (Zig Engelmann’s method for instruction) and thereby praising direct instruction in general, there are indications that others may be following suit. I just read a blog piece by a math teacher who has reached the eye-opening conclusion that conceptual understanding doesn’t always have to precede procedural fluency. In fact, procedures may not be the bogeyman that math reformers have been saying they are for the past hundred years or so.
Pondiscio's article, to which Garelick links here, has drawn the attention of traditionalists. (Even Beals, who's no longer an active blogger, took the time to tweet about Pondiscio this week.) Now this article is about a particular scripted curriculum, "Direct Instruction." While Garelick and the other traditionalists don't necessarily endorse (capital) Direct Instruction, they strongly recommend (lowercase) direct instruction -- that is, traditional instruction from a "sage on the stage."
Garelick continues:
And it isn’t as if math teachers have routinely refused to teach the conceptual understanding. It’s just that if you’ve spent any time at all in a classroom, you will have noticed that your students glom on to the procedures. And unless the conceptual understanding piece was part and parcel of the procedure (as is the case with adding and subtracting with regrouping) few if any remember the underlying concepts. This has led to math texts now “drilling understanding” by making students do the conceptual understanding piece as if it were the algorithm itself; i.e., 3 by 5 rectangles and shading the appropriate parts to represent 2/3 x 4/5 as a means to “understand” what fractional multiplication is.
Here Garelick is writing about a particular peeve here. Some fifth graders in Common Core classes are asked to multiply 2/3 * 4/5 by drawing a diagram. That is, they're given a worksheet with 20 fraction multiplication problems and are required to draw 20 diagrams. If I recall correctly, his story is that if a student chooses to multiply the fractions using the standard algorithm -- without drawing any diagrams -- then the student receives zero credit.
Then one day, some teacher -- a traditionalist hero -- comes in and tells the students that they can do the 20 problems without drawing the diagrams. The students cheer, since they're tired of drawing the diagrams over and over. And so this "proves" that even the students prefer traditionalist direct instruction to anything the progressives or constructivists try to teach.
And indeed, notice the title of this post -- "We've always been at war with Eurasia." This is an allusion to George Orwell's 1984. The comparison Garelick makes is obvious -- he's arguing that the world of constructivism is like Orwell's Oceania, with the progressive pedagogues as Big Brother and the teachers just parroting the Party line.
So far, this post has drawn six comments, with two of them written by SteveH:
SteveH:
[W]hen my son was growing up in a differentiated instruction full inclusion classroom, we quickly learned that their DI was really “differentiated learning (DL). They even called it that. “Trust the spiral” and process. Reaching your highest level of educational development and opportunity was thought to be natural. I can’t tell you how many times we heard that “Kids will learn when they are ready.” This is another way of blaming the student, peers, parents, society and poverty. This philosophy leads them to what I call their two+ generational solution. They are thrilled if little Urban Suzie is the first in her family to get to the community college. Apparently, she will then have the knowledge to do for her kids what we parents had to do with our kids at home or with tutors to achieve a one-generation education. El Sistema clearly shows that the solution can be one-generational if you push and focus on mastery of content and skills from the earliest grades. Almost all of the All-State musicians in the US have taken years of private music lessons. It wasn’t PBL band that made the difference. Band is neither necessary or sufficient. Now, math requires home or tutor private lessons for mastery of skills. I HAD to do that for my “math brain” son.
In this post SteveH repeats some of his favorite stories. "Little Urban Suzie" is a student who attends a community college. "Two-generational solution" refers to the idea that if Suzie can advance beyond her parents (with only a high school education or less), then maybe Suzie's own children can advance one step further and attend an elite four-year school. His "one-generation education" is the idea that the goal should be for Suzie herself to attend an Ivy League or other elite school. And the way to achieve this goal, according to SteveH, is to teach her using traditionalist direct instruction.
"El Sistema" is another one of SteveH's favorite examples. Some traditionalists applaud music instruction as a successful example of direct instruction (with its "Bb-A-Bb" drill and practice) and lament that we can't teach math that way. But according to SteveH, even music instruction is too progressive for his taste ("PBL band"). Perhaps to complete his analogy, band class is like drilling the students to multiply fractions -- with the algorithm, not pictures, but in eighth grade. To SteveH, eighth graders should be learning Algebra I en route to senior-year Calculus. Likewise, SteveH's band students are playing "Bb-A-Bb" at school and playing real songs in private lessons on the weekend -- and these private lessons are what SteveH means by "El Sistema."
SteveH:
Hello?!? Constructivists first have to explain why DI still dominates high schools and colleges and why it all changes for K-6. In our schools, parental (and real life) push back caused the elimination of CMP math in middle school to be replaced by direct instruction using proper Glencoe textbooks. They are losing the war and the boundary between reality and fantasy land is now between 6th and 7th grades.
So apparently "CMP math" is a constructivist math curriculum. I've mentioned the Glencoe texts in previous posts, mostly from three years ago when I was still tutoring. Those were high school texts -- I haven't seen the middle school Glencoe texts to which SteveH is referring here.
Of course, by "reality," SteveH means traditionalism, while constructivism is "fantasy land."
SteveH:
Later, in algebra, understanding is not possible without mastery of skills. Words don’t show understanding. Homework and tests where you have to do all variations of problems show understanding. Invert and multiply is never rote – there are very simple explanations. The proof of understanding is not whether you can explain that in words or even in a proof, but whether you can show how it applies to all variations of rational terms. Simplify (a/b + c)/(1/bc). In my algebra class, everyone had to do only one step at a time and had to put the rule or identity next to each line to justify the change. This was for direct instruction.
Here SteveH attacks problems where students are asked to explain their work. He provides an Algebra II example of what he prefers instead of a paragraph of explanation -- just write down the rule used to simplify a rational expression. Presumably, a proof in Geometry would qualify, since this is exactly what we do with our proofs.
Ironically, I put an example of what SteveH might like in today's post -- when discussing yesterday's Pappas question and how we could write a similar question for today's test. Let's rewrite it so that no picture is even needed to state the question:
A box and a pyramid share the same height, 2 cm, and the same square base. The volume of the pyramid is 6 cm^3. What is the volume of the box?
Then we can check for understanding by looking at how the student solves the problem. We know that the student understands the relationship between two volume formulas if he or she solves the problem in a single step -- multiplying the given pyramid formula by three -- and that the student doesn't understand this relationship if he or she calculates the side of the square first, even if the correct answer is obtained. So this is a "variation of a problem" that assesses understanding. (Indeed, some, though not all, Pappas problems are like this.)
SteveH (second comment):
It took me years and years, but I finally realized that the reason I had trouble learning something was that I did not have the right teacher or textbook. I also realized that really learning something required incremental mastery of the material, as in textbooks with unit chapters and homework variations. I wonder why constructivists never refer to homework and p-sets as hands-on learning? Do they really think that practice is only about speed? We all love engagement, but engagement really has to support mastery. It doesn’t automatically drive mastery after the fact. Mastery generates engagement, not the other way around.
OK, let's talk about SteveH's hypothetical student, "Little Urban Suzie." We'll begin with Suzie as a fifth grader, and her teacher is assigning her a multiplying fractions p-set. The teacher, a traditionalist hero, tells Suzie that she doesn't have to draw the diagrams to multiply. Therefore Suzie is now more likely to want to do the worksheet, right?
Well, that's not true for my Little Urban Suzie. You see, my Suzie hates all math -- and she especially hates fractions. So when the teacher assigns the worksheet, the girl has no intention of doing it -- and she doesn't even bother putting it in her backpack to take home. The fifth grade teacher finds the paper after school, on the floor next to Suzie's desk. Obviously, it makes no difference whether drawing the diagrams is required if the student doesn't even look at Question #1.
The only difference that a private tutor can make is that the teacher, with 30 other students, doesn't notice that Suzie leaves without her paper until it's too late. But when the private tutor sits right next to Suzie and hands her the fractions worksheet, there's no way for the girl to escape. She has no choice but to work on fractions. Only then will she be grateful that the tutor isn't making her draw twenty diagrams along with the fraction problems. But I assume that SteveH uses the name "Little Urban Suzie" to imply that her family can't afford a tutor. Yet as I said before, without a tutor looking over her should, Suzie doesn't even begin Question #1.
Now suppose that somehow, Suzie makes it to eighth grade Algebra I anyway. Does this mean that Suzie is now headed for success? After all, she's now reached the "land of reality" and is no longer in "fantasy land." Her teacher assigns her real p-sets -- page 215, #2-40 even -- in a proper Glencoe Algebra I text. Does this mean that Suzie is bound for the traditional AP Calculus track and headed for Harvard or Yale, just as SteveH wants?
Well, that's not true for my Little Urban Suzie. She still hates all math. When her teacher shows her the Quadratic Formula or rational expressions, she asks her teacher questions. And these aren't questions like "How do you do this?" or "What do you do with the c?" but more like "Why do I have to do this?" and "When will we use this in real life?" When she gets home, she doesn't even crack open her proper Glencoe text, and so she never even sees Question #2. It will be difficult for Suzie to get to AP Calculus and the Ivy League.
Indeed, my Suzie is grateful to attend the community college. I don't know what her major will be, but she will choose a path which requires as little math as possible. SteveH will tell us that many doors are now closed to Suzie, but she doesn't care -- she'd much rather keep the doors closed than take even more class of her least favorite subject than she has to.
SteveH tells us that without mastery, there can be no engagement. But I say that without engagement, Suzie doesn't even look at the first problem on a p-set. On the other hand, she's more likely to participate in an engaging activity or project. Yes, she might only learn a little math from doing the project, but that beats the zero math she learns by leaving a p-set blank.
In the comment thread, one commenter gave the following reply to SteveH:
Oh. You mean they just want to get the work done like generations of students before them. Who’d have guessed?
The Suzie in my story definitely does not want to get the work done. Indeed, she doesn't even want to start it, much less complete it.
Finally, I think back to Garelick's original problem, 2/3 * 4/5. Yes, students don't really need to draw the pictures if they can just multiply the numerators and denominators. But what if we were to change the problem to addition, 2/3 + 4/5? Many students -- whether taught under a traditional or a progressive pedagogy -- want to add the denominators.
My question is, is there any method of teaching fractions, traditional or otherwise, that rids students of the misconception that they should add the denominators? I'd be impressed if Garelick, or any other traditionalist, demonstrates that directly instructed students are less likely to write 8 as the denominator to 2/3 + 4/5 than students taught in a different manner. At the very least, this would be evidence to justify eliminating many of the diagrams and pictures we make students draw to solve their fraction problems.
I conclude this post by pointing out that today is the new moon. As I mentioned back in my August 21st post, there is often a solar eclipse very close to a lunar eclipse -- the former during the new moon and the latter during the new moon. So with the recent "super blue blood moon," an associated solar eclipse is expected. Today's partial solar eclipse, though, was only visible in parts of South America and most of Antarctica.
On the other hand, this new moon also marks Chinese New Year -- observed by many more people than witnessed the solar eclipse. Some people prefer the name "Lunar New Year" because many Asian cultures celebrate it, not just the Chinese. Indeed, Lunar New Year is celebrated in South Korea, where the Winter Olympics are currently being held. This is the Year of the Dog.
Even though I consider the new moon date to be today, February 15th, Lunar New Year is officially tomorrow, February 16th. This is because it's based on the new moon date in the time zone of China (or Korea), where it's already the 16th. (And yes, since Korea is one hour ahead of China, Lunar New Year could actually fall on different dates in the two countries.)
In my new district, tomorrow starts the four-day President's Day weekend. But this blog follows the old district calendar, and so tomorrow is a posting day (Lesson 11-1).
Meanwhile, in New York, there is a February break of a week in length. Actually, schools will be closed tomorrow -- not for Lincoln's Birthday, but for Chinese New Year. Yes, the Lunar New Year is one of the cultural holidays observed in Big Apple schools. This year, Chinese New Year happens to lead directly into the President's week break.
(Yes, I'm avoiding the elephant in the room -- the Florida high school massacre. I have nothing to say about the tragedy -- words can't express how we all feel about what happened. The school there, of course, is closed today and tomorrow, a long weekend that no one wanted.)
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