1. Pappas Problem of the Day
2. The Tweeter "Common Core Math (even if your state renamed it)"
3. The Remaining Two-Year Gap
4. The Path Plan Revisited
5. Another Geometry Problem
6. The Tracking Debate
7. An Analogy
Pappas Problem of the Day
Today on her Mathematics Calendar 2018, Theoni Pappas writes:
Arc ABC is 320 degrees. Find the area of the shaded sector.
(The circle's radius is 15sqrt(pi)/pi. The shaded sector is bounded by minor arc AC, not a major arc.)
Oddly enough, Lesson 8-9 gives only the area of an entire circle, not a sector -- even though Lesson 8-8 does teach arc length. There are a few examples on sector area in the corresponding lesson of the modern Third Edition of the text. The idea, of course, is to multiply the area of the circle by 40/360 -- not 320/360, since the arc is bounded by minor arc AC = 360 - 320 = 40 degrees:
Area of the shaded sector = 40/360 * (area of Circle O)
= 1/9 * pi(15sqrt(pi)/pi)^2
So the area is 25 square units -- and of course, today's date is the 25th. This is another one of those questions where we must start with a unnatural radius (15sqrt(pi)/pi) in order to make the answer come out to a natural number.
This is my first spring break post. As I often do during vacations -- but seldom do during the school year -- I post on the weekend. Today is Sunday the before Easter -- also known as Palm Sunday.
No, I didn't choose today to post because of Pappas. As it turns out, Pappas actually has a whopping five Geometry posts this week, and I won't post five times this week just to get them all. Indeed, five posts would make this feel more like a regular school week, not a vacation week.
And besides, I already skipped posting the Pappas question on a school day. Back on March 14th, Pappas had a Distance Formula question. To me, it's a borderline case whether to count a Distance Formula problem as Geometry. Of course, we do see this formula in Lesson 11-2 of the U of Chicago text, but then again, it's studied in several other courses, not just Geometry.
In the end, I chose not to post it because that day was Pi Day. My post was already jam-packed with so much other stuff (the Pi Day songs I played in class that day, the walkout that occurred on Pi Day, the regularly scheduled lesson). And so I didn't want to add something else to that post that wasn't directly related to pi. Had today's problem occurred on Pi Day instead (with the final answer changed from 25 to 14, of course), then I would have posted it that day after all. Thus I'll decide which Pappas problems to write about and which ones not to write about.
The Tweeter "Common Core Math (even if your state renamed it)"
Most Blogger and Twitter users who use the labels "MTBoS" or "I teach math" aren't traditionalists. I do know of exception -- "Common Core Math (even if your state renamed it)." This tweeter (whose gender isn't public) is highly critical of the standards -- and, as the username implies, that criticism begins with states who no longer use the name "Common Core," yet made only insignificant modifications to the standards:
It definitely appears that the user CCSSIMath (so I don't keep writing the entire username over and over, I'll just use the name in the URL instead) -- is a traditionalist. Let's look at CCSSIMath's pinned tweet, dated January 8th:
On the left, a "released" 2016 @PARCCPlace 10th grade geometry test question. On the right, two pages from a Japanese 5th grade #math textbook. Now, your best arguments how #CommonCore is: (1) internationally benchmarked (2) rigorous
In case you didn't click the Twitter link above, let me describe it. Both lessons are nearly identical -- they show a derivation of the circle area formula (another tie to today's Pappas question). A similar diagram appears in Lesson 8-9 of the U of Chicago text. The difference, according to CCSSIMath, is that the American question is for sophomores, while the Japanese question is for fifth graders.
Now, of course, we see what the intended answers for (1) and (2) are above. We're to conclude that the Common Core Standards, despite claims to the contrary, are neither internationally benchmarked nor rigorous. And if internationally benchmarked and rigorous standards are desired, then the first thing to do is immediately include derivation of the circle area formula as a fifth grade standard under the Common Core.
We already know about other standards which traditionalists believe should be taught in lower grades than specified under the Common Core. These include the standard algorithms for arithmetic. And we already know the solution for this -- replace all mention of "strategies involving place value" with "the standard algorithm" and skip the corresponding standard in the higher grade.
But this one is trickier, since it's Geometry, not arithmetic. In fact, just before January 8th, the tweeter CCSSIMath made several other posts comparing Common Core Math to Japanese math. In each case, the Asian nation teaches in upper elementary school what the Common Core teaches in high school -- and more often than not, this material is related to Geometry. Since this is a Geometry blog, I want to discuss this in more detail.
Let's look at the pinned tweet again. First of all, notice that the circle area formula isn't a high school standard, but a seventh grade standard:
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Indeed, the U of Chicago Transition Math text for seventh grade derives this in Lesson 12-5. So there's really only a two-year gap between the U.S. and Japan, not a five-year gap. So why then does the circle area derivation appear in high school Geometry?
(By way, notice that Pi Day was Day 128 on the blog calendar. If we were following the seventh grade text on the blog, then Lesson 12-5 would have been taught on Day 125, just in time for Pi Day.)
There are several things going on here. First of all, let's what David Joyce says about this:
Theorem 5-12 states that the area of a circle is pi times the square of the radius. A "work together" has students cutting pie-shaped pieces from a circle and arranging them alternately to form a rough rectangle. That idea is the best justification that can be given without using advanced techniques.
By "advanced techniques," Joyce means Calculus. So he has no problem with this informal proof being given in Geometry (and indeed the Common Core admits that it's an informal derivation). The problem CCSSIMath has is that the derivation is given several years too late.
I think the problem here is that Geometry is all about proofs. This means that, as David Joyce puts it, "Every theorem should be proved, or left as an exercise, or noted as having a proof beyond the scope of the course. " But he says nothing about results from before the scope of the course. If a Geometry text leaves out the derivation of the circle area formula -- or more to the point, the triangle area formula -- then the text isn't rigorous, even if the result was proved in an earlier grade. In fact, let's give this as the answer to CCSSIMath's second question:
Now, your best arguments how #CommonCore is: (2) rigorous
Answer: Common Core Geometry is rigorous because all theorems require proof. This includes proofs of results learned in earlier grades. The derivation shown on the PARCC is admittedly informal, but a true derivation requires ideas from Calculus and beyond. And so this proof is, as Joyce puts it, the best justification that can be given at the level of high school Geometry.
The Remaining Two-Year Gap
OK, so now we still have CCSSIMath's question (1) to answer. Even if we admit that the circle area derivation is a seventh grade, not a tenth grade, standard, then there's still a two-year gap between the American and Japanese lessons.
I think back to SteveH and his notions that K-6 math is a "fantasy land" based on a "low slope," while high school math makes the "nonlinear jump" to AP Calculus. Of course, there's some truth to this.
Arithmetic is generally taught by elementary school teachers with Multiple Subject credentials. Many of these teachers choose their profession because they like kids -- not because they like math. These teachers, back when they were young students, might have earned grades of C (or even lower) in their own math classes, and now they must teach rigorous math to their own students.
So the Common Core and other state standards try to take the burden off of elementary teachers by delaying rigorous topics such as the area of a circle until seventh grade, when they're more likely to be taught by a math specialist. But if we want to follow CCSSIMath's suggestion that the standards should be "internationally benchmarked" to Japan, then we must push more difficult math into elementary schools with their Multiple Subject teachers. And here I mean topics like deriving the area of a circle, not just teaching standard algorithms earlier.
Is there a solution to this? Let's double-check what SteveH says about this:
This is a systemic problem for K-8 educators and charter schools are not a complete solution. Most K-8 educators are cut from the same cloth. My son’s first grade teacher admonished my wife and I by saying that our son had a lot of “superficial knowledge.” All teachers should be required to get a degree in a content and skills-rich field, not one where they ironically get directly taught their turf – full inclusion, differentiated and natural learning, and a dislike for “mere” facts and “rote” skills – anything that will justify their social goals that end up having the opposite effect – hiding the skills and knowledge tracking at home and increasing the academic gap.
That doesn't provide a full solution for the elementary school problem. OK, so let's say that all elementary teachers are required to get a degree in a content field. Now suppose that this chosen field is English or history. How would this degree help them teach math any better?
I remember once reading that in Japan and other nations, students begin learning math from specialists somewhat earlier than American students do. In the elementary schools, the students stay in one classroom and the teachers move from class to class -- just like music and P.E. teachers do here at American elementary schools. I wanted to find a link to confirm this, but I can't -- surely some traditionalists have brought this up in the past. If this is true, then it explains why Japanese fifth grade teachers feel comfortable teaching the circle area formula, but not American fifth grade teachers.
The Path Plan Revisited
In past posts, I wrote about the system that my old elementary school introduced -- the path plan. In this system, students are officially divided into "paths," not "grade levels." Here are rough correspondences between paths and grades:
Early Learning Path: Headstart and Kindergarten
Primary Path: Grades 1-2
Transition Path: Grades 3-4
Preparatory Path: Grades 5-6
When I was a young student in the Preparatory Path, our classes stayed in homeroom (for English and history) until lunch. Then after lunch, we attended two more classes. One was math, and the other was an exploratory wheel that switched every trimester (art, health, and science).
Under this plan, only half of the Grade 5-6 teachers actually taught math -- so of course, the half that was stronger in math would teach the subject. My fifth grade teacher was strong in math, so I remained in her class for math. My sixth grade teacher taught science instead, so I went to another teacher's class for math that year.
A simpler schedule was used in the Transition Path. All Grade 3-4 teachers taught math at the same time, but some students moved to other classrooms during math time. I can easy see how the stronger math teachers could be assigned fourth grade math while the others taught third grade math. Thus the best 50% of teachers would cover math in Grades 4-6.
I once posted a more ambitious version of the Path Plan. We push everything down one path, so that Primary Path students leave homeroom only for math, Transition Path students leave for two classes, and Preparatory Path students leave for three classes. Then this ensures that the best 50% of teachers teach math in Grades 2-6, leaving only K-1 for weaker math teachers. Also, in the Preparatory Path, science can be a whole year, not a trimester. Back when I was a fifth grader, we could get away with less than a year of science. But now with state and NGSS science testing, a full year of fifth grade science is crucial. (When I was a fifth grader, our school actually tried having three classes after lunch, but then it switched to two classes with full implementation of the Path Plan.)
The following chart shows how many teachers each student would have at one time under the more ambitious Path Plan, along with which subjects aren't taught by homeroom teachers:
Path Teachers Subjects not in HR
Early Learning 1
Primary 2 math
Transition 3 math, elective
Preparatory 4 math, elective, science
Here's how I'd make these classes fit a bell schedule -- all grades stay in HR until all grades have completed recess. Then after recess, all grades attend the first extra class. Afterward, Primary Path goes to lunch, and the other two paths attend the second extra class. Afterward, Transition Path goes to lunch, and Preparatory Path attends the third extra class, followed by lunch. After lunch, all classes return to homeroom. The length of each class therefore is the same as the length of lunch -- 40 minutes might be a good length of time for both lunch and math.
Notice how this path pattern can be continued beyond elementary school, to a Middle School Path (Grades 7-8) and Early High School Path (Grades 9-10):
Path Teachers Subjects not in HR
Middle 5 math, elective, science, P.E.
Early High 6 typical high school schedule
So in middle school, students have the same teacher for English and history -- which is what my old middle school referred to as "Core class." After Grades 9-10, the pattern continues, but the number of classes goes down every two years:
Path Teachers Comments
Late High 5 only 22 year credits often needed to graduate
AA/Lower Div. 4 (typical college schedule)
BA/Upper Div. 3
PhD 1 dissertation adviser
Here's what we do to address SteveH's ideas about teacher training - there could be two types of Multiple Subject Credential, namely math and no-math. Elementary schools should strive to hire about half math and half no-math for Grades 1-6. All students in Grades 2-6 would have a math credential for their math classes, while no-math teaches either K-1 math or no math at all. (Under the less ambitious Path Plan, it's half math for Grades 3-6, and Grades 4-6 are guaranteed to have a math credential for their math classes. No-math teaches either K-3 math or no math at all.)
Another Geometry Problem
Here is another Geometry problem that CCSSIMath tweets about -- questions that are taught in high school under Common Core, but in elementary school in Japan.
March 6th tweet -- Given:
- Semicircle O of diameter AB = 12
- Arc AC = 135 degrees
- D midpoint of chord
- S: shaded area bounded by CD, BD, and arc CB
Here's how I solved the problem:
First, let's do the easy part -- the area of the entire semicircle. The diameter is 12, so the radius is 6, so the area of the circle is 36pi, so the area of the semicircle is 18pi.
Now we must subtract the area of both Triangle ABD and the segment bounded by the chord
Area of sector: (135/360)(36pi) = 27pi/2
Area of triangle: (1/2)(6)(6)sin(135) = 9sqrt(2)
Area of segment: 27pi/2 - 9sqrt(2)
Now we need the area of Triangle ABD. The base of this triangle is easy to find --
Area of triangle: (1/2)(12)(3sqrt(2)/2) = 9sqrt(2)
So the area of the segment and triangle together is 27pi/2 - 9sqrt(2) + 9sqrt(2) = 27pi/2. This we subtract from the area of the semicircle to find S:
Area of S = 18pi - 27pi/2 = 9pi/2
I admit that this question took me 15 minutes to figure out -- and even then I made a mistake. I didn't realize that the two 9sqrt(2) terms had opposite sign and so I had a more complicated answer.
Now here's what CCSSIMath writes about the question:
To compare: this is a fairly typical 6th grade problem in top #math nations' schools, but it would be done in a high school geometry class in the US, if it's covered at all.
So this is supposed to be a sixth grade problem? I had to use the sine function twice -- one in the formula A = (1/2)ab sin C, and the other to find the y-coordinate of C. And look at the mistake I made earlier -- a sign error. Sixth graders usually aren't proficient with signed numbers yet, so if I made this mistake, surely sixth graders would make the same error.
At the link, a tweeter named "eylem gercek boss" had a different solution. This tweeter added in two extra segments --
Region A: Triangle ADO
Region B: Triangle DFO (where F is the intersection of
Region C: Triangle BFO
Region D: Triangle CDF
Region E: "Triangle" BCF (actually, BC is an arc, not a segment, but you know what I mean)
Now eylem -- uh, let's just say "boss" -- writes:
A = B + D = B + C
I see what the boss is thinking here -- the median of a triangle divides the triangle into two smaller triangles of equal area. This is because the two smaller triangles each have the same base (half that of the large triangle) and height (same as that of the large triangle).
D = C
OK, so the boss subtracted B from both sides.
==> S = D + E = C + E.
OK, so S is given as the sum of D and E, and we substitute to get S = C + E. But notice that C + E is just the sector of the circle! This is easy to find -- it's just like the Pappas problem I wrote about in today's post:
Area of the shaded sector = 45/360 * (area of Circle O)
= 1/8 * 36pi
So now I see what's going on here -- S is indeed an oddly shaped region, but the boss here makes a clever transformation to show that S has the same area as some other much simpler region, in this case a sector. To those who spot the clever transformation, the problem is easy, but to those who don't, the problem is tricky. I never saw the clever transformation and so I struggled -- and I'm a trained math teacher. How much more unlikely then will it be for sixth graders to spot the clever move to make -- or even the sixth grade math teachers?
At the very least, a sixth grade teacher who wants his or her students to solve these problems would have at least pointed out that a median divides a triangle into two equal-area triangles. Not every teacher or textbook emphasizes this.
Also, when we push problems like this down into sixth grade, we expect complaints such as "Why do we have to learn this?" and "When will we use this in real life?" to appear. Also, here's another line I heard my sixth graders say last year -- "This problem takes too long." To many students, any problem that takes more than 30 seconds to solve isn't worth solving. Math isn't something to become proficient at -- it's a barrier to get past in order to reach more enjoyable, non-mathematical activities.
I could go on with CCSSIMath's Twitter page forever, but there are other things I wish to discuss in today's post.
The Tracking Debate
In the most recent Barry Garelick post, the traditionalist teacher mentioned tracking. And where there's tracking, there's race. I wrote a little about race in that post, but as you already know, I save long, extensive discussions about tracking and race for the bottom of vacation posts, like this one.
By the way, recall that the Path Plan is also a mild form of tracking. Students weren't blindly assigned to tracks based on their grade level -- instead, students above grade level might be assigned to a higher path, while those below grade level would be on a lower path. So advanced fourth graders were placed in Preparatory Path while below basic fifth graders were placed in Transition Path. And likewise, advanced second graders were placed in Transition Path, while below basic third graders were placed in Primary Path. (I'm not sure whether the K-1 boundary was crossed this way -- at the time there was only half-day kindergarten, so crossing this line was awkward. Indeed, the K classes were almost always called "kindergarten" -- the name "Early Learning Path" was seldom used.)
Oh, and notice that students are assigned to paths based on their reading ability -- since after all, English (and history) is taught during homeroom. The students' math ability is taken into account when assigning the students to tracks. I know that fifth graders in the Preparatory Path might be assigned to sixth grade math, and so it's possible for fourth graders to be assigned to fifth grade math, even if their reading level keeps them in the Transition Path. I set up the bell schedule earlier so that it's easy to move to a different path during math time.
As for race, notice that even tweeters like CCSSIMath mention race from time to time:
Extensive Data Shows Punishing Reach of Racism for Black Boys https://nyti.ms/2GFoJ7L via @UpshotNYT
I'll only quote a small part of this New York Times article that CCSSIMath links to here:
The disparities that remain also can’t be explained by differences in cognitive ability, an argument made by people who cite racial gaps in test scores that appear for both black boys and girls. If such inherent differences existed by race, “you’ve got to explain to me why these putative ability differences aren’t handicapping women,” said David Grusky, a Stanford sociologist who has reviewed the research.
A more likely possibility, the authors suggest, is that test scores don’t accurately measure the abilities of black children in the first place.
I quote this part of the article because it mentions test scores -- which is the most relevant part for the tracking debate. It explains why tracking disappeared, and why many people are wary of trying to bring it back -- even without tracking, disparities in income exist. How much worse, then, would the gaps be if we tried to bring back tracking?
If it wasn't for race, tracking might still exist today. We might have something similar to a nationwide Path Plan -- and even the Common Core Standards could be set up to assign standards to paths, not grade levels.
But as long as tracking has a racial problem, it'll never be completely brought back.
Here's another analogy -- a thought experiment, mind you -- to discuss the tracking debate further.
Imagine that there is a magic red button. Here's how it works -- as soon as the magic red button is pressed, the income of every black person immediately doubles. Actually, let me make this precise -- I already know how economists might say, "Sure, income doubles, but prices double as well, so no one is better off." If the red button is pressed, then the purchasing power of every black person doubles -- anything a black person can buy now, he or she can buy two of now.
So far, this sounds good. You might think that if we repeated the NY Times graphs in a world where the red button exists, then they'd be more favorable for blacks. More blue squares (representing blacks) would land on the paths to higher income and fewer would be on the paths to lower income.
There's just one problem -- I didn't say what effect the red button has on whites yet. And so let me do so now -- if the red button is pressed, then the income of every white person triples. Again, here I mean that the purchasing power of every white person triples -- anything a white person can buy now, he or she can buy three of now.
And as for Asians, Hispanics, Native Americans, and mixed-race individuals, let's say that the red button increasing the purchasing power by a factor between 2 and 3. Lighter-skinned individuals have a multiplier closer to three, while darker-skinned individuals have a multiplier closer to two. So if the red button is pressed, everyone will have greater purchasing power.
So what effect would the red button have on the NY Times graphs? Notice that the five income paths on the graphs are based on percentiles, so the highest path is the richest 20%. The red button triples whites' income while only doubling blacks' income. So the red button increases gaps -- there would be even more yellow squares on the highest path and blue squares on the lowest path.
But someone who is pro-red button could easily argue the following -- each black person has twice the purchasing power he or she would have without the red button. And so even with more blacks on the lowest path, they are better off with the red button than without it. And so if you oppose the red button, you are actually anti-black.
Skeptics can counter that relative wealth matters, and so blacks aren't truly better off in the world of the red button even if they have twice the purchasing power. Again, we can declare it's built into the magic of the red button that blacks are automatically twice as well-off if the button is pressed. In the end, it doesn't really matter because this is just a thought experiment -- there is no magic red button.
So far, I've written much about race and income, but didn't I say this analogy would have something to do with tracking?
Well, here's the kicker. According to tracking advocates, tracking is the magic red button. They concede that if tracking were brought back, more whites would find themselves on the higher track and blacks would wind up on the lower track. Yet somehow, everyone, regardless of race, would be better off with tracking.
Years ago, I recall quoting a traditionalist who defended tracking. This is what he wrote -- an Asian girl is placed on the highest track. She can take higher-level courses and learn the material at a faster rate than if her teachers had to slow down for the other students. When she grows up, she broader knowledge ultimately allows her to discover a cure for cancer. She's able to sell the medicine and become rich -- so her purchasing power triples, just as I claimed the red button would do.
In the same analogy, suppose a black boy is placed on the lowest track. From this track, he's qualified only for low-paying, blue-collar jobs. One day while working hard on the job, he catches cancer -- exactly the cancer that the Asian woman has found a cure for. The money he saves on hospital bills can be spent on other things -- so his purchasing power doubles, just as I claimed the button does.
And so everyone is better off with the red button, even though the gap between races increases -- and even though the NY Times graph looks worse. Ironically, it's another NY Times article that argues that tracking benefits everyone:
Notice that the study cited in the article takes place in Kenya, where presumably most of the students involved are black.
The reason that I can't wholeheartedly embrace tracking is that I'm skeptical that it will benefit everyone of all races as much as its advocates claim it will. I'm all for pressing the red button and helping everyone, but I'm not completely sold that tracking is the red button. For example, the black man placed on the lower track might not be able to get any job at all. He might not be able to afford the cure created by the Asian woman at all. And so he dies of the curable cancer. In the end, there's no point developing a cure for cancer if those stricken with cancer can't afford the cure.
Many of the commenters in the old 2014 NY Times article above seem to agree with tracking. Let's look at some of these comments:
I don't understand why everything is designed to force faux equality. When did it become wrong to recognize and encourage people for their talents? There is no racial or income requirement to get into one of these programs. Some people are just smarter than others. It's equally unfair to take someone who, can simply "get" 3rd, 4th, 5th, 6th, 7th, 8th or whatever math by looking at it to sit through endless lectures on how a fraction works.
It's obvious that placement on higher tracks benefit those who are so placed. What I don't believe yet is that those placed on lower tracks benefit. Well, here's another commenter:
For years I taught special needs students at their ability, they flourished and grew.
Then the "equality" fascists required that my students be distributed in the mainstream classrooms and I watched in horror as they failed. They retreated into themselves. They were being punished because they were not "normal."
OK, then, so let's imagine what would have happened if the equality "fascists" hadn't intervened. The students would continue to flourish and grow, and feel comfortable in their classes -- that is, until they turn 18. Then when it's time to be hired for a job, the lower-track students still don't have as many skills as those on higher tracks. And when they get their first paychecks, they see a lower number written after the dollar sign than those placed on the higher tracks. These students have become just another blue square falling to the lowest income level.
And also, consider what this placement looks like from the parents' perspective. When they see their children being placed on a lower track, they tend to object to the placement. They might wonder, is their child being placed low due to their race, as the NY Times graphs seem to suggest? Is the school condemning their child to a lifetime of low earnings? I suspect that a major reason that tracking is parental complaining.
Canis Scot's students feel that they are being punished in the gen ed class because they aren't "normal" -- but if tracking remained, they would have felt punished years later with a lower paycheck because they aren't "normal." Am I a "fascist" just because I want the students to have a job that pays well enough for them to have a roof over their head and food on their table?
Later on, another commenter writes that gifted students should be placed on higher tracks because we "need kids who master higher mathematics and design airplanes." But then the students placed on lower tracks are headed for low-paying jobs, where the pay is so low that they can't afford going on vacation or buying plane tickets. So who's going to ride on all those airplanes being built by the higher-track students?
When I was in second grade, my teacher allowed me to study Pre-Algebra independently. I also spent much of my early years reading above grade level. And so I know firsthand the benefits of helping out the students who are above grade level. (On the other hand, I didn't really benefit from the Path Plan because I was already in sixth grade by the time it was fully implemented.)
But despite this, I can't fully support the tracking movement. I simply am afraid that the students -- and races -- who end up on the lowest tracks are set up for lower lifelong earnings.
What I want is a world where the graphs on the original NY Times article -- the one where yellow squares tend to float up to the highest level and blue squares tend to sink -- no longer exist. If it can be shown that tracking truly helps everyone regardless of race, I'd favor it. But until then, I must remain a skeptic.
Once again, I trust the parents who complain about their children being placed on the low track. Only in a different world -- one where I see parents, of all races, actually celebrating that their students are being placed on the low track to get the help they need to be successful -- will I ever accept that tracking works. Then we will have truly pressed the red button.