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?
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 four 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 past years every time I write about Cavalieri, I keep dragging up her name and the old debates.
Instead let's look back at the California Science Test results that I mentioned in yesterday's post:
https://edsource.org/2020/california-science-test-results-2019/623501
OK, today I promised that I'll give a more detailed breakdown of the test scores for the students at my old charter school. Let's start with the overall scores:
Overall:
Fours -- 2
Threes -- 2
Twos -- 39
Ones -- 26
It's typical across the state for twos to be more common than ones. It's mainly in math -- especially secondary math -- where ones are more common than twos. On the other hand, the scores were quite dismal compared to the surrounding district, LAUSD. In the district, nearly 23% of test takers scored threes or fours, while here only about 6% did so.
At my school last year, fifth and eighth graders took the test. Those eighth graders were the kids I had as sixth graders, while I never taught those fifth graders. So let's isolate the eighth grade scores:
Eighth Graders:
Fours -- 0
Threes -- 2
Twos -- 25
Ones -- 9
We notice that there were more extreme scores in fifth grade than eighth grade. While fifth grade produced the only fours at our school, it also produced more ones than twos.
The gender breakdown is also important to me. Sometimes as a male teacher, I worry about whether I'm teaching the girls equally and fairly:
Eighth Grade Girls:
Threes -- 0
Twos -- 17
Ones -- 3
We notice now that there were more extreme scores among the guys than the girls. Both of the eighth grade threes were male, as were most of the eighth grade ones.
Now for the most important breakdown -- by race (which is why I saved this topic for today's traditionalists' post). My old charter school had two campuses, but these are never separated in the test score reports. The sister charter school was almost completely black, while my school had Hispanic as well as black students. Thus I use "Hispanic" as a proxy for "our charter rather than the sister charter." I can be sure that I taught most of these students as sixth graders:
Eighth Grade Hispanics:
Threes -- 1
Twos -- 12
Ones -- 1
The good news is that only a single student in this group scored a one. The bad news is that only a single student in this group scored a three.
And indeed, we compare the scores by race to the state average given in the Edsource article:
The scores also show a wide gap between black and Latino students and their white and Asian peers: Across all grades, 14 percent of black students and 19 percent of Latino students met or exceeded standards [...]
As bad as those percentages are, the scores for our eighth graders were even worse. We would have needed a trio of eighth graders in each race to score threes to reach those statewide percentages (14% black, 19% Latino) -- instead, only one student in each race did so.
The test scores also show the breakdown into earth, life, and physical sciences. For all groups, most students scored the strongest in earth science. The Hispanic eighth graders were the only subgroup whose physical science scores were higher than life science -- for everyone else, physical science scores were at the bottom.
So what does this mean as far as my (lack of) teaching science is concerned? On one hand, my failure to teach the sixth graders much science meant an extra burden on their teachers in seventh and eighth grade to make up what they missed. On the other hand, the scores for the entire charter were low, including the fifth graders (whom I had nothing to do with).
Thus I'm not sure how much impact I would have had if I'd taught science properly that year. Recall that on the day the scores were released, I subbed in a high school Chemistry class where the students were working on a group "test" (or assignment). I'm not quite sure whether giving my sixth graders group science assignments would have helped them score higher on the eighth grade state test.
As for why earth science produced the highest scores, it could simply be because earth science is the easiest of the three sciences (which is why the old California standards had earth in sixth, life in seventh, and physical in eighth). With the California/NGSS standards confusion, I can't be sure exactly what science they were taught in seventh and eighth grades.
On the other hand, it could be that NGSS emphasizes earth science a little more. One thing I noticed about last week's Chemistry class is its full name -- "Chemistry of the Living Earth." I believe that this name change for the science classes in that district is driven by Pearson -- the new consumable texts from Pearson divide each year into semesters, and the title of the second semester test is indeed "Chemistry of the Living Earth."
This thread has generated several comments. Two posters, Allison Nofzinger and Trish Williams, take opposite sides of the traditionalists' debate. Nofzinger recommends "direct instruction" for science, while Williams considers the NGSS to be a "welcome change." (I point out that it's awkward to define "direct instruction" for science where colleges expect incoming students to have laboratory experience, as opposed to math where "direct instruction" is more obvious.)
Notice that one of our main traditionalists, Bruce William Smith, posted in the comment thread here at Edsource about this very issue (the three separate disciplines):
Bruce William Smith:
While the Next Generation Science Standards have some value, I suggest Californians and other Americans look into the separate science tests in physics, chemistry, and biology of Cambridge Assessment International Education if they want to assure themselves that their children are achieving at levels that will leave them internationally competitive in the STEM fields to which so many families aspire, and yet so few qualify.
Notice that while I had Integrated Science as a young high school student, most high schools still have separate science classes for Physics, Chem, and Bio. Even the Chem class listed above incorporates some Earth Science yet is still separate from Physics and Bio. It's only in eighth grade and below where Integrated Science is becoming more prevalent.
Recall that Smith is the traditionalist who likes to push seventh grade Algebra I and junior year Calculus, which he says aligns with other countries. So it's not unexpected for him to compare American to international science as well.
Another interesting commenter here is Lisa Disbrow, who mentions "school choice" (which would include the charter school at which I used to work):
Lisa Disbrow:
I’m a 30 plus year veteran CA teacher. For 30 years, this California education tragedy has been used to empower and protect politicians while our schools have deteriorated.
It’s time to reject the unions, administrators and those who’ve played taxpayers and parents. It’s time to enact school choice and let parents decide how to flee failing schools and find educational options they approve of.
Meanwhile, let's get back to math and the traditionalists. Our main traditionalist, Barry Garelick, posted twice recently, including over the weekend. Here is his first post:
https://traditionalmath.wordpress.com/2020/02/04/awaiting-word-dept/
In Jo Boaler’s “Mathematical Mindsets”, the following paragraph appears in Chapter 2:
“In workshops with Carol Dweck I often hear her tell parents to communicate to their children that it is not impressive to get work correct, as that shows they were not learning. Carol suggests that if children come home saying they got all their questions right in class or on a test, parents should say: “Oh, I’m sorry; that means you were not given opportunities to learn anything.” This is a radical message , but we need to give students strong messages to override an idea they often get in school — that it is most important to get everything correct, and that correctness is a sign of intelligence. Both Carol and I try to reorient teachers so they value correct work less and mistakes more.”
Here are the comments, starting with our usual suspect:SteveH:
This is classic. Come out with a statement designed to cause a stir, but then claim misinterpretations later on. “Growth Mindset” is a process and not an end result, but education needs to be a scaffolded feedback loop that ensures a proper level of correct answers before moving on. That’s what happens in all high school AP/IB classes that align with the reality of colleges and the real world. Anything that dismisses or fuzzifies the connection of learning with mastery of skills and facts is fundamentally flawed. They need to be open and honest with parents about these ideas, and more importantly, offer students and parents the ability to opt out. Unfortunately, their turf vanity, arrogance, and need for control do not allow that to happen. They even fight against urban parents who are desperate to get their kids into alternate charter school choices. It’s truly astounding.
OK, so now both Lisa Disbrow and SteveH have invoked charter schools. By the way, I've noticed that several traditionalists have recent jumped into the charter debate. Last year, the catalysts for the debate were the big teachers' strikes. This year, it's apparently a new anti-charter book written by education historian Diane Ravitch. (For example, several pro-charter posts appeared at the Joanne Jacobs site the week after Ravitch's book was published.)
Of course, SteveH's suggestion that charter schools are a haven for traditionalists' preferred pedagogy and higher test scores appears laughable after I revealed my charter's scores above in this post. Then again, those were science grades, not math, but the point remains.
Then again, the traditionalists would counter that my charter used the anti-traditionalist Illinois State curriculum, so of course their test scores were low. And that charter school has closed down -- which is what traditionalists want to see. Ideally to them, all low-scoring schools will close and all that will be left will be the high-scoring schools -- which they contend will be the traditionalists' schools.
EB:
Boaler’s claim also doesn’t take account of the fact that often, by the time the learner is finished with a problem, s/he has already made a mistake, seen it, and corrected it. A homework assignment or test that seems to be without mistake is actually a bunch of mistakes overcome. Teachers are right to encourage students to ask themselves, all along, “am I using the effective way to approach this problem?” And if memory, or common sense, or estimating, or getting stuck tells the student that they’re on the wrong track, then switching approaches is the right thing to do.
This is tricky. I can't really speak for Boaler or Dweck (so I await a response from them as well), but I assume that all this discussion about mistake making is to prevent students who make mistakes from becoming too discouraged. In a traditionalist classroom, students who make too many mistakes will get tired of being called "wrong" and will be inclined to leave the p-sets blank.
And here is the Garelick's recent post:
“Traditionalists” (as they like to style themselves) are incapable of grasping the fact that high school math exists, and that most high school math teachers aren’t constructivists.
The above quote was from a blog written by a math teacher, and was a post about an article that Katharine Beals and I wrote which was published in online Atlantic in 2015. It caused a stir among those who don’t like what “traditionalists” have to say about teaching math.
What we “traditionalists” do notice about high school math is that many entering freshman do not know basic computation rules, and are dependent on calculators. In an eighth grade algebra class which I teach (and which is equivalent to 9th grade high school algebra), I had a student who had great difficulty multiplying two-digit numbers. He used a convoluted method that took up much time. Thirty years ago, most entering high school freshmen had the mastery of such elementary procedures.
Wow, so suddenly there's that name Katharine Beals again! And I can't believe it's already been five years since that infamous Atlantic article that launched the whole traditionalists' debate.
At the time of the Atlantic article (to which I devoted several blog posts), I considered Beals to be the main traditionalist. Now that she no longer posts about math, her co-author Garelick is now the most important traditionalist.
And here are the comments. Stephanie Sawyer isn't a frequent commenter here -- I believe this is the first comment of hers that I've quoted:
Stephanie Sawyer:
Also, I went and checked out the blogger referenced.
His issues with traditional math-teaching are valid, but he doesn’t understand that traditional math teachers are onto the whole “spit back with no understanding,” that we structure tests so that those who really understand will get the A, and the spit-back kid won’t. Maybe once he/she teaches an AP math class he/she will see this.
Responding to Sawyer is, of course, SteveH:
SteveH:
Exactly!
I’ve said before that when I gave math tests to my college algebra students, even slight variations from the homework tripped up those who only followed a process seen in a previous problem. No good teacher just changes the numbers for test questions. That’s just plain bad teaching. Homework sets are filled with different problem variations and my tests had different variations of those.
They could just say that their techniques work best for those who aren’t “math brains”, but they show no results. The College Board now offers its Pre-AP Algebra course in 9th grade (which is anything but inquiry-based) and one would expect to see these better understanding K-8 kids take off and accelerate to cover four years of math in three years. Do even THEY believe that’s possible? That would be tough for even properly prepared math brains.
My son’s piano teacher once held his hand low and told him that he was trying to have too much fun “down here.” Then he raised his hand high. However, if you work really, really hard, you will have much more fun “up here.” K-6 educators can find ways to make the down low hard skill work easier and perhaps more fun, but they rather flip the process around and assume that the hard work will be driven by engagement and conceptual understanding. They are in dreamland and show absolutely no proof.
SteveH brings up several ideas. First, he often mentions that his homework consists of "different problem variations," and this is how traditionalists assess for understanding. The only problem is that different problem variations on the HW serves to frustrate students, who then respond by leaving the homework blank.
Then he mentions that perhaps progressive methods work for those who aren't "math brains," but they show no "results," by which he means higher test scores (presumably on the AP Calculus exam, since he mentions AP). Of course, traditionalists' p-sets don't result in higher test scores for the students who leave them blank. In other words, "math brains" and "math zombies" refer to those who are willing to work on the traditionalists' p-sets, while my concern is with those who leave them blank.
Finally, SteveH compares learning math to learning the piano. I assume that "up here" refers to a high level of musical performance, the ability to play real songs on the piano. The problem is that students are more willing to "work really, really hard" (that is, make sacrifices) for high performances in music (or sports) than in math.
This is why I incorporated this into one of my new rules -- the need to make sacrifices in order to be successful in math. I expect that it will be difficult for me to convince students to do so -- they won't suddenly make sacrifices just because the traditionalists say so.
Then he mentions that perhaps progressive methods work for those who aren't "math brains," but they show no "results," by which he means higher test scores (presumably on the AP Calculus exam, since he mentions AP). Of course, traditionalists' p-sets don't result in higher test scores for the students who leave them blank. In other words, "math brains" and "math zombies" refer to those who are willing to work on the traditionalists' p-sets, while my concern is with those who leave them blank.
Finally, SteveH compares learning math to learning the piano. I assume that "up here" refers to a high level of musical performance, the ability to play real songs on the piano. The problem is that students are more willing to "work really, really hard" (that is, make sacrifices) for high performances in music (or sports) than in math.
This is why I incorporated this into one of my new rules -- the need to make sacrifices in order to be successful in math. I expect that it will be difficult for me to convince students to do so -- they won't suddenly make sacrifices just because the traditionalists say so.
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