This past weekend, I finally receive my second dose of the Moderna coronavirus vaccine. All that remains is the two-week waiting period after the second dose to be considered fully inoculated.
This is finals week in all three districts whose calendars the blog is following. And so as usual, I'll post the second semester final today. And also as usual, I'll post a traditionalists topic on test day.
There is definitely one thing on the mind of most traditionalists right now -- the proposed new California math framework:
https://www.cde.ca.gov/ci/ma/cf/
Before you ask, I'll say it right now -- no, California is not getting rid of the Common Core. That alone is disappointing to traditionalists, who tend to oppose the Core.
But there are new problems with the proposal that traditionalists don't like. They fear that the framework will eliminate, or at least discourage, their favorite classes (that is, eighth grade Algebra I and senior AP Calculus) even more than the Common Core already does.
The FAQ under the new standards attempt to allay the traditionalists' fears:
https://www.cde.ca.gov/ci/ma/cf/mathfwfaqs.asp
Does the draft Mathematics Framework eliminate middle school mathematics acceleration programs?
No. The draft Mathematics Framework does not eliminate middle school mathematics acceleration programs (including programs that offer Integrated Math 1 or Algebra 1 courses to grade eight students). The draft Mathematics Framework emphasizes the importance of following the sequenced progression of topics laid out in the Common Core State Standards for Mathematics (CCSSM) and considers the latest research on the impact of skipping grades or undermining the sequences progression. Additionally, the CA CCSSM are significantly more rigorous than those from previous grade eight content standards. They address the foundations of algebra and geometry by including content that was previously part of the Algebra I course, including but not limited to a more in-depth study of linear relationships and equations, a more formal treatment of functions, and the exploration of irrational numbers.
But the traditionalists remain skeptical -- and besides, they never accepted that "Common Core Math 8 is more rigorous than pre-Core Algebra I" line from back when the Core was first proposed anyway.
Does the draft Mathematics Framework remove high school calculus?
No. The draft Mathematics Framework includes calculus in the possible high school pathways, while also presenting research that the “rush to calculus” without the depth of understanding is not helpful to students’ long-term mathematics preparation. Data shows that about one-half of all high school students who take calculus repeat the course in college or take pre-calculus in college.
This is a tricky one, and I'm of two minds on this one. On one hand, colleges are decreasingly awarding equivalency, or even credit, to those who pass the AP exams these days. My own alma mater, UCLA, gives Calc equivalency only to 5's. I recently mentioned that USC doesn't give equivalency at all. And I found out that last year, Harvard stopped giving credits for all AP exams, period.
But on the other hand, it's often mentioned that unless students take Calculus, they won't receive fat envelopes (that is, acceptance letters) from certain schools (Ivies or other competitive schools). That is, they won't grant credit if you take AP Calc, but you can't get in without it. You have to take Calc just to get the fat envelope and be admitted to the school, where you must take Calc again.
That's why this is such a hard one for me. I wouldn't want to take a class in high school just to retake it in college, but I'd rather take it twice than not at all because I didn't get into college. For this reason it's difficult to come up with a recommendation for senior year math, which in turn impacts math levels in lower high school and middle school.
What does the draft Mathematics Framework say about Tracking?
Many California schools and districts struggle with questions of how and when to offer different pathways through K–12 mathematics. These pathways of sequential required courses can create unintentional barriers that prevent all students from accessing courses necessary to achieve the math skills and competency needed to graduate ready for college and/or later careers in California’s growing science, technology, engineering, and mathematics (STEM) employment fields.
Here we go -- as soon as the word "tracking" appears, we know the word "race" is soon to follow. Even though the word "race" doesn't appear on this page:
As a result of the documented negative outcomes on certain student populations found in both tracking and ability grouping, the National Council of Teachers of Mathematics (NCTM) strongly advocates for creating a middle school mathematics that will “dismantle inequitable structures, including tracking teachers as well as the practice of ability grouping and tracking students into qualitatively different courses” (NCTM, 2020).
both proponents and opponents of the proposed framework understand "certain student populations" to be races (and possibly genders), so there's no reason to beat around the bush.
There are two words that have been thrown around a lot lately -- "equality" and "equity." Traditionalists consider themselves to favor "equality" in that all students must cross the same bridges in order to reach the same track. For example, if attainment to the higher track requires a high score on a placement exam, then all students with high enough scores should be placed on the high track, even if all of them are of the same race (or gender). To do anything else is to oppose "equality."
On the other hand, the authors of the California framework prefer "equity." Those who favor "equity" believe that all races (and genders) should be allowed to advance. If students are required to cross a bridge in order to advance, but not all races (and genders) are able to cross it, then the bridge itself must be bad -- to them, it's not a bridge but a barrier.
(As an aside, how would we translate "equality" and "equity" into Anglish? The Anglish Moot lists several words for "equality." The first is fairness -- but advocates of both equality and equity consider their own worldview to be fair and their opponents' to be unfair, so that's a nonstarter. Other words listed for "equality" are sameness and alikeness. The word "equity" doesn't appear there, but "equitable" does -- and equity advocates tend to use "equitable" as the adjectival form of "equity." The Anglish word listed for "equitable" is evenhanded, thus suggesting evenness or evenhandedness for "equity.")
As much as I'd like to avoid politics here, the easiest example of equality vs. equity is political -- and has been discussed all over the news lately. Should a photo ID be required to vote? Those on the equality side say yes, since all citizens are equally able to obtain such an ID. Those on the equity side say no, since whenever a requirement is implemented, black voter turnout drops. (Major League Baseball's All-Star Game has been moved due to disagreement over an ID requirement.) I don't pretend to know what the best ID policy is -- but if we can't even agree on voter ID, how can we expect to agree on educational standards, which are more complicated?
A few weeks ago, I linked to David Kristofferson's blog to discuss Physics classes. Well, he's also written about the proposed California framework:
https://eduissues.com/2021/05/10/the-battle-over-the-california-math-framework-revision/
He's written about acceleration in his own district, including the existence of a Multivariable Calculus) class (that is, a class beyond AP Calc BC) at his school. (I say, why should we teach Multivariable Calc in high school -- another class colleges will refuse to grant credit for?) Then again, he disagrees with certain forms of acceleration, such as jumping directly from Pre-Calc into BC. He also suggests offering a Calc class that's not AP-level. This sounds like a good idea if colleges won't grant AP credit anyway -- but then again, will colleges give fat envelopes to those who take non-AP Calc?
Some traditionalists worry that while acceleration will still be offered, it won't be until junior year in high school (which is when San Francisco currently offers it). I've suggested in past posts when a good time to introduce acceleration is -- eighth grade, at least in Integrated Math districts. That's because there's lots of overlap between Common Core Math 8 and Integrated Math I, so we can combine them into one accelerated class. A student on this path can take Integrated II as a freshman, III in sophomore year, Pre-Calc as a junior, and then AP Calc AB during senior year.
But of course, what if we wish to get a student to Calc BC? Considering that I took BC as a senior, I at least want to offer this option, but I'm not quite sure how. Even trickier is trying to accomplish this using the traditional Algebra I-Geometry-Algebra II path -- while Math 8 and Integrated I overlap a lot, Math 8 and Algebra I overlap much less. So a combined Math 8/Algebra I class may be too difficult for our eighth graders.
We can continue to work on this acceleration plan, but it's just as important to work on the equality vs. equity debate as well. I've mentioned several types of tracking plans on the blog, including the Path Plan, based on a mild form of tracking that my old elementary school offered. But even this sort of tracking is open to criticism that it's not equitable enough.
Here's a possible solution -- use placement exams (either statewide on in-house assessments) for placement onto a track or path, but then have a discussion with parents -- especially the parents of those placed on the lower track. The teacher/school can show the parents the placement scores and argue that their students won't be successful on the higher track -- but then if the parents, after hearing this, still believe that their children should be placed higher, then the teachers honor the parents' request.
This is a compromise between equality and equity. Equality is satisfied because higher scoring students (including whites/Asians, as well as males in math) are placed on the higher track, while equity is satisfied because parents who believe that the placement tests are biased against their children get their voices heard, and can place their children on the higher track.
If we apply this to the Path Plan, this doesn't mean that the parent of a second grader can simply request that the child be placed on the Preparatory Path, skipping Transition completely. (Recall that normative grade levels corresponding to paths is Primary 1-2, Transition 3-4, and Preparatory 5-6). Students must spend at least one year in each path. The right of a parent to overrule the placement tests mainly refers to third graders placed in Primary and fifth graders place in Transition, with some consideration also given to second and fourth graders who are on the borderline of making the next higher path but didn't quite make it on scores alone.
(Notice that Path Placement is mainly based on reading scores, with a separate math class. Since girls tend to score higher in reading than boys, there will be more girls moving up than boys. For example, in my Preparatory Path class there were three fourth graders -- and all of them were girls.)
Even though today's Rapoport problem isn't Geometry, I'll mention today's problem only because it contains an error that was caught on Twitter:
Let p(x) be a polynomial with integer coefficients. If p(0) = 17 and n is an integer such that p(n) = n^2, what is n?
To solve this problem, consider factoring x out of all terms except the constant term:
p(x) = xP(x) + k
where P is a polynomial of degree one less than p, and k is a constant. Actually, since p(0) = 17, we know that the constant is 17:
p(x) = xP(x) + 17
Now let's plug in x = n:
n^2 = nP(n) + 17
And since n^2 and nP(n) are both multiples of n, 17 must also be a multiple of n -- that is, n must be a factor of 17. That's all we can conclude about n -- it can be any of those four factors, 1, 17, -1, -17. In particular, we can't conclude that n = 1, even though today's date is the first.
In fact, it's easy to find a polynomial of degree 1 for each of the four possibilities of n:
n = 1 or n = -17: P(n) = -16n + 17
n = 17 or n = -1: P(n) = 16n + 17
This is what I wrote two years about today's test:
This is finals week at the school district whose schedule I'm following on the blog. And so today I am posting my version of the second semester final exam.
As usual, let me give my rationale for choosing these particular questions. When I wrote this final, I wanted it to serve not only as an in-classroom final, but what my vision of an ideal Common Core test, like PARCC or SBAC, should look like.
I've talked several times about the traditionalists who prefer that test questions focus more on content and less on labels. The questions at the end of each chapter of the U of Chicago are divided into four sections, Skills, Properties, Uses, and Representations (SPUR). So we conclude that the traditionalists prefer tests that are heavy on Skills (where most of the content is), and light on Properties (where most of the labels are).
I don't agree completely with the traditionalists here -- especially not in Geometry class. Geometry, after all, is all about proofs, and the reasons that appear in proofs are labels and properties. So if one isn't learning about labels and properties, then one isn't really doing Geometry.
A test that selects from the questions in the U of Chicago text would naturally have mostly Properties and Representation questions, and this is what I started to write. But one traditionalist argument for having more Skills than Properties questions is that with a Skills-based test, students who have the necessary Skills can take the test cold, without having to study for a long time, and still get an excellent grade. But a test that contains many labels and properties would require even the smartest students to spend time learning the particular names of the labels and properties. This is significant considering that one major argument against standardized tests like the Common Core tests is that they require so much time for test prep.
I spent lots of time on this blog preparing for the PARCC test -- not my final exam. Yet I didn't want my test to be just PARCC problems. And so I took questions from the U of Chicago text -- and since I didn't post test review for these question on the blog, they ended up being Skills questions, just as the traditionalists desire.
So here's how I wrote the test. This is a cumulative exam covering the whole text. But it was hard for me to find some good problems for Chapters 1 and 2, and I did just post a review sheet last month for some of the angle theorems from Chapter 3, so I began with Chapter 3. I decided to include numbered questions from the text that were multiples of five, starting with Question 5 and stopping at the end of the Skills section. For Chapter 3, there are six questions that would be included, Questions 5, 10, 15, 20, 25, and 30. But I had to drop Questions 10 and 30 because the particular skill for those questions involve drawing, which isn't easy to do on either a multiple-choice final or a computerized Common Core test.
Here is the chapter breakdown: for Chapter 3, I included four questions, but for Chapter 4, I included just one question. For Chapter 5, I included four questions, but for Chapter 6, I included just one question again. We notice that Chapters 4 and 6, where the transformations are taught, have very few Skills questions, since the main skill in both chapters is drawing the images, and I've already decided to drop all drawing questions. This is in accord with the traditionalist distaste for the Common Core transformations like reflections and translations. But Chapter 7 has just two included questions -- for questions on SSS, SAS, and ASA are also just Property questions.
Chapter 8 has the most included questions, with a whopping ten of them. Seven of these questions are from the Skills section. But after I wrote this test, I've having second thoughts about these. Many of these questions are not straightforward. For example, students are asked to find the perimeter of a square given its area or vice versa, as opposed to finding either of them given the side length. Now traditionalists like these types of problems because they require students to think deeply about the problem -- and I agree, but only up to a point. I have no problem with some of the questions requiring students to think outside the box, but when every question is this difficult, students will eventually become frustrated. But unfortunately, I ended up choosing the multiples of five, and these just happen to be the more difficult problems.
No matter what anyone else says, I want to include some problems from the Uses section, since I still want to demonstrate how math can be applied to the real world. So this means that I include questions 50, 55 and 60 from the Uses section of Chapter 8.
Chapter 9 is a tough chapter, since we covered Chapter 9 only briefly so we could get to Chapter 10. I included three questions (one from Uses) for Chapter 9. Chapter 10 is, of course, a big chapter, and so I included six questions (two from Uses) for this chapter.
Chapter 11, on coordinate geometry contains no Skills problems at all. Dr. David Joyce criticizes coordinate geometry, and so I include only two Uses questions from this chapter. From Chapter 12 I included five questions, with two of them from the Uses section.
Chapter 13 contains very few Skills or Uses questions -- it's a chapter focusing mainly on Properties, just like Chapter 2. So I included no questions from this chapter -- and recall that Chapter 13 will be broken up for my curriculum next year. From Chapter 14 I included six questions, with one of them from the Uses section. Since I covered Chapter 15 only briefly, I was only able to include one question from this chapter -- otherwise Chapter 15 would be a great Skills-based chapter.
This leaves five questions from the PARCC Practice test. I decided to continue the pattern and stick to multiples of five, so I included questions 10, 15, 20, 25, and 30. As we expect for PARCC questions, of course these are mostly Properties questions. These are already set up to be multiple choice -- still I had to set up the questions from the U of Chicago text so that they could be multiple choice as well.
Of course, I set up the questions to be multiple-choice for the purposes of the final. If this really were a computer-based exam, then I would have more free-response questions -- especially those requiring students to enter only a numerical answer.
Notice that Representations has been completely shut out of this test -- and Representations includes graphs and coordinate geometry. One problem with graphing questions on the computer is that they often require students to drag the graph to the correct location -- and this confuses them. I believe that there should be more graphing questions, but it's not clear to me how to make them so that more students can draw them on the computer easily. The only question that involves a graph is officially a Skills-based question from Chapter 8 -- students are to estimate the area of an irregular region.
I still like the idea of a computer-based test, though. Many people say that they oppose Common Core because it's "one size fits all." But the whole point of a computer-adaptive test like the SBAC is to avoid being "one size fits all" -- the same test for every student. I can easily imagine a computerized test asking a question such as to find the area of a square given its perimeter. A student who answers this incorrectly (say, by simply squaring the perimeter). can get an easier question such as to find the area of a square given its side length. Those who answer correctly, on the other hand, can get more difficult questions such as to find the area of a circle given its diameter or circumference, then move on to difficult volume questions, and so on.
Indeed, students who get many questions right could move on to some above-grade-level questions, if time allows. Unfortunately, I doubt that the actual SBAC does this. So SBAC fails to use the full power of having a computer-adaptive test. I wonder whether more traditionalists would be in favor of a computer-adaptive test like the SBAC if students could jump to above-grade-level questions.
Here are the answers to my posted final exam:
CAADA ACDDD ABADB ADCAC ABCBC BAADC ACACB ADBDA ADBCD ACCDC
Once again, I don't post a Form B for this exam.
No comments:
Post a Comment