Last year, I established a tradition of posting my first winter break post on Christmas Eve, and I'm doing so again this year. Last year, I posted an alternative to Common Core which I called "my gift to you," and I'm doing so again this year.
Last year, I proposed that we replace Common Core Math with Singapore Math -- a curriculum that is popular with many traditionalists. Singapore Math pushes more advanced math down into lower grades so that students can take Calculus as seniors.
I also wrote about why so many traditionalists are so obsessed with senior year Calculus. I linked to the websites of certain STEM colleges in this area. such as the California Institute of Technology (Caltech) and Harvey Mudd College. Let's link to those websites once again:
https://www.admissions.caltech.edu/content/how-apply-first-year-applicant
https://www.math.hmc.edu/program/placement/
I even pointed out the irony in that Harvey Mudd calls its first-year math curriculum "Common Core" yet the Common Core Standards, as written, don't adequately prepare students for the Harvey Mudd Common Core.
So we see our dilemma here. Unless we want schools like Caltech and Harvey Mudd to have zero applicants, we need to allow high schools to teach senior year Calculus and middle schools to teach eighth grade Algebra I, not Common Core 8. But we don't really want to force students who have no intention of becoming Mudders or Techers into eighth grade Algebra I and senior year Calculus.
Last year on Christmas Eve I gave everyone the gift of Singapore Math. But since then, I've had to take that gift right back. The Singapore Math course for the equivalent of eighth grade is a super advanced course -- students there learn the equivalent of a full Algebra I course, yet there's as much geometry in the course as Common Core 8. My home state of California nearly recommended a course similar to Singapore's eighth grade course, with enough geometry to pass the SBAC test based on Common Core 8, yet enough algebra to make it to Calculus by senior year. The class was deemed too difficult for eighth graders to pass, and so the proposal was dropped. Maybe Singaporeans can pass such a class, but it's not realistic to expect American eighth graders to pass the course.
It's apparent that we'd want some sort of tracking system so that those headed to Harvey Mudd or Caltech can take higher math than those headed for non-STEM careers. There are two ways to distinguish higher-track math from lower-track math -- depth and speed. The Presidential Birthday and Consistency plan that I proposed recently -- based on the Sidwell Friends curriculum -- emphasizes depth much more than speed. In particular, we see that freshmen on both the middle and high tracks take some form of Geometry (as the Integrated Math I course on the high track is mostly geometry), while seniors on both tracks take Calculus, of either the AP or non-AP variety. We see how speed distinguishes the low track from the higher two tracks -- students on the low track don't take Geometry until sophomore year and therefore don't take Calculus at all.
But many traditionalists want to see the highest track distinguished from the others by speed. They feel that students can get into Algebra and Calculus much faster than the Common Core and other curricula allow, if only teachers and administrators would let them.
Using speed-based tracking, everyone learns the same material to the same depth, but they learn them at much different speeds. I've hinted at speed-based tracking here on the blog before, but I didn't include it in the Presidential Birthday plan because speed-based tracking deviates from Sidwell's curriculum, and as soon as we deviate from Sidwell, charges of hypocrisy may appear (as the president would have his own children learn from a different curriculum from the one that he is imposing on the rest of the country).
So let me propose a new speed-based tracking plan. What should I call this new plan? Instead of Presidential Birthday plan, let's change "Birthday" to "Christmas" since this is actually my Christmas gift to you this year. I also want to change "Presidential" to "Gubernatorial." With the passage of the Every Student Succeeds Act, the states will have more power in selecting a curriculum. So we can think of this as what I'd propose if I were elected governor, rather than president.
Speaking of states, West Virginia is the latest to abolish Common Core. The proposed standards to replace Common Core in the Mountain State include many things traditionalists like -- memorizing the times tables by third grade is emphasized more strongly in West Virginia than in Common Core, and, for a non-math example, West Virginia has new standards for cursive writing.
And there are explicit standards in West Virginia for Calculus. Indeed, here is a link to the standards for high school math:
http://apps.sos.wv.gov/adlaw/csr/readfile.aspx?DocId=26209&Format=PDF
The West Virginia standards do contain some form of tracking. In particular, we see that both West Virginia and Sidwell Friends have an Algebra III course for students who barely pass Algebra II and so aren't ready for Pre-calculus. By contrast, Algebra III classes aren't common here in California.
Of course, since I'm here in California, I shouldn't discuss what's happening in other states. The Gubernatorial Christmas plan that I'm about to describe only applies to my home state of California.
Oh, and here's one last thing to discuss before I can describe the plan. I keep mentioning Presidential Consistency to ensure that students get as good an education as the president's own daughters. But there's no need for me to mention Gubernatorial Consistency -- our current governor, Jerry Brown, never had any children. Our lieutenant governor, Gavin Newsom, does have three children -- his oldest child was born in September 2009. So she just barely missed California's new kindergarten cutoff by a few weeks last year -- instead, she may have been in our state's Transitional Kindergarten, to begin true kindergarten this year. And our State Superintendent is Tom Torlakson -- he has two daughters, but considering his age, he's likely has school-age grandchildren. These are the state officials I'd consider if I wanted to incorporate Consistency Core into possible state standards.
And so let's begin the Gubernatorial Christmas standards. Our goal is to divide the students into tracks and then teach the same content at different speeds depending on the track. But before we can determine who's "above grade level" or "below grade level," we must define "grade level."
Now even though these Gubernatorial Christmas standards are not compatible with the Presidential Birthday standards, I'm tempted to define the middle track of the Presidential Birthday (i.e., Sidwell) standards as on grade level. Or -- since this is California after all -- we can define the old pre-Core California standards to be grade level. Notice that both middle-track Sidwell and pre-Core California consider Algebra I to be on grade level for eighth grade.
But -- no matter what the traditionalists may want -- I still don't like the idea of declaring any senior who can't do Calculus to be "below grade level" -- not even if we take it to be "non-AP Calculus" (which I still don't know exactly what that is -- my guess that it focuses on polynomials and e^x only is just that, a guess). For these standards, I prefer declaring Algebra I to be on grade level for freshmen, so seniors only have to make it to Precalculus to be considered on grade level. This isn't terrible, since there would be many opportunities for students to accelerate and thus make it to Calculus, Caltech, and Harvey Mudd.
Now the heart of these new standards is the testing. I have mentioned a scoring system similar to this proposal a few times here on the blog, but now I'm posting it again. The following is based on computer testing just like the PARCC or SBAC, but students don't simply receive scores on a scale of 1-4 as the SBAC provides. Instead, students receive a three- or four-digit score that's much more descriptive as to whether a student is below, on, or above grade level:
400: Ready to begin 4th grade math
500: Ready to begin 5th grade math
600: Ready to begin 6th grade math
700: Ready to begin 7th grade math
800: Ready to begin 8th grade math
900: Ready to begin Algebra I
1000: Ready to begin Geometry
1100: Ready to begin Algebra II
1200: Ready to begin Pre-calculus
1300: Ready to begin Calculus
Many people complain about computer-adaptive testing, but this scoring system actually justifies the use of computers in testing -- the order of the test questions can be tailor-made to each student based on their responses. Suddenly, the need to test on the computer is justified.
Now this isn't compatible with the Presidential Birthday plan, but we ask ourselves, is this compatible with the Every Student Succeeds Act? The new version of NCLB still requires states to test their students every year from third to eighth grades, as well as once in high school (typically during the junior year). In previous posts, I often proposed that eighth graders (and if there are any seventh graders) in Algebra I shouldn't have to take the Common Core test -- that immediately solves the problem of trying to shoehorn the geometry of Common Core 8 into the Algebra I class. Under this plan, eighth graders can still take a computerized test which would allow them to test either eighth grade math, Algebra I, or whatever level math they have learned.
The computer test should max out at 1300 or Calculus, since the students already have a Calculus test they can take -- the AP Calculus test.
Now the results of the Gubernatorial Christmas tests should be used for placement into math classes for the following year -- which implies that the scores should be given promptly. What I expect is that tracks will ultimately develop based on patterns in the various student scores. For example, the middle track would consist of students who gain about 100 points per year, as per the above chart.
There's likely to be a group of students who can't gain 100 points per year -- the lower track. Let's say that they end eighth grade with 900 points, as they're supposed to, but once they reach Algebra I, they struggle and can gain only 80 points per year rather than 100. So let's look at their scores:
Freshman year: 900-980
Sophomore year: 980-1060
Junior year: 1060-1140
Senior year: 1140-1220
So we see that these low-track students cover most of Algebra I as freshmen. Then as sophomores, they finish Algebra I and start Geometry. In junior year, the first semester is devoted to Geometry and the second semester to Algebra II. Finally, as seniors they finish Algebra II and begin just a little bit of Pre-calculus -- a course that's approximately equivalent to the Algebra III courses that we can find on the East Coast. So in a way, the low-track students are following a sort of Integrated Math plan, since they see both Algebra and Geometry during their sophomore and junior years. This justifies Integrated Math in a way that the Common Core doesn't.
It may be tempting to come up with an accelerated schedule for high-track students which would allow them to reach Calculus and beyond. But let's keep in mind that since high school math is harder than elementary math, we expect most acceleration to occur in the early years (but neither Common Core nor my Presidential Birthday plan make such acceleration easy).
This means that a student is more likely to reach Calculus by reaching Algebra I in eighth -- maybe seventh, perhaps even sixth -- grade and proceed one year at a time until they reach Calculus, than by starting Algebra I as a freshman and accelerating to Calculus. This plan allows students to accomplish this by scoring high enough in elementary school to reach Algebra I in middle school.
So there is less need for Integrated Math on the high tracks, but it's still possible. The eighth grader I tutored last year covered Algebra I the first quarter and then Geometry the rest, so this would be like a student who began the year near 980 and worked his way up to around 1060 by the end of the year.
In fact, I like to divide each course into ten units and have each unit correspond to 10 points on the scoring scale. So a score of 1060 means that a student has completed eight units of Geometry and is now ready to begin Unit 7. If we base this on the middle path Geometry course that I posted last week, notice that there were eleven units listed there. Well, the Introduction to Geometry unit is probably hard to test, so there are only ten testable units:
1. Triangle Congruence
2. Using Tools of Geometry and Triangle Properties
3. Parallel Lines
4. Polygon Properties and Circles
5. Area Formulas
6. Transformations, Tessellations, and Area
7. Volume Formulas
8. Pythagorean Theorem and Volume
9. Triangle Similarity
10. Similarity and Trigonometry
Under this plan, it's possible for a student to start Algebra I as early as sixth grade, which would allow a student to reach Calculus as a sophomore. That way, the student can take the state test as a freshman (thereby meeting the "once in high school" test that the Every Student Succeeds Act needs) and then the AP Calculus AB test the following year. Calculus as a ninth grader would be awkward since then there would be no ESSA test in high school -- but Calculus as a freshman works backward to Algebra I in fifth grade, which is itself awkward since it would be difficult for elementary schools to offer Algebra I.
Indeed, as long as students begin taking ESSA tests in third grade, this allows students to accelerate as early as fourth grade. It may be difficult to offer accelerated classes in elementary school unless the school adopts something like the path plan that I discussed earlier. And yes, this testing and tracking plan works for both ELA and math, but this blog always focuses on math. (Below third grade, tracking may be difficult as long as students aren't required to test.)
Now I've posted two mutually incompatible plans to replace Common Core -- Presidential Birthday and Gubernatorial Christmas. You may be wondering what's next -- the Mayoral New Year's plan, or should it be the King's St. Patrick's Day plan? I can keep writing plans I'd implement if I were the president, governor, pope, or emperor of the universe, but none of those will ever happen. But there is, in fact, a way for me to implement the Gubernatorial Christmas plan as a classroom teacher.
I remember last year subbing in a sixth grade class. The students were working on a worksheet on one-step equations. One of the equations was awkward: 5 - x = 2. The correct answer is x = 3, but many students were convinced that the answer was x = 7. The problem is that obtaining the correct answer x = 3 requires manipulation with signed integers that sixth graders haven't mastered yet -- they'd just barely seen negative integers if at all. and they certainly hadn't performed much arithmetic with negative numbers yet. In short, the equation 5 - x = 2 is inappropriate for sixth graders.
And so here's what I did -- when explaining the question, I just called it a "seventh grade equation," and then I explained how to solve it. This allows me to teach the students who might understand the signed integers how to solve it (so that they are accelerating somewhat towards seventh grade math, if only for that one question) without alienating those who don't get it (since the question was labeled as 7th grade, so it's OK if they didn't understand it for another year).
So far I haven't given such problems on the blog, but I may in the future. Notice that this is actually easier to implement in an Integrated Math class (since if students don't understand, say, a geometry question in Math I, they'll see geometry again in Math II), yet I based the Gubernatoral Christmas on the traditionalist Algebra I-Geometry-Algebra II sequence.
With any tracking plan, I always worry about those who would using tracking to enforce segregation, but this test -- just like the special Geometry test I mentioned last week -- allows students to challenge their placement simply by scoring higher on the test, since the computer-adaptive test automatically gives more difficult questions if the easier ones are answers correctly.
But still, tracking leads to controversy. In the news last week, there were some comments made by Justice Antonin Scalia on the topic of affirmative action. Just as it's impossible to discuss Common Core without getting into politics, it's impossible to discuss tracking without into demographics, especially race. I try to avoid politics on the blog unless the post is labeled "traditionalists," and I want to discuss racial demographics even less, unless necessary when talking about tracking. This is why I'm burying this discussion deep in a post in the middle of the holidays rather than dirtying up a Geometry-labeled post with this sort of discussion.
Here is a link to an article referring to Justice Scalia's comments:
http://www.thecollegefix.com/post/25585/
The fundamental argument for mismatch theory is that non-academic preferences in admissions to a higher education institution does not properly provide beneficial service to its intended receivers.
Since I want to discuss race as little as possible on the blog, let's go back to the old analogy that I posted back in October, about monkeys, fish, and trees. Let's say that there are two schools in our world -- one teaches how to climb trees, and the other teaches how to swim in a pond. Obviously, the monkey will do well at the tree-climbing school, while the fish will obviously fare much better at the pond-swimming school. To admit the fish to the tree-climbing school would clearly be a mismatch, hence the name of the theory, "mismatch theory."
So the fish will struggle at the tree-climbing school. In the pond-swimming school, of course, the fish will swim much better. It will be much happier when it is actually passing the swimming tests as opposed to failing the climbing tests, and it will be much happier when it is graduating from the swimming school and finding a job where its swimming skills will come in handy.
That is, the fish will be happy 29 out of 30 days of the month. The problem is that the one day of the month when the fish isn't happy is payday, when -- if you remember from October -- it finds out that the monkey is making ten times as much money as it is.
And that's the problem I have with mismatch theory. To me, all sorts of large demographic gaps are acceptable (tracking, college admissions, etc.) as long as they disappear by as soon as we reach the dollar sign on the paycheck. We admitted the fish to the climbing school because we wanted to give the fish a chance for a high-paying job. If we don't admit the fish, then it will complain that we never gave it a chance to make ten times the money.
But this problem has no easy solution. A school system can't change the fact that the job that the monkey is best suited for pays only one-tenth as much as the job the fish is best suited for.
Yet this is the problem with Justice Scalia's comments. He prefers that students who would do better in the college be the ones admitted, even if this leads to huge demographic gaps. Notice that this is ultimately related to tracking in the K-12 system, if members of certain demographics are placed on tracks that don't prepare them well for elite colleges and high-paying jobs. (Notice that Scalia refers to "classes that are too fast for them" -- the speed-based tracking that I'm discussing in today's post.)
To me, since the race gap has no easy solution, I'd focus on fixing the gender gap first. So let's discuss this gap in a little more detail.
When the Disney movie Beauty and the Beast first came out, I remember thinking how Gaston, the villain, tells the female lead Belle that it's not proper for a woman to read. The story after all takes place in the 18th or 19th century, and so Gaston's attitude is typical of his era.
Let's contrast this with my favorite TV show, The Simpsons, and recall how Lisa is the smart one while Bart is, let's say more street smart than book smart. Bart's attitude is the direct opposite of Gaston's -- to Bart, it's not proper for him, a male, to read. And indeed, Bart's attitude is more typical of his era -- in the 20th and 21st century, we are in a situation where in high schools, there are more females than males on tracks that lead to college.
Sure, another trend is that males tend to be better in math and science than in reading. So we see that in the STEM subjects, these two opposing forces cancel each other out -- the California SBAC results bear this out, as there is no significant gap between boys and girls in math. ELA, of course, is a different matter, the two trends reinforce each other, and so girls scored significantly better than boys on the ELA SBAC tests.
So how did we get from the Gaston-Belle trend in the 19th century to Bart-Lisa in the 21st? My theory is that males have always preferred action -- doing things. In the 19th century when most women were housewives, reading was considered doing things. But by the 21st century, as technology and sports have advanced, reading is not considered doing something, and so boys decide that they don't like to read. Notice in both the Gaston-Belle and Bart-Lisa cases, the males are the ones who decide to be the gender that reads or not.
So women outnumber men in colleges, but males may prefer vocational training and working with the hands -- and some of these jobs may pay just as well as those that require college. This is something to consider when considering the only gaps that matters to me -- the wage gaps.
Most of this post criticizes a comment made by Justice Scalia, a Republican-nominated judge. I will keep this post as balanced as possible and criticize a comment made by a Democrat.
Former First Lady and presidential candidate Hillary Clinton recently said that all schools that are below average should be closed down. Of course this is silly, since we expect about half of schools to be below average simply by chance. This is a great time to me to remind the readers that for either the Presidential Birthday or Gubernatorial Christmas plans, teachers are only judged on test scores to the extent that students are -- for example, if test scores count as 50% of a teacher's evaluation, then they should count as 50% of a student's grade.
So it goes without saying that if test scores are to play any role at all in whether a school remains open or shut down, then they should play a proportionate role in a student's grade. Closing down a whole school affects all the teachers at the school, whether effective or not. And so I must strongly disagree with Clinton's statement here.
This concludes my post. I wish the readers of this blog a very Merry Christmas!
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