Yesterday, I mentioned Darren Miller's post about Integrated Math. While I don't agree with everything that he wrote, there's one thing I must point out. He is also opposed to his district's new requirement that students take three years of math in order to graduate. He fears that the Integrated Math classes would be watered-down in order to get non-mathematically inclined students to pass them and graduate, making it even harder for any students to reach AP Calculus.
With this, I agree. I certainly don't like the trend of having many school districts -- including not only Miller's district, but also the LAUSD right here in Southern California -- requiring the equivalent of Algebra II just to graduate high school. Yes, Algebra II is required to enter college, but one shouldn't have to be college-bound just to graduate high school. According to California's master plan, the UC system is for the upper eighth, or 12.5%, of high school graduates, while the Cal State system is for the upper third, or 33%, of high school grads. So I agree with Miller, and disagree with the districts that require 100% of students to take what only 33% of them need.
But then this raises the question, which students should take the college-bound classes? This leads naturally to that highly controversial topic -- tracking.
I wrote that I wanted to devote an entire post to tracking, and this post is it. The idea of tracking sounds good in theory, but in practice, it often leads to segregation among demographic groups, and the leaders might show that they only care about the outcomes of those students with whom they share a demongraphic group.
So before stating a tracking plan, one should emphasize that one cares about the outcomes of all students, including those on higher and those on lower tracks, and regardless of the demographics such as income and -- the big one -- race. I don't want to have to make this about race, but any discussion of tracking inevitably leads to race. On this Martin Luther King Jr. weekend, let's maintain the MLK ideal and provide opportunities for all students, independent of race, to attain a comfortable middle-class life.
Earlier on this blog, I proposed part of a tracking plan -- the Path Plan -- based on a scheme that my own elementary school had for a few years. Before I return to discussing that plan, let me put my money where my mouth is and state what I just said that anyone should state before attempting to propose a tracking plan:
I care about the outcomes of all students, including those on higher and those on lower tracks, and regardless of the demographics such as income and race. I want to provide opportunities for all students, independent of race, to attain a comfortable middle-class life.
Now, let's review the plan. Students are not divided into grade levels based strictly on age, but instead into various paths. The normative span of each track is two years, and they correspond roughly to grades K, 1-2, 3-4, 5-6, 7-8, 9-10, and 11-12. Students are placed into paths based on their reading or ELA skills, but may take other subjects, most notably math, on a different path. Students have more than one teacher per day -- the exact number depends on the path. Here are the classes that I proposed to be taught by a teacher other than the homeroom teacher (defined here as the ELA teacher):
Path K: none
Path 1-2: Math
Path 3-4: Math, Elective
Path 5-6: Math, Elective, Science
Path 7-8: Math, Elective, Science, PE
Path 9-10: All except ELA
The closest known plan to this Path Plan is the Joplin Plan. The difference between the Joplin Plan and the Path Plan is that with Joplin, ELA and math are the two classes taught by another teacher for all grades, and so all other subjects are given in the homeroom.
Both the Joplin and Path Plans should be less prone to demographic profiling than a pure tracking plan where within each grade level, there is a high-track and a low-track class. With the Joplin and Path Plans, an above-average second grader and a below-average fifth grader may end up in the same 3-4 math class. The teacher can't simply give the below-average fifth grader an inferior education -- as a fifth grader on a low-track might receive -- because doing so would hurt the above-average second grader in the same class.
The tradeoff, of course, is that students of vastly different ages are in the same class. The above scenario places students three years apart in age in the same class. I consider three years to be the borderline between acceptable and unacceptable. Above a three-year difference, we run the risk of having the older students in the class beat up the younger kids -- especially since an older student in such a class is likely a below-average "bully" and the younger an above-average "nerd."
Some people who propose tracking plans say that, for the higher grades, all the below-average students can do in class is disrupt it, since they are hopelessly behind the other students. Instead, they say that such students should simply be expelled from school. If attending school were no longer mandatory for students beyond a certain age, school can be made more rigorous for those who remain in the school. One can have full Algebra I in eighth grade and AP Calculus in 12th, because the students who can't pass those classes unless watered-down would no longer even be in school. And instead, those children can get jobs and gain work experience (for in such a world, lack of a high school diploma would no longer be an impediment to getting a full-time job).
The problem with this plan is Charles Dickens. That is, his 19th-century novels describe a world in which businesses would exploit their youngest employees. Our language has a word, Dickensian, to describe the bleak working conditions these children face. And so I don't like the idea of children working 100 hours a week for very low pay in some factory at such a young age just because they don't do well in Algebra I.
I define the Dickens age to be the age at which a person can be an employee at a business for his or her own benefit, rather be exploited by a business as in a Dickens novel. Then school should be mandatory until a student reaches the Dickens age. For my blog, I set the Dickens age to be around 16 years old, the conclusion of sophomore year. I wouldn't mind setting it a little lower, to 15 years, but I'm basing this on the Singapore standards, where students attend school up until they finish their O-level exams, at the end of the equivalent of 10th grade.
Some traditionalists say that the high school diploma has become devalued, because it has been made easier to obtain in order to allow more to obtain it. To make a diploma worth something, they say, one has to accept the fact that some people aren['t qualified to get it -- but then these people would be unable to get jobs or a comfotable middle-class lifestyle.
So how easy should it be to obtain the 10th-grade "diploma" that I'm proposing here? Well, during presidential elections, pundits discuss how hard it is for an incumbent (or his party) to be reelected if the unemployment rate is high. Usually, an acceptable unemployment rate is around 5% -- that is, about 95% of the people have jobs. And so this is what I want for my 10th-grade diploma -- it should be attainable by around 95% of the population.
We can fit this into our proposed math standards as follows -- an exit exam can be devised to that a certain amount of math is required to earn this diploma. In particular, the level of math required should be such that 95% of the population can learn the material by the Dickens age. If we line this up with the proposed Singapore math standards, then I'd set it to be the equivalent of the Secondary One standards -- the equivalent of seventh-grade pre-algebra. The Secondary Two eighth grade standards contain too much Algebra I to expect 95% to master them by the Dickens age. (Under the old California standards, I might have said up to the first semester of eighth grade Algebra I, but Secondary Two contains many difficult topics right at the start of the year.)
Now let's think about how this test would be given. There are already two existing tests, the PARCC exam, given to grades 3-8 and at the end of every math course, and the Smarter Balanced or SBAC exam, given to grades 3-8 and 11. I live here in California, an SBAC state. If the equivalent of the SBAC were given, then passing the seventh grade test would be sufficient for graduation.
But recall that the Smarter Balanced exam is supposed to be computer-adaptive -- that is, questions are tailored to each student based on answers to earlier questions. But the way the SBAC is currently set up is not designed to take full advantage of its computer adaptability! There's actually a way to set up a computer-adaptive test so that it supports grade-level (or path-level, using the paths as described above) acceleration. Simply put, if a student answers all of, say, the third grade questions correctly, then the student should start answering fourth grade questions, and if all of these are answered, then the computer should move on to fifth grade questions, and so on.
Then the scores can be determined so, say 300 represents the attainment of an average third-grader, 400 the attainment of an average fourth-grader, 500 a fifth-grader, and so on. Actually, I'd would add 150 points to each of those levels, so that 450 is the average third-grader. This way, any score within fifty points of this -- in the four hundreds -- should be the range of the entering fourth graders. The score therefore can represent placement of a student into a particular grade level or path. Then the following year, the computer-adaptive test can begin by asking grade-level questions based on the student's score the previous year -- so a student scoring in the five hundreds the previous year should
begin receiving fifth grade questions.
A student scoring 800 or above should be placed into eighth-grade math, which, as we stated, contains much Algebra I. So this would end the debate over who should take Algebra I. Since seventh grade math is the highest required for the diploma, the score of 800 would be the score required for receiving our "diploma." (The score of 800 is already a magical number -- it represents a perfect score on a section of the SAT and a proficient score on our old California State Tests.)
Students who have earned an 800 shouldn't have to take an annual SBAC, but instead only the comprehensive high school SBAC exam. This should be given at the Dickens age -- so that means the end of sophomore, not junior, year. Therefore students should keep on taking higher and higher math courses until reaching the Dickens age. The type of student who reaches 800 early is the same type of student who should be taking higher and higher math. The test given at the Dickens age would be integrated, containing both algebra and geometry. But a school or district could still have traditionial pathway classes (Algebra I, Geometry, Algebra II), since such a student can still get the Algebra I classes even if he or she doesn't reach Geometry by the Dickens age.
This was a problem when California first adopted the Common Core. Under the Common Core, states were allowed to augment the standards by 15%. Naturally, California wanted to add Algebra I to the eighth grade math standards. The problem was that the eighth graders would have been still required to take the SBAC for Common Core Grade 8 -- so they would've been in a class that covers all of the Algebra I standards missing in Common Core Grade 8 (such as polynomials and factoring) plus all of the geometry standards from Common Core Grade 8 (such as dilations). The class would no longer have been reasonable for eighth graders to take. So California dropped the idea of adding Algebra I to the Common Core Standards.
The simple solution would have been to say that eighth graders in Algebra I don't have to take the SBAC, just as ninth graders in Algebra I don't have to take the SBAC. But unfortunately, No Child Left Behind requires a test every year from third to eighth grade. Furthermore, I can easy see schools putting their lowest eighth graders into Algebra I, so that they wouldn't have to take the SBAC and so wouldn't lower the school score. This is why the plan that I'm discussing describes how things would be if I ruled the world -- where I can change SBAC to support acceleration and ignore the NCLB requirements however I please.
We should also set the lowest age at which a student could take our computer-adaptive exam. The current SBAC begins at the third grade. Recall that for me, third grade marks the end of pure traditionalism, and pure traditionalism emphasizes the multiplication tables. And so the SBAC should first be given as soon as a student has memorized the times tables. I even argued earlier that if a student attempts to take the test but fails to answer the simple one-digit multiplication problems correctly, the test could immediately end and a failing score be given.
What would the lowest score on the exam be? I once read about someone who argued that even the SAT supports grade/score inflation, because the lowest score on each section is 200, not zero. The score of 200 was chosen because it represents three standard deviations (each s.d. being 100 points) below the theoretical mean (500 points), and only a statistically insignificant number of students would score beyond three s.d. above or below the mean -- hence the scale is 200 to 800. On my scale, a score of 0 would represent the lowest kindergartner. If one wishes, instead of having the test end after a student fails to multiply, several addition and subtraction questions can be given so that scores between 0 and 400 can now be distinguished.
Sometimes, I wonder whether I should actually fit the standards into the path system, so that a given set of standards corresponds to two grade levels. The Singapore Secondary Math Standards and the Common Core ELA Standards already combine grades 9-10, so one can simply work the way down and combine grades 7-8, 5-6, 3-4, and 1-2 into paths.
But this would be a bad idea, for several reasons. First of all, we definitely want to separate seventh and eighth grades and not combine them into a single set of standards, because we want to require seventh, but not eighth, grade math for our "diploma." Second, these paths were based on my own elementary school, which was a K-6 school. Many elementary schools are now K-5, so combining fifth and sixth grades into a single path would be awkward at such schools -- even moreso in my current district, which has a few upper elementary schools for grades 4-5 only.
A third reason is that it's in our human nature to oppose something that we would have originally supported if we feel that we're being forced to support it, especially if it's the federal government doing the forcing. Many traditionalists say that they oppose the Common Core Standards because they don't like its standards being divided into grades and forcing all students in the same grade -- that is, age -- to learn the same thing. If my proposed standards were divided into my proposed paths instead of, then it would separate these standards from being tied down to a specific age -- which is exactly what the traditionalists want. But then, I suspect, they would criticize my standards for forcing this new "path plan" on everyone -- especially if I started using my old school's unfamiliar names like "Primary Path" and "Transition Path." Human nature would cause them to defend the grade plan that they formerly opposed -- going right from Why should all ten-year-olds be forced into the fifth grade? to I know what the third grade is, but what exactly is this "Preparatory Path" supposed to mean?
So my plan is to keep the standards tied down to age, but take advantage of the SBAC's computer adaptibility to allow students to test at different ages.
Here are the answers to the test that I am posting:
1.-3. These are drawings. (Hopefully you can figure out what the dilations should look like!)
4. 22. (No, not 17.6. 88 is the length of NR, not OR!)
5. XY = 336. (No, not 21. The scale factor is 4, not 1/4. This was hard for me to draw accurately while still getting it to fit on the page.)
6. 5.4, to the nearest tenth. (Happy MLK Day!)
8. a. This is a drawing.
c. SU = 9, TU = 12.
9. Answers may vary. One possibility is t/m = h/a. (More complex answers look bad here in ASCII.)
11. Yes, by AA Similarity. (The angles of a triangle add up to 180 degrees.)
12. Yes, by SAS Similarity. (The two sides of length 4 don't correspond to each other.)
13. Hint: Use Corresponding Angles Consequence and AA Similarity.
14. Hint: Use Reflexive Angles Property and AA Similarity.
15. 3000 ft. (No, not 5250 ft. It's 7000 ft. from Euclid to Menelaus, not Euclid to Pythagoras.)
16. 9 in. (No, not 4 in. 6 in. is the shorter dimension, not the longer.)
17. 2.6 m, to the nearest tenth. (No, not 1.5 m. 2 m is the height, not the length.)
18. 10 m. (No, not 40 m. 20 m is the height, not the length.)
19. $3.60. (No, not $2.50. $3 is for five pounds, not six.)
20. 32 in. (No, not 24.5 in. 28 in. is the width, not the diagonal. I had to change this question because HD TV's didn't exist when the U of Chicago text was written. My own TV is a 32 in. model!)