On this Christmas Eve, I post my gift to you -- my vision of how mathematics should be taught in the highest grades, 8-12 -- the grade span that I want to teach, and the focus of this blog. In this grade span there are two major issues that are commonly debated:
Integrated vs. Traditional Pathway -- The traditional sequence of courses goes Algebra I, Geometry, Algebra II, Pre-calculus, and then Calculus. But many states, districts, and schools have been turning towards an integrated approach more common outside the United States. Even though Common Core supports both pathways, some states (such as Utah) and districts (such as my own) use the new standards as a reason to move to an integrated pathway. I've discussed this here on this blog before.
Eighth Grade Algebra and Twelfth Grade Calculus -- Some people criticize the Common Core Standards because they feel that it doesn't adequately prepare students for STEM -- that is, a college major and ultimately a career in Science, Technology, Engineering, and Mathematics. Notice that AP Calculus is not required for admission at most schools -- but one exception right here in Southern California is Harvey Mudd College:
"A year of calculus at the high-school level is an entrance requirement for HMC, so familiarity with limits, differentiation and integration is assumed in all mathematics courses."
Harvey Mudd is, technically speaking, a liberal arts college -- its one of the five undergraduate Claremont Colleges -- yet it focuses on STEM. All of its majors are STEM majors, and therefore, all of its students are required to take three semesters of math. Ironically, Harvey Mudd calls these classes the Common Core math classes, but they have nothing to do with the Common Core State Standards that we discuss on this blog. Indeed, the question we ask is, are students who take the high school Common Core math classes well-prepared to take the Mudd Common Core math classes?
Oh, and it goes without saying that many Mudders take the Putnam exam that I mentioned on the blog earlier this month:
I have the pleasure of having known a high school classmate of mine who attended Harvey Mudd, and he is now a math teacher at our high school.
Of course, most colleges are not Harvey Mudd. But the argument is that many selective colleges, while not literally requiring AP Calculus, have the class as a de facto requirement in that the students who are admitted and are successful in STEM majors have taken calculus in high school. And so this is why many are concerned that Common Core doesn't adequately prepare students for calculus.
Unlike the old California State Standards, where seventh grade math is the last pre-algebra class (so that students can take Algebra I in eighth grade), the Common Core Standards have eighth grade math as the last pre-algebra class, so that Algebra I is taken as a freshman. This means that students would only reach pre-calculus as seniors -- and so they couldn't be admitted into Harvey Mudd or any competitive STEM program.
Some schools since the adoption of Common Core have discouraged their students from taking any class higher than Common Core 8 in eighth grade. An argument is that the Common Core 8 class contains some topics previously taught in Algebra I, such as systems of equations:
Analyze and solve pairs of simultaneous linear equations.
while Algebra I contains some topics previously taught in Algebra II, such as exponential functions.
But recall that it's not really eighth grade Algebra I that matters to the STEM-minded traditionalists, but twelfth-grade calculus. Algebra I in 8th grade is significant only in that a student needs to be in Algebra I by eighth grade in order to reach calculus by senior year -- unless that student is willing to take two math classes in one year, or a math class in the summer. Pushing down a couple of Algebra I and II topics to Pre-Algebra and Algebra I, respectively, and then using that as a reason to delay Algebra I to ninth grade doesn't allow a student to reach calculus in high school. And so, to the traditionalists, this is a poor excuse to water down the math standards. To them, anything less than calculus in senior year constitutes watered-down standards.
And so here are the standards that I propose for secondary school. I decided to compromise between the traditionalist and progressive ways of thinking. For these standards, I took the traditionalist approach of reaching calculus by senior year, and combined it with the integrated pathway often favored by progressives. These are not mutually exclusive -- for example, a local magnet program, the California Academy of Math and Science, used to do this. They required incoming freshmen to have taken Algebra I in eighth grade, then offered three years of integrated math, and finally as seniors, the students took calculus -- not AP Calculus, but an actual college calculus course offered on the adjacent college campus, California State University, Dominguez Hills. (But I've heard that this integrated program, ironically, no longer exists.)
As I mentioned before, the integrated program is more common internationally -- and of course, other countries are known for offering algebra early. And so I look toward other countries for standards that may be acceptable. In particular, I've already discussed the Singapore standards, which are often held by traditionalists in high regard:
Four textbooks have been written for the Singapore standards -- New Elementary Mathematics.
For some reason, the third and fourth textbooks are no longer in print. The link above is for the second textbook, and there's also a link for the first textbook there. The first textbook, intended for seventh graders, reads like a traditional pre-algebra text. But the second textbook, for eighth graders, is an integrated course. The above link gives the 14 chapters as follows:
1. Indices (i.e., exponents)
2. Algebraic Manipulations (i.e., factoring)
3, Literal and Quadratic Equations
4. Word Problems
6. Simultaneous Linear Equations
8. Congruent and Similar Triangles
9. Mensuration (i.e., volumes of pyramids, cones, and spheres)
10. Pythagoras' Theorem and Trigonometry
11. Motion Geometry (i.e., reflections, rotations, and translations)
12. Statistics I
13. Statistics II
14. More Algebraic Manipulations
Just looking at the beginning of this course is daunting. Not only are eighth graders expected to know these Algebra I topics, but they are some of the more difficult topics in the book. In the Glencoe Algebra I text, with which I'm familiar from my student teaching days, exponents appear in Chapter 7, and factoring appears in Chapter 8. But by "algebraic manipulations," Singapore includes algebraic fractions, which don't appear in Glencoe until Chapter 11, and are often skipped by teachers who run out of time at the end of the year.
But believe it or not, I know of someone who teaches her Algebra I class exponents, polynomials, and factoring right at the beginning of the year. I've mentioned her before on the blog -- Sarah Hagan:
I mentioned earlier that Hagan had a problem with her students overgeneralizing -- thinking that two negatives make a positive even when adding, so -3 plus -5 is +8 -- and ignoring "the sage on the stage" when she told them that they were wrong. But, as it turns out, Hagan's solution to this problem is to move on to exponents and polynomials early in the school year! As she writes:
The same students who have been struggling with all of the above have been rocking our last few lessons on naming polynomials and multiplying polynomials. Why? My current theory is that multiplying polynomials is something they've never been exposed to before. So, they actually found it necessary to listen to my explanation...
Somehow, those same students who added -3 to -5 to get +8 were able to multiply (x - 3)(x - 5) to obtain x^2 - 8x + 15 -- with the correct sign on both the -8x and +15 terms!
Apparently, Hagan begins the year with the equivalent of Glencoe's Chapter 1 -- basically a review chapter (and that's when her students added -3 to -5 to get +8), then jumps right into the equivalent of Glencoe Chapters 7 and 8. Unwittingly, Hagan is following the Singapore text order in her class, at least up to Singapore Chapter 7 (after which the text moves to geometry). Also, I'm not quite sure about algebraic fractions -- rational expressions. Many teachers skip this lesson, and I see no evidence that Hagan teaches this to her Algebra I students at all. Then again, the Singapore chapters listed above don't distinguish between Chapters 2 and 14. It could be that the algebraic fractions don't actually appear until Chapter 14. There's no way for me to know for sure, since I don't have a copy of the Singapore text.
Hagan's class proves that students can be successful following the Singapore order. The key difference between Hagan and Singapore is that Hagan's class consisted of ninth graders, but the Singapore text listed above is for eighth graders. I'm not necessarily convinced that American eighth graders can be as successful as Hagan's freshmen following the Singapore order.
Sometimes I wonder whether pushing the Singapore texts back a year would be more realistic for American students. Not only would this put Singapore's Secondary Two class in our freshman year -- more in line with Hagan's Algebra I class -- but it would push Singapore's Secondary Four class back to our junior year. I found out from other sources the second half of Secondary Four consists mainly of review. Putting this review section in our Grade 11 would prepare our students for the SBAC -- which, as you recall, is for Grades 3-8 and 11 -- as well as for the SAT exam that many students take during the second semester of junior year.
But recall our goal -- to appease the traditionalists, we need calculus in senior year. As it turns out, Singaporeans who want calculus take the "Additional Mathematics" topics, with an extra class for pre-calculus in Secondary Three and an extra class for calculus in Secondary Four. But we're trying to avoid doubling up on classes -- otherwise, we'd just stick to Common Core, with its Algebra I freshman year and doubling up in math one year in order to reach calculus senior year.
Also, let's recall why Singapore needs its students to reach calculus by the end of Secondary Four -- the equivalent of our sophomore year. In Singapore, just as in Great Britain, students take many national exams at the end of the equivalent of our tenth grade. J.K. Rowling, in her famous Harry Potter series, has her wizard sit the O.W.L.s, or "Ordinary Wizarding Levels," when he is 15 years old (i.e., the same age as an American sophomore). Here Rowling was parodying the O Level exams that she had to take back when she herself was 15. We see the phrase "O Level Mathematics Syllabus" appear right at the top of the page at the first Singapore link above.
Here in the United States we don't have the equivalent of the O Level exams. Instead, many of our major tests, such as the SBAC and SAT, are taken junior year, not sophomore year. And so we don't need to fit the Singapore grade level sequence to the extent of preparing for national tests that we don't have at the end of sophomore year.
Then again, we notice that if we keep the Singapore texts for the years that they were intended for -- Grades 7-10 -- then the Additional Mathematics courses for pre-calculus and calculus fit right into the junior and senior years. There's no doubling up on math, and calculus is reached senior year, just as I promised the traditionalists.
This means that the Grade 11 SBAC (and PARCC) exams would focus on pre-calculus. Notice that there would be no reason to write Common Core (or "David Walker Core," or whatever I plan on calling my standards based on Singapore's) Standards for calculus, since these would already be covered (and therefore determined) by AP. There would be no Core exam for calculus -- but only the AP exam.
Recall that I recommended the Saxon texts for Grades 4-7. In many ways, the Saxon and Singapore curricula -- both highly recommended by traditionalists -- are interchangeable. Saxon, like Singapore, has an integrated pathway for high school. But notice that Saxon's texts are called Algebra 1/2 (i.e., one-half), Algebra 1, Algebra 2, and Advanced Mathematics. Geometry is actually integrated into the so-called Algebra texts. But Algebra 1/2 is pre-algebra, and Advanced Mathematics is essentially a pre-calculus course. So Saxon integrates four years into three. Therefore one could actually take Algebra 1/2 in eighth grade and still make it to Saxon's Calculus text by senior year. But the Saxon texts must move quickly in order to cover four years' worth of material in three.
But my concern is that so much Algebra I in eighth grade and pre-calculus for juniors is too advanced for many students. I wouldn't mind pushing back the classes so that so many students wouldn't have to fail such classes. But this is where the traditionalists argue that this is a zero-sum game -- if we drop pre-calculus and calculus, the slowest students may pass, but the brightest students are being held back. But if we stimulate the brightest students, the slowest students will fail.
Many traditionalists argue that it's the brightest students who have priority. Indeed, especially at the high school level, they sometimes say that students who can't keep up with a rigorous curriculum should be expelled as they would only disrupt the bright students who can keep up. Of course, simply expelling the disruptive students then opens up a can of worms, because then politics, class, and ultimately race suddenly become factors -- especially if the expelled students end up being disproportionately members of some demographic group.
One argument that the traditionalists come up with is the cure cancer argument -- or maybe I should say Ebola as this is more newsworthy. It goes like this -- let's say we have two students, a bright student who can keep up with a rigorous student, and one who can't. The latter student can only disrupt the gifted student, and so must be expelled. The gifted student, free from distractions and armed with a rigorous curriculum, is able to learn much advanced material and ultimately becomes a scientist who discovers a cure for cancer or Ebola.
But what about the student who was kicked out of school? Let's say this student, with an incomplete education, is lucky enough to avoid prison and find a minimum-wage job at -- well, you can probably figure out what companies come to mind here. But then this person ends up getting fired, let's say for taking too many sick days. For as it turns out, this person is diagnosed with Ebola -- that is, exactly the disease for which our scientist has found a cure!
And now the traditionalists will say "Aha!" The student who was expelled is actually better off, since after that expulsion, the gifted student was free to learn how to cure the disease with which the dropout is now infected. But I say otherwise. For this person now has no job, no money, and no access to the cure. That person now ends up dying of Ebola.
In other words, I have no problem with providing opportunities to the gifted students, provided that some opportunity is given to the slower students. Notice that I didn't say college -- I agree that trying to send everyone to college is a foolish idea. All I want is for all students to have the opportunity to get a job that would allow them to live a comfortable, middle-class life. If the gifted students discover cures for cancer, I want the slower students to be able have enough income and enough health care to access the cure should they end up catching that disease. If everyone has that opportunity, then things like politics and race would never even come up.
This leads, of course, to the tracking debate. I would support tracking only if it allows students on all tracks to attain, eventually, a comfortable, middle-class life. Simply expelling students, to me, doesn't constitute an acceptable form of tracking. But the tracking debate is very complex, and deserves a blog entry of its own.