My Algebra II student is working out of a text published by McDougal-Littell. Here are the four chapters he had to study for the final. One thing about Algebra II texts is that almost every text divides its chapters nearly the same way:
Chapter 6: Polynomial Functions
Chapter 7: Radical Functions
Chapter 8: Exponential and Logarithmic Functions
Chapter 9: Rational Functions
I found it unusual that my student didn't reach Chapter 10 on conic sections -- and I told him that he should consider himself lucky, as conic sections is not an easy topic.
I've mentioned earlier that in my ideal Integrated Math III course, I would focus on three dimensions, which means that both volume and cubic polynomials would be covered. And Chapter 6 of this McDougal-Littell text shows why these two topics are related. One can probably imagine how one can justify the formula (a + b)^3 = a^3 + 3ba^2 + 3ab^2 + b^3 using volume -- we take a cube of side length a + b and see that it is made up of a cube of side length a, three boxes of dimensions a, a, b, three boxes of dimensions a, b, b, and a cube of side length b.
Now Chapter 6 of McDougal-Littell is the first time that I've ever seen anyone justify the factoring formula a^3 - b^3 = (a - b)(a^2 + ab + b^2) using volume. We begin by taking a cube of side length a and remove from one corner a cube of side length b. The remaining figure can be divided into three boxes -- the largest with dimensions a - b, a, a, the medium with dimensions a - b, a, b, and the smallest with dimensions a - b, b, b. We can rotate these so that a - b is the height of all of them, then the three bases become a square of side length a, a rectangle of dimensions a, b, and then a square of side length b. I found this interesting because the difference of cubes formula is one that Algebra II students usually have trouble remembering.
And of course, many of the other polynomial functions mentioned in Chapter 6 model the volumes of boxes, cones, and other 3D figures. So we can clearly see a connection between volume and cubic polynomial functions.
There is another obvious connection between 3D Geometry and Algebra II -- conic sections are the intersection of a cone and a plane. This also reminds me of another topic that sometimes appears in Geometry classes -- the idea of locus.
In Geometry, a locus is the set of all points satisfying a given condition. The most obvious example of a locus is a circle. Here's a common definition:
A circle is the set of points in a plane that are equidistant from a given point O.
We can rewrite this using the word "locus":
A circle is the locus of points in the plane that are equidistant from a given point O (the center).
As it turns out, the other conic sections can also be defined as loci as well:
A parabola is the locus of points in the plane that are equidistant from a given line L (the directrix) and a given point F (the focus) not on the line.
An ellipse is the locus of points in the plane the sum of whose distances from two fixed points F1 and F2 (the foci) is a given positive constant.
An hyperbola is the locus of points in the plane the difference of whose distances from two fixed points F1 and F2 (the foci) is a given positive constant.
It's because of this definition that the graph of y = x^4 isn't really a parabola. But to the typical high school student, the idea that the conic sections can be defined as loci involving one or more foci is so mysterious as to seem like hocus-pocus. (Incidentally, the four words hocus, pocus, focus, locus,
currently return several Google hits, the first few of which having nothing to do with math.) It's another reason why I'd like to connect Geometry and Algebra II -- so that students can make the connections more easily.
The other reason that this doesn't really feel like a "summer" post is the second school district where I sub had a Labor Day Start and thus still has two full weeks of school left. Yes, I mentioned in my earlier post that this district will have a Middle Start next year, but this year is still a Labor Day Start with a later finish. And I actually subbed today. I covered mostly sixth grade English, but during the conference period I was given an eighth grade math class.
This class is officially a Common Core Math 8 class. The regular teacher gave a review worksheet for the final -- which is unusual, a final in a middle school class. Recall that the eighth grader I'm tutoring in Geometry didn't take a final this week at all. This practice final contains 34 questions -- 14 of them being geometry questions and the other 20 being more algebraic in nature. Since this is a geometry blog, let me describe the 14 geometry questions in more detail:
1-2. Angle measures, parallel lines, vertical angles
3-5. Angle measures in triangles
6-7. Pythagorean Theorem
8-10. Volumes of cylinders, spheres, and cones
11-14. Transformations on a plane
As for the algebraic questions, the students really struggled on Question 15:
"Write an equation and solve. Five more than a twice a number is 17, find the number?"
Not a single student wrote down the correct equation. Several students wrote "2x = 17," which would have been correct save the "five more than" part. They also had trouble with Question 23:
"Write an equation in slope intercept form [for the line that] passes through (3, -5) and (-2, 4)."
It's always hard to write an equation for a line given just two points -- especially when the slope turns out to be a fraction. So it's understandable that they would struggle with this question -- but I did wish that more students had gotten Question 23.
I really wanted to get to these harder algebraic questions because I knew that they'd have trouble with them on the final. So I ended up skipping over Questions 11-14. Perhaps I should have asked the eighth graders how well they understood these transformation questions, since transformations are, as we already know, the cornerstone of Common Core Geometry. It might have been interesting to see how well eighth graders actually understand transformations -- if they'd had trouble with them, one way to fix Common Core might have been to delay transformations if not drop them altogether. And that takes us to the main topic of this post -- "How to Fix Common Core."
This class is officially a Common Core Math 8 class. The regular teacher gave a review worksheet for the final -- which is unusual, a final in a middle school class. Recall that the eighth grader I'm tutoring in Geometry didn't take a final this week at all. This practice final contains 34 questions -- 14 of them being geometry questions and the other 20 being more algebraic in nature. Since this is a geometry blog, let me describe the 14 geometry questions in more detail:
1-2. Angle measures, parallel lines, vertical angles
3-5. Angle measures in triangles
6-7. Pythagorean Theorem
8-10. Volumes of cylinders, spheres, and cones
11-14. Transformations on a plane
As for the algebraic questions, the students really struggled on Question 15:
"Write an equation and solve. Five more than a twice a number is 17, find the number?"
Not a single student wrote down the correct equation. Several students wrote "2x = 17," which would have been correct save the "five more than" part. They also had trouble with Question 23:
"Write an equation in slope intercept form [for the line that] passes through (3, -5) and (-2, 4)."
It's always hard to write an equation for a line given just two points -- especially when the slope turns out to be a fraction. So it's understandable that they would struggle with this question -- but I did wish that more students had gotten Question 23.
I really wanted to get to these harder algebraic questions because I knew that they'd have trouble with them on the final. So I ended up skipping over Questions 11-14. Perhaps I should have asked the eighth graders how well they understood these transformation questions, since transformations are, as we already know, the cornerstone of Common Core Geometry. It might have been interesting to see how well eighth graders actually understand transformations -- if they'd had trouble with them, one way to fix Common Core might have been to delay transformations if not drop them altogether. And that takes us to the main topic of this post -- "How to Fix Common Core."
I've stated how the biggest complaint among traditionalists regarding the high school math standards is that they don't lead easily to Calculus in high school -- and not taking Calculus in high school is said to close the door leading to a STEM career. I've discussed how the poster SteveH -- who writes comments on several education blogs -- gave a simple plan to ensure that the door to STEM remains open for students. One surveys students who passed either AP Calculus test with a score of 3 or better and found out what math they took leading up to the AP. SteveH expects most to reply that they took Algebra I in eighth grade, and also in elementary school, if they attended schools with progressive curricula (such as the U of Chicago elementary curriculum), they supplemented this with a traditionalist math education from either their parents or tutors.
Before Common Core, the old California State Standards encouraged Algebra I in 8th grade. Now the state had the possibility of supplementing the Common Core Standards with some of its own, and naturally keeping Algebra I standards in eighth grade was considered. The roadblock was that 8th graders in Algebra I still have to take the SBAC 8th grade math test -- this test, while including the first semester of Algebra I, also contains some geometry standards. Just think about what the final for such a class would look like: it would start out just like the final the eighth graders today were studying for, with geometry questions like 1-14, then first semester Algebra I questions like 15-34, and then how many second semester Algebra I questions would there be -- perhaps these would take us from 35 to 50? I posted a 50-question final for my Geometry class, but this is intended for high school students, not eighth graders.
Then again, Singapore's New Elementary Math 2 for eighth graders seems to contain as much algebra as an Algebra I class, yet still contains about as much geometry as Common Core 8 -- with time to spare for statistics! If California had gone along with the Algebra I/Common Core 8 plan, the class would have looked a lot like New Elementary Math 2. It sounds difficult, yet it must be doable since Singaporean eighth graders can pass it. Other integrated courses such as Saxon Algebra 1 (3rd edition) may have also been suitable for eighth graders in California's proposed class.
So 8th graders would have to take a full Algebra I course plus some geometry in order to do well on the SBAC -- had it not been for the SBAC, California would still have kept its Algebra I standards in eighth grade. So this leads to my first suggestion regarding how to fix Common Core:
Then again, Singapore's New Elementary Math 2 for eighth graders seems to contain as much algebra as an Algebra I class, yet still contains about as much geometry as Common Core 8 -- with time to spare for statistics! If California had gone along with the Algebra I/Common Core 8 plan, the class would have looked a lot like New Elementary Math 2. It sounds difficult, yet it must be doable since Singaporean eighth graders can pass it. Other integrated courses such as Saxon Algebra 1 (3rd edition) may have also been suitable for eighth graders in California's proposed class.
So 8th graders would have to take a full Algebra I course plus some geometry in order to do well on the SBAC -- had it not been for the SBAC, California would still have kept its Algebra I standards in eighth grade. So this leads to my first suggestion regarding how to fix Common Core:
Suggestion #1: Eighth graders enrolled in Algebra I should not be required to take any Common Core exams.
After all, if 9th and 10th graders in Algebra I don't take the SBAC, why should 8th graders? Once 8th graders in Algebra I no longer have to take the SBAC, the reason that California didn't keep 8th grade Algebra I instantly disappears.
But what if an eighth grader were to take Integrated Math I instead of Algebra I. Notice that both Common Core 8 and Integrated Math I contain both algebra and geometry -- indeed, in some ways one can't distinguish between Common Core 8 and Integrated Math I. The final that the eighth graders are taking looks like something I'd give in my own ideal Integrated Math I class, except that I'd save the volume questions for Integrated Math III, as I mentioned above. (Oh, and maybe I'd make the equation of line questions slightly easier.) This raises the possibility of a student taking Common Core 8 and then jumping directly to Integrated Math II as a freshman, which then opens the door to Calculus again.
What I want to do to fix Common Core is somehow to have tests that parents want their students to take rather than opt out. Some states pre-Common Core actually had the ACT as the class that all juniors were required to take. Nowadays, most ACT states have juniors take the PARCC or SBAC in addition to the ACT.
What I want to do to fix Common Core is somehow to have tests that parents want their students to take rather than opt out. Some states pre-Common Core actually had the ACT as the class that all juniors were required to take. Nowadays, most ACT states have juniors take the PARCC or SBAC in addition to the ACT.
Furthermore, notice that the ACT math tests go up to Algebra II. We can encourage the ACT states to get students through the door to Calculus by giving the ACT to sophomores rather than juniors.
Suggestion #2: Let the ACT be the NCLB/Common Core test required in high school -- and let it be given in the 10th grade.
Since states would obviously want their students to do well on the ACT, this will immediately boost Algebra I enrollment in 8th grade and Algebra II enrollment in 10th grade. And since the ACT is used for college admissions as well, parents are less likely to have their students opt out of it.
I also like the idea of having math tests in grades 7 and 10 but not 8 or 9. This would then allow schools and states to choose either a traditionalist or an integrated path without worrying about creating good tests for both paths.
Notice that the ACT Math exam is divided into three sections -- Pre-Algebra/Elementary Algebra, Intermediate Algebra/Coordinate Geometry, and Plane Geometry/Trigonometry. So a state using ACT on a traditionalist path may see that these line up approximately with the courses Algebra I, Algebra II, and Geometry.
But there are still a few problems with this idea. First of all, the states that use the ACT often give the test several months before the end of the year. I noticed that this year, many states gave the ACT on March 3rd, with the 17th as a make-up date. This would mean that all of the Intermediate Algebra material that appears on the ACT must be taught by the end of February. Perhaps the ACT could be given early junior year rather than late sophomore year -- late enough for sophomores to complete Algebra II but too early for states to consider pushing Algebra II back to junior year.
Another problem is that I want to decrease the number of tests students have to take. Before the Common Core, many high schools gave only the ACT, but nowadays, these schools give both the ACT and PARCC (or SBAC). Meanwhile, for lower grades, the ACT seeks to develop a new Common Core exam for Grades 3-10 to replace the PARCC and SBAC exams. This new test will be called ACT Aspire. I don't know enough about Aspire to make a judgment on it. In particular, I cannot easily tell whether the Grade 9-10 exams (which they call "Early High School") favor the traditionalist or integrated paths for math.
There's one more thing about the zeal to prepare students for Calculus. Traditionalists say that they way to prepare students for Calculus because they want to keep the door to a STEM career open. So denying students the opportunity to take Calculus would only close doors.
I say that forcing non-STEM students to remain on the path to Calculus can close doors. I can only imagine the poor sophomore who has no intention of majoring in STEM, forced to take a hard class like Algebra II only to end up getting a D or F in it. This low math grade ends up dropping the GPA below 2.0, and the student is unable to participate in the sports/activities that he/she really likes. Or the student may want to get a part-time job, but the parents won't allow it because they see the report card with the low math grade. They might not understand that their child likely already knows all the math he/she will ever need for a non-STEM career -- all they see is the D or F.
This is what I want -- since we have students take Calculus in order to keep doors open for STEM, this class and the math classes that prepare for it must not be forced upon students in a way that it would close doors for non-STEM.
Perhaps one way that a school can lead students to Calculus without penalizing non-STEM students is to enroll students in the class that leads to Calculus, but if, say, at the end of the first quarter (when the grade is not yet permanent), the student can switch to an easier math class without any penalty to the student's GPA. The sophomore failing Algebra II at the quarter might switch to another course such as Statistics or Financial Math, or even a pre-Algebra II class to prepare the student for success in Algebra II as a junior which, while perhaps unsuitable for STEM, at least can get the student into a good college in a non-STEM major. The only problem with this is that many schedule changes at the quarter-mark can wreak havoc on the school's master schedule and force teachers in all subjects, not just math, to learn the names of new students after the first quarter.
This should give us plenty to think about, though. My next post will be sometime next week.
I also like the idea of having math tests in grades 7 and 10 but not 8 or 9. This would then allow schools and states to choose either a traditionalist or an integrated path without worrying about creating good tests for both paths.
Notice that the ACT Math exam is divided into three sections -- Pre-Algebra/Elementary Algebra, Intermediate Algebra/Coordinate Geometry, and Plane Geometry/Trigonometry. So a state using ACT on a traditionalist path may see that these line up approximately with the courses Algebra I, Algebra II, and Geometry.
But there are still a few problems with this idea. First of all, the states that use the ACT often give the test several months before the end of the year. I noticed that this year, many states gave the ACT on March 3rd, with the 17th as a make-up date. This would mean that all of the Intermediate Algebra material that appears on the ACT must be taught by the end of February. Perhaps the ACT could be given early junior year rather than late sophomore year -- late enough for sophomores to complete Algebra II but too early for states to consider pushing Algebra II back to junior year.
Another problem is that I want to decrease the number of tests students have to take. Before the Common Core, many high schools gave only the ACT, but nowadays, these schools give both the ACT and PARCC (or SBAC). Meanwhile, for lower grades, the ACT seeks to develop a new Common Core exam for Grades 3-10 to replace the PARCC and SBAC exams. This new test will be called ACT Aspire. I don't know enough about Aspire to make a judgment on it. In particular, I cannot easily tell whether the Grade 9-10 exams (which they call "Early High School") favor the traditionalist or integrated paths for math.
There's one more thing about the zeal to prepare students for Calculus. Traditionalists say that they way to prepare students for Calculus because they want to keep the door to a STEM career open. So denying students the opportunity to take Calculus would only close doors.
I say that forcing non-STEM students to remain on the path to Calculus can close doors. I can only imagine the poor sophomore who has no intention of majoring in STEM, forced to take a hard class like Algebra II only to end up getting a D or F in it. This low math grade ends up dropping the GPA below 2.0, and the student is unable to participate in the sports/activities that he/she really likes. Or the student may want to get a part-time job, but the parents won't allow it because they see the report card with the low math grade. They might not understand that their child likely already knows all the math he/she will ever need for a non-STEM career -- all they see is the D or F.
This is what I want -- since we have students take Calculus in order to keep doors open for STEM, this class and the math classes that prepare for it must not be forced upon students in a way that it would close doors for non-STEM.
Perhaps one way that a school can lead students to Calculus without penalizing non-STEM students is to enroll students in the class that leads to Calculus, but if, say, at the end of the first quarter (when the grade is not yet permanent), the student can switch to an easier math class without any penalty to the student's GPA. The sophomore failing Algebra II at the quarter might switch to another course such as Statistics or Financial Math, or even a pre-Algebra II class to prepare the student for success in Algebra II as a junior which, while perhaps unsuitable for STEM, at least can get the student into a good college in a non-STEM major. The only problem with this is that many schedule changes at the quarter-mark can wreak havoc on the school's master schedule and force teachers in all subjects, not just math, to learn the names of new students after the first quarter.
This should give us plenty to think about, though. My next post will be sometime next week.
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