Friday, June 10, 2016

Semester 2 Review and Next Year Preview (Day 180)

This is the 30th and last of my MTBoS30 posts. It's also the day when I reflect upon the semester that ended and begin looking ahead to next year.

Of course, the biggest thing on my mind now is my new charter middle school and the classes that I will be teaching in the fall. So here are some things I know so far about my first teaching assignment:

First, I know that I will be teaching all three middle school grades -- sixth, seventh, and eighth. Let's look at the Common Core standards for these three grades:

Grade 6-7 (same strands for each):
Ratios and Proportional Relationships
The Number System
Expressions & Equations
Statistics & Probability

Grade 8:
The Number System
Expressions & Equations
Statistics & Probability

On this blog, I will write only about the eighth grade class. So let's look at some of those eighth grade standards in more detail:

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27.

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Solve linear equations in one variable.

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Construct a function to model a linear relationship between two quantities. Determine the rate of change  and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Verify experimentally the properties of rotations, reflections, and translations.

Explain a proof of the Pythagorean Theorem and its converse.

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

The most important concept learned throughout middle school is fractions -- or to be more precise, it's ratios. Students learn arithmetic with fractions in sixth grade, the field of rational numbers in the seventh grade, and the application of ratios to similarity in the eighth grade. Indeed, we've seen one Common Core curriculum refer to middle school math as "A Story of Ratios."

This blog is named "Geometry, Common Core Style." Here Geometry refers to the high school class of that name, usually taken between Algebra I and II. When I begin writing about my eighth grade class, this will become a misnomer, since my focus will be on ratios, not geometry. Still, we see that eighth grade is the year when reflections, rotations, translations, and dilations are introduced, so there will still be some geometry. Besides, we found out yesterday that there's a lack of middle school blogs in the MTBoS, so I'm happy that I'll be able to contribute to the MTBoS from that perspective.

In the fall, I will be posting only three days a week. This fits my new school's block schedule. Indeed, this is another reason that I chose to participate in MTBoS30 this year despite writing only 21 posts in May due to skipping weekends -- next year, I'll write only about a dozen posts in May and be even further away from 30 (that is, if there even is an MTBoS30 challenge next May).

Now juggling three preps will certainly be a challenge for any teacher, especially a new teacher. I am already looking ahead to next year's school calendar to figure out how I will avoid having to grade tests for all three preps at the same time.

Last year, the first day of school at my new school was the third Tuesday in August, and so I'm anticipating that Tuesday, August 16th, will be the first day of school. Today is the last day of school at both my old and new schools, and so we can pencil in Friday, June 9th, 2017, as the last day of school next year. Because my new charter school is associated with the LAUSD, the school calendar is very similar to the district calendar.

When considering how to teach my nre eighth grade class, I especially want to look at the Statistics & Probability strand. I mentioned that last week was the final math class I subbed for this year, and that day the students had to take a quiz -- eighth graders on statistics, seventh graders on geometry. I wrote about how much trouble the eighth graders having with their stats quiz:

"I observe how much these students struggle on this quiz -- especially the question where they must identify and write an equation for a trend line. (Considering how much the eighth graders struggled with stats, maybe it's a good thing that the seventh graders didn't make it to their own stats chapter!)"

This is a good time to segue into our traditionalists-labeled topic of the week, because some traditionalists have written about teaching stats in middle and high school.

I've said before that the traditionalist on whom I want to focus the most now is "Bill." Yesterday Bill commented on an article about expanding AP to more students. Let me begin by linking to the article:

The author of this article is Jay Mathews, who first came up with the idea of judging high schools across the country by how many students are taking AP classes. The net result is that schools are encouraged to enroll as many students into AP classes as possible in order to boost their rankings.

Here is Bill's comment:

That’s all well and good, but many colleges today will not accept scores of 3 for credit on AP exams in STEM areas, and in many cases, colleges require a grade of 4 or 5 to get credit and start in say Calculus I, or General Chemistry 2 or General Physics 2, etc…
They’ll dumb it down until it becomes worthless as a measuring stick to get college credit, but at least the CLEP (College Level Examination Program) still exists which does award credit if the exams are passed (most colleges also won’t allow more than 32 credit hours to be earned by examination).

It's hard for me to tell what Bill means by "dumb it down" -- does he mean the AP exams? My question is, how are raw scores on the AP converted into the five bins 1-5? It could be that scores are set so that a certain number of test-takers earn a 1, 2, 3, 4, or 5 (norm-referenced).

If this is the case, then I can sort of see Bill's point here. Students who aren't ready for Calculus shouldn't be placed into AP Calc, but schools place there anyway in order to raise their Mathews ranking. As these students aren't ready for Calculus, they would naturally earn a 1 on the AP exam, but then there would be too many scores of 1. So the cut scores must be lowered -- and now it means that with lower scores needed for a 3, students who earn a 3 don't know as much material as the ones who earned a 3 before the recentering. So colleges must raise their requirements so that only 4's or 5's earn credit -- I can tell you firsthand that my own alma mater, UCLA, is particularly strict about who can earn Calculus equivalency. Only 5's are granted equivalency (though 4's on the BC test are granted AB equivalency). All other 3's and 4's are granted college credit but not equivalency. (I notice that the only 3's that are granted course equivalency are the foreign languages, while in the sciences, not even 5's are granted equivalency.)

Bill's solution, of course, is to admit only the strongest students into AP Calc and other classes. Then the tests can keep their standards high without there being too many 1's. (We've seen that the PARCC has a similar problem with cut scores, but for different reasons, since unlike the AP's, we expect all students to take the PARCC or other state Common Core exam.)

These days I want to focus mostly on the traditionalist Bill, but I can't help but go back to the other traditionalist whose blog I frequently visit -- Dr. Katharine Beals. Last weekend, Beals wrote a post about the need to take Calculus, after reading an article about how students don't need Calculus and ought to take classes like Linear Algebra (which I mentioned earlier this week) instead of Calculus:

Of course Beals, like most traditionalists, wants to see as many students in Calculus as possible:

Economics, engineering, and mathematical statistics, he [Beals's uncle -- dw] pointed out, also require knowledge of calculus. Indeed, my oldest son, who just received a bachelor's in mechanical engineering, reports that differential equations--and not linear algebra--were crucial to his course of study.

Plus, Uncle M points out, calculus is a lot more challenging than the kind of applied linear algebra favored by the authors. They aren't explicit about this, but there's linear algebra, and then there's linear algebra. There's an abstract variety, taught by algebraists, in which one operates in n-dimensional space with complex numbers and proves theorems about domains, ranges, kernels, vectors, and eigenvalues. And there's an applied variety, often taught outside mathematics departments, in which one works with matrices populated with actual numbers and applies these to real-life situations. The two varieties are really completely different subjects. Abstract linear algebra, in my experience, is really tough: riddled with mind-bending abstractions and impossible to visualize. In comparison, applied linear algebra is a piece of cake. It's also easy compared with calculus. And the same is true of applied statistics.

Aha -- there's "statistics," which takes us full circle to the problem I have with getting eighth graders to understand stats. Beals continues:

They're not explicit about this, but it's applied linear algebra and applied statistics that our data science CEO and computer science professor are advocating for.

Since calculus is harder, argues Uncle M, it makes sense to introduce it gradually, starting in high school. Not doing so burns bridges--bridges to math, physics (which still exists), mathematical statistics (as opposed to applied statistics), engineering, and economics. It's far less clear that continuing to mostly not teach linear algebra in high school burns any bridges.

But I'll tell you what does burn bridges, Beals. A sophomore must take Algebra II in order to make it to Calculus as a senior. A weak math student is placed into Algebra II, fails it, and then is kicked off the football team because his GPA is too low. So Calculus burns the bridge preventing this poor sophomore from playing his favorite sport.

A junior must take Pre-Calc in order to make it to Calculus as a senior. A weak math student is placed into Pre-Calc, fails it, and then her parents won't let her get a part-time job after school because she's failing a class. So Calculus burns the bridge preventing this junior from making money after school.

And of course, the senior in Calculus itself fails it -- let's say he earns a 2 on the exam. All that time he had to spend studying for Calculus along the way just to fail it could have been spent studying for and passing exams in non-STEM subjects -- plus the college rejects him for having too many D's and F's on his report card, mostly in math. So Calculus burns the bridge preventing this senior from getting into college and completing a non-STEM major.

So Beals writes that not teaching students Calculus burns bridges from them, but I counter that forcing them to take Calculus can also burn bridges for them.

Let's see what the commenters on Beals's blog have to say. First, here's Auntie Ann:

These arguments seem to me to be ways to mitigate the lack of mathematics rigor in K-12 curricula and mathematics achievement by our young adults. 

Here Auntie Ann implies that any classes other than Calculus "lack mathematical rigor."

They are coming up with excuses and work-arounds for the fact that too many college kids can't manage the same math today that their parents were required to take 20 years ago. Instead of dealing directly with the failure of K-12, they have to find a way to say that the crappy educational outcomes are actually not a bad thing, and might be a good thing.

Here she implies that failure to pass Calculus is a "crappy educational outcome." Let's look at the next commenter, "Niels Henrik Abel" (and no, I won't repeat who the real Abel was again):

Indeed. I have wondered about the apparent fascination that these people seem to have with statistics. It's often included in "quantitative literacy" type courses. Why, I don't know, because you really need a decent grasp of the calculus to understand the how and why of statistics. Otherwise, you just get a bunch of kids who learn to plug and chug with z-scores and t-tables - you know, that "rote learning" stuff they're always bellyaching about.

Three years ago, Bill wrote something similar:

Stats at the college or high school level requires a working
knowledge of basic algebra. I took a course (introductory)
in informatics which had stuff like mean/mode/variance/std deviation, and some probability.

So Bill writes that one can't pass stats without passing "basic algebra" first. We can't be sure what Bill means by "basic algebra" -- does that mean Algebra I, or even the part of Algebra I that is part of Common Core 8 (approximately the first semester of Algebra I)? But Niels Henrik Abel takes Bill's comment a step further -- one can't truly pass stats without understanding Calculus first!

Let's think back to the problem the eighth graders struggled with last week -- finding trend lines. It's possible to find trend lines on a graphing calculator such as the TI-83 or TI-84 -- but what exactly is the calculator doing? If there are only two data points, the calculator simply finds the equation of the line joining them (which students learned earlier in 8th grade). But if there are three or more points, the calculator uses a more complicated formula. It's actually a form of the point-slope formula, where the point is just (mean of x, mean of y), but the slope is very hard to find. It definitely requires advanced algebra -- perhaps even bordering on Calculus, as Abel implies.

Of course, Common Core doesn't expect 8th graders to find the trend line exactly. Students simply estimate the trend line by eyeballing the data points, and write an equation for that line. But I remember Bill once writing that if it isn't exact, it isn't mathematics. Eighth graders haven't had enough Algebra to write trend lines exactly, so they shouldn't be taught trend lines at all -- instead, they should be taught Algebra.

"Abel" continues:

This fits hand-in-glove with what one of the drafters of Common Core (Jason Zimba) admitted: the Common Core sequence will not prepare students for a STEM college career, nor will they prepare students for admission into a selective college.

"I'm really awfully glad I'm a Beta, because I don't work so hard." Common Core is designed to produce not Alphas, but Betas (at best), Gammas, Deltas, and Epsilons....

The last commenter in this thread is anonymous:

"Naturally, few pieces about math education can resist claiming that part of the problem is rote learning"

What about the millions of people who watch NFL games on TV by rote? Couldn't a computer do a much better job?

Actually, a much better analogy (and once used by Beals herself) would be the NFL players who practice drills during the week and memorize the plays by rote.

Of course, we already know the answer to that question. People don't mind doing things by rote if they are easy, fun, or high-status. Watching football games is both easy and fun, and playing the games is high-status. If the traditionalists want students to learn Calculus by rote, they need to find a way to make it easy, fun, or high-status, or it won't happen. Beals (well, her uncle, at least) admits that Calculus is not easy, and Abel calling those who can do Calculus "Alphas" is the closest he can get to making Calculus high-status.

We know that Beals and Abel want more students to take Calculus. But Bill, apparently, implies that fewer students should take AP Calculus. Of course, "traditionalists" isn't a monolithic group, so I shouldn't just assume that Abel, Beals, and Bill are all on the same page. Still, is there a way to reconcile their comments? What exactly do the traditionalists want?

Here's what I think they want. Abel laments that the Common Core was written for "betas" and below, so naturally he'd prefer that the national or state standards be written for "alphas" instead. This means that the eighth grade standards should be Algebra I, period; the freshman standards should be Geometry, period; and the senior standards should be Calculus, period.

But this doesn't mean that all seniors should take Calculus. From Bill's comments, I infer that students should be placed on the Alg I-Geom-Alg II-Precalc-Calc path, but if they fail a course, they should be forced to repeat it. So a sophomore failing Algebra II should just retake Algebra II as a junior -- not placed into Pre-Calculus anyway, nor encourage to take Stats or another course, but Algebra II, period.

According to the traditionalists, no high school student should be placed in any course other than Algebra I, Geometry, Algebra II, Pre-Calculus, Calculus AB, or Calculus BC. Perhaps they might allow a student who passed Calculus as a junior (i.e., who took Algebra I in 7th grade) to take AP Stats as a senior. And of course, traditionalists don't like Integrated Math at all.

This also answers the question about what the traditionalists would do about the eighth graders who have trouble with trend lines. Based on Abel's and Bill's comments, trend lines should be thrown out of the 8th grade curriculum, along with transformations which they also hate. Instead, these units should be replaced with units on polynomials and the Quadratic Formula -- thereby making the Common Core 8 class into a true Algebra I class.

Of course, my hands are tied -- I can't really throw out the Common Core. And besides, I don't necessarily agree with the traditionalists' goals for high school math in the first place. I only quoted the traditionalists to inform you, the readers, their opinion about stats in high (and middle) school. I am curious, though -- how would all those students who struggled with the trend line question have fared if this had been a quiz on polynomials and the Quadratic Formula instead? Unfortunately, I'll never know the answer to that one.

I haven't completely decided what to do about stats in my eighth grade class yet. Perhaps it may be better to teach trend lines and stats earlier in the year -- maybe right after the unit on linear functions (since that's what a trend line really is). And within the unit on linear functions, let's give the students more questions on writing a linear equation from only its graph -- since that's what we expect the students to do with trend lines. We can start with examples where the lines cross the y-axis (so we have a clear y-intercept) and move on to those lines which aren't drawn all the way to the axis (as the data often doesn't begin with x = 0).

Of course, in the past, I've also advocated for teaching the geometry unit earlier in the 8th grade as well -- since one 8th grade standard asks students to use similarity to derive the slope formula, yet geometry is usually taught well after students learn about slope! I tried to rewrite an 8th grade course here on the blog in which similarity is taught before slope -- and it turned out to be a big mess, so I chose to abandon the whole idea.

There are many decisions I must make about how I will teach my course -- but I won't make them until I have a chance to meet with other math teachers at my school in August. I am the newbie, the outsider here -- my job is to provide continuity with the previous teachers in order to make it easy on the students. In particular, I won't make a complete decision on how to assign the students grades until I've spoken to the other teachers.

And so this wraps it up for Day 30 of the MTBoS30 and Day 180 of the school year. This blog is now getting ready to move into summer mode. Here are my plans for the summer:

-- I want to continue -- and hopefully finish -- Legendre's book on spherical geometry. Suddenly I have less desire to write about spherical geometry since I found out I'd be teaching middle school in the fall instead of Geometry, but I do want to finish what I started.
-- I will have a series of posts in which I plan for my new classes in the fall -- including things like rules and expectations. These posts will be labeled "traditionalists" as I will compare how the traditionalists (such as Bill) want cmath lasses to be taught to how I will actually teach them.
-- Meanwhile, I will not continue my "How to Fix Common Core" series from last year. Again, this would be biting the hand that feeds me, as I'm getting ready to teach Common Core classes for fall. I will repeat criticisms of the Core if it is part of a traditionalists' quote, just as I did in today's post.
-- I was planning on creating that senior high school course for Governor Brown's contest, though I was never going to post much about it on the blog. I still plan on working on it, though it's hard to think about a Grade 12 course since I'm preparing to teach Grades 6-8 in the fall.

Just like last year, I'll post sporadically during the summer. In particular, I plan on taking a week off before I write my first summer post. I was considering posting around June 19th, but this may be affected by my transition to a new Internet service next week. I may instead post on the 17th just before the transition, or instead wait until the transition is complete.

It's been a pleasure writing about Geometry for a second year. Have a nice summer, and let's get ready for the new school year to begin on August 16th.

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