Friday, September 25, 2015

Chapter 2 & 3 Quiz (Day 22)

Today I post the quiz for Chapters 2 and 3, based on the test for Chapters 1 and 2 from last year. I don't repeat what I wrote last year, since that was mostly about the school calendar -- and I already discussed the relationship between my units and the calendar in yesterday's post. And besides, what I wrote last year was about the Early Start and Labor Day Start calendars, and not the Middle Start Calendar that the blog is observing this year.

I will say this about the Middle Start Calendar though -- today we are one-fourth of the way through the first semester. Last year, I referred to an eighth of a school year as a quaver -- named after what the British call an eighth note in music. This is another reason why I posted the quiz today -- so that teachers will have another major component to the grades before issuing progress reports for the first five-week period -- the first quaver.

Today I subbed in another eighth grade math class. This class was computer-based as there were a class set of Google Chromebooks. As we already know, math classes on the computer have become more common now that the students have to take the SBAC exam on computers.

As it turns out, the students were working on the very same lesson that the eighth grade class where I subbed last week was on -- Section 1.3 of the California Glencoe text, which is on multiplication and division of monomials. This is a bit behind where the class needs to be if it's to finish the text in a timely manner. We know that Volume I, intended for the first semester, has only four chapters and we're a fourth of the way through the semester, so it would be more logical to be just wrapping up Chapter 1 today. Of course, if one wants to finish Chapter 9 before the SBAC, then one might even need to squeeze in Chapter 5 by the end of the semester, On the other hand, being still midway through the first chapter now means that one would be lucky even to finish Volume I by the end of the school year.

Then again, despite the pressure to move forward in the text, teachers are likely to spend extra time in Chapter 1 if they see that the students are struggling to learn the material. Recall that in a traditionalist Algebra I class, monomials (leading to polynomials) appear at the start of second semester. We've seen teachers successfully start the year with monomials and exponents -- but for ninth graders in an Algebra I class, not eighth graders. Again, I often wonder why so many textbooks have to put so much difficult material in Chapter 1.

This leads to what I said I would often do on quiz and test days -- write about the issues regarding Common Core and traditionalists. Recently I was reading some of my old posts, and I want to link two old blog posts written three months apart.

Back in May, recall that I was reading a book about the two mathematicians -- Abel and Galois -- who proved that it's impossible to solve the quintic equation using radicals. One of the appendices of that book was about another mathematician -- the ancient Greek "Father of Algebra," Diophantus. I wrote about how the mathematician found two numbers whose sum is 20 and whose sum of squares is 208 -- he did so by letting the two numbers be 10 + x and 10 - x. Let's complete the calculation:

(10 + x)^2 + (10 - x)^2 = 208
100 + 20x + x^2 + 100 - 20x + x^2 = 208
200 + 2x^2 = 208
2x^2 = 8
x^2 = 4
x = +/-2

This means that the two numbers are 10 + x, or 12, and 10 - x, or 8. We can see why this works -- letting 10 + x and 10 - x be the numbers leads to the 20x terms cancelling after squaring.

Now in August, I wrote about the traditionalist Dr. Katharine Beals, who lamented that the Common Core tests don't contain problems involving "simultaneous quadratics," as the only Wentworth text from a century ago did. One of the problems on that page is:

The sum of two numbers is 20, and the sum of their cubes is 2060. Find the numbers.

And now we say, hey! Why don't we let the two numbers be 10 + x and 10 - x again, as Diophantus would have done? I know -- it's a bit awkward that a question on cubes appears in the section on simultaneous quadratics, but let's see what happens anyway:

(10 + x)^3 + (10 - x)^3 = 2060
1000 + 30x + 30x^2 + x^3 + 1000 - 30x + 30x^2 - x^3 = 2060
2000 + 60x^2 = 2060
60x^2 = 60
x^2 = 1
x = +/-1

And so the numbers are 10 + x, or 11, and 10 - x, or 9. Just as in the sum of squares problem, terms cancel to leave us with only a quadratic term and a constant.

Let's try using the Diophantus trick with another problem. The first problem on this page is:

The difference of two numbers is 6, and their product exceeds their sum by 39. Find the numbers.

Now had the sum been 6, we would have let the numbers be 3 + x and 3 - x, but notice that since it's the difference that's six, we can let the numbers be x + 3 and x - 3 instead:

(x + 3)(x - 3) = (x + 3) + (x - 3) + 39
x^2 - 9 = 2x + 39
x^2 - 2x - 48 = 0
(x - 8)(x + 6) = 0
x - 8 = 0 or x + 6 = 0
x = 8 or x = -6

For x = 8, we obtain x + 3 = 11 and x - 3 = 5 as the numbers, and for x = -6, we obtain x + 3 = -3 and x - 3 = -9 as the numbers. We can check to see that both pairs (11, 5) and (-3, -9) indeed work out. I admit that this doesn't work quite as cleanly as the case when we're given the sum, and so one might still prefer to solve this one the standard way (where x and x + 6 are the numbers).

Now here's the thing -- I had recommended the book where I read about the greats like Diophantus, Abel, and Galois to math teachers, and implied that telling our students about how these famous mathematicians solved problems would motivate them to learn. But then when discussing the Wentworth problems, I said that students should not see those sorts of problems, since all they'd do when they saw them is complain that they have to solve them.

So what gives? It all goes back to the Common Core debate. Traditionalists want to see more Calculus in high school, and in the courses leading up to Calculus, they want the types of problems that will prepare the students for Calculus, such as the Wentworth problems -- as opposed to the "pseudo-algebra" that currently appears on the Common Core tests.

But recall that a few weeks ago, New York parents and student were complaining about how difficult the Common Core tests were in Algebra I. And we saw that they weren't complaining about problems of Wentworth difficulty, but mere Quadratic Formula questions. One parent had even written that the Quadratic Formula belongs in Algebra II or even Precalculus -- not Algebra I.

So let's summarize. The New Yorkers believe that the Quadratic Formula is inappropriate for freshmen, and that they should wait until junior or senior year to learn it. On the other hand, the traditionalists also believe that the Quadratic Formula is inappropriate for freshmen, but that they should learn the formula earlier, in eighth or even seventh grade!

I believe that in some ways, the status quo -- and the Common Core is ultimately an extension of the status quo of the late 20th century -- is a compromise between the traditionalists who want the students to reach Calculus in high school and the progressives who think that even parts of Algebra I are too much to reach in high school. But notice that if Calculus is needed for STEM types and Algebra I at the most is needed for non-STEM types, this compromise -- "pseudo-Algebra II," isn't exactly right for either group.

And so we see the problem even with having two math tracks, where (Common Core) Algebra I is offered to eighth graders on the upper track and to freshmen on the lower track. The upper track is still missing topics that are needed for success in Calculus, while the lower track is still forced to learn topics that aren't needed for success in a non-STEM career. The two tracks that we want -- STEM and non-STEM -- would need to be several years apart, not just a single year.

Indeed, I think back to my own education. When I was first completing my B.S. in Math at UCLA, I decided to apply to grad school to work towards a Masters degree. The counselors then told me that in some ways, a M.A. in math is useless. To become a mathematician or a math professor, of course a doctorate would be required, whereas to teach math in high school or work other math-related jobs in industry, a mere B.S. is enough. There are so few jobs for which the M.A. in math is both necessary and sufficient. So I applied to the Ph.D. program and got in, but I knew that I was quickly reaching my mathematical peak -- the most math I could learn without struggling to pass. After I failed my first Ph.D. qualifying exam, I settled for the M.A. and left UCLA.

So it is with the Common Core's "pseudo-Algebra II" -- the high school math equivalent of the Masters degree. If a student wants a STEM career then Calculus is required, whereas for a student who wants a non-STEM career, Algebra I is enough. For no one is "pseudo-Algebra II" both necessary and sufficient.

Notice that this traditionalist vs. progressive struggle applies to the younger grades as well. A traditionalist would say that a six-year-old, just starting the first grade, should be memorizing simple addition and subtraction facts and preparing to use the standard algorithm to add and subtract simple two-digit numbers. A progressive might point to Finland -- a country where formal schooling doesn't begin until age seven -- and say that the six-year-old shouldn't be learning formal math at all.

So neither traditionalists nor progressives say that we should teach six-year-olds something else and call it math, yet this is exactly what happens with the Common Core compromise. As traditionalists want standard algorithms for first graders and progressives want no algorithms, the Common Core provides nonstandard algorithms -- such as those which emphasize "making ten." This is similar to the debate with older students -- the traditionalists want rigorous Precalculus for sixteen-year-olds, the progressives want no rigorous math for non-STEM sixteen-year-olds, and the Common Core teaches sixteen-year-olds something else ("pseudo-Algebra II") and calls it rigorous math.

This week, a photo of a "Common Core check" has gone viral. A certain father, upset that his young child was learning Common Core math, decided to write a check using Common Core math -- that is, instead of using digits for the amount, he used obscure symbols. At the following link, Hemant Mehta, a high school math teacher, explains what the symbols on the check actually mean. (Note: the following website is a religious -- or more precisely, an anti-religious -- website. Fortunately, this article and the discussion that follows it have nothing to do with religion or anti-religion.)

Herrmann, in particular, was frustrated with something called “ten-frames.” What are those? They’re a handy way to think about numbers in groups of ten.

Mehta's post has drawn thousands of comments. I didn't read all of them, nor will I quote specific comments here on the blog as I normally do. But two types of comments jump out at me -- some comments argue that Common Core is more rigorous than pre-Core standards and pushes advanced topics like algebra and statistics down into seventh grade. Other comments argue that Common Core is less rigorous and fails to prepare students for STEM. (There were also side discussions about whether people even use checks any more in the 21st century.)

It may be possible to design a curriculum with two tracks -- one leading to Calculus, the other stopping at some Algebra I. But this could be politically dangerous, since as soon as we have two tracks, we immediately worry about the demographics of the students who are placed on the tracks.

One can argue that the current "pseudo-Algebra II" compromise is the worst position of all. If we are worried more about the non-STEM students who have no intention of taking Calculus, then we should protect them from having learn higher abstract math. This would be easy to implement now that we've already given Common Core tests -- we take any math question or topic that more than half of the test takers answered incorrectly and throw it out of the test. I would expect the Quadratic Formula to be dropped from Algebra I, and monomials to be dropped from eighth grade Prealgebra, under this plan.

After one year of doing this, the remaining questions are fixed as the new, simpler curriculum. This remainder should consist of basic math questions that we expect most students to learn. I admit this may be tricky, though, since I bet many students in middle school got the fraction questions wrong. Which sort of fraction questions are fair but difficult and which ones are unfair and need to be thrown out? This change would have zero effect on strong STEM students -- they find the current test too easy and would find this test too easy -- but it would allow non-STEM students to graduate and teachers of non-STEM students to be labeled as good teachers (in states where Common Core tests are used for graduation or teacher evaluation).

On the other hand, if we are more worried that not enough students are prepared for STEM, then we need to go whole hog and set up the required standards so that they lead to Calculus. We can implement this by giving the traditionalist SteveH survey -- we take students who received 3 or better in AP Calculus and find out what they did to get there, including what textbooks were used and what outside tutoring was required.

Progressives, under the SteveH plan, may object to having to learn so much math so early -- whether it's first grade addition or eighth grade algebra. But what they would never do is set up Jeopardy categories or write checks making fun of SteveH math.

I'm between a rock and a hard place here. Once again, I took Calculus myself in high school and want to see as many students pass it as possible, but when I see students such as some of the eighth graders today working hard to learn about monomials, it's just hard to imagine them being forced into a full Algebra I course as the traditionalists would prefer.

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