Friday, September 4, 2015

Chapter 1 Test (Day 8)

Today was my first day as a substitute teacher this year. Although I mostly subbed for seventh grade history, during my conference period I covered a special ed sixth grade math class. As this was special ed, the students were working mostly on two-digit addition. For the most part, the students used the standard algorithm. The real issue, of course, is not whether sixth graders use the standard algorithm, but whether third graders use it, since the Common Core pushes the standard algorithm for addition back to fourth grade. There were also a few questions where students added multiples of 10, so that they can see the relationship among 29 + 14, 290 + 140, and 2900 + 1400.

I am now posting my first test. It is actually the Chapter 1 Quiz that I posted last year, but now I'm considering it to be a "test." This is mainly because in my semester plan at the start of the year, I refer to the first day of school up to Labor Day as the first "unit," and then the month starting with Labor Day as the second "unit." Each test that I post corresponds to one of these "units." Still, I don't want to overburden the students with a hard test at the start of the year, so this still has only 10 questions.

Even though my series "How to Fix Common Core" is over, I will often use these quiz and test days to post links to articles about the Common Core debate, including recent traditionalist arguments. I will start by rewriting what I wrote last year about today's test (including my rationale for including the questions that I did), and then link it back to the Common Core debate.

There is a Progress Self-Test included in the book. But even if I threw out the questions based on sections 1-6 through 1-8, there are some questions that I chose not to include.

For example, the first question on the Self-Test asks the students to find AB using a number line. This is very similar to some of the questions that I gave on the Wednesday and Thursday worksheets. But there is one crucial difference -- this one is the first in which both A and B have negative coefficients.

Now I know what the test writers are thinking here. The test writers want to know whether the students understand a concept. There's not enough room on the test to give both easier and harder questions. If a student gets a harder question correct, we can be sure that the student will probably get a much easier question right as well. But if the student only answers an easier question correctly, we can never be sure whether the student understands the more difficult question. Therefore, the test should contain only harder questions, since anyone who gets these right understands the simpler concepts too.

But now let's think about this from the perspective of the test taker, not the test maker. Let's consider the following sequence of hypothetical conversations:

Student: The distance between 4 and 5 is 9.
Teacher: Wrong. You're supposed to subtract the coordinates, not add them. The distance is 5 - 4 = 1.
Student: Oh.

Student: The distance between -4 and 2 is 2.
Teacher: Wrong. When subtracting, change the sign. The distance is 2 - (-4) = 6.
Student: Oh.

Student: The distance between -8 and -4 is 12.
Teacher: Wrong. You forgot the negative in front of the 4. The distance is -4 - (-8) = 4.
Student: Oh.

And we can see the problem here. The teacher wants the student to be able to find the distance no matter what the sign of the coordinates are -- not just when they're positive. But the problem is that the instant that a student finally understands how to solve the first problem, the teacher suddenly makes the problem slightly harder, and the student becomes confused.

Of course, you might be asking, why only give one problem on Wednesday? Why can't we give more problems to check for student understanding of the all-positive case, then move on to negatives? But you see, I'm imagining the above hypothetical conversations as occurring during, say, a warm-up given during the first few minutes of class -- and warm-ups typically contain no more than one or two questions. The student is never allowed to taste success, because each day a little something is added to the problem (like a negative sign) that's preventing the student's answer from being completely correct. The student never hears the words "You're right." And that's just with negative signs -- the U of Chicago text includes questions with decimals as well. I immediately threw all decimals out of my problems -- since decimals confuse the students even more, most notably when we draw number lines and mark only the integers.

Well, I don't want this to happen, especially not on the quiz or test where most of the points are earned. I want the student to taste success -- and this includes the student who's coming off of a tough second semester of Algebra I and is now in Geometry. Sure, if you feel that some students need to be challenged, then challenge them with all the negatives and decimals you want. But I don't want to dangle the carrot of success in front of a student (making them think that they've understood a concept and will get the next test question right), only to jerk it away at the last moment (by adding extra negative signs that will make the student get the next test question wrong), all in the name of challenging the students.

And so my test questions are basically review questions rewritten with different numbers. My rule of thumb is that the test contains exactly the same number of negative signs as the review. Some teachers may see this as spoon-feeding, but I see it as setting the students up for success. Any student who works hard to prepare for the test by studying the review will get the corresponding questions correct on the test.

Of course, some questions about the properties are hard to rewrite. I considered using the question from the U of Chicago text, to get from "3x > 11" to "3x + 6 > 17." But notice that the correct answer -- Addition Property of Inequality -- is difficult to remember and will result in many students getting it wrong. So even here I changed it to the Addition Property of Equality. After all, the whole point of learning the properties is to be able to use them in proofs. The Addition Property of Equality is much more likely to appear than the corresponding Property of Inequality. All including Inequality on the test accomplishes is increasing student frustration over a property that rarely even appears in proofs.

So last year, I wrote that I didn't want the students to feel frustrated during the test. Now here's the connection to Common Core -- many students felt exactly the frustration that I want them to avoid when taking the Common Core tests!

Here's a link to an anti-Core website. This site describes the Algebra I test taken in New York State:

Now this refers to the tests taken in New York State, which are neither PARCC nor SBAC. But based on what I've seen on the Algebra I PARCC exam, many of the complaints about the New York test apply to PARCC -- and so I wouldn't be surprised if they describe SBAC as well.

Let's look at some of these specific complaints:

Problems that ask a student to first subtract two quadratic equations, and then multiply by a 2nd degree monomial with a fractional coefficient (for 2 pts), or asking about completing the square with fractions (2 pts), only serves to reinforce the message that math is hard.

Yes, we've discussed how some of Algebra II now appears in Algebra I. So far, this link has nothing to do with the traditionalists, but we've seen the traditionalists complain about this before. In particular, they're upset first that the Core pushes Algebra II into Algebra I, then the schools point out that the new Algebra I is too difficult for eighth graders, so the students have to wait until ninth grade to take the harder Algebra I -- which doesn't lead to Calculus as a senior.

Indeed, this next comment comes from the parent of a student who has to take Geometry in the summer in order to reach Calculus as a senior:

“My daughter also came home extremely aggravated with the CC Algebra regents test today. She said that there were others who cried and were just lost. My daughter said that she couldn’t understand what some of the questions were asking to even attempt to try to solve it (in pen no less). She said there was a lot of material on the test that was not covered during the year. This is a high honor student who is challenging the geometry course this summer so she can set herself up with trig next year in her all 11th grade courses (in 10th grade). What is happening here??? Why is this being allowed?? Common Core NEEDS to go!!!”

Here's another comment from the parent of a student trying to accelerate:

“Friends here with double and single accelerated kids (7th taking algebra 1) reported their kids saying test was more difficult than any of the practice materials and many did not finish.”

This is exactly what I'm trying to avoid on this blog -- following an easy practice worksheet like yesterday's with a harder test today. Yet that's exactly what we see with the Common Core.

Here lies the problem -- I don't want my tests to be hard, with a huge jump between the practice and the real test. But by making the tests easier, now there's a huge jump between my tests and the actual PARCC, SBAC, or New York Regents exam. We even saw how last year, I had to spend the last month and a half making sure that I covered everything that appears on the PARCC Practice Exam.

Here's a very long rant that appears at the link:

“I’m not a math teacher but I tutored about 10 kids from 4 different districts for this CC Algebra Regents.
Algebra should be a challenging yet positive introduction to upper level math instruction. Common core will make many kids (and parents) join the “I hate math” club. I’ve taken all the available tests, real and actual, and have a good feel for what challenges the kids faced. I do not think parents or students were prepared for the difficulty of this test. The generous curve makes it so that kids will pass, but kids want to do more than pass.
Scoring 30 points will pass with 65.
How is having kids pass a Regents with 35% correct helping them be “college and career ready”?
For the CC math “Guinea pigs” it’s a huge disadvantage to have to take a brand new course and brand new Regents with no full actual practice exams that is scheduled 2 weeks prior to normal Regents week. These leading edge kids had 2 weeks less with curriculum and practice, no Barron’s book to refer to.
I’d rather kids know 100% of less and would love to see a quarter or even a third of CC algebra eliminated so kids could focus on mastery, proficiency and long term retention of the most important areas instead of being spread thin and not really understanding upper level concepts.
What I saw that was interesting from helping kids from different districts, was that while most things were the same, different districts approached certain topics differently and there were some concepts covered in one district but completely left out in others.
(Poor) Wording can mask what’s being asked on this test. It’s possible the questions were posed in an unfamiliar or unusual way that made kids think they didn’t know something when they actually did. This happened over and over when I worked on practice exams with kids.
Kids had gaps through no fault of their own. After all, they couldn’t learn something that wasn’t part of a curriculum when they had the course. Unfortunately their current class assumed they learned it.
Non IEP and very strong math students feel frustrated by the CC Algebra Regents also.
In 2013, the first year of the new CC Algebra standards, the entire curriculum (modules) wasn’t even fully ready before school started. Teachers were taken out of class for professional development as the math airplane was “being built as it was being flown.”
CC math should not have been dumped on every grade at once but gradually phased in, starting with K. Changing the approach to math in the “middle of the movie” isn’t helpful or in students best interests.
And before that, it should have been tested on a small scale to know it would be successful and work out the kinks, develop solid materials, etc. (And), where’s the proof that CC is better, even if it starts in K?
My district instituted “extra help” on the elementary level mostly to help with math
Taking parents out of the math equation will not lead to success in math. Pockets are only so deep. The kids who will succeed and do better are those whose families can afford tutors
CC redefined algebra into something that made classic Algebra textbooks obsolete. Algebra is the new Algebra 2. It’s very demanding and is difficult for many students to do well in.
To end a challenging year with a overly difficult Regents is wrong.
I hope kids still have opportunities to be successful in high school math despite this terrific burden placed on them. It’s disappointing that so many kids have lost their self confidence in math. ”
There are many issues in this long letter:

-- As I mentioned earlier this week, many students will say "I hate math" around the time they take Algebra I with or without the Common Core.
-- The letter mentions scores of 30-35% being considered passing. Of course, I've discussed this here on the blog before. On one hand, remember that a batting average of .300 to .350 is considered excellent, and so if answering these questions are as hard as hitting a baseball, there's nothing wrong with declaring 30-35% to be passing. On the other hand, students are used to getting scores like 70%, 80%, and 90% on the tests. This means not only that 70% is a more acceptable cut score than 35%, but also that any test that doesn't have the smartest students easily clearing 70% must be flawed.
-- The writer wants to eliminate one-fourth to one-third of the curriculum. Considering the reference to Algebra I being like Algebra II later on, I suspect it's the Algebra II part that should be cut from the Algebra I curriculum.
-- We already know that many students grandfather students who were in the middle of high school at the time of the conversion to the old curriculum. Well, this writer would have preferred that anyone older than kindergarten be grandfathered in, which would mean that the Common Core Algebra I test wouldn't be given for nearly another decade.
-- The writer wished that the Common Core had been tested on a small scale. I've already mentioned the existence of the U of Chicago Lab School to test out that curriculum -- a similar Common Core Lab School could have been created to test out the Core. Then this solves the "guinea pig" problem that the writer mentioned earlier as well -- parents and students would have chosen to attend the Common Core Lab School. (Recall that the traditionalist SteveH has already come up with a way to test out potential standards -- just survey students who received a 3 or higher in AP Calculus, and include all schooling, homeschooling, and tutoring in the survey. In SteveH's exact words, the STEM track is to be "driven downwards by AP Calculus.")

The letters continue on, discussing how the top students in the class -- who should ace the test -- are instead crying out the door at the end of the test.

Traditionalists say that Common Core Algebra I is more difficult than the pre-Core class -- but in all the wrong ways. The traditionalist Dr. Katharine Beals is fond of comparing the old Wentworth Algebra questions from a century ago to the Common Core questions. Of course, she remarks that the current questions are too much about "labels" and not enough about "content." But the test question mentioned in the link was about subtracting and multiplying polynomials and completing the square, which sound just like Wentworth questions.

So here I ask, suppose the New York Regents exam were replaced with some from Wentworth. Now, would the top students still be crying after the text, or are the Wentworth questions so much better that the top students would be smiling out the door, confident that they have a raw score of 70% or 80% or even 90%?

For example, one student specifically complained about #12:

“I was particularly upset about #12, the graph of the polynomial function. Is that even in the CC standards for Algebra 1?! I haven’t taught CC Algebra1 yet, but on know polynomial functions are part of the Algebra 2/Trig curriculum. Isn’t there enough to teach and test the students on without tossing things on there that aren’t even part of the curriculum?”

Following a trail of links, I was able to find this question on the blog of a Big Apple math teacher, Chris Burke. (I show both the question and the answer, since I can't show the graph.)

12. Which equation(s) represent the graph below?
I. y = (x + 2)(x2 - 4x - 12)
II. y = (x - 3)(x2 + x - 2)
III. y = (x - 1)(x2 - 5x - 6)
(2). II, only. The zeroes are at -2, 1 and 3. Each equation shows one of those zeros. You have to factor the trinomials to find out if they will reveal the other two zeroes. If you factor you get
I. (x + 2)(x - 6)(x + 2)
II. (x - 3)(x + 2)(x - 1)
III. (x - 1)(x - 6)(x + 1)
Only choice II fits the graph.
You also could have put the three equations in your graphing calculator and graphed them or checked the Tables of Values.

Now let's compare that to a graphing problem from Wentworth:

Would the commenter who was upset at #12 have preferred to see a Wentworth problem instead? Yet this is the sort of problem the traditionalists would want to see more of.

Finally, here's a comment that I suspect traditionalists will agree with:

“I do not think the multiple choice were that difficult, and thought most free response questions were fair. I thought the part IV was an annoying question as it simply tested their ability to use a calculator, and many students voiced how it took them a long time to graph it because they were using multiple points rather than integer points along the parabola.
I also think there were too many explain how you arrived at your answer when the algebraic solutions were self explanatory.
I do not understand how certain questions are only worth 2 points in Part II when they require multiple steps.
Also think the curve is ridiculous for this exam since in the past it really hurts the stronger students and rewards those who score low, which is a whole other monster of issues with the Regents System.”

Before I leave the traditionalists, here's a recent article from Atlantic about Common Core Math:

Two known traditionalists, Dr. Barry Garelick and SteveH, posted in the discussion thread. Here is what SteveH wrote in the thread:

Most high schools go on the same with their proper STEM tracks driven downwards by AP Calculus. "Integrated" math in high school has mostly lost the battle with the reality of college admissions and the real world. And middle schools become the transition battle ground open to the sensibilities and meddling of local educators. CC is giving some of them cover for eliminating a proper algebra I in eighth grade track. For other schools, life goes on the same with the math track split happening in seventh grade and offering a proper algebra class. The problem is how do students get on that top track which is the only likely way to open up STEM opportunities in college and beyond? When I was in school, I was able to do that with absolutely no help from my parents. That was impossible to do with my son. I wasn't teaching him grand or complex ideas in math. I was ensuring mastery of basic skills and knowledge. I was not "trusting the spiral" as in his Everyday Math classes. I was just doing what K-6 schools have not done in decades - ensuring mastery of skills on a grade-by-grade basis. Incredibly, his schools used to send home notes to us parents to practice "math facts" at home. That's because their pedagogy loathes the task. Go ahead and learn counting up (which we did when I was growing up) and the lattice method, but ensure mastery. Ah, that's the missing link in all of this. No mastery. K-6 educators just want to "trust the spiral" and talk about understanding. They completely miss all of the understanding that is buried in mastery. It's not rote. This is their fundamental pedagogical flaw.

Here SteveH has a point about the U of Chicago elementary text (Everyday Math), but I disagree with him about integrated math. Colleges admit many students who took integrated math -- and I'm talking about the foreign students, from countries that teach integrated math. Once again, the U.S. is an outlier with its "traditionalist pathway" of Algebra I, Geometry, and Algebra II. Of course, recall what I wrote earlier -- there's integrated math and there's integrated math. It could be that the best pathway is the "integrated sandwich" -- traditionalist Algebra I in eighth grade, integrated math in Grades 9-11, and then AP Calculus as a senior.

This is what Garelick wrote in the thread. Here he refers to elementary school math -- the level with which I agree with the traditionalists the most:

With the adoption and implementation of the Common Core math standards, there has been increased emphasis and focus on students showing “understanding” of the conceptual underpinnings of algorithms and problem-solving procedures. Instead of adding multi-digit numbers using the standard algorithm and learning alternative strategies after mastery of that algorithm is achieved, students must do the opposite. That is, they are required to use inefficient strategies that purport to provide the “deep understanding” when they are finally taught to use the more efficient standard algorithm. The prevailing belief is that to do otherwise is to teach by rote without understanding. Students are also being taught to reproduce explanations that make it appear they possess understanding – and more importantly, to make such demonstrations on the standardized tests that require them to do so.

I compare this to what I saw the sixth graders do today. As I don't sub in elementary schools, my only contact with arithmetic is in special ed classrooms. One thing that many of the students had trouble with today was the difference between an even number and an odd number. Indeed, one can point out that if students can't even describe the difference between even and odd numbers, how can we expect them to give the necessary explanations required for the Common Core tests? But once again, I have no problem with Garelick's full traditionalism in the earliest elementary years.

Here's how another poster, Peter Ford, responded to Garelick:

In the words of Jim Rome: "rack it."
The fundamental error being made by CC implementers is ignoring or glossing over the importance of KNOWLEDGE to math success.

I hope that everyone enjoys the Labor Day Weekend. Next week marks the release of the SBAC scores in California. Some California websites such as EdSource are already preparing for the occasion, and of course I'll discuss the results here -- even though it won't be a test day on the blog.

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