Lesson 2-2: If-then Statements (Day 10)

Today is the long-anticipated reveal of the Common Core results here in California. There is much discussion about what these results mean.

My focus is on the difference between ELA and math scores in each grade level. Here are the percentages of students who scored proficient (3 or 4) on the Smarter Balanced exam:

Third Grade: ELA 38%, Math 40%
Fourth Grade: ELA 40%, Math 35%
Fifth Grade: ELA 44%, Math 30%
Sixth Grade: ELA 43%, Math 33%
Seventh Grade: ELA 44%, Math 34%
Eighth Grade: ELA 45%, Math 33%
High School: ELA 56%, Math 29%

As it turns out, the 56% ELA proficiency of the high school juniors was a pleasant surprise. The California website EdSource, which is live-blogging the results, writes:

http://edsource.org/2015/most-california-students-below-standards-on-common-core-aligned-tests/86341

Scores fall within one of four achievement levels tied to the Common Core: “standard not met,” “standard nearly met,” “standard met,” and “standard exceeded.” (See what each level means.) With a key exception, California’s overall scores closely followed what Smarter Balanced test developers predicted last fall when they set the achievement level scores, based on a 1,000-point scale. In grade 11, 56 percent of students met or exceeded the standard in English language arts; Smarter Balanced had projected 41 percent. Since most community colleges and the California State University system are using the “standard met” level as an indicator that students are on track for college, the high score awards a tangible benefit.

Math was a different story, however, with only 29 percent of 11th graders meeting or exceeding the standard, the benchmark for college readiness, 45 percent not meeting the standard and 25 percent nearly meeting the standard. More analysis is needed to determine if the disparity between performance on 11th grade math and English language arts tests reflected the quality of instruction, a comparably easier English language arts test or challenges in measuring high school math knowledge with one test for all students.

No, I strongly disagree here. To me, the reason for the disparity between ELA and math in high school is not due to the quality of instruction, but because algebra is hard. Even though overall, the scores in California were lower than those in Washington (almost certainly due to the presence of more English learners in California), the pattern in both states are identical. Only third graders performed better in math than in English. As the students get older, their ELA percentages tend to increase (because they now know how to read), while the math scores decreased (with first fractions, and then algebra appearing). The better than expected ELA scores in California mean that the gap between ELA and math proficiency in high school is 27 percentage points.

Once again, just as during the "How to Fix Common Core" series, we look at these scores and think about the traditionalists who push for eighth grade Algebra I and senior year Calculus. They are upset that the Common Core tests only go up to Algebra II -- and to them, it's not even real Algebra II, but some sort of "pseudo-Algebra II." Well, only 29% of juniors in both Washington and California are proficient in "pseudo-Algebra II" -- yet the traditionalists want Calculus for the other 71%? I can only wonder how low the proficiency levels would be if the math test the juniors had to take were at the level preferred by traditionalists.

Indeed, this is exactly what one commenter at the EdSource website wrote:

http://edsource.org/2015/poll-california-voters-still-unsure-about-common-core/86224

Bruce William Smith:
Too much time is spent on testing whether pupils are achieving the Common Core standards, because those standards are not internationally competitive, and so publishing results on those tests will give a misleading impression to Americans about where their children truly stand in achievement compared with their peers overseas, who are increasingly landing on American shores and taking American university places away from those whose parents paid to build those universities but who are themselves disadvantaged in the competition for admissions by ignorant, credulous adults. The upcoming results will probably look bad, given how much reduction of expectations has been messaged through the media by state education leaders; but since there are no questions, for example, requiring calculus of high school pupils, unlike the tests of young Americans’ peers in Asia and Europe, or even of precalculus, or of trigonometry, we can assume many American youth will not have been taught such subjects and so would be scoring zero with regard to them; but questions on what is not taught don’t appear on these inferior tests, so the relative gap between Common Core-learning children and their competitors overseas appears reduced, although it will continue to exist, and likely even grow, in reality, which won’t escape the notice of future employers.

Based on this post, Bruce William Smith is certainly one I'd call a "traditionalist." Like most other traditionalists, Smith recommends that juniors take a test with Precalculus, Trigonometry, and some Calculus on it, while the 71% of test takers whose scores were deficient in math probably feel that it's horrible that they're forced to take a test with mere "pseudo-Algebra II." Notice that Smith goes beyond most other traditionalists in that to him, even senior year is too late to start Calculus, as he wants to see Calculus on the junior-year test.

I just hope that the "future employers" Smith mentions are STEM employers. Once again, we think back to the "I hate math" images from last week -- images created by those who say that even Algebra I is much more math then they'll ever use in real life -- and compare it to the "future employers" who won't hire anyone who isn't proficient in Trig or Calculus.

This is why I had so much trouble trying to improve the Common Core. The only way to get students into AP Calculus and prepare them for STEM careers is to have them take Algebra I as eighth graders, but when we do that, we have large number of students fail, and students -- especially those who have no intention of going into STEM -- wondering why they are forced to take highly abstract, symbolic math classes that they will never use in real life.

This is what I wrote last year about today's lesson. I have updated a comment I made last year about the proof of a certain conjecture.

Lesson 2-2 of the U of Chicago text continues the study of logic by focusing on "if-then" statements. I certainly agree with the text when it writes:

"The small word 'if' is among the most important words in the language of logic and reasoning."

There are a few changes that I will make to the text. First of all, the text refers to the two parts of a conditional statement as the antecedent and the consequent -- although it does mention hypothesis and conclusion as acceptable alternatives. I'm going to follow what the majority of texts do and just use the words hypothesis and conclusion. Actually, Dr. Franklin Mason doesn't even use the word hypothesis -- he simply uses the word given -- since after all, the hypothesis of a theorem corresponds to the "given" statement in a two-column proof.

When I teach or tutor students in geometry, one of my favorite examples is "if a pencil is in my right hand, then it is yellow." So I pick up three yellow pencils, and we observe that the conditional is true. But let's suppose that I pick up a blue pencil in addition to the three yellow pencils. Now the conditional is false, since we can find a counterexample -- the blue pencil, since that's a pencil in my right hand yet isn't yellow.

Notice that I decided to replace the word instance with the word example -- so that the connection between examples and counterexamples becomes evident.

The text has to go back to an example from that dreaded algebra again. Of course, it's an important example, since students often forget that 9 has two square roots, 3 and -3. But I decided to include it anyway since it's simple -- it's not as if I'm making students use the quadratic formula or anything like that.

Then the book moves on to a famous mathematical statement: Goldbach's conjecture, named after the German mathematician Christian Goldbach who lived 300 years ago:

If n is an even number greater than 2, then there are always two primes whose sum is n.

At the time the book was written, the conjecture had been verified up to 100 million, but the conjecture had yet to be proved. But what about now -- has anyone proved Goldbach's conjecture yet? As it turns out, the answer is still no -- but now the conjecture has been verified up to four quintillion -- that is, the number 4 followed by 18 zeros.

But there has been work on a similar statement, called Goldbach's weak conjecture:

If n is an odd number greater than 5, then there are always three primes whose sum is n.

This is called weak because if the better-known (or strong) conjecture is true, the weak is automatically true because we can always let the third prime just be 3. Ironically, when Goldbach himself actually stated his conjecture, he stated the weak version of the conjecture. It was a letter from Euler -- you know, the same Euler who solved the bridge problem that we discussed as an Opening Activity -- that convinced Goldbach to state the strong conjecture instead.

Now as it turns out, someone has claimed a proof of Goldbach's weak conjecture -- namely the Peruvian mathematician Harald Helfgott. Last year, Helfgott's proof was still being peer-reviewed -- that is, checked by other mathematicians to find out whether the proof is correct. By now, Helfgott's proof has finally been verified. Yes, mathematicians are still proving new theorems everyday.

Dr. M also mentions Goldbach's conjecture, on a worksheet for his Lesson 2-1. Often students are fascinated when they hear about conjectures that take centuries to prove, such as Goldbach's conjecture or Fermat's Last Theorem. I often use these examples to motivate students to be persistent when trying to come up with proofs in geometry -- if mathematicians Helfgott and Wiles didn't give up even after centuries of trying to prove these conjectures, then why should they give up after minutes?

The final example in this section has students rewrite statements into if-then form. With the newly released Common Core scores still fresh on my mind, I still can't help but think of the conditionals:

If a school is an elementary school, then its math scores are higher than its ELA scores.
If a school is a high school, then its math scores are lower than its ELA scores.

These two statements undoubtedly have counterexamples, but these hold more often than not, judging by the released scores.

I've found that oftentimes, English learners struggle with this part of the lesson. The teacher must point out why, for example, the "something" in the example "all triangles have three sides" must be a figure: "if a figure is a triangle, then it has three sides." So not only must we appease algebra haters when we include algebra in the geometry lesson, but we must also consider English learners when including English in the geometry lesson (again, no wonder that so many students performed poorly on the Common Core tests).

Once again, I decided to include some review questions. Notice that the most of the review questions in this section are from yesterday's lesson, Lesson 2-1. We skipped Lesson 1-9 so I threw out the Triangle Inequality question. Once again, that question marked Previous course is an Algebra I question, and so once again, I rewrote it so that the solution is a whole number. Finally, I decided to avoid that inequality question completely.