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## Tuesday, June 20, 2017

### The Cubic Regression Rules!

The actual exam for the June 2017 New York Algebra 1 (Common Core) has not been released, and teachers are strictly forbidden to communicate via the web on the exam until after June 30, so as I begin my yearly review (as a NY citizen who happens to be a retired teacher), all I can comment on is the conversion chart. Here I have plotted it and included (dotted line if you look closely) the cubic regression curve based on the data contained in the conversion chart itself.
I have always looked at level 1 as the failing range, and level 2 as the "safety net" where Special Ed passes but non-Special Ed does not. That might have changed by now. Needless to say, a regular ed- student has to get to level 3 to be considered "passing"

I noticed that, in comparison to the conversion chart for last June, every raw score from 4 to 21 received a scaled score 1 or 2 points higher this year.  No raw score from 24 and up got any boost this year, and a few lost a point on the conversion side.

Here are June 2016 and June 2017 plotted together:

Needless to say, the cubic regression is pretty much the same. I do wonder why NYSED prefers it so much.

Here I have added (in green) the conversion chart for Algebra II from 2016:

Ever since NYSED started using the raw-score to converted-score approach to scoring these exams, there has been battle between those who liked the old "straight-line" approach and NYSED itself. In case you are wondering, the straight-line approach is based on the concept that if you answer 50% of the test correct, your score is 50. Answer 75% correct, get a 75, etc. If I added the graph of the straight-line approach, it would be a straight line from bottom left (0,0) to upper right (86, 100).
What the heck- here it is:
It is easy to see that almost everywhere the scaled score exceeds the 'straight-line" score, with a minimal gain at the top end, and increasingly large gains as one heads from the bottom towards that magical mystical number known as "65".

I am partially from the "old school". When I took the Regents Exam in algebra 1 (I believe it was properly called "9th Year Mathematics" back then) the exam consisted of 30 short-answer questions (2 points each) and 7 Part 2 ten-point questions, of which you had to answer 4.  There was a bit of a top end hammer since a student capable of answering all 7 part 2 questions could get credit for no more than 4 of them. (If a student answered more than 4, only the first 4 answered would count)

Thinking back to those tests, the typical 9th Year Mathematics exam contained a dozen multiple choice questions, each with 4 choices. The random guesser would, on average, get 3 correct out of 12, contributing 6 points towards passing. That student would have to earn 59 points out of the remaining 76. In last year's Algebra I exam there were 24 multiple choice questions, with 4 choices each, so the random guesser would average out with 6 correct, for 12 raw score points. To get to the minimum passing score that student would need 15 out of the 36 remaining possible points. Which is more difficult, 59 out of 76 or 15 out of 36? Hard to tell, recognizing that a student who has to guess on all of them probably doesn't know much of the course.

The other end does intrigue me: the student who gets all the multiple choice correct, be it by guessing or by knowledge or some combination of the two.

In 9th Year Mathematics the multiple choice got you 24 points on your way to a minimum passing grade of 65. In Algebra I last year, the multiple choice gets you 48 points on your way to a minimum passing score of ....27. For that matter, a student who can successfully answer 11 out of the 30 multiple choices gets 22 points. Now that student guesses  on the other 13 multiple choice questions will, on average, get 3 or 4 correct. Even at 3, those student now has 6 more points, and has now earned a passing raw score of 28. the student has "passed" the exam without going beyond the multiple choice questions. If that student had guessed by filling in choice 2 or choice 3 for all guesses, he/she had a solid advantage. (There were 5 answers of "1", 7 answers of "2", 7 answers of "3", and 5 answers of "4". I will have to see if it is a trend to put correct choices in the middle slots.)

What does appear to be a trend to me is a trend to make exams in such a manner that passing scores are in easier reach while top scores are harder to obtain. Some may say it appears to narrow the "achievement gap", but appearances can be deceiving.

## Thursday, June 8, 2017

### Pictures worth 1000 words.

Last week I happened to see a car in Vienna with mathematics all over it. Due to a lot of pedestrians the best picture I could take was of the spare tire case. This picture was taken right near a jewelry store and a Bitcoin store, so in my mind a Tom Clancy novel could start with it. The Casino Wien is just around the corner, the Hotel Sacher is a block away (Sacher of Sacher torte fame). The Vienna Opera is across the street from the Hotel Sacher. The options for starting a novel of intrigue are plentiful. (I erased out the license plate for the privacy of the car owner, although I would question anyone driving a Mercedes covered in math who had a desire for privacy.)
Here is another picture taken, this time in Kunsthistorisches Museum in Vienna. I believe a middle school math lesson could be constructed around these buttons, starting with the task of placing the buttons in numerical order. For some reason the U.S. seems to avoid negative numbers in elevators, let alone decimals. Could there be a greater comfort level with numbers over there "across the pond" than here at home?