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?

Thursday, May 4, 2017

Bezier for Young People

Bezier Curves are generally not confronted by k-12 students at all. Their equations can be complex, and in the absence of dynamic geometry, the topic in general can be mind boggling. Historically, algebra has been the driving force for graphing. With dynamic geometry, that can be reversed.

If you can mentally stomach (how's that phrase?) the idea of a point on a line steadily sliding from one location on that line to another location on that line, you've mastered the necessary skills. Visually, a point sliding from one place to another would look like this:

Being able to picture this is all you need to know. Here it is

Tuesday, May 2, 2017

Jack and Jill

Here is a short Geogebra example I put together as an example of how GeoGebra could be used to introduce young students to mathematics that they might never see unless they got into a precalculus class that included polar graphs.

Among the concepts used (but not named) are midpoint, rotation, rotational speed, and 3-leafed rose.
The only geometric term used is "circle".

Even so, the situation can be used as a springboard lots of questions.

Among those questions could be:

  • What if they walked the same speed?
  • What if Jill walked faster?
  • What if they walked in the same direction around the circle?
Denying younger students the opportunity to ask these questions and explore their solutions is to do a disservice to those students.

Saturday, April 15, 2017

Why? Why? Why?

Here is question 3 from the January 2017 New York Regents Exam in Algebra I (Common Core),
This question shocked me when I first saw it.

To get a taste of what such a correlation coefficient is, I suggest you take a look at Wolfram. New York State's EngageNY has an "introduction" to correlation coefficients (find it here) that says "It is not necessary for students to compute the correlation coefficient by hand, but if they want to know how this is done, you can share the formula" and then shows

It should be noted that this formula the bounds of summation and variable subscripts are omitted, so it is a meaningful formula only for those "in the know", and I suspect that very few Algebra I students fit that description. 

Is this a case of presenting a "magic button" on a calculator as a key to answering a complex question? Is that how math should be taught?

New York's modules include the following:

Take note of the phrase "use technology".  That essentially means the student should plug in the numbers, hit the necessary buttons, and find a result. It is more like following a recipe, and most people recognize that a recipe becomes unnecessary after it has been followed a number of times: not because it is not being followed and not because it is understood, but because it is remembered.

Here is a question from the August 2016 Algebra I Regents;
I would suspect that after enough repetitions of the recipe some students might have caught on and realized that the choices offered make this question a bit easier than it would have been had the data been a bit different and the choices involved some seemingly random 4-digit decimals.  Perhaps that might be why questions such as this have, over the years, been categorized as "cookbook" problems.

The next question here is from the Algebra I Regents exam from August 2015:
There is absolutely no way any student will successfully answer this question without following a recipe on their graphic calculator. I do not know why NYS left all the space on the page as all that is asked of the student is to write down one equation and then a two digit decimal together with one word.

Here is a similar question from the January 2015 Algebra I exam:
The big difference here is the 60% increase in time entering the data (16 values instead of 10) and the explanation of part (b). But again, it's enter two columns of data, hit a couple more keys on the calculator, and read off a result.

A similar process is involved in the next question, the big difference is that it asks the student to pick an item from a different line in the calculator's display:
Do all these questions belong in Algebra I?

Are these items here only because of the lobbying efforts of the companies that sell the graphic calculators?

I would much rather see students get the flavor of least squares analysis by using FREE software such as GeoGebra. For an example, check out the Least Squares Demonstration here.

While correlation coefficient seems to be an easy testable item (when proper calculator is present), it should be recognized that it is merely a measurement of how well a regression line actually "fits" the data.  The true mathematical questions begin with the regression line itself: what it is, why it is, why it is useful, when is it useful,  how we know it is the "best" fit, what do we do when this is not useful, and many others. For a sense of the current state of regression analysis, just visit Wikipedia.

Please take note: correlation coefficients and a slew of its tag-a-longs could fit in high school mathematics, but not in Algebra I.  Dynamic geometric approaches to concepts such as least squares could be well developed in an Algebra I course, using software such as Desmos or Geogebra or other equivalent packages. 

The biggest problems with high school mathematics occur when students are expected to know and do things mindlessly. Recipes need to be avoided. Magic buttons on calculators need to be avoided like poinson needs to be avoided.

The true value and meaning of the quadratic formula comes only to one who tires of completing the square. Completing the square obtains its true value and meaning when one tires of trying to find factors (especially when they do not exist!). The real meaning of factoring comes to one who is fed up with constantly having to guess and check. There is a hierarchy to the knowledge and skills of mathematics. Jumping too quickly to a higher level does a disservice to students.

Imagine what would happen if we limited single variable quadratics to finding intersections with the x-axis on a graphic calculator, and skipped over everything mentioned in the previous paragraph! It makes just as much sense as tossing regression analysis into Algebra I only because it can be done on a graphic calculator. Jumping too quickly to a higher level does a disservice to students, especially when that jump takes place largely do to the mere availability of a handheld calculator.

Tuesday, March 28, 2017

Is Kindergarten too old?

Here is a very simple dynamic graph, showing a point sliding down a radius in the same amount of time that it takes the circle to do one rotation around its center. Simple to see and explain.

It is the explanation part that makes this valuable. No matter what grade level, it could be used. From simple use as an introduction to the words circle, radius, rotation, and speed (imagine if children started learning these words in kindergarten) to a PreCalculus class being asked to generate parametric equations so that they could continue the graph on their graphic calculators.

The file itself can be found here.

If you wish to read more about this type of structure, check out this at Wikipedia.