How can New York State test kids in math when it can no longer consistently write appropriate questions? This gaffe is almost two years old, but it looks like no one noticed the problem, until it showed up on the Association of Mathematics Teachers of New York State (AMTNYS) listserve this week.

On the August 2013 geometry regents, students were asked to find the slant height of a cone, given the lateral area.

It’s easier than it sounds. There is a formula sheet in the back that gives

L = πrl,

where L = lateral area and l = slant height and r = radius.

Heres’s the question:

14.  The lateral area of a right circular cone is equal to 120π cm. If the base of the cone has a diameter of 24 cm, what is the length of the slant height, in centimeters? (1) 2.5 (3) 10 (2) 5 (4) 15.7

Since radius is half the diameter, r = 12 and plugging in:   120π = π(12)l, or l = 10, choice 3.

But wait. The height of the cone (like a flagpole from the base to the highest point), the radius (like a stripe from the base of the flagpole to the edge of the cone), and the “slant height” form a right triangle, with the slant height being the hypotenuse. So how is the hypotenuse (10) shorter than the base (12)? Can’t happen in the real world… but in New York State?

Here is an insightful comment from the listserve:

This is another example of the type of error that has been occurring on Regents exams since the early 1990s when the math bureau of NYSED was downsized from 7 very experienced and talented people (a bureau chief + 6 math specialists) to an inexperienced few.  It is also a product of contracting out the writing of exams to rich companies that had no experience in this area.
The errors often occur from the creation of questions that require substitution into formulas without looking at a drawing to see if the numbers are possible.

1. May 14, 2015 am31 9:36 am 9:36 am

Not directly on point, but this reminds me of an activity I did creating a psuedosphere model with my kids (blogged here) We used a bunch of disks with the same radius and then cut out different sized arcs to make our cones. While this point didn’t come up directly, it is a good experience to see and feel the relationship between the slant height and the circumference of the base circle. Also, it gives an intuitive picture of what would happen if the slant height is too small relative to the base radius.

For the Regents question, is there a path to the answer that gets stuck because this cone is impossible? I’m intrigued because, while I feel uncomfortable about a question involving an impossible object, I can’t quite articulate why that makes a difference to whether this is a bad or good question. The quality of the question seems to hinge on what you think about assessing the following: does the student know the definitions (slant height, lateral area, diameter, radius), can they remember or look for the formula, can they do the algebra and arithmetic to complete the calculation?

Finally, here we have a situation where –> . While the scope domain of validity of the geometric relationship is limited, the algebraic relationship isn’t. Is there a sensible way to reverse the arrow to create a geometric picture associated with triplets outside the original domain? I guess you could associate it with an object that looks like a ruffled dress?

• May 16, 2015 am31 8:45 am 8:45 am

How can we make this work? Interesting challenge! I’m playing with various ways of “inverting” the cone. But each process mauls the lateral area formula! Still trying….

2. May 14, 2015 pm31 1:44 pm 1:44 pm

Imagine how many math “errors”, ambiguities, etc will occur on a new test with items not released for inspection, aka, PARCC. Even the SAT had errors until questions were published. Now “bad” items are weeded out from experimental sections and there is high quality control/editing. Privatization and staff cutbacks inevitably lead to loss of quality and our children are the victims, not to mention, educators whose evaluations are linked to these. “Garbage in, garbage out” — would be laughable if not so sad. John Oliver’s scathing expose of the Testing Industry 2 weeks ago on Last Week Tonight should be required viewing by politicians and Board members. I’m waiting for a rebuttal from Pearson, etc!!

3. May 28, 2015 pm31 4:39 pm 4:39 pm

They (NY STATE dept of Ed. and the governor appointed commissioner of Ed., who was approved by the Board of Regents) really messed up the Regents program.
The testing metric experts told us that a 100 point test with student choice
was not a reliable exam. When I went up to write for the text book company
which was in charge with the state ed. reps, I complained and that is what they
insisted was the way they were going. Math teachers, especially, complained. “This is a farce” is what they said. The professional organizations of the math teachers complained loud and often. The costs to produce the NEW exams were huge compared to the OLD exams. The old regents committees were given honorariums and stipends and hotel. travel and food expenses. We did it on the cheap as a service to the students and schools of the state. Now we have book publishers, with plenty of staff, collecting lots of revenue to do the job., at taxpayer expense.
Consider the Regents that you took when you were a student. (See the Regents archives,
a bunch of my questions are on the 2000 Course II exam)
30 out of 35 part 1 questions, worth 2 points each ( 70 points to choose from, pick your best 60 points. Some were open ended questions, right or wrong; some were multiple choice)
The part 2 questions were 10 points each, choose 4 out of 7. (70 points available, choose, your best 40 points). partial credit was awarded for good progress toward a solution.
Teachers had professional discretion for grading. Were there abuses of this discretion? Yes, but most teachers were FAIR, and some marginal students got breaks that they might be deserving of, based on the quality of their work during the year.
So in total there were 140 points “on the table”, and a kid choose to answer 100 points of questions for his score. There was no curve, or scoring table determined after the test was administered. Mastery was 85. Failure was 65. There was no phony business to disguise results to parents, students, administrators, and the public. So, in reality, a passing grade was really 65/140, or 46%, based on the number of points “on the table”
Compare that system to what we have now. The passing percentage is around 35% of the total points, before the conversion. Parents don’t know anything about what is going on.

The Common Core….basically a good idea , for a national syllabus… has been a fraud for the math program., at least, at the high school level. They said that the math program taught too much, in not enough depth.
The promised to slim down the curriculum, and get better mastery of the math topics that they left in.
Did they do that? Not based on what I hear from former math colleagues. Too much material, not enough time to teach it well. I haven’t examined but a couple of Algebra 1
exams, but that is what I’ve seen so far. The questions seem ok, moving towards more mature function notation. That seems ok to me, as long as the kids have been brought along to that level in the middle schools.
An opinion from someone who has been there….before and after.

The above comment was written by my father, a 40 year veteran of teaching kids how to do their best. “Will I ever need this?” asked an 11th grade math student, after a lesson on solving quadratic trigonometric equations. “Never”, I answered honestly. “Then, why are we learning this?”, she questioned me. “Oh, you must be mistaken. You think I am teaching you math. Oh no, I am teaching you how to do your best. When will you ever need that? Just all the time!”
Two years teaching math, science, and English as a Peace Corps Volunteer in Nepal. Twenty nine years teaching at a 7-12 public high school. Elmont Memorial HS. Three years at New Hyde Park Memorial HS, 6 years at Sacred Heart Academy, an all girls private HS in Hempstead, Long Island.
I wrote questions and served on the exam committee for the NYS Regents Course II exams for 3 years, and wrote and edited questions for the “new” Regents Algebra and Geometry exams. I have taught every level of math from 7th grade through AP Calculus.
“Teaching the world. One student at a time”, is how one of my students once referred to me. I had the greatest job in the world. Everyone should be so lucky.