Having spent a great deal of time dissecting and analyzing the 2011 New York Math Regents Exams, I was quite interested to see the January 2012 tests.

The same kinds of issues are generally present. There are instances of mathematical errors, poorly constructed questions, underrepresented topics, and 9th-grade questions on 11th-grade exams. Here is a quick overview of the Algebra 2 / Trigonometry Regents, the highest-level state math exam in New York.

**Mathematical Errors / Poorly Constructed Questions**

The exam writers for the New York Regents exams continue to find new and innovative ways to construct erroneous questions. Here is number 23 from the multiple choice section:

*Which calculator output shows the strongest linear relationship between x and y?*

This is a bad question to begin with, in that it really isn’t a *math *question. The student isn’t being asked to ponder anything mathematical; instead, the student must recognize a step in an artificial procedure about which they have no real mathematical understanding. (The mathematical technique for finding linear regression equations is not taught in this course; it is expected that the student will use the calculator to generate the equation).

What’s remarkable here lies in the “answer”. The way you assess the relative strengths of regression equations is by comparing their correlation coefficients (the *r *values). Generally speaking, the closer *r* is to 1 or -1, the stronger the correlation. In answer choice (4), the value of *r* is closer to -1 than, say, the value of *r* is to 1 in answer choice (1). Thus, (4) is the correct answer. Pretty easy, right?

Amazingly, the situation represented in answer choice (4) is a logical impossibility**. ** Since the *r* value is negative, this means the correlation between the two sets of data is negative; but the regression line’s slope (the *b* value) is positive! This cannot happen.

As a result, a scoring correction was issued (after the exams had most likely been graded) and all students were to be given credit for this problem regardless of what answer they put. And so another flawed question makes it through the draft-revision-publish cycle and into the hands of thousands of students.

**Underrepresented Topics**

By my count, only one question on this exam (worth 2 out of 87 total points) required the use of either the Law of Sines or the Law of Cosines. Now, I don’t know how many points should be allocated for these particular techniques, but they are fundamental ideas in trigonometry: they should be a non-trivial part of the course.

Indeed, a quick look at the exam’s reference sheet is illuminating: less than half the formulas that are traditionally provided for the student relate to questions on this particular test.

**9th Grade Questions on 11th Grade Exams**

The final question, and the highest-valued question (6 points) on this Algebra 2 / Trigonometry exam, asked the student to simplify the following expression:

Once again we see a problem from the 9th-grade Algebra curriculum playing a significant role in the 11th-grade Math Regents exam. A quick look at the official Integrated Algebra Pacing Guide shows that dividing and simplifying rational expressions, the techniques that this problem requires, are part of the 9th-grade course.

This isn’t necessarily an easy problem, and it does require dealing with a cubic polynomial (although factoring by grouping is an optional part of the 9th-grade curriculum). But this is yet another example in the evolution of this exam showing that, year after year, the hardest questions seem to get easier.

To be completely fair, it was nice to see the exam writers include asymptotes on their graphs this time (to avoid fake asymptotes like these), and they did demonstrate a little more understanding about 1-1 functions (perhaps they did a little studying after this disaster). But overall, it seems to be business as usual.

Back when Brooklyn Tech had math placement exams for the incoming freshman(I think the Class of 2010 was the last year that occurred), I remember answering a question like factoring and simplifying polynomials. Sad that every year the Regents get easier.

Isn’t the slope the b value?

Ha! Yes, it is; thanks for pointing it out. I’ve made the correction.

Actually, I think I may have fallen into the same trap as the exam writers: since linear regression equations are usually given in the form

ax + b, I just expectedato be the slope.Thanks for the edit!

Nice to know that NYSED is listening to the experience of our math teachers in thinking about how to revise and improve these tests.

[…] math teacher at New York’s prestigious Brooklyn Tech High School analyzed multiple weaknesses in the Algebra II/Trigonometry Regents exams from 2011 and 2012. His take on […]

I remember hearing of a badly designed regents problem concerning an octahedron and a tetrahedron, both having edges of the same length:

If one of the triangular faces of the tetrahedron is glued onto a triangular face of the octahedron to form a compound polyhedron, how many faces does the newly-formed polyhedron have?

The regents answer was 10, but the correct answer is 7, as the dihedral angles are supplementary.

Have you heard of this problem? I’m looking for a reference.

Thanks!

I think you are recalling a PSAT question which asked about the result of gluing a tetrahedron and a square pyramid together.

http://news.google.com/newspapers?nid=1129&dat=19810318&id=TYFIAAAAIBAJ&sjid=qW0DAAAAIBAJ&pg=2032,2757555

Yes – that’s it! Thanks for the reference. Interesting, about how I assumed the worst about the regents!

An r value does not indicate a linear relationship. It is quite easy to create a plot that is clearly curved but has an r value very close to 1 as well as plots that are linear with low r-values. This is not a nit-picky issue. Anyone with experience in statistics will tell you that it is a must to look at the plot before declaring a relationship to be linear.

I would be surprised if anyone with a hand in writing this exam actually knew how to compute a correlation coefficient, much less address the criticisms you’ve posted here.