Regents Recap — June 2012: Erroneous Questions

Here is another installment from my review of the June 2012 New York State Math Regents exams.

Mathematically erroneous questions consistently appear on these exams.  Here are two recent examples, both from the Algebra 2 / Trigonometry exam.

According to the scoring key, the correct answer is (4).  This would be the correct answer if the angle were given as -50 degrees.  Notice, however, that no degree symbol is present.  This means the angle is actually -50 radians.  In degrees, -50 radians is equivalent to roughly -2864.8 degrees, which itself is equivalent to roughly 15 degrees.  Thus, the actual correct answer is (3).

The above problem could be considered a typo (although no correction was ever issued), but the most erroneous Regents questions demonstrate a real lack of mathematical understanding on the part of the exam creators.   Consider the following question on complex numbers.

None of these answers are correct.

The exam writers believe that (3) is the correct answer.  Given a complex number a + bi,  the conjugate is indeed a – bi, provided that and b are real numbers.  But x is a variable, and there is no reason to assume that x has to be a real number.  If x = i, for example, (3) is not the complex conjugate of  -5x + 4i.  In this case, the conjugate of the original expression is i, while (3) evaluates to – 9i.

As emphasis on standardized exam performance continues to grow, a few points here or there can make a big difference in the lives of students, teachers, and schools.  The consistent appearance of erroneous mathematics on these exams calls into question their validity as a measurement of “student achievement”.

Regents Exam Recap: June 2012

Below is a collection of posts analyzing the June 2012 New York State Math Regents exams.

The purpose of these analyses is to bring critical attention to the quality and utility of these standardized exams, which play an increasingly substantial role in today’s educational environment.

But, as the role of standardized tests continues to grow in the evaluation of students, teachers, schools, and districts, one question rarely gets asked:  “Are these tests any good?”.

Analyses of June 2012 New York State Math Regents Exams

June 2012 Math Regents Exams — Mathematically Erroneous Questions

June 2012 Math Regents Exams — Poorly Constructed Questions

June 2012 Math Regents Exams — Unscaled Graphs

June 2012 Math Regents Exams — Throwing Darts

June 2012 Math Regents Exams — Spot the Function

June 2012 Math Regents Exams — Some Improvement

 

Click here to see recaps of other New York State Math Regents exams.

Math Quiz: NYT Learning Network

Through Math for America, I am part of an on-going collaboration with the New York Times Learning Network.  My latest contribution, a Test Yourself quiz-question, can be found here:

https://learning.blogs.nytimes.com/2012/08/06/test-yourself-math-aug-6-2012/

This question is based on the rare tie that occurred in a Olympic qualifying race.  In how many different ways could eight racers finish first, second, and third?

Guest Post for Moebius Noodles: Mathematical Weaving

It was an honor to contribute a guest post to Moebius Noodles, a wonderful project focused on creating resources that provide rich mathematical experiences for young children.

After seeing my TED Talk on Creativity and Mathematics, Maria Droujkova, one of the authors of Moebius Noodles, contacted me and asked if I would put together a piece about how I use weaving to explore mathematical ideas.

My piece is titled “Weaving Your Way Through Mathematics,” and can be found on the Moebius Noodles website.

http://www.moebiusnoodles.com/2012/07/weaving-mathematics/

More resources on mathematics and weaving can be found here.

https://mrhonner.com/weaving/

Replace Algebra with … Algebra?

linear functionsArguments that suggest we over-emphasize mathematics in education don’t bother me.  I love math and see its utility in every aspect of my life, but I understand not everyone feels this way.  Also, when someone says we shouldn’t teach math, or we should teach less, it encourages me to reflect on my own beliefs about math and teaching.  This is usually a valuable experience.

So I read “Is Algebra Necessary?“, Andrew Hacker’s OpEd in the New York Times, knowing that I probably wouldn’t agree with much of it, but still prepared to examine my beliefs.  Hacker offers up a few of the standard reasons why students shouldn’t be forced to take “Algebra” in high school (“it’s too hard”; “it turns kids off”; “not everyone’s going to be an engineer”), but he doesn’t really bring anything new or substantial to the discussion.

One thing I did find interesting, however, was Hacker’s suggested fix:  instead of “Algebra”, we should be teaching courses like “Citizen Statistics”.

It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted — and include discussion about which items should be included and what weights they should be given.

This is indeed a good idea.  Ironically, exploring the mathematics of the CPI is largely an algebraic activity.

Discussing which items should be included means creating a mathematical model and declaring variables for the unknown quantities we wish to investigate.  These fundamental skills are taught and emphasized in high school algebra.

Determining the weights that these items should be given essentially amounts to finding the coefficients of some function of those variables and exploring the consequences of those choices.  In their basic forms, these skills are also taught and developed in high school algebra.

Thus, it seems to me that Hacker is suggesting we replace Algebra, with, well, Algebra.

There are worthwhile discussions to be had about what we are teaching, why we are teaching, and how we are teaching.  But those discussions should be led by people who really understand what’s going on.  If Andrew Hacker thinks we should replace Algebra with Algebra, then someone else should be leading the discussion.

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