Archive of posts filed under the Testing category.

## Regents Recap — August 2017: Yes, You Can Work on Both Sides of an Identity

In a controversial post last year, I argued that it’s perfectly acceptable to work on both sides of an equation in proving an algebraic identity. While it’s common to tell students “You can’t cross the equal sign” in this situation, doing so is mathematically legitimate as long as the new equation is true under exactly the same circumstances as the original.

For example, when proving an algebraic identity, multiplying both sides of an equation by 2 is permissible, because = y and 2x 2y are true under exactly the same conditions on x and y. Squaring both sides of an equation however, is not, since

$x^2 = y^2$

can be true under conditions that make y false, say, when x and y-2.

The post in question, “Algebra is Hard”, was a response to a June 2016 Regents scoring guide that deducted a point from a student who, in proving an algebraic identity, multiplied both sides of the equation by a non-zero quantity. The student was penalized for “not manipulating expressions independently in an algebraic proof“, a vague and meaningless criticism.

“Algebra is Hard” received quite a bit of attention, and while many agreed with me, I was genuinely surprised at how many readers disagreed. Which was terrific! Of course my argument makes perfect sense to me, but it was great to have so many constructive conversations with teachers and mathematicians who saw things differently.

But my argument recently received support from the most unlikely of sources: another Regents exam.

Take a look at this exemplar full-credit student response to an algebraic identity on the August 2017 Algebra 2 exam.

Notice that the student works on both sides of the equation and subtracts the same quantity from both sides. Even though the student did not manipulate expressions independently in an algebraic proof, full credit was awarded.

The note here about domain restrictions is an amusing touch, given that it was the explicit domain restriction in the problem from 2016 that ensured the student wasn’t doing something impermissible (namely, multiplying both sides of an equation by 0).

So in 2016 this work gets half credit, and in 2017 this work gets full credit.

While it’s nice to see mathematically valid work finally receiving full credit on this type of problem, it’s no consolation to the many students who lost points for doing the same thing the year before. What’s especially frustrating is that, as usual, those responsible for creating these exams will admit no error nor accept any responsibility for it.

Be sure to read “Algebra is Hard” (and some of the 40+ comments!) for more of the backstory on this problem.

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## Regents Recap, August 2017: How Do You Explain that Two Things are Equal?

Sue believes these two cylinders from the August, 2017 New York Regents Geometry exam have equal volumes. Is Sue correct? Explain why.

Yes, Sue, you are correct: the two cylinders have equal volumes. I computed both volumes and clearly indicated that they are the same. Take a look!

Wait. Why did I only get half-credit? What’s the problem, Sue? You don’t think this is an “explanation”? The two volumes are equal. The explanation for why they are equal is that I computed both volumes and got the same number. I don’t know of any better explanation for two things being equal than that.

What’s that? You wanted me to say “Cavalieri’s Principle”? But if I compute the two volumes and show that they are equal, why would I need to say they are equal because of some other reason?  Oh, never mind, Sue. See you in Algebra 2.

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## Regents Recap, June 2017 — Assessing Irrationality

Despite its shortcomings, this kind of question keeps appearing on New York State math exams.  This is number 27 from the June, 2017 Common Core Algebra exam.

Here’s an example of a full credit response according to the official model response set provided by the state.

There is no explanation here.  The argument is simply It’s True Because It’s True:  the difference between a rational number and an irrational number is irrational because the difference between a rational number and an irrational number is irrational.  All the student has done is identified one number as rational and one number as irrational (without even identifying which is which) and recited the frequently-tested property.

As scored, this question is designed to test recall of a specific, incidental fact while intentionally avoiding the relevant mathematical content, namely, what it means for a number to be rational or irrational.  A second model response that actually demonstrates some mathematical knowledge about irrational numbers earns only partial credit.

Unlike the student in the first response, or the test makers for that matter, the student here recognizes that the irrationality of the square root of 2 should be established.  The explanation isn’t completely correct, but it demonstrates much more understanding than the first response.  Unfortunately, as long as questions like this keep appearing on these exams, students and teachers will continue to be rewarded for mindlessly regurgitating what the test makers want to hear.

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## Regents Recap — June, 2017: More Trouble With Statistics

High school math courses contain more statistics than ever, which means more statistics questions on end-of-year exams.  Sometimes these questions make me wonder what test makers think we are supposed to be teaching.  Here are two examples from the June, 2017 exams.

First, number 15 from the June, 2017 Common Core Algebra exam.

This question puzzled me.  The only unambiguous answer choice is (3), which can be quickly eliminated.  The other answer choices all involve descriptors that are not clearly defined:  “evenly spread”, “skewed”, and “outlier”.

The correct answer is (4).  I agree that “79 is an outlier” is the best available answer, but it’s curious that the exam writers pointed out that an outlier would affect the standard deviation of a set of data.  Of course, every piece of data affects the standard deviation of a data set, not just outliers.

From the Common Core Algebra 2 exam, here is an excerpt from number 35, a question about simulation, inference, and confidence intervals.

I can’t say I understand the vision for statistics in New York’s Algebra 2 course, but I know one thing we definitely don’t want to do is propagate dangerous misunderstandings like “A 95% confidence interval means we are 95% confident of our results”.  We must expect better from our exams.

UPDATE: Amy Hogan (@alittlestats) has written a nice follow up post here.

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## Regents Recap — June, 2017: Three Students Solve a Math Problem

I will never understand why so many exam questions are written like this (question 5 from the June, 2017 Algebra exam):

What is the purpose of the artificial context?  Why must the question be framed as though three people are comparing their answers?  Why not just write a math question?This question not only addresses the same mathematical content, it makes the mathematics the explicit focus.  This would seem to be a desirable quality in a mathematical assessment item.

Instead of wasting time concocting absurd scenarios for these problems, let’s focus on making sure the questions that end up on these exams are mathematically correct.

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