For over 10 years I have been writing and speaking about erroneous math test questions and their consequences. Question 25 from the June 2022 New York State Algebra 2 exam offers a clear and simple picture of those consequences.

The student is asked if the equation has “imaginary solutions”, that is, if the solutions to this equation, 2 +3i and 2 – 3i, are imaginary numbers. These solutions are complex but not imaginary, because imaginary numbers are multiples of i, the imaginary unit. Therefore the answer should be no, this equation does not have imaginary solutions.

As you might have guessed, that’s not the answer they were looking for.

In this “complete and correct” response from the state’s official model response set, the student identifies these solutions as imaginary. These numbers are not real, but they are not imaginary, a subtle but meaningful distinction that neither the student nor the exam creators seem to understand.

Is the distinction important? Maybe not. But what is important is that this student’s lack of understanding of complex numbers will only be amplified by this exam. Even worse, teachers around the state might themselves be confused after reading this model response set. What will they teach their students about imaginary numbers next year?

Worst of all, what about the students who actually do know the difference between imaginary numbers and non-real complex numbers? They’re caught in a trap: Should they give the correct answer and possibly lose points, or should they try to guess what the exam creators really meant to ask? These tests put students in this trap over and over and over again, and ultimately students learn that details don’t matter and that thinking too much is a hazard. Students, and their teachers, deserve better.

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On the one hand, it’s good that standardized math tests are trying to include more examples of mathematical modeling, one of the true applications of math to the real world. On the other hand, if these tests promote a false, even dangerous, idea of what a mathematical model is, then they shouldn’t bother trying.

This question from the New York State Algebra 2 Regents exam commits a fundamental error of mathematical modeling: it confuses the model for the phenomenon itself.

Is the maximum depth of the water 12 feet? We don’t know. The model of the water’s depth, d(t), takes a maximum value of 12 feet, but the model is only an approximation of reality. The actual maximum depth of the water is likely to differ from the model, as are the times of high and low tide. We can’t draw specific conclusions like (1), (2), or (4), we can only approximate. This means that all these statements are probably false.

Oddly enough, answer choice (3) seems to understand that models are just approximations, which makes the other answer choices even less defensible. (And all of this ignores the question of whether or not students have the requisite domain-specific knowledge of oceanography to understand what high- and low- tides are.)

In the grand scheme of these exam errors, this is a minor footnote. But as I’ve argued in these posts, and in my talk g = 4, and Other Lies the Test Told Me, these kinds of errors have a cumulative effect of training students to stop thinking when doing and applying math and instead just try to guess what the question writer wants to hear. We should expect more from our assessments.

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