Teaching

Is 2+3i an Imaginary Number?

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 $x^2 + 4x-13=0$ 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.

When a Model Isn’t a Model

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.

Related Posts

Regents Recaps
June 2022 Algebra 2 Regents Exam (Twitter Thread)
Regents Recap — January 2018: Problems with Pre-Calculus
Regents Recap — August 2015: Modeling Data

By MrHonner, 4 yearsJuly 14, 2022 ago

Resources Teaching Writing

What a Math Party Game Tells Us About Graph Theory — Quanta Magazine

My latest column for Quanta Magazine explores some deep (and recent!) results in graph theory using a simple mathematical party game. Trying to get your entire group of friends to each shake an odd number of hands leads to some fundamental and surprising results, like the impossibility of some simple configurations.

This also ties in to some recent research that has determined new bounds on the way a graph can be partitioned into subgraphs. You can read the full article here, which includes some fun and challenging exercises at the end.

By MrHonner, 4 yearsMarch 25, 2022 ago

Resources Statistics Teaching

The All 1s Vector

Here’s a short post based on a Twitter thread I wrote about a very underappreciated vector: The all 1s vector!

Every vector whose components are all equal is a scalar multiple of the all 1’s vector. These vectors form a “subspace”, and the all 1’s vector is the “basis” vector.

Let’s say you have a list of data — like 4, 7, -3, 6, and 1 — and you put that data in a vector v. An important question turns out to be “What vector with equal components is most like my vector v?”

To answer that question you can *project* your vector onto the all 1’s vector. You can think of this geometrically — it’s kind of like the shadow your vector casts on the all 1’s vector. There’s also a formula for it that uses dot products.

Because of the way the dot product works and the special nature of the all 1’s vector, v•a is the sum of the elements of v and a•a is the number of elements in v. This makes (v•a)/(a•a) the mean of the data in v!

Since 3 is the mean of your data, the vector with equal components that is most like your vector is the all 3’s vector. This makes sense, since if you’re going to replace your list of data with a single number, you’d probably choose the mean.

Now the cool part. Look at the difference in these two vectors: These are the individual deviations from mean for each of your data points, in vector form!

And geometrically this vector of deviations is perpendicular to the all 1’s vector! You can check this using the dot product.

So data can be decomposed into two vector pieces: one parallel to the all 1’s vector with the mean in every component, and one perpendicular to that with all the deviations. You can see hints of independence, variation, standard deviation lurking in this decomposition.

You can check out the original thread on Twitter here, including some very interesting replies!

Teaching

Making Math by Design — Queen’s College

I’m excited to be visiting Queen’s College on Tuesday to speak to students in QC’s TIME 2000 program. TIME 2000 prepares future teachers by having cohorts of undergraduates study math and education together. The program also puts on a great conference that shares the fun and beauty of math with high school students.

I’ve participated in the conference several times, but on Tuesday I’ll be speaking at the TIME 2000 Spring Seminar series. In Making Math by Design I’ll talk about the decisions that teachers make, and the consequences of those decisions, when we design and implement mathematical tasks for our students. I’m looking forward to doing some math together and having a good conversation about it afterward.

It’s been several years since I’ve visited Queen’s College, and I’m excited to be heading back, especially since this talk was originally scheduled for 2020 and was my first in-person talk cancelled by the pandemic! Let’s hope nothing else happens before Tuesday.

Is 2+3i an Imaginary Number?

When a Model Isn’t a Model

What a Math Party Game Tells Us About Graph Theory — Quanta Magazine

The All 1s Vector

Making Math by Design — Queen’s College

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