Teaching

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.

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By MrHonner, ago

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, ago

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!

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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.

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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!

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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!

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

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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!

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By MrHonner, ago

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.

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By MrHonner, ago

Pitfalls for Parents — MoMath

I’m excited to be joining Steven Strogatz this coming Monday for QED: Pitfalls for Parents, hosted by the National Museum of Mathematics.

In this event, Steve — a renowned mathematician, author, and speaker, as well as the museum’s current mathematician-in-residence — answers questions from the audience about K -12 mathematics. Steve has been inviting classroom teachers along as special guests, and I’m thrilled to be joining him on February 7th at 7 pm.

You can find out more about the event, and register, here.

By MrHonner, ago
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