Welcome to Matt Salomone’s Mathematics Home

Welcome to my newly-updated web home at matthematics.com .

I enjoy thinking, tweeting, and sometimes blogging about emerging issues in the teaching and learning of mathematics and quantitative literacy. Some of what you’ll find here includes:

Open Courses/Text Projects:

Continue reading “Welcome to Matt Salomone’s Mathematics Home”

The MPG Illusion, Revisited

In their paper on using cognitive illusion to improve quantitative literacy:

Numeracy Infusion Course for Higher Education (NICHE), 1: Teaching Faculty How to Improve Students’ Quantitative Reasoning Skills through Cognitive Illusions Wang, F., & Wilder, E. I. (2015).  Numeracy, 8(2), 6.

the authors describe the following scenario, originally presented inScience magazine article and summarized in the above slide from my Grow Up, Branch Out interactive video.

Consider a family that has an SUV that gets 10 MPG and a sedan that gets 25 MPG. Both are driven equal distances in a year. Is the family better off replacing the SUV with a minivan that gets 20 MPG or replacing the sedan with a hybrid that gets 50 MPG?

This problem is described as a cognitive illusion because many of us, thinking heuristically (that is, without stepping back and working out the details), are drawn to either conclude that:

  1. The 25-to-50-mpg upgrade saves more fuel, since 50 mpg is qualitatively much more efficient. (It’s a “big number” in this context.)
  2. Or, the two upgrades save equal amounts of fuel, since each represents the same doubling of fuel economy in miles per gallon. (Our preferred, and very often quite useful, proportional-reasoning heuristic.)

However, neither is true. Doubling the miles-per-gallon economy of the lesser efficient vehicle results in greater fuel savings. Why is this the case? We’ll look at how arithmetic exposes this to be an issue of denominator neglect, and how the way Americans think about fuel economy is essentially shaped by cultural choices.

The Arithmetic

The key observation here is about unit conversion. Converting a quantity from one set of units into another is always a matter of either multiplying or dividing by the number 1, by representing 1 as a clever fraction whose numerator and denominator are equal measurements using unequal units.

Fuel economy, in this view, can be thought of as a conversion factor representing an equality between an amount of fuel consumed (in gallons) and a distance driven (in miles). The SUV making 10 miles per gallon, for instance, can be represented as the “equation”

\( 10 {\rm \; miles} = 1 {\rm \; gallon} \)

This can be made into a conversion factor, then, by dividing this equation by either of its sides:

\( \frac{10 {\rm \; miles}}{1 {\rm\; gallon}} = 1 = \frac{1 {\rm \;gallon}}{10 {\rm \; miles}} \)

Where we get tripped up, when we’re thinking heuristically, is in figuring out which of these two conversion factors is appropriate to the question. That is, are we more worried about multiplying something by 10? Or dividing it by 10?

Asking a crucial quantitative reasoning question — what’s changing and what’s not? — reveals that the number of miles the vehicles are being driven each year is remaining constant. So we will have the same number of miles for the SUV as for the minivan; the question is, what happens to the amount of fuel consumed? We therefore need to convert from miles into gallons, meaning we must “cancel the miles and introduce the gallons.” This means using the \(\frac{1 {\rm \;gallon}}{10 {\rm \; miles}}\) factor, or dividing by the MPG rating.

A little additional thought should persuade us that the actual number of miles driven per year will not affect our answer, so we’ll choose a convenient, round number: say, 10,000 miles. Here, then, are the results of dividing that number by each of the MPG figures to determine the amount of fuel each of these four vehicles would need to drive that far.


SUV (10 mpg):

\(10\,000 {\rm \; miles} \cdot \frac{1 {\rm \; gallon}}{10 {\rm \; miles}} = 1\, 000 {\rm \; gallons}\)

Sedan (25 mpg):

\(10\,000 {\rm \; miles} \cdot \frac{1 {\rm \; gallon}}{25 {\rm \; miles}} = 400 {\rm \; gallons}\)

Minivan (20 mpg):

\(10\,000 {\rm \; miles} \cdot \frac{1 {\rm \; gallon}}{20 {\rm \; miles}} = 500 {\rm \; gallons}\)

Hybrid (50 mpg):

\(10\,000 {\rm \; miles} \cdot \frac{1 {\rm \; gallon}}{50 {\rm \; miles}} =200 {\rm \; gallons}\)

This upgrade saves: \(1\, 000 – 500 = 500\) gallons This upgrade saves: \(400 – 200 = 200\) gallons

So, the mere fact that the numbers we’re seeing on the screen (the MPG ratings) belong in the denominator of the conversion not only thwarts our proportional reasoning, it inverts it. Denominators, as I often tell my students, are Bizarro World: up is down, bigger is smaller, and “nothing could mean anything.”

The Gravamen of Culture

Quantitative reasoning is a human expression of mathematical thought. As such, QR is inescapably socially constructed and culturally informed.

In this example, the cultural information that obscures our reasoning is the distinctly American habit of reporting fuel economy with the fuel in the denominator (miles per gallon). Cognitively, this has several effects.

  1. On the plus side, it supports “more is better” cognition. Except for trolls who roll coal, most of us consider more fuel-efficient vehicles to be more preferable when shopping for a car. Situating the fuel in the denominator means (again, Bizarro World) higher values mean less fuel: higher MPG rating means more fuel efficiency.
  2. On the down side, it centers our cognition on miles instead of gallonsNot only is this the reason for our thwarted intuition in this problem, it also invites us — whether we realize it or not — to think of our fuel consumption as fixed, and imagine instead how much further we could drive on that same amount of fuel. By playing into our tendency toward denominator neglect, this framing also plays into the fossil-fuel industry’s hands because, for most drivers, annual mileage is more likely to remain constant when they purchase a new car. The real impact they are making when they upgrade is that they are (gasp!) consuming less fuel.

It’s just like the classic advertising story about A&W’s failed third-pound burger:

Denominator neglect is the source of all manner of false cognitive illusions. So, why do I describe this as a cultural phenomenon in the U.S.?

Because European regulatory agencies have chosen to avoid it. European agencies report fuel economy with fuel in the numerator instead of the denominator, for instance, turning the dimensional “equation”

\(100\; {\rm kilometers} = 24 \; {\rm liters}\)

into the conversion factor

\( \frac{24\; {\rm liters}}{100\; {\rm kilometers}} \)

instead of the reciprocal. This approach sacrifices the “more is better” cognition, because a smaller figure — using less fuel over an equal distance — represents greater efficiency. But it avoids denominator neglect, it shines the spotlight on the amount of fuel consumed as the driver’s main independent variable, and it re-energizes our proportional reasoning.

In the EU, for example, this exercise would not create a cognitive illusion.

Consider a family that has an SUV that uses 24 L per 100 km and a sedan that uses 10 L per 100 km. Both are driven equal distances in a year. Is the family better off replacing the SUV with a minivan that uses 12 L per 100 km, or replacing the sedan with a hybrid that uses 5 L per 100 km?

Now, because the fuel usage is in the numerator instead of the denominator, and we are multiplying by these numbers in our conversion rather than dividing, our proportional reasoning should give us a more sensible insight. Let’s imagine 16,000 km of annual driving, which is approximately the same as 10,000 miles (though again, that specific amount will not affect the conclusion):

SUV (24 L / 100 km):

\(16\,000 {\rm \; km} \cdot \frac{24 {\rm \; L}}{100 {\rm \; km}} = 3\, 840 {\rm \; L}\)

Sedan (10 L / 100 km):

\(16\,000 {\rm \; km} \cdot \frac{10 {\rm \; L}}{100{\rm \; km}} = 1\, 600{\rm \; L}\)

Minivan (12 L / 100 km):

\(16\,000 {\rm \; km} \cdot \frac{12 {\rm \; L}}{100 {\rm \; km}} = 1\, 920 {\rm \; L}\)

Hybrid (5 L / 100 km):

\(16\,000 {\rm \; km} \cdot \frac{5 {\rm \; L}}{100 {\rm \; km}} = 800 {\rm \; L}\)

This upgrade saves: \(3\, 840- 1\, 920= 1\, 920 \) L This upgrade saves: \(1\, 600 – 800= 800\) L

Shockingly, the answer is still the same as our MPG example! (In part, this is because the fuel economy figures I chose are approximately equivalent to the originals.) But the proportional reasoning “works” in that we can clearly see, in the last row, that halving the fuel consumption rates has indeed halved the amount of fuel consumed in either case. It’s just that the SUV is using so much more fuel in a year compared to the sedan, since both are being driven equal distances, so half of its total consumption is still significantly greater than half of the sedan’s. We’ve traded denominator neglect for a form of base rate neglect instead. But at least there’s a glimmer of valid proportional reasoning here.

The Moral

Because of the precise nature with which numbers carry meaning, correct quantitative insights almost always require engaging our slower, analytical cognitive machinery — rather than relying on the quick, reflexive answers our relational minds provide by default.

But our “System 2” thinking requires significant effort to activate, and that means that the lazy, heuristic System 1 can occasionally catch out even more numerate people. Trying to power quantitative skills with heuristic thinking is like trying to build a campfire with only newspaper and lighter fluid: bound to generate more light than heat, and leave everyone out in the cold.

Grow Up, Branch Out: Quantitative Literacy for the 21st Century

This interactive video and set of resources was developed to support a faculty workshop offered by the Massachusetts Department of Higher Education in 2019. Special thanks to Robert Awkward, DHE’s Director of Learning Outcomes Assessment, for organizing and providing financial and logistical support for these workshops; and to Mary-Ann Winkelmes, Director of Teaching and Learning at Brandeis University, for co-facilitating the workshops by engaging participants in designing transparent assignments. Continue reading “Grow Up, Branch Out: Quantitative Literacy for the 21st Century”

Anti-Numeracy: Valid, But Not Okay

The only thing worse than our pervasive cultural misbeliefs about numeracy is when mathematicians give them cover.

Here’s an example, with author omitted. (These gags are ubiquitous and I’m not trying to “cancel” anyone!)

I am a trained mathematician right up until I have to calculate a restaurant tip. (In reply to the below)

And while I know these are tongue-in-cheek funny jokes (so please don’t @ me), I have to ask: Who laughs? Who’s supposed to laugh? And what happens when they do?

Continue reading “Anti-Numeracy: Valid, But Not Okay”

Boom, Bust, Hockey Stick: Unanimity in the U.S. Supreme Court since 1945

Earlier this month, I did a preliminary assessment of Andrew Torrez’s speculation on the Opening Arguments podcast that the Roberts Court has ushered in a new era of polarization on the U.S. Supreme Court. The answer, looking at 20 years of history, seemed to be no. A wider view of 75 years of history, meanwhile, suggests the answer is… still no. The Roberts Court is not significantly more polarized in its merit case votes than any other Court in this history.

Does the defendant’s case hold water? “No. Andrew. Was. Wrong!”

But, the data suggest two interesting trends in Supreme Court unanimity over the past 75 years: a steady boom-and-bust cycle about every decade, and a significant Roberts Court uptick in the second derivative suggesting that year-over-year, the consensus about consensus may be disappearing.

Continue reading “Boom, Bust, Hockey Stick: Unanimity in the U.S. Supreme Court since 1945”

The Visual Syllabus (2019 National IBL Conference Poster)

“If my teaching is ‘different,’ my syllabi should be too.”

About three years ago, concomitant with my wholesale switch to standards-based grading, I also set aside the well-worn course syllabus template that I’d used for all my courses and set out, from a blank page, to design a syllabus my students would find worth reading. The result is a colorful, four-page visual syllabus that is now the key artifact of my teaching. Continue reading “The Visual Syllabus (2019 National IBL Conference Poster)”

Teaching on Twitch II: My Rig

I’ve made this much longer than usual because I have not had time to make it shorter.
– Blaise Pascal

Since I started making instructional videos for the open web about 7 years ago, and especially since I started live streaming problem-solving sessions and office hours via Twitch, I’ve been meaning to give a run-down on the hows of the particular kind of video capture I do. (For some examples of the videos my streaming and capture produce, see my YouTube group theory playlist from this semester.)

This is a quick snapshot of my setup to highlight some of the amazing hardware and software that make my videos and my streaming possible.

My home office desk, set up for a live streaming abstract algebra problem session. Click for an enlarged view.

On My Desk (Hardware):

  1. PC. I’ve been a Mac user since graduate school, but crossed back over to PC for my home office this year because I wanted something easier to customize and upgrade. I prioritized graphics card power, RAM, and a speedy solid-state hard drive, and for less than half the price of my work-issued MacBook Pro, this handles real-time video compositing far, far better. Plus, plenty of screen real estate is super helpful for streaming.
  2. A “nice” microphone. Most important part of any educational video setup. Mine’s a Behringer B-2 Pro, connected via a Shure XLR-to-USB audio capture. This very microphone may or may not also have been part of the production of the Klein Four’s double-platinum 2005 album, Musical Fruitcake.
  3. An “okay” webcam. Not pictured here – it’s mounted above my monitor – is the ubiquitous Logitech HD Pro C920, one of the best values in a 1080p webcam I’ve seen. Since we’re using the separate microphone, all we need the webcam to do is feed video – not audio.
  4. iPad Pro. An iPad has been the one constant in my setup since day one, because it is a great platform for hand-written whiteboard work and it runs the killer app Doceri (see below) for the purpose. I started on an iPad 2 (long ago!) so fancy Apple Pencil support isn’t necessary – though it is nice.
  5. Personal green screen. ( http://thewebaround.com ) I don’t always use it, but this nifty tool lets me key out (delete) the background behind me so I can float neatly over my screen, just like a TV meteorologist. See this live stream for an example of the effect.
  6. Mechanical keyboard. Optional. There’s an open conjecture that any keyboard will suffice. But ask me about mine. (Plural.)
A view of my PC desktop during a streaming session. Click here for a full view.

On My Desktop (Software):

This is where I’ve stirred together the most different tools over the years, and where the difference between recording for asynchronous viewing and sharing for live streaming is the biggest.

  1. Doceri interactive whiteboard app ( http://doceri.com ). Capturing & streaming. Hands down, the top of my list. Doceri is the one tool that I’ve run with the most in transforming my teaching. As you can see in the two pictures above, Doceri is an app on my iPad, connected via Wi-Fi to the PC that’s mirroring its display and feeding that display into a live stream. And mirroring whiteboard work onto a PC is only a fraction of what Doceri can do. It can also capture videos natively on its iPad app and share them to a variety of destinations including direct to YouTube, which I use to capture parts of my lectures during face-to-face classes. It can also, when connected via Wi-Fi to a PC, control that PC remotely from the app. Seriously – this app was worth every one of the few pennies I spent for it seven years ago. They’re not paying me to say that. But I wouldn’t turn them down if they wanted to.
  2. OBS Open Broadcasting Software ( http://obsproject.com ). Streaming only, though it’s capable of capture too. OBS is the Grand Central Station into which all my hardware and software feed, which composites the video and audio and pushes out a live stream. OBS is the software of choice for what I believe is a clear majority of streamers – most of whom stream video gameplay to places like Twitch, though it can also be connected to other live video services such as Periscope, Instagram Video, and Facebook Live. Because it’s open-source software, it comes with no cost and no developer support, but because it’s so widely used there is a large community of users who support one another’s technical questions in places like the OBS subreddit.
  3. Screencast-O-Matic ( http://screencast-o-matic.com ). Capture only. When I’m capturing a simpler video with only whiteboard and webcam, Screencast-O-Matic is a more lightweight and user-friendly alternative to OBS. Because of its ease of use and ability to quickly composite webcam and screen capture videos, this is a popular tool with many instructors of online and hybrid courses.
  4. Streaming canvas elements. (I got mine from the free site TwitchOverlay.) Streaming only. This package of image files includes things like the screen and textbox backgrounds, webcam frames, and other visual elements that inhabit my streaming canvas. Getting these all set up just so took a little bit of work, but once I had the basics down in OBS I could just duplicate that “scene” and tweak it to vary what’s being displayed (whiteboard, web browser, both, just webcam, etc.).
  5. Text editor. Streaming only. The “agenda” list of topics on my streaming canvas is read from a text file so that I can update it in real-time from a simple text editor.
  6. Live chat platform. Streaming only. If you’re streaming to a platform where your students can already log in and chat (e.g., using Facebook Live and they all are on Facebook) this is a given. But most of my students lack the Twitch account needed to engage in live stream chat natively through the platform. So instead, I monitor our course’s Slack channel that we already use as a learning management tool during live streams for student questions.
  7. Maine coon mix cat. Sleeping on the blanket in the background. Again, optional equipment but I don’t trust being catless on the internet.
  8. Social media presences. Streaming & capture. This goes without saying, but the more connected you are via social media and the web, the more open, searchable, and accessible your content can be. My accounts on Twitch (to host a public stream), YouTube (to catch and curate all my videos, including archives of live streams), Twitter (to share with the wider #AcademicTwitter and #MTBoS audiences), IFTTT (an automation tool that does things like send automatic tweets and Slack messages to students when I go live), and this very blog, add up – I hope – to give access to my stuff to whomever is interested. Including my students, yes, but by no means limited to them.

At least, that’s how I do it right now. Questions? Want to try some of this out for yourself? Looking for one single tool to test out to move your teaching into the video realm? ( Doceri. ) Hit me up here or on Twitter.

The Infinite Series Sorting Hat

Looking for a way to keep all those convergence tests for infinite series straight? Looking for a cultural reference that some of your calculus II students will still find timely and relevant for a few more years? Look no further than the infinite series sorting hat.

Get the printable versions:

(Thanks to the many folks on Twitter who gave me suggestions to improve the original version!)