In the book and movie The Martian, astronaut Mark Watney is stranded on Mars when the rest of his crew barely escapes a sandstorm in which Watney is injured, and left for dead. Surviving in a hostile alien environment takes every ounce of courage, ingenuity and humor he can muster, invoking biology, chemistry, physics, and duct tape. The author Andy Weir, the scriptwriter Drew Goddard, and director Ridley Scott do a magnificent of avoiding (in Andy’s words) “sounding like a Wikipedia article”, and instead create a rousing (in Drew’s words) “love letter to science”.

After seeing the film, it dawned on me that the entire film is a glorious example of how to teach mathematical problem solving through story telling — something I believe is much needed in mathematics education. And indeed I found several articles online discussing the Math of The Martian, and The Martian is full of Word Problems. Here are five lessons all math teachers can learn from The Martian. (Warning: Spoilers ahead.)

Make the viewer care about the problem. Word problems in math class are lousy stories — mere equations with names attached — so students race through them without bothering to think carefully. In the Martian, you care about the problems Watney faces because you want him to survive, and because you are intrigued by the unfamiliar other-worldly setting. The classic educational game Oregon Trail uses a similar plot device — you are a pioneer leading a team of settlers out to the American West — to draw students into a singularly memorable educational experience.

A good way to make students care about word problems is to put student names and interests into the problem. It’s a simple trick, but it helps. The app and web site Bedtime Math captures student interest by leading with a popular image or intriguing news story, then asking questions about the story. You can do something similar by basing word problems on current events, or topics you are studying in class. You can also have students make up problems for others to solve — students are more motivated to solve problems that came from other students.

Let the problems lead. Problem solving is naturally dramatic. You encounter a problem you don’t know how to solve. You try something. It doesn’t work. You try something else. Eventually you have an insight and triumph. That’s the dramatic arc of story telling, and also of problem solving.

Author Andy Weir constructed The Martian by solving the unfolding sequence of problems that he put in front of his protagonist. Watney solves his food problem by farming potatoes. But where will he get water on Mars? Andy had to call on his rusty knowledge of chemistry to solve that one, and early readers corrected his mistakes. He let the actual science guide the plot, which led to a delightfully plausible story.

Just as Andy did not in advance where his plot was going, you as a teacher do not need to know in advance all the answers to the problems you pose. Instead, show your students that it is okay to struggle, to not know, and follow the problem wherever it leads.

Make thoughts visible. Every problem Watney solves has visible consequences. Plants grow. Warmed vehicle interiors allow him to wear less clothes. Rocket fuel explodes when the chemical mixture isn’t quite right. In contrast, solving a math problem on a worksheet has no visible consequence, so it is hard to care about the result. In Dan Meyer’s TED talk Math Class Needs a Makeover, he shows how to turn a drab word problem into an arresting visual experience.

Show the emotion. The Martian includes plenty of technical detail without ever losing narrative momentum. How? First, there is plenty of drama in the numbers. To survive Watney needs 1,500 calories a day. Growing potatoes will provide him with 115,500 calories. That will extend his food supply 76 days, which alas is far short of the thousand or so days he needs to last.

But more importantly, Watney is an astronaut, a human with a preternatural ability to focus on solving problems as they come up, even in the face of death. Especially in the face of death. So his palpable enthusiasm for numbers is an expression of his character. And of course what we’re really seeing here is author Andy Weir’s infectious zest for problem solving. By showing us the full range of Watney’s emotions, from joy to despair, the author keeps us engaged in the story.

Many teachers are self-conscious about showing the full emotion of problem solving, because that means admitting that you don’t always know what to do, can be bored or intimidated by problems, and sometimes make mistakes. But if we want students to be able to feel comfortable with uncertainty we need to start by modeling it ourselves. If we always act like we know exactly what we are doing, then students will do the same, covering up their confusion by pretending they understand, and never correcting their own misunderstandings.

A good way to make everyone feel safe with expressing emotion is to present puzzles that you have not tried yourself, and that require original thinking. That puts you and the class in the same position. Your role as teacher is to coach students to ask good questions. And you can do this confidently even if you don’t know the answers yourself.

Show the messiness. This is where The Martian really shines. Drama is about things going wrong. And in The Martian just about every type of thing that can go wrong does: from faulty weather prediction, freak accidents and mistaken assumptions (calculating the oxygen in the habitat without accounting for his own breathing), to political maneuvering (cooperating with China), logistical tradeoffs (shaving days off a production schedule), and ethical decisions (save Watney or save the other five crew members?).

And this is where schools get it all wrong. Traditional math classes prepare students to answer highly manicured, abstract problems like “find solutions to the equation x^2 – 5x + 6 = 0”. To this end, worksheets remove all messiness: no story, no context, no multiple solutions, only problems that fit a specific format. The result is students who memorize formulas without understanding what they mean, and cannot apply their learning to situations that differ even slightly from the forms they see in textbooks. By protecting students from the messiness of problem solving, we cripple them, and give them the false impression that problem solving is neat and efficient.

For students to master mathematics, they need to be experience math in the wild — full-bodied situations with real consequences and no neatly prepared roadmaps. Only by encountering meaningful problems, failing, and recovering, can students learn what it means to survive in the world mathematically. In the future I hope to see more stories like The Martian woven into mathematics education.

Shown below is the first voice of a new three-voice canon, based on the theme of Bach’s work  The Musical Offering. Your challenge is to figure out how to transform the first voice to make the second and third voices. To transform a voice you may delay when it starts by some number of bars, transpose it up or down by some interval, invert the intervals so the melody goes up where the original went down, reverse it in time (retrograde), or change the speed (augmentation or diminution, usually by a factor of two). The three voices, played simultaneously will make a harmonious whole, as they do in familiar canons like Frère Jacques. For a full introduction to canons, see Norman McLaren’s wonderful animation Canon or my talk for the Museum of Mathematics about symmetry in music and art. Please email me if you find the solution; I’m curious how hard it is to solve. I will post the solution on Bach’s birthday, March 31. (For those of you who think Bach’s birthday is on March 21, you are correct, but scholars analyzing how the calendar system changed between then and now have calculated that his birthday actually fell on what we would now call March 31.)

Musical Offering Canon 1: by augmentation

I have now written three canons on the theme of the Musical Offering. Here is the story of the first Musical Offering canon.

In 1975 I had the pleasure of meeting Doug Hofstadter at Stanford University. I was an undergraduate enamored of Escher’s art, Bach’s music, and Gödel’s mathematics (I had read Nagel and Newman’s book Gödel’s Proof the previous summer, in Hawaii). The encounter was an uncanny meeting of minds; as Doug recalls we were a bit suspicious of each other, having encountered wacky people with similarly eccentric interests. 

But the connection was real and we became fast friends. I spent many happy evenings working on my homework at a computer terminal alongside Doug, who was writing his magnum opus Gödel, Escher, Bach. To help him write his book he employed a classic professor trick: teach a course in order to motivate you to write the book. Each week new dialogs and chapters would appear hot off the dot matrix printer.

A couple years later he taught the course again, and I helped out as a teaching assistant. We decided to open the class with a re-enactment of the fateful meeting of Bach and King Frederick the Great of Prussia, in which the king sat down at a new-fangled pianoforte and challenged “Old” Bach to improvise a fugue on a long chromatic theme he had composed. Bach not only improvised a 3-voice fugue on the spot; three months later he sent the king an incredible manuscript called The Musical Offering, which included a six-voice fugue, a trio sonata, and ten canons, all on the royal theme. The full story is told in the dramatic the book Evening in the Palace of Reason. Here is the original theme, a sinuously chromatic melody with a somewhat clumsy closing cadence.

I had already learned the six-voice fugue when Doug’s 1977 GEB class began. Over the preceding winter break I learned the three-voice fugue as well. Not content with that accomplishment, I told Doug I would surprise him with a new canon on the Musical Offering theme. It was with great pleasure that I performed an original canon by augmentation (the faster voice plays the score twice in the same time as the slower voice plays it once) for the class. Here is the first voice. 

Voice 1 of Canon by Augmentation on the Theme of the Musical Offering

Note that the first half of the composition is an embellished version of the original Musical Offering theme. On the next two pages are the full canon written out for two voices. Both voices start at the same time. The second voice is an octave lower and twice as slow, which means by the time the first voice ends, the second voice is only half done. To fill in the remaining time, the first voice then repeats, until the two voices catch up with each other at the very end (actually they end a bar or so shy of completion). Note that every note in the canon appears three times, and must play three different roles — a bit of musical algebra that I found quite absorbing to solve.

Some of Doug’s friends took my canon further, playing it in four voices with two people seated at one piano. The additional two parts play the theme even slower at quarter speed and eighth speed in lower octaves. The canon was not intended to work this way, so there are a few clashing dissonances, but the overall effect is quite entertaining.

Musical Offering Canon 2: by 50% delay at the octave

Years later I presented Doug with a second Musical Offering canon. In this brief eight-bar composition one voice starts four bars after the other, so the two voices are 50% out of phase with each other. I gave it to Doug originally as a “puzzle” canon, meaning I gave him one voice, and asked him how to transform the theme to produce a second accompanying voice. Here is the first voice of the canon.

And here is the full two-voice canon.

To compose this canon, I first played the theme against itself at the desired 4-bar delay. Here is the combined effect using the original theme without modification.

Rough first approximation of the 180° Canon on the Theme of the Musical Offering

The light gray areas of the canon work decently without modification; I added passing notes to give them more movement. The first two dark gray areas fall flat because both voices sit on the same note. To solve this problem I shifted some notes in time — in the first region I shifted the E flat in the bass voice earlier by a quarter beat, and introduced a scale-wise descent, so that by the time E flat sounds in the treble voice, the bass has moved down to a harmonious C. To enliven the chromatic descent in the third dark gray region, I delayed the treble voice by a quarter beat, creating a pleasing staggered pattern.

Because of the way this canon is structured, the last four bars of the piece are the same as the first four bars but with the voices reversed, so modifications to the first four bars work almost automatically in the last four bars. As I do when creating an ambigram, I created half the composition, then copied it to make the other half. In fact, the whole process of creating a canon is quite similar to creating an ambigrams: you start with a theme (word), choose a transformation (musical or geometric), modify the theme (word) so it both obeys the strict transformation and sounds (looks) harmonious (legible). 

The finishing touch in composing both canons and ambigrams is to rationalize the quirks of the composition by making them part of a consistent style. In this canon, for instance, the alternating pattern of intervals in the first bar creates an angular marching rhythm that I continued throughout the piece. The goal is to compose a piece that works both mathemtatically and musically. I enjoy composing canons because the music seems to write itself, shaped by the pressures of the formal constraints.

Finally, here is a comparison of the original Musical Offering theme and the variations that appear in each of the canons, so you can hear how the theme has been altered.