Durable Mathematics is a project, in progress, on making mathematical ideas memorable through story, motion, humor, physical experience, and visual anchors.
The larger project collects classroom demonstrations, public math writing, interactive models, and teaching routines that help students remember mathematics as something they have seen, heard, moved through, laughed about, and returned to.
Students often leave a math course remembering fragments: a formula, a test problem, a rule that worked once but no longer has a living context. Durable Mathematics begins from a different question: what would make an idea hard to forget?
The goal is not to replace precision with entertainment. The goal is to build memorable entry points that lead back to precise mathematics. A good anchor gives students something they can retrieve later: a physical action, a visual, a phrase, a story, a sound, or a surprising classroom moment.
In the full project, I plan to connect this framework to research on retrieval practice, dual coding, embodied cognition, multiple representations, and active learning, while keeping the center of the work close to the classroom: what students actually see, say, remember, and use.
A durable anchor is not a trick, and it is not entertainment sprinkled on top of the real class. It is a memorable place for an idea to live: a story, picture, graph, object, joke, video, phrase, physical action, or public example that helps students return to the mathematics later.
The goal is still serious mathematics. Students need definitions, techniques, practice, and fluency. But they also need moments that help those techniques attach to meaning. A durable anchor gives a formula a path back into memory.
This project grows out of more than 20 years teaching over 20,000 freshman and sophomore math students in large gateway courses at the University of Washington. Those courses are not disposable prerequisite machines. For many students, they are where they decide what mathematics is, whether they belong near it, and whether struggle means failure.
One of my favorite examples begins with a tennis ball, a phone camera, and a walk across campus.
If you throw a tennis ball straight up beside a sculpture and time how long it stays in the air before returning to the height of your hand, you can estimate the height of the throw with a wonderfully simple rule:
Time the toss. Double it. Square it.
If the ball is in the air for T seconds, then the height above your hand is approximately (2T)2 feet.
That little rule is not a trick. It is a compact version of projectile motion near Earth, rewritten so students can use it in their heads. It turns a campus sculpture into a live calculus and physics question: How high did the ball go? How fast was it moving? What assumptions are we making? How much does air resistance matter? Why does the graph look like a parabola?
I came across this rule only by being curious. I had not heard it before; I found it while playing with the idea, trying to make the activity fun, and asking what could be done mentally from a simple video. I have not found a reference to this exact rule of thumb in the small amount of searching I have done. It is very unlikely that I am the first person to notice it, but maybe; either way, it is a useful example of what instructors can find when we let ourselves engage with the material playfully and seriously at the same time.
The method also gives the mathematics a story. A student can remember standing near George Washington or the Broken Obelisk, watching the ball rise and fall, timing the motion, graphing the height, and seeing a formula become an estimate about a real object in a real place.
Learn more about the sculpture measurements and videos.
The tennis-ball example begins with a visible classroom moment, but it quickly becomes a real mathematical model. Ignoring air resistance, the acceleration is approximately a(t) = -32 ft/sec2, so the height above the hand is modeled by h(t) = v0t - 16t2. The top of the flight occurs where h'(t) = v(t) = 0, which gives a physical version of the first derivative test.
The same example also makes assumptions visible. If we include quadratic drag for a tennis ball, the velocity model becomes dv/dt = -g - Kv|v|. That turns the original classroom demonstration into a path toward differential equations, numerical solving, model comparison, and error analysis.
In the draft, the George Washington toss lasted about 2.83 seconds. The simple double-and-square rule gives about 32.04 feet above the hand, while the quadratic-drag model gives about 32.01 feet. For sculpture-scale tosses, the memorable rule is not only simple; it is surprisingly accurate.
The tennis ball example carries several anchors at once:
This is the kind of example I want Durable Mathematics to collect: not just a clever activity, but a reusable anchor that can support formal calculation, interpretation, modeling, and memory.
I have examples like this for nearly every topic in all the courses I teach: movie quotes for unit conversion, monkey faces for algebra, money examples for exponentials and logarithms, Disneyland teacup animations for circular motion, ocean waves as a way to discuss what sinusoidal models do and do not capture, catenaries for exponentials, and many, many tangent line and derivative stories, including headlights on a car and the meaning of concavity.
There are related rates examples that can be genuinely fun, optimization examples that feel like puzzles instead of chores, integration examples that include students dropping water balloons on me, 3D prints of solids that make volumes feel physical, historic and important applications of each integration method, the cutting pancakes project for trigonometric substitution, center of mass applications, arc length used to estimate the actual distance traveled by a baseball during a home run, and differential equations applications such as time-lapse videos of a snowball melting or coffee cooling experiments done in class.
And that list does not even reach 3D calculus. When I taught multivariable calculus online during the pandemic, I told my students that every lecture would include a new visual. I kept that promise for over a year. That course now has multiple durable anchors in every single lecture.
So I have many things to share, and I suspect other teachers do too. I think we should create more spaces where instructors can share the quirky, memorable, mathematically honest applications they have found, then collect them into a new kind of engaging curriculum. Yes, students need skills. But skills without fresh, engaging, memorable applications often do not mean much to students. We need to spend at least as much time building meaningful applications as we do drilling techniques.
I want this kind of work to be something instructors are allowed to share, study, and focus on as a real research area in mathematics education: not a side hobby, not a bag of tricks, but a serious form of mathematical teaching, curriculum design, and public scholarship.