Building Trust Through
Transparency

Students today use more computational tools than any generation before them — and understand fewer of them.

• 🧮  They reach for CAS, graphing calculators, homework-system widgets, and now LLMs
• 📦  Each tool is a black box — an answer comes out, and the machinery stays hidden
• 📉  Trust in the tool becomes a substitute for understanding the mathematics
• 🎯  I want to argue: transparency is what turns a computational tool into a learning tool

Textbooks routinely ask students to do a very hard thing: run a mental movie of a physical system.

• 📖  "Now imagine the pendulum at small amplitude, with light damping, being driven at resonance…"
• 🎞️  Mental simulation is the skill students are trying to develop — not a skill they already have
• ⚠️  Symbolic notation alone is a thin channel; equations and intuition drift apart when the dynamics stay invisible
• 💡  If we want understanding, we have to give the dynamics somewhere to show up

The default tools students encounter in math courses today hide their mechanism by design.

• 🔒  CAS and symbolic engines: you type an integral, you get an answer — no steps, no derivation
• 📱  Graphing calculators and online homework systems: closed code, proprietary formats, black-box grading
• 💬  LLM chatbots: confident prose, no reproducibility, no way to inspect why a particular step appeared
• 🧪  The student cannot audit the reasoning, and neither can the instructor — trust is taken on faith

• 👀  View source: every formula, every numerical method, every visual choice is readable JavaScript
• 🔁  Reproducible: same input produces the same output, on any device, with no hidden seed or remote call
• 🏠  Local: runs entirely in the browser — no login, no account, no proprietary server
• 🛠️  Forkable: a colleague can copy the tool, modify it for their own course, and host their own version
• 🎓  This is what invites trust — not a promise, but an audit trail

Sixteen months of building, freely shared on GitHub Pages. Analytics snapshot, April 2026:

Notable traffic from UC Berkeley, UC Santa Cruz, Carleton College, the University of Tsukuba (Japan), Izhevsk (Russia), and Brown University — where Prof. Peyam Tabrizian has used the tools in instruction.

Engineering students learn best when the math has a physical referent — and when they can poke that referent.

Tools already built for Math 308 include:

• ⚙️  Mass-spring-damper, coupled multi-mass systems, forced oscillations with resonance sweep
• 📈  Phase portraits: drop initial conditions, watch saddles, spirals, and centers emerge live
• 🌀  Van der Pol, Duffing, logistic map, cobweb & bifurcation diagrams
• 🔣  Step-by-step tutorial tools: integrating factor, Wronskian, exact equations, undetermined coefficients

The same equation of motion scales from a single damped oscillator to a chain of coupled masses with three normal modes — every step is something the student can drive with a slider.

Single damped mass on a vertical spring — the canonical mẍ + cẋ + kx = 0. Slide damping → exponential envelope visibly contracts. Resonance sweep makes the peak appear instead of being derived.

• 🎚️  One mass: drag damping, drag stiffness, watch the envelope reshape in real time
• 🔁  Add a forcing term and sweep ω — the resonance peak emerges, not derived
• 🔗  Couple two masses → beating; four masses → three normal modes you can pick out by eye
• 📐  Each mass's displacement xᵢ(t) prints live underneath the picture
• 🧮  Integrator and force calculation are readable JavaScript — no opaque physics engine

Hand-drawing phase portraits used to be the lesson. With a live tool, it becomes a five-minute warmup — and the time goes to interpretation.

• 🖱️  Click anywhere — a trajectory unrolls from that initial condition in real time
• ↗️  Eigenvector lines update live as students change the matrix entries
• 🎚️  Sweep one parameter and watch a saddle morph into a spiral as eigenvalues turn complex
• ✍️  The hand-drawing exercise becomes a warmup, not the lesson
• 🧭  Time freed up for the real question: what does each picture say about stability?

The Method of Corners stops being a checklist of algebra and starts being something the student can see: a feasible region, its vertices, and the objective sliding across them.

• 🧱  Type any constraints — the half-planes are shaded and intersected into a feasible region in real time
• 📍  Corner points are computed automatically and the objective is evaluated at each one
• 🎯  The maximum (or minimum) is highlighted, including the degenerate case where the optimum lies along an entire edge
• 🧮  Business-math and finite-math students get a geometric foothold before they ever meet the simplex algorithm

If any topic in the curriculum is intrinsically visual, it is dynamical systems — and yet on paper it lives as algebra. The browser is where the algebra and the picture finally meet.

• 🪨  Physical view: a mass moving in a double-well potential — the geometry students should picture before any equation
• 🎯  Phase portrait (x, ẋ): the strange attractor emerges as the trajectory threads between the two wells
• 📈  Displacement vs time: chaotic time series, color-coded by speed, links the picture to the data students would actually measure
• 🎚️  Drag a slider — damping, forcing, frequency — and watch limit cycles, period doubling, and chaos appear

The language choice matters because the question isn't "what is fastest to build?" — it's "what will still run in ten years, on any device, without friction?"

For core mathematics the tools have zero runtime dependencies — the math is JavaScript, not a library call.

A subtle but enormous shift in what a single instructor can maintain. The tools did not change. The way they get edited did.

A tool isn't transparent if it's only transparent to sighted users with a mouse. WCAG 2.1 AA is the honest benchmark.

• 👁️  Visual: high-contrast palette, scalable typography, visible keyboard focus rings throughout
• ⌨️  Keyboard: every slider, every button, every control reachable without a pointing device
• 🔊  Assistive: MathJax + MathCAT pipeline so equations are read aloud correctly by screen readers
• 🏷️  Semantic: ARIA labels, table captions, landmark regions — propagated across the portfolio with agentic edits
• 📚  This aligns with Provost OER priorities — the Math 308 question bank I'm building now is being designed accessibility-first