Harness the power of ‘what if?’ in STEM teaching

Most STEM subjects are based on a simple idea: using mathematical models to describe how systems behave. Whether we study a moving spacecraft, a static structure or a heat flow, we rely on their mathematical models. A mathematical model is a way to describe the behaviour of a dynamical system so we can predict things. In practice, it is usually a set of equations that describe how a system evolves over time or responds to changes in conditions. 

The life sequence of a mathematical model

A mathematical model has a life sequence: first, we derive equations; then we simulate them, often getting a lot of numbers as a result. Finally, we visualise our messy numbers and interpret the result. 

Students are expected to connect these separate parts themselves, but many do not. Each stage of this sequence introduces a specific barrier, which the instructor needs to address. 

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One solution is handling all of them at once within a single computational environment that supports symbolic mathematics, numerical computation and interactive visualisation. This makes it possible to replace static PDF slides with computable, interactive presentations that can also serve as shared lecture notes. Here is how each stage of a model’s life sequence can be addressed.

Derivation of equations

STEM textbooks often present long chains of algebra and calculus. They can be overwhelming and difficult to verify, and can have typos. Students either tend to follow steps without fully understanding them or simply skip the derivations completely, missing an opportunity to note sophisticated but useful tricks that lead to the nice final formulae they typically work with.

A solution is to carry out symbolic derivations step by step right in the lecture room. But not on the board – with modern computer algebra systems, it can be done programmatically, right inside the presentation. 

In this case, each derivation becomes explicit and checkable. Students can reproduce the result and see how assumptions affect the final equations. Derivations turn into something they can explore and verify.

Numerical simulation

Once the equations are derived, the next step is simulation. Here, many students face a different problem: limited coding experience. Textbooks often provide little or no usable code, which turns simulation into a technical obstacle.

A computable presentation naturally removes this barrier, as working codes are directly included in the lecture material. Students can run the model, then inspect the code and modify it. All the parameters are visible throughout. The focus shifts toward understanding what drives the results.

Visualisation and analysis of results

In traditional lecture slides and notes, results of numerical simulations are usually presented as static figures, tables or plots. These show what happened, but give little insight into what happens if we change something.

Adding interactivity – sliders and buttons – into presentations is a game-changer. When students are able to adjust parameters and see the result immediately, the plot becomes an experiment. They can ask “what if?” and test their intuition right away. For example, let students vary the spacecraft thrust and observe how the orbit changes. They can quite literally see causality. 

Single-window service

A nice bonus for students: addressing the above three challenges in a single computational environment means they avoid constant switching between slides, code editors and tables. 

This helps to reduce their cognitive load. They no longer need to reconnect pieces of information each time the tool changes. The lecture develops as a continuous process, rather than a sequence of disconnected steps.

It also allows for more flexible teaching. When a coding question arises, the instructor not only gives the answer, but shows it on the spot.

The cost of interactivity and its long-term value

While this is an effective approach, it isn’t effortless. It will take time to develop interactive, computable presentations.

It’s important to maintain a balance, especially for students with limited coding experience. Make sure to be there to provide them with guidance from the start. 

But rest assured, the effort will be worth it. Over time, interactive, computable presentations become a long-term asset, which can be adapted for different types of courses.

Start small, but start

To do this, you don’t need to completely redesign the course. Start with one model, where students usually struggle.

Make the derivation symbolic and reproducible. Provide a working simulation. Add an interactive visualisation. Ask students to change something and predict the outcome.

This is enough to change the learning dynamic. In STEM subjects, action often brings clarity.

The lecture itself evolves into a space where ideas are not only delivered but also tested. “What if?” – that is the question.

Dmitry Sizov is an instructor at the School of Engineering at Nazarbayev University.

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