I'm suggesting teaching the foundations of mathematics. I would pick a system like:

• ETCS (Elementary Theory of the Category of Sets),
• HoTT (Homotopy Type Theory),
• Dependent Type Theory,
• ZFC?

I suggest this can conclude with one of:

• The theory of cardinal numbers.
• Defining the arithmetic operations on $\mathbb N, \mathbb Q, \mathbb R, \mathbb C$ and proving their identities.
• I'm perhaps overly idealistic, but perhaps some funky topos theory: Nonstandard Analysis? Synthetic Differential Geometry? Synthetic Computability?

Advantages:

• They'll learn mathematical notation and terminology. People who don't learn this notation and terminology might otherwise struggle with more advanced maths. Examples of terms they'll learn: Functions, tuples, Cartesian product, sets, subsets, natural number, etc.

• They'll learn a proof calculus like Natural Deduction, along with proof by induction. These are idealised, general and rigorous models (imitations?) of the proofs constructed by actual human beings.

• Has some link to topics like Functional Programming.

Disadvantages:

• Not sure how this can help most engineers.

• No overlap with geometry.

• No help with calculus, except clarifying some basic terms.

• Could be accused of being pedantic.

Questions:

• Will students or teachers find this easy to learn or teach?

• Where are the problems to solve? ---- Here I suggest cardinal numbers might provide a small problem list.