I think the calculus of variations might be a better approach to introducing ODEs in first year.

You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of dy/dx = 0, you end up with an ODE (viz. the Euler-Lagrange equation).

This also motivates Lagrange multipliers which are usually taught around the same time as ODEs. They are similar to the Hamiltonian, which is a synonym for energy and is derived from the Euler-Lagrange equations of a system.

Of course you would brush over most of this mechanics stuff in a single lecture (60 min). But now you've motivated ODEs and given the students are reason to solve ODEs with constant coefficients.

You don't need a Lagrangian to invent mechanical ODEs. You could talk about mixing tanks, objects under complicated forces, and so on with a lot less background information.

We had a great professor and this was one of the most enjoyable classes I've ever taken. One particular assignment was a group paper where we were supposed to essentially explain and use the SIR model. We extended the model to an SIRZ model and effectively argued that zombie apocalypses in fiction are essentially impossible unless they include some supernatural elements. Under a wide range of assumptions (e.g., zombies rot/zombies don't rot) the infection always stopped before it spread significantly. (We got an A.)

I use zombies in my epidemic modeling class - one of my favorite results is, with a semi-complex model, you can replicate the script of most zombie movies mathematically (lots of people die, the survivors take shelter somewhere and are safe for awhile, attrition starts to take hold, things collapse and then you're left with a small surviving fraction of protagonists at the end).

You need to learn the math before applying it. You don't apply things that you don't understand.

There are certainly people who can learn that way, but it's not effective for most people. Most people have a limit to the amount of abstraction they can operate under before they need a tangible connection. Once that connection is made, most people can continue on to higher levels of abstraction.

The problem is math is entirely abstract. So you must learn the abstraction before applying it. Otherwise, you learn how to add 2 + 3, but you don't learn how to add n + m.

That strikes me as a strange way of looking at it. Most math taught to people who aren't pure mathematicians is taught precisely because of application to concrete situations. We value pure math largely because of the potential for future concrete applications.

In case you aren't aware (and I wasn't until I trained to be a teacher, so this isn't meant to be condescending), there are alternative methods of teaching besides abstraction-first. See https://en.m.wikipedia.org/wiki/Inquiry-based_learning

I'm aware. Examples are given to demonstrate abstract concepts. However, that doesn't preclude you from needing to learn the abstract concepts eventually. Otherwise, you go back to knowing times tables for 2's and 3's, but not n's and m's.

I need the concrete before the abstract just so my mind knows that what I’m seeing is not BS. Because in finance, 95% of the math is BS formulas that have no connection to reality.

Except for compounding interest, percent change, percent difference, multi-variate statistics... I'm quite dumbfounded to read this comment considering nearly all of finance is done through computers. Computers need algorithms in order to make predictions and algorithms are glorified formulas.