Originally Posted by

**Randy**
Nearly everyone in the sciences learns how to solve differential equations before they learn the fundamental theorem of calculus. Nearly everyone in computer science learns to write a high level programming language before they learn about Turing Machines.

I guess it's down to this wonderful phenomena that's pretty much exclusive to STEM fields where higher levels of complexity can often be simpler to understand. Whereas if you're writing a book on the subject, **you really should start at first principles.**

I have to disagree.

The difficultly is that most STEM subjects are more like a web of knowledge than a nice linear progression.

A nice illustration exists in one of calc 2's most dreaded topics: series. Ask people why they care about series, and nobody really knows until maybe their senior year. The problem is that the topic begins with first principles: defining what a series is, going through divergence and convergence, and then something about Taylor Series at the end. But because of that "first principles" method, many people miss the entire chapter is designed around Taylor series. Loosely stated, Taylor series are the basis for nearly all forms of systematic approximation in math; however, whether or not the approximation can exist depends on whether or not the series diverges/converges. Hence why 80% of the section is spent on series tests.

The organization would be to briefly introduce Taylor series and clearly establish the goals so that way people learn that checking for convergence/divergence is just a step taken to solve a bigger problem. Science becomes exponentially worse in this regard: in physics, there's a reason we don't start with relativity and quantum mechanics despite being first principles. Even classically, No mechanics course starts with principle of least action, despite it being arguably the most fundamental law of law (or its quantum analog, Feynmann Path Integral).

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A better statement is probably this: we should start with models that has a clear immediate use. Students can better understand, for example, why the octet rule can be useful in explaining chemical reactions such as simple combustion reactions. But then, as we begin to find issues with that model, introduce what the "next step" is in improving the model. Rinse and repeat. This method, however, has the odd benefit of saving first principles for last since only once you've encountered the most complete model you've encountered the first principles. However, this builds a web in a more logical way, where students better understand why they're learning what they're learning.