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In the United States, some universities offer "Introduction to Logic" courses. These courses are often offered to undergraduates who are not majoring in mathematics, as a way that the undergraduates can satisfy requirements that they take a certain number of "math" and/or "philosophy" classes. Some of these courses teach:

A notation for representing logical statements. This notation is very compact, with single characters representing things like "is a member of", "is not true", et cetera. Unfortunately, this notation is not the same as the notation used by computer programmers, nor is much of this notation used by many non-mathematicians.

How to set up simple logic puzzles using the notation.

Some proofs of basic concepts.

Similar length Math and Computer Science courses cover a lot more.

In practice, a good introductory university-level course in relational databases covers about four times as much material, and may cover (in an incidental manner) all of the logic puzzles and basic concepts taught in the "Introduction to Logic" course.

Similarly, many secondary schools teach "Geometry" as a "how to write a mathematical proof" class. The proofs are at a level that non-mathematicians consider thorough; mathematicians often have much more detailed proofs for the same concepts. In the process of doing these proofs, the students learn most of the logic concepts taught in the "Introduction to Logic" course discussed above.

Statistics courses teach approaches that allow drawing inferences with less than 100% confidence.

A good math course teaches "ways of thinking through problems" that are not taught in the "Introduction to Logic" course discussed above. For example:

Given a story problem (written in words that talk about a real-life problem), how to identify aspects of the problem that can be measured and compared.

How to identify what you want to know about a problem.

How to identify things you already know about a problem.

Drawing a picture of the problem.

Labelling variables.

Identifying directions in which variables increase.

Identifying relationships between variables.

Identifying the precision to which variables can be accurately measured.

Identifying "field conditions" and "boundary conditions" that restrict the possible answers.

The process of using the relationships (found above) to solve for what you want to know follows a logical process. This process often includes naming intermediate concepts, and looking at things from multiple points of view.

Explaining what the answer means.

Confirming that the answer is consistent with the original information. (A.k.a. "Check by substitution" and "sanity checks".)

"Checks by substitution" follow a logical process.