Interesting paper by Maria Bako includes discussion of pitfalls of teaching advanced mathematics (undergraduate level) without due attention to formal logic. Following is an excerpt:
Mathematics without logic
We can continue the examples of the previous section. Despite this logic has a hard fate. In textbooks, we can find careless definitions; for example:
f + g is continuous at a point, provided f and g are.
Of course it is harder to note the extended version of the previous definition:
For all functions f , all functions g, and all real numbers a, if f is continuous at a and g is continuous at a, then f + g is continuous at a.
It is clear why we teach the former version, but in this case even talented students cannot extend the definitions and theorems noted in short form. According to Selden just only 8.5 percent of his mid-level undergraduate mathematics majors could “unpack” informally written mathematical statements into their logically equivalent formal statements. Some textbook-writers keep quiet about quantifiers, because in their opinion students cannot understand them.
Hence if students do not practice this kind of extensions, they have only superficial knowledge about theorems and definitions.
With this superficial knowledge students can solve simple problems, but they cannot solve harder ones, e.g. proofs. In the case of indirect proofs we negate the extended form of the theorems, and there are problems with negation, too. According to Selden undergraduate students believe, that the negation of “All x is y.” is “All x is not y.” (No x is y). The combination of the quantifiers and implication are similarly problematic : some student believe, that “the relation R is symmetric if xRy and yRx." Similar problems arise if they need to define the subset property with the elements.
My notes: 1.The negation of "All x is y" is "Some x are not y." 2. A relation R is symmetric iff "All x All y (if xRy then yRx)." It is not enough to show that for some x and y, we have xRy and yRx.
As I read the paper, the author advocates teaching logic in secondary school. (I agree). The author advocates uses games and puzzles and is of the opinion that emphasizing algorithms is not good. ( I don't entirely agree with this. Part of the enjoyment of mathematics is due to the feeling of power that you get when you can crank out the answers to hard problems without seeming to think about them too much. And I find little puzzles and riddles repulsive. But I suppose they are no more oppressive than typical algebra "word problems" like "If Sally is 5 years older than Ed..." )
The example of "f + g is continuous at a point provided f and g are" doesn't completely horrify me. I do agree that a secondary school program that taught quantifiers and negation would greatly improve a student's experience in undergraduate mathematics courses. For example, it would enable the typical calculus student understand the definition of limit. It would also be nice if the calculus teacher wasn't saddled simultaneously with the job of teaching topics in math (like the summation of finite series) and also the related logic ( like mathematical induction).
How advanced is the advanced mathematics we are addressing? At the graduate level mathematics discussions are not conduced as line after line of propositional logic. Nor are books and papers often written in such a style. (An exception would be papers about logic itself and other material in the study of "Foundations".) I think the main contribution of teaching logic in secondary school would be to improve undergraduate mathematics education.
The more I at look at this paper, the more problems I see. (As someone else has rightly pointed out, the example given in my excerpt is actually a theorem, not a definition. And, like you, I didn't like the games and puzzles.) Nevertheless, there is an important idea here: It may be that students' lack of facility in formal logic is holding them back from a deeper understanding of higher mathematics and being able to apply mathematical theory to more difficult problems. It may be that you have to know what "the rules" are -- to have internalized them in some sense -- before you can break them with impunity.
BTW, I have since started a thread, "Math without Logic" at sci.math and sci.logic on this topic.