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A logical fallacy refers to an error in reasoning that undermines the logic of an argument. In critical thinking, specifically within the context of mathematics and philosophy, it's essential to avoid such fallacies to maintain the integrity and validity of arguments.

For instance, in the provided example, we could incorrectly assume that because the premises and conclusion are all true, the argument itself must be valid. However, this is not the case. Logical fallacies can be deceptively convincing, and students may often accept conclusions if the individual statements seem accurate on their own. A well-known logical fallacy is the non sequitur, Latin for 'it does not follow', where the conclusion does not logically follow from the premises, as clearly demonstrated in the exercise's argument.

It is crucial for students to identify and understand common logical fallacies in order to critically assess arguments. Some common types include the straw man, ad hominem, false dilemma, and hasty generalization, among others. Recognizing these fallacies when they occur in arguments can prevent the acceptance of faulty reasoning and contribute to better analysis and understanding.

Argument validity is a fundamental concept in critical thinking and mathematics that pertains to the logical connection between premises and conclusion. A valid argument is one where if the premises are true, the conclusion must also be true. It's not about the actual truth of the premises or conclusion, but about the logical structure that links them.

In our exercise, the argument presented is structurally invalid because the truth of the conclusion ('The Earth revolves around the Sun') is not supported by the premises provided ('All dogs are mammals' and 'Apples are fruits'). Even if the statements are individually true, they fail to establish a logical connection that would necessitate the truth of the conclusion based on the truth of the premises.

To improve the understanding of argument validity, it's helpful to introduce students to the concepts of deductive reasoning, where the conclusion is guaranteed by the premises, and inductive reasoning, where the premises make the conclusion likely. Formal logic and truth tables can also serve as valuable tools for students to test and determine validity in various types of arguments.

Critical thinking in mathematics requires more than just numerical skills; it involves the ability to logically reason and to rigorously question assumptions and conclusions. This is integral for solving problems as well as for the broader application of mathematical principles.

In practicing critical thinking, students should be encouraged to question how premises relate to each other and to the conclusion, to consider the significance of each statement in an argument, and to explore alternative methods of arriving at solutions. The exercise provided is a straightforward example of why critical analysis is vital: The argument has true premises and a true conclusion but fails in conveying a proper logical connection.

Ultimately, building skills in critical thinking allows students to approach complex problems methodically, recognize patterns, construct and deconstruct arguments, and apply mathematical frameworks to various scenarios. Encouraging students to debate, justify their reasoning, and explore the structure of arguments can significantly enhance their analytical capabilities both in the realms of mathematics and beyond.