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Discrete Mathematics is a branch of mathematics that deals with discrete elements. Unlike continuous mathematics, which studies concepts such as calculus and real numbers that can vary smoothly, discrete mathematics involves objects that can only take on distinct, separate values.

For example, when you roll a die, the outcome is a discrete number between 1 and 6. There are no in-betweens. Similarly, discrete mathematics covers topics such as logic, algorithms, graph theory, and combinatorics - all of which are essential for computer science and cryptography. It is integral to understanding the scaffolding behind decisions, whether in computer programs or in constructing logical arguments.

Concrete objects refer to things that are tangible and can be perceived by our senses. These are the things that exist in the physical world, like trees, buildings, and books. When teaching complex subjects, using concrete objects can be beneficial. It allows students to form a sensory connection to a concept, making it easier to grasp and remember.

For instance, physical models or blocks can be used to teach mathematical concepts such as volume or to represent entities in an equation. By touching and manipulating these objects, students are able to understand abstract mathematical principles in a more concrete way.

Abstract concepts, on the other hand, involve thoughts, ideas, and qualities that are not physically present. These can include emotions like love or concepts such as justice and freedom. In mathematics and other sciences, abstract concepts often involve intangible theories or ideas that require a higher level of thinking and cannot be touched or seen.

In discrete mathematics, for instance, the concept of infinity is abstract – it represents an idea rather than something concrete. Abstract concepts can be challenging to teach and understand because they often depend on individual interpretation and there's no physical reference to rely on. To improve comprehension, educators can tie these concepts to concrete objects or familiar experiences to create a bridge between the tangible and intangible.

Intangible theory refers to theories and knowledge that are not able to be touched or seen but can be understood and applied through intellectual processes. These theories are often the foundations of scientific principles, underlying laws of nature, or theorems in mathematics.

An example of intangible theory in discrete mathematics is the set theory, which talks about the collection of distinct objects considered as a whole. While the individual elements might be concrete, the theory discussing the relationships and operations between sets is abstract and intangible. Effective teaching of intangible theories involves creating analogies or hypothetical scenarios where these theories can be visualized or experienced in the context of real-world applications.