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How you can teach students about the importance of maps?

Updated: 8/18/2019
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13y ago

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This is a very good question. Maps turn up everywhere in mathematics and are a very fundamental thing that needs the right appreciation by students from the very beginning.

Now what is all of science and specifically mathematics about: Information.

The fundamental entity of information exchange is to select a state in a system by another system. To select you need a set of states to select from. We are here at the fundamental set concept that turns up in the foundation of mathematics.

The procedure of selection can be identified by the map.

Examples:

- A human that talks to another human, selects concepts in that other human by using the language. Mathematically this event could be described as two sets of concepts and a map (the talking) between them.

- A light beam of a certain wave length falling on the retina excites a certain state in a sensor cell there. Mathematically: We have a set of wave lengths and a set of states in the sensor cell. The map is the physical event.

So in information processing entities (animals including us humans, computers, plants, ...) as well as in physical systems we have sets and maps.

If the map is a one to one map (injective) including all elements of the target set (surjective) then there is no freedom to select any further elements from the target set, i.e. no additional information needed. Therefore this kind of map is important.

Quantum mechanics tells us that when looking at the small scale there is no such ideal map in reality. It also tells us that when we reach a certain scale, things cannot be divided further (quantum) contrary to the idealized numbers in mathematics.

The set concept also gives us an idea why numbers are so important. You can imagine to give every element of the set a number tag (notice: this is also a map) and from then on you use numbers to identify the elements of the set. And maps between the real sets are replaced by maps between number sets.

For the same reason computers work with numbers.

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13y ago
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Q: How you can teach students about the importance of maps?
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