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- Jan 26, 2012

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Baloney.

I want to present a little table. This table is meant to illustrate which representations can be accurately inferred from other representations. The vertical column on the left is the origin representation, and the horizontal row on the top is the target representation. In other words, this table is attempting to answer the question, "Which representations can accurately generate which other representations?"

$\implies$ | Algebraic | Graphical | Numerical | Verbal |

Algebraic | Of course. | Yes. | Yes. | Yes. |

Graphical | With difficulty. Often impossible without additional information. | Of course. | Yes, but requires much effort. | Yes, if enough data is present. |

Numerical | With difficulty. Often impossible without additional information. | Yes, if enough data is present. | Of course. | Yes, if enough data is present. |

Verbal | Yes. | Yes, but often must be done through another representation. | Yes, but often must be done through another representation. | Of course. |

So one of these origin representations can generate the other three with very little difficulty: the algebraic/analytical representation. Therefore, when it is available, the algebraic/analytical representation is the most superior.

Am I saying that the other representations are useless? By no means. Of course they're not. Sometimes, all you get is numerical, like when you're doing data acquisition in a control setting. Other times, all you have is graphical, like one time when I needed the VI curve for a diode and was given a graph of it in the diode's data sheet.

What I am saying is that the algebraic/analytical is not equal in value to the other three - it is superior. Therefore, in teaching calculus, I emphasize the algebraic/analytical approach whenever it is available.