Mathematical Reasoning?

We, as mathematicians, really appreciate definitions — concise and consistent definitions.

What is ‘mathematical reasoning’?  How does it differ (if it does) from ‘quantitative literacy’?

This post focuses on ‘mathematical reasoning’ to clarify my own thinking.  Mathematics is the science of quantities, perhaps better stated in the plural — the sciences of quantities.  A science (singular) refers to a field in which there are shared concepts and theories.  In mathematics, we have several basic domains which have their own concepts and theories — geometry, statistics, arithmetic, calculus, algebra (a vague term), and more.  Within the context of general college mathematics, the first four listed are the most likely sciences involved.

If ‘mathematics’ is plural, what meaning does ‘mathematical’ have?  It might simply mean ‘related to one or more of the mathematics’.  Should ONE of them be sufficient?  What does ‘reasoning’ mean if there is more than one mathematics involved?

The more I ponder this problem, the more I am drawn to ‘literacy’ instead of ‘reasoning’.  My expertise is not that deep in all of the mathematics; however, it seems to me that the ‘reasoning’ involved is unique to each mathematics.  I can hear some of the readers saying “but, they are all LOGICAL!”, and that is true … but not sufficient.  Labeling something as ‘logical’ simply means that there is some systematic process involved in the reasoning, and I again suggest that there are many substantive differences in this reasoning between the mathematics involved.

For example, geometry involves both formal and informal logic; the reasoning often is based on identifying basic shapes and objects within different configurations and after different transformations.  We use phrases like “spatial sense” and “part-whole”, which also come up in calculus.  On the other hand, statistics involves descriptive work and inferential work; ‘hypothesis’ is used differently than we do in geometry, and nothing is ever proven … it’s all a matter of probability.

Could ONE mathematics be sufficient for ‘mathematical reasoning’, in the context of general mathematics at college?  I hope not.  There is little value in providing one science only in mathematics, just as there is little value in providing one science only in the ‘hard sciences’, for general education.  Specializing has value for advanced work.  General education needs to focus on a broader view, both to show the nature of the field of mathematics and to provide a set of ideas that students are likely to find useful.

I think I would rather use the name ‘mathematical reasonings’ (plural).  A course in ‘mathematical reasonings’ would likely be a more advanced general education course than we normally offer.  When I look at courses labeled ‘reasoning’, what they really focus on is ‘problem solving’; this is laudable, and I have such a course that I love to teach. 

My conclusion is that we should not use the label ‘mathematical reasoning’, both because the mathematics involved being plural and because we do not really focus on the reasoning.

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