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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Quantitative literacy?

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

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

This post focuses on ‘quantitative literacy’ to clarify my own thinking.  Since mathematics is the set of sciences of quantities, using ‘quantitative’ instead of ‘mathematical’ does not necessarily change the meaning.  However, the use of the word ‘quantitative’ implies that we might emphasize more the application of mathematics, rather than the structure of the sciences of mathematics.

To many people, ‘quantitative’ will tend to suggest the science of arithmetic (known quantities) rather than other mathematics.   When I look at courses that include quantitative in the title, I generally see applications of arithmetic … with perhaps a little basic geometry.  Only occasionally do I see statistics in such a course, and I have yet to see calculus included.  Since the science of calculus involves quantities under change, this seems ironic.  Are the concepts of calculus so advanced or obscure that students in a general education math class can not understand them?

I am concluding that I would prefer ‘mathematical’ to the ‘quantitative’ — not that I want to have the theory of mathematics exclude the application of the mathematics.  Rather, I want us to focus on multiple mathematics, not just arithmetic and some geometry.

How about the word ‘literacy’?  This word is problematic, since the synonyms include ‘knowledge’, ‘learning’ and ‘education’.   However, we can overcome this problem by being precise and consistent in our definition.  Perhaps we can define ‘literacy’ to mean ‘understands and can apply basic concepts’, as a parallel to the language literacy definition (‘can read and write’).  With that definition, I rather like the word ‘literacy’ appended to mathematical.

Of course, we have much work to do before we KNOW what ‘mathematical literacy’ means.  Which mathematics? What are the basic concepts of the ones we include?  Our professional community needs to deal with these questions, as many of our colleges have shifted away from a pre-calculus/calculus type of general education course … and towards a reasoning/literacy type of course.  Much valuable and creative work is being done; however, we need to develop some shared conceptions of this type of curriculum.  A lack of shared curricular concepts creates problems for articulation and transfer, and causes us to develop this part of the profession in more isolation than would be ideal.

 
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New Online Presentations

Take a look at the Instant Presentations page; this week, two presentations were added dealing with the big questions of changing to a New Life model … placement of students, and instructional materials.

The Instant Presentations page is at https://www.devmathrevival.net/?page_id=116

Online — “instant presentations” now include MLCS and ‘Transitions’

The Instant Presentation page now includes several videos (4 or 5 minutes each) on the New Life curriculum.  Today’s additions are the two videos on the Transitions course … the powerful second course in the New Life model.

Take a look, and enjoy!!