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.

Join Dev Math Revival on Facebook:

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.

 
Join Dev Math Revival on Facebook:

Modules in Developmental Mathematics

Are modules a good thing in developmental mathematics?  They are certainly popular these days, with quite a few colleges … and some entire state systems … putting their entire developmental math program in modules.

The word ‘modules’ overlaps in meaning with some more generic labels such as ‘chapters’ or ‘units’.  However, the word module implies a planned independence — you install only the modules needed, or only repair the one that is ‘broken’.  In the case of developmental mathematics, one of the benefits attributed to modules is that students only have to study what they ‘need’ (based on some assessment).

The design implies that there are discrete skills necessary, that developmental mathematics is all about students demonstrating accurate procedures in the modules prescribed for them, and that we should not expect any other ‘value added’ for our students.  Briefly, here are arguments against each of these implications.

A number of studies have looked at the actual mathematics used in various occupations.  This list often includes arithmetic skills (especially with percents), some formula work, and a few other items.  Rather than discrete skills, the occupational need is for the problem solving and ‘STEM-like skill set’ (see http://www.insidehighered.com/news/2011/10/20/study-analyzes-science-work-force-through-different-lens for example).  Our 2010 developmental mathematics courses emphasize discrete skills because of history, not because that is what students need.  By constructing a series of independent modules, we are creating a wider gap between our curriculum and the needs of our students. 

Much of the technology behind the current module ‘frenzy’ is heavily procedural, with the main thing being correct answers.  There are two significant shortcomings of this approach.  First, getting a correct answer has only an indirect connection to knowing something; we have all seen students get a ‘correct’ answer with either multiple errors or no comprehension of what they are doing.  Second, procedural details are notorious for being forgotten — the old ‘use it or lose it’ syndrome; if we focus on procedures, we are essentially saying that developmental mathematics has no lasting benefit for the students.  If this is true, we would be more professional to take the student’s money and give them their grade without them going through the game of producing the correct answers for us.

The last implication, that we should not expect any other value-added for our students, goes beyond the prior concern.  Are students in developmental mathematics classes so limited intellectually that we should give up on their capacity to learn mathematics in the academic sense?  By setting the standard so low, we are not only limiting our students — we are actively reinforcing every bad attitude about mathematics.  Many of these attitudes are based on perceptions of mathematics as dealing only with specific procedures and correct answers, coupled with a belief that normal people are not capable of understanding mathematics. 

You may be wondering if I somehow believe that the existing developmental mathematics curriculum is better than what I describe; in general, it is not much better.  There might be a better problem solving component, and a little bit of conceptual understanding.  However, we have inherited a curriculum that has been fixated on algebraic procedures (and a particular collection of procedures). 

No, it is not that modules are breaking something good.  Rather, modules are giving the illusion that we are fixing something because we can point to a change.  Change is not progress, not unless the change results in achieving what the community of professionals sees as something of basic and intrinsic value.  When I have conversations with mathematicians, I do not hear people describe the outcomes of modules as being of value … we value concepts and connections, thinking and problem solving.

Modules are an easy ‘solution’ to the wrong problem, and I suggest to you that modules create additional problems.  Let us not get distracted by the technological appeal of modules; instead, let us look critically at the mathematics we deliver … and how we can actually help our students.

Join Dev Math Revival on Facebook:

Completion in Developmental Math

There is always an excitement in the finish line — whether we are talking about a 5K run, a horse race, or a college degree.

We are facing unprecedented pressure to focus on “completion”, especially as it concerns developmental mathematics.  Sometimes, this goes beyond pressure … as when we are directed to use a certain ‘solution’ to raise completion rates (and save money as well).  Many of us are approaching this ‘completion’ issue as professionals, and we ask many questions.  Answers are elusive, and may be impossible in the scientific sense.

Of course, the issue is not just ‘completion’ … rather ‘completion of what’.  If we re-package a curriculum into a series of independent modules, we may have a chance of raising the ‘completion’ of the series; however, this improvement is not certain, and depends upon instructional and institutional policies.  At the same time, creating independent modules assures that we will not be delivering a coherent course to our students; our students will continue to view mathematics as a random set of procedures used to achieve answers to questions that nobody seems to really care about (besides us).

So, ‘completion of WHAT’ is critical.  As mathematicians, we value the concepts of our fields.  We value the connections that exist among concepts.  We value the recognition of these concepts within multiple problem situations.  We value the clear thinking and insights that are signs of mathematical reasoning.  For many years, we have been distracted from core values such as these; we (for too long) have delivered a curriculum that seems more based on ‘right answers’ than any meaningful mathematics.

We are at a point in history where there is an opportunity for us to create our change.  Rather than allowing a re-packaging of an unsatisfactory product (look! new, improved … now in a smaller package!!), we can look at designing a basic curriculum that reflects our core values.  I am not saying that students in developmental mathematics should be pretending to be real mathematicians; however, all of our students can understand some basic concepts … can see connections … recognize different uses of these concepts … and improve their reasoning.  This may seem to be a more difficult instructional goal than the old curriculum focusing on procedures.  However, I would remind you that the best thinking about cognition emphasizes concepts and connections; retaining details of procedures is more difficult than retaining concepts and connections. 

We can improve our completion rates, and improve our curriculum.  These goals fit together; looking at one without the other will not lead to a solution nor to progress.  We can create change just by doing something — our job as professionals is to create progress based on our core values.  One model that reflects our core values is the “New Life” model, and I encourage you to become familiar with that design. 

Our ‘completion’ goal is a curriculum serving the needs of our students, resulting in high rates of achievement based on our core values.

Join Dev Math Revival on Facebook: