Mathematical Reasoning … Can We Recognize It? Do We Allow It?

My department has been discussing the concept of ‘rigor’, which usually invokes some variant of ‘mathematical reasoning’.  Definitions of either concept often involve communication and flexibility, though our practices may not encourage any of this as much as we would like.

In general, if a learner is simply showing the same behaviors (and mathematical analysis) that have been described … justified … and demonstrated during the class then I do not see much rigor.  Building mathematical reasoning involves exploring something new, and sometimes shows in failed attempts to solve a problem

Of course, the label ‘failed’ is an artificial description based on some arbitrary standard (like a correct ‘answer’).

Recently,  I graded my final batches of final exams.  I want to share two examples of mathematical reasoning which pleased me, in spite of the fact that the solutions submitted by the students were at odds with the expectations on our grading rubric.

Here is the first, in our Math Lit course:

Our rubric called for students to use the area formula once — since that information is shown in the problem.  We actually did very little work in this class with compound shapes.  I was pleased with this student’s analysis, which exceeded anything shown in class.

The second is from our Intermediate Algebra course (which is extinct as of next month):

In this case, our rubric was based on the expectation that students would ‘clear fractions’ (which ain’t the “good thing” it once was).  A handful of students did a reasoning process like that shown above AND recognized that the equation was a contradiction.  [The specific work above is from a student who only got a 2.5 grade in the class.  She has a lot more potential, which I did tell her.]  Although I don’t have any evidence, I can be hopeful that the emphasis on reasoning during my classes contributed to these students seeing a ‘different way’ to solve these problems.

In the algebraic situation, which is more important — to ‘always check solutions’ (awful advice!) or to ‘recognize a contradiction’?  Mathematically, there is no contest; connecting type of statement (contradiction in this case) with the solution set (none) is a fundamental concept in basic algebra … a concept actively discouraged by much of our teaching.

Using the phrase ‘mathematical reasoning’ does not mean that we build any mathematical reasoning in our students.  Our courses cover too much mechanics; “a thousand answers and a cloud of dust’.  Let go of trivial procedures (like extreme factoring of polynomials and simplifying radicals of varying indices with complicated radicands, or memorizing a hundred trig identities).

The easy choice is to emphasize procedures and answers.  The fun choice is to emphasize reasoning and analysis — even in basic courses.

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