Using Mathematics: It’s Not Always About ME !!

In the traditional college mathematics curriculum, mathematics is used to solve problems which students do not care about.  Some reform curricula involve mathematics only for problems which most students care about.  Is one of these extremes naturally superior to the other?

Perhaps some researchers are already working on experiments to test that hypotheses.  My own conjecture on this might surprise a few people:

The net gain for students is higher in a curriculum which solves problems which students do not care about, compared to a curriculum focusing on problems students do care about.

The traditional curriculum normally focuses on individual students creating a symbolic statement (equation or function) for the problem, and then using this symbolic statement to determine all answers.  The reform curricula often engage students with informal group work around a context, looking for alternative strategies to find the answer; symbolic work comes later (often on a different class day).

Most reformers will assert that the group work in a context provides definite advantages in student learning.  The etymology of this assertion often has its roots in a constructivist point of view; the original researchers in this area were more interested in the social context and juvenile development.  We often conflate the issue by speaking of a ‘constructivist theory’ — there is no constructivist theory (since a theory provides predictions that can be tested with either positive or negative results); I’ve never seen research supporting constructivism in learning mathematics with adults.

However, there is a non-trivial advantage to the reform work with work on problems which students care about:

Students having the novel experience of working on problems they care about is exciting and motivating.

Seeing that process in class is exciting for instructors; sometimes, we become addicted to this experience to the point that we think students have to be dealt with in this manner all of the time.

Is a math class all about ME?? (a student)

Of course it isn’t.  Students are in college to either get an education or training (or both).  Getting an education is all about “not me” — understanding other points of view, analyzing problems, and solving … often with the person deliberately left out (objective point of view).  We might think that ‘training’ should deal with just problems which students care about … this view has two fatal flaws.  First, let’s assume that training exists to get a job (employment); how much of any job is something that the student personally cares about?  Sure, the student picks a program that they care about in general — but their job is going to involve a large portion of specifics which they don’t care about.

The second fatal flaw in the training point of view is ‘stability’ (or lack there of).  How many workers deal with the same types of problems for years at a time?  We are hearing from business and industry that they need a flexible work force — not one constrained by ‘it’s important to me’.

When I teach our traditional algebra courses (beginning & intermediate) I almost always make a statement such as the following:

Passing this math course means that you can apply mathematics to problems which you don’t care about, but you did so because somebody else said they were important.

The main downfall of the traditional curriculum is that it does not modify the pre-existing negative attitudes about mathematics [though I try 🙂 ],  Students have a negative attitude about mathematics and especially about ‘word problems.  Using problems which students care about can provide some scaffolding to get students out of their negative attitudes.

We can’t stop there.  For each problem students care about, we should have them deal with 2 or 3 which they don’t care about.  We need to make the connections between the processing done on the ‘care about’ problems and the symbolic tools of the trade (expressions; functions; known relationships [such as D=rt]).

At the developmental level, students will be proceeding to college courses.  College courses have a general expectation of dealing with symbolic statements.  Being able to determine solution to a specific problem is often a trivial exercise in itself.  Students need to see quantitative relationships and use appropriate symbolism to state that relationship.  We have no confidence that the majority of these situations will be innately important to the student; we do them a diservice to imply that the only mathematics they need is to find solutions to problems they care about.

We need to get rid of the traditional curriculum, recognizing that we achieved some good results within that.  We also need to moderate our use of ‘problems students care about’, and we need to make sure that we always keep the focus on the tools of the trade (relationships, symbolic statements, representations).

 
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1 Comment

  • By schremmer, June 4, 2016 @ 7:05 pm

    At this level, mathematics is emphatically not “about solving problems”.

    Pretending otherwise is a deceptive practice, regardless of whether the students care or does not care for said problem.

    When a basketball player goes to the weight room, it is emphatically not “about solving problems”, it is an investment towards a still distant goal. It is only then that s/he will have to “solve problems”.

    And, to quote Hung-Hsi Wu in the response of the Notices of the American Mathematical Society to Why Do Americans Stink at Math?, Elizabeth Green’s New York Times article,

    If Americans do “stink” at math, clearly it is because they find the math in school to be unlearnable. […] For the past four decades or so the mathematics contained in standard textbooks has played havoc with the teaching and learning of school mathematics.

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