Points, Lines, and Mental Maps

Do we consistently use different directions for problems involving discrete points as opposed to equations or functions?  Does “plot” mean something different than “graph”?  Most importantly, what mental maps do beginning students create for graphical representations of points and lines?

I am observing some patterns in my students’ behavior:

1. “Plotting” two distinct points is treated as if they represented a straight line.
2. Graphing a linear equation is done inflexibly, often by the first or last method seen or practiced.
3. Comparing and contrasting basic methods of graphing linear equations is seen as ‘confusing’.

We are doing our test today on linear equations in two variables, in our beginning algebra class.  The very first item on the test is plotting two distinct ordered pairs.  Some students have difficulty with the concept of ordered pairs, but most plot the two points accurately … and the majority of those connect the two points with a line.  This issue came up in our discussions, based on student questions … still, the programming seems too strong to resist: got two points?  Draw a line!!

Our course is quite traditional in coverage and outcomes at this time.  This chapter includes graphing equations by using specified input values, then by intercepts along with a 3rd point; we use a table of values via a calculator, and then graph using slope.  We’ve done quizzes with directions about a method, and worksheets with both method directions and ‘choose the most efficient method’.  However, each student tends to graph every problem by the same method.  Many students are using slope to graph every problem, even when intercepts would be easier … and even when the method is misapplied to a special case (missing x or missing y); graphing y=4 often results in a graph of y=4x.

Although a comparison of methods was part of our sequence of activities in class (as a secondary point), we tried to have a discussion of the methods as part of our review process in class.  This was not popular … most students did not want to think about how methods compared, only about how they can work problems correctly.  This is the same challenge in metacognition that students face when asked about what their improvement plan is … “I will do better” is confused with thinking about how to do better.

I’ve been working at this type of teaching and learning for quite a while now.  More students seem to have a simplistic view of learning in which having one method that gets mostly correct answers is seen as better than understanding how to choose the tool for the problem at hand; consistently getting one type of problem wrong is not seen as feedback about the learning process … it’s seen as an acceptable price to pay for getting other problems correct.

One way to look at these issues is to view them through a literacy framework.  Instead of listing behavioral outcomes for linear equations in two variables, we would ask the question: “How would we determine that students understand linear equations in two variables?”  We certainly include too many topics in the existing courses to assess understanding of most topics, and are left with simpler performance measures.  The mental map developed by each student responds to the conditions of learning, including assessment.

My own teaching and assessment is part of the problem I am seeing.  In some zones, I have limited flexibility … the basic content of the course is fixed.  However, I need to see how I can change the conditions of learning to encourage the more complex mental map that I think is important.

 Join Dev Math Revival on Facebook:

No Comments

No comments yet.

RSS feed for comments on this post. TrackBack URI

Leave a comment

You must be logged in to post a comment.