Student Difficulties with Graphs

I teach College Math, and we start out the semester with a lesson on functions.  One of the things we ask students to do is look at the graph of a function and decide if it is possible or impossible.  This task is surprisingly difficult for some students.  I got the activity from Shodor, a nonprofit organization serving students and educators by providing materials and instruction relating to computational science, at http://www.shodor.org/interactivate/activities/PossibleOrNot/.  Here is the first graph that students discuss:

Some students immediately discern that this graph implies traveling backwards in time and is impossible, but others see the graph as a path an object is traveling and think that it is possible.  Some students have trouble interpreting the first two linear segments as representing an object traveling away from something (distance is increasing) and then traveling back towards something (distance is decreasing).

The other graph that students struggle with is this one:

Many students resist the conclusion that this graph is impossible, even after I ask them to interpret the points lying on the vertical segments on the graph.  Many students persist in seeing this graph as a path in two dimensions, as if the grid represented a blue print of the floor of a room.

I recently read an article about student difficulties with using and reading graphs ("Examining Students' Reluctance to Use Graphs" by Frances Van Dyke and Alexander White in Mathematics Teacher, Sept. 2004, p. 110).  One of the types of difficulties discussed in the article relates to the Possible/Impossible activity that I use.  Students seem to do well with interpreting graphs of value as a function of time, but struggle with distance as a function of time:

I showed these two graphs to 12 students, some in a Trigonometry class and some in a College Math class.  Every single one got the correct answer for the first question, but only 4 got the correct answer for the second one (every other student chose 'd').  I also asked my 13-year-old son to look at them.  He got the first one right, and also chose letter 'd' for the second one.  I asked him to explain his answer, and he said that the problem didn't state that Chris stood at a distance from the statue for any amount of time (basically he was arguing that the problem didn't explicitly say that any time passed).  It is possible that students may have chosen a different answer if the problem stated that Chris stood at a distance from the statue for a few minutes, but I suspect that some students would still choose letter 'd'.  To improve, students need activities where they see/experience motion happening through time and then create the graph.  This will help strengthen their skill in interpreting these types of graphs.  Dan Meyer (in conjunction with BuzzMath) has collected several videos for this purpose (http://graphingstories.com/1Pp).  These videos are a great way for students to process and represent the relationship between distance and time.

The article also discussed the difficulty that students have with understanding the connection between a graph and an equation.  I have seen evidence of this in my students, even the ones at the top level of achievement.  An example from the article is a problem that attempts to determine whether students understand the connection between the equation of a line, the graph of the line, and a point on the line being a solution to the equation:

I gave this to a handful of students in my Trigonometry class, a group that is fairly accomplished in mathematics (certainly not remedial), and many of them did not choose the correct answer.  I was amazed at students' confusion with the wording of the question.  One student even substituted x=12 and y=1 into the equation and got 29, but still chose letter "b" as the answer!  I also showed the problem to my 15-year-old son, who is an advanced math student, and he responded that he didn't understand what a solution to the given equation looked like.  I suspect that when asked to find a solution to an equation, he is used to having an equation with only one variable in it.  He is currently in Algebra 2 in high school.  Out of the 12 students that I asked, only 4 got the correct answer.

In the College Algebra course that I teach, the unit on Systems of Equations offers an opportunity for students to practice describing solution sets that contain more than one point (when there are a certain number of variables in the system, but not enough linearly independent equations to generate a unique solution).  Many students struggle with how to express the solution set - both the idea that there are an infinite number of solutions, but also with the notation.  In the future, I could introduce students to these concepts in the earlier unit on linear equations by presenting the type of problem above and discussing how to express the solution set to the equation 3x-7y=29.

Another type of question that my College Algebra students struggle with is the type below (also from the article cited above):

Another question that I ask my students about this type of graph is to find when f(x) is greater than g(x) and also when g(x) is greater than f(x).  Students struggle with how to give the answer and usually want to define the intervals in terms of the y-coordinates on the graph (the range) rather than in terms of the input of the function (the domain).

Students need more practice in understanding how the graph of an equation relates to the set of solutions to the equation.  Students need to better understand that solving equations are finding the conditions that make it true.  Too often finding a solution means finding one number that makes the equation true, and students don't get enough practice with finding solution sets that contain more than one discrete value.  Students also need more practice in relating a graph to function notation.  Many students struggle with function notation, and I think that using graphs when teaching function notation would be a help to students.  Finally, students need to practice creating and interpreting graphs of distance versus time.