Analysis of Change: Graphs as a Literal Picture

Contents

Theme
Common Core State Standards
Introduction
Symbolic Representation
Interviews with students
Mathematical Issues
Teaching Strategies
References
Your Experience

Theme

Interpreting graphs

Common Core State Standards

8.F.5

Introduction

To understand graphing, students must understand that a coordinate graph is a visual representation of a numeric relationship between two changing quantities. They must go beyond interpreting or constructing the graph of a situation as a literal picture of the situation (as in a velocity-time graph when they describe the positive slope as the bicycle climbing a hill instead of accelerating). This entry will describe common student misconceptions and teaching strategies associated with coordinate graphing as a literal picture instead of a description of a relationship between two changing quantities.

Symbolic Representation

Imagine the following ball on a hill:

Imagine I let this ball go, it rolls down the large incline, up the small incline, and onto the flat surface.

Group 1: Draw a graph that represents the speed of the ball from where I let it go to the end of the flat surface at the top of the small incline.

 

The students got 55% correct (as indicated by the four types of responses listed along the top row). 24% had responses that matched the pictorial situation (as indicated by the second row). 7% had responses in the 3rd row, and 15% in the fourth row or miscellaneous. (Berg 1994)

Interviews with students

(I: Interviewer, S: Student)

I: Why did you draw what you did? (for one of the incorrect drawings—2nd row, 1st column)
S: First I drew a line going down to show where the ball is going down. Then the ball goes up a little, so I drew a line that went up a little bit on the graph. Then the ball goes flat so I drew a flat line.

(This student was drawing the position of the ball, not its speed vs time)

I: Why did you draw what you did? (for the correct drawing—1st row, 1st column)
S: When the ball goes downhill it goes faster so I drew my line going up to show that the ball is going faster. Then the ball goes uphill. I drew my line going down to show that the ball is slowing down. Then the ball is rolling horizontally so it keeps the same speed so I drew my line horizontal.

Mathematical Issues

In the first interview based on the sample problem above, the student used their everyday knowledge of a ball rolling but demonstrated a common misconception by interpreting the graph as a literal picture. Another example of this misconception is when the point of intersection of two lines on a speed vs. time graph of two cars is frequently interpreted as the point that the cars crash.

The student in the second interview has made a connection between the two variables in the situation and how the relationship is represented on the coordinate graph.

Students' understanding and experience of reality has a strong influence on their thinking as they examine graphs. Often, they incorrectly interpret the interaction between the axes because their experience leads them to interpret the image as a representation of the reality visually rather then information from the action. For instance, in the following videos students struggle to counteract their experience as they interpret the graphs. In the first, the boy uses his experience of walking rather than thinking of the interaction between time and distance. In the second, the girl uses her experience of driving to Oklahoma to interpret the situation. (password: algebrathinking)

Problem: Which of the graphs below represent journeys?  Describe what happens in each case. Why do you think that?

 

 

Problem: The Ramirez family’s whole holiday is shown on the graph. The vertical axis shows the distance in kilometers away from home. The horizontal axis shows the time in days since the start of their trip. a) During which days did the Ramirez family travel fastest? b) They stayed with friends for a few days. Which days were these? c) On average, how fast did the Ramirez family travel to get to their destination?

 

 

 

Teaching Strategies

At the Center for Algebraic Thinking you can find the "Action Grapher" app for iOS that is built with research on iconic translation in mind. One of the main issues with this concept is that students think of a real situation and imagine a graph as reflecting that situation. The app asks students to consider a biker going over different shaped terrains and what the graph might look like. They are provided with three different ways to graph that situation: Distance vs. Time, Height vs. Time, and Speed vs. Time. They estimate what they think the graphs will look like by drawing the graphs with their finger. Then students can animate the graphs to see how they take shape as the biker travels the terrain. This way students can see how graphs can be related to real life, as the height graph does look like the terrain, but also how the graph communicates something very different, as in the other two graphs. 

Elementary-aged students seem to be the most influenced by a graph’s content. They often interpret the abstract representation of a graph as a picture of the actual event. A minimal amount of instruction with graphing can help them overcome this error. Instruction may include tasks in which students collect their own data, create a graph, and are then asked questions about the graph they created (e.g. Where on the graph does it show...?). Activities that use a Calculator Based Ranger (CBR) to examine distance v. time graphs is another way students can see how motion and time are represented graphically.

Students seem to have the most difficulty with graphs depicting change, especially acceleration and velocity. They have the easiest time comprehending graphs that depict distance or position (see Motion Graphs). Because background context is so important for successful graph reading, it is important to provide graphical representations that are culturally accessible to students. It is better to start learning to read graphs within contexts that are familiar to students, and then move toward traditional curricular topics once internal graph reading competency has been established. For example, students can collect data on the drop height and bounce height of a ball, or height of a plant over time. See also Student Background Knowledge.

References

Ainley, J. (2000). Exploring the Transparency of Graphs and Graphing. 18-27.

Berg, C. A., & Smith, P. (1994). Assessing Students' Abilities to Construct and Interpret Line Graphs: Disparities between Multiple-Choice and Free-Response Instruments. Science Education, 78(6), 527-554. 

Billings, E., & Klanderman, D. (2000). Graphical representations of speed: obstacles preservice K-8 teachers experience. School Science and Mathematics, 100(8), 440-450.

Friel, Susan N; Bright, George W. (1996). Building a Theory of Graphicacy: How Do Students Read Graphs? Paper presented at the annual meeting of the American Educational Research Association, New York, NY.

Hadjidemetriou, C., & Williams, J. (2002). Children's graphical conceptions. Research in Mathematics Education, 4, 69-87.

Mitchelmore, M., & Cavanagh, M. (2000). Students' difficulties in operating a graphics calculator. Mathematics Education Research Journal, 12(3), 254-268.

Monk, S. (1992). Students’ understanding of a function given by a physical model. In G. H. E. Dubinsky (Ed.), The Concept of Function: Aspects of Epistemology and Pedagogy. USA: Mathematical Association of America.

Monteiro, Carlos, & Ainley, Janet. (2004). Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education. Universidade Federal de Pernambuco/Brazil - University of Warwick/UK.

Nemirovsky, R. (1998). Body motion and graphing. Cognition and Instruction, 16(2), 119-172.

Nemirovsky, R., & Monk, S. (1996). "If you look at it the other way...": An exploration into the nature of symbolizing. In P. Cobb, E. Yackel & K. McClain (Eds.), Symbolizing and Communicating in Mathematics Classrooms Perspectives on Discourse (pp. 177-221): Routledge.

Roth, Wolff-Michael (2002). Reading graphs: Contributions to an integrative concept of literacy. Journal of Curriculum Studies, 34(1), 1-24

J. Clement (1989). The Concept of Variation and Misconceptions in Cartesian Graphing. Focus on Learning Problems in Mathematics, Volume 11, 77-87.

Your Experience

  • What have you learned about your students’ thinking regarding this topic?
  • What have you learned that is an effective teaching strategy to help students understand this topic?