Analysis of Change: Interpreting Graphs

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.EE.5,6 8.F.1,2,3,4,5

Introduction

Many students have difficulty interpreting graphs. This entry describes different stages of student thinking about interpreting graphs and teaching strategies that can help students understanding of graphs as they learn to interpret them.

Symbolic Representation

Problem Response

Tell me everything you can about this graph.

 

Students may respond at an Explicit, Implicit, or Conceptual level (see interviews below)

Interviews with students

(I: Interviewer, S: Student)

I: What can you tell me about this graph?
S: Time is on the x axis and Distance is on the y axis. It has straight lines and they go up.

(Explicit: superficial reading that focuses on the details of the graph such as title, type and names of variables, type of phenomenon represented. This type of processing does not involve interpreting meaning for the data represented.)

S2: It appears that this object traveled at a constant rate over time, stopped briefly, continued at the same rate as before, stopped again for a short time and then continued at the same rate. (Implicit: the student can interpret the graph beyond its isolated elements. They can identify trends and relationships between the variables)
S3: This is basically a linear relationship, meaning the time and distance seem to remain constant except for the two places there is no distance being covered for a short amount of time. Whatever the ‘object’ is, it appears to be able to go from 0 to its travel speed instantaneously since I don’t see any place on the graph indicating a speeding up or slowing down. (Conceptual: an overall analysis of the structure of the information. Interpreting, explaining, or predicting the phenomenon represented)

Mathematical Issues

One study proposes three levels of analysis in the processing of graphical information. Those levels are:

1. Explicit: superficial reading that focuses on the details of the graph such as title, type and names of variables, type of phenomenon represented. This type of processing does not involve interpreting meaning for the data represented.

2. Implicit: This involves interpretation of the graph beyond the reading of discrete elements of the graph. It emphasizes the translating of information that is present, but via symbols and involves knowing certain “codes” that the reader must have mastered. It involves going beyond the reading of coordinate points in isolation and includes noticing patterns between the values.

3. Conceptual Information: This level emphasizes the relationships between the variables and also draws on prior knowledge to interpret the phenomenon being represented. Students base the conceptual information on analyzing the structure of the graph and use their knowledge of the content to produce interpretations, explanations, or predictions.

Although these three levels of processing are increasingly more complex, they should not always be considered sequential. The way a student responds to a graph will also depend on the questions or tasks they are given in association with the graph.

A challenge for students can be understanding the interaction between axes. Initially, students tend to focus on a single dimension.  In the following video, the student is struggles to consider time and distance simultaneously. (password: algebrathinking)

Problem: Which of the following are journeys? Describe what happens in each case. Why do you think that?

Teaching Strategies

Teaching graph comprehension should take place at three successive stages. At the elementary stage, students should be able to extract information from the data, or “read the data.” Teacher questions such as “What is the value of X at time 12:00?” is an appropriate cue for this level. To be successful at the elementary stage, students must understand the conventions of the graph design, including the scale and labels.

At the intermediate stage students find relationships within the data, or “read between the data.” Teacher questions to encourage this type of thinking might include condensing data categories (such as “How many boxes have more than 30 raisins?”), identification of trends in parts of the data (“How did the value of the stock change over the first 5 days?”) and observing relationships in visual displays without reference to the contextual meaning (“How do the changes in these two curves compare?”). At the intermediate stage, students must be able to make comparisons and perform the appropriate computations.

The final stage “reads beyond the data,” and requires interpolation/extrapolation such as prediction or generation. Teacher prompts to develop this type of thinking might include reduction of the data to a single statement, (“From June 15 to June 30, what was the trend in the value of stock X?”), extending the data to form predictions, (“If students opened another box of raisins, how many might they expect to find?”). and using the data as evidence to support arguments, (“If this graph was offered as a piece of evidence to prove true the statement, “the economy is improving,” how would you describe the connection between the graph and the attempt to prove the statement true?”). To be successful in the final stage, students must also be able to relate the data into the context of the situation.

One suggestion is to begin with instruction of picture graphs, line plots and bar graphs in the early grades, progressing to stem plots and pie graphs in grades 3-5, and adding histograms, box plots and line graphs in grades 6-8. Picture graphs should precede bar graphs, and as an intermediate stage, the bar graph can be composed of towers of rectangles, so that the individual data elements are still visible in the graph. When introducing more complex forms of data, the number of data points, number of categories, and the variation or spread in the data should be considered. As data sets increase in size, the concept of scaling needs to be introduced. Students frequently struggle with histogram representations because of the “disappearance” of data points.

An integrated approach that emphasizes both the mechanical interpretation of graphs AND the power of graphs as a model for studying relationships between variables is necessary. Student difficulties appear to arise because variable relationships presented symbolically are too abstract for most students in the 11- 13 age range. Students should be involved in the entire process - gathering the data, creating the graph, analyzing variables. It is possible to teach graphing skills in isolation if all we expect is for students to perform algorithmically.

References

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

Berg, C. A., & Phillips, D. G. (1994). An Investigation of the Relationship between Logical Thinking Structures and the Ability to Construct and Interpret Line Graphs. Journal of Research in Science Teaching, 31(4), 323-344.

Friel, S., Curcio, F., & Bright, G. (2001). Making sense of graphs: Critical factors influencing comprehension and instructional implications. Journal for Research in Mathematics Education, 32(2), 124-158.

McClain (Eds.), Symbolizing and Communicating in Mathematics Classrooms Perspectives on Discourse (pp. 177-221): Routledge.

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.

Postigo, Yolanda and Pozo, Juan (2004). On the Road to Graphicacy: The learning of graphical representation systems. Educational Psychology, Vol.24, No.5.

Swatton, P and Taylor, RM (1994). Pupil Performance in Graphical Tasks and Its Relationship to the Ability to Handle Variables. British Educational Research Journal, Vol 20, No.2 (1994), pp. 227-243.

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?