6.EE.9 and 8.F.5
This entry examines students’ initial thinking about graphically representing real-world data. Teaching students the mechanical skills of graphing such as plotting points, drawing the axis, and labeling is reasonably straightforward. What is more challenging is to help them understand the relationship between the variables they are plotting.
Problem 1 and responses from Sherin, B. L. (2000). How students invent representations of motion: A genetic account. The Journal of Mathematical Behavior, 19(4), 399-441. Problem 2 and responses from Mevarech, Z. R., & Kramarski, B. (1997). From verbal descriptions to graphic representations: stability and change in students' alternative conceptions. Educational Studies in Mathematics, 32(3), 229-263.
Typical student responses are included below.
PROBLEM #1 Response labeled figure 2 below
|I:||Tell me about your picture.|
|S:||It said the car was speeding, then he stopped short and took a drink and then went slowly after that. I drew a car to show each part of the trip and how fast he was going. I had to make up the speeds. So, you can see on my picture where he stopped to drink the cactus water because his speed is 0 and I drew a person by the cactus.|
PROBLEM #1 Response labeled figure 18 below
|I:||Tell me about your picture.|
|S:||The man starts out on the left and the arrow shows that he is going fast. The dot shows he stopped. Then the short arrows mean he went slowly and then got faster until he stopped again and drank. After that he started out slowly again and then sped up.|
PROBLEM #2 Response labeled figure 15 below
|I:||Tell me about the graphs you drew.|
|S:||It said to make a graph of the different girls so I knew I needed 4 different graphs to show each one. I put the number of hours they studied on the bottom and the test scores on the side. Sarah said the more she studied the better her scores were, so I made a graph to show she studied a lot of hours and had the most success. I thought about the others the same way - I showed the hours they studied and what their test scores were.**
Students are not likely to choose a coordinate graph to represent a situation if they have not had experience with this type of model. The first problem and set of interviews shows how students find other ways to represent a contextual situation.
Once students have had even limited experience seeing situations modeled using a set of axes they begin to think about ways to represent data on them. Based on the work of Clement (1986), here is a proposed model consisting of four different levels of representation used to model a real-world situation on a graph. Take, for example, the modeling of the shape of a speed vs time graph for a bike rider coasting over the edge of a hill:
Level 1: A practical representation incorporating everyday knowledge of the subject and connecting it to concrete experience. The student would draw a picture shaped like a hill. The student does not connect the two variables of speed and time.
Level 2: The student understands that at any particular time, the bicycle is at a particular speed. The student understands that there are two variables (speed and time) and understands that they are happening simultaneously. They may depict the speed and time in an alternate way (as in the Motion Picture Problem - figure 2 and figure 18 above) .
Level 3: The student understands there is a connection between speed and time and is beginning to develop ways to represent the relationship graphically. The student may draw a separate graph for various times during the ride (similar to the ‘homework’ graph above in figure 15).
Level 4: The student can use an x and y axis to map speed and time and understands the representation of placing points a certain distance from the x and y axis to represent this.
When constructing graphs, some students consider individual points, and others consider the over-all shape of the graph. When teaching construction of graphs, it is important to attend to both ideas. Thinking about the shape of the graph from the context of the situation allows students to select points to plot more carefully, and plotting points may provide scaffolding to help students understand the over-all shape of the graph. Students do better on graphing tasks that they have familiarity with and can relate to (such as hours spent studying and grades) than more abstract tasks with formal mathematical language.
If the teacher over-emphasizes one type of relationship (i.e. an increasing relationship) or one type of graph (i.e. line graph), it may lead to a student perception that all graphs have that form. Teachers should make sure that students are exposed to different types of graphs (e.g. linear, increasing, decreasing, non-linear, those that go through the origin and those that don’t). Only using graphs that are increasing in nature may lead to the common mistake of making all graphs increasing, even when this does not fit the situation.
Traditional approaches to teaching graphing that begin by breaking the process into step-by-step items in a procedure may in fact perpetuate the perceived problem that graphing is a difficult topic. Instead, teachers can present students with a purposeful task in a familiar context, and students will be able to act intuitively to use line graphs. Using computers to help construct the graphs can help students look beyond the specific procedure of creating the graph and spend more time considering the meaning of the graph as it pertains to data.
Intentional discussion of common conceptual errors is more effective than teaching styles that avoid or ignore student misconceptions. Using problem situations intentionally designed to evoke common errors provides opportunities for students to confront these misconceptions. For instance, if students agree that the graph of a bicycle going over a hill should look like a hill, start with this representation and ask students to think about how their speed would change (draw on their experience with bicycles). Summarize their thinking in words: “So what I am hearing is that you would start out with some speed going into the hill but then you would slow down as you climb the hill. Then, at the top you would coast a little and then speed up as you go down the other side. Is that correct? So lets take a look at this graph to see if it matches what you told me...”
The app Action Grapher is an excellent tool for exploring the graph of a bicyclist riding up a hill. The user is first presented with a terrain with a bicycle and the option to graph height, speed, and distance versus time on three different graphs. Once the user has sketched these graphs, tapping "Go" animates the biker and correctly draws the three graphs. This app can be a good possible strategy for helping students learn to construct graphs of simulations of real-world data.
Ainley, Janet (1995). Re-viewing Graphing: Traditional and Intuitive Approaches. For the Learning of Mathematics volume 15 number 2 June 1995. Pages 10-16
Bell, A., Brekke, G., & Swann, M. (1987). Diagnostic teaching: 5 graphical interpretation teaching styles and their effects. Mathematics Teaching, 120, 50-57.
Clement, John. The concept of Variation and Misconceptions in Cartesian Graphing. Focus on Learning Problems in Mathematics, 1989.
**Mevarech, Z. R., & Kramarski, B. (1997). From verbal descriptions to graphic representations: stability and change in students' alternative conceptions. Educational Studies in Mathematics, 32(3), 229-263.
*Sherin, B. L. (2000). How students invent representations of motion: A genetic account. The Journal of Mathematical Behavior, 19(4), 399-441.
Vergnaud, G., & Errecalde, P. (1980). Some steps in the understanding and the use of scales and axis by 10-13 year old students. Paper presented at the Proceedings of the fourth international conference for the psychology of mathematics education.
- 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?