Analysis of Change: Scaling

Contents

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

Theme

Creating graphs

Common Core State Standards

8.EE.5 8.F.5

Introduction

Understanding and interpreting scale is an ongoing problem for students when they are graphing. Scale is not typically something that they pay attention to, and with the increasing use of technology (graphing programs, applcations, and graphing calculators) scales often need to be changed to correctly interpret the data. This entry will look at ways that these new technology tools can improve student understanding of scale.

Symbolic Representation

Problem Response
Using a graphing calculator, plot y=2x+3 and y=-0.5x - 2.5. Explain why the graphs of the lines do not appear at right angles on the screen. What could you do to make the lines look more perpendicular?

Only four of the students (16%) in one study explained without prompting from the interviewer that unequal scaling of the axes would distort the graph. (Mitchelmore & Cavanagh, 2000)

Display the graph of y=x2 - 2x + 3 on a graphing calculator. Can you change the window settings of the calculator so that the graph appears as a horizontal line?  

Interviews with students

(I: Interviewer, S: Student)

I: Why do the lines not appear at right angles on the screen?
S: Because the lines are not perpendicular to each other. The graphing calculator shows it correctly.
I: What is the slope of the two lines?
S: 2 and - ½. Oh...those should be perpendicular. I wonder why they aren’t
I: Can you change the scale on the calculator to make them perpendicular?

Mathematical Issues

Because students rarely pay attention to scale, they often make mistakes about the format of the window of a graphing calculator. Very few (less than 10% according to one study) recognised that the apparent slope of a linear graph depends on the scaling of the axes. Simply changing the scale on a graph can have a great impact on the difficulty of the task for students. In many cases students continue to read the scale as if it was in single units even when it has been changed to count by 2’s, 5’s, or other increments.

Graphing software allows students to enter data and make several similar graphs promoting discussion regarding which type of graph and scale best portrays the trends and patterns associated with the task being investigated.

Should mathematics teachers attempt to structure their lessons and choose examples so that students avoid misconceptions regarding the technology limitations of their graphing calculator? Or should they deliberately plan activities which ask students to face them? Students may benefit from confronting the limitations of the technology and attempting to explain them. Students basic mathematical understanding could be strengthened and this could lead to interesting mathematical discussions . Care needs to be taken in how student misconceptions about the technology are addressed. It might be wise to avoid difficulties in the early stages and then structure challenges to draw attention to old misconceptions or new developments.

Teaching Strategies

To help students pay better attention to scales and scaling, teachers should use better examples. Do not choose functions that fit neatly into a grid that spans from -10 to 10 on the x and y-axis. Students also need to realize that the shape of the graph depends on the window you are viewing it through. Using graphing utilities is an excellent way to explore different views of the same graph.

When dealing with problems of shape and scale avoid linear functions. They are easy to graph but do not contain enough features to develop students’ visual intuition. Although counter intuitive for teachers, it may be more effective to analyze the graphs of linear functions after more complicated algebraic functions. This is possible with graphing calculator technology.

Listen to students talk about graphs. Do not assume they “see” graphs the same way a teacher does. Language is key to helping students internalize the connections between symbolic and graphical representations.

Recognize when using graphing technology, students might not know what features to attend to. Just because expert mathematicians know that the slope and the y-intercept are salient and defining features on a graph, that may not be the feature that students pay closest attention to. For example, when students were presented with the graphs of a series of functions (y=2x+1, y=3x+1, y=4x+1), very few students noticed that the y-intercept was constant. Instead, many students described the lines with higher slope values as “more jagged”, because the pixels on their computer screen readout did not form a perfectly smooth line.

The Microcomputer-Based Laboratory (MBL), Calculator-Based Laboratory (CBL), and Tablet Enhanced Learning Environments for Mathematics are all technologies that allow students to physically model position, speed and acceleration. A student can attach a probe to an object, move the object, and the computer program can generate a graphical display of the position, speed and acceleration. Hands on experience can help foster student-student discussion and can also help to form conceptual understanding.

Graphing calculators can be a helpful tool in teaching graphing concepts when used correctly. For example, when investigating quadratic functions, it is important for the teacher to prepare the student by introducing appropriate vocabulary. The teacher presents a quadratic function and has the students brainstorm what they think are “important points” on the graph of the parabola (highest point, x-intercepts, y-intercepts), and the teacher helps provide technical vocabulary for these ideas. Then when students investigate different graphs on their graphing calculators, they have an idea of what to look for and they have the terminology to be able to communicate their ideas.

Students are able to listen to a graph and detect the overall shape of the graph. Through music, the up and down nature of the graph can be abstracted and students can understand the mathematical relationship underlying it (Nisbet & Bain, 2000). This may also help students who prefer an auditory learning style. With advances in computer software it is feasible that technology can produce auditory output from a mathematical function in order to help visually impaired students with their understanding.

To improve LD student success rate when constructing and interpreting bar and line graphs, more emphasis should be placed on graph label, scale, information being presented

References

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

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

Demana, Frank; Schoen, Harold L; Waits, Bert (1993). Graphing in the K-12 Curriculum: The Impact of the Graphing Calculator.

Sherin, B. L. (2000). How students invent representations of motion: A genetic account. The Journal of Mathematical Behavior, 19(4), 399-441.

Ainley, Janet; Nardi, Elena; and Pratt, Dave. The Construction of Meanings for Trend in Active Graphing. International Journal of Computers for Mathematical Learning5: 85-114, 2000.

Moschkovich, J., Schoenfeld, A., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 69-100). Hillsdale, NJ: Lawrence Erlbaum Associates Publishers.

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

Mok, I. A. C., Johnson, D. C., & Cheung, J. Y. H. (2000). Introducing technology in algebra in Hong Kong: addressing issues in learning. International Journal of Mathematical Education in Science and Technology, 31(4), 553-567.

Nisbet, S., & Bain, J. D. (2000). Listen to the graph: Childrens matching of melodies with their visual representations. In 24th conference of the International group for the psychology of Mathematics education.

Ruthven, K., Deaney, R., & Hennessy, S. (2009). Using graphing software to teach about algebraic forms: a study of technology-supported practice in secondary-school mathematics. [Feature FTI: Y; P]. Educational Studies in Mathematics, 71(3), 279-297.

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.

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?