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The Research Basis for Graphing and Data Analysis Software Tools to Support Instruction in Grades K–8

This article excerpted from a Scholastic Research Paper.
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In our information-driven world, the ability to graph and interpret data is a critical skill. Every day, we are confronted with graphs and tables conveying important information—from election results to social trends to fluctuations in the stock market. In school, students evaluate data not only in math class but across a range of subject areas—whether analyzing graphs of historical trends or charting results from a science experiment.

Not surprisingly, statistical literacy has become "a major goal of the school curriculum" (Ben-Zvi, 2000). Once the province of college curricula, statistics and data analysis are now considered important topics in all grades, beginning in the primary years. The National Council of Teachers of Mathematics has recommended greater emphasis on data analysis within the mathematics curriculum at every grade level, and national standardized tests, such as the National Assessment of Educational Progress, have increased the proportion of test items in this area (Shaughnessy & Zawojewski, 1999).

However, data analysis remains a challenge for students (Lajoie & Romberg, 1998). At the primary level, students typically have trouble making the transition from concrete "object graphs" to abstract representations of data. In later elementary and middle school, they often fail to develop the deep conceptual understanding necessary to perform higher-level data analysis skills. Many students have difficulty drawing inferences or making predictions, identifying appropriate ways to display data, or recognizing when a graph’s scale has been manipulated to mislead. In short, they struggle with "exactly the [skills] that people need in their everyday lives" (Shaughnessy & Zawojewski, 1999).

Developing data analysis skills takes time. These skills must be learned early and reinforced often. Research reveals several important areas of the graphing and data analysis curriculum where elementary students struggle:

  • The transition from concrete to abstract
  • Data representations
  • Classification
  • Scale
  • Graph interpretation
  • Mean, Median, and Mode

The Transition from Concrete to Abstract

When primary students are introduced to graphing, they typically begin by creating “object graphs” in which physical objects (e.g., shoes, milk cartons, attribute blocks) are sorted and lined up to form a real-life picture graph. However, students often have trouble connecting these real objects to abstract representations (Friel, Curcio, & Bright, 2001).

Research shows that computer software plays an important role in bridging the gap between concrete and abstract representations. In contrast to physical manipulatives, software can provide immediate visual or auditory feedback—linking students’ physical experiences with corresponding symbolic representations (Clements & McMillen, 1996).

In The Graph Club 2.0, for example, each category of data is represented by a graphic symbol that functions much like the physical objects in an object graph. Students drag these symbols one at a time from the “symbol bins” to the appropriate location on the graph. The program reinforces the addition of each new symbol by counting aloud. This concrete approach to manipulating data in a graph helps students connect the real objects in an object graph to abstract representations (pictures of objects in a picture graph, or the height of a bar in a bar graph). As students begin to understand these representations, The Graph Club 2.0 lets them progress to more abstract and efficient methods of adding data, such as dragging the top of a bar to the desired height, or simply typing a numerical value in a table.

Data Representations

Studies show that students have difficulty recognizing the same data when it is presented in different ways (Kerslake, 1981; Bright & Friel, 1998). To address this problem, teachers should provide students with numerous opportunities “to compare multiple representations of the same data set” and engage students in “rich discourse on what each of the representations shows” (Bright & Friel, 1998). Older students have particular difficulty understanding the relationship between raw and grouped representations of data (Bright & Friel, 1998). Tallying raw data helps students better understand the transition between raw and grouped data (Bright & Friel, 1998).

Tom Snyder Production’s The Graph Club 2.0 and GraphMaster make it easy for students to view and compare different representations of the same data. Whenever students open a new file using The Graph Club 2.0, they are presented with two side-by-side representations of the same data. As students make a change to one graph (for example, by dragging additional symbols into a picture graph), the other graph changes dynamically, helping students see the relationship between the two representations. This feature also supports students as they make the transition from more concrete to more abstract types of graphs (e.g., picture to bar). The GraphMaster program’s Tally feature allows students to tally columns of raw data. The resulting Tallied Data display provides a conceptual link between the data table and the final graph.

Classification

Students who develop sorting and classification skills during kindergarten (along with related skills such as sequencing) are significantly more likely than their peers to succeed academically throughout elementary school (Dudek, Strobel, & Thomas, 1987; Silliphant, 1983). Teachers can help students develop classification skills by giving them many practice opportunities with a wide variety of objects.

The Graph Club 2.0 reinforces sorting and classification skills. Unlike some other elementary graphing programs, which sort data automatically, The Graph Club 2.0 provides students with a hands-on sorting experience each time they make a graph. When students create a graph using the program, they add data to a category by dragging symbols into the corresponding column (for example, the cat symbol must be dragged to the cat column). A reinforcing sound confirms that students have dragged a symbol to the correct location. The program’s extensive library of symbols allows students to sort and graph everything from animals or food to hair color or weather — reinforcing classification skills in a wide variety of contexts.

Scale

Understanding a graph’s scale is essential to making sense of the data it displays, yet many students assume that every “tick mark” on a scale represents a single unit and misread graphs in which the scale uses a different increment (Dunham & Osborne, 1991). Increments of more than one unit can be particularly challenging for young students, who often have a limited ability to count by 2s, 5s, 10s, etc. (Friel, Curcio, & Bright, 2001).

Students also frequently fail to understand how the proportions of a graph are affected by scale (Goldenberg et al., 1988; Kerslake, 1981). When a graph’s proportions change, students assume that the data has changed. They don’t realize that the same data can look radically different when displayed with a different scale.

Research shows that extensive practice exploring and describing the effects of scale helps students develop “scale sensitivity” and become more critical of misleading graphs (Ben-Zvi, 2000). The Graph Club 2.0 allows students to change a graph’s scale easily, making it simple for students to explore how scale changes can affect a graph’s appearance and can distort trends. Graph Master requires students to make a decision about scale whenever they create a graph. In contrast to Excel and other standard spreadsheet programs, Graph Master can be set so that default values for scale minimum, maximum, and step size are not provided; students must identify appropriate values on their own. This focuses students’ attention on scale and reinforces the idea that a graph’s step size can be greater than one.

Graph Interpretation

A critical goal of the data analysis curriculum is to teach students to interpret data in order to make decisions and solve problems. However, studies show that students struggle with all but the most basic interpretive tasks.

Researchers have identified three levels of graph-interpretation skill that students should master (Curcio, 1987;Wainer, 1992). The first level involves extracting specific values from a graph: How many books did Jessie read in November? The second level requires integrating information from different parts of the graph: How many more books did Jessie read in February than in September? The third level involves understanding the data set as a whole, making predictions or noticing trends: How many books do you think Jessie might read in March, and why? While students generally can extract data values from a graph (a lower level task), they perform poorly on higher-level tasks that involve additional computation or inference (Shaughnessy & Zawojewski, 1999).

Students also have difficulty expressing their mathematical ideas in writing. This is particularly true for younger students. One study found that “the quality of students’ written work [was] inferior to their verbal reasoning” on statistics questions (Jacobs, 1993). Providing opportunities to speak as well as write about their statistical ideas helps all students demonstrate their understanding.

The Graph Club 2.0 and GraphMaster were designed to support students as they build skills in data analysis and interpretation. The programs include a built-in notebook feature, encouraging students to interpret and write about their graphs. This process makes student thinking explicit and external, helping to build conceptual understanding. Notebook text can be printed with graphs and assessed. The notebook also includes an audio feature that lets students record their ideas orally. This provides a scaffold for students to express more complex ideas, as well as an alternative for younger students or those whose writing skills are less developed. As students write and talk about their graphs, they develop a better understanding of their data.

Center: Mean, Median, Mode

An important statistical skill identified by the NCTM Principles and Standards (2000) is the ability to interpret and summarize what is “typical” in a set of data using measures of center—mean, median, and mode. Many students display poor conceptual understanding of these measures (Mokros & Russell, 1995). Although they may have memorized algorithms for calculating the mean or median, students fail to understand what these values communicate about the data, and view them as arbitrary procedures rather than useful analytical tools.

Graph Master allows students to display statistics (including mean, median, and mode) for graphs containing numerical data. The program displays these values in both numerical and graphical form. This display helps students make a visual connection between the values of the data and the mean or median. For example, students may note that the mean is never greater than the highest data value (an easy mistake to make when applying the algorithm without understanding). Students can view a data table and graph side by side and explore how changing the data will affect the mean, median, and mode. As they do so, students build a conceptual, rather than formulaic, understanding of these measures.

Sources

Ben-Zvi, D. (2000). Toward understanding the role of technological tools in statistical learning. Mathematical Thinking and Learning, 2, 127–155.
Bright, G.W., & Friel, S. N. (1998). Graphical representations: Helping students interpret data. In S. P. Lajoie (Ed.), Reflections on statistics: Learning, teaching, and assessment in grades K–12 (pp. 63–88). Mahwah, NJ: Lawrence Erlbaum Associates.
Clements, D. H., & McMillen, S. (1996). Rethinking "concrete" manipulatives. Teaching Children Mathematics, 2, 270–279.
Curcio, F. R. (1987). Comprehension of mathematical relationships expressed in graphs. Journal for Research in Mathematics Education, 18, 382–393.
Dudek, S. E., Strobel, M., & Thomas, A. D., (1987). Chronic learning problems and maturation. Perceptual and Motor Skills, 64, 407–429.
Dunham, P., & Osborne,A. (1991). Learning how to see: Students' graphing difficulties. Focus on Learning Problems in Mathematics, 13, 35–49.
Friel, S. N., Curcio, F. R., & Bright, G.W. (2001). Making sense of graphs: Critical factors influencing comprehension and instructional implications. Journal for Research in Mathematics Education, 32, 124–158.
Goldenberg, E. P., Harvey,W., Lewis, P. G., Umiker, R. J.,West, J., & Zodhiates, P. (1988). Mathematical, technical, and pedagogical challenges in the graphical representation of functions (Tech. Rep. No. 88-4). Cambridge, MA: Educational Technology Center, Harvard Graduate School of Education.
Jacobs,V. R. (1993). Stochastics in middle school: An exploration of students’ informal knowledge. Unpublished master’s thesis, University of Wisconsin, Madison, Wisconsin.
Kerslake, D. (1981). Graphs. In K. M. Hart (Ed.), Children's understanding of mathematics concepts (pp. 120–136). London: John Murray.
Lajoie, S. P., & Romberg,T. A. (1998). Identifying an agenda for statistics instruction and assessment in K–12. In S. P. Lajoie (Ed.),
Reflections on statistics: Learning, teaching, and assessment in grades K–12 (pp. vii–xxi). Mahwah, NJ: Lawrence Erlbaum Associates.
McGatha, M., Cobb, P., & McClain, K. (1998). An analysis of students' statistical understandings. Washington, D.C.: Office of Educational Research and Improvement, U.S. Department of Education.
Mokros, J., & Russell, S. J. (1995). Children's concepts of average and representativeness. Journal for Research in Mathematics Education, 26, 20–39.
National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston,VA:
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston,VA
Pasnak, R., Holt, R., Campbell, J.W., & McCutcheon, L. (1991). Cognitive and achievement gains for kindergartners instructed in piagetian operations. Journal of Educational Research, 85, 5–13.
Scheaffer, R. L.,Watkins, A. E., & Landwehr, J. M. (1998).What every high school graduate should know about statistics. In S. P. Lajoie (Ed.), Reflections on statistics: Learning, teaching, and assessment in grades K–12 (pp. 3–32). Mahwah, NJ: Lawrence Erlbaum Associates.
Shaughnessy, J. M., & Zawojewski, J. S. (1999). Secondary students’ performance on data and chance in the 1996 NAEP. The Mathematics Teacher, 92, 713–718.
Silliphant,V. M. (1983). Kindergarten reasoning and achievement in grades K–3. Psychology in the Schools, 20, 289–294.
Wainer, H. (1992). Understanding graphs and tables. Educational Researcher, 21, 14–22.

Tom Snyder Productions

Tom Snyder Productions, a Scholastic company, is a leading developer and publisher of educational software for K–12 classrooms. The company was founded over 20 years ago by Tom Snyder, a former science and music teacher who pioneered the use of technology in the classroom to enhance teaching and learning. Today we are proud to carry over 140 award-winning software titles covering each curriculum area, developed with strict adherence to our high standards for quality and innovation. Our products help teachers meet curriculum goals in over 400,000 classrooms, improving student performance and understanding. Our mission is to create innovative products and services to inspire great teaching and improve student learning.

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