Data management is the topic of the fifth Ideas
and Resources for Teachers of Mathematicsnewsletter. At the elementary level this
curriculum strand is called Data Management
and Analysis, at the middle level, Data
Management, and at the secondary level, Data
Analysis and Consumer Mathematics.
Throughout the new Saskatchewan mathematics
curricula data management includes graphical
representations, statistics, analysis and
interpretation of data, and probability.
Statistics is derived from Latin meaning "of the state". Originally, statistics referred to numerical information about state or political territories. Dolan, Williamson and Muri (1997) define statistics as "the science of collecting, organizing, and interpreting data"(p. 175). Billstein, Libeskind, and Lott (1993, p. 439) outline the following historical note concerning the first recorded study in data collection and analysis.
The Curriculum and Evaluation Standards for School Mathematics (1989) stress the need for students to be able to critically analyze data.
A graph is a visual representation of data, showing comparison between variables. It communicates information about relationships in a concise, appealing and easy to understand format. This information, to be useful, needs interpretation. Data collection is best done with a purpose and a question--what is it you want to investigate? Do children want to know what TV shows grade fives like, or what they ate for breakfast, or what form of transportation they used today to come to school, ... , then conducting a survey and collecting real data from the class is the way to go. Children should first be asked to predict or hypothesize the outcome and then develop a plan to test their prediction, execute their plan, analyze the data and determine the appropriateness of their prediction, and then decide what they should now do with this information--how they should display it--perhaps in graphical form. Data can be collected for the purpose of teaching children how to organize data and construct graphs, but it is more normal for data to be collected for a real purpose, because a question needs to be answered or a hypothesis tested. The NCTM 1982 Yearbook has three excellent articles depicting statistics topics of interest to children in the middle years--one on sports cards (Kuhl), where baseball cards are used to compute averages and to construct graphs, one on opinion polls (Vance), and one on using graphics to represent statistics (Sanok).
A developmental progression of graphing for the elementary grades would look something like the following stages. These stages have been developed and expanded from the work of the Nuffield Foundation (1969), Irmis (1971), and from the work of Cathcart, Pothier and Vance (1997). An example of data collected from an elementary class might be ways of getting to school.
Graphs can be used and abused. The newspaper sometimes uses the wrong type of graph to depict information, thus deliberately skewing the data in favor of one variable. Examples of newspaper graphs and statistics should be collected (by teachers and by children), and classes can use these examples to discuss aspects of graphing.
Ways of Teaching Graphing
Think about how you might teach graphing (isolated unit; integrated with other math/school subjects; sporadically/daily etc. as situations arise). Would you only teach graphing in math class? Think also about why you might teach graphing--what for you, other than the fact that it's in the curriculum, would make you want to teach it--what are the advantages of teaching graphing or using graphing? Of all mathematical topics, graphing is perhaps the most easily integrated across the curriculum: it can be integrated within mathematics itself, within other subject areas, and with children's experiences outside school. Where possible real data should be used--that is data generated from questions the children ask, and data collected by and analyzed by children. Children's response to this graphical data may be in the form of a story, a discussion, oral or written questions, or a play. Data collected, managed, visually presented, and interpreted by children will provide them with authentic experiences similar to the production of graphs in society. All graphs have a purpose and a value (albeit sometimes skewed) in how they are presented. Children need to be aware of the different types of graph, which data is most suitable for which type, and the intended and real meaning of graphical representations.
Students (appropriate to their grade level) should be involved in all steps of statistical 'production'--from "formulating key questions; collecting and organizing data; representing the data using graphs, tables, frequency distributions, and summary statistics; analyzing the data; making conjectures; and communicating information in a convincing way" (NCTM, p. 105). Students in grades 9 through 12 should consolidate, and extend their earlier statistical understandings through studies in mathematics (e.g., curve fitting, through social studies opinion polls, plant growth records in biology, and generally through any experience in their school or personal lives that integrates with any form of statistical analysis). Secondary students need to understand the concepts of randomness, representation, bias in sampling, regression lines and scatter plots, central tendencies, margins of error, and what is meant by a normal distribution. These concepts can best be studied through statistical samples that portray information of interest to secondary students.
MathFINDER (Kreindler & Zahm, 1992), a sourcebook of lessons to illustrate the NCTM Standards, suggests examples of statistical lessons or activities for children to work on. At the K-4 level this book suggests that students sprout seeds and conduct a plant-growth experiment (p. 21); at the grades 5-8 level an appropriate activity might be to study the ages of presidents at their death (p. 48) or in Canada-- our prime ministers; at the grades 9-12 level, students might study ice cream cone prices and construct and draw inferences from a summary chart and to apply measures of central tendency, variability, and correlation (p. 76).
Middle and secondary students can access statistical data available in the world wide web and use this data in various ways to analyze and interpret . Dixon and Falba (1997) have listed a number of world wide web sites that address statistical data and outline some class activities at the middle level that use this data.
Billstein, R., Libeskind, S., and Lott, J. (1993). A Problem Solving Approach to Mathematics for Elementary School Teachers. Don Mills, ON: Addison-Wesley, Inc.
Cathcart, G. W., Pothier, Y. M., and Vance, J. H. (1997). Learning Mathematics in Elementary and Middle Schools, Second Edition. Scarborough, ON: Prentice Hall Allyn and Bacon.
Dixon, J.K., and Falba, C.J. (1997). Graphing in the Information Age: Using Data from the Worldwide web. Mathematics Teaching in the Middle School, 2, 5, pp 298-304.
Dolan, D., Williamson, J., and Muri, M. (1997). Mathematics Activities for Elementary School Teachers, A Problem Solving Approach. Don Mills, ON: Addison-Wesley, Inc.
Irmis, E. (1971, April). Graphing. A presentation made at the 49th Annual Metting of the National Council of Teachers of Mathematics, Anaheim, CA.
Kreindler, L., and Zahm, B. (1992). MathFINDERTM Sourcebook: A Collection of Resources for Mathematics Reform. Armonk, NY: The Learning Team, Inc.
Kuhl, O. (1982). Sports card math. In L. Silvey and J. R. Smart (Eds.) Mathematics for the Middle Grades (5-9), pp. 162-163. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM, Inc.
Nuffield Foundation Mathematics Project Publications (1969). Pictorial Representation. Don Mills, ON: Longman Canada Ltd.
Sanok, G. (1982). Using graphs to represent statistics. In L. Silvey and J. R. Smart (Eds.). Mathematics for the Middle Grades (5-9), pp. 172-176. Reston, VA: National Council of Teachers of Mathematics.
Vance, J. H. (1982). An opinion poll: A percent activity for all students. In L. Silvey and J. R. Smart (Eds.). Mathematics for the Middle Grades (5-9), pp. 166-171. Reston, VA: National Council of Teachers of Mathematics.
The above section has served to introduce the topic of statistics and to develop the idea that statistical inquiry by students is very important to develop--through authentic developmentally appropriate and curriculum-relevant activities.
The following three sections of this newsletter will present activities and samples of children's work in data management at each of the elementary, middle and secondary levels. Following the activities and work samples you will find a data management resource list, which suggests resources for internet sites, software, children's literature, audio/visual and published material (i.e., trade books) that have ideas for teaching and learning data management.
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