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.
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"John Graunt (1620-1674) was an English haberdasher who studied birth and death records and discovered that more boys were born than girls. He also found that, because men were more subject to death from occupational accidents, diseases, and war, the number of men and women of marriageable age were about equal.
In 1662, Graunt's book Natural and Polotical Observations upon the bills of Mortality was published. His work was te first to analyze statistics and draw conclusions on ths basis of such analysis. Graunt's work led to the development of actuarial science, which is used by life insurance companies" |
The Curriculum and Evaluation Standards for School Mathematics (1989) stress the need for students to be able to critically analyze data.
| "In this age of information and technology, an ever-increasing need exists to understand how information is processed and translated into usable knowledge. Because of society's expanding use of data for prediction and decision-making, it is important that students develop an understanding of the concepts and processes used in analyzing data. A knowledge of statistics is necessary if students are to become intelligent consumers who can make critical and informed decisions" (p. 105) |
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.
| Stage 1: | Living Graphs |
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Here the human body is used--a human
histogram. A large plastic floor graph would be
divided into columns and rows, with each square
big enough for a child to stand on. Children
would then stand on squares, one child to a
square, and thus the class would be able to tell
the favorite mode of travel, how many came by
car, etc., how many more came by bus than by
car, and so on.
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| Stage 2: | Concrete Representations |
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This is a concrete representational type of
graph. Examples are a toy car representing the
person who came by car, a small plastic animal
representing a pet, a lino tile representing
choice of cereal, a life saver on a knitting needle
representing choice of candy, and clips or
clothes pins on cards representing method of
transportation. The large plastic floor graph can
again be used, but this time the object
represents the child and the class can then sit
around the graph and talk about it.
In stages 1 and 2 it is important that each child either is the object or has an object and that the comparison of relationships is made initially between two things. This will extend and solidify such concepts as more than/less than/the same as and will also address adding on (if there are 6 red shoes and 4 white then the difference between them is determined by counting on from the 4). |
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| Stage 3: | Introduction to Grid: Pictogram/Pictograph |
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This type of graph may use a 10 cm milk carton
cube or any object where there are multiple
copies of the same object (e.g., match boxes;
film containers, etc.) and the plastic floor graph,
probably on a smaller scale than the one used in
stages 1 and 2. Children's pictures are glued to
the bottom of the object and choice of object is
signified by placing an object in the appropriate
column. This type of graph occurs on the floor
for everyone to see. Once made and discussed
you may wish to make it more permanent by
having children draw how they came to school
and paste it on a pre-marked wall graph--where
each segment is the same size--children would
need to be given paper that would fit the graph
segments.
Here a pictorial symbol is used as a representation of a concrete/living object. Each child makes a picture of, for example, mode of transportation and pastes it on a square of the graph. This kind of graph can handle more data (i.e., more than 2 sets of relationships). |
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| Stage 4: | Block Chart/Graph |
| This is a more abstract form of graphing where pre-cut squares are chosen to signify an appropriate choice and these squares are pasted onto a wall graph or chart. Again each child has his/her own square. The children cannot simply stick their "square" anywhere. They must begin at the bottom of the graph (horizontal axis) and work up. | |
| Stage 5: | Squared Paper Graph |
| This is like # 3 & # 4 only now the complete format of the graph is pre-made and the children have to "fill in" a small section. It doesn't matter any more who fills in which section, but again they must begin at the bottom of the appropriate axis. At this stage children may also make their own complete graph, using squared paper and data supplied. They are able now to distance themselves personally from the data (it doesn't matter if it is not about them). | |
| Stage 6: | Abstract Representations |
| These graphs include Bar Lines, Pie Charts, Line Graphs, Line Plots, Stem and Leaf Plots, etc. Frequency tables and frequency polygons are also used at more advanced levels of graphing. Depending on the age and ability of your children with graphing you will use an appropriate type of graph. Later on, when children are more versed in the types of graphs, you will use the best kind of graph to depict the information you have and what you want to know from your information. All graphs need to be interpreted and the task of teachers, once children know the basics of graphing appropriate for their level, is to use the graphs for interpretation and critical thinking. |
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.
Bibliography
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.