Data Analysis in Elementary Mathematics Teaching

1. CHAPTER 21Developing Concepts of Data Analysis Elementary and Middle School Mathematics Teaching Developmentally Ninth Edition Van de Walle, Karp and Bay-Williams Developed by E. Todd Brown /Professor Emeritus University of Louisville

2. Big Ideas • Statistics is its own field distinct from mathematics; one key difference is focus on variability of data in statistical reasoning. • Doing statistics involves a four-step process: formulating questions, collecting data, analyzing data, and interpreting results. • Data are gathered and organized in order to answer questions about the populations from which the data come. • Different types of graphs and other data representations provide different information about the data and, hence, the population from which the data were taken. • Measures that describe data with numbers are called statistics. • Both graphs and statistics can provide a sense of the shape of the data.

3. What Does It Mean to Do Statistics? • Statistical literacy is needed by all students to interpret the world. • Statistics and mathematics are two different fields. • The shape of the data: • How data is spread out or grouped • Characteristics about the data set as whole can be described

4. Process of doing statistics Formulate Questions

5. Formulate Questions Data collection should be for a purpose, to answer a question. • Students should have opportunities to generate their own questions. • Student-generated questions make the data collection more meaningful. • Student- or teacher-initiated questions should be well defined. • Questions that can be answered using statistics.

6. Data Collection • Two types of data- • Categorical-data grouped by labels • Favorite ice cream, color of car, etc. Numerical-data that counts things or measures on a continuous scale How many miles to school, temperature over time, weight of student backpacks.

7. Sampling • Statistics DOES NOT involve gathering from the “whole” population. • Uses a representative sample. • Sampling takes into consideration-variability • Variability means gender, time of day when surveyed, culture, etc. • Students need to • Consider how they will gather data that will include a representative sample • Asking- • What is the population for your question? • Who or what is the subject of your question?

8. Using Existing Data Sources • Print resources • Newspaper • Almanacs • Sports record book • Maps • Children’s literature • Web Resources • USDA Economic Research Service Food Consumption • Google Public Data Explorer • Better World Flux • U.S. Census Bureau

9. Data Analysis: Classifications • Making decisions about how to categorize things • Attribute materials

10. Try this oneActivity 21. 5 Guess My Rule • Materials-Use students’ data in the class. • Directions-Decide on an attribute, e.g., wearing jeans, glasses, hat, etc. • Tell the students “I have a rule.” • Call a person to the front that meets your rule and one that does not meet the rule. • Call up more students and ask the students to predict whether the person meets or does not meet the rule. • Before announcing the rule, give all students a chance to consider the possibilities.

11. Data Analysis: Graphical Representations • Students should be involved in deciding how they want to represent their data. • Creating graphs requires skill and precision. Choosing appropriate scale and labels

12. Data Analysis: Graphical Representations • Object graph is a small step from sorting; actual articles are used as the graph, e.g., types of shoes, favorite fruit, M&Ms!!! • Picture graph moves up a level of abstraction and uses drawing or pictures to represent data; e.g., a book drawing could mean 5 books. • Bar graph-Something is used to represent the data; e.g., sticky notes, multilink or Unifix cubes, etc.

13. Data Analysis: Graphical Representations • Pie charts/Circle graphs are generally used to show percentages/parts in their relation to the whole. • Early pie chart Ratio table with percent

14. Continuous Data Graphs • Line and dot plots • Stem and Leaf plots • Histogram • Box Plots

15. Bivariate Graphs • Line Graph- coordinate axis for plotting bivariate data • Scatter plot - best fit is determined by the line you select that defines the observed relationship

16. Data Analysis: Measures of Center and Variability

17. Measures of Center • Try this one • Activity 21.18 You Be the Judge

18. Variability • Focusing only on outliers or extremes • Considering change over time • Examining variability as the full range of data • Considering variability as the likely range or expected value • Looking at how far points are from the center • Looking at how far off a set of data is from some fixed value

19. Variability • Range-difference between highest and lowest data points • Mean absolute deviation-related to mean-tells how the spread of data. High MAD indicates a lot of deviation between the data points and mean.

20. Analysis and Interpretation of Statistics • Questioning and assessment should focus on how effectively the graphical representations communicate the findings. • Difference between actual facts and inferences that go beyond the data. • Questions should focus on the mathematical ideas as well as the statistical ideas. • Context of the situation • What can be learned or inferred from the data

21. Ideas for Meaningful Discussion about Interpreting Data • What do the numbers (symbols) tell us about our class (or other population)? • How do the numbers in this graph (population) compare to this graph (population)? • Where are the data “clustering”? How much of the data are in the cluster? How much are not in the cluster? • What does the graph not tell us? What might we infer? • What new questions arise from these data? • What is the maker of the graph trying to tell us?