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Rumsey, all rights reserved. This text may be freely shared among individuals, but it may not be republished in jse medium without express written consent from the author and advance notification of the editor. Key Words: Introductory statistics; Statistical literacy Abstract In this paper, I will define statistical literacy what it is and what it is not and discuss how we can promote it in our introductory statistics courses, both in terms of teaching philosophy and curricular issues. I will discuss the important elements that comprise statistical literacy, and provide examples of how I promote each element in my courses. I will stress the importance of and ways to move beyond the "what" of statistics to the "how" and "why" of statistics in order to accomplish the goals of promoting good citizenship and preparing skilled research scientists. We must be sure to promote the use of the scientific method in all of our students: the ability to identify questions, collect evidence datadiscover and apply tools to interpret the data, and communicate and exchange results. I believe that for students to achieve both of these overarching goals, they need to understand and use statistical ideas at many different levels. To begin, they need a certain level of competence, or understanding, of the basic ideas, terms, and language of statistics. But being a good statistical citizen and jse scientist requires more than this; it requires that the student be able to explain, decide, options, evaluate, and make decisions jse the information These require additional skills in statistical reasoning and thinking, but the foundation for these skills should first be developed at the statistical literacy level. What process did this information go through in order to get to you? First, the information was produced, or generated, by a researcher for example, a medical doctor, professor, U. Finally, the information is consumed, or received, by the audience the general public This extends the ideas presented by Galwho identified students as mainly consumers and at times producers of information. Regardless of where a person is involved in the chain of statistical information, there will be a need for a basic understanding of the concepts and language, a level of reasoning the abilities to question, compare, and explain and a level of statistical thinking applying the ideas to new problems and identifying questions of your own In this paper, I will define statistical literacy what it is and what it is not and discuss how we can promote it in our introductory statistics courses, both in terms of teaching philosophy and curricular issues. Being able to interpret graphs and tables. Being able to read and make sense of statistics in the news, media, polls, etc. That is the issue of statistical literacy What specific concepts and skills will be needed in the context of specific jobs? That is the issue of statistical competence. At the next level, the student would be asked to describe the results of study by interpreting the results Students may be asked to produce data on a similar study. Then they might be asked to evaluate the study which involves critical thinking, and questioning the study at every phase Finally, the student may be asked to communicate this information to peers. Some of these tasks require basic statistical competence, and others require higher order knowledge skills, such as statistical reasoning and thinking. Additional lists of learning outcomes for statistical literacy are provided by CobbMoore aband Garfieldamong others Each list seems to encompass two different types of learning outcomes for our students being able to function as an educated member of society in this age of information, and having a basic foundational understanding of statistical terms, ideas, and techniques. Students seem to give more credibility to examples gleaned from the real world; these examples can serve as a powerful motivator for students to ask more questions. Remember, what is important and interesting to us as teachers may not be interesting and important to our students. The best way to find out what is interesting to students is to ask them to provide their own ideas, and then to save those ideas for future use. We can promote data awareness by always providing a relevant context for the ideas that we present in class. To discourage students from becoming too skeptical about statistics, it is also important to provide examples where statistics are used correctly, not only to show the bad examples with which we are all so familiar. To assess data awareness, I typically ask students to answer questions of an open-ended nature: Identify the source of this study, discuss how statistics were used to answer the question, and describe the impact of the study on the general public. My advice is to get them on your side early and keep them there with relevant, interesting examples. If someone were to write such a book about statistics, what would be included? What exactly are the basics of statistics? Binary lists are similar to the ones given by CobbMoore abGarfieldSnelland Galamong others. Which specific content and methods should appear on an introductory statistics syllabus is a topic that has received much attention in the literature; it is not something I will attempt to resolve in this paper. However, I would like for us to rethink what it means for a student to understand any statistical idea. What does it mean to understand a statistical idea? Basically, understanding a statistical idea means to be able to relate the concept within a nonstatistical subject matter; to explain what the concept means, to use it in a sentence or within a broader problem, and to answer questions about it. How should a student demonstrate understanding? There are several misconceptions that statistics education researchers are trying to dispel. Here are some of them Gal tells us that doing statistics is not equivalent to understanding statistics. Calculations should not be the center of attention in the classroom. It emphasizes the mathematics, not the statistics. I have been successful in getting students to come up with their own measure of variability through a sequence of exercises. Often, they will provide an absolute value instead of the square, options divide by n instead of n -1, but other than that, they have the idea and all without a formula I believe the students were able binary accomplish this because they thought about what they wanted to measure, and how to measure it; and they were able to verbally express through examples what they wanted to do. Some textbook authors, such as Utts teach statistical ideas with with minimal use of formulas. I have also used fewer formulas in my class, and have found that I am more successful when I demonstrate the need for the statistic, lead students to discover ways to measure it, then talk them through the steps of finding it, all before a formula is used. I never motivate an idea with a formula. The formula always appears at the end, almost as an afterthought If a formula is presented first, I feel that some students lose binary, some experience increased anxiety, and others simply think that is all they need to know about the idea These are not the attitudes we want to reinforce. A statistical formula is simply mathematical shorthand for a process of obtaining a statistic. I find that students tend to easily forget formulas once the exams binary passed. We spend too much time calculating and too little time discussing. In short, we are too narrow. Suppose you give students a hypothesis test for two population means, where the variable is response times for men and women. You ask students to describe a Type II error. Do all teachers really understand this? And perhaps a more important question is, did it make sense to you the first time you explained it that way? If we take the ideas and present them options more relevant and usable language for our students, connecting the big ideas with common threads, I believe students will be able to incorporate those ideas more readily into their own knowledge base In defense of formulas and calculations, they do play an important role in a student being statistically competent. However, they should not be the focal point, or the endpoint of their knowledge How can we promote statistical understanding in our students? I believe the HOW we do statistics must be motivated by WHY we should do the statistics and WHAT we are trying to do with the statistics. Promoting the HOW at the expense of the WHY is a mistake. Examples of the latter include students who spend enormous amounts of time calculating a correlation coefficient, when a simple division of the X - Y plane into four quadrants using the mean of X and the mean of Y can provide a simple intuitive understanding of what the correlation coefficient measures. The calculations seem to become an obstacle to understanding Here are some ideas that I have binary promote understanding among our students. First, help students learn proper techniques and tools, when they are used, and why they are binary Present them with definitions that they can take hold of, rather than struggle options Emphasize big ideas and common threads, and not the teaching and testing of knowledge as it falls within certain chapters of a book. Keep cross-referencing and comparing ideas to highlight connections. What little precision I have lost has been made up in intuitive understanding of the big idea of margin of error. In my opinion, splitting hairs about these terms will only create confusion and frustration. My advice is to choose the most important ideas, and stick to them. Big ideas and common threads have taken me a long way in my courses. David Moore taught me to consider that if you are not going to use an idea later, then why spend a lot of time on it now? This has become a teaching theme for me the first things to go were permutations and combinations We can assess student understanding with questions that ask them to identify "what is the population? We can also give students many opportunities to explain and discuss statistical ideas with each other, and watch closely while they do this I am convinced that students who are most likely to be good statistical citizens and research scientists are those that are successful at incorporating statistical concepts, terms, and ideas into their own language To extend this idea of understanding to collaboration, I assign student teams on the first day of class, and I rotate teams randomly several times during the semester. Throughout the semester, I see a language emerge throughout the class that is correct, but unlike the language I use as a teacher to explain ideas. This simulates a true collaborative, team-based, student-driven environment, much like the environment that many students will encounter in the workplace. If I lectured all the time, think of how much my students and I, myself would be missing! My advice is jse not assume too much. For example, after discussing histograms in the traditional way, along with the concepts of descriptive statistics that help us discuss shape, center, and spread, I asked students to choose from three possibilities a picture of a data set that had jse least amount of variability I gave them no numbers. The options were a bell-shaped curve, a flat histogram, and a histogram where all the values were located in two bars that were very close together. Almost of my students chose the uniform distribution. Binary asking them to explain, they said the data showed no change from left to right in the uniform distribution. I then realized they had the wrong concept of a histogram, and that it was probably based on their experiences from the media -- which shows mostly line graphs values tracked over time. When we give students opportunities to produce their own data and find basic statistical results, I think we are helping them to gain ownership of their own learning. We also promote their skills to take charge of a problem involving data, much like they will have to do in the workplace. When I teach a specific technique, I try to develop the big ideas that underlie the technique, and then use it to generalize to other situations. For example, binary students see how a confidence interval is created to estimate a population mean in a general way, they can quickly see how it applies to a proportion, to two samples, and even to small samples, with small variations. Students are so much more motivated by research questions. Statistics are the tools that help them to answer the questions. Once they realize that, they appreciate statistics so much more. Given results statistics, graphs, computer output, tables, or raw data can a student explain in options or her own words what the results mean? Again, a research question will provide a proper context. When I ask students to binary the conclusion of a hypothesis test, I stress that they must tell me jse the decision was reject H or do not reject Hand why they decided that p -value and test statistic. They also need to explain what it means in the context of the original research question since we rejected Hwe conclude that there is a home field advantage in Major League baseball, for example. Moreover, this is the truly fun part of statistics for students -- seeing it used to answer questions in which they are options interested. How can we assess the ability of students to interpret results? I think the best thing we can do is give them opportunities to interpret their own results, using their own data I have had jse best success with this; it gives students ownership, and allows them to really focus on what information they want to tell us, and what it means to them. Data ownership goes a long way here. I also try to provide situations in which teams of students work together, addressing different pieces of a bigger problem within a common context. I think this helps students to simulate a collaborative work environment that is focused, yet offers opportunities for individual choices. On exams and homework, I like to ask questions that specifically focus on interpretation not on the whole process of conducting the hypothesis test, for example. As an illustration, suppose a researcher wants options know if there is any difference in average grade point average between male and female students. His test statistic, based on male and female students, is The p -value is less than What is your conclusion? I also notice that when students create statistical results themselves, they seem to develop a better understanding of how to interpret the results. For example, when covering Chi-Square tests as a tool for determining whether two qualitative variables are associated, I used to find that students had difficulty reading the two-way tables that I gave them. Students were unable to discern which direction the conditional probabilities were in; they seem to get rows and columns mixed up. In one class of 50 students, not one of them missed the interpretation when they organized the tables themselves. From here, they went on to understand how a given table can be interpreted. Interpretation should also involve some basic ability to assess the correctness of the technique used. For example, a bar chart shows how often certain numbers are chosen in a state lottery. This is an entirely different skill. Among recruiters who hire new graduates, one of the top criteria they look for is the ability to communicate their ideas to others. Certainly this is a skill worth developing in our students when it comes to statistical information. The key to developing good communication skills in our students is to expose them to different styles. I use team teaching and team learning in my introductory classes, and it really helps students to see another viewpoint. They also see that teachers might not always be in perfect agreement. I believe that this is healthy for students. Exposure to alternative notations, symbols, and definitions is essential for good statistical citizenship, in my opinion. How many times have you felt bound by a textbook to use its notation? Broadened exposure will help minimize these problems. I also stress communication skills through simulations of real life jse. Examples include the following Write a letter to the options explaining why a graph showing the number of jse for versus should have also included the population size for each year. Suppose your friend is a journalist trying to figure out why two polls regarding public opinion on campaign finance form do not agree, because she has to submit an article on it What would you tell her? Organize and conduct a debate over whether or not city funds should be spent to hire more police officers. Use statistics to back up your points. How do you make your point to your manager? Find someone who has never had a statistics class, and explain the idea of margin of error to them. The goals of our introductory statistics courses are two-fold. First we want to promote and develop good statistical citizenship, and we also want to produce good research scientists on whatever level students will be involved. In order to do that, we begin by developing a basic foundation of knowledge of statistical concepts and ideas, which I call statistical competence. Statistical competence promotes and develops skills in data awareness, production, understanding, options, and communication. In my opinion, the answer is no. It is a good beginning, but it is not the end. Once students have a basic functional knowledge, they need the ability to question, to inquire, to probe, to compare and contrast, to explain, and to evaluate at a higher level. They might know what a matched pairs experiment is, and that it reduces variability, but they need to be able to explain HOW this results in a more powerful hypothesis test, and WHY we are able to draw a cause-effect relationship in some cases, and not in others. They also need to be able to think on their own, to identify their own questions, and come up with their own solutions using statistics. And that requires statistical reasoning and thinking. However, it is important to note that statistical competence is a requirement for statistical reasoning and thinking. Finally, I do not want to imply that in a statistics course, all the statistical competence is built first, then all the reasoning, then all the thinking. I think that it is important to always present a statistical problem in a relevant context with a legitimate, and relevant, research question. I think that as students learn more, they will ask more involved questions, and we can revisit and reinforce the scientific method over and over again throughout the course. Each time they go through the process, they will reinforce their understanding of terms and concepts, and their reasoning and thinking skills Chance, B. Steen, Washington, DC Mathematical Association of America, Gal, I. SStatistics: Concepts and Controversies 4th ed. Garfield, Amsterdam: IOS Press and International Statistical Institute.
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