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Improve Test Scores and Student Engagement by Starting Your Year Teaching Statistics

ByAnna Cannelongo

Introduction

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“I believe that we owe it to our children to prepare them for the world that they will encounter—a world driven by data. Basic data fluency is a requirement not just for most good jobs, but also for navigating life more generally, whether it is in terms of financial literacy, making good choices about our own health, or knowing who and what to believe.”

~ Steve Levitt, co-author of Freakonomics on his October 2, 2019 podcast

Testing

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Statistics and Probability is a significant component of Ohio’s state math tests (and many other states as well.) For example, in Ohio the statistics and probability standards are about 18-22% of the test questions in Algebra 1 and Geometry. In 7th grade 22-29% of the test questions are about statistics and probability. Yet many teachers put off teaching those topics until after testing (if they teach at all), and then they wonder why their test scores are bad. Or they may try to squeeze all the concepts in a week or so before testing. I know I did when I taught Algebra 1!

Student Engagement

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Statistics is fun! Or at least it should be. There are so many real-world statistics problems that students will find interesting. In the classroom, statistics should be student-centered. Students need to learn how to ask statistical questions, collect and analyze data, and interpret their results. Students should be active participants doing statistics instead of just learning rotely about statistical concepts. When students are driving the learning by pursuing answers to questions they are interested in, student engagement soars! A good statistics lesson will have students asking just as many questions when it’s over as they did when the lesson started (and perhaps even making global generalizations).

This does not mean that class is a free for all, and any statistical question or topic will be explored. But rather the teacher provides some guidance, structure, and parameters for student thinking to take place. For example, the teacher may want to teach about two-way frequency tables, so he/she chooses the topic of social media. Then he/she may steer the classroom discussion towards investigating a question that will lend itself to using two-way tables.

Statistics is a Life Skill

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In today’s world of big data, statistical thinking is one of the most important skills that people can have.  As adults we encounter statistics every day, yet we rarely encounter complex algebraic equations. Statistics can help us understand our social media feeds, political polls, and how to assess risk. Understanding it can help us be aware of the tactics companies are using to persuade us to buy something. Therefore, all adults need to be statistically literate.

Statistical literacy is the ability to read and interpret data presented in graphs, charts, tables, etc. People use and manipulate data, and it is important not only to be able to interpret the data but also evaluate it. People need to look at data with a critical eye. Being statistically literate will help people make good decisions and informed choices in the real-world.

Data is changing

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The way we tell stories with data is changing. Data can now include text on social media or a collection of pictures or sounds. In today’s world, data is available almost instantaneously, and through the use of technology we can gather really large amounts of it. Because of the large quantity of data, it is more vital than ever before that all people learn how to reason statistically.

Statistical Problem Solving: The GAISE Model

The GAISE Model is the authority of statistical modeling for the K-12 classroom (especially 6th-12th grade) for the Common Core states. Students need to formulate statistical investigative questions; collect/consider the data; analyze the data; and interpret the results. Although they do not need to do all 4 steps for every classroom exercise, they need to have some investigations which require them to do all four steps throughout the year. There are also 3 levels (A, B, and C) that students need to progress through in order to become proficient statistical thinkers.

The GAISE model is a MUST read for educators who are teaching statistics and/or probability. However, it is a little difficult to read as its organization is not intuitive. My recommendation is that everyone should start by reading the overview and framework on pages 13-19. Then decipher at which level(s) A, B, or C that your students are. ( Ohio’s Model Curriculum WITH Instruction Supports grades 6 and above can also help you decipher the GAISE level by statistics standard as the document explicitly calls it out.) Then find the topic you want to teach such as scatterplots and read the examples for the appropriate level and the levels that precede your focus standard’s level. Note: Although Ohio’s standards align mostly to the Common Core during the last revision, Ohio put more explicit language reflecting the GAISE model which is helpful.

Summary of Levels A, B, and C

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As students progress through the levels the reasoning process, the investiagations will become less teacher-focused and more student-driven. Here is a general breakdown of the levels.

Level A

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Students at level A are just beginning to understand the role of questioning and how to ask statistical questions. Developmentally it is important that they understand that data can vary. Students are at the beginning of multi-variate thinking and learning how to attend to multiple variables simultaneously. Additionally, they should be able to identify the four steps of the GAISE model. They need to understand that the same data can be displayed in different ways, but some ways are more useful than others. Students at this level should focus on center as mean (equal share) and median, variability as the range, and dispersion as how many units a data point is from the mean (equal share). They should be able to quantify the number of clusters, identify whether the data is symmetric or not and explain if there are any gaps present in the data. For more information about this level, see the GAISE model. For more information about this level and how it relates to most of the Common Core Standards, see Ohio’s Model Curriculum WITH Instructional Supports Grades 6 and 7. Note: Although Ohio’s standards align mostly to the Common Core during the last revision, Ohio put more explicit language reflecting the GAISE model which is helpful.

Level B

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Most of the Common Core statistics standards are at level B (but not all). Students at this level understand that statistical reasoning is a problem-solving process and are able to go through all four steps of the GAISE model. They are able to use a variety of graphical models to summarize the data and choose the best one to communicate their data based on the context. Students should be able to interpret the center using the mean (balance point) and/or median deciding which is better for the question of interest.  Variability focuses on Interquartile Range (IQR) and Mean Absolute Deviation (MAD). Students should be able to interpret whether the shape is symmetric or asymmetric and identify the number of modes in the data. At this level association focuses on measures of correlation such as the quadrant count ratio (QCR) and a comparison of conditional proportions across categorical variables. They are starting to understand how statistics is not something that is just done in school, but it is something that affects their lives and allows them to make informed decisions. For more information about this level, see the GAISE model. For more information about this level and how it relates to most of the Common Core Standards, see Ohio’s Model Curriculum WITH Instructional Supports Grades 6 through Algebra 2. Note: Although Ohio’s standards align mostly to the Common Core during the last revision, Ohio put more explicit language reflecting the GAISE model which is helpful.

Step 1: Formulate the Question • Students should pose their own statistical question of interest (Level C). • Students are starting to form questions that allow for generalizations of a population (Level B-C). Step 2: Collect Data • Students should begin to use random selection or random assignment (Level B). Step 3: Analyze Data • Students measure variability within a single group using MAD, IQR, and/or standard deviation (Level B). • Students compare measures of center and spread between groups using displays and values (Level B). • Students describe potential sources of error (Level B). • Students understand and use particular properties of distributions as tools of analysis moving toward using global characteristics of distributions (Level B-C). Step 4: Interpret Results • Students acknowledge that looking beyond the data is feasible by interpreting differences in shape, center, and spread (Level B). • Students determine if a sample is representative of a population and start to move toward generalization (Level B-C). • Students note the difference between two groups with different conditions (Level B).
Image from Ohio’s Algebra 1 Model Curriculum with Instructional Supports

Level C

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Students at Level C should be able to answer questions about associations and relationships among multiple variables as well as making predictions. They will be able to distinguish between surveys, observational studies, and experiments. Students at this level will also take a deeper dive into randomness. Their interpretations of the data will be deeper and more insightful than those at levels A and B. Additionally, they should also be able to articulate the limitations of their conclusions based on data, and they should be aware that others should be able to and may want to reproduce their analysis. For more information about this level, see the GAISE model. For more information about this level and how it relates to many of the Common Core Standards, see Ohio’s Model Curriculum WITH Instructional Supports Algebra 1, Geometry, and Algebra 2 or use the document that focuses on the conceptual category of high school statistics and probability. Note: Although Ohio’s standards align mostly to the Common Core during the last revision, Ohio put more explicit language reflecting the GAISE model which is helpful.

Expectations for Learning-Algebra 2/Math 3 Previously, students have been informally introduced to data collection methods and bias. In this cluster, the concept of randomization is introduced in data collection methods. Students are also introduced to the concept of margin of error, and they begin to formalize the concept of statistical significance. The GAISE Model The GAISE Model is a framework for all statistical problem solving and should not be taught in isolation. For this cluster, the focus is on Steps 2, 3, and 4 at Level C. Students are building on the framework developed in earlier grades. Algebra 2/Math 3 students use more in-depth reasoning and a greater level of precision and complexity. Step 1: Formulate the Question • Students should be fluent in posing their own statistical question of interest. • Students should form questions to allow generalizations be made about a population Step 2: Collect Data • Students should purposefully design for differences through random selection or random assignment. • Students design samples through selection. • Students design experiments through randomization. Step 3: Analyze Data • Students understand and use global characteristics of distributions in analysis. • Students compare group to group using displays and measures of variability. • Students describe and quantify sampling error. Step 4: Interpret Variability • Students are able to look beyond the data in some contexts. • Students are able to generalize from a sample to population. • Students are aware of the effects of randomization on the results of experiments. • Students understand and distinguish between observational studies and experiments.
Image taken from Ohio’s Statistics and Probability Model Curriculum with Instructional Supports by Conceptual Category.

Data Collection

Oftentimes, data collection takes time. Unfortunately, if you save the statistics and probability standards until the end of the year, there isn’t much time for data collection. However, if you start the year by teaching statistics and probability standards, you will give students plenty of time to collect good data. This will allow you to do several high-quality investigations throughout the year that go through all steps of the GAISE model.

Teacher Confidence and Professional Development

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Many math teachers struggle with statistics, as they may never have had a statistics course. Or if they did, it was a one-off course that was so long ago that they forgot most of the content. Therefore, it’s important to get professional development around teaching statistics. I would recommend looking at how concepts progress across grade bands. If you were once a Common Core state, you may find Ohio’s Grade-8-Geometry-Learning-Progressions-by-Topic.pdf.aspx (ohio.gov) helpful along with Ohio’s Model Curriculum with Instructional Supports. Some of the resources I listed in the section below also have great teaching notes if you need to refresh your knowledge of statistics.

When providing professional development around statistics and probability at the Ohio Department of Education and Workforce, we focused on these aspects:

  • The GAISE Model Framework
  • The expectations at various levels (A, B, and C) of the GAISE model framework at each grade level.
  • The development of the concept of mean across the grade bands (mean as fair share, mean as a balance point, and sample mean)
  • Measure of Spread across different grade bands (Range, Interquartile Range, Mean Absolute Variation, and Standard Deviation)
  • Association in Scatterplots (informal fitting of the trend line, Quadrant Count Ratio, Mean-Mean Line Method, and Pearson’s Correlation Coefficient)
  • Categorical Bivariate Data (Two-Way Frequency Tables, Two-Way Relative Frequency Tables, Agreement-Disagreement Ratio, Phi Coefficient)

For more information on Using Ohio’s Model Curriculum To Teach Statistics and Probability check on my blog post and don’t forget to subscribe to my blog for more great ideas!

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Resources

Here are some resources that might prove useful when teaching Statistics using the GAISE model:

  • Ohio’s Model Curriculum WITH Instructional Supports: There are good examples under the statistics standards of each grade level document. There are also links to lessons/resources by cluster. Note: Although Ohio’s standards align mostly to the Common Core during the last revision, Ohio put more explicit language reflecting the GAISE model which is helpful.
  • Statistics Education Web (STEW): This website has great, engaging lessons aligned to the GAISE model.
  • Census at School: This is a great tool to find real data for statistics investigations.
  • Statistics in School by the US Census: This resource has great math/statistics lessons using Census data.
  • EngageNY Math Curriculum: The EngageNY Curriculum has become Eureka, so they have buried their free curriculum on their website, but you can still find it in the archives. The great thing about this resource is that there are great teacher notes if you as a teacher need to refresh your knowledge of statistics. Here is their Curriculum Overview for Grades 6-8 and their Curriculum Overview for Grades 9-12 to help you navigate the modules.
  • The Georgia Department of Education Inspire Course Content Collections: They have created entire units with tasks and teacher notes that are mostly aligned to the Common Core standards. This is another great resource if you need to refresh on how to teach statistical concepts. Their ew hosting site seems very interactive and easy to use which is a bonus.
  • Mathematics Vision Project: It has great standards-based lessons that are free for Algebra 1, Geometry, and High School. It’s easy to see the student-facing lessons, but I think you now need to create an account to get the teacher lessons.
  • Illustrative Mathematics by Kendall Hunt: This is another great curriculum that’s standards-based. It’s an open education resource (OER), which means that it’s mostly free. You will have to create an account, and there may be a cost with some print materials, but I think the online stuff is free. I’m not fully sure as I no longer have an account, but you can access the resources here. I believe it’s intended to be completed online.

My Personal Recommendations

If it were me, I would start each quarter with some type of statistical investigation. In first quarter, regardless of the grade, I would start with an introduction to the statistical problem-solving model as laid out in the GAISE framework. Then I would focus on asking statistical questions and collecting data. In first quarter I would choose one type of statistical model to focus on. For example in 6th grade, I might start off with mean (fair share), range, and an investigation involving dot plots. Then in 2nd quarter I might introduce an investigation where a histogram is more useful for showing the data.

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Comment

Let me know what statistics lesson you decide to start the year with!

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