Main Questions:
What types of data are there?
- Students will learn vocabulary around data collection, including variable, categorical variable, quantitative variable, count, frequency, relative frequency.
- Students will be able to organize data into tables with cases and columns of variables; paying attention to types of variables and asking meaningful questions about quality of the data collection and the intended use of the variables.
How can we organize the data numerically and visually?
- Students will understand that there are both visual and numerical ways to summarize and display statistics.
- Students will be able to identify the type of variable and the appropriate numerical and visual summary strategies for that variable.
- Students will be able to verbally describe the relationships displayed in a graph.
How can we collect data while avoiding bias?
- Students will be critical about the documentation or lack thereof about how data was collected.
- Students will be critical about the meaning and use of variables that have been defined in a study.
- Students will begin to recognize the role of representation in data and generalizability.
Unit
Notes on Pedagogy and Social Justice Integration for this unit.
This introductory unit is designed with both the ICUCARE protocol in mind, using the protocol in all lessons. In addition, it is planned with the advice of Tonya Gau Bartell in integrating social justice teaching with yearlong and not lesson long design. I am sketching five lessons here along with a summative task that introduce the statistical unit objectives of recognizing and characterizing variables, as well as laying the foundation for students to take a questioning approach to the methodology of data collection and the intended and unintended implications of the strategic decisions that researchers make when designing data collection.
ICUCARE Protocol in Unit
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Protocol step
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Strategies
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I
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Include others as experts
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Surveying students for applications in math within their life and their family and integration into summative projects. Using group work that relies on expertise and individual contributions. Selective reinforcement by teacher when listening to group discussion. Build classroom trust through group work and share out discussions so that all students are respected as experts.
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C
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Critical Consciousness
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“Be color-conscious not color blind” Explicit teaching about racial statistics, looking at historical and present census categories, demographic data. We will examine the culture in which statistics arise, paying attention to the way that data is collected and directed. Ensuring that students take time to wonder why data is being collected and who benefits from the outcome of a set of statistics. Exploring context through listening, reading and discussion in history of statistics and demography (lesson 5) Exploring demographic data from WEB DuBois.
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U
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Understanding students
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Surveying students to understand connections to mathematics and potential areas of projects. Having students review each other’s data to create displays and summarize ideas. Students working in groups to relate to each other and learn from each other. Through collective information sharing and knowledge building, allowing for humanity and connection to combat the idea of one person or group holding all knowledge/ to lessen stereotype threat.
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C
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Culturally relevant curriculum
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Allow opportunities for cultures to be seen (non-assimilationist approach) Examination of racial categories, worldwide and in the US. Examination of the history of racial categories in the US. Time for discussion and reaction. Summative projects will be built in the actual lives of students. Throughout this and other units, we will turn to the technique of measuring our own worlds to build mathematical models that describe our own lives. Students will be exposed to explicit teaching about how the origination of statistics is tied to the eugenics movement and will take time to discuss and process these ideas. In addition, students will explore datasets produced by resistance mathematicians and celebrate the accomplishments of these culturally relevant organizations and individuals.
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A
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Assess Prior Knowledge
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Lesson building design, allow for group discussion to access and assess knowledge base that the class is building together. Incorporate a prior knowledge engagement as part of any group work and activity.
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R
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Release control
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Project based work, allowing students to create dynamic work in response to class lessons and materials (group work, collaborative graphs, summary project) Working as teams to analyze and communicate their findings, thereby allowing students to create models and be the owners of knowledge in the classroom.
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E
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Expect more
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Define and assess student work on Common Core practice standards within lessons
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Lesson 1 - Opening/ Centering activity and discussion
Narrative and rationale: Poster walk, collaborative analysis and discussion:
Students will react to statements in the posters, allowing open ended data to be produced that describes student reactions to Math in school and on the Job as well as assessing prior knowledge about the ideas of statistics, graphs, measurements. The data will be used by the students to practice organizing it. However, the data is relevant in that it will provide individual and group-wide information about areas in which mathematics intersects in students’ lives. This information will be used to build future lessons or projects that relate to the knowledge base of students and are embedded in students’ lived experiences in order to give student expertise a main role in building classroom models. The group work will allow for students to collaboratively build the understanding of the notions of quantitative and categorical data while building rapport and community around examining shared interests or differences in reactions to the posters.
Content Objective:
Students will identify variables as categorical or quantitative
Students will collaboratively to collect data and sort into categories
Students will create a collaborative dotplot
Practice standards:
1) Look for and make sense of structure: Students will be able to select categorical and quantitative variables from the data gathered. Students will use these as a way to collate and describe the data.
2) Use tools strategically: Students will use discussion, sticky notes and posters to create numerical and visual displays of data.
Activities:
Create posters around the room with headings:
Math in School Statistics Math on the Job Graphs Measurements
Students will be given 10 minutes to interact with the posters. No specific direction is given about how to interpret the headings. Students are given 3 sticky notes and can have as many more as needed. Expectation that students use all 3 sticky notes and to add comments to at least 3 posters. At the end of 10 minutes, students will be assigned a group and a poster:
The group has 4 tasks (each person has a task) works for 10 minutes
- Counter: Count the number of comments that the poster received and post the number on the board dotplot
- Sorter: Create as many categories as needed by moving the sticky notes around the poster and sorting the comments into categories
- Summarizer: Writes a sentence or two to describe the responses on the posters including the number of comments and the types and sizes of categories
- Reporter: Will report summary out for group
Students will report out after 10 minutes with a 2 minute report per group.
Definition of Categorical and Quantitative variables will be given. Example of a quantitative variable and graph is number of comments/ dotplot. Example of categorical variable is the sorted categories with percentage of comments in each category (bar graph can be formed from sticky notes)
Students can take time to re-sort their poster data by quantitative and categorical data.
Lesson 2
Narrative and rationale: Pairs reading and collating demographic information.
In groups, students will examine census forms from countries around the world to assess their understanding of quantitative and categorical variables. Students will collaboratively build a bar graph using the data that they have sorted to examine as a whole group which types of questions are universal across countries, and how countries may differ in the types of questions asked. Students are working in pairs to provide the opportunity that all students are engaged in the sorting and all have the chance to share their expertise about their assigned country. Varying between groups and pairs allows students to build collaborative skills. Students will be deconstructing census questions to begin to think critically about the collection of variables about people in diverse societies. We will follow up in the next lesson to interrogate the purpose of data classifications and implications of these categories.
Content Objective:
- Students will identify variables as categorical or quantitative
- Students will collaboratively build categorical data display: Bar graph
- Students will work in small groupings to analyze bar graph and to actively interrogate sources and purposes of data
Practice standards:
1) Make sense of problems and persevere to solve them: Students will work as a whole class to compile raw information about the purpose of census data by comparing questions from forms from a number of diverse countries.
2) use tools strategically: Students will use census forms, sticky notes and conversation to build a visual display of data.
3) to look for and make use of structure: Students will use their understanding of categorical and quantitative data to summarize world-wide census data.
Activities:
- Pairs will be given a (in some cases translated) set of questions or categories from census forms worldwide (using at least 2 from each continent). Students will be given 2 piles of small sticky notes. The color represents Categorical or Quantitative, for example yellow Categorical and Blue Quantitative.
- Teacher will model using the US form, noting on sticky notes the name of the category for example: Age (blue), Gender or Sex (yellow), married or single (yellow)
- Students will work in pairs to create a label for each data point in blue or yellow. Once the group is done, they will bring their sticky notes to a class graph on chart paper. New Categories will be added as needed, and if a category already exists; the students will add their sticky note to the chart.
Once the groups have completed their work, students will be asked to jot answers to the following questions in their notes:
- What are the most common categories worldwide (how can you tell from the graph?)
- What are the least common categories?
- Which categories does the US Census collect that your country's example did not?
- Which categories did your country example collect that the US does not?
Students will turn and talk, comparing answers and share out by pairs
Lesson 3
Narrative and rationale: Introduction to data collection design for studies analyzing race and census.
In groups and as a class, students will be introduced to the idea of designing measurements of variables as part of data collection and the idea of how those variables are analyzed and then lead to inferences. Students will compare outcomes from the census study in the previous class and look at racial and ethnic classifications across the examples. Students will discuss the similarities and differences found and explore the implicit meaning of the outcomes. Group discussion and individual reflection will examine the question of whether racial classification implies racial stratification. Students will reflect on whether these numbers can neutrally describe experience or whether they carry the implicit meanings of racial domination within the data displays.
Content Objective:
- Students will interrogate purpose and use of data collection
- Students will understand that the design of a survey determines the way variables are measured and that these measurements may affect how the data is interpreted and expressed.
Practice standards:
Construct viable arguments and critique the reasoning of others: Students will analyze the choices made in worldwide census questionnaires to determine if they are measuring similar or different attributes in the populations of each country. Students will formulate ideas about potential inferences that may form from these data and compare and contrast the outcomes.
Activities:
Teacher will post the outline of a bar graph on the board. Students will be asked to put an x in the category that best answers the question What color is your shirt?
As questions arise in the data collection those will be written on the board: Some of these questions may be:
What if I have more than one color on my shirt? What if I have 2 layers/ 2 shirts?
POST DIAGRAM of DATA COLLECTION DESIGN
As a whole group, and while using the flowchart we will add steps to the shirt color question to examine the decisions that were made in the process.
Working in groups students will look at the bar chart from the day before.
- Director: Leads the group through the questions and steps of activity allots 2 minutes or so per question.
- Reader: Reads each question for group members to respond to
- Notetaker: collects answers/ types or writes them
- Reporter: Will report summary out for group
Answer Questions:
- Are Race/ ethnicity/ religion/ language-spoken the same category or different?
- What use would the government have to collect this info?
- How is it important?
- How accurately are these categories measured?
- What are some potential sources for mis-measurement?
Teacher shows the recent test questions by US Census (ethnicity questions)
- What are the purposes of these questions?
- What information is being collected? How can it be used in positive and negative ways?
- Are there ways that the data empowers or disempowers people across demographic groups?
- What inferences could be drawn from data that divides people into racial categories?
Individual reflection closure:
Do you agree that “racial classification (sorting individuals by race) and racial stratification (ordering races into a hierarchy) are inseparable in the US”? - Zuberi
Lesson 4:
Narrative and rationale: Historical context of race and census/race and statistics
As a class, students will listen to a portion of a podcast about the history of race in the census. Students will watch a slide deck about the history of racial classification and the history of eugenics and statistics. Students will be given time to discuss and write about these materials.
Content Objective:
Students will use a critically conscious approach to examine the collection of data about race, in a historical and in the current context.
Practice standards:
1) Construct viable arguments and critique the reasoning of others: students will look at data critically to notice the explicit and implicit messaging along with the numbers.
Activities:
Slideshow galton and eugenics
Handout Pew history of census and race
Partner questions: choices to discuss:
What surprised you? How does this knowledge affect your trust of statistics?
What is the relationship between Eugenics, racial stratification, white dominance?
Are the ideas put forth by Eugenicists still a part of the statistics we examine in the census?
Lesson 5
Narrative and rationale: analyzing categorical graphs using examples from W.E.B. DuBois data visualizations from 1899.
In groups, students will compare and contrast categorical graphs of data. Students will work in groups to build collaboration skills and give all students an opportunity to provide expertise in the analysis. Students will compare graphs from W.E.B. DuBois with contemporary US Census graphs. W.E.B. DuBois worked with his students at Atlanta University to measure aspects of the lives of African-American residents of Georgia. The data and graphs were collected to be shown at the Paris World Fair in 1900. The resulting graphs are both historically unique, beautifully conceived and rendered and present an accurate and compellingly positive vision of the lives of African Americans in the late nineteenth century. We will contrast this with the minimal information that was presented in the US Census.
Content Objective:
Students will examine data visualizations looking for physical features in the graphs that express categories
Students will examine data visualizations looking for physical features in the graphs that express quantities
Students will compare and describe effective features of graphs
Practice standards:
1) Look for and make sense of structure: Students will compare physical features of graphs and identify how color, line shape and proportion describe numerical data
2) Attend to precision: Students will compare the effectiveness of the graphical displays.
3) Construct viable arguments and critique the reasoning of others: Students will compare the implicit and explicit meanings of the data displays
Activities:
Students will be broken into groups. Each group receives two graphs with similar contents. Students receive sticky notes to jot answers on.
The group has 4 tasks (each person has a task) works for 10 minutes
- Director: Leads the group through the questions and steps of activity allotted 2 minutes or so per question.
- Reader: Reads each question for group members to respond to
- Notetaker: collects information and writes it down
- Reporter: Will report summary out for group
Group will compare these two graphs.
What is the content of these two graphs?
How does the graph tell the story (visual language) (color/ line/ area/ proportion)?
What does the graph emphasize?
What story do you think the author is conveying with this graph?
Group share out and class discussion. Discussion of graph authors and audience and how/where data was presented.
Lesson 6
Narrative and rationale: preparing for a summative project: creating a graph of categorical data.
As a class, students will re-examine responses to the posters of the first class (posted in the room). Students can choose a categorical variable that they would like to examine from the topics or ideas posted on posters or they can create another idea. Using the flowchart of planning for variable measurement, students can plan how they intend to collect data about the chosen topic. Students can prepare the measurement tool (survey or observational checklist) and plan for the type of intended visual display (pie chart, bar graph, segmented bar graph)
Content Objective:
Students will collect categorical data and display in a graph
Practice standards:
Model with mathematics: Students will create a model of the distribution of answers from their topic of choice
Use appropriate tools strategically: Students will select and create an appropriate graphAttend to precision: Students will collect data accurately and display it correctly.
Activities:
Students will complete a graphic organizer to prepare for a summative project that they will submit for feedback.
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Your information
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Another person’s information
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Categorical variable (or variables) that you would like to measure. You can reference the posters we did on the first day for ideas about math, reference census questions or invent a new topic.
Categorical variables take on values that are names or labels
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How will you collect the data?
Survey questions
Observations
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Who will you survey? or
who/what will you observe?
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What tool will you use?
Survey questions ( platform: paper/text/social media)
Observational checklist
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Have you clearly defined your variable?
Are there cases you have to rule in or out as you measure? (think of shirt color example)
What meanings can occur that make your variables non-neutral?
What other meanings or inferences can be drawn from your variables?
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What type of data display will you use?
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