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Using Technology to Address Diversity in the Classroom - I

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Using Technology

to Address

Diversity in the Classroom - I

Valentina Aguilar, Ph.D. Applied Mathematics

Department of Mathematics

Western Michigan University

Katya Gallegos, Specialist

Teaching, Learning and Educational Studies

Western Michigan University

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- Introduction
- Activities for the Classroom
- Recommendations and Literature

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- Use of technology to explore Diversity through mathematical activities in the middle school classroom
- Consider practices for promoting inclusive learning environments

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Mathematical content

- Use data to find a linear model
- Use a linear model to predict values of variables
- Simple interest
- Compound interest
Related to diversity

- Reflect about gender disparities

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- Analyze information using statistical tools appropriate for their grade level
- Use a histogram to summarize and represent data
- Use measures of center and dispersion to analyze data
- Use technology available on the world wide web to represent and analyze data

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Goals on mathematical content:

- Represent data with a pie chart
- Understand categorical data
- Use of percentage
Goals on diversity

- Reflect about the different compositions of families
- Promote awareness and acceptance of the different family types

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INTRODUCTION

Mathematics:

a tool to understand our Diverse World

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I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.

- James Caballero,

mathematics teacher and author of ‘A Geometry Game’

Indeed,

mathematics is

an amazing and

powerful tool to

understand the

world around us …

To the wonders of nature and science…

Swarm Intelligence, mathematical study of collective behavior …

www.ams.org/mathmoments Photo by Jose Luis Gomez de Francisco

Richardson’s and Kolmogorov’s laws to mathematically explain turbulence in aerodynamics …

www.ams.org/mathmomentsPhoto courtesy of NASA Ames Data Analysis Group

GPS’s functionality derived from arithmetic, algebra and geometry …

www.ams.org/mathmomentsPhoto courtesy of the Aerospace Corporation

Wavelets for faster storage and retrieval of FBI’s current fingerprint files: 200,000,000,000,000 bytes!

200,000,000,000,000 bytes!

www.ams.org/mathmomentsPhoto courtesy of Christopher Brislawn, Los Alamos National Lab

Forecasting the weather with Mathematical Models…

www.ams.org/mathmomentsPhoto courtesy of Lloyd Treinish, IBM Research Center.

Bernoulli’s principle, algebra and geometry to determine the right shape for soccer balls…

www.ams.org/mathmomentsPhoto courtesy of University of Sheffield and Fluent, Inc

Animation …

Geometry and Music…

Cooking tonight’s dinner …

Playing video games with your sister…

and

Saving for retirement …

Losing those 10 extra pounds …

Planning your next trip …

Math Modeling and Medicine…

And the list goes on and on …

The United States is and has always been a nation of immigrants.

We know that, one way or another, math is part of our own unique life experience…

And our lives are different because WE are different. We live in a very and increasingly DIVERSE world.

The following online tool can be used to mathematically exploreimmigration data since 1880.Move the cursor along the time line to visualize color-differentiated data in the map.

Data analysis of the graphic display

Interpreting and inferring conclusions from histograms and maps

Finding data relationships and equivalent data representations

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Let’s start thinking about one of the aspects of diversity:

Gender

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Activity 1: Gender Gap

We can use this activity to reflect about gender disparities and increase students’ awareness to promote an equitable environment in the classroom.

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Average income ofmen and women in the US

This graph was created using 2005 US Census Bureau data, released for 2006.

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- What is the average income of an Asian female?
- What is the average income of an Asian male?
- What is the minimum average income among all?
- What do you think this chart tells us about women and men?
- Why do you think there are such differences in income?
- What impact in every day life does earning less than other people have?
- Add more questions you can ask your students.

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To study this question, let’s say John and Joanna (both white) save 10% of their average income as stated in the chart in a bank account that pays a fixed rate of 5% simple interest.

How much money will they have in ten years?

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Now find the amount each person will have after t years if they deposit their 10% now in a bank that pays 5% simple interest.

See the solution

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The equations should be

M = 210 t + 4200

and

M = 140 t + 2800

respectively for John and Johanna.

Graph these two equations on the same

Cartesian plane using a line graph applet:

Recommended applet:

http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

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If you tried to graph the equations, you would notice the graphs do not appear because the coefficients from the equations are large numbers.

There are two solutions for this issue:

- either change the setting of the graphs and redefine the ranges of both variables, or
- change the units involved in the equations.

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What units can be used to make the graph easier to visualize?

If we choose one hundred dollars as 1 unit, then the equations are:

M = 1.4t + 28

and

M = 2.1t + 42

where M is given in hundreds of dollars and t is the number of years.

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Now answer the questions:

Does the gap grow?

By how much?

See the answer

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Now let’s investigate the same question with the more realistic situation of the amounts subject to compound interest.

Given the fact that the mathematical model is different for compound interest and simple interest would you expect that the difference between John and Joanna to stay the same or to be bigger?

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The mathematical model is an exponential function of the form

Here, M is the amount after n years of depositing P dollars at a rate of r% compounded k times a year. The number of years is the variable n.

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Let’s assume that the interest is fixed and it is compounded monthly, that is, k=12. So:

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The equations for John and Johanna are respectively:

Graph these two equations on the same Cartesian plane using an online graph applet. Recommended applet:

http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

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Now, again answer the questions:

Does the gap grow?

By how much?

See the answer

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Wrapping up the activity:

Conducting a discussion about gender expectations and stereotyping wraps up this activity. Examples of questions to be asked are:

- Are girls good in math and science?
- Can girls play all sports?
- Can boys do ballet?
- How do our own attitudes contribute to fostering an equitable environment where both boys and girls feel they are treated fairly?
- Write other possible questions for such discussion.

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Sex

Gender identity

Sexual orientation

Socioeconomic status

Abilities and skills

Ethnicity

Age

Physical attributes

Religious beliefs

Cultural practices

Political affiliation

Intellectual ideologies

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- Recognizes and respects the presence of all diverse groups in an organization or society
- Acknowledges and values their socio-cultural differences
- Encourages and enables their continued contribution within an inclusive cultural context
- Empowers everybody within the organization or society

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- Values cultural differences
- Permeates all aspects of school practices, policies and organization
- Affirms the pluralism that students, their communities and teachers reflect
- Provides the knowledge, dispositions, and skills to promote fair distribution of power and income.

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Racism

Sexism

Classism

Linguicism

Ablism

Ageism

Heterosexism

Religious intolerance

Xenophobia

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Activity 2:

THE NAMES IN YOUR CLASSROOM

The teacher can use this activity to familiarize students’ backgrounds and identify their particular needs that might come from cultural, ethnic, gender or socio-economic situations.

Teachers can include themselves in the activity to build trust within the classroom.

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- Explore students’ names and their origins and differences - the history of their names.
- Use statistical tools to describe the difference or similitude of students’ names.
- Reflect about what the mathematics tells and how to apply it.

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Your students will take this opportunity to communicate about their homes, parents, siblings, and care givers. Names in many cases reflect cultural background. So the activity provides a time for reflecting on ethnic, gender and race differences. Some names have meanings in English or in other languages.

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This activity provides a context that engages students (nobody is indifferent to his or her own name) and their curiosity to learn more.

The teacher can talk about his or her name in order to build trust.

Mathematics provides a tool for analyzing similarities and differences in names.

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Reflecting about our names is an engaging activity. Each name is personal and usually is linked to family preferences but also reflects ethnic and cultural background.

Some names are gender specific; some are not. Some names have meanings in English or in other languages. Explore these issues with your students and you will soon have a lively conversation going.

Think about what questions you can ask to the class so cultural and ethnic differences stand out in positive ways.

See suggestions

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This is the roster of Mr. Potter’s class:

Read the names to yourself and think about what comes to your mind. Do people make judgments after looking at a name?

- Look at what social researches think.

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Names have many qualities; one of them is the length of the name. The number of letters in a name is the variable that we will use to represent the data.

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Now we display the table with the values of the variable “length of names in the list”. Some of the data has been done; please fill in the rest.

Length of names in the list

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- There are many ways to represent numerical data. One of them is a histogram.
- First, we need to construct a frequency table for these data. The range is
10 – 3 = 7. We can decide the number of groups to create. If we want 5 groups, the length of each interval is 7/5 = 1.4.

If we choose 8 groups, the length will be

7/8 = 0.88.

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Peek at the answer

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- Explore the resources in your school. Software, like Tinkerplot, lets you create the most common statistical displays.
- Handheld technology.
- The web offers many free applets.

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- The following link will lead you to the tool to input the data with no need to create the frequency table.
- Input the data. When you are done, select the interval width by sliding the bar. Set it to 0.88.
Histogram Generator

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- This site lets you choose the length of the intervals. You can explore the effect of changing the length. Describe the effect in terms of the number of intervals created.
- The tool also gives you the mean - the arithmetical average, computed with the formula
- The standard deviation is a measure of dispersion of the data. It is computed as an average of the “squared” distances from the mean using the formula

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- Is the histogram symmetric?
- Does the data have gaps?
- If you change the length of the intervals, does the mean change?
- If you change the length of the intervals, does the median change?
- If you change the length of the intervals, does the deviation change?

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- Open a new document in the application.
- From the tool bar, drag a table to your page.
- Where the word new appears, write the categorical variable “name”. A new column will be created next to it. Write “length.”
- Under “name” write all the names from Mr. Potter’s classroom. Under “length” write the number of letters in each name. When you copy the table, a case stack will appear. Each card of the stack represents one data set: a name and the corresponding length.
- From the tool bar, drag a plot to your page. Then select the length attribute and drag it to the horizontal axis of the plot.

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- The plot is showing each individual case with a circle icon. Select the length attribute and drag it to the horizontal axis of the plot.
- From the tool bar, select separate. Then drag the outermost circle to the edge until there is no separation line (or bins). Click on hat on the toolbar and choose box plot from the hat menu.
- Finally, to get rid of the circle icons, for the circle icon menu in the plot, choose hide icons.
Now, think about a situation where a new student with a very long name is added to the list. Add a long name to your table and watch what happens with the box and whiskers plot.

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- If a very long name was added to the list, would the mean change?
- If a very long name was added to the list, would the median change?
- Which of the two measures of center is more resistant to changes in the data? Why?

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Use the mathematical ideas to come back to the context of the problem and answer the following:

- What measure of center, the mean or the median, would be better for representing the “average” name length of the class?
- Does the “average” tell the whole story about the differences in the names?
- What number of letters would make a name unusually long?
- In what situations would the average length be important to know?

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Learning Style

LEARNING

PROFILES

Intelligence

Preference

Gender

and

Culture

Tomlinson, 2004 “How to differentiate instruction in mixed-ability classrooms”

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A. Learning Style

Environmental or personal factors that

influence learning results

Group

Orientation

Physical

Environment

Cognitive

Style

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1. Group Orientation

- Independent / self-orientation
- Group / peer orientation
- Adult orientation
- Combination

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2. Cognitive Style

•creative / conforming

•essence / facts

•whole-to-part / part-to-whole

•expressive / controlled

•nonlinear / linear

•inductive / deductive

•people-oriented / task or object-oriented

•concrete / abstract

•collaboration / competition

•interpersonal / introspective

•easily distracted / long attention span

•group achievement / personal achievement

•oral / visual / kinesthetic

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3. Physical Environment

•quiet / noise

•warm / cool

•still / mobile

•flexible / fixed

•“busy” / “spare”

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Analytic

Naturalist

Practical

Intrapersonal

Creative

B. Intelligence

Preference

Interpersonal

Verbal

Linguistic

Musical

Rhythmic

Logical

Mathematical

Bodily

Kinesthetic

Spatial

Visual

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Technology provides many tools to implement several representational approaches to a mathematical concept. These representations can fit the cognitive preference of diverse learners.

For those students with more developed musical/rhythmical skills, one can use the following applets to let them discover the different mathematical aspects of a linear function:

http://illuminations.nctm.org/ActivityDetail.aspx?ID=36

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If we have a student that has a cognitive style that relies on visual activities, the same concept of linear function can be approached with the following tool:

http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/lsif.html

Or open http://science.kennesaw.edu/~plaval/tools/linear.html

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If we have a student that has a cognitive style that relies on kinesthetic activities, the same concept of linear function can be approached with the following tool:

http://www.ies.co.jp/math/java/geo/lin_line/lin_line.html

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For students that like to find patterns and are capable of easily inducing rules from particular cases, the teacher might provide games like the following:

http://nlvm.usu.edu/en/nav/frames_asid_191_g_4_t_1.html?from=grade_g_4.html

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- Use this activity to reflect with the
class on how different families can

be.

- Each family composition brings
specific conditions to children.

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It is important to acknowledge different family models without the label of necessarily “dysfunctional”.

Start the activity by asking the students to write a small paragraph describing his or her family, or,

Alternatively, the teacher can open a discussion about how family units are composed.

Discuss the categories used by the US Census to classify family members.

Use the following online tool to explore the data about demographics:

http://www.socialexplorer.com/pub/home/home.aspx

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Do a query of the household composition in the U.S. Some of the data is included in the following chart. Please complete the information:

Peek at the answer

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Create a chart using, for instance, one of the applets found among the NCTM online resources. The pie chart will look like the graph on the right.

http://illuminations.nctm.org/ActivityDetail.aspx?ID=60

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Pose “interpret” type questions such as:

- Why is the green portion bigger than the blue?
- What percentages of households have a grandparent as a householder?
- Write your own interpret questions.
Pose questions that require further computations like:

- What percentage of households consist of a “single”
householder?

- Write more “derive” type questions.

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FINAL RECOMMENDATIONS:

- Honor the diversity of your students: anthropologically, generalizations can be useful to understand common socio- cultural trends, but be careful to avoid stereotyping. There are always differences between individuals.
- Know your students: consider their everyday actual experiences and build confidence through caring.
- Provide opportunities for mathematical discussions: encourage sharing and debate of students’ solutions. Insert new problems and questions connected to their interests.
- Avoid deficit models: value the resources that diverse students bring to the mathematics classroom. Differences can be perceived as opportunities and not necessarily as obstacles and deficiencies.

All this material and more can be read on the MODULE SUMMARY, including a list of Online resources and Bibliography.

- Consider how you presently incorporate diversity into your mathematics classroom. Compare and contrast your present practices with those discussed in this session.
- Share any favorite diversity applications you currently use with others at this time.

- Write a statement or two that summarized what you have learned during this session.
- What would you like to have added to this session on diversity in the mathematics classroom?
- E-mail your responses for the above to
ruth.a.meyer@wmich.edu