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Virginia Council of Teachers of Mathematics | www.vctm.org

VIRGINIA MATHEMATICS

TEACHER

Vol. 48 No. 1

Different Perspectives on Teaching Mathematics!

Virginia Mathematics Teacher vol. 48, no. 1

Editorial Staff

Dr. Agida Manizade Editor - in - Chief Radford University

Dr. Jean Mistele Associate Editor Radford University

Ms. Grace Chaffin Assistant Editor Radford University

Ms. Mia Bialobreski Assistant Editor Radford University

Printed by Wordsprint Blacksburg, 2200 Kraft Drive, Suite 2050 Blacksburg, Virginia 24060

Many Thanks to our Reviewers for 48(1)

Virginia Council of Teachers of Mathematics

Agida Manizade, Radford University

President: Agida Manizade

Jean Mistele, Radford University

Ann Wallace, James Madison University

Past Presidents: Lynn Foshee Reed

Tracey Proffitt, Payne Elementary School

Secretary: Kim Bender

Maria Timmerman, Longwood University

Kristin Rojas, Prince William Public School

Historian: Beth Williams

Alexander Moore, Virginia Tech

Harold Mick, Virginia Tech

Treasurer: Virginia Lewis

Padmanabhan Seshaiyer, George Mason University

Brianna Kurtz, Mary Baldwin University

Webmaster: Ian Shenk

Toni Sorrell, Longwood University

NCTM Representative: Skip Tyler

Complete VMT Editorial Staff

Elementary Representatives: Sarah Matthews, Scarlett Kibler

Agida Manizade, Editor - in - Chief, Radford University

Jean Mistele, Associate Editor, Radford University

Middle School Representatives: Kathleen Stoebe, Jennifer Evans

Darryl Corey, Section Editor, Radford University

Secondary Representatives: Javier Cabezas, Fallon Graham

Alexander Moore, Section Editor, Virginia Tech

John Adam, Section Editor, Old Dominion University

Math Specialist Representative: Michelle DeLoach

Eric Choate, Section Editor, Radford University

Two - Year College: Nikki Harris

Grace Chaffin, Assistant Editor, Radford University

Mia Bialobreski, Assistant Editor, Radford University

Four - Year College: Toni Sorrell, Darryl Corey

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Table of Contents:

Featured Awards ................................................... 5

Note from the Editors ............................................ 9

Message from the President ................................ 10

The Mathematics of Congressional Reapportionment: Rounding in Action ................ 12

Math Jokes ........................................................... 20

Technology Review: Using Social Media in Teaching Mathematics: Applications of Facebook and TikTok... ....................................... 21

Organization Membership Information ............... 24

Busting Blockbusters ........................................... 25

Meet Your Representatives .................................. 26

Elementary Preservice Teachers Engaging in a Task for Statistical Problem - Solving …. .............. 28

NCTM Affiliate Information ................................ 34

Call for Manuscripts ............................................ 35

The Exponential Function and What it Can Tell us about Consumption of Non - Renewable Resources ............................................................. 36 Comprehension and solving math word Problems .............................................................. 43 Data Visualization - A tool for solving unsolved mysteries ............................................... 49

Good Reads ......................................................... 54

HEXA Challenge ................................................. 56

NASA Resources Linked to Different Perspectives on Teaching Mathematics .............. 58

Grant and Scholarship Opportunities ................. 61

Key to the 47(2) Puzzlemaker .............................. 62

Solutions to 47(2) HEXA Challenge .................... 63

Puzzlemaker ......................................................... 67

Conferences of Interest ........................................ 68

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Featured Awards: 2022

Virginia Council of Teachers of Mathematics

Jennifer Evans Vernon Johns Middle School, Petersburg City Public Schools Michalowicz First Timer Grant Award Winner

Maria Fuentes James Madison University Edward Anderson Scholarship Recipient

Sarah Flippen Madison Elementary School, Caroline County Public Schools Virginia finalists for the Presidential Award for Excel lence in Mathematics and Science Teaching

Megan Landis University of Mary Washington

Edward Anderson Scholarship Recipient

Damien Ettere Armstrong Elementary School, Fairfax County Public Schools Virginia finalists for the Presidential Award for Excel lence in Mathematics and Science Teaching

Sabrina Khan Virginia Commonwealth Uni versity

Edward Anderson Scholarship Recipient

Farr Quinn (Jefferson - Houston PreK - 8 IB School, Alexandria City Public Schools)

Stephanie Kessinger Charles Barrett Elementary School, Alexandria City Public Schools

William C. Lowry Educators of the Year

William C. Lowry Educators of the Year

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Featured Awards: 2022 & 2023

Virginia Council of Teachers of Mathematics

Nicole Justice L. C. Bird High School, Ches terfield County Public Schools

Holly Tate Ferdinand T. Day Elementary School, Alexandria City Public Schools Professional Development Grant Award Winner

William C. Lowry Educators of the Year

Sara Kofalt Math Specialist

Kate Roscioli Neabsco Elementary School, Prince William County Public Schools

William C. Lowry Educators of the Year 2023

William C. Lowry Educators of the Year

Theresa Wills University

Melody Locher J. A. Chalkley Elementary School, Chesterfield County Public Schools Professional Development Grant Award Winner

William C. Lowry Educators of the Year 2023

Kate Roscioli Neabsco Elementary School, Prince William County Public Schools Professional Development Grant Award Winner

Aziz Zahraoui High School

William C. Lowry Educators of the Year 2023

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Featured Awards: 2023

Virginia Council of Teachers of Mathematics

John Barclay Middle School

Lowery Guy Van Buskirk Middlesex County School District

William C. Lowry Educators of the Year

Flannagan Grant Recipient

Nicole Nuske Middlesex County School District

Megan Worrell Elementary

William C. Lowry Educators of the Year

Flannagan Grant Recipient

Tammy Sanford A.R. Ware Elementary Professional Development Grant

Kyungwon Kim Virginia Tech

Edward A. Anderson Schol arship Winner

Elijah Morris University of Virginia

Dr. Carol Fraley Walsh Middlesex County School District

Edward A. Anderson Schol arship Winner

Flannagan Grant Recipient

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Featured Awards: 2023

Virginia Council of Teachers of Mathematics

Katherine Garcia - Larner Arlington Public Schools Ena Gross Professional Development

Emily Burrell South Lakes High School Scholarship Winner— Virginia Finalists Presidential Award

Yvette Lee JR Tucker High School Scholarship Winner— Virginia Finalists Presidential Award

Ashley Prickett Jefferson Houston PreK - 8 Scholarship Winner— Virginia Finalists Presidential Award

Joseph Fuoco NCTM Michalowicz First Timers Grant

Iesha Samuels NCTM Michalowicz First Timers Grant

Katie Johnson VCTM Michalowicz First Timers Grant

Virginia Council of Teachers of Mathematics

Mission Statement

The Virginia Council of Teachers of Mathematics is the public voice of mathematics educators in Virginia, supporting teachers to ensure equitable mathematics learning at the highest quality for all students through leadership, professional development, and research.

Virginia Council of Teachers of Mathematics

Vision Statement

The Virginia Council of Teachers of Mathematics is the state ’ s leading community of educators, by educators, for the benefit of all K - 16 mathematics educators. VCTM members are connected, valued, and supported to ensure all VA students access to the highest quality of mathematics teaching and learning. We envision Vir ginia students who are inspired by the usefulness and beauty of mathematics, empowered by the opportunities mathematics affords, and prepared as confident doers of mathematics in the changing Digital Era.

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Note From the Editors

As we end our journey as the editors of the Virgin ia Mathematics Teacher Journal, we reflect on all the ways teachers across our state and nation have taught and thought about teaching mathematics and our discipline, mathematics education, in general. The interdisciplinary nature of mathematics lends itself to multiple perspectives – modeling mathe matics in nature, tackling equity in the classroom, sharing innovative ways to engage students as we, mathematics teachers, develop our students ’ math ematical thinking, and providing new ways to use the latest technology in our classroom. Today we celebrate these multiple perspectives in our last edition, Different Perspectives on Teaching Mathe matics. In this issue, we explore rounding, an under appreciated concept, and procedure, until we con sider its impact on congressional reapportionment. Linking mathematics to our political system with the number of representatives each state sends to Congress has a large impact on party control. Since the beginning of our nation, major debates ensued as our founding fathers struggled to answer the question, “ How do we handle the fractional lefto vers? ” Eric Choate shares the history with us and helps us understand how complex the issue be comes when considering rounding. During our term as editors, we witnessed the shift to including more statistics in the mathematics cur riculum as the world becomes drenched in numbers to explain perspectives on all aspects of our lives. Zareen Rahman and Ann Wallace share a statistical activity in their pre - service teachers ’ classroom. The hands - on task is engaging and promotes deep er thinking when examining data through the statis tical lens. Questions arise that need answers, and

clarity about the meaning of the data comes to the forefront. This all highlights what it means to en gage in statistical problem - solving. John Adam shares a connection between mathe matics and nature when he explores the exponen tial function to explain the consumption of non renewable resources. However, the outcome is not bleak, as we see how renewable resources can sup port our energy needs if our nation has the political will. A new perspective links literacy with mathematics in Jean Mistele ’ s article, Reading Comprehension and Mathematics Word Problems. Reading the word problem and understanding the word problem is the first step in the problem - solving process. Yet, many students struggle to translate the text from English into mathematics language, which is critical to identify an appropriate mathematical strategy to solve the problem. We close our last issue with gratitude to the many people who contributed to the journal during our time as editors. We are grateful to all the authors who shared mathematics and mathematics teaching practices from many different perspectives. We are grateful to the reviewers and their feedback that was educational to the authors and resulted in out standing articles for the readers. We are grateful to our editorial staff who made each volume possible. We look forward to the new editorial team as they bring new and different mathematics perspectives to the readers in the years ahead.

Dr. Agida Manizade and Dr. Jean Mistele

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Message From the President

Agida G. Manizade

This is my last issue as the Editor and my first is sue as the elected President of the VCTM. I was invited to join the Virginia Council of Teachers of Mathematics (VCTM) Board of Directors in 2015 as the Editor - in - Chief of the Virginia Mathematics Teacher Journal (VMT) . Due to the hard work and dedication of the entire editorial team, this journal was recognized as an Outstanding Publication by the National Council of Teachers of Mathematics (NCTM) in 2018. It has been an honor and privi lege to serve as the Editor of this outstanding prac titioner - oriented publication. I want to thank the entire editorial team for all their hard work and in credible professionalism. I want to welcome our new editorial team, including Brianna Kurtz and Toni Sorrell. I am excited to see the amazing work that they have in store for us. It is no secret that the past several years were chal lenging for all teachers across the world. The Pan demic has been a commonly shared experience that did not recognize borders, socioeconomic status, ethnicity, or race. All mathematics teachers across the globe had to quickly learn how to adapt and do what is needed for their students. Through this ex perience we have learned some important lessons. Coming back into in - person classes and settings, we, as teachers, had to grapple with another chal lenge: the realization that regardless of the amount of effort we put forward, students did not learn as much and as deeply as we hoped. Teachers had to adapt yet again, by being the cheerleaders for our students, differentiating our instruction to accom modate for students ’ developmental levels in math ematics, providing additional instruction when needed, and utilizing the best practices and availa ble technological tools and manipulatives.

At the Virginia Council of Teachers of Mathemat ics, we too felt the toll of the pandemic. As well as many other state and national teacher organiza tions, our membership numbers were at an all - time low. In order to connect to our audience and en gage with mathematics educators who were over worked and in need of support, we revised our strategies. Since the beginning of my presidency in July 2022, our board: approved the new strategic plan; adapted a new mission, vision, and diversity position statement as presented on our website; and implemented a new membership outreach strategy. At the VCTM, we also changed our traditional fi nancial model that had been in place since 1976. For the first time this year, we partnered with cor porations across the Commonwealth through our Elite Sponsorship Program, and we generated enough funding through corporate partnerships and exhibitors to cover the cost of the annual confer ence. This allowed us to obtain funding that can now be used to support novice and experienced teachers across the Commonwealth and advance our strategic plan. I look forward to connecting with you during the next two years and invite you to join the VCTM Board by participating through the election process this Spring and next Spring. I also would like to invite you to join us at the Annu al NCTM Conference this October in Washington, DC. The VCTM will serve as an affiliate cohost this year. Last but not least, our next Annual Con ference will be held in early March of 2024. We hope to see you. Once again, it is a great pleasure to serve as the President of this wonderful volun teer run organization. Dr. Agida Manizade

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The Mathematics of Congressional Reapportionment: Rounding in Action

Eric P. Choate

The US Constitution awards state representatives “ according to their respective Numbers, ” but is si lent about how this process is done. Rounding makes this problem harder than it seems like it should be, and throughout history, political machi nations have muddied these calculations. This pro vides a great opportunity for secondary teachers to apply computational thinking to a relatively large data set that comes from the American government and history perspective instead of the typical sci ences. Article 1, Section 2, of the Constitution of the United States, requires that members of the House of Representatives shall be apportioned to the states “ according to their respective Numbers. ” It initializes the House with 65 representatives for the original thirteen states and directs an “ Enumeration ” be made within three years of the first meeting of Congress and within every ten years thereafter to reapportion representatives. Nei ther the total number of representatives nor the method of apportionment is specified, and so the first Congress had to decide how to do this. Since their first attempt led to the first presidential veto of an act of Congress, this problem must be harder than computing simple proportions. Why? The answer is, of course, that since each state ’ s share of representatives must be a positive integer, we must round the proportions. We can illustrate the issue with a simple fictitious problem in Table Introduction

1. A fake country splits ten representatives among three states with 10.2, 10, and 9.8 million people. Their quotas —the exact non - integer number of seats each state should get—is between 3.2 and 3.4, so each state rounds down to three representa tives. Reasonable, yes, but it has an important problem, it only awards nine representatives, not ten. Or, if the total increases to twenty, now each state rounds up to seven so that one too many seats are awarded. How do we fix this? Since the first

census, Congress has used four different methods to correct the sum. All four start with the same quotas with their differences lying in how they round. They can lead to different apportionments, but sometimes the size of the House was chosen precisely because that value showed the same re sults for two different methods. You may notice that here we started with the sim plest possible scenario illustrating the problem. We can do this one in our heads. This is sufficient to illustrate how the methods are implemented. How ever, to reproduce the apportionment based on the 2020 Census, we need to apply the method to 50 Table 1: Simple apportionment of a fake country with three states

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states, splitting 435 representatives. We cannot do these calculations in our heads or even with pencil, paper, and a calculator, not because they are of a harder type, but because the number of calcula tions grows so rapidly with the number of states. We must think about this problem algorithmically to instruct a computer to do these calculations for us. I have taught this material in a college class on the mathematics of democracy, and it motivates students to learn a programming language like Matlab or R. This may not be the best approach for high school classes. Our focus here is to do this algorithmic thinking in a spreadsheet, such as Ex cel or the sample Google Sheet referenced in this article, which I encourage you to copy from https:// tinyurl.com/y3artfkk. Formulas without the ability to apply them to real - world data are not helpful. We begin with a historical summary of the four different methods Congress has used to reappor tion the House of Representatives, including dis cussions of what motivated changes when they happened and an explanation of why we chose the current system that does not round like we teach our elementary school students to round. We con clude with suggested classroom activities or as signments in which students compare the different methods, see how they differ from each other, and apply the methods to various hypothetical scenari os based on populations of the states and our choice for the number of representatives. A note on resources: The best mathematical re source for information on apportionment is Fair Representation (Balinski & Young, 2001). The first half is written for a less technical audience and takes more of a narrative approach to the histo ry of the methods used by Congress to reapportion itself, whereas the second half takes a more theo retical approach, proving theorems about how vari ous apportionment methods work or perhaps do not work as we would hope. Also of note, is The American Census: A Social History (Anderson, 2015), which focuses more on the history of the census but also discusses how it has been used in reapportionment. Additional presentations of this problem are in Bradberry (1992), Caulfield (2010), Malkevitch (2002a, 2002b), and Swenson (2022).

1790: The First Apportionment

Counting a nation of nearly four million people in 1790 was a daunting task. The census date was August 2, 1790, but the US marshals tasked with this duty were not able to report their results until 1792. Vermont and Kentucky became states during the course of this tabulation, and so they were also included as states. The enumeration counted the entire free population plus three - fifths of the en slaved population, as it would until the passage of the 14th Amendment in 1868. The total enumera tion was P = 3,615,924, and state - by - state results are given on the page titled “1790 Census ” of the Google Sheet. Next, Congress had to decide N , the size of the House of Representatives, and then , the number of representatives for state i. There are just three constitutional constraints: 1) Each state gets at least one representative, 2) Seats are awarded proportionally with the enumeration, and 3) There are at most 30,000 people per repre sentative, which caps the size of the House. In 1790, the largest value of N that kept P/N greater than 30,000 was 120. Secretary of the Treasury Alexander Hamilton proposed using that value and the following method to round the quotas so that the total was correct.

The Method of Hamilton

1. Compute each state ’ s quota

, where

is the enumeration of state i.

2. Round each quota down so that state i is awarded seats, where [] is the floor or greatest integer function. The sum of the is now less than or equal to the total N .

3. Compute each state ’ s remainder which would be between 0 and 1.

,

4. Rank the states from largest remainder to smallest, which establishes the priority for awarding the remaining seats.

For

example,

Virginia ’ s

quota

was

so that _______ with the remainder _______ . This was the second largest remainder after New Jersey.

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The sum of the __ was only 111, and so the states with the nine largest remainders were given one more seat. Thus, Virginia increased to ________ . Congress approved this apportionment, but George Washington used the first presidential veto to re ject it for two reasons: 1) He applied the 30,000 people - per - representative minimum to each state instead of the national total, and eight states violat ed this criterion. 2) He felt that using the largest remainders did not award seats proportionally— New York ’ s population was more than five times Delaware ’ s, but since Delaware had a larger re mainder, it received an additional representative instead of New York. Therefore, Congress had to start over with both a smaller total and a different method. Luckily, Sec retary of State Thomas Jefferson had already pro posed a different method.

Virginia received 19 out of 105 representatives. Congress did not codify the method so future Con gresses would have to debate this again. The meth od of Jefferson was used again following the 1800, 1810, and 1820 Censuses, with the size of the House increasing so that no state lost a seat. However, as more data went into this method, a bias toward larger states began to emerge. From 1790 to 1830, Delaware ’ s quota was always be tween 1.52 and 1.95, but only once did it round up to two seats. However, New York always rounded up, and worse, in 1820, its quota was 32.50 but Jefferson ’ s method “ rounded ” up past the next in teger to 34, and in 1830, its 38.59 rounded up to 40. Congressmen from smaller states demanded an alternative method. Jefferson ’ s method is one example of a class of methods called Divisor Methods . All Divisor Methods have the same basic structure of adjusting the value of D until the correct sum is awarded, with the differences being how the methods round. In 1830, Rep. John Quincy Adams of Massachu setts and former president proposed a different di visor method that rounded all the quotas up instead of down. Predictably, this was biased toward smaller states and was rejected. Sen. Daniel Web ster of Massachusetts who introduced the natural compromise, Quotas, which are rounded in the standard manner, round down if the quota is below ______ and up otherwise. That is, the rounding point is the arithmetic mean of the integers above and below the quota. 1. Start with the divisor D=P/N. Compute each state ’ s quota 2. If __________, then round down so that _______ , but if __________, round up so that _________. 3. If the sum of the __ is less than N , decrease the divisor D until the number of seats awarded increases to the desired N . If the sum of the __ is greater than N, increase D until the number of seats awarded decreases to the desired N . The Method of Webster

The Method of Jefferson

1. Choose a divisor D to be approximately the number of people per representative for the na tion. Start by setting D = P/N. Compute each state ’ s quota in terms of D as

2. Round each quota down so that

3. If the sum of the __ is less than N , decrease the divisor D and repeat until the number of seats awarded increases to the desired N . They found N = 105 to be the largest value that made each state have at least 30,000 people per representative. For this N , the “ true ” divisor was P/N = 34,437, but that awarded only 97 seats after the quotas were rounded down. The correct num ber of seats would be awarded for any divisor in the interval _______________ and so D = 33,000 was chosen. Under this method, Virginia received

representatives. Congress passed this in the Appor tionment Act of 1792, and President Washington signed it into law.

1800 - 1910: Exploration of Methods

The Apportionment Act of 1792 only stated that

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Table 2: The “ Alabama Paradox ” in 1880

Congress chose this method in 1830. The page ti tled, “1830 Census ” compares the methods of Jef ferson, Adams, and Webster for the 1830 census. All three methods agree for 9 of the 24 states and each method gives a different value for the two largest states. Webster gives New York 39, where as Jefferson gives it 40 and Adams only 37. Penn sylvania gets 27, 28, and 26, respectively. Web ster ’ s method was also chosen for the 1840 Cen sus. In 1850, Rep. Samuel F. Vinton of Ohio proposed a different approach—choosing a method and House size before the Census results became avail able to prevent arguing over each methods ’ impact on individual states. His arguments were persua sive to a point, but he was apparently unaware that his proposed “ new ” method was actually just the method of Hamilton. “ Vinton ’ s ” method became law, but his idea of capping the size of the House was quickly abandoned. In 1850, 1860, and 1870, some states were given extra seats after the method was applied for political reasons. This was to justi fy, the 1850 Census that gave California only one seat, but Congress knew California would grow very rapidly during the coming decade, so they gave it an extra seat. In 1870, Congress applied Hamilton ’ s method with N = 283, but months later gave nine states an extra seat for reasons that were not immediately obvious.

reasonable, it suffers from subtle paradoxes that were not immediately apparent. In 1880, as they usually did, Congress asked the Census Office to prepare tables using Hamilton ’ s method with val ues of N from 275 to 350. All this data revealed a problem to the Census Office. In the words of its chief clerk C.W. Seaton, “ I met with the so - called ‘ Alabama ’ paradox where Alabama was allotted 8 Representatives out of a total of 299, receiving but 7 when the total becomes 300,” which he felt was “ conclusive proof that the process employed in ob taining it is defective. ” Increasing the total number of representatives can make a state lose a seat. How can that happen? Ta ble 2 shows the details. When increasing from N = 299 to N = 300, each state ’ s quota increases by a factor of 300/299, which means a larger state ’ s re mainder increases by a larger amount than a small er state ’ s. Specifically, when N = 299, 279 seats are awarded based on the initial rounding down. Since Alabama has the 20th largest remainder, it gets the last seat followed by the larger states of Texas and Illinois. When N increases to 300, Tex as ’ s quota increases by 0.032, whereas Alabama ’ s quota increases by 0.025, which is enough for Tex as to pass Alabama in the remainder rankings. The fourth - largest state Illinois ’ s increase of 0.062 is enough to pass both Alabama and Texas in the re mainder rankings. Therefore, adding a new seat to the total shifts a seat away from Alabama to Illi nois or Texas.

While the method of Hamilton does not seem un

Table 3: The number of seats for Maine and Colorado under the method of Hamilton as a function of the House Size

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The Census Office reported this problem to Con gress, but Congress was not persuaded to abandon the method. In fact, they chose N = 325 precisely because the methods of Hamilton and Webster re turned the same apportionment at this value. This process repeated itself in 1890. The Census Office said the method of Hamilton is flawed, but enough of the representation in Congress still liked it, and they compromised with N = 356, because the two methods agreed. The breaking point came in 1900. The House Se lect Committee on the Twelfth Census asked the Census Office to use Hamilton ’ s method from N = 350 to 400 seats. Table 3 demonstrate the problems that were evident to the committee. The committee, wary of the populist leanings of Colorado, chose N = 357. However, this caused Maine to lose a seat, prompting Maine Rep. John E. Littlefield to say, “ God help the State of Maine when mathematics reach for her and undertake to strike her down. ” The full House rejected the com mittees ’ proposal. Instead, they chose the Webster Method with N = 386, which was large enough so no state lost a seat. Two other interesting paradoxes may appear with Hamilton ’ s method. In the “ New States Paradox, ” adding a new state and sufficient seats for its ap portionment, may shift a seat from one existing state to another. In the “ Population Paradox, ” two states may both grow in population, but the faster growing state may lose a seat to the slower growing state. These are both signs of a method that seems suboptimal. Divisor Methods like Jef ferson and Webster do not have these paradoxes. However, Hamilton Method does have one ad vantage over the Divisor Methods. It “ stays within the quota. ” That is, Hamilton ’ s method, a state will always gets an integer above or below its quota. We saw earlier with New York in 1820 that, Jef ferson rounded its quota of 32.50 up beyond the next integer to 34 seats. Balinski & Young (2001) demonstrate that it is not possible for a method to both be free from these three paradoxes and stay within the quota.

Hill Method

In 1902, the Census Bureau became a standing part of the federal government instead of a temporary office for each census. They began to collaborate with university mathematicians, statisticians, and economists to lay a stronger theoretical foundation for their work. Cornell professor and future presi dent of the American Statistical Association Wal ter F. Willcox, worked with them to convince Con gress that Webster ’ s Method was the soundest, and it was used in 1910 with N = 433 so that no state lost a seat. Provisions were made so that Arizona and New Mexico would each get one seat if they became states, which happened in 1912, pushing the House of Representatives to 435 members. In 1910, Joseph A. Hill, the chief statistician in the Division of Revision and Results at the Census Bu reau, started to push back against Webster ’ s Meth od. Back in 1830, University of Vermont professor James Dean submitted another divisor method to Congress. Congress promptly ignored it, but it is worth noting here. Dean recommended rounding the quota by comparing each state ’ s divisor ________ to the overall D and choosing to minimize ______. Equivalently, if _____ , this can be accomplished by rounding not with respect to n +0.5, the arithmetic mean of n and n +1 but with respect to the harmonic mean of n and n +1. The reciprocal of the harmonic mean is the arithmetic mean of and ____, or This number is between n and n + 1, and so it can also be used as a rounding threshold, but it is slightly smaller than n + 0.5. For example, to round 4.48, we would look at the harmonic mean of 4 and 5, which is ___________, and so 4.48 would round up to 5 using this method. This may not seem like an obvious way to round, but it has a justification—maximizing equity of representation. Ideally, the people per representa tive for each state would be same as the nation as a whole, but due to rounding, some states have greater representation than others. .

1920 - present: Deadlock and the Huntington

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Minimizing ______ prioritizes each state ’ s number of representation to be as close to the national rate as possible versus each state ’ s representative of representatives to be closest to its quota. Dean ’ s method was never used. However, it is im portant to mention that Hill appreciated Dean ’ s views on equity in representation, but he proposed a slight modification. While Dean minimized the absolute difference between the representations ______ , Hill felt it was more appropriate to mini mize the relative difference. That is, minimize ______________. This ensures that if a seat were transferred from one state to another, the minimum relative deviation from the national representation of the pair would increase. While the absolute and relative differences will move in tandem, they can differ. Two examples help us understand the dif ference between these two measures. Hill used a hypothetical example. Consider two pairs of states. In the first pair, one state has 100,000 people per representative and the other state has 50,000 people. In the second pair, one state has 75,000 people and the other 25,000 peo ple. Both pairs exhibit the same absolute differ ence. This Dean would say the underrepresented states share the same measure of inequality. How ever, the underrepresented state in the second pair has a relative difference of

quota was 3.51. If these two states were to split 19 representatives, should they be split 15 - 4 or 16 - 3? As shown in Table 4, when Massachusetts has 15 seats, one Massachusetts representative represents 36,273 more people than one Florida representa tive. If Massachusetts increases to 16, now Florida is underrepresented with an absolute difference of 40,472 people per representative. Dean chooses the smaller of these two values and gives Massachu setts 15 and Florida 4. Hill, however, looks at the relative difference: when Massachusetts has 15 seats, one Massachusetts representative represents 19.3% more people than one of Florida ’ s repre sentatives. Giving one of Florida ’ s seats to Massa chusetts makes Florida underrepresented, but by only 19.2%. Thus, Hill would disagree with Dean and say it is preferable to give Massachusetts the 16 seats. Hill ’ s presentation to Congress contained an error in his algorithm, but Edward V. Huntington, Hill ’ s Harvard classmate who became a Harvard profes sor and statistical consultant to the military in World War I, caught this error, and so the correct ed method is known as the Huntington - Hill meth od. The Hill - Huntington method is also known as the method of equal proportions. Whereas Dean ’ s method uses harmonic mean rounding, the Huntington - Hill method is achieved by rounding the quota with respect to the geomet ric mean of the integers above and below the quo ta. That is, _______ . This number is always great er than the harmonic means of n and n +1 but less than their arithmetic mean. Continuing our previ ous example of rounding 4.48, the geometric mean of 4 and 5 is _________ , and so the Huntington Hill method would round up like Dean ’ s method on the other hand, Webster ’ s Method would round down.

This is worse than the underrepresented state in the first pair, which has a difference of only

Balinski & Young (2001) compare two states from the 1910 Census with N = 425 representatives. Massachusetts had a quota of 15.71 and Florida ’ s

Table 4: Balinski & Young ’ s example comparing the methods of Dean and Huntington - Hill.

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As n becomes large, the harmonic and geometric means approach the arithmetic mean from below. For the large states, there is little difference be tween the three methods: Dean, Huntington - Hill, and Webster. However, for smaller states, lower rounding points become more noticeable so that the Huntington - Hill Method shows a slight bias in favor of smaller states compared to Webster, but not as strong of a bias as Dean ’ s Method shows. 1. Start with the divisor D = P/N . Compute each state ’ s quota 2. If ______________ , then round down so that . Otherwise round up so that _______ . is less than N , decrease the divisor D until the number of seats awarded increases to the desired N . If the sum of the is greater than N , increase D until the num ber of seats awarded decreases to the desired N . In 1920, this new method was advanced by the Census Bureau and competed with Webster ’ s Method in Congress. However, this competition was overshadowed by something else—changing demographics. Rapid industrialization before and during World War I led to a dramatic internal mi gration. Prior to 1910, internal migration was mainly east - to - west. Between 1910 and 1920, this shifted from rural - to - urban. Rural areas lost five million people while urban areas gained fourteen million. For the first time in our history, a majority of the population lived in urban areas. Congress debated the new apportionment method that overrepresented the rural population, which would soon lose influence. The House subcommittee proposed the Webster ’ s method and increasing N by 48 to 483 so that no state would lose a seat. However, smaller states balked at how many of these new seats would go to states with large urban areas. Plus, there was concern that this number of representatives would not actually fit into the House chamber. The small er states favored the Huntington - Hill Method, be cause of its small - state bias, but the larger states favored the Webster Method. This deadlock con 3. If the sum of the Huntington - Hill Method

tinued for nine years.

In June 1929, with the 1930 Census looming, a grand bargain was finally reached with the Reap portionment Act of 1929. It did not actually ap prove an apportionment but, instead focused on establishing a process. The 1920 Census would not be used for apportionment, and they would just wait for the 1930 Census. The House number of representatives would become fixed at the current size. It could increase temporarily when new states were added, but that would only last until the next Census cycle to returned the size to N = 435. The president would transmit to Congress apportion ments methods of Webster and Huntington - Hill. If Congress took no action to change the method, the most recently used method would be used. In 1930, the two methods gave the same apportion ment, and so without specifying the method, Con gress chose Webster ’ s by default since it had been used in 1910. However, after the 1940 Census, the two methods did not agree. Webster gave Michigan 18 and Ar kansas 6, whereas Huntington - Hill gave Michigan 17 and Arkansas 7. With Arkansas more likely to send another Democrat to Washington, the Demo cratically controlled Congress decided to switch to the Huntington - Hill method, and it has been used for each reapportionment since. After the 1990 Census, Montana lost a seat and sued in federal court saying that the Huntington - Hill method did not provide equal representation. A panel of three district court judges agreed, but in US Department of Commerce v. Montana , the Supreme Court re versed that decision and said that the Huntington Hill method was constitutional. Though delayed, the 2020 Census results were eventually released in April 2021, and the results based on the Huntington - Hill method, to round with respect to the geometric mean, the algorithm uses the if() function. Its first argument is a logical test. The second argument is returned if the test is true, and the third argument is returned if the test if false. We use this logical statement in the exam ples that follow.

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the Sainte - Laguë method or the method of ma jor fractions. It is used in Germany, Norway, Sweden, and New Zealand, to name a few. Recreating the results of another country ’ s re cent election can help give our students a great er understanding of how the world wrestles with different approaches to apportionment. 5. In addition to the decennial census, the Census Bureau computes official annual estimates of state populations every July 1. The page titled “ Hypothetical Apportionments ” has the July 1, 2021, estimates as well as the official 2020 Census populations. There are several different questions that are worthy of consideration based on these numbers: A. Do the 2021 estimated values give a different apportionment? B. Would the method of Webster give a different apportionment? C. How many more people did Virginia need to gain another seat? D. If ten more seats were added, which states would get them? E. What is the maximum number of repre sentatives allowed by the Constitution based on the 2020 Census? F. The populations of the District of Co lumbia and Puerto Rico are included. How many seats would they have got ten had they been states in 2020 or 2021? G. The Greater Idaho Project encourages parts of eastern Oregon and northern California to join Idaho. What effect would this have on the apportionment? Experiments like these help us better understand how our government operates, and they also re mind us that not all aspects of its structure are stat ic but, change over time. As people who like to calculate, we should be experimenting with differ ent governments in silico . It is fitting to close with a quote from Thomas Jefferson from a letter he Conclusion

Experiments

We now understand the history of apportionment for the House of Representatives to the states, but Congress only does this once every ten years. Knowing how this process works allows us to ap ply the current method (or others) to any number of hypothetical situations that can help us to better understand our country, its history, and where people live, provided we have the computational skills to answer various “ What if …” questions. Some suggested classroom activities are shown below. 1. Examine the effect of increasing or decreasing the number of representatives or changing the divisor on the title page “1790 Census. ” 2. The 1790 Census page shows the free and en slaved populations of each state. The Three Fifths Compromise was negotiated between states that wanted to count only the free popu lation and the states that wanted to count the entire population. Had either of these perspe cives been enacted, how would it change the apportionment and the maximum number of representatives? 3. Each reapportionment determines the number of electoral votes each state receives for the next two to three presidential election cycles. Could some presidential winners have been different based on the apportionment? A. 1796 and 1800: Hamilton ’ s vs Jeffer son ’ s? Three - Fifths, free only, or entire population? B. 1876: Nine states received extra repre sentatives after Hamilton ’ s method was applied. 4. Proportional representation occurs in other countries that award parliamentary seats to par ties based on their proportion of the vote. Jef ferson ’ s method is commonly known as the d ’ Hondt method, and it is used in Belgium, the Netherlands, Spain, and Israel, among many others. Typically, a minimum percentage of votes acts as a threshold for a party to qualify for seats. Webster ’ s method is often known as

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duction. Mathematical Association of America. www.maa.org/press/periodicals/ convergence/apportioning - representatives - in - the - united - states - congress - introduction Malkevitch, J. (2002a). Apportionment I. American Mathematical Society. www.ams.org/ publicoutreach/feature - column/fcarc - appor tion1 Malkevitch, J. (2002b). Apportionment II . Ameri can Mathematical Society. www.ams.org/ publicoutreach/feature - column/fcarc - appor tionii1 Swenson, D. (2022). A Statistics - Related Ap proach to the Legislative Apportionment Problem … and also, a Second One. The College Mathematics Journal, 53 (5), 382 391.

wrote in 1816 to Samuel Kercheval. He wrote one of the five passages that is carved in the Jefferson Memorial in Washington, DC, "I am not an advo cate for frequent changes in laws and constitutions, but laws and institutions must go hand in hand with the progress of the human mind. As that becomes more developed, more enlightened, as new discov eries are made, new truths discovered and manners and opinions change, with the change of circum stances, institutions must advance also to keep pace with the times. We might as well require a man to wear still the coat which fitted him when a boy as a civilized society to remain ever under the regimen of their barbarous ancestors." Anderson, M. J. (2015). The American Census: A Social History (2nd ed.). Yale University Press. Balinski, M. L., & Young, H. P. (2001). Fair Rep resentation: Meeting the Ideal of One Man, One Vote. (2nd ed.). Brookings Institution Press. Bradberry, B. A. (1992). A Geometric View of Some Apportionment Paradoxes. Mathe matics Magazine, 65 (1), 3 - 17. Caulfield, M.J. (2010). Apportioning Representa tives in the United States Congress— Intro References

Eric P. Choate Associate Professor Radford University echoate2@radford.edu

Math Jokes

What did the triangle say to the circle? “ You ’ re pointless. ”

Why wasn ’ t the geometry teacher at school? Because she sprained her angle.

Are monsters good at math? No, unless you Count Dracula.

Which king loved fractions? Henry the ⅛.

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Technology Review Section Editor: Alex Moore

In this section, we feature websites, online manipulatives, and web - based applications that are appropriate for K - 12 mathematics instruction. We are looking for critical reviews of technologies which focus on both the benefits and limitations of using these tools in a K - 12 mathematics classroom.

Using Social Media in Teaching Mathematics:

Applications of Facebook and TikTok

Amanda Sawyer & Austin Evans

During the COVID - 19 pandemic, what were stu dents doing outside of their schoolwork? They were messaging their friends, learning new dances, and developing acting skills using social media platforms like Facebook and TikTok. Social media is an expanding apparatus that has been integrated into the everyday lives of our students. We, as teachers, can use these tools to assist them in their academic lives as well. Both Facebook and TikTok enable users to watch, share, and, most important ly, communicate their thoughts and perspectives. While students were stuck at home, they spent hours creating content and learning from their peers through these platforms. As a mathematics teacher educator and a practicing classroom teach er, we believe these same platforms can be used in our mathematics classrooms to strengthen commu nication and mathematical engagement with stu dents. In this article, we present some possibilities for how students can use Facebook and TikTok to support mathematical communication. Originally designed for college students, Facebook is a social networking website that allows you to connect with friends, family, and other community groups online through status updates, photos, and videos. On Facebook, individuals can post mathe matics problems asking their followers to comment on solutions with the expectation that they will an Facebook

swer the problem incorrectly. Figure 1 shows a basic system of equations problem using everyday objects. We thought this would be a unique and engaging way to bring a real - world context into my mathematics instruction while teaching an intro ductory lesson on how and why we solve system of equations in algebra. Students can see comments on the mathematics problem post and attempts by different people to solve the task. In most cases on Facebook, these comments will have been made by adults. By providing students with adults ’ misun derstandings, students see the value of making mis takes, can read different people ’ s attempts to solve the task, evaluate for themselves whether an at tempt is correct or not, and build algebraic reason ing. Overall, posting a mathematics problem on Facebook allows students to see multiple represen tations and solutions as well as common miscon ceptions in solving a system of equations. Teachers can post mathematical questions to their Facebook page to help facilitate discussions about multiple representations of solutions and individual misconceptions. For example, using the problem in Figure 1, students can ask their friends the follow ing questions: Can you please solve this problem for my class, and explain your thinking? (Prompt continues on next page.)

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