Inside An Early 21st-Century Geometry Classroom Michael McKinley
Foreword This is a collection of artifacts that was found in what was a 21st-century Wisconsin math classroom. I hope that their discovery provides insight into how a typical math class was operated at that time. -Michael McKinley, author
A Definition We found this entry in the journal of the classroom teacher. It outlines his definition of “mathematical literacy”. August 30, 2014 I think that I am getting closer to what it means to be “mathematically literate” (which I will call “numerate”). To be numerate, a person needs to be able to do more than simply working equations or solving algebra problems. A calculator or computer can do these things, but it wouldn’t be called “numerate”. Being numerate requires a much deeper level of understanding. A lot of math is logical thinking – figuring out a proof, solving equations, etc. – so it makes sense that developing mathematical literacy helps develop logical thinking. However, that is not all a student needs to be numerate. He needs to be able to follow proofs as well as write his own. He needs to be able to interpret charts and diagrams and create accurate ones based on data and mathematical descriptions. A student needs to be able to justify the steps he takes and explain his process to others.
Students The MAP scores for math suggest that around half of the students are performing below grade level, with one student performing at an elementary school level. However, while MAP scores are a useful tool to gauge student performance, it is not the be-all and end-all. A poor score on the MAP test could be the result of a bad night’s sleep, the student being distracted by home life, or other issues. Thus, it is important to use a combination of other tools to assess student knowledge and ability.
Students Further insight into creating a classroom responsive to students’ needs could be attained by asking them to respond to written prompts concerning, among other things, their feelings about and experiences with math and their interests outside the classroom.
Self-Concept Check A self-concept check is a good way to pre-assess student knowledge and ability. It takes less time than giving students a more formal diagnostic test and students generally don’t feel as pressured.
RSQC2Match the step with its definition Step Definition Write something specific about the lesson Share with a peer Write something you’d like to share about the lesson Name main points Ask about something you don’t understand • Recall • Summarize • Question • Comment • Connect
Exit/Admit Slips While allowing students to ask questions in class was a beneficial way for teachers to assess student understanding, a student might feel embarrassed to ask a question or otherwise unwilling. Asking students to anonymously give feedback on what they do and do not understand was considered a valuable tool in this classroom.
TPS • Think – students consider the problem individually • Pair – students discuss the problem and solutions in partners • Share – students share their findings as a class
Marzano’s Six Steps New Term
Frayer Model Across • A list of things that fit the word • Goes in the center of the chart Down • A list of related, but different concepts (with justification) • A list of properties that the word has • The student writes the meaning of the word
RAFTS You are a high-school math teacher. Your task is to create a writing assignment for a class of high school geometry students that will assess their knowledge about triangle congruence. • R – role • A – audience • F – format • T – topic • S – strong verb
Word Problem Roulette Source: Mathematics teaching in the Middle school Vol. 18, No. 8, April 2013
3-Way Tie A way for comparing three distinct, but related topics. Students write the words in three corners of the triangle. On each edge, they relate how the two words relate. In the center, they summarize and explain how all three fit together. Writing 3-Way Tie Comparison Relate sectors, arcs, and circles Relate slope, speed, and rates Relate perimeter, area, and surface area Synthesis
Source: Mathematics teaching in the Middle school Vol. 18, No. 8, April 2013
Strategies and MI Note: mathematical/logical applies to all, so I omitted the column to save space
Kinesthetic Learning Even though kinesthetic intelligence isn’t well-represented in the strategies, that does not mean that it is forgotten or has no place in a math classroom. There are many learning activities that can incorporate students moving around / working with manipulative.
Abbott, E. A. (1884). Flatland, a romance of many dimensions. “Classic of science (and mathematical) fiction — charmingly illustrated by author — describes the journeys of A. Square and his adventures in Spaceland (three dimensions), Lineland (one dimension) and Pointland (no dimensions). A. Square also entertains thoughts of visiting a land of four dimensions — a revolutionary idea for which he is banished from Spaceland.” I chose this book because it provides a somewhat humorous discussion of geometry and the relationship between dimensions. While the book also serves as a satire of society at the time, the mathematical content is valuable.
Paulos, J. A. (1988). Innumeracy: Mathematical illiteracy and its consequences. New York, NY: Farrar, Straus and Giroux. “Why do even well-educated people understand so little about mathematics? And what are the costs of our innumeracy? John Allen Paulos, in his celebrated bestseller first published in 1988, argues that our inability to deal rationally with very large numbers and the probabilities associated with them results in misinformed governmental policies, confused personal decisions, and an increased susceptibility to pseudoscience of all kinds. Innumeracy lets us know what we're missing, and how we can do something about it. Sprinkling his discussion of numbers and probabilities with quirky stories and anecdotes, Paulos ranges freely over many aspects of modern life, from contested elections to sports stats, from stock scams and newspaper psychics to diet and medical claims, sex discrimination, insurance, lotteries, and drug testing. Readers of Innumeracy will be rewarded with scores of astonishing facts, a fistful of powerful ideas, and, most important, a clearer, more quantitative way of looking at their world.” I chose this book because it explores the problem of poor “mathematical literacy” and the applicability of math outside the classroom. The books relates math to many subjects, making it very adaptable to a diverse range of interests.
Hogben, L. (1968). Mathematics for the million: How to master the magic of numbers. W. W. Norton & Company. “Taking only the most elementary knowledge for granted, Lancelot Hogben leads readers of this famous book through the whole course from simple arithmetic to calculus. His illuminating explanation is addressed to the person who wants to understand the place of mathematics in modern civilization but who has been intimidated by its supposed difficulty. Mathematics is the language of size, shape, and order—a language Hogben shows one can both master and enjoy.” I chose this book because it could serve as a good supplement, especially for students who are struggling with earlier concepts. It could also give students who are more advanced and bored additional topics to explore.
Bruce, C. (2002). Conned again, Watson!. Basic Books “In Conned Again, Watson!, Colin Bruce re-creates the atmosphere of the original Sherlock Holmes stories to shed light on an enduring truth: Our reliance on common sense-and ignorance of mathematics-often gets us into trouble. In these cautionary tales of greedy gamblers, reckless businessmen, and ruthless con men, Sherlock Holmes uses his deep understanding of probability, statistics, decision theory, and game theory to solve crimes and protect the innocent. But it's not just the characters in these well-crafted stories that are deceived by statistics or fall prey to gambling fallacies. We all suffer from the results of poor decisions. In this illuminating collection, Bruce entertains while teaching us to avoid similar blunders. From "The Execution of Andrews" to "The Case of the Gambling Nobleman," there has never been a more exciting way to learn when to take a calculated risk-and how to spot a scam.” I chose this book because it intertwines math with a narrative. Presenting math in this way could help students get more engaged than when presented with a “traditional” math book.
Lewis, M. (2003). Moneyball: The art of winning an unfair game. (1st ed.). New York: W.W. Norton & Company Inc. “Billy Beane, the Oakland A’s general manager, is leading a revolution. Reinventing his team on a budget, he needs to outsmart the richer teams. He signs undervalued players whom the scouts consider flawed but who have a knack for getting on base, scoring runs, and winning games. Moneyball is a quest for the secret of success in baseball and a tale of the search for new baseball knowledge—insights that will give the little guy who is willing to discard old wisdom the edge over big money.” I chose this book because it might help engage students who are interested in sports. Like the other books, it presents a “real-world” use of math in a place the students might not expect.
Derbyshire, J. (2003). Prime obsession: Bernhard Riemann and the greatest unsolved problem in mathematics. Washington D.C.: Joseph Henry Press. “In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled “On the Number of Prime Numbers Less Than a Given Quantity.” Today, after 150 years of careful research and exhaustive study, the Riemann Hypothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it.” I chose this book to challenge bright students who might be bored in class. Despite the somewhat daunting-sounding topic, the book assumes the reader is not a mathematician and thus presents the topics in a very accessible way.
Downing, D. (2003). Algebra, the easy way. Barron's Educational Series. “This book tells of the adventures that took place in the faraway land of Carmorra. During the course of the adventures, we discovered algebra. This book covers the topics that are covered in high school algebra courses. However, it is not written as a conventional mathematics book. It is written as an adventure novel. None of the characters in the story know algebra at the beginning of the book. However, like you, they will learn it.” I chose this book as a supplement for students who still might be struggling with algebra. I think that because the book is written as a narrative, the students might find it more engaging.
Flannery, S. (2001).In code: A mathematical journey. Workman Publishing Company. “In January 1999, Sarah Flannery, a sports-loving teenager from Blarney in County Cork, Ireland was awarded Ireland's Young Scientist of the Year for her extraordinary research and discoveries in Internet cryptography. The following day, her story began appearing in Irish papers and soon after was splashed across the front page of the London Times, complete with a photo of Sarah and a caption calling her "brilliant." Just 16, she was a mathematician with an international reputation.In Code is a heartwarming story that will have readers cheering Sarah on. Originally published in England and co-written with her mathematician father, David Flannery, In Code is "a wonderfully moving story . . . about the thrill of the mathematical chase" (Nature) and "a paean to intellectual adventure" (Times Educational Supplement). A memoir in mathematics, it is all about how a girl next door, nurtured by her family, moved from the simple math puzzles that were the staple of dinnertime conversation to prime numbers, the Sieve of Eratosthenes, Fermat's Little Theorem, Googols-- and finally into her breathtaking algorithm. Parallel with each step is a modest girl's own self-discovery--her values, her burning curiosity, the joy of persistence, and, above all, her love for her family.” I chose this book because it is about a high-school girl. Because the book is about somebody the students’ age and about a girl, I hope some might be able to relate to her.
Seife, C. (2000). Zero: The biography of a dangerous idea. Penguin Books. “In Zero, Science Journalist Charles Seife follows this innocent-looking number from its birth as an Eastern philosophical concept to its struggle for acceptance in Europe, its rise and transcendence in the West, and its ever-present threat to modern physics. Here are the legendary thinkers—from Pythagoras to Newton to Heisenberg, from the Kabalists to today's astrophysicists—who have tried to understand it and whose clashes shook the foundations of philosophy, science, mathematics, and religion. Zero has pitted East against West and faith against reason, and its intransigence persists in the dark core of a black hole and the brilliant flash of the Big Bang. Today, zero lies at the heart of one of the biggest scientific controversies of all time: the quest for a theory of everything.” I chose this book because I think it helps to dispel the notion that math is static and unchanging. I also hope that it would encourage students to think about math in a different way.
McKellar, D. (2012). Girls Get Curves: Geometry Takes Shape. Hudson Street Press. “In her three previous bestselling books Math Doesn't Suck, Kiss My Math, and Hot X: Algebra Exposed!, actress and math genius Danica McKellar shattered the “math nerd” stereotype by showing girls how to ace their math classes and feel cool while doing it. Sizzling with Danica's trademark sass and style, her fourth book, Girls Get Curves, shows her readers how to feel confident, get in the driver's seat, and master the core concepts of high school geometry, including congruent triangles, quadrilaterals, circles, proofs, theorems, and more! Combining reader favorites like personality quizzes, fun doodles, real-life testimonials from successful women, and stories about her own experiences with illuminating step-by-step math lessons, Girls Get Curves will make girls feel like Danica is their own personal tutor. As hundreds of thousands of girls already know, Danica's irreverent, lighthearted approach opens the door to math success and higher scores, while also boosting their self-esteem in all areas of life. Girls Get Curves makes geometry understandable, relevant, and maybe even a little (gasp!) fun for girls.” I chose this book because I hope that it would help girls get more interested in math and help to dispel the idea that “math isn’t for girls” or “girls aren’t good at math”.
Standards We uncovered a set of documents that we believe outline what 21st-century educators believed was essential knowledge. These topics can be divided into five broad categories.
Congruence Students studying geometry were expected to • Experiment with transformations in the plane • Understand congruence in terms of rigid motions • Prove geometric theorems • Make geometric constructions
Congruence (A) • Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. • Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). • Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. • Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. • Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.