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Inside An Early 21st-Century Geometry ClassroomMichael McKinleyThis 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, authorForewordA DefinitionWe found this entry in the journal of the classroom teacher. It outlines his definition of mathematical literacy.

August 30, 2014I 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 wouldnt 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.

Modes of CommunicationThe StudentsNameMAP MathMAP ReadingMAP LanguageKatharine Beams232232223Steve Lopez225214207James Keller215214199Cynthia Antke234245234Victoria Barnwell222222208Dana Perez234232225Louis Reis262243236Ricky Fleming247226232MAP DataLogicalSpatialLinguisticKinestheticMusicalInterpersonalIntrapersonalKatharine BeamsSteve LopezJames KellerCynthia AntkeVictoria BarnwellDana PerezLouis ReisRicky FlemingMI Survey ResultsThe 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 nights 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.StudentsFurther 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.StudentsAssessment StrategiesA 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 dont feel as pressured. Self-Concept CheckRSQC2Match the step with its definitionStepRecallSummarizeQuestionCommentConnectDefinitionWrite something specific about the lessonShare with a peerWrite something youd like to share about the lessonName main pointsAsk about something you dont understand

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.Exit/Admit SlipsBuilding ComprehensionK-W-FIDEALTPSThink students consider the problem individually

Pair students discuss the problem and solutions in partners

Share students share their findings as a classVocabulary InstructionMarzanos Six StepsNew TermFrayer Model

AcrossA list of things that fit the wordGoes in the center of the chartDownA list of related, but different concepts (with justification)A list of properties that the word hasThe student writes the meaning of the wordCompares multiple termsVisual/SpatialExamines properties of the termProvides examplesExplicitly defines termsMarzanos six steps-+++Frayer model+++Semantic feature analysis+-+--Semantic Feature AnalysisIncorporating WritingRAFTSYou 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 roleA audienceF formatT topicS strong verbWord Problem Roulette

Word Problem RouletteSource: Mathematics teaching in the Middle school Vol. 18, No. 8, April 2013

3-Way Tie3-Way TieA 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. ComparisonWritingSynthesisRelate sectors, arcs, and circlesRelate slope, speed, and ratesRelate perimeter, area, and surface area

Source: Mathematics teaching in the Middle school Vol. 18, No. 8, April 2013 Read.LVWSRepr.Self-concept checkXXRSQC2XAdmit/Exit SlipXKWLXXXTPSXXXXXIDEALXXXXXXStrategies and ModesRead.LVWSRepr.Semantic Feature AnalysisXXMarzanoXXXXFrayer ModelXXRAFTSXWord Problem RouletteXXXXX3-Way TieXXStrategies and ModesSpat.Ling.Kine.Mus.Inter.Intra.Self-concept checkXRSQC2XAdmit/Exit SlipXXKWLXTPSXXIDEALXXXStrategies and MINote: mathematical/logical applies to all, so I omitted the column to save spaceSpat.Ling.Kine.Mus.Inter.Intra.Semantic Feature AnalysisMarzanoXFrayer ModelXRAFTSXXWord Problem RouletteXX3-Way TieXXStrategies and MIEven though kinesthetic intelligence isnt 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.Kinesthetic Learning21st Century TechnologyProduct / WebsiteURLDescriptionGeometers Sketchpadhttp://www.keycurriculum.com/products/sketchpad Interactive software for letting students explore geometry. (Not free).NCTM Congruence Theoremshttp://illuminations.nctm.org/ActivityDetail.aspx?ID=4Helps students discover triangle congruence theoremsKhan Academyhttps://www.khanacademy.org/Videos to provide additional support for students outside the classroomTechnologyProduct / WebsiteURLDescriptionGeogebrahttp://www.geogebra.org/cms/en/Interactive software for letting students explore geometry. Not as easy-to-use as Geometers Skethpad, but free.Geometry PadApp for Android and iPad tabletsPBS Teachershttp://www.pbs.org/teachers/classroom/9-12/math/resources/Activities and ideas for teachers. Some activities are interactiveTechnologyAnnotated BibliographyClassic 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.

Abbott, E. A. (1884). Flatland, a romance of many dimensions.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.Paulos, J. A. (1988). Innumeracy: Mathematical illiteracy and its consequences. New York, NY: Farrar, Straus and Giroux.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 ordera 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.Hogben, L. (1968). Mathematics for the million: How to master the magic of numbers. W. W. Norton & Company.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.

Bruce, C. (2002). Conned again, Watson!. Basic BooksBilly Beane, the Oakland As 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 knowledgeinsights 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.Lewis, M. (2003). Moneyball: The art of winning an unfair game. (1st ed.). New York: W.W. Norton & Company Inc.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.Derbyshire, J. (2003). Prime obsession: Bernhard Riemann and the greatest unsolved problem in mathematics. Washington D.C.: Joseph Henry Press.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.Downing, D. (2003). Algebra, the easy way. Barron's Educational Series.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 Codeis a heartwarming story that will have readers cheering Sarah on. Originally published in England and co-written with her mathematician father, David Flannery,In Codeis "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.Flannery, S. (2001). In code: A mathematical journey. Workman Publishing Company.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 thinkersfrom Pythagoras to Newton to Heisenberg, from the Kabalists to today's astrophysicistswho 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.Seife, C. (2000). Zero: The biography of a dangerous idea. Penguin Books.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 isnt for girls or girls arent good at math.McKellar, D. (2012). Girls Get Curves: Geometry Takes Shape. Hudson Street Press.Math Education Standards in 21st Century WisconsinStandardsWe 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.Students studying geometry were expected toExperiment with transformations in the planeUnderstand congruence in terms of rigid motionsProve geometric theoremsMake geometric constructionsCongruenceKnow 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.Congruence (A)Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.Congruence (B)Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segments endpoints.Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Congruence (C)Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Congruence (D)While learning about the similarity, right triangles, and trigonometry, students were expected to:Understand similarity in terms of similarity transformationsProve theorems involving similarityDefine trigonometric ratios and solve problems involving right trianglesSimilarity, etc.Verify experimentally the properties of dilations given by a center and a scale factor:A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.The dilation of a line segment is longer or shorter in the ratio given by the scale factor.Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Similarity, etc. (A)Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.Similarity, etc. (B)Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.Explain and use the relationship between the sine and cosine of complementary angles.Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Similarity, etc. (C)As with previous topics, students were expected to understand and apply theorems about circles (A). Students were also expected to calculate the arc lengths and areas of sectors (B).CirclesProve that all circles are similar.Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.Circles (A)Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.Circles (B)Educators in the 21st century considered it important that students be able to link geometry with algebra. Students were expected to write equations for conic sections based on the geometric description and vice versa (A). While learning geometry, students were expected to construct geometric proofs. They were also expected to use coordinates and algebra to prove simple theorems (B).Expressing Geometric Properties with EquationsDerive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.Derive the equation of a parabola given a focus and directrix.Expressing Geometric Properties (A)Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2).Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).Find the point on a directed line segment between two given points that partitions the segment in a given ratio.Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.Expressing Geometric Properties (B)In addition to working with two-dimensional objects, high school students were expected to understand some facts about three-dimensional objects, such as volume (A). Students were also expected to understand how two-dimensional and three-dimensional objects are related (B).Geometric Measurement & DimensionGive an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieris principle, and informal limit arguments.Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.Geometric Measurement & Dimension (A)Note: Standard 2 is deliberately omitted65Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.Geometric Measurement & Dimension (B)