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Mathematics: What’s the Big Idea?
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- Teaching Math: A Video Library, K-4
Mathematics: What’s the Big Idea?A video workshop on teaching mathematics for K-8 teachers; 8 ninety-minute video programs, workshop guide, and Web site; graduate credit available What do quilts have to do with palaces? When is a third more than a half? K-8 teachers of mathematics will contemplate these and other provocative questions in the workshop, which offers motivation and tools for teachers who want to explore new ways of teaching math. Using a variety of models, activities, and video clips, this workshop encourages participants to reflect upon their own practice and discuss ideas for innovation in teaching. Produced by the Harvard-Smithsonian Center for Astrophysics and Massachusetts Corporation for Educational Telecommunications (MCET). 1997. ISBN: 1-57680-131-4 Individual Program Descriptions1. Workshop 1. Patterns and Functions: What Comes Next? (90 min.) Mathematics is about patterns waiting to be found. This workshop demonstrates how students' explorations of patterns can grow richer and more complex as they move through the grades. 2. Workshop 2. Data: Posing Questions and Finding Answers (90 min.) From the earliest grades, students learn to connect situations, data, and graphs. This workshops shows data displays that can be developed through the grades. 3. Workshop 3. Geometry: Castles and Shadows (90 min.) Shadows give two-dimensional representation to three-dimensional objects. Teachers discuss the intriguing relationship between two- and three-dimensional objects that are at the heart of geometry. 4. Workshop 4. More Geometry: Quilts and Palaces (90 min.) Geometry appears in works of art, architectural wonders, and physical structures. This workshop explores geometrical figures, transformation, and connections to art and science. 5. Workshop 5. Whole Numbers: Memory and Discovery (90 min.) What does it take to develop fluency with whole number calculations? This workshop compares algorithms and explores mental math strategies. 6. Workshop 6. Ratio and Proportion: When Is a Third More Than a Half? (90 min.) Helps identify students' misconceptions about fractions that hinder their understanding of later concepts. Program participants work with rational numbers and activities delaing with ratio, proportions, and equivalent fractions. 7. Workshop 7. Algebra: It Begins in Kindergarten (90 min.) This workshop traces the fundamental concepts of algebra that students can develop through the grades. 8. Workshop 8. The Future of Mathematics: Ferns and Galaxies (90 min.) The advent of new technologies allows for amazing mathematics that could not exist without computers. This program looks to future directions for mathematics in the 21st Century. Teaching Math: A Video Library, K-4A video library for K-4 math teachers; 52 video programs (from 15 to 30 minutes in length) and guide See how the National Council of Teachers of Mathematics standards are used in elementary classrooms across America. Elementary teachers tap the excitement and energy of children from kindergarten through grade 4 as they solve problems, learn to make connections between concepts, and communicate and reason mathematically. Teaching Math K-4 documents effective teaching and learning in many schools: small, large, rural, suburban, and inner-city. Produced by WGBH Boston. 1997. ISBN: 1-55946-896-3 Individual Program Descriptions1. Introduction This overview shows the different parts of the library and their uses. 2. Ants Go Marching Kindergarteners develop number sense by exploring number concepts and number relationships. The number sequence is emphasized and the distinction is made between cardinal and ordinal meanings for number. NCTM standards: number sense and numeration, patterns and relationships, communication, connections 3. Math Buddies Kindergarteners matched with sixth-grade coaches experience the numbers 1 through 50 through various activities involving hands-on experiences and common objects. NCTM standards: number sense and numeration, communication, connections. 4. Place-Value Centers First-graders develop an understanding of the numeration system by relating counting, grouping, and place-value concepts. Activities include measuring with Unifix cubes and using base-ten blocks. NCTM standards: number sense and numeration, measurement, connections. 5. Pumpkin Seeds A grade 1-2 class, working in groups, develops their sense of larger numbers by estimating and counting the number of seeds in pumpkins. NCTM standards: number sense and numeration, estimation, communication, reasoning. 6. Animals in Yellowstone Fourth- and fifth-graders develop number sense and meaning for large numbers by estimating how many bison, elk, and pronghorn they saw on a field trip to Yellowstone National Park. Students debate and justify their estimates verbally and in writing. NCTM standards: number sense and numeration, estimation, problem solving, connections. 7. Cubes and Containers Kindergarteners sort Unifix cubes in various ways, focusing on the properties of the objects’ similarities and differences. By creating patterns, children develop an early understanding of geometry. NCTM standards: concepts of whole number operations, number sense and numeration, communication, reasoning. 8. Amazing Equations Using the day of the month, first- and second-graders investigate the concepts of addition and subtraction as they share story problems that relate to the date. Teams use everyday language and experiences to connect to the mathematical language and symbolism for operations. NCTM standards: concepts of whole number operations, whole number computation, communication, connections. 9. Domino Math First- and second-graders investigate number relationships and explore the concept of addition in a part-whole model using dominoes. They develop mathematical communication as they represent mathematical ideas with physical materials, words, diagrams, and symbols. NCTM standards: concepts of whole number operations, whole number computation, problem solving, communication. 10. Marshmallows Second-graders create and discuss a bar graph based on the number of marshmallows they estimate each person in their class would eat on a camping trip. After discussing their results, students determine how many bags of marshmallows to take. NCTM standards: concepts of whole number operations, statistics and probability, reasoning, problem solving. 11. What’s the Price? Third-graders use problem-solving approaches--such as role playing or drawing pictures—to investigate and understand division. They make connections to everyday life and use calculators as they determine unit costs for two different boxes of cereal. NCTM Standards: concepts of whole number operation, fractions and decimals, problem solving, communication. 12. Dino Math Kindergarteners team up in buddy pairs to explore addition combinations using a "dinosaur math" mat. They work with basic addition facts and use a part-whole concept to perform the addition. NCTM standards: whole number computation, number sense and numeration, communication. 13. Window Puzzle First-graders investigate number combinations by working with a window puzzle (a square divided into four equal squares). In this task students explore both addition and subtraction employing physical materials to aid in computation as well as using mental math. NCTM standards: whole number computation, communication, reasoning. 14. Wheel Problem First-graders are asked how many vehicles could be in a parking lot if the total number of wheels is 24. Students review each step of the problem-solving process before they decide on which materials to use and develop strategies. NCTM standards: whole number computation, concepts of whole number operations, problem solving, communication. 15. Bean Sprouts Pairs of second-graders explore subtraction, based on the number of plants sprouting from the bean-seeds they have planted. The importance of context for student understanding, and various approaches to problem-solving, are exemplified. NCTM Standards: concepts of whole number operations, reasoning, problem solving, communication. 16. This Small House Second- and third-graders use calculators, paper and pencil, and mental math within a realistic task. Students plan the decorating of their milk carton houses using spatial sense to select appropriate furnishings while staying within their allocated budget. NCTM standards: whole number computation, geometry and spacial sense, connections, communication. 17. Choose a Method A fourth-grade class shares their reasoning in evaluating the appropriateness of different computational methods (base-ten blocks, calculators, mental math, or paper and pencil) to specific problems. NCTM standards: whole number computation, estimation, communication, reasoning. 18. Thanksgiving Quilt Creating quilt squares from construction paper, first graders develop spatial sense as they discuss and handle different shapes. They connect geometric ideas to number ideas as they cut squares into congruent triangles. NCTM standards: geometry and spacial sense, patterns and relationships, communication, connections. 19. Pattern Blocks Second-graders learn the mathematical terms for pattern-block pieces: hexagon, trapezoid, square, triangle, and rhombus. Ideas about fractions emerge as students spot size relationships between shapes. NCTM standards: geometry and spacial sense, number sense and numeration, reasoning, connections. 20. Shapes From Squares A second/third-grade class develops spatial sense as they subdivide and change squares to create different shapes. The language of geometry—square, trapezoid, hexagon, etc.—grows naturally from their explorations. NCTM standards: geometry and spacial sense, communication, reasoning. 21. A Rocket Shape Second- and third-graders experiment to subdivide a square to recreate a rocket shape. After completing their rockets, they reconvene as a class to discuss their difficulties and problem solving strategies. NCTM standards: geometry and spacial sense, measurement, problem solving, reasoning. 22. Circumference/Diameter After reviewing the meaning of radius, diameter, center, and circumference, fourth-graders working in teams measure circular objects throughout the room. They are then challenged to find the relationship between the circumference and the diameter. NCTM standards: geometry and spatial sense, measurement, connections, reasoning. 23. Windows, Dinos, and Ants First-graders are engaged in problem solving and measuring with both standard and nonstandard units. Students work in groups to measure three different distances: ant farm tunnels, dinosaurs, and the length from their classroom window to the playground below. NCTM standards: measurement, number sense and numeration, problem solving, reasoning. 24. How Long Is a Minute? Having already studied the concept of an hour, first-graders investigate time as a measure of duration. They list activities, such as writing their name, that could be accomplished within a minute, then estimate how many times in one minute they could do it. NCTM standards: measurement, estimation, reasoning, connections. 25. Balloon Travel In an integrated math/science lesson, second- and third-graders collect data to answer questions such as, "What is the farthest a balloon can travel before falling?" To answer the question, they must understand distance, volume, capacity, and time. NCTM standards: measurement, estimation, connection, problem solving. 26. Meter Cords Third- and fourth-graders use linear measurement in learning about decimals. Students measure different items with a meter divided into 10 parts then learn to write their measurements using decimal notation. NCTM standards: measurement, fractions and decimals, connections, communications. 27. Pencil Box Staining Fourth-graders are faced with the task of finding out how much stain to buy from the hardware store and encounter problems as they work with many mathematical ideas in the context of a real application. Students work in groups with pencil box pieces, a ruler, calculator, and instruction sheet. NCTM standards: measurement, fractions and decimals, problem solving, reasoning. 28. Ladybugs First-graders choose ladybugs as a topic for learning. Based on their observations, students make bar graphs and a class chart to record the number of heads, wings, feet and antennas ladybugs have. They make connections among real objects, diagrams, and numerals. NCTM standards: statistics and probability, number sense and numeration, connections, communication. 29. Woodpecker Habitat First- and second-graders apply probability and sampling techniques to their study of the habitat of the endangered red-cockaded woodpecker. Using colored cubes to represent elements in the environment, students simulate factors that might harm or help the birds. NCTM standards: statistics and probability, whole number computation, reasoning, connections. 30. Bubble Gum Contest Third-graders stage a bubble gum blowing contest using sampling to determine the ratio of winners to entrants. They enlarge their sample, collecting data from all the third-graders in their school and use fractions to interpret the data. NCTM standards: statistics and probability, fractions and decimals, connections, communication. 31. Dice Toss Fourth-graders work with statistics, probability, fractions and decimals while conducting an experiment to see which sum comes up most often when rolling two dice. Once the groups complete their experiments, they compile their findings on a class bar graph and analyze the graph. NCTM standards: statistics and probability, fractions and decimals, communication, reasoning. 32. Questioning Data A fourth- through sixth-grade class takes data collected from surveys on questions of personal interest. They then represent the data in a graph, and write about what the graph interprets and the questions they still have about the survey subject. NCTM standards: statistics and probability, connections, communication, reasoning. 33. Fraction Strips First- and second-graders make fraction pieces from paper strips and play a game that involves covering a whole strip with fractional pieces. As they play they informally add fractions and make connections from objects and actions to symbols. NCTM standards: fractions and decimals, number sense and numeration, reasoning, communication. 34. Arrays and Fractions A first- through third-grade class investigates fractional parts of a set by building arrays that represent wholes of different sizes. In their task they use mathematical language and symbols and form mathematical connections among concepts of addition, area, multiplications, division, and fractions. NCTM standards: fractions and decimals, number sense and numeration, reasoning, communication. 35. Everyday Decimals Second- and third-graders extend their understanding of common fractions to notation for decimal fractions and to the numeration system. They interpret the use of decimals in the real world by bringing to class items that have decimals or fractions written on them. NCTM standards: fractions and decimals, number sense and numeration, connections. 36. Cookies To Share Through a story about sharing cookies, fourth-graders investigate a the problem of dividing eight cookies among 12 children. It helps them develop meaning for the concept of division and leads to the use of fractions. NCTM standards: fractions and decimals, number sense and numeration, communication, connections. 37. Fractions With Geoboards Fourth- and fifth-graders investigate the concept of halves using the geoboard as an area model. They learn that one-half means two equal-sized parts with equal areas, but that are not necessarily congurent. NCTM standards: fractions and decimals, geometry and spacial sense, reasoning, problem solving. 38. People Patterns One of several lessons on patterns, individual kindergarteners are lined up to represent different patterns to the class. In groups they create their own patterns from simple two-element patterns or more complex six-element patterns to share with the class. NCTM Standards: patterns and relationships, number sense and numeration, communication, reasoning. 39. All Sorts of Buttons Kindergarteners and first-graders hear a story about buttons, then sort their own collection of buttons to develop skills of classification--observing likenesses and differences. Students see that objects can be looked at in a number of ways and develop a sense of pattern and regularity. NCTM standards: patterns and relationships, number sense and numeration, reasoning, connections. 40. Story-Based Centers Second-graders work at learning centers around their classroom that are based on the story "Caps for Sale." At the centers students construct graphs, figure out coin combinations to equal 50 cents, use a computer to explore patterns, and write story problems and number sentences. NCTM standards: patterns and relationships, whole number computation, connections, communications. 41. Products and Sums Exploring relationships between addition and multiplication, fourth-graders represent numbers, sums, and products with symbols and with geometric patterns on paper. NCTM standards: patterns and relationships, number sense and numeration, reasoning, communication. 42. Valentine Exchange A bilingual fourth-grade class uses a Valentine’s Day card exchange problem to explore mathematical relationships and share problem solving strategies. NCTM standards: patterns and relationships, number sense and numeration, problem solving, reasoning. 43. Beans, Beans, Beans Kindergarteners sharpen their estimation skills and number sense as part of a unit on plants and seeds. Using three known quantities for reference, student groups must estimate the number of an unknown quantity of beans, and then count the beans by tens. NCTM standards: estimation, number sense and numeration, reasoning, communication. 44. How Many People Will Fit? First-graders investigate the concept of area by figuring out how many people will fit in four different areas in the school building. They organize and record solutions and use measuring, counting, addition to find the total number of people. NCTM standards: estimation, measurement, reasoning, problem solving. 45. Cranberry Estimation Second-graders in Massachusetts estimate the number of scoops of cranberries that will fit in a jar. They report, graph, and discuss group estimates with the class as the concepts of range, mode, and median emerge. NCTM standards: estimation, statistics and probability, reasoning, connections. 46. Buffalo Estimation A third-grade class that has been studying the Oregon Trail explores estimation by figuring out how many buffalo would fit in their classroom and on their playground. Students develop number sense as they think about the reasonableness of estimates and use a referent to adjust their estimates. NCTM standards: estimation, number sense and numeration, reasoning, connections. 47. The White Pages This lesson helps fourth-graders develop number sense by exploring magnitudes of large numbers (the number of listings in the phone book) and reasonableness of estimates. Students mark their estimates on a number line and justify their estimates verbally and in writing. NCTM standards: estimation, number sense and numeration, problem solving, connections. 48. Problem Solving The half-hour program includes 13 classroom excerpts from the content standards lessons which illustrate students investigating and learning mathematics through problem solving. Teachers share their approaches and observations. 49. Communication The half-hour program includes 18 classroom excerpts from the content standards lessons which show students representing, discussing, reading, writing, and listening as vital parts of learning and using mathematics. It shows how communication fosters an understanding of mathematical ideas and the language of mathematics. 50. Reasoning The half-hour program includes 16 classroom excerpts from the content standards lessons which illustrate the central role of reasoning in mathematics. As students explain and justify their thinking and solutions throughout the excerpts, teachers emphasize that how a problem is solved is as important as its answer. 51. Mathematical Connections The half-hour program includes more than a dozen classroom excerpts from the content standards lessons that illustrate mathematical connections. Connections are made among different topics in mathematics, to other curriculum areas, and to students' daily lives. 52. Classrooms Over Time: Problem Solvers Fall and Spring 53. Classrooms Over Time: Long Term Projects Long Term Projects - two segments with Steven Levy's class at the Bowman School, Lexington, MA Segment 1 - Pencil Box Staining Segment 2 - Bike Path Usage. 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