Published on August 7, 2007
Measurement and Geometry: Building Conceptual Understanding in Young Children: Measurement and Geometry: Building Conceptual Understanding in Young Children Presenter: Cindy Stone Goals of Today’s Session: Goals of Today’s Session Understand what the research says Understand the big mathematical ideas in the activities and why we teach what we teach Form a network of colleagues and a support system to continue growth in mathematical content knowledge Research: Research Students need to 'mentally structure and revise their construction of space….' National Research Council, 2001 We are not doing an adequate job of teaching measurement; perimeter, area and volume, and geometry. Schmidt, Houang, andamp; Cogan, 2002 As students study increasingly sophisticated mathematics and science, those who are able to construct and analyze physical and mental models will be at an advantage. Consider the use of spatial reasoning in the fields of computer networking, communications technology, architecture, and the development of CAD. Mokros, Russell andamp; Economopoulos, 1995 Research: Research Modeling, generalization, and justification are not learned in the abstract…these practices are embedded in the process of learning important mathematics and science ideas. Carpenter andamp; Romberg, 2004 Approaching mathematics content through investigations helps students develop flexibility and confidence in approaching problems, fluency in using mathematical skills and tools, and proficiency in evaluating their solutions. Kliman andamp; Russell, 1998 Assessment – Not a Snapshot, a Photo Album : Assessment – Not a Snapshot, a Photo Album All concepts develop over time; assessment should show development from naïve to sophisticated understanding How will I know when my students know what I want them to know? What will constitute acceptable evidence? How can I grade a hands-on task? Multiple tasks for assessment – the key word is BALANCE Concrete Semi-concrete Abstract Rubrics: Rubrics A rubric is a scoring guide for assessing student work The rubric describes the criteria for each performance level Rubrics are measuring tools that are usually tailor-made by the person doing the assessing Rubrics can be either holistic or analytic in nature The holistic rubric focuses on the entire response; the assessor evaluates the work as a whole The analytic rubric is more like a checklist and looks at specific aspects of the response Mathematical Tasks: Mathematical Tasks Tasks should connect to prior lessons and real world contexts Tasks should be open enough to provide a challenge to gifted children while allowing for success for at-risk students Tasks should be based on standards, not what the teacher likes to teach Questioning in the Mathematics Classroom: Questioning in the Mathematics Classroom Teacher questioning: Can you draw, explain, or write about your thinking and your solution? Will your solution always work? Will your solution work only in some circumstances? Student questioning: Students should be taught to question each other Students should be provided opportunities to say they don’t understand or to ask for clarification What’s in a Bucket?: What’s in a Bucket? Task 1: Use pieces of yarn or any other tool in the bucket to measure the height, diameter, and circumference of the bucket Record your results Task 2 Use index cards, Post-it® notes, or paper to cover the outside surface of the bucket Record your results What’s in a Bucket?: What’s in a Bucket? Task 3 Hang the bucket from the spring scale. Fill the bucket with 10 items, 20 items, 30 items, and on up to 100 at 10-item increments. Record the weight at each interval. Discuss and record your results Task 4 Fill the bucket with water, sand , or rice. How many cups did it hold? Record your results What’s in a Bucket?: What’s in a Bucket? Class Discussion Were the measurements the same? Why or why not? Explain your thinking in your Mathematics Journal. At what age would these activities be appropriate? Why non-standard units of measure? What standards do they address? What are the mathematical concepts these activities are developing? Quick Images (from Investigations in Number, Data, Space): Quick Images (from Investigations in Number, Data, Space) Task 1 You will be shown a geometric shape on the overhead for a few seconds. You will then draw the shape on your paper. These shapes will use 3 – 10 dots in various formations. Task 2 You will be shown a geometric shape on the overhead again and this time try to replicate it with interlocking cubes. Class Discussion: Class Discussion What concept do these two activities develop? Why do we ask students to share what they saw? Didn’t they all see the same thing? How could this be expanded upon as students master these activities? What Were the Mathematical Concepts?: What Were the Mathematical Concepts? Spatial reasoning Relationships between units of measure and length, diameter, circumference of a container, surface area, and volume State Standards Addressed: State Standards Addressed Kinder Spatial reasoning Objects are alike and different Describe and identify an object by its attributes using informal language Compare two objects based on their attributes Describe and compare real-life objects or models of solids (10) Measurement. Compare and order two or three concrete objects according to length (shorter or longer), capacity (holds more or holds less), or weight (lighter or heavier) State Standards (cont.): State Standards (cont.) 1st grade Geometry and spatial reasoning Describe and identify objects in order to sort them according to a given attribute using informal language Estimate and measure length, capacity, and weight of objects using nonstandard units State Standards (cont.): State Standards (cont.) 2nd grade Geometry and spatial reasoning Identify attributes of any shape or solid The student recognizes and uses models that approximate standard units (metric and customary) of length, weight, and capacity References: References Carpenter, Thomas, andamp; Romberg, Thomas, (2004). Powerful Practices. University of Wisconsin. Madison. Kliman, Marlene, and Russell, Susan Jo, (1998). Building Number Sense. TERC. Cambridge. Mokros, Jan, Russell, Susan Jo, andamp; Economopoulos, (1995). Beyond Arithmetic. Dale Seymour. Palo Alto. National Research Council, (2001). Adding It Up. National Academy Press. Washington, D.C. Schmidt, William, Houang, Richard, and Cogan, Leland, (2002). A Coherent Curriculum: The Case of Mathematics. Research for Better Schools. Philadelphia.