1. M EASUREMENT Jonathan L. Brendefur, Ph.D. Sam Strother, MA.E. Boise State University

2. M EASUREMENT S UMMARY

3. W HY MEASUREMENT TASKS MATTER Build working memory capacity Develop spatial reasoning skills Provide foundational concepts of rational numbers, particularly fractions and decimals Build multiplicative reasoning skills Increase computational fluency Explain, model, and justify a variety of mathematical concepts Battista 2006; Clements, 1999; NCTM, 2000; Stephan & Clements, 2003, Van de Walle, 2006

4. C HALLENGES Measurement is an incredibly complex and varied topic Misleading culturally-influenced vocabulary Development is a strong factor Low-SES students are at a disadvantage due to lack of experience Most textbooks are not aligned to research Lack of conceptual focus in teacher preparation Ball & Fieman-Nemser, 1988; Battista, 2006; Clements, 1999; Lehrer, 2003

5. W HAT IS MEASUREMENT ? Comparing an object’s attribute to a unit Other views… Describing space in terms of quantity and quality Assigning order to physical and theoretical environments Imagine qualities of the world Freudenthal, 1975; Lehrer, 2003; Lehrer & Chazan, 1998; NCTM, 2000

6. B IG I DEAS OF M EASUREMENT Units Zero Transitivity Conservation

7. U NITS Iteration - consistently repeating a unit with no gaps or overlaps to create a whole Partitioning - ’splitting’ a whole into consistently sized units of measure Equivalence - consistency and precision Unit-based reasoning - What unit should be used? What attribute is to be measured? What will my measurement tell me? Proportional relationships (e.g. 1 foot is 12 inches and 24 half-inches) Quantitative Reasoning* Accumulation - the number of units used constitute the ‘measure’ (related to the Cardinal Principle of counting) IteratingPartitioning

8. Z ERO Origin -‘Zero’ units of measure indicates the starting point for the measurement. Zero is arbitrary and can ‘begin’ at any place on the object. (also referred to as the zero-point ) Absolute value -distance from zero (most common meaning) 2 = -2 0 -22 Lehrer, 2003

9. W HEN STUDENTS DON ’ T UNDERSTAND ZERO …. Kamii, 2006

10. 3 12 7 6 6 ? 7 7 T RANSITIVITY Comparisons - using known measurements to find unknown measurements Quantitative* - using a comparison of an attribute to a unit. The number of units is used to solve the problem. (“This pencil is 4 units long”) Qualitative - making general comparisons between objects’ common attribute to solve a problem (“This pencil is the same as this side of the rectangle.”) Battista, 2006; Lehrer, 2003 * If a=b and b=c, then a=c (One view of transitivity ) The transitive property is about the relationships between a, b, and c…not necessarily relationships of equality.

11. C ONSERVATION Measurements can stay the same even when the object is moved, decomposed, or rearranged. 4 L 2 L

12. C OMMON A PPROACHES F OUND IN T EXTBOOKS TraditionalReformWhat research says… Length Show students how to use a ruler Combine various standard units in measurements (e.g. feet and inches) Use non-standard, informal units; create the ‘need’ for standard units 1.Comparisons 2.Non-standard units 3.Standard manipulatives 4.Standard units 1.Comparisons w/justification 2.Numerical measures (both standard and non- standard). By 2 nd and 3 rd grade-necessity for standard units 3.Change informal tools to symbols and units of measure (e.g. paces to foot strips) Build common reference points Estimation and spatial reasoning (construction of rulers) Area Show students the formula Practice computing the answer Teach perimeter and area simultaneously ‘Tile’ the figure with square units or non- standard units “Cover the figure and count the squares.” Begin informally Emphasize the construction of arrays --Students must ‘see’ columns and rows and iterate these as unit groups Reconcile the issue of ‘double-counting’ corners of rectangles Why square units? Teach perimeter as ‘surrounding’ and area as ‘covering’---teach separately Volume Show students the formula Practice computing the answer 2d representations of 3d objects Build figures with cubes Count the cubes Structure objects as ‘layers of arrays’ Discuss, reason, justify, and model (3d to 2d) Nets and spatial reasoning Filling space and filling objects (not just capacity) Angle Labeling vocabulary (e.g. acute, obtuse, and right) Use a protractor to measure and compare angles Comparison Informal comparisons leading to formal vocabulary Angle as movement, shape, and measure Sweep and bending Portion of a circle Battista, 1998; Kamii, 2006; Lehrer, 2003;Outhred & Mitchelmoore 2000; Simon, 1995; Stephan & Clements, 2003

13. E FFECTIVE INSTRUCTIONAL TOOLS AND TECHNIQUES Broken rulers Hand-made tools (e.g. rulers, protractors, tiles) Graph paper Realistic contexts (e.g. string for art projects, painting a wall, filling a bucket, building a model, navigation) Modeling, discussion, justification Reference point estimation (e.g. partitioning) Initially--avoid conversions within the same measurement task (e.g. 1 ½ ft. as opposed to 1 ft. 6 inches) Ratio tables (conversions) Battista, 1998; Kamii, 2006; Lehrer, 2003;Lamon, 2005; Outhred & Mitchelmoore 2000; Simon, 1995; Stephan & Clements, 2003; Van de Walle, 2006