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## Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity (2009) Center for Education (CFE)

### Citation Manager

. "6 The Teaching-Learning Paths for Geometry, Spatial Thinking, and Measurement." Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Washington, DC: The National Academies Press, 2009.

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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity

to these attributes and also helps them apply seriation abilities. Teachers should listen carefully to see how they are interpreting and using language (e.g., length as the distance between end points or as “one end sticking out”).

Children should be given a variety of experiences comparing the size of objects. Once they can do so by direct comparison, they should compare several items to a single item, such as finding all the objects in the classroom longer than their forearm. Ideas of transitivity can then be explicitly discussed. Next, children should engage in experiences that allow them to connect number to length. Teachers should provide children with both conventional rulers and manipulative units using standard units of length, such as centimeter cubes (specifically labeled “length-units”; from Dougherty and Slovin, 2004). As they explore with these tools, the ideas of length-unit iteration (e.g., not leaving space between successive length-units), correct alignment (with a ruler), and the zero-point concept can be developed. Having older (or more advanced) children draw, cut out, and use their own rulers can be used to discuss these aspects explicitly.

In all activities, teachers should focus on the meaning that the numerals on the ruler have for children, such as enumerating lengths rather than discrete numbers. In other words, classroom discussions should focus on “What are you counting?” with the answer in length-units. Given that counting discrete items often correctly teaches children that the length-unit size does not matter, teachers should plan experiences and reflections on the nature of properties of the length-unit in various discrete counting and measurement contexts. Comparing results of measuring the same object with manipulatives and with rulers and using manipulative length-units to make their own rulers help children connect their experiences and ideas.

In second or third grade, teachers might introduce the need for standard length-units and the relation between the size and number of length-units. The relationship between the size and number of length-units, the need for standardization of length-units, and additional measuring devices can be explored at this time. The early use of multiple nonstandard length-units would not be used until this point (see Carpenter and Lewis, 1976). Instruction focusing on children’s interpretations of their measuring activity can enable them to use flexible starting points on a ruler to indicate measures successfully (Lubinski and Thiessen, 1996). Without such attention, children are just reading off whatever ruler number aligns with the end of the object into the intermediate grades (Lehrer, Jenkins, and Osana, 1998).

By kindergarten, length is used in other areas, such as understanding addition and graphing. For example, bar graphs use length to represent counts or measures. Kindergartners can answer such questions as “more” and “less,” as well as simple trends, using length of the bars.

Emphasis on children’s solving real measurement problems and, in so

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 Front Matter (R1-R12) Summary (1-4) Part I: Introduction and Research on Learning (5-6) 1: Introduction (7-20) 2 Foundational Mathematics Content (21-58) 3 Cognitive Foundations for Early Mathematics Learning (59-94) 4 Developmental Variation, Sociocultural Influences, and Difficulties in Mathematics (95-120) Part II: Teaching-Learning Paths (121-126) 5 The Teaching-Learning Paths for Number, Relations, and Operations (127-174) 6 The Teaching-Learning Paths for Geometry, Spatial Thinking, and Measurement (175-222) Part III: Contexts for Teaching and Learning (223-224) 7 Standards, Curriculum, Instruction, and Assessment (225-288) 8 The Early Childhood Workforce and Its Professional Development (289-328) Part IV: Future Directions for Policy, Practice, and Research (329-330) 9 Conclusions and Recommendations (331-350) Appendix A: Glossary (351-358) Appendix B: Concepts of Measurement (359-362) Appendix C: Biographical Sketches of Committee Members and Staff (363-370) Index (371-386)