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• Index 371-386

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Appendix A Glossary Accumulator mechanism refers to the nonverbal counting mechanism of infants that generates mental magnitudes for sets by adding a fixed magnitude for each unit that is enumerated. This system is inherently inexact, and its inexactness increases with increasing number. It pro- vides an approximate numerical representation that does not preserve any representation of the items. Hence, it does not provide a way to distinguish successive numbers, such as 10 and 11. Additive comparison situations are those in which two quantities are com- pared to find out how much more or how much less one is than the other. Analog magnitude system refers to approximate representations of large numbers beginning with toddler and preschool-age children. Attribute blocks refer to collections of blocks in which attributes (e.g., color, shape, size, thickness) are systematically varied so that children can sort them in multiple ways. Cardinality refers to the number of items in the set. Change plus/change minus situations refer to addition and subtraction situations in which there are three quantitative steps over time, a start quantity, a change, and a result. Change plus situations can be formu- lated with an equation of the form start quantity + change quantity = result quantity. Change minus situations can be formulated with an equation of the form start quantity − change quantity = result quantity. 351

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352 MATHEMATICS LEARNING IN EARLY CHILDHOOD Child-guided experiences refer to experiences in which children acquire knowledge and skills through their own exploration and through inter- actions with objects and with peers. Composing/decomposing refers to putting together and taking apart and applies to numbers as well as to geometry and measurement. For ex- ample, 10 ones are composed to form one group of 10 and 6 can be decomposed into 5 + 1. Two identical right triangles can be composed to form a rectangle, and a hexagon can be decomposed into six trian- gles. Measurement itself requires viewing the attribute to be measured as composed of units. Computational fluency refers to accurate, efficient, and flexible computa- tion with basic operations. Credentialing refers to the process of demonstrating and receiving formal recognition from an organization for achieving a pre-defined level of expertise in education. Direct instruction refers to situations in which teachers give information or present content directly to children. Early childhood education (ECE) teachers refer to all personnel whose pri- mary role is to provide direct instructional services for young children. Included in this category are lead teachers, assistant teachers, aides, and family child care providers. ECE teaching workforce refers to those who carry out both instructional and noninstructional roles in ECE settings. The term is an inclusive one that embraces teachers, others who work in the ECE settings and whose primary responsibility is not instructional (e.g., administrators), and individuals who work in settings that support ECE (e.g., resource and referral coordinators). Encouragement and affirmation refers to feedback that relates to teachers’ abilities to motivate children to sustain their efforts and engagement. Explicit instruction refers to all of a teachers’ instructional actions and interactions that are not unplanned or incidental. Feedback loops refer to sustained exchanges between a teacher and child (or group of children) that leads the child to a better or deeper under- standing of a particular idea. Finding a pattern refers to looking for structures and organizing and clas- sifying information. It is a mathematical process used throughout mathematics. Focused curriculum (primary mathematics) refers to a curriculum that is designed and has the primary goal to teach mathematics with meaning- ful connections to children’s interest and prior knowledge.

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APPENDIX A 353 Formal education refers to the amount of credit-bearing coursework a teacher has completed at an accredited institution, including two- or four-year colleges and universities. Formative assessment refers to the process of gaining insight into children’s learning and thinking in the classroom and of using that information to guide instruction. It entails the use of several methods—observation, task, and flexible interview—that help the teacher develop ideas about children’s thinking and learning and about teaching methods that can help them learn. Formative assessment is often inseparable from teach- ing and usually not distinctly identified as assessment, but formative assessment can also be used in a deliberate and organized format. Geometry refers to the study of shapes and space, including flat, two- d ­ imensional space as well as three-dimensional space. In-service education refers to the formal education and training that one may receive while having formal responsibility for a group of children. Instruction/pedagogy refers to intentional teaching. Instructional feedback refers to a response where the teacher provides stu- dents with specific information about the content or process of learning and provides the opportunity to practice and master knowledge and skill. Instructional supports refer to concept development, quality of feedback, and language modeling. Integration refers to the blending together of two or more content areas in one activity or learning experience with the purpose of making content meaningful and accessible but also allowing more content to be covered during the instructional period. Intentional teaching refers to holding a clear learning target as a goal and adapting teaching to the content and type of learning experience for the individual child, along with the use of formative assessment to de- termine the child’s development in relation to the goal. Language modeling refers to a practice by adults when they converse with children, ask open-ended questions, repeat or extend children’s re- sponses, and use a variety of words, including more advanced language and building on words the children already know. Manipulatives refer to concrete objects—including blocks, geometric shapes, and items for counting—to support children’s mathematical thinking. Mathematics teaching-learning path refers to the significant steps in learn- ing a particular mathematical topic with each new step building on the earlier steps. Teaching-learning paths are often referred to as learning

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354 MATHEMATICS LEARNING IN EARLY CHILDHOOD trajectories, a term that emphasizes the sequential and direct route from one skill level to the next. The sources of a teaching-learning path are: (1) the subject matter being taught—what skills and knowledge provide the foundation for later learning, and (2) what is achievable/­ understandable for children at a certain age given their prior knowl- edge. Teaching-learning paths also provide a basis for targeting the curriculum, assessing children’s progress along the path, and adapting their instruction to help children make continued progress. Mathematizing refers to reinventing, redescribing, reorganizing, quantify- ing, structuring, abstracting, and generalizing concepts and situations first understood on an intuitive and informal level in the context of ev- eryday activity into mathematical terms. This process allows children to create models of situations using mathematical objects or actions and their relationships to solve problems, including the use of increasingly abstract representations. Measurement refers to the process of determining the size of an object with respect to a chosen attribute (such as length, area, or volume) and a chosen unit of measure (such as an inch, a square foot, or a gallon). Morphological marker refers to the word element that signifies quantity, such as whether the word is singular or plural. For example, the s on the end of dogs, which indicates that the word is plural, is the morpho- logical marker. The term quantifier morphology is used interchangeably with morphological marker. Number competencies refer equally to both the knowledge and skills con- cerning number and operations that can be taught and learned. Number sense refers to the interconnected knowledge of numbers and op- erations. It is a combination of early preverbal number sense and the increasingly important influence of experience and instruction. Numeral refers to the symbol used to represent a number. Numerosity refers to the quantity of a set. Object file system refers to the representation of each object in a set com- prised of very small numbers, but no representation of set size. For this form of representation, the objects in a small set are in 1-to-1 corre- spondence with each mental symbol. Thus, a set of three items is rep- resented as “this” “this” “this” rather than “a set of three things.” One-to-one (1-to-1) correspondence refers to correspondence between two collections if every member of each collection is paired with exactly one member of the other collection and no members of either collection is unpaired or is paired with more than one member.

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APPENDIX A 355 Place value refers to the meaning of a digit in a written number as deter- mined by its placement within the number. Pre-service education refers to the formal education and training that one receives prior to having formal responsibility for a group of children. Primary mathematics/focused mathematics time refers to a dedicated in- structional time focused on mathematics as the primary goal. Professional development is an umbrella term including both formal educa- tion and training. Prompting thought processes refers to a particular feedback strategy for mathematics instruction that asks students to explain their thinking or actions. Providing information refers to clarifying incorrect answers or providing very specific information about the correct answer. Put together situations refer to addition/subtraction situations in which two quantities are put together to make a third quantity. Relating and ordering refers to mathematical processes of comparing and placing in order. Relating parts and wholes level refers to a level of thinking that occurs when children combine pattern block shapes to make composites that they recognize as new shapes and to fill puzzles, with growing inten- tionality and anticipation. Scaffolding refers to an instructional strategy in which the teacher provides information and assistance that allow children to perform at a higher level than they might be able to do on their own. It extends knowledge rather than verifying prior or existing knowledge. Secondary (embedded) mathematics refers to a form of integration through which teaching and exposure to mathematics content is an ancillary activity. One or more subjects other than mathematics, such as literacy or science, are the primary goals of the activity. Spatial orientation refers to knowing where one is and how to get around in the world. Children have cognitive systems that are based on their own position and their movements through space, as well as external references. They can learn to represent spatial relations and movement through space using both of these systems, eventually mathematizing their knowledge. Spatial visualization/imagery refers to the process that occurs when there is understanding and performing imagined movements of two-dimen- sional and three-dimensional objects. To do this requires creating a mental image and manipulating it, showing the close relationship be- tween these two cognitive abilities.

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356 MATHEMATICS LEARNING IN EARLY CHILDHOOD Subitizing is the process of recognizing and naming the number of objects in a set. Conceptual subitizing refers to using pattern recognition to quickly determine the number of objects in a set, such as seeing 2 things and 2 things and knowing this makes 4 things in all. Perceptual subitizing refers to instantly recognizing and naming the number of objects in a set. Superposition is the act of placing one item on top of another. Take apart situations refer to addition/subtraction situations in which a total quantity is taken apart to make two quantities (which do not have to be equal). These situations generally have several solutions. For example: Joey has 5 marbles to put in his 2 pockets. How many can he put in his left pocket and how many in his right pocket? Tangram is a puzzle consisting of seven flat shapes, called tans, which are put together in different ways to form distinct geometric shapes. Teacher effectiveness refers to the impact of teachers’ actions and behaviors on the accomplishments and/or learning outcomes of the children they teach. Teacher-guided instruction refers to teachers’ planning and implement- ing experiences in which they provide explicit information, model or demonstrate skills, and use other teaching strategies in which they take the lead. Teacher-initiated learning experiences refer to classroom experiences that are determined by the teacher’s goals and direction, but ideally also reflect children’s active engagement. Teacher quality refers to the positive actions and behaviors of teachers, particularly with regard to their interactions with young children. Thinking about parts level refers to a level of thinking that occurs when preschoolers can place shapes contiguously to form pictures in which several shapes play a single role (e.g., a leg might be created from three contiguous squares) but use trial and error and do not anticipate cre- ation of new geometric shapes. Training refers to the educational activities that take place outside of the formal education process. Such efforts may include coaching, mentor- ing, or workshops. Unitizing refers to finding or creating a mathematical unit as it occurs in numerical, geometric, and spatial contexts. Virtual manipulatives refer to manipulatives accessed through learning soft- ware and composed of digital “objects” that resemble physical objects and can be manipulated, usually with a mouse, in the same ways as

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APPENDIX A 357 their authentic counterparts. Virtual versions of concrete manipulatives typically used in mathematics education include base 10 blocks, Cui- senaire rods, and tangrams. Many available virtual manipulatives are paired with structured activities or suggestions to aid implementation in the classroom. Visual/holistic level refers to a level of thinking that occurs when children have formed schemes, or mental “patterns,” for these shape catego- ries. It refers to the ability of preschoolers to learn to recognize a wide variety of shapes, including shapes that are different sizes and are presented at different orientations. They also learn to name common three-­dimensional shapes informally and with mathematical names (“ball”/sphere, “box” or rectangular prism, “rectangular block” or “triangular block,” “can”/cylinder). They name and describe these shapes, first using their own descriptions and increasingly adopting mathematical language.

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