Move the cursor to the right cell.THE MATHEMATICAL KNOWLEDGE CHILDREN BRING TO SCHOOLFor example, the sum of J first whole numbers can be represented in the following. With the cursor in the left cell, press Alt+ to insert an equation container, but leave it empty (with the placeholder text showing). Drag the column edge from the middle of the table over to the right, making it just a bit wider than needed for equation numbers. Press Ctrl+Alt+U to remove its borders.

If your app isnt full screen, you may need to select Symbol again before More Symbols. Go to Insert > Symbol > More Symbols. On the slide, click where you want to add the symbol.

The state of children’s mathematical development as they begin school both determines what they must learn to achieve mathematical proficiency and points toward how that proficiency can be acquired.In which I run through a useful tip that Microsoft managed to just miss explaining. Starting from infancy and continuing throughout the preschool period, they develop a base of skills, concepts, and misconceptions about numbers and mathematics. Its the first item on the list.Children begin learning mathematics well before they enter elementary school.

In this chapter we describe the current state of knowledge concerning the proficiency that children bring to school, some of the factors that account for limitations in their mathematical competence, and current understanding about what can be done to ensure that all children enter school prepared for the mathematical demands of formal education.The most fundamental concept in elementary school mathematics is that of number, specifically whole number. During the last 25 years, developmental psychologists and mathematics educators have made substantial progress in understanding the ways in which these strands interact. Preschoolers’ mathematical thinking rests on a combination of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Applying the framework to research on preschoolers’ mathematical thinking also provides a good example of the way in which the strands of proficiency are interwoven and interdependent. That framework is useful in thinking about the skills and knowledge that children bring to school, as well as the limitations of preschoolers’ mathematical competence.

5 At first, these items are perceptual they might be, for example, beads, marbles, fingers, taps, steps, or drumbeats. There must be items to count and there must be a procedure to make each utterance of a number word correspond with one of the items to be counted. Counting is not simply reciting the number word sequence. This separation is somewhat artificial because counting is a good example of the way in which the different strands of mathematical proficiency are interwoven.As children learn to count, their thinking changes in a way that shapes their concept of number. In this section, we discuss children’s understanding of the conceptual aspects of counting. We discuss issues related to competent counting, including the learning of number names, in the section on procedural fluency below.

Counting in the absence of perceivable objects is the culmination of a rather intricate developmental process. This process of creation is clearly demonstrated when a child appears to count specific items in a situation where no such items are visible, audible, or tangible. 6 The counter must always create a mental representation of the items that are counted. Later, children become able to count sets of things (e.g., “how many different colors of buttons are there?”) as well as items that may not be readily perceivable.

12 The many errors preschoolers make when counting could indicate that they fail to understand the importance of the counting principles. 11One reason it has been hard to resolve contrasting claims about how children come to understand the conceptual basis for counting is that preschoolers’ performance when they count is often quite variable, as it is with most other tasks. 10 An intermediate view is that conceptual and procedural knowledge of counting develop interactively, with small changes in one contributing to small changes in the other. Other studies indicate that much of children’s conceptual understanding of counting follows (and may be based on) an initial mastery of conventional counting procedures. According to one viewpoint, 9 children’s emerging understanding of these counting principles organizes and motivates their acquisition of conventional counting procedures. 7Early research on children’s understanding of the mathematical basis for counting focused on five principles their thinking must follow if their counting is to be mathematically useful: 8One-to-one: there must be a one-to-one relation between counting words and objects Stable order (of the counting words): these counting words must be recited in a consistent, reproducible order Cardinal: the last counting word spoken indicates how many objects are in the set as a whole (rather than being a property of a particular object in the set) Abstraction: any kinds of objects can be collected together for purposes of a count andOrder irrelevance (for the objects counted): objects can be counted in any sequence without altering the outcome.The first three principles define rules for how one ought to go about counting the last two define circumstances under which such counting procedures should apply.The relation between children’s conceptual understanding of counting and their mastery of conventional counting remains controversial.

Four-year-olds were better than 3-year-olds at rejecting true errors. Three-year-olds showed perfect acceptance of correct counting, 96% acceptance of unconventional but correct counting, and 67% rejection of real errors. For example, asked to judge the accuracy of counting by a puppet who counted either correctly, incorrectly, or unconventionally (e.g., starting from an unusual starting point but counting all of a set of items), 3- to 5-year-olds demonstrated very good performance. A child’s difficulty in managing the complex processes involved in counting could mask a real understanding of its conceptual basis.One way of circumventing the ambiguity of children’s counting behavior involves asking them to judge the adequacy someone else’s counting rather than perform the activity themselves.

Thus, children’s own counting activity might form the basis for their judgments of what constitutes successful counting.There are also important limits on children’s ability to use counting in problem solving. Finally, and most relevant to the relation between counting skill and judgment of another’s counting, the only children who failed to meet a criterion of 75% correct in rejecting the puppet’s counting errors also failed to meet the same criterion in their own counting. 14 In fact, 3-year-olds’ acceptance of unconventional correct counting was actually higher than that of 4-year-olds, suggesting that some of the acceptance of unconventional correct counting came from a blanket acceptance of the puppet’s performance.

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