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Teaching Methods - Mathematics 1. In this paper we prepare a brief description of the method to teach a group of ten first grade students to count rationally to 15. We assume that all of these students can count rationally to 10. To begin with we have to note that these students need to possess some skeletal set of counting principles, which guide interaction with the environment and the development of counting skills. The development of counting within the early school period involves “the perfection of counting procedures rather than the emergence of new or firmer principles” (Gelman & Meck, 1983, p. 352). Gelman and Gallistel (1978) depicted three how-to-count principles (procedural rules for counting) and two permissibility principles (statements concerning what can be counted and how items can be counted): 1.

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The stable-order principle decrees that count tags must be generated in the same sequence on every count. (E.g. explaining students that 1,2,3 is similar to 11, 12, 13 with the only difference that the latter have 1 in front of the number). 2. The one-to-one principle specifies that every item in a collection must be marked with one and only one unique tag. (E.g. explaining children that they can’t say 11 or 12 more than once. If they want to say a new number, they have to move on to the next one in a row). 3. The cardinality principle dictates that the tag used to mark the last item in a collection represents the total number of items in the collection. (E.g. showing students that once they’ve counted 15 items, they have a total of 15 items). 4. The abstraction principle states that various kinds of items may be collected together for the purpose of counting.

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5. The order-irrelevance principle specifies that the items of a collection can be marked in any order and, as long as the one-to-one principle is observed, the outcome (the last tag) will be the same. It is important to note that count tags need not be number words or even verbal. Gelman and Gallistel (1978) pointed out that a number of languages have used the alphabet as tags. Thus, in our case, we have to teach students that they need to use the ‘count-to-ten’ technique when counting to 15. We can draw numbers 11 through 15 and 1 through 5 on the board (two rows, one below the other) in order to show that the same structure is employed in both cases. 2. An individual assessment would appropriate in our case.

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We can ask each of the ten students to count from 1 to 15. It can be done either individually (one person speaking from his or her place rather than in front of the whole group to avoid excessive pressure on children) or in groups of 2-3 people (when students work in groups and then share their results with the rest of the children). Regardless of the method used, the instructor takes notes on every student’s performance and makes decisions on how to improve the situation if the student fails to adequately count to 15. 3. Gelman and Gallistel (1978) suggested that a set of unique mental markers of some type might be used by English Language Learners and students with learning disabilities to determine numerosities. Thus, the counting principles just described apply to any counting system, whether it involves number words (what we commonly think of as counting) or other types of tags (e.

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g., the electronic-based counting of a computer). It is the five principles just described or some subset of these principles that Gelman and Meck (1986) proposed as children's initial or skeletal conceptual competence--the understandings that children bring to the task of acquiring counting skills.

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