數學基礎認識與理解 Mathematical Literacy and Understanding,數量的認識和數數,兒童從多大開始有數量的觀念呢? 數量的認識是怎麼? 是先天的還是後天培養的? 兒童對數量的認識是是透過聽覺, 視覺, 還是其他呢? 兒童何時懂得加數,減數? 他們懂得做多大的數量的加減? 兒童何時懂得數數? 數數過程中包含什麼概念?,Humans are born with a fundamental sense of quantity?,Same number of marbles?,4/5 years old: Left, yes; Right, no. they can answer correctly until 7-8 years.,Younger children do not posses a conceptual understanding of numbers and that any number re-related activities are learned by rote?,2 1/2 years to 4 1/2 years olds,Take the row you want to eat, and eat all the M&Ms in that row.,Children understand more than and less than.,Numerical Competencies,Numerosity (數量感): 對數量的認識 Ordinality(序列感): 對順序的認識 Arithmetic: 基本數學運算,思考問題,你以為兒童能否認識簡單的數量呢? 如果可以, 他們可以認識到多大的數量? 這種能力是與生俱來的(Innate)還是後天學習得到的? 如果你是一個研究人員, 你將怎樣設計你的實驗以找出這些問題的答案呢?,Numerosity,Habituation procedure: dishabituation means notificaiton of number of presented dots changed. 4 months to 7 1/2 months: discriminate 2 from 3 items but not 4 from 6 itms. 10 - 12 months: 2 from 3, not 4 from 5 infants: look longer when 2 dots presented with 2 drum bits, (abstracts codes for numerosities up to 3 or 4 items),Infants abilities in numerosity,Not dependent on a specific modality not influenced by factors such as: whether dots or household items are presented, whether the presented items are static or moving, density of the displays,Ordinality,Infants sensitive to numerosity implies they understand larger than or less than? Ordinality before or after numerosity? Why?,思考問題,Ordinality,Infants represent numerosity do not imply they can rank the representations. Developed during the first 1 1/2 years of life.,討論問題,兒童懂得加數和減數嗎, 何時開始懂得呢? 如果他們年紀這麼少便懂得加減數, 這顯示了什麼呢?,Arithmetic,Infants have a preliminary sense of addition and subtraction at 5 months of age.,Objects take out or add in,Infant looks in this direction,討論題目,你怎樣看出一組物件的數目?,Development of Early Numerical Abilities,Subitizing Counting Estimating,Reaction Time Patters for Making Numerosity Judgments,討論題目,試解釋上圖背後的原因,Error Rates,Rare for arrays with 4 or fewer items 50% for arrays with more than 7 items,討論題目,兒童何時開始數數? 什麼原因令他開始? 數目有什麼特質? 數數是什麼? 包括什麼過程? 兒童要懂得什麼才能開始數數? 數數過程中包含許多概念, 兒童是先有概念才數數,還是從數數中學習概念?,Properties of Number System,Each number word is unique and represent a unique quantity numbers are serially ordered each number reflects a group of smaller numbers.,Counting: basic skills involved,One-to-one correspondence between number names and the counted items order the number names in the correct sequence the last number named in the count (the cardinal number) represents the total number of counted items.,Cardinality and Ordinality,Cardinality: number word assigned to the last counted object can be used to represent the total number of the counted objects Ordinality: successive number words represent successively larger quantities.,Cardinality and Last-word Rule,Test of cardinality: A child is asked to count his or her fingers asked, “How many fingers do you have?” understand the concept of cardinality or just use the Last-word Rule? Cardinality is always confused by how the items are arranged.,Ordinality,Refers mostly to the childs knowledge of equivalence and greater than and less than. Children as young as 2 1/2 years of age have an understanding of ordinal relationships.,Developmental Mechanisms for Counting and Number Knowledge,Most researchers agree that a sensitivity to numerical information is, at least in part, inborn, there is considerable disagreement over the relative importance of this innate sensitivity. Tow general positions: principals-first: innate principles guide the development, procedures-first: first count by rote and gradually induce counting concepts.,Principles-First,Behavior of young children guided by 5 principles: one-one correspondence: one number word one counted object stable order: same sequence of number words for counting cardinality: the number word associated with the last counted item has a special meaning. Abstraction: awareness of what is countable (skill at counting mixed sets too) order irrelevance:,The principles guide and structure the childs counting behavior, serve as a reference against which the child can evaluate this actual counting behavior, and motivate that behavior.,Procedure First,Children first learn to count largely by rote through the imitation of parents or siblings, for example. Induces basic principles by noticing regularities in the outcome of counting.,