Logico-mathematical Knowledge Research Articles

The construction of geometric mental relationships in a pattern block puzzle activity at age four

Using constructivist scholar Constance Kamii’s perspective on autonomy, physical, social, and logico-Mathematical knowledge, this research investigates a curriculum method developed to engage young children in the construction of geometric knowledge. To study this method, I designed a seven-week pattern block puzzle intervention with four-year-old children in a constructivist classroom. All children were considered at-risk for school failure. The investigation is rooted in my unsatisfying experiences with pattern blocks in my preschool classrooms. Data were obtained from pre and posttests assessment administered to ascertain effects of the intervention on all children and from observations of children’s behavior as they engaged in the pattern block activity. Piaget’s (1936/1952), Piaget and Inhelder, 1948/1956) theory of knowledge and intelligence was the framework for detailed qualitative analysis of one exemplar’s progress during the intervention. Results of pre and posttests and microanalysis indicate children made significant progress in their construction of geometric knowledge. Children learned to match shapes with corresponding spaces and distinguish among and coordinate the sizes of angles and spaces. I describe using a series of detailed drawings, one exemplar’s actions during the activity. I conclude with suggestions for further research and educational implications.

Jean Piaget’s Genetic Epistemology as a Theory of Knowledge Based on Epigenesis

This article aims to highlight Jean Piaget’s theory of knowledge and situate it in this context since its beginnings in Ancient Greece where, in Plato, we already find this seminal idea: knowledge is acquired in successive and upward moments (dialektikê), starting from an opinion on the sensible world (doxa) towards the épistêmê of the intelligible world, the world of Ideas or concepts. Piaget’s Theory of Knowledge, we believe, was determined by four moments: 1) his research as a malacologist under the guidance of Godet and Raymond, 2) the acquaintance with Kant’s philosophy at age of 21, 3) his internship at the Binet/Simon laboratory, 4) his studies on the Limnaea Stagnalis. His core idea: it is possible for human beings to attain the necessary and universal knowledge due to the exchange processes of their organisms with the environment, which give rise to the epigenetic ontogenesis of their specific organic mental structures, framed for the specific act of knowing. Epigenetic ontogenesis begins with the infans first actions in the world, from the very moment of birth. Around two years of age, these actions will be represented and organized in groups linked to empirical experience, until the brain be able to perform the operations of the Abelian Group. The physiological development ends here, and the logico-mathematical knowledge becomes possible.

Dialektyka wiedzy logikomatematycznej w ujęciu Jarosława Ładosza

The aim of the article is an analysis of the early works of Jarosław Ładosz, a Polish philosopher and mathematician, who in the 1960s conducted a thorough examination of the most important scientific accomplishments in the field of logic and mathematics from the perspective of Marxist philosophy. Being nowadays assessed as a symbol of dogmatism and orthodoxy in Polish Marxism, Ładosz revised most of the superstitions on the relationship between mathematical logic and dialectics, which have been legitimized in official Marxist philosophy since the times of Marx and Engels, in his early works. Having rejected the claims of some Marxists for the formalization of dialectics, he presented the original concept of dialectics as a methodological tool for studying the sources of logical knowledge. Combining dialectical materialism with Jean Piaget’s epistemology, he formulated an elaborate, original hypothesis of the social construction of logico-mathematical knowledge, while at the same time transcending the subject-object division as-sumed in Marxist dogmatic epistemology.

Elapsed Time: Why Is It So Difficult to Teach?

Based on Piaget's theory of logico-mathematical knowledge, 126 students in grades 2–5 were asked 6 questions about elapsed time. The main reason found for difficulty with elapsed time is children's inability to coordinate hierarchical units (hours and minutes). For example, many students answered that the duration between 8:30 and 11:00 was 3 hours 30 minutes (because from 8:00 to 11:00 is 3 hours, and 30 more minutes is 3 hours 30 minutes). Coordination was found to begin among logicomathematically advanced students, through reflective (constructive) abstraction from within. The educational implications drawn are that students must be encouraged to think about durations in daily living and to do their own thinking rather than being taught procedures for producing correct answers to elapsed-time questions.

Giving Change When Payment Is Made With a Dime: The Difficulty of Tens and Ones

ION AND REPRESENTATION Piaget made a distinction between empirical (or simple) and (or or reflecting) abstraction. In empirical abstraction, we focus on one or more properties of objects and ignore the others. For example, when we categorize objects by color, we focus on color (physical knowledge) and ignore all the other properties. In abstraction, by contrast, we create mental relationships (logico-mathematical knowledge) that do not exist in objects. For example, when we are presented with a dime and a penny and we say that they are different, we are creating a mental relationship that is not in the objects in the external world. This relationship is created by our thinking. The numbers 2, 8, 10, or any other numbers are likewise created by each individual through abstraction. (We prefer use the term constructive abstraction rather than reflective (or reflecting) abstraction because the former seems easier understand.) When the Romans wrote XL VI instead of 46, their system of representation (social knowledge) was different from ours. However, their logico-mathematical knowledge of tens and ones, created by abstraction, was exactly the same as ours. In our Arabic system of writing, we use the (position) of the and the differently from the Romans. When we say that place value concepts are hard for children, we must differentiate between the difficulty that children have in representation from the difficulty they have in abstraction. The errors they make in representation usually occur because of their low-level abstraction. The verb to is often used carelessly, as when we hear that base10 blocks represent the base10 system. Base10 blocks do not represent anything by themselves. Representing is what an individual does. When adults are presented with 4 ten-blocks and 6 one-blocks, we can represent 46 ourselves because we have already abstracted a system of tens out of our system of ones. When young children see the same blocks, however, they cannot represent forty-six themselves unless they have constructed a system of tens through abstraction. In Piaget's theory (1945/1962), representing refers (a) internally evoking or externally expressing an idea and (b) interpreting (comprehending) an external representation. An example of an internal evocation is our thinking about 5 cookies when the cookies are not present. An example of an external expression of this idea is our drawing of 5 circles represent 5 cookies (which may or may not be present). When we understand the 5 circles thus drawn as being 5 cookies, this is an example of interpreting an external representation. Another example is adults' and children's interpretations of 4 ten-blocks and 6 one-blocks. We can give these objects only the meaning that our level of permits. A study by Kato, Kamii, Ozaki, and Nagahiro (2002) demonstrated that young children represent numerical quantities at or below their level of but not above. In Japanese day-care centers, the authors individually showed 60 children who were from 3 7 years old 4 plastic dishes, 3 metal spoons, 6 pencils, and 8 wooden blocks. Their request each child was put something on a piece of paper so that your mother will be able tell what I showed you (p. 35). The children This content downloaded from 207.46.13.58 on Sat, 17 Sep 2016 05:41:53 UTC All use subject http://about.jstor.org/terms

Fostering the Development of Logico-Mathematical Thinking in a Card Game at Ages 5–6

A class of 14 kindergartners in Japan was videotaped while playing a card game in groups of three involving the placement of cards in numerical order. The children were followed up in first grade, and it was found that development in one area of logico-mathematical knowledge (for example, the making of temporal relationships) stimulates development in other areas (such as classification and numerical reasoning). This specific, interrelated mode of structuring was not expected and suggested that it is unwise for adults to try to plan a sequence of development. The implication of the findings is that it may be better to provide "natural" play activities that encourage children to think logico-mathematically than to conceptualize specific standards for 3-to-6-year-olds' mathematics education.

The Development of Logico-Mathematical Thinking at Ages 1–3 in Play With Blocks and an Incline

Fifty 1- to 3-year-olds were asked to imitate 1) the rolling of a cylinder down an incline and 2) the making of an incline with a board and a block. Their incorrect imitations and progress were videotaped and analyzed, and interpreted in light of Piaget's theory about the elaboration of logico-mathematical relationships in an interrelated way. The educational implication drawn is that the development of logico-mathematical knowledge as a network of interrelated mental relationships is a better goal for preschool mathematics education than isolated objectives, such as ability to sort, recognize shapes, and understand words like “first” and “second.” Piaget (1936/1963, 1937/1954) observed babies' development of logico-mathematical knowledge during the first two years of life and later studied children's cognitive development after the age of four. What happens between ages 2 and 4 is not well understood, and the purpose of this study is to fill part of this gap.

The Development of Logico-Mathematical Knowledge in a Block-Building Activity at Ages 1–4

To study the developmental interrelationships among various aspects of logico-mathematical knowledge, 80 one- to 4-year-olds were individually asked to build “something tall” with 20 blocks. Percentages of new and significant behaviors increased with age and were analyzed in terms of the development of logico-mathematical relationships. It was found, for example, that the new spatial relationships children made as they grew older also changed the classificatory, seriational, numerical, and temporal relationships they made. The educational implication drawn is that it is better to define objectives for preschool mathematics education in terms of the development of logico-mathematical knowledge rather than the learning of specific aspects of elementary school mathematics. More broadly, an argument is made in favor of encouraging preschool children to think and make many mental relationships rather than to teach them specific subject matter.

Manipulatives: when are they useful?

This article examines the usefulness of manipulatives in light of Piaget's theory of how children acquire logicomathematical knowledge. It argues that since children construct logicomathematical knowledge through their own thinking, manipulatives are desirable when they encourage children to think (i.e., to make relationships through constructive abstraction) in problem solving. A specific object can therefore be beneficial if used in certain ways but not in others. The same object can also be useful at a certain time in the child's development but not at others. We conclude by pointing out that mathematical relationships do not exist in objects and that children do not acquire these relationships through empirical abstraction from objects.

Logic of meanings and situated cognition

Inspired by the views of Piaget & Garcia (1991)that logico-mathematical knowledge has its foundations in a logic of meanings, this paper aims at a better understanding of how knowledge develops in everyday contexts, by taking into account situational aspects, individual reasoning processes, and logico-mathematical understanding. We first provide a re-analysis of data on the understanding of proportionality developed outside of school, in buying and selling activities. The set of studies we discuss shows how procedures to solve proportionality problems in specific contexts later develop as general and flexible approaches to solve problems in different contexts. Then, in an interview with an adult with restricted schooling, we show how understanding of graphical representation is achieved through the attribution of meanings to a graph on the basis of personal wishes, opinions, and knowledge about the situation and by a continuous search for logical coherence.

Children's understanding of sugar water solutions

A total of 270 children aged four to 13 were tested in order to explore their physical and logicomathematical knowledge of sugar water solutions, and the relationship between the development of these two types of knowledge. Physical knowledge development followed the hypothesized sequence of non‐preservation to preservation to liquefaction. There was some evidence that atomism is a more mature construction than liquefaction, supporting Piaget and Inhelder (1941/1974) and refuting Slone (1987). Development was slower, relative to Piaget and Inhelder's sample. The scarcity of atomistic responses was interpreted in terms of the notion that atomism is a function of schooling. The hypothesized logicomathematical developmental sequence was strongly supported, from direct relations to inverse relations to proportions, thereby confirming Slone's (1987) results. However, no evidence of a U‐shaped developmental curve for proportions was found. In general the hypothesis that physical knowledge constructions precede ...

Essay Reviews: Making Mathematics: Children's Learning and the Constructivist Tradition

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter LinkedIn Tools Icon Tools Get Permissions Cite Icon Cite Search Site Citation Paul Cobb; Essay Reviews: Making Mathematics: Children's Learning and the Constructivist Tradition. Harvard Educational Review 1 September 1986; 56 (3): 301–307. doi: https://doi.org/10.17763/haer.56.3.r2840370x17g3201 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu nav search search input Search input auto suggest Search