Arithmetic Practice Research Articles

FARMERS, POLITICS, AND ACCOUNTING: THE HISTORY OF STANDARD VALUES – AN ACCOUNTING CONVENIENCE OR POLITICAL ARITHMETIC?

ABSTRACT This paper examines accounting in the social, political, and economic context within which it operates. Specifically, the farming sector in New Zealand provides the context for studying the history of standard-value accounting. This accounting practice emerged with the support of accountants, farmers, and the state as the tax regime in New Zealand slowly moved to an income tax for farmers from 1915. The paper examines how accounting became a practice of political arithmetic, mediating the economic power of the farmers with the rest of the tax base of New Zealand. Standard-value accounting for livestock became a device that represented the power of farmers to receive favorable tax treatment compared with other New Zealanders, while still demonstrating they carried their fair share of the country's tax burden.

My‐Mini‐Pet: a handheld pet‐nurturing game to engage students in arithmetic practices

AbstractIn the last decade, more and more games have been developed for handheld devices. Furthermore, the popularity of handheld devices and increase of wireless computing can be taken advantage of to provide students with more learning opportunities. Games also could bring promising benefits – specifically, motivating students to learn/play, sustaining their interest, reflecting their learning/playing status, and facilitating the learning/playing progress. However, most of these have been designed for entertainment rather than education. Hence, in this study we incorporate game elements into a learning environment. The My‐Mini‐Pet system is a handheld pet‐nurturing game environment, in which students learn with an animal learning companion, their My‐Mini‐Pet. Three design strategies are adopted. First, the pet‐nurturing strategy, which simulates the relationship between the pet and its owner, the My‐Mini‐Pet becomes a motivator/sustainer of learning. Second, the pet appearance‐changing strategy, which externalizes the learning status of the student. In other words, the My‐Mini‐Pet plays the role of a reflector. Third, the pet feedback strategy, which links the behaviours of the student and his/her pet, the My‐Mini‐Pet acts as a facilitator of learning. A pilot study was also conducted to preliminarily investigate the effectiveness and experiences of the strategies on allowing the student to understand arithmetic practices. The results showed that the strategy was effective, encouraging the students to engage in learning activities. Furthermore, the game attracted the students’ attention and stimulated discussion between peers. Some implications about the further developments are also discussed.

School mathematics and its everyday other? Revisiting Lave’s ‘Cognition in Practice’

In the last three decades there have been a variety of studies of what is often referred to as ‘everyday’ or ‘street’ mathematics. These studies have documented a rich variety of arithmetic practices involved in activities such as tailoring, carpet laying, dieting, or grocery shopping. More importantly, these studies have helped to rectify outmoded models of rationality, cognition, and (school) instruction. Despite these important achievements, doubts can be raised about the ways in which theoretical conclusions have been drawn from empirical materials. Furthermore, while these studies rightly criticised prevalent theories of rationality and cognition as too simplistic to account for everyday activities, it seems that some of the proposed alternatives suffer from similar flaws (i.e., are straightforward inversions of the to-be-opposed theories, rather than more nuanced views on complicated issues). In this article we illustrate our sceptical view by discussing four case studies in Jean Lave’s pioneering and influential ‘Cognition in Practice’ (1988). By looking at the case studies in detail, we investigate how Lave’s conclusions relate to the empirical materials and offer alternative characterisations. In particular, we question whether the empirical studies demonstrate the existence of two different kinds of mathematics (‘everyday’ and ‘school,’ or ‘formal’ and ‘informal’) and whether school instruction tries to replace the former with the latter.

The influence of problem features and individual differences on strategic performance in simple arithmetic

The present study examined the influence of features differing across problems (problem size and operation) and across individuals (gender, amount of daily arithmetic practice, calculator use, and arithmetic skill) on simple arithmetic performance. Regression analyses were used to investigate the role of these variables in both strategy selection and strategy efficiency. Results show that more skilled and highly practiced students used memory retrieval more often and executed their strategies more efficiently than did less skilled and practiced students. Furthermore, calculator use correlated with both retrieval and procedural strategy efficiency but not with strategy selection. Only very small associations with gender were observed, with boys retrieving slightly faster than girls. Implications of the present findings for models of mental arithmetic are discussed.

Learning about ‘learning identities’ in the school arithmetic practice: The experience of two young minority Gypsy girls in the Greek context of education

The present study explores the experience of two young Gypsy girls in solving school arithmetic tasks in interaction with an adult who supports their participation. Along with learning the use of arithmetic tools, a basic element concerning the experience of the two girls as they try to gain entry into the school practice is learning about developing relevant ‘learning identities’. Analysis of the girls’ interaction with an adult identifies that the type of ‘learning identities’ required involve at least two dimensions: (a) learning to value the use of formal arithmetic tools, and (b) learning about relevant ways of behaving in the school mathematics practice.

Extensive Practice in Mental Arithmetic and Practice Transfer Over a Ten-month Retention Interval

Participants were trained on nine single-digit multiplication problems over three sessions. Transfer of practice was tested with nine different multiplication problems in the fourth session, for half the participants. Retention of problems was examined 10 months later. In addition, memory ratings were taken after practice and before retention. Response initiation time (RTinitiate), a measure of mental calculation time, decreased during the first two practice sessions, becoming asymptotic in the third. The problem-size effect decreased with practice but was not eliminated even at asymptotic response times. Response execution time (RTexecute), a measure of motor response execution, was not affected by problem size, indicating a dissociation between both measures. No overall transfer advantage from practised to new problems was observed for RTinitiate. However, learning curves within practice and transfer sessions suggested positive transfer at the beginning of the transfer session and negative transfer at t...

When 1+1≠2: making mathematics in central Brazil

In this ethnographic account of mathematical activity among the Juruna, Kayabi, and Suya of central Brazil, I show how arithmetic practices are fashioned in a specific social setting. Values and symbolic properties of both the gift exchange and capitalist economics structure arithmetic dilemmas in the Xingu Indian Park. Within a broad social field that transcends the boundaries of the park to include prospecting sites and cattle ranches, economic calculations are extended to all kinds of goods, both material and symbolic. The distribution and circulation of these different forms of capital are discussed in view of the constitution of particular arenas of exchange. Practice theories highlight the ways in which mathematical knowledge is constituted in everyday activities, challenging functional assumptions about cognition and schooling. By articulating principles of the gift with those of capitalist exchange, mathematics is construed by the juruna, Kayabi, and Suyá as a product of social work and symbolic fashioning.

Recantation or Any old ω-sequence would do after all

Abstract I once argued (in ‘What Numbers Could Not Be’) that an adequate account of the numbers and our arithmetic practice must satisfy not only the conditions usually recognized to be necessary: (a) identify some w-sequence as the numbers, and (b) correctly characterize the cardinality relation that relates a set to a member of that sequence as its cardinal number—it must also satisfy a third condition: the ‘<’ of the sequence must be recursive. This paper argues that adding this further condition was a mistake—any ω-sequence would do, no matter how undecidable its ‘<’ relation.

What you design is not what you get (WYDINWYG): Cognitive, affective, and social impacts of learning with ils — An integration of findings from six-years of qualitative and quantitative studies

Despite the wide application of ILSs in schools today, little research is available to inform us of their actual effects on teaching and learning and to guide us how to design an optimal ILS environment. This chapter integrates results of six years of qualitative/naturalistic and quantitative studies of arithmetic practice in four ILSs (two in the U.S. and two in Israel), in order to provide us with this important information. These studies examined a variety of cognitive, behavioral, sociological, affective, and instructional issues identified in the ILS work of students and their teachers, and the effects of a variety of ILS design issues related to hardware, software, and method of operation on students' and teachers' behavior in the ILS environment.

Good Students beat the computer system: Strategies for self learning from computerized practice in arithmetic

Naturalistic methods ofinquiry were used to investigate learning processes ofabove-average second, third, and fourth graders while practicing arithmetic with a computer. Because the sofn-vare enabled the better students to accelerate through the practice material, they receivedpractice in topics that had not yet been covered in class, and thus were attempting material which was new to them. It also happened that they encountered exercises that they had learned in class but hadforgotten how to solve. This study reveals that when confronted by exercises they do not know how to solve, above average students use a variety of strategies that lead to their identification of solution algorithms, while not always understanding the underlying concepts. The article identifies the different problem solving strategies that students used, sorts them into categories, and illustrates them with examplesfrom students' protocols. On the basis of the findings, suggestions are made for designing computer softwarefor arithmeticpractice that promotes student problem solving strategies along with mathematical understanding. One of the major advantages expected of the computer as a curriculum learning tool is that it enables each child to interact individually with the software and to learn at his or her own pace, style, and ability level. In learning or practicing mathematics with computers, the material is arranged sequentially according to either increasing difficulty or to the curriculum topics. Each student is able to advance through this sequence at hislher own rate. However, if students in the same classroom, but with heterogeneous aptitudes are allowed to advance each at their own rate through the curriculum material, we may expect that after a while they will develop large discrepancies among their placement in the curriculum units. Indeed, several studies (e.g., Osin & Nesher, 1979; Hativa, 1986, 1988) reveal large differences in the mean level of arithmetic practice between high and low achievers of the same classroom. Such differences grow with the length of use of the computerized work. The better students advance in the arithmetic computer curriculum levels at a much faster rate than the low achievers. In addition, there can be large initial discrepancies in the starting level of practice among students as a result of an initial diagnosis administered at the beginning

An Ethnographic Study of the Mathematical Ideas of a Group of Carpenters

Abstract Mathematical knowledge, primarily the practice of arithmetic, which is developed outside of educational institutions by people engaged in their daily working and domestic activities, has been documented. The idea that different forms of mathematics may develop as a result of different modes of thinking and different environmental demands across cultural groups, has been suggested by research in ethnomathematics. However, there exist no clear guidelines for recognising and describing such culture-specific mathematics, which may differ significantly from the conventional view of mathematics. This article describes a study of the mathematical ideas of a group of carpenters, focusing mainly on their geometric ideas, and provides an account of the methodological strategy which was used. The researcher conducted an ethnographic study over the period of six months, working as an apprentice with a group of carpenters in Cape Town, South Africa. One descriptive example of an interaction with two carpenters involved in problem solving is analyzed in this article. The results show that the carpenters engaged in a wide variety of mathematizing activities in order to solve the problem.

The computer as a tutor ? can it adapt to the individual learner?

Educational literature indicates that computers have not yet completely fulfilled the promise for systematic individualization of instruction. We offer here a partial explanation for this, based on observations of elementary-school children doing computer-managed drill and practice in arithmetic in four different CAI systems. A general model for adaptation of instruction to the learner, using computerized management of practice, is presented. Our data show that the key elements of this process — the software-based students' evaluations and the subsequent decisions — are sometimes inaccurate or even totally wrong and result in presentation of inappropriate material or feedback to the learner. We suggest the incorporation of human teachers into the evaluation and decision-making process.

An ethnographic study of the mathematical ideas of a group of carpenters

Mathematical knowledge, primarily the practice of arithmetic, which is developed outside of educational institutions by people engaged in their daily working and domestic activities, has been documented. The idea that different forms of mathematics may develop as a result of different modes of thinking and different environmental demands across cultural groups, has been suggested by research in ethnomathematics. However, there exist no clear guidelines for recognising and describing such culture-specific mathematics, which may differ significantly from the conventional view of mathematics. This article describes a study of the mathematical ideas of a group of carpenters, focusing mainly on their geometric ideas, and provides an account of the methodological strategy which was used. The researcher conducted an ethnographic study over the period of six months, working as an apprentice with a group of carpenters in Cape Town, South Africa. One descriptive example of an interaction with two carpenters involved in problem solving is analyzed in this article. The results show that the carpenters engaged in a wide variety of mathematizing activities in order to solve the problem.

Computer-Based Drill and Practice in Arithmetic: Widening the Gap Between High- and Low-Achieving Students

Precollege schools use their computers predominantly for drill and practice in arithmetic. It has been shown by research to be the most effective application of CAI (Computer-Assisted Instruction). The present study used naturalistic methods to examine differential effectiveness of CAI in arithmetic for high- and low-achieving students. Seven elementary school students with varied backgrounds with regard to socioeconomic status (SES), grade level, type of school, and achievements in arithmetic participated in the study. The study identifies strategies that students use to advance effectively through the levels of the CAI practice with material not yet studied in class. The study also discusses software- and hardware-related errors that students make while working with the CAI system. In many cases, these errors are unrelated to the knowledge of arithmetic. It seems that the low achievers, more than the high achievers, are prone to make these types of errors, which are partly responsible for ineffective CAI work. The study concludes that the higher achieving students observed were much more able than the observed low achievers to adjust to the special environment of computer work and to derive great benefits from it.

Sigal's Ineffective Computer-Based Practice of Arithmetic: A Case Study

Sigal's Ineffective Computer-Based Practice of Arithmetic: A Case Study

Sigal's Ineffective Computer-Based Practice of Arithmetic: A Case Study

Sigal, a second-grade girl in Israel, was observed for several months while practicing arithmetic with a structured and controlled computer-assisted instruction (CAI) system. Sigal was average in arithmetic in her class, enjoyed the computer practice, and believed that it helped her in arithmetic. However, observations showed that she consistently failed to solve problems on the computer that she could do successfully with paper and pencil. She was therefore given CAI practice well below her real arithmetic ability. She often oscillated among a few types of exercises, sometimes regressing rather than advancing upward in the hierarchical levels of practice. Several kinds of interventions did not bring about substantial improvement in her work with the computer. Continued observations in the following 2 years showed almost no change in her CAI-related behavior. This paper suggests the reasons for Sigal's problems with the CAI practice and draws implications for CAI design and implementation.

Computer-Based Drill and Practice in Arithmetic: Widening the Gap between High- and Low-Achieving Students

Precollege schools use their computers predominantly for drill and practice in arithmetic. It has been shown by research to be the most effective application of CAI (Computer-Assisted Instruction). The present study used naturalistic methods to examine differential effectiveness of CAI in arithmetic for high- and low-achieving students. Seven elementary school students with varied backgrounds with regard to socioeconomic status (SES), grade level, type of school, and achievements in arithmetic participated in the study. The study identifies strategies that students use to advance effectively through the levels of the CAI practice with material not yet studied in class. The study also discusses software- and hardware-related errors that students make while working with the CAI system. In many cases, these errors are unrelated to the knowledge of arithmetic. It seems that the low achievers, more than the high achievers, are prone to make these types of errors, which are partly responsible for ineffective CAI work. The study concludes that the higher achieving students observed were much more able than the observed low achievers to adjust to the special environment of computer work and to derive great benefits from it.

The Practice of Arithmetic in Liberian Schools

The Practice of Arithmetic in Liberian Schools

The Practice of Arithmetic by American Grocery Shoppers

This study examines the arithmetic procedures people use when shopping for groceries in American supermarkets. The data were gathered by following people around supermarkets while observing and recording their behavior. It is much easier to investigate arithmetic practice in controlled environments that simulate a few theoretically interesting aspects of the much more complex outside world. But there is one important feature of grocery shopping behavior that cannot be simulated easily. In the supermarket, people generate their own problems as well as solve them, whereas in the laboratory, problems are formulated by the researcher. How problems originate is an interesting question in its own right: What kinds of problems do people ordinarily encounter in the course of grocery shopping that require the manipulation of quantitative information? But there is another reason to study problem-formation, one that has more theoretical importance. It may be that the way people generate problems largely determines how they go on to solve them. The relationship between problem-formation and problem-solving in supermarket arithmetic is the central focus of this study.

The Practice of Arithmetic in Liberian Schools

The examination of arithmetic practices in schools has proven to be as problematic as research on arithmetic in everyday settings described by Lave in the Introduction to this Symposium. Previous American research on schools, and cross-cultural research on cognition, provide incomplete but suggestive models for examining the practice of arithmetic in classrooms. Data from research in elementary classrooms in Liberia may be used to demonstrate that a contextually based study of arithmetic practice gives new insight as to how children actually accomplish some mastery of the school curriculum.