Toward a student-centred process of teaching arithmetic

This article describes a way toward a student-centred process of teaching arithmetic, where the content is harmonized with the students’ conceptual levels. At school start, one classroom teacher is guided in recurrent teaching development meetings in order to develop teaching based on the students’ prerequisites and to successively learn the students’ arithmetic. The students are assessed in interviews. Two special teachers participate and their current models of each student's arithmetic are tested when assessing the students. The students’ conceptual diversity and the consequent different content in teaching are shown. Further, the special teachers’ assessments and the class teacher's opinion of the new way of teaching are reported. A wide range both of the students’ conceptual levels and of the kinds of relevant problems was found. The special teachers manage their duties well and the classroom teacher has so far been satisfied with the new teaching process.     

Neither even nor odd: Sixth grade students’ dilemmas regarding the parity of zero

This study investigates two sixth grade students’ dilemmas regarding the parity of zero. Both students originally claimed that zero was neither even nor odd. Interviews revealed a conflict between students’ formal definitions of even numbers and their concept images of even numbers, zero, and division. These images were supported by practically based explanations relying on everyday contexts. By using mathematically based explanations that rely solely on mathematical notions, students were able to correctly conclude that zero is an even number. Extending the natural number system in elementary school to include zero can be used as springboard to encourage the use of mathematically based explanations.     

A model of students’ combinatorial thinking

Combinatorial topics have become increasingly prevalent in K-12 and undergraduate curricula, yet research on combinatorics education indicates that students face difficulties when solving counting problems. The research community has not yet addressed students’ ways of thinking at a level that facilitates deeper understanding of how students conceptualize counting problems. To this end, a model of students’ combinatorial thinking was empirically and theoretically developed; it represents a conceptual analysis of students’ thinking related to counting and has been refined through analyzing students’ counting activity. In this paper, the model is presented, and relationships between formulas/expressions, counting processes, and sets of outcomes are elaborated. Additionally, the usefulness and potential explanatory power of the model are demonstrated through examining data both from a study the author conducted, and from existing literature on combinatorics education.     

Helping students build a path of understanding from ratio and proportion to decimal notation

Various studies have shown that students of all levels struggle to understand decimal numbers. This paper discusses a novel approach to increasing students’ conceptual understanding of decimal numbers. Rather than approach decimal notation as a discrete and separate mathematical topic, this approach enables students to work with contextual problems to gain a solid understanding of ratio and proportion. Using their understanding of ratio and proportion as a foundation, students can then build connected and related understandings of fractions, decimals and percents. The study discussed in this paper illustrates that grounding decimal instruction in the broader context of ratio can help students gain deeper conceptual understandings of decimal notation as well as fractions and percents.     

Prevailing educational practices for area measurement and students’ failure in measuring areas

The present research was carried out with the participation of 106 students in their last grade in Elementary School and revealed certain problems that these students faced in understanding the concept of area measurement. The students in the sample persisted on using measurement strategies that often led to failure.Our research plan comprises a comparison between the strategies for area measurement strategies used by two groups: the experimental group (E.G.) and the control group (C.G.). The experimental group attended a special teaching course, which stressed the conceptual characteristics of the area measurement process.The present research aims at revealing the students’ understanding, strategies, and misconceptions regarding area measurement. In addition to that, we examine whether the special teaching course and the use of different measurement tools may lead the two research groups to adopt different measurement strategies.     

Conceptual variation and coordination in probability reasoning

This study investigates students’ conceptual variation and coordination among theoretical and experimental interpretations of probability. In the analysis we follow how Swedish students (12-13 years old) interact with a dice game, specifically designed to offer the students opportunities to elaborate on the logic of sample space, physical/geometrical considerations and experimental evidence when trying to develop their understanding of compound random phenomena.The analytical construct of contextualization was used as a means to provide structure to the qualitative analysis performed. Within the frame of the students’ problem encounters during the game and how they contextualized the solutions of the problems in personal contexts for interpretations, the analysis finds four main forms of appearance, or of limitations in appearance, of conceptual variation and coordination among theoretical and experimental interpretations of probability.     

Looking at the complexity of two young children's understanding of number

This paper presents a qualitative study that investigated two third-grade students’ understanding of number. The children were videotaped while they worked to record everything they knew about the number, 72. Their artifacts and conversations were then analyzed using the Pirie-Kieren dynamical theory for the growth of mathematical understanding as the theoretical framework. The results of the video analysis revealed the two students’ understanding of natural numbers as being conceptually complex and existing in several different realms of the Pirie-Kieren model. Significant instances of Primitive Knowing, Image Making, Image Having, Property Noticing, Formalizing, Observing, as well as how the children's understanding existed beyond “Don’t Need” Boundaries are identified and examined in detail. Other features of the model — Structuring, Inventising, Folding Back, and Connected Understanding - are also explained and possible examples illustrating these kinds of mathematical thinking in relation to the two children's understanding of number are offered.     

High school students’ understanding of the function concept

This paper is a study of part of the Algebra Project's program for underrepresented high school students from the lowest quartile of academic achievement, social and economic status. The study focuses on students’ learning the concept of function. The curriculum and pedagogy are part of an innovative, experimental approach designed and implemented by the Algebra Project. The instructional treatment took place over 7 weeks during the Junior Year of 15 students from our target population. Immediately after instruction, a written instrument was administered followed, several weeks later, by in-depth interviews. The results are that many of our participants achieved a level of knowledge and understanding of functions on a par with beginning college students, including preservice teachers, as reported in the literature. Many conceptual difficulties that have been reported in the research literature were not as prevalent for our participants and some of them were capable of solving difficult problems involving composition of functions. We conclude that, with appropriate pedagogy, it is possible for students in the Algebra Project's target population to learn substantial and non-trivial mathematics at the high school level, and that the Algebra Project approach is one example of such a pedagogy.     

Promoting productive mathematical classroom discourse with diverse students

Productive mathematical classroom discourse allows students to concentrate on sense making and reasoning; it allows teachers to reflect on students’ understanding and to stimulate mathematical thinking. The focus of the paper is to describe, through classroom vignettes of two teachers, the importance of including all students in classroom discourse and its influence on students’ mathematical thinking. Each classroom vignette illustrates one of four themes that emerged from the classroom discourse: (a) valuing students’ ideas, (b) exploring students’ answers, (c) incorporating students’ background knowledge, and (d) encouraging student-to-student communication. Recommendations for further research on classroom discourse in diverse settings are offered.     

Making sense of the traditional long division algorithm

This classroom scholarship report presents a group of elementary students’ experiences learning the traditional long division algorithm. The traditional long division algorithm is often taught mechanically, resulting in the student's performance of step-by-step procedures with no or weak understanding of the concept. While noting some initial difficulties, the class episodes in this article provide examples of internalization that highlight the active role of the learner in transforming concrete representations into an abstract algorithm. Several factors encouraged students to be deeply engaged in making sense of the long division algorithm: meaningful tasks based on a theoretically well-articulated curriculum, effective pedagogical measures, and dynamic class discussions.     

Double negative: The necessity principle,commognitive conflict,and negative number operations

We use the DNR framework to analyze a classroom episode introducing negative integer exponents, comparing and contrasting our analysis with Sfard's recent commognitive analysis of a similar episode concerning multiplication of signed numbers. Students in both episodes objected to the standard rules for integer products or exponents, and they persisted in preferring their own rules even after the teacher justified the standard ones. We examine how pattern-based justifications may not address students’ intellectual needs, and we suggest other pedagogical strategies that promote student reasoning.     

Exploring college students’ mental representations of inferential statistics

We report a case study that explored how three college students mentally represented the knowledge they held of inferential statistics, how this knowledge was connected, and how it was applied in two problem solving situations. A concept map task and two problem categorization tasks were used along with interviews to gather the data. We found that the students’ representations were based on incomplete statistical understanding. Although they grasped various concepts and inferential tests, the students rarely linked key concepts together or to tests nor did they accurately apply that knowledge to categorize word problems. We suggest that one reason the students had difficulty applying their knowledge is that it was not sufficiently integrated. In addition, we found that varying the instruction for the categorization task elicited different mental representations. One instruction was particularly effective in revealing students’ partial understandings. This finding suggests that modifying the task format as we have done could be a useful diagnostic tool.     

Issues in theorizing mathematics learning and teaching: A contrast between learning through activity and DNR research programs

By continuing a contrast with the DNR research program, begun in Harel and Koichu (2010), I discuss several important issues with respect to teaching and learning mathematics that have emerged from our research program which studies learning that occurs through students’ mathematical activity and indicate issues of complementarity between DNR and our research program. I make distinctions about what we mean by inquiring into the mechanisms of conceptual learning and how it differs from work that elucidates steps in the development of a mathematical concept. I argue that the construct of disequilibrium is neither necessary nor sufficient to explain mathematics conceptual learning. I describe an emerging approach to instruction aimed at particular mathematical understandings that fosters reinvention of mathematical concepts without depending on students’ success solving novel problems.     

Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes

This article reports on the activity of two pairs of sixth grade students who participated in an 8-month teaching experiment that investigated the students’ construction of fraction composition schemes. A fraction composition scheme consists of the operations and concepts used to determine, for example, the size of 1/3 of 1/5 of a whole in relation to the whole. Students’ whole number multiplicative concepts were found to be critical constructive resources for students’ fraction composition schemes. Specifically, the interiorization of two levels of units, a particular multiplicative concept, was found to be necessary for the construction of a unit fraction composition scheme, while the interiorization of three levels of units was necessary for the construction of a general fraction composition scheme. These findings contribute to previous research on students’ construction of fraction multiplication that has emphasized partitioning and conceptualizing quantitative units. Implications of the findings for teaching are considered.     

The fractional knowledge and algebraic reasoning of students with the first multiplicative concept

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of three different multiplicative concepts participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed.     

An exploration of students’ errors in derivatives in a university of technology

This paper reports on an exploration of errors that were displayed by students who studied mathematics in chemical engineering in derivatives of various functions such as algebraic, exponential, logarithmic and trigonometric functions. The participants of this study were a group of twenty students who were at risk in an extended curriculum programme in a university of technology in Western Cape, South Africa. The researcher used a qualitative case study approach and collected data from students’ written work. This research uses action, process, object, and schema (APOS) theory to classify errors into categories and to analyse and interpret the data collected. The students displayed five different kinds of errors, namely, conceptual, interpretation, linear extrapolation, procedural and arbitrary. The use of APOS theory as a framework revealed that several students’ errors might be caused by over-generalisation of mathematical rules and properties such as the power rule of differentiation and distributive property in manipulation of algebraic expressions. This study suggests that teaching of the standard rules of differentiation should put emphasis on its restrictions to eliminate common errors that normally crop up due to over-generalisation of certain differentiation rules.     

Algebra word problem solving approaches in a chemistry context: Equation worked examples versus text editing

Text editing directs students’ attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three equation steps each of which comprises essential units of relevant information. In an experiment, students were randomly assigned to either text editing or equation worked examples condition in a regular classroom setting to learn how to solve algebra word problems in a chemistry context. The equation worked examples group outperformed the text editing group for molarity problems, which were more difficult than dilution problems. Empirical evidence supports the theoretical rationale in using equation worked examples to facilitate students’ construction of a coherent problem structure so as to develop problem skills and expertise to solve molarity problems.     

Why do U.S. and Chinese students think differently in mathematical problem solving?: Impact of early algebra learning and teachers’ beliefs

This paper reports two studies that examined the impact of early algebra learning and teachers’ beliefs on U.S. and Chinese students’ thinking. The first study examined the extent to which U.S. and Chinese students’ selection of solution strategies and representations is related to their opportunity to learn algebra. The second study examined the impact of teachers’ beliefs on their students’ thinking through analyzing U.S. and Chinese teachers’ scoring of student responses. The results of the first study showed that, for the U.S. sample, students who have formally learned algebraic concepts are as likely to use visual representations as those who have not formally learned algebraic concepts in their problem solving. For the Chinese sample, students rarely used visual representations whether or not they had formally learned algebraic concepts. The findings of the second study clearly showed that U.S. and Chinese teachers view students’ responses involving concrete strategies and visual representations differently. Moreover, although both U.S. and Chinese teachers value responses involving more generalized strategies and symbolic representations equally high, Chinese teachers expect 6th graders to use the generalized strategies to solve problems while U.S. teachers do not. The research reported in this paper contributed to our understanding of the differences between U.S. and Chinese students’ mathematical thinking. This research also established the feasibility of using teachers’ scoring of student responses as an alternative and effective way of examining teachers’ beliefs.     

The impact of using multiple modalities on students’ acquisition of fractional knowledge: An international study in embodied mathematics across semiotic cultures

Principled by the Embodied, Situated, and Distributed Cognition paradigm, the study investigated the impact of using a research-based curriculum that employs multiple modalities on the performance of grade 5 students on 3 subscales: concept of unit, fraction equivalence, and fraction comparison. The sample included five schools randomly selected from a population of 14 schools in Lebanon. Eighteen 5th grade classrooms were randomly assigned to experimental (using multimodal curriculum) and control (using a monomodal curriculum) groups. Three data sources were used to collect quantitative and qualitative data: tests, interviews, and classroom observations. Quantitative data were analyzed using two methods: reliability and MANOVA. Results of the quantitative data show that students taught using the multimodal curriculum outperformed their counterparts who were instructed using a monomodal curriculum on the three aforementioned subscales (at an alpha level = .001). Additionally, fine-grained analysis using the semiotic bundle model revealed different semiotic systems across experimental and control groups. The study findings support the multimodal approach to teaching fractions as it facilitates students’ conceptual understanding.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.