Accurate and Inaccurate Conceptions About Osmosis That Accompanied Meaningful Problem Solving

This study focused on the knowledge of six outstanding science students who solved an osmosis problem meaningfully. That is, they used appropriate and substantially accurate conceptual knowledge to generate an answer. Three generated a correct answer; three, an incorrect answer. This paper identifies both the accurate and inaccurate conceptions about osmosis of each correct and incorrect solver. The investigation consisted of a presolving clinical interview, think-aloud solving of the problem, and retrospective report of the solving. Of the 12 accurate conceptions identified here, two were especially important in enabling these solvers to generate a correct answer. Of the 8 inaccurate conceptions, either of 2 blocked a correct answer. Four, however, accompanied (and could therefore be concealed by) a correct answer. Teachers could use this information to make a meaningful solving of this problem accessible to more students and to identify more effectively students' inaccurate conceptions about osmosis.     

High School Biology Students' Knowledge and Certainty about Diffusion and Osmosis Concepts

The purpose of this study was to investigate students' understanding about scientifically acceptable content knowledge by exploring the relationship between knowledge of diffusion and osmosis and the student's certainty in their content knowledge. Data was collected from a high school biology class with the Diffusion and Osmosis Diagnostic Test (DODT) and Certainty of Response (CRI) scale. All data was collected after completion of a unit of study on diffusion and osmosis. The results of the DODT were dichotomized into correct and incorrect answers, and CRI values were dichotomized into certain and uncertain. Values were used to construct a series of 2 × 2 contingency tables for each item on the DODT and corresponding CRI. High certainty in incorrect answers on the DODT indicated tenacious misconceptions about diffusion and osmosis concepts. Low certainty in incorrect or correct answers on the DODT indicated possible guessing; and, therefore no understanding, or confusion about their understanding. Chi‐square analyses revealed that significantly more students had misconceptions than desired knowledge on content covering the Influence of Life Forces on Diffusion and Osmosis, Membranes, the Particulate and Random Nature of Matter, and the Processes of Diffusion and Osmosis. Most students were either guessing or had misconceptions about every item related to the concepts osmosis and tonicity. Osmosis and diffusion are important to understanding fundamental biology concepts, but the concept of tonicity not be introduced to high school biology students until effective instructional approaches can be identified by researchers.     

Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).     

Fifth graders’ additive and multiplicative reasoning: Establishing connections across conceptual fields using a graph

This paper looks at 21 fifth grade students as they discuss a linear graph in the Cartesian plane. The problem presented to students depicted a graph showing distance as a function of elapsed time for a person walking at a constant rate of 5 miles/h. The question asked students to consider how many more hours, after having already walked 4 h, would be required to reach 35 miles. To answer this question, the students needed to extend the graph that was presented, either mentally or on paper, as the axes did not go up to 7 h or 35 miles. They also needed to be able to consider not only the total number of hours to reach 35 miles, but also the interval of time after 4 h. The purpose of this paper is to consider the student responses from the viewpoint of multiplicative and additive reasoning, and specifically within Vergnaud's framework of multiplicative and additive conceptual fields and scalar and functional approaches to linear relationships (Vergnaud, 1994). The analysis shows that: some student answers cannot be classified as either scalar or functional; some students combined several kinds of approaches in their explanations; and that the representation of the problem using a graph may have facilitated responses that are different from those typically found when the representation presented is a function table.     

Examining middle school pre-service teachers’ knowledge of fraction division interpretations

This study investigated 11 pre-service middle school teachers’ solution strategies for exploring their knowledge of fraction division interpretations. Each participant solved six fraction division problems. The problems were organized into two sets: symbolic problems (involving numbers only) and contextual problems (involving measurement interpretation and the determination of unit rate interpretation). Results showed that most of the participants exhibited a great amount of procedural knowledge as they applied algorithms to obtain the correct answers to the symbolic problems. They also exhibited a great amount of conceptual understanding of how and why they obtained the correct answers to the contextual problems. However, the pre-service middle school teachers neither provided interpretations to the symbolic problems nor accepted that the contextual problems involved fraction division operation. The results suggest that the measurement and rate concepts were often unlinked to fraction division.     

Carpenter,tractors and microbes for the development of logical–mathematical thinking – the way 10th graders and pre-service teachers solve thinking challenges

The objective of this case study was to investigate the ability of 10th graders and pre-service teachers to solve logical–mathematical thinking challenges. The challenges do not require mathematical knowledge beyond that of primary school but rather an informed use of the problem representation. The percentage of correct answers given by the 10th graders was higher than that of the pre-service teachers. Unlike the 10th graders, some of whom used various strategies for representing the problem, most of the pre-service teachers’ answers were based on a technical algorithm, without using control processes. The obvious conclusion drawn from the findings supports and recommends expanding and enhancing the development of logical–mathematical thinking, both in specific lessons and as an integral part of other lessons in pre-service frameworks.     

On the continuity of functions

In this paper, results of a questionnaire about continuity of functions are analysed. The tested students attend Novi Sad grammar school. The aim of this test was to check the student's theoretical and visual knowledge of continuous functions at the end of their high school education. The questions were given with and without the graphs of functions. The main conclusion is that the graph of a function has a significant influence on the students opinion about continuity. Looking at the graphs of functions mostly led to correct answers, but sometimes it causes problems, if it has a break. The idea came from a 1981 paper by Tall and Vinner, where the “concept image” of a continuous functions was examined.     

Examining epistemic beliefs across conceptual and procedural knowledge in statistics

The purpose of this paper was to examine whether students’ epistemic beliefs differed as a function of variations in procedural versus conceptual knowledge in statistics. Students completed Hofer’s (Contem Edu Psychol 25:378–405, 2000) Discipline-Focused Epistemological Beliefs Questionnaire five times over the course of a semester. Differences were explored between students’ initial beliefs about statistics knowledge and their specific beliefs about conceptual knowledge and procedural knowledge in statistics. Results revealed differences across these contexts; students’ beliefs differed between procedural versus conceptual knowledge. Moreover, students’ initial beliefs about statistics knowledge were more similar to their beliefs about conceptual knowledge rather than procedural knowledge. Finally, regression analyses revealed that students’ beliefs about the justification of knowledge, attainability of truth and source of knowledge were significant predictors of examination performance, depending on the examination. These results have important theoretical, methodological and pedagogical implications.     

Middle School Mathematics Teachers Learning to Teach with Calculators and Computers

This paper reports the results of a project in which experienced middle grades mathematics teachers immersed themselves in calculator and computer use for both doing and teaching mathematics and prepared themselves as leaders for communicating their knowledge to colleagues. Project evaluation included interviews with participants at the beginning and end of the project and evaluation forms completed at the end of the project. Pre-interviews indicated that virtually all of the participants had no experience using technology to teach mathematics. Many felt that technology was not likely to be as effective in helping students learn mathematics as other teaching techniques. Post-interviews indicated that all teachers were confident of their abilities to use some technologies in teaching mathematics. They acknowledged that technology was useful in developing conceptual understanding and that their role was to guide this conceptual development. The differences in participants' perceptions about how the project affected them yielded suggestions for future inservice efforts about technology.     

Analogies and Reconstruction of Probability Knowledge

The effectiveness of utilizing analogies to effect conceptual change in students' alternative probability concepts was investigated. Forty-one senior high school mathematics students were engaged in a knowledge reconstruction process regarding their beliefs about common everyday probability situations, such as sports events or lotteries. The students were given situations similar to those shown in previous research to reveal alternative mathematical conceptions. They were also given analogous researcher-generated anchoring situations that had been pretested and found to elicit mathematically acceptable responses. The cognitive dissonance produced by the conflicting responses motivated students to reconstruct their knowledge. The results of the investigation showed that analogies can be effective in producing a desired conceptual change in high school students' probability concepts.     

Students’ experiences with mathematics teaching and learning: listening to unheard voices

This study documents students’ views about the nature of mathematics, the mathematics learning process and factors within the classroom that are perceived to impact upon the learning of mathematics. The participants were senior secondary school students. Qualitative and quantitative methods were used to understand the students’ views about their experiences with mathematics learning and mathematics classroom environment. Interviews of students and mathematics lesson observations were analysed to understand how students view their mathematics classes. A questionnaire was used to solicit students’ views with regards to teaching approaches in mathematics classes. The results suggest that students consider learning and understanding mathematics to mean being successful in getting the correct answers. Students reported that in the majority of cases, the teaching of mathematics was lecture-oriented. Mathematics language was considered a barrier in learning some topics in mathematics. The use of informal language was also evident during mathematics class lessons.     

Evaluating students’ conceptual and procedural understanding of sampling distributions

Introductory statistics courses, which are important in preparing students for their daily lives, generally derive inferential statistics from informal knowledge. In this transition process, sampling distributions have an important place, yet research has shown that students often have difficulties with this concept. In order to increase their understanding of sampling distributions, students should have a strong conceptual foundation that is balanced with procedural knowledge. To address this issue, this study was designed to examine the relationship between college students’ procedural and conceptual knowledge of sampling distributions. With this aim in mind, an achievement test consisting of two sections – procedural and conceptual knowledge – was prepared. In answering the questions related to procedural knowledge, the participants were more successful in identifying the relationship between standard deviation of a population and sample means. However, they lacked theoretical knowledge about statements that they had heard or knew intuitively. Simulation activities provided in statistics courses may support students in developing their conceptual understanding in this regard.     

Students' Concepts‐ and Theorems‐in‐Action on a Novel Task About Similarity

Similarity is a fundamental concept in the middle grades. In this study, we applied Vergnaud's theory of conceptual fields to answer the following questions: What concepts‐in‐action and theorems‐in‐action about similarity surfaced when students worked in a novel task that required them to enlarge a puzzle piece? How did students use geometric and multiplicative reasoning at the same time in order to construct similar figures? We found that students used concepts of scaling and proportional reasoning, as well as the concept of circle and theorems about similar triangles, in their work on the problem. Students relied not only on visual perception, but also on numeric reasoning. Moreover, students' use of multiplicative and proportional concepts supported their geometric constructions. Knowledge of the concepts and ideas that students have available when working on a task about similarity can inform instruction by helping to ground formal introduction of new concepts in students' informal prior experiences and knowledge.     

Promoting Percent as a Proportion in Eighth‐Grade Mathematics

The literature provides many and varied suggestions for promoting conceptual understanding of percent and performing percent calculations. The diversity of ideas provides a wide selection but offers little clarity on the true nature of percent. From the premise that percent is fundamentally a proportion, this study incorporated a proportional approach for percent problem solving within an instructional program on percent. Classroom research with eighth‐grade students indicated that the method was readily adopted by students and helped them experience success in percent problem solving, with percent problem solving proficiency maintained over a delayed period. It is hypothesized that the method has the potential to promote students' conceptual knowledge of percent as a proportion and the multiplicative structure of percent, as well as to build proportional knowledge.     

Discovering and addressing errors during mathematics problem-solving—A productive struggle?

The present study investigates students’ struggles when encountering errors in problem-solving. The focus is students’ problem-solving activities that lead to productive struggle and what the students might gain therefrom. Twenty-four students between the ages of 16 and 17 worked in pairs to solve a linear function problem using GeoGebra, a dynamic software application. Data in the form of recorded conversations, computer activities and post-interviews were analyzed using Hiebert and Grouws’ (2007. Second handbook of research on mathematics teaching and learning (Vol. 1). 404) concept of productive struggles and Schoenfeld's (1985. Mathematical problem solving: ERIC) framework for problem-solving. The study showed that all students made errors concerning incorrect prior knowledge and erroneously constructed new knowledge. All participants engaged in superficial, unproductive struggles moving between a couple of Schoenfeld's episodes. However, a majority of the students managed to transform their efforts into productive struggle. They engaged in several of Schoenfeld's episodes and succeeded in reconstructing useful prior knowledge and constructing correct new knowledge—i.e., solving the problem.     

The role of problem representation and feature knowledge in algebraic equation-solving

Domain experts have two major advantages over novices with regard to problem solving: experts more accurately encode deep problem features (feature encoding) and demonstrate better conceptual understanding of critical problem features (feature knowledge). In the current study, we explore the relative contributions of encoding and knowledge of problem features (e.g., negative signs, the equals sign, variables) when beginning algebra students solve simple algebraic equations. Thirty-two students completed problems designed to measure feature encoding, feature knowledge and equation solving. Results indicate that though both feature encoding and feature knowledge were correlated with equation-solving success, only feature knowledge independently predicted success. These results have implications for the design of instruction in algebra, and suggest that helping students to develop feature knowledge within a meaningful conceptual context may improve both encoding and problem-solving performance.     

A conceptual pathway to confidence intervals

Finding ways for the majority of students to better understand conventional normal theory-based statistical inference seems to be an intractable problem area for researchers. In this paper we propose a conceptual pathway for developing confidence interval ideas for the one-sample situation only from an intuitive sense to bootstrapping for students from about age 14 to first-year university. We make the case that conceptual development should start early; that probability and statistical instruction should change so that both orientate students towards interconnected stochastic conceptions; and that the use of visual imagery has the potential to stimulate students towards such a perspective. We analyse our conceptual pathway based on a theoretical framework for a stochastic conception of statistical inference based on imagery and some research evidence. Our analysis suggests that the pathway has the potential for students to become conversant with the concepts underpinning inference, to view statistics probabilistically and to integrate concepts into a coherent comprehension of inference.     

Comparing German and Taiwanese secondary school students’ knowledge in solving mathematical modelling tasks requiring their assumptions

Mathematization is critical in providing students with challenges for solving modelling tasks. Inadequate assumptions in a modelling task lead to an inadequate situational model, and to an inadequate mathematical model for the problem situation. However, the role of assumptions in solving modelling problems has been investigated only rarely. In this study, we intentionally designed two types of assumptions in two modelling tasks, namely, one task that requires non-numerical assumptions only and another that requires both non-numerical and numerical assumptions. Moreover, conceptual knowledge and procedural knowledge are also two factors influencing students’ modelling performance. However, current studies comparing modelling performance between Western and non-Western students do not consider the differences in students’ knowledge. This gap in research intrigued us and prompted us to investigate whether Taiwanese students can still perform better than German students if students’ mathematical knowledge in solving modelling tasks is differentiated. The results of our study showed that the Taiwanese students had significantly higher mathematical knowledge than did the German students with regard to either conceptual knowledge or procedural knowledge. However, if students of both countries were on the same level of mathematical knowledge, the German students were found to have higher modelling performance compared to the Taiwanese students in solving the same modelling tasks, whether such tasks required non-numerical assumptions only, or both non-numerical and numerical assumptions. This study provides evidence that making assumptions is a strength of German students compared to Taiwanese students. Our findings imply that Western mathematics education may be more effective in improving students’ ability to solve holistic modelling problems.