Teaching algorithmic problem solving or conceptual understanding: Role of developmental level,mental capacity,and cognitive style

It has been shown previously that many students solve chemistry problems using only algorithmic strategies and do not understand the chemical concepts on which the problems are based. It is plausible to suggest that if the information is presented in differing formats, the cognitive demand of a problem changes. The main objective of this study is to investigate the degree to which cognitive variables, such as developmental level, mental capacity, and disembedding ability explain student performance on problems which: (1) could be addressed by algorithms or (2) require conceptual understanding. All conceptual problems used in this study were based on a figurative format. The results obtained show that in all four problems requiring algorithmic strategies, developmental level of the students is the best predictor of success. This could be attributed to the fact that these are basically computational problems, requiring mathematical transformations. Although all three problems requiring conceptual understanding had an important aspect in common (the figurative format), in all three the best predictor of success is a different cognitive variable. It was concluded that: (1) the ability to solve computational problems (based on algorithms) is not the major factor in predicting success in solving problems that require conceptual understanding; (2) solving problems based on algorithmic strategies requires formal operational reasoning to a certain degree; and (3) student difficulty in solving problems that require conceptual understanding could be attributed to different cognitive variables.     

Reasoning strategies of students in solving chemistry problems as a function of developmental level,functional M‐capacity and disembedding ability

Achievement in science depends among other factors on hypothetico‐deductive reasoning ability, that is, developmental level of the students. Recent research indicates that the developmental level of students should be studied along with individual difference variables, such as Pascual‐Leone's M‐capacity (information processing) and Witkin's Cognitive Style (disembedding ability). The purpose of this study is to investigate reasoning strategies of students in solving chemistry problems as a function of developmental level, functional M‐capacity and disembedding ability. A sample of 109 freshman students were administered tests of formal operational reasoning, functional M‐capacity, disembedding ability and chemistry problems (limiting reagent, mole, gas laws). Results obtained show that students who scored higher on cognitive predictor variables not only have a better chance of solving chemistry problems, but also demonstrated greater understanding and used reasoning strategies indicative of explicit problem‐solving procedures based on the hypothetico‐deductive method, manipulation of essential information and sensitivity to misleading information. It was also observed that students who score higher on cognitive predictor variables tend to anticipate important aspects of the problem situation by constructing general figurative and operative models, leading to a greater understanding. Students scoring low on cognitive predictor variables tended to circumvent cognitively more demanding strategies and adopt others that helped them to overcome the constraints of formal reasoning, information processing and disembedding ability.     

Facilitating problem solving in high school chemistry

The major purpose for conducting this study was to determine whether certain instructional strategies were superior to others in teaching high school chemistry students problem solving. The effectiveness of four instructional strategies for teaching problem solving to students of various proportional reasoning ability, verbal and visual preference, and mathematics anxiety were compared in this aptitude by treatment interaction study. The strategies used were the factor-label method, analogies, diagrams, and proportionality. Six hundred and nine high school students in eight schools were randomly assigned to one of four teaching strategies within each classroom. Students used programmed booklets to study the mole concept, the gas laws, stoichiometry, and molarity. Problem-solving ability was measured by a series of immediate posttests, delayed posttests and the ACS-NSTA Examination in High School Chemistry. Results showed that mathematics anxiety is negatively correlated with science achievement and that problem solving is dependent on students' proportional reasoning ability. The factor-label method was found to be the most desirable method and proportionality the least desirable method for teaching the mole concept. However, the proportionality method was best for teaching the gas laws. Several second-order interactions were found to be significant when mathematics anxiety was one of the aptitudes involved.     

Cognitive development and problem solving of Afro-American students in chemistry

The aims of this study were to investigate the level of cognitive development of Afro-American students enrolled in general chemistry courses at the college level and to determine the strategies used by both successful and unsuccessful Afro-American students in solving specific types of stoichiometric problems. It was found that the choice of a strategy is not significantly related to cognitive development of the student in specific types of stoichiometric problems. However, the following trend was noted: Students who are formal-operational in thought are more likely to be successful when solving mole-volume problems and complex mole-mole problems than are their concrete-operational counterparts. Additionally, a systematic strategy proved to be successful for the students, regardless of the cognitive development, when balancing simple and complex chemical equations. Also, algorithmic/reasoning strategies were needed to solve the mole-volume problem. A higher level of cognitive development and reasoning may be crucial factors in solving the more sophisticated types of problems in stoichiometry.     

Development and validation of a path analytic model of students' performance in chemistry

This article reports the development and validation of an integrated model of performance on a chemical concept - volumetric analysis. From the chemical literature a path-analytic model of performance on volumetric analysis calculation was postulated based on studies utilizing the proportional reasoning schema of Piaget and the Cumulative learning theory of Gagne. This integrated model hypothesized some relationships among the variables: direct proportional reasoning, inverse proportional reasoning, prerequisite concepts (content) and performance on volumetric analysis calculations. This model was postulated for the two groups of students involved in the study - that is those who use algorithms with understanding and those who use algorithms without understanding. Two hundred and sixty-five grade twelve chemistry students in eight schools (14 classes) in the lower mainland of British Columbia, Canada participated fully in the study. With the exception of the test on volumetric analysis calculations all the other tests were administered prior to the teaching of the unit on volumetric analysis. The results of the study indicate that for subjects using algorithms without understanding, their performance on VA problems is not influenced by proportional reasoning strategies while for those who use algorithms with understanding, their performance is influenced by proportional reasoning strategies.     

An Integrative Perspective on Students’ Proportional Reasoning in High School Physics in a West African Context

This study examines students’ use of proportional reasoning in high school physics problem‐solving in a West African school setting. An in‐depth, constructivist, and interpretive case study was carried out with six physics students from a co‐educational senior secondary school in Nigeria over a period of five months. The study aimed to elicit students’ meanings, claims, concerns, constructions, and interpretations of their difficulty with proportional reasoning as they worked on a series of 18 high school physics tasks. Multiple qualitative research techniques were employed to generate, analyse, and interpret data. Results indicated that several socio‐cultural, psychosocial, cognitive, and mathematical issues were associated with students’ use of proportional reasoning in physics. Students’ capacity to reason proportionally was not only linked to their difficulty with the concept, structure, and strategies of proportional reasoning as a learning and problem‐solving skill, but was also embedded in the social, cultural, cognitive, and contextual elements involved in the learning of physics. The study concludes with a discussion of the implications for teaching high school physics.     

Instructional influence of a molecular-level pictorial presentation of matter on students' conceptions and problem-solving ability

The instructional influence upon students' conceptions and problem-solving ability of presenting pictures at the molecular level when introducing chemistry concepts and solving chemistry problems was investigated. Before instruction, the Group Assessment of Logical Thinking (GALT) was administered and its score was used as a covariate. For the treatment group, 31 pictorial materials were used during 21 hours of Korean academic high school chemistry classes. For the control group, traditional instruction was used. Six classroom observations (1 hour each in duration) for each group were made. After instruction, the Chemistry Conceptions Test, and the Chemistry Problem-Solving Test (CPST) consisting of 10 pairs of pictorial and algorithmic problems, were administered. Korean students' success on pictorial questions from the CPST was higher than that reported in the literature for college students; however, Korean students did very poorly on algorithmic questions. The GALT score was significantly correlated with students' conceptions and problem-solving ability. Analysis of covariance results indicated that instruction with pictorial materials at the molecular level helped students construct more scientifically correct conceptions than traditional instruction. However, use of the pictorial materials had no facilitating effect on problem-solving ability. © 1997 John Wiley & Sons, Inc. J Res Sci Teach 34: 199–217, 1997.     

Differences in problem solving by nonscience majors in introductory chemistry on paired algorithmic-conceptual problems

The purpose of this investigation was to identify and describe the differences in the methods used by faculty teaching introductory chemistry and students enrolled in an introductory chemistry course at the university level to solve paired algorithmic and conceptual problems. Of the 180 students involved, the problem-solving schemas of 20 selected students and 2 professors were evaluated using a graphical method to dissect their think-aloud interviews into episodes indicative of solutions to paired problems on density, stoichiometry, bonding, and gas laws. The interviewed students were classified into four different problem-solving categories (i.e., high algorithmic/high conceptual, high algorithmic/low conceptual, low algorithmic/high conceptual, and low algorithmic/low conceptual), and composite graphs of their problem-solving schemas were compared to those representative of members of the faculty experts' category. Results of these comparisons indicated that as the students' ability to solve both algorithmic and conceptual problems improved, less time and fewer transitions between episodes of the problem-solving schemas were required to complete the problems. Regardless of the students' problem-solving ability, algorithmic-mode problems always required more time and a greater number of transitions for completion than did the paired conceptual-mode problems. However, regardless of topic, all students more frequently correctly solved the algorithmic-mode problems than the corresponding paired conceptual-mode problems. © 1997 John Wiley & Sons, Inc. J Res Sci Teach 34: 905–923, 1997.     

初中化学物质鉴别题的解答探究

在初中化学学习中,物质鉴别类的题目是考试中的热点与难点。提高学生解决这类题目的能力,对学生化学素养的培养有重要作用。文章从物质鉴别例题出发,为学生提出一些解题思路和技巧,以提升学生的解题速度,培养学生良好的化学推理思维。     

How Swedish pupils,aged 12‐15 years,understand light and its properties‡

The reasoning patterns used by a sample of Western Australian secondary school students aged 13‐16 were investigated with regard to the following reasoning modes: proportional reasoning, controlling variables, probabilistic reasoning, correlational reasoning, and combinatorial reasoning.

There was a wide range in students’ reasoning abilities at all year levels. Large percentages of students did not use formal operational reasoning patterns when they attempted to solve problems assessing their ability to use each of the five reasoning modes. Commonly used, but incorrect reasoning patterns were identified for each reasoning mode.

The students’ ability to use formal reasoning patterns was found to be an important factor in determining student achievement in lower secondary science, in their selection of year 11 science subjects, and their achievement in these subjects.

The results of the study indicate that it is important for teachers to be aware of the reasoning patterns of their students and the cognitive demands of course content, so that they can optimally match the content and their teaching strategies with the abilities of their students. Further research is needed to establish the nature of instruction which might best facilitate cognitive growth.     

Heuristic Reasoning in Chemistry: Making decisions about acid strength

The characterization of students’ reasoning strategies is of central importance in the development of instructional strategies that foster meaningful learning. In particular, the identification of shortcut reasoning procedures (heuristics) used by students to reduce cognitive load can help us devise strategies to facilitate the development of more analytical ways of thinking. The central goal of this qualitative study was thus to investigate heuristic reasoning as used by organic chemistry college students, focusing our attention on their ability to predict the relative acid strength of chemical compounds represented using explicit composition and structural features (i.e., structural formulas). Our results indicated that many study participants relied heavily on one or more of the following heuristics to make most of their decisions: reduction, representativeness, and lexicographic. Despite having visual access to reach structural information about the substances included in each ranking task, many students relied on isolated composition features to make their decisions. However, the specific characteristics of the tasks seemed to trigger heuristic reasoning in different ways. Although the use of heuristics allowed students to simplify some components of the ranking tasks and generate correct responses, it often led them astray. Very few study participants predicted the correct trends based on scientifically acceptable arguments. Our results suggest the need for instructional interventions that explicitly develop college chemistry students’ abilities to monitor their thinking and evaluate the effectiveness of analytical versus heuristic reasoning strategies in different contexts.     

Preservice Middle and High School Mathematics Teachers’ Strategies when Solving Proportion Problems

The purpose of this study was to investigate eight preservice middle and high school mathematics teachers’ solution strategies when solving single and multiple proportion problems. Real-world missing-value word problems were used in an interview setting to collect information about preservice teachers’ (PSTs) reasoning about proportional relationships. An explanatory case study methodology with multiple cases was used to make comparisons within and across cases. Analysis of the semi-structured interviews with each PST revealed that using practical problems, in which plastic gears and a mini balance system were provided, and multiple proportion problems facilitated the PSTs’ recognition of the proportional relationships in their solutions. Therefore, they avoided using cross-multiplication and erroneous strategies in those problems. Among the strategies that the PSTs used in solving single and multiple proportion problems, the ratio table strategy was the most frequent and effective strategy. The ratio table strategy enabled the PSTs to recognize the constant ratio and product relationships more than the other strategies. The results of this study illuminate how PSTs reason about proportional relationships when they cannot rely on computation methods like cross-multiplication.     

Cognitive strategies used by chemistry students to solve volumetric analysis problems

The study investigated the strategies used by 47 high school students to solve volumetric analysis problems in chemistry. Using the talking-aloud technique, the students were required to calculate the concentration of hydrochloric acid used in a titration with NaOH after having performed the titration themselves. Students were met individually and their verbalization audiotaped. After making this calculation, each student was asked to use the same data to predict the concentration of acid in three situations involving different mole ratios. It was found that two main strategies, Formula Approach and Proportional Approach with their variants, were employed by the students during the problem solving process. The Formula Approach was found to be used mainly by the students in the high ability group while students in the low ability group used the Proportional Approach. It was also found that problems involving 2:1 stoichiometric ratios presented a number of conceptual problems to the students. These conceptual problems were found to be related to their inability to write balanced equations or write correct formulas, focusing on only the strength of acid, inability to use the mole ratios in the calulations and deriving the mole ratios from the formulas of reactants.     

Middle School Students’ Reasoning in Nonlinear Proportional Problems in Geometry

In this study, we investigate sixth, seventh, and eighth grade students’ achievement in nonlinear (quadratic or cubic) proportional problems regarding length, area, and volume of enlarged figures. In addition, we examine students’ solution strategies for the problems and obstacles that prevent students from answering the problems correctly by using a mixed method research design. A total of 935 middle school students were given a paper-pencil test and 12 of them were interviewed. Findings indicated that achievement of the participants were low and that students used a limited number of strategies for solving the problems. In addition, these strategies were found to have lacked the argument of the linear proportional and nonlinear proportional relationships among length, area, and volume concepts for most of the participants’ answers. Moreover, analysis revealed that the confusion of linear proportional and nonlinear proportional relationships and misinterpretation of additive and multiplicative relationships were serious obstacles while solving the nonlinear proportional problems related to the area and volume of enlarged figures.     

Students’ Understanding of Conservation of Matter,Stoichiometry and Balancing Equations in Indonesia

This study examines Indonesian students’ understanding of conservation of matter, balancing of equations and stoichiometry. Eight hundred and sixty‐seven Grade 12 students from 22 schools across four different cities in two developed provinces in Indonesia participated in the study. Nineteen teachers also participated in order to validate the 25‐question survey used with all students. Significant differences in student success in answering specific questions occurred when comparing high‐achievement and low‐achievement schools. However, in general, student understanding of this fundamental principle in chemistry was low. The study found that the average score for all students on the survey was 41%. The findings suggest that students are most successful in solving problems used by teachers and textbooks that are algorithmic‐based (i.e., stoichiometry). As there were no strong positive correlations between student performance on conceptual questions and algorithmic questions, we suggest that further research should focus on teaching practices and curricula that support the development of the students’ conceptual understanding.     

Proportional Reasoning among 7th Grade Students with Different Curricular Experiences

Contextual problems involving rational numbers and proportional reasoning were presented to seventh grade students with different curricular experiences. There is strong evidence that students in reform curricula, who are encouraged to construct their own conceptual and procedural knowledge of proportionality through collaborative problem solving activities, perform better than students with more traditional, teacher-directed instructional experiences. Seventh grade students, especially those who study the new curricula, are capable of developing their own repertoire of sense-making tools to help them to produce creative solutions and explanations. This is demonstrated through analysis of solution strategies applied by students to a variety of rate problems.     

Factors in the development of proportional reasoning strategies by concrete operational college students

This study was designed as a test for two neo-Piagetian theories. More specifically, this research examined the relationships between the development of proportional reasoning strategies and three cognitive variables from Pascual-Leone's and Case's neo-Piagetian theories. A priori hypotheses linked the number of problems students worked until they induced a proportional reasoning strategy to the variables of M-space, degree of field dependence, and short-term storage space. The subjects consisted of students enrolled in Physical Science I, a science course for nonscience majors at the University of Southern Mississippi. Of the 34 subjects in the study, 23 were classified as concrete operational on the basis of eight ratio tasks. Problems corresponding to five developmental levels of proportional reasoning (according to Piagetian and neo-Piagetian theory), were presented by a microcomputer to the 23 subjects who had been classified as concrete operational. After a maximum of 6 hours of treatment, 17 of the 23 subjects had induced ratio schemata at the upper formal level (IIIB), while the remaining subjects used lower formal level (IIIA) schemata. The data analyses showed that neither M-space and degree of field-dependence, either alone or in combination, nor short-term storage predicted the number of problems students need to do until they induce an appropriate problem-solving strategy. However, there were significant differences in the short-term storage space of those subjects who mastered ratio problems at the highest level and those who did not. Also, the subjects' degree of field-dependence was not a predictor of either the ability to transfer problem-solving strategies to a new setting or the reuse of inappropriate strategies. The results of this study also suggest that short-term storage space is a variable with high correlations to a number of aspects of learning such as transfer and choice of strategy after feedback.     

Refinements in mathematics undergraduate students’ reasoning on completed infinite iterative processes

This paper reports a teaching experiment in which two students engaged in tasks that challenged them to describe a final state for a variety of infinite iterative processes. The results from the study indicate that the students used multiple reasoning strategies for addressing these tasks. Refinements in the students’ reasoning occurred as students constructed relationships between the problems they were solving and problems they had solved previously, applying some of the reasoning strategies that they used for one problem to make sense of or solve another problem. We discuss how these findings relate to the existing body of research on infinite iterative processes.     

Investigating Upper Secondary School Teachers’ Conceptions: Is Mathematical Reasoning Considered Gendered?

This study examines Swedish upper secondary school teachers’ gendered conceptions about students’ mathematical reasoning: whether reasoning was considered gendered and, if so, which type of reasoning was attributed to girls and boys. The sample consisted of 62 teachers from six different schools from four different locations in Sweden. The results showed that boys were significantly more often attributed to memorised reasoning and delimiting algorithmic reasoning. Girls were connected to gamiliar algorithmic reasoning, a reasoning type where you use standard method when solving a mathematical task. Creative mathematical founded reasoning, which is novel, plausible and founded in mathematical properties, was not considered gendered.