Types of reasoning required in university exams in mathematics

Empirical research shows that students often use reasoning founded on copying algorithms or recalling facts (imitative reasoning) when solving mathematical tasks. Research also indicate that a focus on this type of reasoning might weaken the students’ understanding of the underlying mathematical concepts. It is therefore important to study the types of reasoning students have to perform in order to solve exam tasks and pass exams. The purpose of this study is to examine what types of reasoning students taking introductory calculus courses are required to perform. Tasks from 16 exams produced at four different Swedish universities were analyzed and sorted into task classes. The analysis resulted in several examples of tasks demanding different types of mathematical reasoning. The results also show that about 70% of the tasks were solvable by imitative reasoning and that 15 of the exams could be passed using only imitative reasoning.     

Understanding the concepts of proportion and ratio among grade nine students in Malaysia

The purpose of the study was to investigate the concepts of ratio and proportion constructed by grade nine students by investigating grade nine students proportional reasoning schemes and procedures on three types of tasks: missing value, numerical comparison and qualitative reasoning. Comparisons among the different categories were made and the strategies used in solving these problems were identified. The relationship between student grades on a national examination and their knowledge of proportional reasoning was determined. The results of the quantitative analysis indicated that students performed generally well on the missing value tasks but their scores on the numerical comparison and qualitative tasks were much lower. The results indicate that only a small percentage of students who did well on the national exams were able to solve complex proportional problems and the grades obtained were not indicative of their knowledge of ratio and proportion. The difficulty experienced by the ninth graders indicated that students frequently used additive reasoning, that is a comparison of two numbers by subtraction rather than division. It appears that students cannot begin to understand the functional and scalar relationship inherent in a proportion until they first develop multiplicative reasoning.     

Mathematical Dimensions of Students' Use of Proportional Reasoning In High School Physics

This study examines the mathematical processes used by students when solving physics tasks requiring proportional reasoning. The study investigates students' understanding and explanations of their mathematical processes. A qualitative and interpretive case study was conducted with 6 students from a coeducational urban high school for 5 months. Students were engaged with some high school physics tasks requiring proportional reasoning, during which a hermeneutic dialectic design was used to investigate their processes, understandings, and difficulties. Research techniques such as interviews, dialectical discourses, journal dialogue, and video and audio recordings were employed to generate, analyze, and interpret data. Results of the study indicate that the students employed mathematical proportional reasoning patterns and algorithms which they could not explain. Students also had difficulties translating physics tasks into mathematical statements, symbols, and relations. Students could not perform mathematical operations that were not directly obvious from the physics tasks, and some had difficulty with division. Students did not have adequate understanding of the mathematical processes involved in proportional reasoning.     

The curriculum’s view of knowledge transferred to national tests in mathematics in Sweden

This article provides a report on the testing system in Sweden. The Swedish assessment system today is goal and knowledge related. Important for this system is the view of knowledge in the curriculum. In this article the authors give an overview of the view of knowledge in the curriculum and in the syllabi in mathematics and they describe how this influences the tasks in the national tests. The change from a norm-referenced to a goal and criteria referenced grading system influences how students' solutions of different tasks will be marked and assessed. In this article the authors also describe different assessment methods e.g. the assessment matrix, which focuses on different aspects of knowledge based on the syllabi in mathematics.     

Investigating written expressions of mathematical reasoning for students with learning disabilities

Assessment results from two open-construction response mathematical tasks involving fractions and decimals were used to investigate written expression of mathematical reasoning for students with learning disabilities. The solutions and written responses of 51 students with learning disabilities in fourth and fifth grade were analyzed on four primary dimensions: (a) accuracy, (b) five elements of mathematical reasoning, (c) five elements of mathematical writing, and (d) vocabulary use. Results indicate most students were not accurate in their problem solution and communicated minimal mathematical reasoning in their written expression. In addition, students tended to use general vocabulary rather than academic precise math vocabulary and students who provided a visual representation were more likely to answer accurately. To further clarify the students struggles with mathematical reasoning, error analysis indicated a variety of error patterns existed and tended to vary widely by problem type. Our findings call for more instruction and intervention focused on supporting students mathematical reasoning through written expression. Implications for research and practice are presented.     

The State of Proof in Finnish and Swedish Mathematics Textbooks—Capturing Differences in Approaches to Upper-Secondary Integral Calculus

Students’ difficulties with proof, scholars’ calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related reasoning in integral calculus, a key subject in transitioning from secondary to tertiary education. We analyze expository sections and nearly 2000 students’ exercises in the four most frequently used Finnish and Swedish textbook series. Results indicate that Finnish textbooks offer more opportunities for learning proof than do Swedish textbooks. Proofs are also more visible in Finnish textbooks than in Swedish materials, but the tasks in the latter reflect a higher variation in nature of proof-related reasoning. Our results are compared with methodologically similar U.S. studies. Consequences for learning and transition to university mathematics, as well as directions for future research, are discussed.     

From fractions to proportional reasoning: a cognitive schemes of operation approach

Four seventh grade students participated in a constructivist teaching experiment in which manipulatives within a computer microworld were used to solve fractional reasoning tasks followed by tasks that involve concepts of rate, ratio and proportion. Through a retrospective analysis of video tapes, their thinking processes were analyzed from the perspective of the types of cognitive schemes of operation used as they engaged in the given problem situations. One result of the study indicates that the modifications of the students’ available schemes of operation when solving the fractional reasoning tasks formed a basis for the cognitive schemes of operation used in their solutions of tasks involving proportionality.     

Analyzing Student Work as a Professional Development Activity

When worthwhile mathematical tasks are used in classrooms, they should also become a crucial element of assessment. For teachers, using these tasks in classrooms requires a different way to analyze student thinking than the traditional assessment model. Looking carefully at students' written work on worthwhile mathematical tasks and listening carefully while students explore these worthwhile tasks can contribute to a teacher's professional development. This paper reports on a professional development activity in which teachers analyzed mathematical tasks, predicted students' achievement on tasks, evaluated students' written work, listened to students' reasoning, and assessed students' understanding. Teachers' engagement in this way can help them develop flexibility and proficiency in the evaluation of their own students' work. These experiences allow teachers the opportunity to recognize students' potential, strengthen their own mathematical understanding, and engage in conversations with peers about assessment and instruction.     

Assessing Students' Mathematical Communication

Assessment of students' mathematical communication through the use of open-ended tasks and scoring procedures is addressed, as is the use of open-ended tasks to assess students' mathematical communication by providing students opportunities to display their mathematical thinking and reasoning. Also, two scoring procedures (quantitative holistic scoring procedure and qualitative analytic scoring procedure) are described for examining students' communication skills.     

Analyzing students’ communication and representation of mathematical fluency during group tasks

Findings discussed in this paper are from a larger research project exploring mathematical fluency characteristics, and teacher noticing and interpreting of mathematical fluency. The current study involved students from seven primary classes (Kindergarten – Grade 6, N = 63 students) and investigated students’ written work samples and oral discussions as they collaborated in small groups to solve mathematical tasks. Students displayed mathematical fluency both orally and in written/drawn form. Certain aspects of mathematical fluency were easier to identify orally (adaptive reasoning) particularly for younger students and when students did not provide any written reasoning. Analyzing the oral responses was often needed to identify mathematical fluency beyond knowledge of a correct procedure (strategic competence). Findings suggested that the various representations students used were valuable for observing mathematical fluency. These results suggest that oral assessments as a means to understand and interpret students’ mathematical fluency are necessary.     

The effect of explanations on mathematical reasoning tasks

Studies in mathematics education often point to the necessity for students to engage in more cognitively demanding activities than just solving tasks by applying given solution methods. Previous studies have shown that students that engage in creative mathematically founded reasoning to construct a solution method, perform significantly better in follow up tests than students that are given a solution method and engage in algorithmic reasoning. However, teachers and textbooks, at least occasionally, provide explanations together with an algorithmic method, and this could possibly be more efficient than creative reasoning. In this study, three matched groups practiced with either creative, algorithmic, or explained algorithmic tasks. The main finding was that students that practiced with creative tasks did, outperform the students that practiced with explained algorithmic tasks in a post-test, despite a much lower practice score. The two groups that got a solution method presented, performed similarly in both practice and post-test, even though one group got an explanation to the given solution method. Additionally, there were some differences between the groups in which variables predicted the post-test score.     

Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).     

Mathematical Thinking Involved in U.S. and Chinese Students' Solving of Process-Constrained and Process-Open Problems

This study examined U.S. and Chinese 6th-grade students' mathematical thinking and reasoning involved in solving 6 process-constrained and 6 process-open problems. The Chinese sample (from Guiyang, Guizhou) had a significantly higher mean score than the U.S. sample (from Milwaukee, Wisconsin) on the process-constrained tasks, but the sample of U.S. students had a significantly higher mean score than the sample of the Chinese students on the process-open tasks. A qualitative analysis of students' responses was conducted to understand the mathematical thinking and reasoning involved in solving these problems. The qualitative results indicate that the Chinese sample preferred to use routine algorithms and symbolic representations, whereas the U.S. sample preferred to use concrete visual representations. Such a qualitative analysis of students' responses provided insights into U.S. and Chinese students' mathematical thinking, thereby facilitating interpretation of the cross-national differences in solving the process-constrained and process-open problems.     

Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).     

Mathematical Thinking Involved in U.S. and Chinese Students' Solving of Process-Constrained and Process-Open Problems

This study examined U.S. and Chinese 6th-grade students' mathematical thinking and reasoning involved in solving 6 process-constrained and 6 process-open problems. The Chinese sample (from Guiyang, Guizhou) had a significantly higher mean score than the U.S. sample (from Milwaukee, Wisconsin) on the process-constrained tasks, but the sample of U.S. students had a significantly higher mean score than the sample of the Chinese students on the process-open tasks. A qualitative analysis of students' responses was conducted to understand the mathematical thinking and reasoning involved in solving these problems. The qualitative results indicate that the Chinese sample preferred to use routine algorithms and symbolic representations, whereas the U.S. sample preferred to use concrete visual representations. Such a qualitative analysis of students' responses provided insights into U.S. and Chinese students' mathematical thinking, thereby facilitating interpretation of the cross-national differences in solving the process-constrained and process-open problems.     

Promoting uncertainty to support preservice teachers’ reasoning about the tangent relationship

Including opportunities for students to experience uncertainty in solving mathematical tasks can prompt learners to resolve the uncertainty, leading to mathematical understanding. In this article, we examine how preservice secondary mathematics teachers’ thinking about a trigonometric relationship was impacted by a series of tasks that prompted uncertainty. Using dynamic geometry software, we asked preservice teachers to compare angle measures of lines on a coordinate grid to their slope values, beginning by investigating lines whose angle measures were in a near-linear relationship to their slopes. After encountering and resolving the uncertainty of the exact relationship between the values, preservice teachers connected what they learned to the tangent relationship and demonstrated new ways of thinking that entail quantitative and covariational reasoning about this trigonometric relationship. We argue that strategically using uncertainty can be an effective way of promoting preservice teachers’ reasoning about the tangent relationship.     

Robot Manipulations: A Synergy of Visualization, Computation and Action for Spatial Instruction

This article considers the use of a learning environment, RoboCell, where manipulations of objects are performed by robot operations specified through the learner's application of mathematical and spatial reasoning. A curriculum is proposed relating to robot kinematics and point-to-point motion, rotation of objects, and robotic assembly of spatial puzzles. Various instructional methods are supported by the RoboCell robot system, such as interactive demonstrations, modeling, computer simulations and robot operations, providing diverse activities in spatial perception, mental rotation and visualization. Pre-course and post-course tests in two middle schools and a high school indicated significant student progress in the tasks related to the categories of spatial ability which were practiced in the course. This revised version was published online in July 2006 with corrections to the Cover Date.     

Creative and algorithmic mathematical reasoning: effects of transfer-appropriate processing and effortful struggle

Two separate studies, Jonsson et al. (J. Math Behav. 2014;36: 20–32) and Karlsson Wirebring et al. (Trends Neurosci Educ. 2015;4(1–2):6–14), showed that learning mathematics using creative mathematical reasoning and constructing their own solution methods can be more efficient than if students use algorithmic reasoning and are given the solution procedures. It was argued that effortful struggle was the key that explained this difference. It was also argued that the results could not be explained by the effects of transfer-appropriate processing, although this was not empirically investigated. This study evaluated the hypotheses of transfer-appropriate processing and effortful struggle in relation to the specific characteristics associated with algorithmic reasoning task and creative mathematical reasoning task. In a between-subjects design, upper-secondary students were matched according to their working memory capacity.

The main finding was that the superior performance associated with practicing creative mathematical reasoning was mainly supported by effortful struggle, however, there was also an effect of transfer-appropriate processing. It is argued that students need to struggle with important mathematics that in turn facilitates the construction of knowledge. It is further argued that the way we construct mathematical tasks have consequences for how much effort students allocate to their task-solving attempt.     

Gender differences in the mathematics achievements of German primary school students: results from a German large-scale study

In Germany, national standards for mathematics for the end of primary school were established in 2004. In the present study, data were collected to evaluate these standards, and were used to compare the mathematical skills of girls and boys. Many studies have shown that gender differences are strongest at the highest levels of education. The findings from primary school are less consistent. Thus, in our study we analyzed achievement differences in a sample of approximately 10,000 third and fourth graders, representative of the German elementary school population. Gender-specific competencies were compared in the different content domains, both for the general mathematical competence, and for the cognitive levels of the tasks. Overall, boys outperformed girls, but substantial variation was found between the content domains and general mathematical achievement. Differences were higher in grade three than in grade four. The proportion of boys in the classroom did not appear to affect the individual level of performance. Analysis of the items on which boys or girls clearly outperformed each other reproduced a pattern of specific item characteristics predicting gender bias consistent with those reported in previous studies in other countries.