Diagnosing misconceptions: Revealing changing decimal fraction knowledge

Conceptual change is a gradual process that occurs as students integrate new information into their existing conceptions. Throughout this process, assessing learning requires measures to diagnose misconceptions and understand how knowledge is changing. We developed three measures of misconceptions to assess students' knowledge early in instruction on decimals that measured the: 1) prevalence of misconception errors based on response patterns, 2) existence of misconceptions in a more abstract context, and 3) strength of misconceptions using confidence ratings. Students ages 9–11 (N = 297) completed the assessment at three time points. These measures revealed that whole number and role of zero misconceptions decreased and fraction misconceptions increased over time. The current measures also differentiated between weaker misconceptions that were changed after brief instruction and strongly held misconceptions. The current measures can create a more complete picture of knowledge than only measuring students' accuracy, providing a window into the conceptual change process.     

Using heuristic worked examples to promote inquiry-based learning

Inquiry learning can be facilitated by having students investigate the domain through a computer simulation and express their acquired understanding in a runnable computer model. This study investigated whether heuristic worked examples can further enhance students' inquiry behaviour, the quality of the models they create, and their domain knowledge. High-school students were offered a simulation of an electrical circuit and a modelling tool. Students in the experimental condition (n = 46) could consult heuristic worked examples that explained what activities were needed and how they should be performed. Students in the control condition (n = 36) did not receive this support. Cross-condition comparisons confirmed that heuristic worked examples improved students' inquiry behaviour and enhanced the quality of their models. However, few students created a model that reflected full understanding of the electrical circuit, and the expected between-group difference in posttest scores failed to appear. Based on these findings, improvements to the design of heuristic worked examples are proposed.     

Using learner generated examples to introduce new concepts

In this paper we describe learners being asked to generate examples of new mathematical concepts, thus developing and exploring example spaces. First we elaborate the theoretical background for learner generated examples (LGEs) in learning new concepts. The data we then present provides evidence of the possibility of learning new concepts through a symbiosis of induction and abduction from experience and deduction from the relationships generated in exemplification. In other words, experience can be organised in such a way that shifts of understanding take place as a result of learners’ own actions. Actions, in this context, include mental acts of organisational reflection.

Example-based learning: The benefits of prompting organization before providing examples

Example-based learning often follows a design in which learners first receive instructional explanations that communicate new principles and concepts and second examples thereof. In this setting, using the knowledge components of the instructional explanations to explain the examples (i.e., generating principle-based self-explanations) is considered to be a highly important learning process. However, a potential suboptimality of this learning process is that it scarcely requires learners to organize the content of the instructional explanations into coherent mental representations. Thus, in two experiments we investigated whether prompting learners to organize the content of the instructional explanations before providing them with the examples (and self-explanation prompts) enhances the effectiveness of example-based learning. We consistently found that organization prompts fostered learning regardless of whether the learners also received self-explanation prompts. Hence, in example-based learning, learners should be prompted to not only generate principle-based self-explanations but also to organize the content of the instructional explanations.     

Enhancing scientific discovery learning through metacognitive support

Using a virtual physics lab, we analyzed the impact of metacognitive support on simulation-based scientific discovery learning (SDL). The dependent variables for learning outcome were the immediate conceptual knowledge gain and the retained conceptual knowledge three weeks later. Additional dependent variables were the actual use of a domain-specific cognitive strategy, motivation, emotions, and cognitive load. To contrast the effects of metacognitive support with possible effects of goal specificity, the experimental study followed a 2 × 2 design with a sample of N = 129 ninth grade students and with metacognitive support (yes vs. no) and learning goals (specific vs. nonspecific) as factors. The results showed positive effects of metacognitive support on learning outcome, on actual cognitive strategy use, and on learning emotions. No interaction effect of metacognitive support and goal specificity on learning outcome was observed.     

Example-based learning: Effects of different types of examples on student performance,cognitive load and self-efficacy in a statistical learning task

Previous research has indicated the disconnect between example-based research focusing on worked examples (WEs) and that focusing on modeling examples. The purpose of this study was to examine and compare the effect of four different types of examples from the two separate lines of research, including standard WEs, erroneous WEs, expert (masterly) modeling examples, and peer (coping) modeling examples, on student performance (knowledge retention, near transfer, and far transfer), cognitive load, and self-efficacy. One hundred and sixteen students participated in the study by undergoing computer-based instruction in one of the four versions differing in how examples were provided. The results showed that, overall, expert modeling examples were most effective in promoting knowledge retention, near transfer, and far transfer, while peer modeling examples were shown to be superior in fostering self-efficacy among the four different types of examples.     

Effects of worked examples, example-problem, and problem-example pairs on novices’ learning

Research has demonstrated that instruction that relies more heavily on example study is more effective for novices’ learning than instruction consisting of problem solving. However, ‘a heavier reliance on example study’ has been implemented in different ways. For example, worked examples only (WE), example-problem pairs (WE-PS), or problem-example pairs (PS-WE) have been used. This study investigated the effectiveness of all three strategies compared to problem solving only (PS), using electrical circuits troubleshooting tasks; participants were secondary education students who were novices concerning those tasks. Based on prior research, it was hypothesized and confirmed that WE and WE-PS would lead to lower cognitive load during learning and higher learning outcomes than PS. In addition, the open questions of whether there would be any differences between WE and WE-PS, and whether there would be any differences between PS-WE and PS were explored. Results showed no differences between WE and WE-PS or between PS-WE and PS. This study can inform instructional designers on which example-based learning strategies to implement: it does not seem necessary to alternate example study and problem solving, but when doing so, example-problem pairs should be used rather than problem-example pairs.     

Profiles of algebraic competence

The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map algebraic symbols onto word problems was evaluated. Two weeks later, unassisted algebraic problem-solving tests were completed. Cluster analyses revealed four meaningful, relatively well-ordered, arithmetic/algebraic competence profiles reflecting partial knowledge states associated with the acquisition of algebraic understanding. The findings are discussed in terms of the conceptual changes associated with the acquisition of algebraic competence.     

The diffusion of the learning pyramid myths in academia: an exploratory study

This article examines the diffusion and present day status of a family of unsubstantiated learning-retention myths, some of which are referred to as ‘the learning pyramid’. We demonstrate through an extensive search in academic journals and field-specific encyclopaedias that these myths are indeed widely publicised in academia and that they have gained a considerable level of authority. We also argue that the academic publishing of these myths is potentially harmful to both professional as well as political deliberations on educational issues, and therefore should be criticized and counteracted.     

Effects of different sequences of examples and problems on motivation and learning

Recent research has shown that example study only (EE) and example-problem pairs (EP) were more effective (i.e., higher test performance) and efficient (i.e., attained with less effort invested in learning and/or test tasks) than problem-example pairs (PE) and problem solving only (PP). We conducted two experiments to investigate how different example and problem-solving sequences would affect motivational (i.e., self-efficacy, perceived competence, and topic interest) and cognitive (i.e., effectiveness and efficiency) aspects of learning. In Experiment 1, 124 technical students learned a mathematical task with the help of EEEE, EPEP, PEPE, or PPPP and then completed a posttest. Students in the EEEE Condition showed higher posttest performance, self-efficacy, and perceived competence, attained with less effort investment, than students in the EPEP and PPPP Condition. Surprisingly, there were no differences between the EPEP and PEPE Condition on any of the outcome measures. We hypothesized that, because the tasks were relevant for technical students, starting with a problem might not have negatively affected their motivation. Therefore, we replicated the experiment with a different sample of 81 teacher training students. Experiment 2 showed an efficiency benefit of EEEE over EPEP, PEPE, and PPPP. However, only EEEE resulted in greater posttest performance, self-efficacy, and perceived competence than PPPP. We again did not find any differences between the EPEP and PEPE Condition. These results suggest that, at least when short training phases are used, studying examples (only) is more preferable than problem solving only for learning. Moreover, this study showed that example study (only) also enhances motivational aspects of learning whereas problem solving only does not positively affect students’ motivation at all.     

Heuristic for learning common emitter amplification with bipolar transistors

Mathematics in engineering education causes many thresholds in the courses because of the demand of abstract conceptualisation. Electronics depend heavily on more or less complex mathematics. Therefore the concepts of analogue electronics are hard to learn since a great deal of students struggle with the calculations and procedures needed. A survey was done focusing on students’ struggle to pass a course in analogue electronics by introducing a top-down perspective and the revised taxonomy of Bloom. From a top-down perspective you can create learning environments from any spot in the taxonomy using a step-by-step approach of the verbs understand and apply. Three textbooks with a top-down perspective on analogue electronics are analysed on the concept of amplifying with a transistor circuit. The study claims issues when losing the top-down perspective to present concepts and procedures of the content to be learned.     

Delayed benefits of learning elementary algebraic transformations through contrasted comparisons

Students studying algebra often make mistakes because of superficial similarities between addition and multiplication problems. In two experiments, we investigated whether these errors can be prevented by presenting addition and multiplication problems in such a way that students are encouraged to compare the problems at a deeper level. In Experiment 1, 72 sixth graders were assigned to two self-learning programs. In the contrast program, addition and multiplication were mixed and juxtaposed. In the sequential program, students first received only addition problems followed by multiplication problems. The results revealed that during the training, students performed worse under the contrast condition. However, in the follow-up tests (1-day, 1-week, 3-months), these findings were reversed: the contrast group clearly outperformed the sequential group. The findings were replicated under improved methodological conditions in Experiment 2 with 154 sixth graders. These experiments show that contrasted comparison of superficially similar but conceptually different material results in improved long-term learning.     

Conceptual and procedural approaches to mathematics in the engineering curriculum: views of qualified engineers*

The research interest underpinning this paper concerns the type of mathematical knowledge engineering students may acquire during their specialised education in terms of the conceptual and procedural dimensions of doing and using mathematics. This study draws on interviews with 25 qualified engineers from South Africa and Sweden regarding their views on the role of mathematics in engineering education, with special focus on the conceptual and procedural aspects of mathematical knowledge. A thematic analysis of the interview data led to the identification of two main themes. According to the conceptual view a predominantly conceptual approach is needed and valued more than procedural skills, while the balanced view emphasises a balance of conceptual understanding and procedural fluency as well as links between them. It is suggested that the mathematical education of engineers would need to be more conceptually oriented to prepare for the demands at the workplace.     

An investigation of student teachers' knowledge about their own learning

Twenty-nine student teachers from a large metropolitan university in Queensland, Australia were interviewed at the beginning (Time 1) and end (Time 2) of a year-long graduate diploma in education to investigate the nature of their knowledge about learning and changes in such knowledge over the year. At Time 1 and Time 2 most students thought learning should be meaningful and preferred to use transformative learning approaches. However, students indicated a willingness to engage in reproductive approaches to learning if the content to be learned was uninteresting, workloads were high, or assessment was examination-focussed. The results also indicated that while many students did not experience significant changes in their knowledge about learning over the year, they believed that transformative learning had become more of a focus for them. Investigating student teachers' knowledge about learning has implications for effective learning in teacher education programs.     

Innovation in postgraduate teaching: mixed methods to enhance learning and learning about learning

Growing pressure to restructure and reform tertiary education is encouraging university academics to use innovative practices that assist students to develop ‘employable’ skills. The hybrid approach described in this paper stimulated students to be self‐directed adult learners who maximized their learning of content and skills by means of problem‐based learning and action research strategies. The lecturer also operated as a reflective practitioner and role model by using an action research approach. This paper demonstrates the value of student empowerment, communication and leadership in autonomous learning groups. It outlines methods by which academic teaching staff can build continuous improvement into a university unit’s curriculum design and processes. These can be powerful additions to lecturers’ teaching strategies and to students’ learning experiences.     

Primary special school teachers' knowledge and beliefs about supporting learning in numeracy

This paper presents findings from a qualitative study of a group of 12 teachers in primary special schools in Scotland for children with moderate learning difficulties. It sets out an analysis of classroom observations and interviews that explored teachers' knowledge and beliefs about teaching and learning in mathematics with children with moderate learning difficulties. The teachers were interviewed pre‐ and post‐intervention; this was a research‐based professional development programme in children's mathematical thinking (Cognitively Guided Instruction) which teachers then developed in their classrooms. The findings showed that prior to the professional development, the teachers had a limited knowledge of children's mathematical development with teaching frequently informed by intuitive beliefs and dated and sometimes discredited practices. Most teachers had low expectations of children with learning difficulties. Post‐intervention, the teachers reviewed this stance and affirmed that a deeper understanding of children's mathematical thinking provided a more secure knowledge base for instruction. They also recognised the extent to which learners were constrained by existing classroom practices. The paper argues for the commonality of this knowledge base and considers the problematic nature of viewing such knowledge as sector specific.     

Understanding teacher learning in secondary education: The relations of teacher activities to changed beliefs about teaching and learning

In this study, relations between learning activities of teachers and changes in their beliefs were examined. Thirty-four teachers in Dutch secondary education were asked to complete a questionnaire regarding their beliefs about teaching and learning on two occasions. They were also asked to report on learning activities that they undertook. Teachers who had changed their beliefs in a direction congruent with the aims of recent educational reforms often reported experimentation with colleagues' teaching methods. Teachers who changed their beliefs in a direction that was not congruent with the reform often reported experimentation with alternative methods due to discontent with the effectiveness of current methods.     

Experimental evidence of the relative effectiveness of problem-based learning for knowledge acquisition and retention

This study investigated the effects of problem-based learning (PBL) on knowledge acquisition and knowledge retention in a controlled experiment in a lab setting. Eighty-eight first-year psychology students were randomly assigned to either a PBL condition, a lecture condition, or a self-study condition. All participants had the opportunity to study the same text. Their knowledge acquisition was tested on both an immediate post-test and a delayed post-test, one week later. The results showed beneficial effects for the PBL group on the immediate and the delayed test compared to the lecture group. There were no differences between knowledge acquisition scores for the self-study condition and either of the other conditions. Moreover, no differences on knowledge retention were shown, because the decline in performance over time was equal in all conditions. In conclusion, the present study provides experimentally based evidence of the disadvantageous effects of instruction through lectures compared to PBL.     

学习方法的探索

本文探讨了如何提高学习效率,进而对学习过程中的入门、攻坚、整合、融通、应用、创新、协作等实际问题提出了实用的方法。