Diagnostic Assessments in Mathematics to Support Instructional Decision Making

Diagnosis is an integral part of instructional decision-making. As the bridge between identification of students who may be at-risk for failure and delivery of carefully designed supplemental interventions, diagnosis provides valuable information about students’ persistent misconceptions in the targeted domain. In this paper, we discuss current approaches to diagnosis in mathematics and highlight the strengths and limitations of each approach for making instructional decisions. We point to cognitive diagnostic assessments as an emerging solution for providing detailed and precise information about students’ thinking that is needed to provide appropriate educational opportunities for students struggling in mathematics. Accessed 41,983 times on https://pareonline.net from October 12, 2009 to December 31, 2019. For downloads from January 1, 2020 forward, please click on the PlumX Metrics link to the right

Various ways to determine rational number size: an exploration across primary and secondary education

Understanding rational numbers is a complex task for primary and secondary school students. Previous research has shown that a possible reason is students’ tendency to apply the properties of natural numbers (inappropriately) when they are working with rational numbers (a phenomenon called natural number bias). Focusing on rational number comparison tasks, recent research has shown that other incorrect strategies such as gap thinking or reverse bias can also explain these difficulties. The present study aims to investigate students’ different ways of thinking when working on fraction and decimal comparison tasks. The participants were 1,262 primary and secondary school students. A TwoStep Cluster Analysis revealed six different student profiles according to their way of thinking. Results showed that while students’ reasoning based on the properties of natural numbers decreased along primary and secondary school, almost disappearing at the end of secondary school, students’ reasoning based on gap thinking increased along these grades. This result seems to indicate that when students overcome their reliance on natural numbers, they enter a stage of qualitatively different errors before finally reaching the stage of correct understanding.This research was carried out with the support of Conselleria d’Educació, Investigació, Cultura i Esport (Generalitat Valenciana, Spain) (PROMETEO/2017/135) and with the support of the University of Alicante (UAFPU2018-035)

Students' difficulties, conceptions and attitudes towards learning algebra : an intervention study to improve teaching and learning

The skills necessary to identify and analyse errors and misconceptions made by students are needed by teachers of all levels especially at the lower secondary school level in Malaysia. If students are to be successful in tackling mathematical problems later in their schooling, the one prerequisite is the mastery of basic concepts in algebra. Despite the best efforts of the teachers, students still develop algebra misconceptions. Is it possible to reduce or eliminate these misconceptions? The research involved a survey of 14 year-old students in Form 2 (Grade 8) in the Penampang district of Sabah, East Malaysia. The focus of this study lies in students’ difficulties, conceptions and attitudes towards learning algebra in the framework of conceptual change. A possible way to help students overcome their learning difficulties and misconceptions is by implementing diagnostic teaching involving conflict to foster conceptual change. The study involved evaluating the efficacy of a conceptual change instructional programme involving cognitive conflict in (1) facilitating Form 2 students’ understanding of algebra concepts, and (2) assessing changes in students’ attitudes towards learning mathematics, in a mixed quantitativequalitative research design.A 24-item Algebra Diagnostic Test and a 20-item Test of Mathematics-Related Attitudes (TOMRA) questionnaire were administered as a pretest and a posttest to 39 students in each of a heterogeneous high-achieving class and a below-average achieving class. In addition 9 students were purposefully selected to participate in the interview.The results of the study indicated that students’ difficulties and misconceptions from both classes fell into five broad areas: (1) basic understanding of letters and their place in mathematics, (2) manipulation of these letters or variables, (3) use of rules of manipulation to solve equations, (4) use of knowledge of algebraic structure and syntax to form equations, and (5) generalisation of rule for repetitive patterns or sequences of shapes.The results also showed that there was significant improvement in students’ achievement in mathematics. Further, students’ attitude towards inquiry of mathematics lessons showed significant positive improvement. Enjoyment remained high even though enjoyment of mathematics lesson showed no change. Also, changes in students’ understanding (from unintelligible to intelligible, intelligible to plausible, plausible to fruitful) illustrated the extent of changes in their conceptions.Different pedagogies can affect how conceptual change and challenge of misconceptions occurs. Therefore, knowledge of the origin of different types of misconceptions can be useful in selecting more effective pedagogical techniques for challenging particular misconceptions. Also, for teachers to create an effective learning experience they should be aware of and acknowledge students’ prior knowledge acquired from academic settings and from everyday previous personal experiences. Since all learning involves transfer from prior knowledge and previous experiences, an awareness and understanding of a student’s initial conceptual framework and/or topic can be used to formulate more effective teaching strategies. If this idea is taken a step further, it could be said that, because misconceptions comprise part of a conceptual framework, then understanding origins of misconceptions would further facilitate development of effective teaching strategies.Further research is needed to help teachers to understand how students experience conflict, how students feel when they experience conflict, and how these experiences are related to their final responses because cognitive conflict has both constructive and destructive potential. Thus, by being able to interpret, recognise and manage cognitive conflict, a teacher can then successfully interpret his/her students’ cognitive conflict and be able to make conceptual change more likely or help students to have meaningful learning experiences in secondary school algebra

Students' difficulties, conceptions and attitudes towards learning algebra : an intervention study to improve teaching and learning

The skills necessary to identify and analyse errors and misconceptions made by students are needed by teachers of all levels especially at the lower secondary school level in Malaysia. If students are to be successful in tackling mathematical problems later in their schooling, the one prerequisite is the mastery of basic concepts in algebra. Despite the best efforts of the teachers, students still develop algebra misconceptions. Is it possible to reduce or eliminate these misconceptions? The research involved a survey of 14 year-old students in Form 2 (Grade 8) in the Penampang district of Sabah, East Malaysia. The focus of this study lies in students’ difficulties, conceptions and attitudes towards learning algebra in the framework of conceptual change. A possible way to help students overcome their learning difficulties and misconceptions is by implementing diagnostic teaching involving conflict to foster conceptual change. The study involved evaluating the efficacy of a conceptual change instructional programme involving cognitive conflict in (1) facilitating Form 2 students’ understanding of algebra concepts, and (2) assessing changes in students’ attitudes towards learning mathematics, in a mixed quantitativequalitative research design.A 24-item Algebra Diagnostic Test and a 20-item Test of Mathematics-Related Attitudes (TOMRA) questionnaire were administered as a pretest and a posttest to 39 students in each of a heterogeneous high-achieving class and a below-average achieving class. In addition 9 students were purposefully selected to participate in the interview.The results of the study indicated that students’ difficulties and misconceptions from both classes fell into five broad areas: (1) basic understanding of letters and their place in mathematics, (2) manipulation of these letters or variables, (3) use of rules of manipulation to solve equations, (4) use of knowledge of algebraic structure and syntax to form equations, and (5) generalisation of rule for repetitive patterns or sequences of shapes.The results also showed that there was significant improvement in students’ achievement in mathematics. Further, students’ attitude towards inquiry of mathematics lessons showed significant positive improvement. Enjoyment remained high even though enjoyment of mathematics lesson showed no change. Also, changes in students’ understanding (from unintelligible to intelligible, intelligible to plausible, plausible to fruitful) illustrated the extent of changes in their conceptions.Different pedagogies can affect how conceptual change and challenge of misconceptions occurs. Therefore, knowledge of the origin of different types of misconceptions can be useful in selecting more effective pedagogical techniques for challenging particular misconceptions. Also, for teachers to create an effective learning experience they should be aware of and acknowledge students’ prior knowledge acquired from academic settings and from everyday previous personal experiences. Since all learning involves transfer from prior knowledge and previous experiences, an awareness and understanding of a student’s initial conceptual framework and/or topic can be used to formulate more effective teaching strategies. If this idea is taken a step further, it could be said that, because misconceptions comprise part of a conceptual framework, then understanding origins of misconceptions would further facilitate development of effective teaching strategies.Further research is needed to help teachers to understand how students experience conflict, how students feel when they experience conflict, and how these experiences are related to their final responses because cognitive conflict has both constructive and destructive potential. Thus, by being able to interpret, recognise and manage cognitive conflict, a teacher can then successfully interpret his/her students’ cognitive conflict and be able to make conceptual change more likely or help students to have meaningful learning experiences in secondary school algebra

Validation of the Item-Attribute Matrix in TIMSS-Mathematics Using Multiple Regression and the LSDM

For many cognitive diagnostic models, the item-attribute matrix (or Q-matrix) is an essential component which displays the relationship between items and their latent attributes or skills in knowledge and cognitive processes. However, it is a challenge to develop an effective Q-matrix.The purposes of this study were (1) to validate of the item-attribute matrix using two levels of attributes (Level 1 attributes and Level 2 sub-attributes), and (2) through retrofitting the diagnostic models to the mathematics test of the Trends in International Mathematics and Science Study (TIMSS), to evaluate the construct validity of TIMSS mathematics assessment by comparing the results of two assessment booklets. Item data were extracted from Booklets 2 and 3 for the 8th grade in TIMSS 2007, which included a total of 49 mathematics items and every student\u27s response to every item. The study developed three categories of attributes at two levels: content, cognitive process (TIMSS or new), and comprehensive cognitive process (or IT) based on the TIMSS assessment framework, cognitive procedures, and item type. At level one, there were 4 content attributes (number, algebra, geometry, and data and chance), 3 TIMSS process attributes (knowing, applying, and reasoning), and 4 new process attributes (identifying, computing, judging, and reasoning). At level two, the level 1 attributes were further divided into 32 sub-attributes. There was only one level of IT attributes (multiple steps/responses, complexity, and constructed-response). Twelve Q-matrices (4 originally specified, 4 random, and 4 revised) were investigated with eleven Q-matrix models (QM1 ~ QM11) using multiple regression and the least squares distance method (LSDM). Comprehensive analyses indicated that the proposed Q-matrices explained most of the variance in item difficulty (i.e., 64% to 81%). The cognitive process attributes contributed to the item difficulties more than the content attributes, and the IT attributes contributed much more than both the content and process attributes. The new retrofitted process attributes explained the items better than the TIMSS process attributes. Results generated from the level 1 attributes and the level 2 attributes were consistent. Most attributes could be used to recover students\u27 performance, but some attributes\u27 probabilities showed unreasonable patterns. The analysis approaches could not demonstrate if the same construct validity was supported across booklets. The proposed attributes and Q-matrices explained the items of Booklet 2 better than the items of Booklet 3. The specified Q-matrices explained the items better than the random Q-matrices

A cognitive diagnostic system for explaining algebra errors

"This thesis presents a new approach to cognitive diagnosis within the domain of algebra that has greater power than existing techniques."Doctor of Philosoph