The Use of Think-Aloud Strategies to Solve Word Problems

In this action research study of my sixth grade mathematics class, I investigated how students’ use of think-aloud strategies impacts their success in solving word problems. My research reveals that the use of think-aloud strategies can play an important role in the students’ abilities to understand and solve word problems. Direct instruction and modeling of think-aloud strategies increased my students’ confidence levels and the likelihood that they would use the strategies on their own. Providing students with a template to use as they solve a word problem helps students to better focus in on the think-aloud strategies I had been modeling for them.

Writing Relevant Word Problems: Seeking to Increase Student Mathematical Achievement

In this action research study of eighth grade mathematics, I investigated my students’ use of writing and solving word problems. I collected data to determine if writing and solving word problems would have a positive effect on students’ abilities to understand and solve word problems. These word problems are grade-level appropriate and are very similar to the problems on the eighth grade online assessment of state standards. Pre- and post-test data, weekly word problems that focus on specific mathematics topics, beginning and end surveys about word problem perceptions, and a teacher journal reveal that student engagement in this weekly practice of writing and solving word problems did influence the students’ overall abilities for, achievement in and attitudes toward solving word problems. Except for some students’ perceptions, the influence was largely positive. This suggests that word problems can be a constructive feature in eighth mathematics instruction.

Written Solutions of Mathematical Word Problems

In this action research study of my classroom of ten ninth grade algebra students, I investigated how my students expressed written solutions of mathematical word problems. I discovered that my students writing and performance improved as they experienced different strategies to attack problem solving. These experiences helped improve the confidence of my students in their problem solving skills and in their mathematical writing. I also discovered that my teaching style changed, as my students took on more responsibility for their learning. As a result of this research, I plan to implement problem solving activities in all my classrooms next year. I also plan to have my students develop their written communication skills by presenting their solutions to their problem solving activities in writing.

Making Sense of Word Problems

In this action research study of my classroom of sixth grade mathematics, I investigated word problems. I discovered that my students did not like to try word problems because they did not understand what was being asked of them. My students also saw no reason for solving word problems or in having the ability to solve them. I used word problems that covered topics that were familiar to the students and that covered the skills necessary at the sixth grade level. I wanted to deepen their understanding of math and its importance. By having my students journal to me about the steps that they had taken along the way to solve the word problem I was able to see where confusion occurred. Consequently I was able to help clarify where my students made mistakes. Also, through writing down the steps taken, students did see more clearly where their errors took place. Each time that my students wrote their explanations to the steps that they used in solving the word problems they did solved them more easily. As I observed my students they took more time in writing their explanations and did not look at it as such a difficult task anymore.

Socially shared metacognitive regulation during collaborative learning processes in student dyads and small groups

Traditionally metacognition has been theorised, methodologically studied and empirically tested from the standpoint mainly of individuals and their learning contexts. In this dissertation the emergence of metacognition is analysed more broadly. The aim of the dissertation was to explore socially shared metacognitive regulation (SSMR) as part of collaborative learning processes taking place in student dyads and small learning groups. The specific aims were to extend the concept of individual metacognition to SSMR, to develop methods to capture and analyse SSMR and to validate the usefulness of the concept of SSMR in two different learning contexts; in face-to-face student dyads solving mathematical word problems and also in small groups taking part in inquiry-based science learning in an asynchronous computer-supported collaborative learning (CSCL) environment. This dissertation is comprised of four studies. In Study I, the main aim was to explore if and how metacognition emerges during problem solving in student dyads and then to develop a method for analysing the social level of awareness, monitoring, and regulatory processes emerging during the problem solving. Two dyads comprised of 10-year-old students who were high-achieving especially in mathematical word problem solving and reading comprehension were involved in the study. An in-depth case analysis was conducted. Data consisted of over 16 (30–45 minutes) videotaped and transcribed face-to-face sessions. The dyads solved altogether 151 mathematical word problems of different difficulty levels in a game-format learning environment. The interaction flowchart was used in the analysis to uncover socially shared metacognition. Interviews (also stimulated recall interviews) were conducted in order to obtain further information about socially shared metacognition. The findings showed the emergence of metacognition in a collaborative learning context in a way that cannot solely be explained by individual conception. The concept of socially-shared metacognition (SSMR) was proposed. The results highlighted the emergence of socially shared metacognition specifically in problems where dyads encountered challenges. Small verbal and nonverbal signals between students also triggered the emergence of socially shared metacognition. Additionally, one dyad implemented a system whereby they shared metacognitive regulation based on their strengths in learning. Overall, the findings suggested that in order to discover patterns of socially shared metacognition, it is important to investigate metacognition over time. However, it was concluded that more research on socially shared metacognition, from larger data sets, is needed. These findings formed the basis of the second study. In Study II, the specific aim was to investigate whether socially shared metacognition can be reliably identified from a large dataset of collaborative face-to-face mathematical word problem solving sessions by student dyads. We specifically examined different difficulty levels of tasks as well as the function and focus of socially shared metacognition. Furthermore, the presence of observable metacognitive experiences at the beginning of socially shared metacognition was explored. Four dyads participated in the study. Each dyad was comprised of high-achieving 10-year-old students, ranked in the top 11% of their fourth grade peers (n=393). Dyads were from the same data set as in Study I. The dyads worked face-to-face in a computer-supported, game-format learning environment. Problem-solving processes for 251 tasks at three difficulty levels taking place during 56 (30–45 minutes) lessons were video-taped and analysed. Baseline data for this study were 14 675 turns of transcribed verbal and nonverbal behaviours observed in four study dyads. The micro-level analysis illustrated how participants moved between different channels of communication (individual and interpersonal). The unit of analysis was a set of turns, referred to as an ‘episode’. The results indicated that socially shared metacognition and its function and focus, as well as the appearance of metacognitive experiences can be defined in a reliable way from a larger data set by independent coders. A comparison of the different difficulty levels of the problems suggested that in order to trigger socially shared metacognition in small groups, the problems should be more difficult, as opposed to moderately difficult or easy. Although socially shared metacognition was found in collaborative face-to-face problem solving among high-achieving student dyads, more research is needed in different contexts. This consideration created the basis of the research on socially shared metacognition in Studies III and IV. In Study III, the aim was to expand the research on SSMR from face-to-face mathematical problem solving in student dyads to inquiry-based science learning among small groups in an asynchronous computer-supported collaborative learning (CSCL) environment. The specific aims were to investigate SSMR’s evolvement and functions in a CSCL environment and to explore how SSMR emerges at different phases of the inquiry process. Finally, individual student participation in SSMR during the process was studied. An in-depth explanatory case study of one small group of four girls aged 12 years was carried out. The girls attended a class that has an entrance examination and conducts a language-enriched curriculum. The small group solved complex science problems in an asynchronous CSCL environment, participating in research-like processes of inquiry during 22 lessons (á 45–minute). Students’ network discussion were recorded in written notes (N=640) which were used as study data. A set of notes, referred to here as a ‘thread’, was used as the unit of analysis. The inter-coder agreement was regarded as substantial. The results indicated that SSMR emerges in a small group’s asynchronous CSCL inquiry process in the science domain. Hence, the results of Study III were in line with the previous Study I and Study II and revealed that metacognition cannot be reduced to the individual level alone. The findings also confirm that SSMR should be examined as a process, since SSMR can evolve during different phases and that different SSMR threads overlapped and intertwined. Although the classification of SSMR’s functions was applicable in the context of CSCL in a small group, the dominant function was different in the asynchronous CSCL inquiry in the small group in a science activity than in mathematical word problem solving among student dyads (Study II). Further, the use of different analytical methods provided complementary findings about students’ participation in SSMR. The findings suggest that it is not enough to code just a single written note or simply to examine who has the largest number of notes in the SSMR thread but also to examine the connections between the notes. As the findings of the present study are based on an in-depth analysis of a single small group, further cases were examined in Study IV, as well as looking at the SSMR’s focus, which was also studied in a face-to-face context. In Study IV, the general aim was to investigate the emergence of SSMR with a larger data set from an asynchronous CSCL inquiry process in small student groups carrying out science activities. The specific aims were to study the emergence of SSMR in the different phases of the process, students’ participation in SSMR, and the relation of SSMR’s focus to the quality of outcomes, which was not explored in previous studies. The participants were 12-year-old students from the same class as in Study III. Five small groups consisting of four students and one of five students (N=25) were involved in the study. The small groups solved ill-defined science problems in an asynchronous CSCL environment, participating in research-like processes of inquiry over a total period of 22 hours. Written notes (N=4088) detailed the network discussions of the small groups and these constituted the study data. With these notes, SSMR threads were explored. As in Study III, the thread was used as the unit of analysis. In total, 332 notes were classified as forming 41 SSMR threads. Inter-coder agreement was assessed by three coders in the different phases of the analysis and found to be reliable. Multiple methods of analysis were used. Results showed that SSMR emerged in all the asynchronous CSCL inquiry processes in the small groups. However, the findings did not reveal any significantly changing trend in the emergence of SSMR during the process. As a main trend, the number of notes included in SSMR threads differed significantly in different phases of the process and small groups differed from each other. Although student participation was seen as highly dispersed between the students, there were differences between students and small groups. Furthermore, the findings indicated that the amount of SSMR during the process or participation structure did not explain the differences in the quality of outcomes for the groups. Rather, when SSMRs were focused on understanding and procedural matters, it was associated with achieving high quality learning outcomes. In turn, when SSMRs were focused on incidental and procedural matters, it was associated with low level learning outcomes. Hence, the findings imply that the focus of any emerging SSMR is crucial to the quality of the learning outcomes. Moreover, the findings encourage the use of multiple research methods for studying SSMR. In total, the four studies convincingly indicate that a phenomenon of socially shared metacognitive regulation also exists. This means that it was possible to define the concept of SSMR theoretically, to investigate it methodologically and to validate it empirically in two different learning contexts across dyads and small groups. In-depth micro-level case analysis in Studies I and III showed the possibility to capture and analyse in detail SSMR during the collaborative process, while in Studies II and IV, the analysis validated the emergence of SSMR in larger data sets. Hence, validation was tested both between two environments and within the same environments with further cases. As a part of this dissertation, SSMR’s detailed functions and foci were revealed. Moreover, the findings showed the important role of observable metacognitive experiences as the starting point of SSMRs. It was apparent that problems dealt with by the groups should be rather difficult if SSMR is to be made clearly visible. Further, individual students’ participation was found to differ between students and groups. The multiple research methods employed revealed supplementary findings regarding SSMR. Finally, when SSMR was focused on understanding and procedural matters, this was seen to lead to higher quality learning outcomes. Socially shared metacognition regulation should therefore be taken into consideration in students’ collaborative learning at school similarly to how an individual’s metacognition is taken into account in individual learning.

The contributions of domain-general and numerical factors to third-grade arithmetic skills and mathematical learning disability

Explanations of the marked individual differences in elementary school mathematical achievement and mathematical learning disability (MLD or dyscalculia) have involved domain-general factors (working memory, reasoning, processing speed and oral language) and numerical factors that include single-digit processing efficiency and multi-digit skills such as number system knowledge and estimation. This study of third graders (N = 258) finds both domain-general and numerical factors contribute independently to explaining variation in three significant arithmetic skills: basic calculation fluency, written multi-digit computation, and arithmetic word problems. Estimation accuracy and number system knowledge show the strongest associations with every skill and their contributions are both independent of each other and other factors. Different domain-general factors independently account for variation in each skill. Numeral comparison, a single digit processing skill, uniquely accounts for variation in basic calculation. Subsamples of children with MLD (at or below 10th percentile, n = 29) are compared with low achievement (LA, 11th to 25th percentiles, n = 42) and typical achievement (above 25th percentile, n = 187). Examination of these and subsets with persistent difficulties supports a multiple deficits view of number difficulties: most children with number difficulties exhibit deficits in both domain-general and numerical factors. The only factor deficit common to all persistent MLD children is in multi-digit skills. These findings indicate that many factors matter but multi-digit skills matter most in third grade mathematical achievement.

Student Problem Solving

The purpose of this study is to determine if students solve math problems using addition, subtraction, multiplication, and division consistently and whether students transfer these skills to other mathematical situations and solutions. In this action research study, a classroom of 6th grade mathematics students was used to investigate how students solve word problems and how they determine which mathematical approach to use to solve a problem. It was discovered that many of the students read and re-read a question before they try to find an answer. Most students will check their answer to determine if it is correct and makes sense. Most students agree that mastering basic math facts is very important for problem solving and prefer mathematics that does not focus on problem solving. As a result of this research, it will be emphasized to the building principal and staff the need for a unified and focused curriculum with a scope and sequence for delivery that is consistently followed. The importance of managing basic math skills and making sure each student is challenged to be a mathematical thinker will be stressed.

Strategies and Precise Vocabulary Knowledge: Exploring the Relationships Among Mathematics Vocabulary, Problem Solving, and Confidence

In this action research study I focused on my eighth grade pre-algebra students’ abilities to attack problems with enthusiasm and self confidence whether they completely understand the concepts or not. I wanted to teach them specific strategies and introduce and use precise vocabulary as a part of the problem solving process in hopes that I would see students’ confidence improve as they worked with mathematics. I used non-routine problems and concept-related open-ended problems to teach and model problem solving strategies. I introduced and practiced communication with specific and precise vocabulary with the goal of increasing student confidence and lowering student anxiety when they were faced with mathematics problem solving. I discovered that although students were working more willingly on problem solving and more inclined to attempt word problems using the strategies introduced in class, they were still reluctant to use specific vocabulary as they communicated to solve problems. As a result of this research, my style of teaching problem solving will evolve so that I focus more specifically on strategies and use precise vocabulary. I will spend more time introducing strategies and necessary vocabulary at the beginning of the year and continue to focus on strategies and process in order to lower my students’ anxiety and thus increase their self confidence when it comes to doing mathematics, especially problem solving.

Using Math Vocabulary Building to Increase Problem Solving Abilities in a 5th Grade Classroom

In this action research study of my 5th grade mathematics class, I investigated how students’ understanding of math vocabulary impacts their understanding of the curriculum. I discovered math vocabulary plays an important role in a student’s ability to understand daily lessons, complete homework, discuss ideas in groups, take tests and be successful on achievement tests. A student’s ability to understand the words around him (or her) in math class seem very related to his or her ability to solve word problems. Word problems are what our national assessments are all about. I also discovered that direct instruction and support of math vocabulary increased test scores and confidence in students as test takers. As a result of this research, I plan to continue to find ways to emphasize the vocabulary used in our current math curriculum. This process will start at the beginning of the year. I will continue to look for strategies that promote math vocabulary retention in my students. And finally, I will share my findings with my colleagues, so my research can be used as part of our School Improvement Goals.

Mathematical adventures,

"This book consists in the main of reprints of articles on the teaching of mathematics and on related topics which the author has written and published in educational journals at various times during the past ten years."--Foreword.

Experimenting with infinite groups, I

A group is termed parafree if it is residually nilpotent and has the same nilpotent quotients as a given free group. Since free groups are residually nilpotent, they are parafree. Nonfree parafree groups abound and they all have many properties in common with free groups. Finitely presented parafree groups have solvable word problems, but little is known about the conjugacy and isomorphism problems. The conjugacy problem plays an important part in determining whether an automorphism is inner, which we term the inner automorphism problem. We will attack these and other problems about parafree groups experimentally, in a series of papers, of which this is the first and which is concerned with the isomorphism problem. The approach that we take here is to distinguish some parafree groups by computing the number of epimorphisms onto selected finite groups. It turns out, rather unexpectedly, that an understanding of the quotients of certain groups leads to some new results about equations in free and relatively free groups. We touch on this only lightly here but will discuss this in more depth in a future paper.

Financial Education Through Mathematics and IT Curricula: Pocket Money Management

Report published in the Proceedings of the National Conference on "Education in the Information Society", Plovdiv, May, 2013

Developing Financial Expertise Through Mathematics and IT School Curricula

Report published in the Proceedings of the National Conference on "Education and Research in the Information Society", Plovdiv, May, 2014

Definizione e validazione di uno strumento per la misurazione dell'impatto delle variabili redazionali di un item matematico sulla performance degli studenti

Questa tesi descrive la ricerca condotta tra l'autunno e l'inverno di quest'anno da un gruppo di ricercatori in didattica della matematica relativamente all'influenza che le variazioni redazionali di un quesito matematico hanno sulle performance degli studenti. Lo scopo della ricerca è quella di strutturare e validare una metodologia e uno strumento che permettano di individuare e quantificare l'influenza delle variazioni del testo sulle prestazioni dello studente. Si è sentita l'esigenza di condurre uno studio di questo tipo poichè è sempre più evidente il profondo legame tra il linguaggio e l'apprendimento della matematica. La messa a punto di questo strumento aprirebbe le porte a una serie di ricerche più approfondite sulle varie tipologie di variazioni numeriche e/o linguistiche finora individuate. Nel primo capitolo è presentato il quadro teorico di riferimento relativo agli studi condotti fino ad ora nell'ambito della didattica della matematica, dai quali emerge la grossa influenza che la componente linguistica ha sulla comprensione e la trasmissione della matematica. Si farà quindi riferimento alle ricerche passate volte all'individuazione e alla schematizzazione delle variazioni redazionali dei Word Problems. Nel secondo capitolo, invece si passerà alla descrizione teorica relativa allo strumento statistico utilizzato. Si tratta del modello di Rasch appartenente alla famiglia dei modelli statistici dell'Item Response Theory, particolarmente utilizzato nella ricerca in didattica. Il terzo capitolo sarà dedicato alla descrizione dettagliata della sperimentazione svolta. Il quarto capitolo sarà il cuore di questa tesi; in esso infatti verrà descritta e validata la nuova metodologia utilizzata. Nel quinto sarà eseguita un analisi puntuale di come lo strumento ha messo in evidenza le differenze per ogni item variato. Infine verranno tratte le conclusioni complessive dello studio condotto.

THE EFFECT OF PURPOSEFUL MATHEMATICS DISCOURSE IN THE CLASSROOM ON STUDENTS’ MATHEMATICS LANGUAGE IN THE CONTEXT OF PROBLEM SOLVING

This study examines how one secondary school teacher’s use of purposeful oral mathematics language impacted her students’ language use and overall communication in written solutions while working with word problems in a grade nine academic mathematics class. Mathematics is often described as a distinct language. As with all languages, students must develop a sense for oral language before developing social practices such as listening, respecting others ideas, and writing. Effective writing is often seen by students that have strong oral language skills. Classroom observations, teacher and student interviews, and collected student work served as evidence to demonstrate the nature of both the teacher’s and the students’ use of oral mathematical language in the classroom, as well as the effect the discourse and language use had on students’ individual written solutions while working on word problems. Inductive coding for themes revealed that the teacher’s purposeful use of oral mathematical language had a positive impact on students’ written solutions. The teacher’s development of a mathematical discourse community created a space for the students to explore mathematical language and concepts that facilitated a deeper level of conceptual understanding of the learned material. The teacher’s oral language appeared to transfer into students written work albeit not with the same complexity of use of the teacher’s oral expression of the mathematical register. Students that learn mathematical language and concepts better appear to have a growth mindset, feel they have ownership over their learning, use reorganizational strategies, and help develop a discourse community.