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Forty-eight Algebra II standard level students were divided into two classes. One was taught using the traditional learning approach of lecture and test; the other was taught using a cooperative learning approach in which students were grouped into teams of four members of mixed ability. A case study approach was adopted for this comparison of cooperative learning and more traditional teaching methods. The case study covers two school years, from October 1991 until June 1992 (which was used as a pilot program) and then October 1992 until June 1993. The subjects were assessed on measures of grade improvement and evaluation. The time has come to change how we teach math. Math must be learned as an active process. New approaches in cooperative learning can increase the level of understanding and appreciation of mathematics and decrease student's anxiety levels. The findings point out clear differences between the cooperative learning classroom and the traditional classroom. The cooperative learning classroom is more flexible as well as creative. Students measure more positive attitudes and feelings toward mathematics in this environment. Results show that the cooperative learning group demonstrates stable gains in math appreciation and achievement as well as improved interracial relationships, some overcoming of math anxiety and improved discipline.

This qualitative content analysis research study examined changes made to K-5 state mathematics standards across the United States from 2012 to 2022. This study aimed to answer the research question: In what ways, if any, do K-5 state mathematics standards differ from the CCSSM? This was accomplished through four additional sub questions which include: (1) In what ways, if any, do K-5 state mathematics student process and practice standards differ from the CCSSM? (2) In what ways, if any, do K-5 state mathematics standards content domains differ from the CCSSM? (3) In what ways, if any, do states describe how learning trajectories are addressed in K-5 state mathematics standards? and (4) In what ways, if any, is the relationship between procedural and conceptual learning outcomes represented in K-5 state mathematics standards? Data collection included state mathematics standards documents and any publicly available relevant supporting documents found on state department of education websites. Of the 21 standard revisions from 15 states considered for the study, revisions from six states were selected for coding. From the coding process, themes were developed regarding patterns in changes that occurred in individual states' standards. The most prominent and common themes of changes included the addition of standards (e.g., personal finance, estimation, patterns, statistics, and probability), the merging of domains, the lack of specific evidence to the inclusion of learning trajectories in the development of revisions, and movement away from a balanced approach to learning outcomes. There were no consistencies in changes across all states that were coded. The results from this study can be used to promote consistency for future considerations for states that are revising their standards or to urge the reconvening of a writing committee for a revision of the Common Core State Standards for Mathematics.

Thls study was designed to Investigate the effects of Introducing a process problem solving component to the Middle School mathematics curriculum. The process problem solving component replaced the tradltlona1 tlme spent in clase on drlll and practice, review and skill building problem sets. One hundred sixty-eight seventh grade students and three hundred eighth grade students were involved over a four year period. The problem solving component introduced included huerlstlc Instruction and the development of various strategies including working backwards from answer to solution. reformulating a problem in various ways, thinking of a simpler problem, diagram drawing, pattern discovery and trlal and error. The eoclallzatlon of the classroom and the dynamics involved ls explained in this etudy and viewed as an effective alternative to 'chalk and talk" methods of classroom Instruction. Student wllllngness to become active participants in lea~nlng mathematics and the Increase ln parental involvement In their child's mathematical education are affective constructs throughout the study. The mean differences for the dependent measures were tested for slgnlflcant differences

Since there are insufficient mathematicians to meet economic and educational demands and many well-qualified, successful mathematics students exhibit signs of disaffection, the student experience of undergraduate mathematics is high on the political agenda. Many undergraduates struggle with the school-university transition, which has been associated with students’ prior experiences of mathematics which, at A-level, are regularly criticised for being too easy and too different to undergraduate mathematics. Furthermore, the University of Oxford administers a Mathematics Admissions Test (OxMAT) as a means of identifying those best prepared beyond the limited demands of A-level. Consequently, a study was conducted into the mathematical enculturation of Oxford undergraduates, specifically in terms of examination questions and students’ approaches to learning. Analysis of the Approaches and Study Skills Inventory for Students (ASSIST) (Tait et al., 1998) revealed the majority of students to adopt strategic approaches to learning (ATLs) in all four year-groups, though the descriptions given by students in interviews of the nature of their ATL highlighted some shortcomings of the ASSIST as the motivation for memorisation appeared to be an important factor. The MATH taxonomy (Smith et al., 1996), revealed that most A-level questions require routine use of procedures, whereas the OxMAT tested a variety of skills from applying familiar mathematics in new situations to justifying and interpreting information to form proofs. This is more in-line with the requirements of undergraduate assessment, although the MATH taxonomy and student interviews revealed that these still allowed for rote memorisation and strategic methods. Thus, the changing nature of mathematics and questions posed to students at the secondary-tertiary interface appears to affect students’ ATLs, though this is not reflected by the ASSIST data.

The purpose of this research study was to determine characteristics of entering vocational students in Kentucky Area/State vocational schools and to develop a computerized diagnostic instrument and tutorial package for assisting students in the mastery of necessary basic skills in mathematics. After specific math skills were identified in which proficiency is required of vocational education students, item pools were constructed for each skill. The skill item pools were validated using approximately 500 public school students throughout the grades of four through eight in public schools of Harlan County, Kentucky, Lee County, Virginia, and Washington County, Tennessee. The items within each item pool were found to be statistically equivalent. Computer programs were coded in the BASIC language using the item pools to randomly select and generate a diagnostic instrument and tutorial program relevant to the basic math skills. Three randomly generated forms of the diagnostic instrument were sent to 100 students in twenty area state vocational schools of Kentucky for normalization and form validation. The diagnostic instrument showed a strong positive coefficient of reliability with an average of.95 over the three forms used in the normalization process. There was no significant difference between the mean raw scores of the three forms. A 67 percentile score was found to be the norm which was to be statistically equivalent to the Tests of Adult Basic Education (TABE) at the 8.75 grade equivalent. An experiment was conducted using vocational students at Hazard State Vocational School as subjects to determine the affects of the tutorial package on basic math skill mastery using equivalent forms of the diagnostic instrument for pretesting and posttesting. Results of the experiment indicated that the computer managed instruction tutorial package had a significant affect in increasing posttest scores of the experimental group over the control group. It was concluded that the problem of constructing a computerized diagnostic math instrument and tutorial package capable of enhancing mastery of basic math skills to assist vocational students in gaining entrance into vocational school was achieved. A recommendation was made for further research and development to use the random item pool model for other development of computer assisted instruction (CAI) software.

Compare and contrast various representations of patterns and relationships. We will describe, analyze, and generalize patterns represented graphically or numerically using words and symbolic rules and connect this to models made with toothpicks and square tiles.

Mathematics is defined as an abstract way of thinking. Abstraction ranks among the least accessible mental activities. In an educational context the encounter with mathematical abstraction is the crucial step of the transition from informal school mathematics to the formalism of university mathematics. This transition is characterised by cognitive tensions. This study aimed at the identification and exploration of the tensions in the novice mathematician's encounter with mathematical abstraction. For this purpose twenty first-year mathematics undergraduates were observed in their weekly tutorials in four Oxford Colleges during Michaelmas and Hilary Term of Year 1. Tutorials were tape-recorded and fieldnotes kept during observation. The students were also interviewed at the end of each term of observation. The recordings of the observed tutorials and the interviews were transcribed and submitted to an analytical process of filtering out episodes that illuminate the novices' cognition. An analytical framework consisting of cognitive and sociocultural theories on learning was applied on sets of episodes within the mathematical areas of Foundational Analysis, Calculus, Linear Algebra and Group Theory. This topical analysis was followed by a cross-topical synthesis of themes that were found to characterise the novices' cognition. The novices' encounter with mathematical abstraction was described as a personal meaning-construction process and as an enculturation process: the new culture is Advanced Mathematics introduced by an expert, the tutor. The novices' interaction with the new concept definitions was obstructed by their unstable previous knowledge. Concept image construction was described as a construction of meaningful metaphors and an exploration of the 'raison-d'-être' of the new concepts and the new reasoning and was characterised by the tension between the Informal/Intuitive/Verbal and the Formal/Abstract/Symbolic — which was discussed in terms of semantics and reasoning. The novices were in difficulty with the mechanics of formal mathematical reasoning as well as with applying these mechanics in a contextualised manner. This decontextualised behaviour was linked to the fragility of their knowledge with regard to the nature of rigour in formal mathematics.

This study looked at what components are in student solutions to computational estimation problems involving fractions. Past computational estimation research has focused on strategies used for estimating with whole numbers and decimals while neglecting those used for fractions. An extensive literature review revealed one study specifically directed toward estimating with fractions (Hanson & Hogan, 2000) that researched adult estimation strategies and not children's strategies. Given the lack of research on estimation strategies that children use to estimate with fractions, this study used qualitative research methods to find which estimation components were in 10 eighth-grade students' solutions to estimation problems involving fractions. Analysis of this data differs from previous estimation studies in that it considers actions as the unit of analysis, providing a smaller grain size that reveals the components used in each estimation solution. The analysis revealed new estimation components as well as a new structure for categorizing the components. The new categories are whole number and decimal estimation components, fraction estimation components, and components used with either fractions or whole numbers and decimals. The results from this study contribute to the field of mathematics education by identifying new components to consider when conducting future studies in computational estimation. The findings also suggest that future research on estimation should use a smaller unit of analysis than a solution response to a task, the typical unit of analysis in previous research. Additionally, these results contribute to mathematics teaching by suggesting that all components of an estimation solution be considered when teaching computational estimation, not just the overarching strategy.