The recent past has seen an increasing frequency of calls for teachers to implement evidence-based practices (Davies, 1999; U.S. Department of Education, 2002). At the same time, it has been noted that teachers see research as largely irrelevant to practice (Lester & William, 2002; Steen, 1999). If one accepts the premise that research holds value for educational practice (Margolinas, 1998; NCTM Research Committee, 2006; Silver, 1990), it is important for teacher educators to develop instructional experiences that bring teachers into the discourse surrounding educational research. At the present time, communities of teachers and researchers are often largely separated by communication-related barriers (Sowder, 2000; Silver, 2003). Lesh and Lovitts (2000) observed the following about the relationship between research and practice: In mathematics and science education, the flow of information between researchers and practitioners is not the kind of one-way process that is suggested by such terms as information dissemination. Instead, to be effective, the flow of information usually must be cyclic, iterative, and interactive (p. 53). An implication of this statement is that a transmission view of familiarizing teachers with research is naive, just as mathematical pedagogy based on such a view is misguided (Kline, 1977). Lesh and Lovitts (2000) went on to state, Although simpleminded, 'delivery-and-reception' metaphors are recognized widely now as being inappropriate for describing the development of students, teachers, or other complex systems, these same machine-based metaphors continue to be applied to the development of programs of instruction (p. 57). Teachers' Conversations as Complex Systems Complexity science provides a framework for designing and analyzing the types of complex systems for the development of teachers mentioned by Lesh and Lovitts (2000). Davis and Simmt (2003) provided a discussion of the implications of complexity science for mathematics education. They defined complex phenomena in the following manner: First, each of these phenomena is adaptive. That is, a complex system can change its own structure ... Second, a complex phenomenon is emergent, meaning that it is composed of and arises in the complicated activities of individual agents. In effect, a complex system is not just the sum of its parts, but the product of the parts and their interactions (Davis & Simmt, 2003, p. 138). These two defining characteristics also apply to complex systems that arise in other disciplines, such as cells, bodily organs, cultures, economics, and ecosystems (Johnson, 2001). Literature pertaining to mathematics teacher education contains empirical examples of complex systems emerging among teachers as they converse with one another. Davis and Simmt (2003) characterized a study group of teachers trying to solve mathematics problems as a complex system. The structure of the conversations among the study group changed as individuals each brought unique contributions to the solutions of problems to the conversation. Smitherman (2005) described similar dynamics among a group of pre-service teachers discussing fraction concepts. She described how she conducted a classroom conversation among pre-service teachers by asking them to share their thoughts on the fraction one-third. By the end of the conversation, many of the aspects in the NCTM (2000) standards connected to fractions had been considered by the group. In each complex system, knowledge was constructed in a non-linear fashion as individuals contributed their perspectives on the objects of study at hand to the conversation. In studying teachers' interactions within complex systems, it is important to keep in mind that complexity science provides a framework for analyzing human interactions, which are never devoid of contextual peculiarities (Stacey, 2003). …
One of the disappointments associated with the mathematical reform movement is the increasing mismatch between the improvements made in curriculum and instruction and prevalent assessment modes (Firestone & Schorr, 2004; Niss, 1993). Despite early calls for in assessment practice, such as the 1989 Curriculum and Evaluation Standards (NCTM), recent research into the teaching and learning of mathematics that has provided detiled consideration of its socially situated nature, has not focused to the same degree on mathematics assessment (Morgan, 1998). Therefore interest has increased in matching assessment methods to developments in curriculum. There is a pressing need to assess a much wider range of abilities than has been the case heretofore, including problem posing and solving, representing, and understanding. Traditional mathematical assessment has frequently relied upon the ability of students to display behavior that matches their assessor's expectations rather than on any underlying understanding (Morgan, 1998). These traditional assessments communicate that mathematics is an endeavor that involves determining a quick answer using a preexisting, memorized method (Bell, 1995; Clarke, Clarke & Lovitt, 1990; Hancock & Kilpatrick, 1993) thus failing to represent the true complexity of mathematics (Galbraith, 1993; Izard, 1993; Wheeler, 1993). In contrast, assessment data that provide direct information about improving the learning experience increase legitimate mathematical learning that is thorough and connected (Black & William, 1998; NCTM, 1995). The measurement of de-contextualized technical skills should be replaced with measures that reflect what is known about what it means to know and do mathematics, i.e., that capture the degree of acquisition of both conceptual and procedural knowledge and the connections between them, and that assess the solving of worthwhile problems, the communication and justification of conjecture, and the representation of mathematical thinking in multiple ways (NCTM, 2000). Ridgeway (1998) states, As an issue of policy, the implementation of standards-based curricula should always be accompanied by the implementation of standards-based assessment. In fact, incremental in assessment systems will foster concurrent improvement in professional and curriculum development (p. 2). The 1989 Standards states, As the curriculum changes, so must the tests. Tests also must because they are one way of communicating what is important for students to know.... In this way tests can effect change (pp. 189, 190). Both the Standards (NCTM, 1995) and the Principles and Standards (NCTM, 2000) state that assessment tasks communicate what type of mathematical knowledge and performance are valued (p. 22). Therefore, standards-based assessment complements standards-based instruction (Dunbar & Witt, 1993). Paralleling reform in mathematics curriculum and instruction have been calls to authenticate student assessment in all subject areas. Terms such as Authentic Assessment, Alternative Assessment, and Assessment have become banners to rally focused efforts to paradigms about the nature and purpose of assessment. According to McMillan (2004), a performance assessment is ... one in which the teacher observes and makes a judgment about the student's demonstration of a skill or competency in creating a product, constructing a response, or making a presentation. They possess several important characteristics: 1. Students perform, create, construct, or produce something 2. Deep understanding and/or reasoning skills are assessed. 3. They involve sustained work. 4. They call on students to explain, justify, and defend. 5. Performance is directly observable. 6. They involve engaging ideas of importance and substance. 7. There is a reliance on trained assessor's judgments for scoring. 8. Multiple criteria and standards are pre-specified and public. …
We describe a teaching experiment about third grade studentsunderstanding of the equal sign and their initial forays into analyzing expressions. We used true/false and open number sentences in forms unfamiliar to the students to cause students to reconsider their conceptions of the equal sign. Our results suggest a sequence of three stages in the evolution of students' understanding of the equal sign with students progressing from a procedural/computational perspective to an analytic perspective. Our data show that as students deepened their conceptions about the equal sign they began to analyze expressions in ways that promoted algebraic thinking. We describe the essential elements of instruction that advanced this learning.
The Hindu-Arabic number system represents amounts of objects by number symbols, without referring to other properties of these objects (e.g., color, size). That is to say that the number symbols, like many other symbols, are linked to the objects that they represent in an arbitrary but agreed upon manner with fixed representation rules (Vygotsky, 1978). Therefore, when the number system is used in a teaching-learning process, the child is required to perform a relatively complicated cognitive process. He has to refer to the meaning behind the symbols and to make the connections between the symbols and the quantities (Bialystok, 1992; DeLoache, Miller & Rosengren, 1997; Dorfler, 2000; Kaput, 1991; Lesh & Doerr, 2000, Thomas, Jolley, Robinson & Champion, 1999). One question that is naturally raised regarding child's knowledge of the Hindu-Arabic number system is: What are the factors that determine the child's grasp of this symbolic system? One of the major factors that ought to be considered is the development of symbolic thinking. The development of symbolic thinking addresses the cognitive processes that take place in the structure of the mental representation during the change from the to that of the (Nemirovsky & Monk, 2000). In the early stages of the development of symbolic thinking, children are at the unity level. At this level, children believe that the symbolic representation reflects the nature of the object it represents. Thus, for example, children will write names of large objects with large letters (Thomas, Jolley, Robinson, & Champion, 1999). When differentiation occurs, the child separates between the object being represented and its symbolic representation. At this differentiation level the child understands that there is no connection between the size of the symbol and the size of the object that the symbol represents. The ability of a child to employ symbols of numbers as symbols representing the mathematical meaning of the number is a result of a developmental process (Bialystok, 1992; Bialystok & Cobb, 1996; Worthington & Carruthers, 2003; Hughes, 1986, Munn, 1998; Worthington, 2003). However, newborns are cognitively equipped from the very outset to recognize objects and their quantities and almost immediately begin to accumulate knowledge about numbers (Butterworth, 2000; Dehaene, 1997; Wynn, 1992, 2002). Moreover, symbolization ability begins to develop in children from the earliest life stages (Kamii, Kirkland & Lewis, 2001; DeLoache, Miller & Rosengren, 1997; Mandler, 1992, Piaget 1962), and as a result, children acquire various types of knowledge about the written symbols of numbers (Bialystok, 1992; Carruthers & Worthington, 2005; Hughes, 1986). Tolshinsky-Landsmann (1986) found, for instance, that four year-olds differentiate between Hebrew letters and numerical symbols and that they consider one numerical symbol to be a number, whereas they do not think of one letter as a word (indeed, in Hebrew, one letter does not constitute a word). Tolshinsky-Landsmann and Karmiloff-Smith (1992) reported that children from England and from Spain at the age of about four distinguish between symbols that belong to the number system and those that do not. As previously stated, the ability to attribute quantities to numerical symbols develops gradually. Two of the prominent researchers that have significantly contributed to our understanding of this developmental process are Bialystok (1992, 2000) who describes the different stages of the development of number symbolic thinking and Hughes (1986), who describes the development of the numerical symbolic representation. Bialystok (1992) describes three hierarchical stages of number symbolic thinking. At the first stage children recite number sequences from their memory and employ the appropriate name for each number in the number sequence. At this stage children understand that counting is a way of describing quantities. …
With the advent of reform-based curricula and recommendations related to the teaching of mathematics from the National Council of Teachers of Mathematics (NCTM, 2000), teachers need access to ongoing professional development which models ways in which teachers are now asked to teach (Ball & Cohen, 1999; M.S. Smith, 2001). An extended professional development program was implemented with teachers of grades 3 through 5. This program sought to model reform-based teaching techniques and advocated hands-on, manipulative based activities for the mathematics classroom. results of this preliminary qualitative study are presented here. Changes in teacher were documented through a Grounded Theory approach to data analysis (Glaser & Strauss, 1967). A framework for identifying stages of teacher change was developed. ********** Professional development of teachers has long been used as an avenue for imparting new teaching techniques to inservice teachers. With the advent of reform-based curricula and recommendations related to the teaching of mathematics from the National Council of Teachers of Mathematics (NCTM) (2000), teachers need access to ongoing professional development which models ways in which teachers are now asked to teach (Ball & Cohen, 1999; M.S. Smith, 2001). The professional culture of mathematics education must be transformed and requires extensive changes in teachers' deeply held beliefs, knowledge, and habits of practice (Kitchen, 2003, p. 3). These changes take place through ongoing professional development. Although the need for teachers to change their practices to be more in line with reform-based ideas and curricula is recognized, little research has investigated the support teachers need to make changes in their (Kitchen, 2003). Hoban (2002) states, clearly we need a new way of thinking about educational change that takes into account the complex nature of teaching, teacher learning, and the change (Hoban, 2002, p. 21). This paper presents the results of a preliminary qualitative study in which an extended professional development program was implemented with teachers of grades 3 through 5 in an effort to effect change in their teaching practices as well as their attitudes toward manipulative use in the classroom. This research study led to the development of stages of teacher change and the identification of perceived barriers to manipulative use in the elementary school classroom. stages of teacher change will be discussed here. primary method for investigating the efficacy of this process was Grounded Theory (Glaser & Strauss, 1967), a qualitative research approach which attempts to generate a theoretical framework through data collection and analysis pertaining to the participants' experiences in order to find themes, in this case related to teacher actions in the classroom. Data analyzed included transcripts from teacher interviews and focus groups, informal interviews with the school principal and mathematics specialist, classroom observations of teachers interacting with their students, and anecdotal notes from model teaching experiences. Content Knowledge of Teachers In her comparison of United States and Chinese elementary school teachers, Ma (1999) found that the U.S. teachers were lacking in their ability to diagnose children's errors and misconceptions to a degree that intervention could take place on a conceptual level. Many U.S. teachers provided explanations for student misconceptions that were procedural in nature while the Chinese teachers provided both procedural and conceptual explanations of student errors and ways of connecting the misconception. Chinese teachers were often more able to provide multiple representations of mathematical ideas whereas many of the U.S. teachers were only able to provide one or two representations of the same mathematical concepts. Additionally, Cohen, Hill, and Kennedy (2002) found that in order for students to understand particular content, their teachers must also have an understanding of the content. …
Abstract The interpretation of histograms is a complex process requiring the integration of understanding about how graphs convey information with knowledge about how statistical constructs are displayed graphically. For this study, students in an introductory statistics class completed three histogram comparison tasks at the end of the course to assess their abilities to identify similar means and standard deviations and to evaluate skewness as represented in histograms. Fewer than 50% of the students completed all three tasks successfully. Common errors included inferring the relative value of the mean according to the center of the x-axis rather than the center of the distribution of data, identifying histograms with greater heights as those having the greater standard deviations, and interpreting skewness as a shift of the center of the distribution along the x-axis rather than an asymmetry of the distribution. Introduction Statistical reasoning includes interpreting numeric descriptive statistical measures and their corresponding graphic displays. Students need to develop their understanding of the information that descriptive statistical measures provide about a set of data. Students should also be able to interpret statistical graphs to assess the distribution, central tendency, and variability of data sets and be able to use these characteristics to compare data sets (Garfield & Gal, 1999). Developing students' abilities to interpret histograms is a common goal for introductory statistics courses. First, histograms serve a role in describing the characteristics of data sets, providing visual depictions of samples, populations, and sampling distributions. Second, an understanding of histograms contributes to the understanding of statistical concepts, including descriptive statistics such as mean, standard deviation and skewness. Comprehension of inferential statistics such as t-tests and the role of p-values in hypothesis testing depending in part upon an understanding of histograms as well. Textbooks rely heavily on histograms in their presentations of statistical concepts. Third, the most commonly taught inferential techniques are based on assumptions of normality, usually described in introductory courses in terms of histogram shape. It is therefore necessary that students understand the concepts represented in histograms and that they are able to extract relative measures of central tendency, variation and symmetry of distribution in order to interpret histograms and make comparisons among them. Although research has documented some of the difficulties students have in understanding basic statistical concepts (Garfield & Ahlgren, 1988; Pollatsek, Lima, & Well, 1981), there is little research documenting college students' interpretations of histograms. In this study, college students enrolled in an introductory statistics course were assessed on their abilities to interpret the concepts of mean, standard deviation, and symmetry of data distribution as represented in histograms. More specifically, students were asked to compare histograms showing symmetric and asymmetric distributions along with varying means and standard deviations. From these, students were directed to identify histograms displaying means and standard deviations similar to a "reference" histogram and to identify a skewed distribution. With the goal of determining specific aspects of students' abilities to extract information from histograms at the end of the statistics course, we constructed a set of noncontextual problems that were included on the final examination and were intended to challenge students to interpret and compare histograms at a high level of abstraction. Students' responses were analyzed so that we could identify common errors and interpret these errors in a way that would inform our pedagogy. Because of the complexity of the histogram interpretation tasks, we first ran a pilot study. …
Introductory college mathematics courses comprise a large percentage of course offerings in postsecondary institutions, serving over half of all students who ever study mathematics in college (Cohen, 1995). In a report of mathematics classes offered in fall, 2000, 14% of the sections were remedial and another 38% were introductory level, including precalculus (Lutzer & Maxwell, 2000). Many students are ill-equipped for introductory college math courses. Many degree programs in non-technical fields require math prerequisites, which are often stumbling blocks for students. A matter of scientific interest is the nature of students' attitudes toward mathematics and the relationship between attitudes and achievement in mathematics, especially as it relates to the achievement gap in mathematics between males and females, and the lack of interest by females in science, technology, engineering, and mathematics majors (STEM). In the past decade the American Association of University Women (AAUW) and the National Science Foundation (NSF) have invested nearly $90 million to fund hundreds of projects aimed at increasing the participation of girls and women in STEM (AAUW, 2004). During the past few years, SAT math scores indicate that the gender gap is narrowing because females on average gained 19 points while males gained 13 (Hoover, 2001). Explanations of the math gender gap have focused on social and cognitive differences. Males do better on multiple choice tests in mathematics, while girls are better on open-ended or essay questions that involve verbal skills (Beller & Gafni, 2000). Boys have better spatial ability (Collins & Kimura, 1997; Nordvik & Amponsah, 1998). Differential treatment of males and females in math classes has also been used to explain the difference, because females are not supported in math aspirations by their instructors and their parents (Hammrich, 2002). Efforts to create equal educational opportunities for females are primarily based on changing the attitudes of females about the study of math and pursuit of technical careers, because there are only social impediments to women entering technical fields and professions. Some researchers maintain that it is important to foster safe and nurturing environments in order to encourage female students' success in science and mathematics (Allen, 1995; Hammrich, 2002; Mann, 1994). Research has cast doubt on explanations that account for cognitive differences, because achievement in mathematics courses in middle school and high school is virtually the same for males and females (Davis-Kean, Eccles, & Linver, 2003). Data from the National Assessment for Educational Progress (NAEP) also confirm that at all grade levels there is little difference in the overall performance of males and females (Campbell, Reese, O'Sullivan, & Dossey, 1996; Kenney & Silver, 1997). Performance in specific content area also reflects little difference between males and females; the only statistically significant gender difference appeared at grade 12 for items in the areas of measurement and geometry, with males having statistically significantly better performance. NAEP (Kenney & Silver, 1997) reported little overall difference between males and females for those who enrolled in core college preparatory courses, with the exception of calculus, which was taken more frequently by males. These data reflect a national trend toward increased course taking by high school students in response to increased graduation requirements, and they attest to a change in the achievement of females. NAEP data regarding affect toward mathematics showed that males in grades 8 and 12 were significantly more likely than females to agree that they liked mathematics, but there was little or no difference between males and females in their perception of being good at mathematics. Students at all grade levels appeared to view mathematics as having considerable social and economic utility. …
A general education teacher completed an action research project in order to serve his students with special needs better by truly incorporating a special education teacher in co-teaching. By doing so with a set routine each day, the students were able to gain more mathematics understanding and experience a more positive learning experience in mathematics The time was split into four distinct segments each day, and a template was provided to each student in order to make taking notes and understanding concepts easier. Both teachers taught important concepts reaching all students. The data is supportive of increased student achievement after t-tests were completed on student test scores from the first year of co-teaching (with little structure and little teaching by the special education teacher) to the second year of co-teaching where the structure and effective co-teaching guidelines were used. Background This study is based on an action research project that focused on using data to improve teaching students with special needs. The instructor who engaged in action research was in his first few years of teaching, and had only experience teaching general education mathematics classes. In his second year of teaching, he was given the opportunity to teach a within a class. The instructor was informed that this type of had a specific balance of students diagnoses with learning disabilities versus students with no diagnosis, as well as a teacher from the Special School District present in to help co-teach the In a typical within a class (CWC) structure, a group of students, some with disabilities and some without, are taught together with a general education teacher and a special education teacher in one classroom. Traditionally co-taught classrooms involve the general education teacher teaching the lesson with the special education teacher assisting by helping keep students on task and answering individual questions as needed (Magiera, Smith, Zigmond, & Gebauer, 2005; Friend & Reising, 1993). In general, the special education teacher often is more supportive while the general education teacher often leads the In many instances, the first year of co-teaching is exactly what is listed above: the mathematics teacher leading the while the special education teacher is the support. However, for co-teaching to be effective, teachers must move beyond this kind of arrangement. There are, in fact, a variety of methods for co-teaching, and the description below explains only two models (see Friend & Reising, 1993, for descriptions of various methods). The first year the instructor taught the within a class, he taught it the same traditional way he always taught his Algebra IB classes. For example, he provided examples on the board, engaged students to work problems at their desks and answer questions, and encouraged students to take notes and write down examples. The only difference was that a CWC teacher helped students one-on-one during lectures, made copies of notes for students who were absent, helped keep students on task, and occasionally made a comment to support what the general education teacher taught. At the end of the year, there was not a noticeable difference in the performance of the students in the CWC classes compared to the previous year (classes with few documented students with disabilities). The discipline was better, but student understanding did not seem affected. The instructor realized that having a second adult in the room did not really impact student achievement. Therefore, the mathematics instructor and special education teacher began to examine the classroom environment with the following questions: 1. How could student understanding and retention improve? 2. How could the CWC teacher's skills be utilized the best? 3. How could both teachers sustain the attention of more students for a greater period during class? …
The number of students with learning disabilities attending institutions of higher education has dramatically increased in the last ten years and will continue to do so. Since federal laws in the United States and other counties require appropriate accommodations for students with learning disabilities, it is important for universities to evaluate the effectiveness of accommodations and services. This study was designed to evaluate the effectiveness of a class reserved for students with learning disabilities and to identify predictors of success based on student documentation. Evaluation of the relationship between student success and specific student documentation will facilitate the development of sound, researched policies for making accommodation and placement decisions. Introduction Increased attention to and legal support for those with learning disabilities in education has led to a dramatic increase in the number of students with learning disabilities attending colleges and universities. Unfortunately, few institutions are monitoring performance, graduation rates, attrition, satisfaction, or other indicators of success for students with learning disabilities in their learning services programs (Vogel & Adelman, 1992). Evaluation of what little information there is available for these programs suggest that there is reason to be concerned with success of students with learning disabilities (Vogel & Adelman, 1992). Most universities have ways to assess academic courses through student evaluations of teaching but few have a system in place for feedback on their learning services programs. This raises serious questions about the assessment and implementation of good accommodation and placement practices and the importance of evaluating the services that are in place. Mathematics learning is crucial to the overall academic success of students with learning disabilities. A significant number of students with learning disabilities have difficulty with mathematics learning (Miller & Mercer, 1997) and other learning disabilities may interfere with testing ability and mathematics learning even if students are not diagnosed with specific mathematics learning impairments (Nolting, 2000). Students with learning disabilities are more likely to fail to organize information, both mentally and physically, in a way that allows for easy retrieval, use, and generalization (Scheid, 1990). Often, students with learning disabilities achieve approximately one year of mathematical understanding for every two years of school attendance (Miller & Mercer, 1997). This progress continues into adulthood, leaving students with learning disabilities behind their contemporaries (Raskind, Goldburg, Higgins, & Herman, 1999). All of this suggests the placement and accommodations associated with mathematics learning are important in the success of students with learning disabilities in higher education. Placement and Accommodations Practices Ofiesh and McAfee (2000) surveyed ninety-one college learning services programs to examine current use of psycho-educational evaluations such as the Wechsler Adult Intelligence Scale (WAIS) and the Woodcock-Johnson Tests of Cognitive Abilities and Tests of Achievement (WJ). They found that these tests were being consistently used for eligibility, placement, and accommodation determinations. For eligibility purposes, a diagnostician has a formal and systematic process for interpreting scores supported by research. However, the process of how test scores are being used to determine accommodations and placement is much less structured and is widely undocumented through survey or empirical data at the postsecondary level. There is a strong need for further research to validate the practice of interpreting specific parts of psycho-educational test scores to make accommodation decisions (Ofiesh & McAfee, 2000). These decisions are often left to a learning services specialist who must make practical connections between test scores and available accommodations or courses. …
This paper reports the results of an action research project that examined the use of interactive guided notes in two sections of freshman level college algebra. This method unifies lecture, in-class guided practice, and cooperative learning into the students' note taking. Student success and satisfaction were dramatically higher in the course sections using the guided notes. The use of guided notes also made it possible to include discussion, inquiry, and group problem solving in a course that is otherwise taught entirely by lecture. The paper also describes how the author used principles from concept and information mapping to inform the development of the guided notes. Introduction In mathematics classrooms at the secondary and college level there are institutional norms and policies that hinder the process of changing to reform-based practices (McDuffie & Graeber, 2003). One of the most entrenched norms found in these classrooms is the emphasis on traditional lecture and student note-taking format. This paper reports the results of an action research project on the use of interactive guided notes as an alternative to the traditional lecture method. The paper also reflects on how this process improved student success and satisfaction in freshman level college mathematics courses and supported the inclusion of discussion, inquiry, and group problem solving into courses that are otherwise taught entirely by lecture. In conclusion, this paper suggests possible ways to encourage the use of guided notes in mathematics courses at the secondary and post-secondary levels and discusses the need for ongoing inquiry into the effectiveness of instructional methods and educational policies as the culture in which they function continues to change rapidly. Action research is a form of investigation designed for use by teachers to solve problems and improve professional practices in their own classrooms. Action research involves systematic observations and data collection, which can then be used by the practitioner-researcher in reflection, decision making and the development of more effective classroom strategies (Parsons & Brown, 2002). The problem addressed by this action research project was the high failure rate in freshman level mathematics classes at a small state university campus in the Midwest. One project had already created a mathematics learning center with developmental courses and required labs for incoming freshmen placed in the program using the placement test developed for this purpose by the Minnesota State Colleges and Universities (MinnSCU) Center for Teaching and Learning. While participation in the math learning center by these students significantly improved their subsequent success rate in college algebra, there were still 20-30% of the students who were unable to successfully complete college algebra on their first try. The impetus for this action research project came from reading the observations of others studying typical mathematics lessons in Japan and Germany as well as personal observations of how student note taking actually interfered with student interaction and learning in the classroom. Trelfa (1998) notes that in all levels of Japanese schools mathematics is normally taught, not directly from the textbook, but from printouts that the instructor makes for each class. The printout, or worksheet, contains the lesson objectives and problems related to each day's lesson. These are typically clear and well organized in order to help students follow the lecture, study and review. They are not typically graded by the teachers but rather kept by the students for reference and review purposes. Additionally, the following four classroom observations contributed to my interest in the development of guided notes of freshman college mathematics courses. First, many students are often unable to write coherent notes while at the same time listening to and thinking about what the instructor is saying. …
This study sought to better understand instructional models that could be expected to improve student understanding of graphs of kinematic variables (distance, velocity and acceleration). The effect of using CBL-instruments and cooperative group structure (alone and in concert) was examined for repairing students' misconceptions. Misconceptions were determined using Nemirovsky and Rubin's (1992) definitions for cues that indicate students' misconceptions. Laboratory activities utilized in the various instructional settings were created that incorporated strategies developed by Kykstra, D. I., Boyle, C. Fl, Monarch, I. A. (1992) to promote conceptual change. Results of the study support previous research that even though students understand the requisite mathematical concepts, they still have misconceptions concerning the interpretations concerning the interpretation of mathematical terms in a physical setting. The most problematic misconception was indicated by students' use of Linguistic Cues; students interpreted mathematical terms using common language interpretations, not mathematical interpretations. Students continued to use common language interpretations even when confronted with physical situations using CBL-tools that contradicted their (incorrect) conclusions. Students needed to not only be confronted by their misconception, but needed the confrontation to be confirmed by the teacher or they maintained that their interpretation was correct and that the physical evidence was wrong. ********** You can not apply mathematics as long as words still becloud reality. (Herman Weyl 1885-1955) Mathematicians and physicists believe that when people communicate mathematics using algebraic symbols, communication is precise and unambiguous. However, when applying the symbols of mathematics many students would agree with Wehl that there is a great deal of ambiguity. For example, many students have difficulty articulating their understanding of the relationship between a function, its derivative, and its graph. One of the principle applications of these concepts is with problems involving distance, velocity and acceleration of a moving object (kinematic variables). Virtually all students come to the classroom with some personal experience with kinematics. The desire to build conceptual understanding of functions, graphs and the physical phenomena that they relate to, using knowledge that students already have, is consistent with widely accepted constructivist principles. Clement states, We assume that it is desirable to be able to ground new material in that portions of the student's intuition which is in agreement with accepted theory. When this is possible, it should help students to understand and believe physical principles at a 'make sense' level instead of only at a more formal one (1989, p. 1). Unfortunately, students' personal understanding of kinematic variables may be incomplete or erroneous. Students' difficulties are often grounded in knowledge based on their personal experiences (Monk, 1990; Nemirovsky, J.R., Monk, S., 1992) Further, many students continue to have difficulty interpreting graphs of kinematic variables even following instruction in mathematics and in physics courses (Beichner, 1994; McDermott, L.C., Rosequist, M. L., Van Zee, E. H., 1986). Students recognize that the slope of a velocity graph is acceleration, but fail to reflect on the physical interpretation of negative acceleration, and whether the interpretation is different when velocity is negative rather than positive. The traditional model of instruction for mathematics and physics courses has been a lecture/homework format, with lectures concentrating on the algebraic interpretation of variables. This traditional format may not be effective for developing understanding of graphs of kinematic variables: Teachers cannot simply tell students what the graphs' appearance should be. …
From research observations of activities during a second-grade mathematics problem-centered learning classroom, synergistic argumentation emerged as a class norm for discussing the students' mathematics. In this paper we analyze the contrast between two students who were participants in this class. Both students were accustomed to sharing during whole-class discussions in this classroom environment. Both were very capable mathematical thinkers. One student, Brett, depicts the use of argumentation successfully while the other, Miriam, depicts the use of argumentation ineffectively. An important aspect of Brett's argumentation was identified as hermeneutic listening. Brett's engagement enhanced the learning environment whereas Miriam's stance was counter-productive, at least for her. ********** Our research over the last several years investigated a second-grade mathematics classroom where the teacher enacted a problem-centered learning environment (Wheatley, 1991). During the last three years, we focused in particular on the quality of discourse and argumentation that occurs during the whole class sharing time in the lesson. The purpose of this study has two fold: (a) to describe the mathematics whole class sharing session and (b) to analyze its function for effective learning occurring through conflict and disagreement in the open explanation of solutions and strategies. In the problem-centered learning environment students are encouraged to express their ideas freely, try to make sense of each other's methods, listen, question, and carry on a conversation between and among themselves. Our overall goal was to analyze the whole class interaction patterns and learning opportunities. In the second year of this investigation, two students in particular provided us with contrasting pictures of discourse and argumentation during their involvement in the whole-class discussions. In this paper we will describe and analyze episodes involving these two students. Examining their contrasting stances of sharing or not sharing of ideas, provides a deeper understanding of the nature of discourse and argumentation and its degrees of success or failure in learning enhancement from the young students' perspective. Framework for this Study In recent years there has been increasing numbers of investigations into classroom environments where children talk openly about mathematics and explain their mathematical solutions. In these environments the teacher's role is different; s/he (a) facilitates the development of students' mathematical understands and (b) acts as a facilitator by organizing instruction so that students are interactively engaged in the doing and talking dynamically about mathematics (NCTM, 2000; Wood, 1999). A number of researchers argue that collaboration and whole-class sharing encourages children to learn with understanding through opportunities to explain, justify, and listen to one another's ideas (Cobb, 1998; Cobb & Yackel, 1996; Cobb, Yackel, & Wood, 1992; Kazemi, 1998; Mevarech, 1999; Wheatley, 1991). Further, they argue that explanations are the best means for students and teachers to elaborate meaning and to make connections to other mathematical topics as well as to their own prior knowledge. These conditions help students to construct rich networks of meaning. As students share their explanations they seek meaningful ways to communicate and elaborate their ideas with each other and the teacher, negotiating meanings as opposed to just reciting facts. As students negotiate, they adjust their interactions by presenting rationales for their strategies while attempting to make sense of each other's ideas. Sfard (2001) contends that this open communication is equivalent to thinking itself. She states that our thinking is a dialogical endeavor as we inform, argue, reflect, and question others (as well as ourselves). Thus, thinking is communication; not necessarily verbal, and not necessarily inner. …
While it is known that the mathematics achievement levels of deaf children are substantially below that of their hearing peers, it is not known when or in what capacity these delays begin. It is possible that differences in achievement are also demonstrated in the early thinking skills of these children, for example, the ability to classify. The purpose of this pilot study was to begin to examine the pre-classification skills demonstrated by young deaf children. Findings from this study indicate a possibility that deaf/hard-of-hearing children experience substantial limitations in pre-classification skills as demonstrated through their performance on free-sorting, abstract tasks of the nature used in this study. Limitations in the development of pre-classification skills could impact deaf children's understanding of hierarchical concepts and part/whole relationships thereby influencing their ability to demonstrate adequate understanding of mathematical concepts. Background The low performance of deaf students in the area of mathematics has been well documented (Wood, Wood, Griffiths, & Howarth, 1986; Traxler, 2000; Luckner & McNeil, 1994, Ansell & Pagliaro, 2006). Yet while it is known that the performance of deaf students is not up to par, the reason behind and solution for this problem are still unknown. Typically, research in this area has focused on the school age population, including students between the ages of 12 years (Wood et al., 1986) and 19 years (Luckner & McNeil, 1994). These studies have been solely focused on deaf children's mathematics learning in the classroom and/or their abilities to problem solve (Luckner & McNeil, 1994; Ansell & Pagliaro, 2006). An area that has not yet been investigated is the thinking skills, including the ability to classify, that deaf children bring to the classroom with them. Defining Classification As defined by Piaget (1962), the first step in classification is an ability to make collections. This differs from formal classification in that membership in a collection is dependent upon perceptions, therefore, the members of a set must be physically present (Phillips & Phillips, 1996). Young children are not capable of classification because they are not yet able to abstract out any one attribute to tie a group together. While it may be possible for them to organize groups that differ by one criterion, (e.g., a group of cards that differ only by color), they are not able to do this if there are multiple differences (Lunzer, 1964). The ability to classify develops through practice in making collections (Phillips & Phillips, 1996). According to Piaget, there are three sequential levels that one must pass through to develop an understanding of classification. The first two levels are pre-classification skills and include the making of graphic and non-graphic collections. True classification is demonstrated in the third level through expression of the understanding of class inclusion and hierarchical relationships (Phillips & Phillips, 1996). In graphic collections the items to be sorted are viewed independently. A common approach to grouping at this level is to sort items into carefully arranged spatial configurations. For example, the child may create pictures or designs out of the materials to be sorted. A child functioning at the graphical level will examine similarities between items; comparisons however, are made between only two items at a time. S/he is unable to establish a relationship of similarity between individual items and the whole group. Properties of items are not considered as criteria for membership in a group, rather what the total arrangement looks like is the child's primary concern (Phillips & Phillips, 1996). Unlike graphic collections, in non-graphic collections items are assigned to piles or groups based on similarity. While this is a more sophisticated level of pre-classification, non-graphic collections differ from true classification in that items still need to be within close proximity to each other and their properties must be directly perceptible (i. …
Perceived self-efficacy beliefs have been found to be a strong predictor of mathematical performance while problem posing is considered fundamental in mathematical learning. In this study we examined the relation among efficacy in problem posing, problem-posing ability, and mathematics achievement. Quantitative data were collected from 176 fifth and sixth grade students, and interview data from six students selected on the basis of hierarchical cluster analysis. Students' perceived efficacy to construct problems was found to be a strong predictor of the respective performance as well as of the general mathematics achievement. A strong correlation was also found between ability in problem posing and general mathematics performance. The students constructed problems of greater variety and complexity on the basis of informal tasks rather than on the basis of formal tasks. Significant differences were found in problem posing ability, between fifth and sixth grade students. The findings provide support to earlier studies indicating the predictive power of context-specific efficacy beliefs. Implications are drawn about strategies for enhancing students' efficacy beliefs and problem-posing ability. Theoretical Background and Aims Research on mathematics teaching and learning has recently focused on affective variables, which were found to play an essential role that influences behavior and learning (Bandura, 1997). The affective domain is a complex structural system consisting of four main components: emotions, attitudes, beliefs, and values (Goldin, 2002). Beliefs can be defined as subjective knowledge, theories, and conceptions and include whatever one considers as true knowledge, although he or she cannot provide convincing evidence to support it (Pehkonen, 2001). Self-beliefs can be described as beliefs regarding personal characteristics and abilities and include dimensions such as self-concept, self-efficacy, and self-esteem. Self-efficacy can be defined as one's belief that he/she is able to organize and apply plans in order to achieve a certain (Bandura, 1997, p. 3). This study focuses on self-efficacy of primary students with respect to problem posing. Self-efficacy is a task-specific construct and there is a correspondence between self-efficacy beliefs and the criterial task being assessed; in contrast, self-concept is the sense of ability with respect to more global goals (Pajares, 2000; Bandura, 1986), while self-esteem is a measure of feeling proud about a certain trait, in comparison with others (Klassen, 2004; Bong & Skaalvik, 2003). The task-specificity of efficacy beliefs implies that related studies are more illuminating when they refer to certain tasks, such as problem posing; the predictive power of self-efficacy is in this case maximized (Pajares & Schunk, 2002). On the other hand, the level of specificity could not be unlimited; as Lent and Hackett (1987) have rightly observed, specificity and precision are often purchased at the expense of practical relevance and validity. The construct self-efficacy is tightly connected to motivation and plays a prominent role in human development since it directly influences behavior. According to Bandura's social cognitive theory, every individual possess a system that exerts control on his/her thoughts, emotions and actions. Among the various mechanisms of human agency, none is more central or pervasive than self-efficacy beliefs (Bandura & Locke, 2003; Pajares, 2000). Research on self-efficacy has recently been accumulated providing among other things notable theoretical advances that reinforce the role attributed to this construct in Bandura's social cognitive theory. Several studies have indicated a strong correlation between mathematics self-efficacy and mathematics achievement (Klassen, 2004). It was further found that mathematics self-efficacy is a good predictor of mathematics performance irrespective of the indicators of performance (Pajares, 1996; Bandura, 1986) and regardless of any other variables (Bandura & Locke, 2003; Pajares & Graham, 1999). …
This study investigated preservice elementary teachers' pictorial representations of how they envision their future classroom and describe their own actions as well as those of their students. Drawings were analyzed for the preservice teachers' self-perceptions and for the language used in descriptions to refer to the teacher in the drawing for indicators of teacher-centeredness vs. student-centeredness. The drawings revealed that 82% of the preservice elementary teachers drew a female teacher and that 71% of them do not yet identify themselves as the teacher. Findings also showed that the majority of preservice elementary teachers still envision a classroom that is more teacher-centered than student-centered despite efforts of teacher preparation programs to effect a change in thinking. Introduction Preservice teachers come into teacher preparation programs with firmly established beliefs, attitudes, and perceptions about teaching that are born out of and nurtured by their previous experiences in school (Minor, Onwuegbuzie, Witcher, & James, 2002). Moreover, these beliefs, which can be held from early childhood (Goodman, cited in Thomas, Pederson, & Finson, 2001) and which are resistant to change, tend to direct teachers' classroom practice (Hart, 2002). As preservice teachers proceed through professional education programs, they are given opportunities, typically though journal writing and assembling their education portfolios, to examine their reasons for choosing to be a teacher, to explore what it means to be a good teacher, and to reflect upon their teaching and attitudes. Some researchers have offered them another avenue by which they may reflect upon their attitudes. Some researchers have offered them another avenue by which they may reflect upon their attitudes about teaching--by drawing their perceptions of what a teacher looks like. Thomas, Pederson, and Finson (2001), recognizing and acknowledging their disparity between the traditional, information-laden, teacher-centered science education experiences that preservice teachers bring with them and the contemporary, inquiry-based, student-centered science teaching that preservice teachers experience in teacher preparation programs, have validated the Draw a Science Teacher Test (DASTT). The DASTT was fashioned after Chambers' Draw-A-Scientist-Test (DAST) (as cited in Thomas, Pederson, & Finson, 2001) which was guided by Goodenough's Draw-A-Man-Test (DAMT) and Goodenow's Draw-A-Person-Test (DAPT) (as cited in Chambers, 1983). While the DAMT and DAPT were designed as intelligence instruments in the case of the former, or as indicators of the drawer's self-image in the case of the latter, Chambers (1983) developed the DAST in order to determine at what age and to what degree children first develop definite images of the scientist. Moseley and Norris (1999), recognizing that preservice teachers could not be expected to invest themselves in science education reform if they will held the traditional, even stereotypical image of a scientist, i.e. white, male, wearing lab coat, eyeglasses, and facial hair, surrounded by lab equipment and books, administered the DAST to preservice teachers. They found that the preservice teachers' perceptions of scientists were similar to those of children, so the DAST became a springboard for discussion between teacher educators and preservice teachers, aimed at adjusting the lens through which preservice teachers view science and the scientific community, widening the focus to a more diverse image. According to Thomas, Pederson, and Finson (2001), the DASTT gives preservice teachers the opportunity to place themselves into a visual image of their future classrooms, whereby they can (a) picture themselves as elementary science teachers, (b) place themselves along a teaching theory continuum, and (c) consider the ways in which they developed their own science teaching beliefs (p. …
To study the roles that the graphing calculator plays in solving problems about functions, a small quasi-experimental study was conducted with four pairs of undergraduate students solving problems with and without the graphing calculator. The analysis of the protocols of the sessions did not reveal major differences that could be attributed to the presence or absence of the tool but indicated differences in strategies used with each problem that could be explained in terms of the nature of the knowledge at stake and to students’ availability of that knowledge. The study suggests a model for conducting research that looks for explaining the effects of technology in learning and instruction. Con el fin de analizar el papel que la calculadora gráfica juega en la resolución de problemas sobre funciones, se hizo un pequeño estudio cuasi-experimental con cuatro pares de estudiantes de pre-grado variando la condición de la disponibilidad de la calculadora. El análisis de los protocolos de las sesiones revela que no hay mayores diferencias que se puedan atribuir a la presencia o ausencia de la calculadora gráfica; sin embargo, las diferencias observadas en el uso de estrategias que se usaron en cada problema pueden explicarse en términos de la naturaleza del conocimiento en juego y de la disponibilidad de tal conocimiento para los estudiantes. El estudio sugiere además un modelo para realizar investigaciones que busquen explicar los efectos de la tecnología en el aprendizaje y en la instrucción.
This study uses concept mapping to investigate the logarithm. Empirical evidence documents the extent to which selected instructors and students command substantive knowledge about logarithms and analyzes their ability to make mindful use of their current understanding. Initial mapping reflects inadequacy and provides the basis for an in-depth search of the cultural and historical context that gave rise to the logarithm concept. The author presents a map that incorporates the essence of the concept from a cultural historical perspective as its central structure. This key ideational relationship, the conceptual cross link between arithmetic and geometric sequences discovered by John Napier, precisely identifies the source of weakness in conceptualization empirically evidenced among faculty and students. The improved map visually provides the understanding necessary for corrective intervention and is the prime reference in the development of a pedagogical approach toward more substantive knowledge and mindful use of the logarithm concept. Study results indicate that concept mapping can provide an epistemological tool for sound curricular and instructional development in mathematics education; one that seeks to locate and build on the essence of conceptual foundations. In a scientific discipline, graphically rendering such substantive cognitive structures maximizes the probability of their mindful use in related mathematical reasoning.
The paper discusses one of the case studies of a multiple-case study teaching experiment conducted to investigate the usefulness of the metacognitive tools of concept maps and vee diagrams (maps/diagrams) in illustrating, communicating and monitoring students' developing conceptual understanding of matrices and systems of linear equations in an undergraduate mathematics course. The study also explored the tools' role in scaffolding and facilitating students' critical and conceptual analyses of problems in order to identify potential methods of solutions. Data collected included students' progressive maps/diagrams, journals of reflections and justifications of revisions, and final reports and researchers' annotated comments on students' maps/diagrams and anecdotal notes from presentations. Findings showed that students developed more enriched, integrated and connected understandings of matrices and systems of linear equations as a result of continually organizing coherent groups of concepts into meaningful networks of propositional links, critically reflecting on the results against feedbacks from critiques and negotiations for shared meanings, and crystallizing these conceptual changes and nuances where appropriate as revised or additional propositional links. Verifying and justifying solutions were greatly facilitated through the combined usage of concept maps and vee diagrams. Findings suggest that students' classroom experiences in working, thinking and communicating mathematically can be enhanced by incorporating these metacognitive tools into students' repertoire of effective learning strategies. Introduction Current emphases in national and state curricular frameworks urge the promotion of deep knowledge and deep conceptual understanding of students as well as enhancing students' abilities and skills in working, thinking and communicating mathematically. To achieve these content and process outcomes, mathematics teachers are encouraged to be innovative, investigative and explorative in their pedagogical approaches to designing and developing learning activities (NCTM, 2000; NSW 2002). External examination reports (MANSW, 2005) indicate that a high proportion of students have difficulties comprehending the meanings of key concepts in the context of problems, justifying solutions, and presenting coherent mathematical arguments. Furthermore, first year university students' mathematical performances (Mays, 2005) in diagnostic tests show that most have mathematical misconceptions with fractions, percentages and multi-digit subtraction. Similarly, national surveys in Samoa confirm that learning by rote-memorization is quite prevalent in most schools (DOE, 1995). Such findings resonate with recurring comments in examiners' reports concerning students' obvious inabilities to effectively apply existing knowledge to successfully answer exam questions (Afamasaga-Fuata'I, 2001, 2002a, 2002b, 2002c, 2003, 2005a, 2005b). In foundation and undergraduate mathematics classes in Samoa, students find it difficult to explain and justify their answers mathematically in terms of the conceptual structure of relevant topics. Instead their verifications are often in terms of sequences of steps of procedures. Whilst this may work for familiar problems, this procedural view constrains them when solving qualitatively and structurally different problems (i.e., novel problems). According to Richards (1991), this manifestation is typically a communication problem resulting from students' inability to understand the meaning of a language (i.e., concepts, principles, theorems and theories) used in mathematical discussions and dialogues of more mathematically literate others. Subsequently, less mathematically literate students are unable to make sense of such conversations, offer conjectures or evaluate mathematical assumptions. When doubtful, students tend to use any procedure to get an answer without really checking whether an algorithm is suitable to the problem (Schoenfeld, 1996). …
We address the question whether to present students with a single method with a number of methods for solving quadratic inequalities. Twenty 10th graders were presented with three differently sequenced methods, i.e., the graphic, the sign-chart, the logical connectives method. We examined participants' preferences when solving related tasks, their success in doing so. Almost all students correctly solved the different inequalities, most liked being presented with several methods. The method most frequently used was the graphic method, many students preferred the method they had studied first. Some conclusions educational implications are drawn. Students' Preferences When Solving Quadratic Inequalities When planning instruction, teachers may face the dilemma of whether to present their students with a single method for solving a specific type of mathematical problem with a number of methods. When a single method is chosen, a question arises about the criteria for this choice. If, on the other hand, several methods are going to be presented, may wonder whether there is any significance to the order of presentation. A related question is, what should such didactic decisions refer to? For example, should we only consider students' success (as defined by the ability to answer correctly) should we also attend to the methods students choose either when solving related mathematical tasks when responding to explicit questions regarding their preference? We address these issues with regard to the process of solving quadratic inequalities. The literature commonly presents three major methods for solving quadratic inequalities: the graphic method, the sign-chart method, the logical-connectives method. The graphic method involves the interpretation of graphic representations, e.g., using parabolas to solve quadratic inequalities. The sign-chart method involves finding the zeros of an equivalent equation using a sign chart to determine the solution of the inequality. The logical-connectives method involves the translation of the inequality into a system of linear inequalities, which are connected to each other by or and connectives (see example in Figure 1). Two approaches for teaching quadratic can be identified in the literature: the single-method approach, presenting the students with only method, the multiple-method approach, presenting the students with any combinations of the two all three above-mentioned methods for solving this type of inequality. For example, Dreyfus Eisenberg (1985) preferred the graphic method provided mathematical didactical argumentation for their claim: [FIGURE 1 OMITTED] this [graphic] approach appears to make the solution of many easier for average weaker than average students who have had some experience with graphing functions. It also provides quite a bit of insight into what it means to solve an inequality in what sense are related to functions. (p. 653) McLaurin (1985) Dobbs Peterson (1991) identified the sign-chart method as the best method for teaching quadratic inequalities. They explained that one of the most appealing aspects of sign charts is that they serve as a uniform relatively easy method for solving what many consider to be more complicated inequalities (p. 664). In fact, the researchers' claims were actually more far-reaching, since both McLaurin Dobbs Peterson suggested that the method they offered provided students with a powerful tool for solving not only quadratic, but any type of inequality. However, their papers report no research that supports their conclusions, which seems to suggest that these conclusions are based on the authors' impressions from their own teaching. Piez Voxman (1997), on the other hand, supported the multiple-method approach for solving inequalities. …
The aim of this paper is to present the results of an experiment carried out in the Mexican Autonomous Metropolitan University (UAM) based on recent theories of educational research on building up knowledge. The research group was studying Calculus 1 following the Programme of Intervention, SAM (Mediated Learning System), an innovative teaching project in mathematics. The Programme of Intervention, SAM, is based on the theories of the cognitive paradigm, and an important element of such Programme is concept mapping. The Programme considers that through the use of concept mapping in the study of mathematics, cognitive abilities can be acquired leading to an improvement in the intelligence coefficient of the students. Through the Programme the teacher is the agent between the curriculum contents and the students. He designs the classroom presentation and the concept mapping according to how the student learns. The teacher considers that concept mapping is an important element to develop skills. Introduction In the education context, the cognitive paradigm considers that education must be oriented towards the achievement of meaningful learning (it must make sense) and towards the development of general and specific strategic abilities of learning. Teaching, concretely in the classroom, from the point of view of this paradigm, must allow for the learning of the contents of the curriculum in the most meaningful way. This has the implication that planning and organization of the didactic processes are vital for the creation of the minimum conditions for meaningful learning. In addition, with regard to teaching within the framework of the cognitive paradigm, the teacher must start from the idea that the student is active and can learn meaningfully, that he can learn to learn and to think. The teacher must concentrate on the task of the combination and organization of didactic experiences in order to achieve learning. In order to bring information that will contribute in some way to aspects related to the teaching and learning of mathematics, the initiative arose of undertaking research concentrating on a Programme of Intervention designed SAM (Mediated Learning System). Such a programme-a special way of acting in the classroom-would consider mathematic contents as a vehicle to develop awareness in the student, understanding cognitive development as the development of a collection of cognitive skills. One of the important elements of the Programme SAM is concept mapping. Programme SAM The Programme SAM takes into consideration the thought processes both of the teacher and the student; processes proper to the analysis of the cognitive paradigm. The program is based on Ausubel's theory of meaningful learning (Ausubel, Novak & Hanesian, 1988). It is a Programme that follows the model Learning-Teaching (how the one who is learning learns, based on which, it is possible to design the teaching) and serves as a tool to achieve student learning where the teacher acts as a mediator for the learning. The Programme, on one hand, considers the abilities and cognitive skills as objectives, and on the other, mathematical contents and method are considered as a means to achieving these objectives. From this point of view, the Programme orients the teaching towards cognitive development, which is why it is considered as a Programme of cognitive intervention. The Programme has objectives by skills: Develop skills of induction and deduction considered as a part of a capacity for logical reasoning and develop the skills of situating, locating and expressing graphically as part of the capacity for spatial orientation. The Programme SAM achieves its objectives when the teacher uses concept mapping as a support material during action in the classroom. The teacher has to arrange and organize the contents of the subject in order to facilitate learning. …
