Using Concept Maps and Interpretive Essays for Assessment in Mathematics

This study explored the use of student-constructed concept maps in conjunction with written interpretive essays as an additional method of assessment in three undergraduate mathematics courses. The primary objectives of this study were to evaluate the benefits of using concept maps and written essays to assess the “connectedness” of students' knowledge; to measure the correlation between students' scores on the concept maps and written essays, course exams, and final grade; and to document students' perception of the effect of this approach on their mathematical knowledge. Results indicated that concept maps, when combined with written essays, are viable tools for assessing students' organization of mathematical knowledge. In addition, students perceive this approach as enhancing their mathematical knowledge.     

The Impact on Incorporating Collaborative Concept Mapping with Coteaching Techniques in Elementary Science Classes

The purpose of this research was to evaluate a collaborative concept‐mapping technique that was integrated into coteaching in fourth‐grade science classes in order to examine students' performance and attitudes toward the experimental teaching method. There are two fourth‐grade science teachers and four classes with a total of 114 students involved in the study. This study used a mixed method design, incorporating both quantitative and qualitative techniques. The findings showed that the two teaching methods obtained significant difference with respect to students' test scores. Using collaborative concept mapping to learn science could increase the opportunity of discussion between peers, thus fostering better organization and understanding the content. In addition, coteaching could enable teachers to share their expertise with one another. It could facilitate the implementation of collaborative concept mapping and the construction of student's concept mapping. Team teachers' attitude could affect the students' learning performance. However, some of the students had negative views on drawing concept maps because they found it was troublesome to write down many words, difficult to draw and arrange proposition, and time‐consuming. Coteachers' instant feedback and students' journal writing could guide and examine the students' concept maps to facilitate their cognitive learning.     

Analyzing Student Work as a Professional Development Activity

When worthwhile mathematical tasks are used in classrooms, they should also become a crucial element of assessment. For teachers, using these tasks in classrooms requires a different way to analyze student thinking than the traditional assessment model. Looking carefully at students' written work on worthwhile mathematical tasks and listening carefully while students explore these worthwhile tasks can contribute to a teacher's professional development. This paper reports on a professional development activity in which teachers analyzed mathematical tasks, predicted students' achievement on tasks, evaluated students' written work, listened to students' reasoning, and assessed students' understanding. Teachers' engagement in this way can help them develop flexibility and proficiency in the evaluation of their own students' work. These experiences allow teachers the opportunity to recognize students' potential, strengthen their own mathematical understanding, and engage in conversations with peers about assessment and instruction.     

On MTL's First Decade

Three experiments used multiple methods—open-ended assessments, multiple-choice questionnaires, and interviews—to investigate the hypothesis that the development of students' understanding of the concept of real variable in algebra may be influenced in fundamental ways by their initial concept of number, which seems to be organized around the notion of natural number. In the first two experiments 91 secondary school students (ranging in age from 12.5 to 14.5 years) were asked to indicate numbers that could or could not be used to substitute literal symbols in algebraic expressions. The results showed that there was a strong tendency on the part of the students to interpret literal symbols to stand for natural numbers and a related tendency to consider the phenomenal sign of the algebraic expressions as their “real” sign. Similar findings were obtained in a third, individual interview study, conducted with tenth grade students. The results were interpreted to support the interpretation that there is a systematic natural number bias on students' substitutions of literal symbols in algebra.     

Development of a Geometric Spatial Visualization Tool

This paper documents the development of the Geometric Spatial Assessment. We detail the development of this instrument which was designed to identify middle school students' strategies and advancement in understanding of four geometric concept domains (geometric spatial visualization, spatial projection, cardinal directions, and periodic patterns) after experiencing a carefully designed integrated lunar unit. Previous research with students using this lunar unit showed students making significant gains on lunar‐related concepts (both scientific and mathematical) on a Lunar Phases Concept Inventory (LPCI) ( Lindell & Olsen, 2002 ). Following the administration of single domain assessments, clinical interviews were conducted to ascertain students' problem solving strategies. Results allowed us to select four suitable multiple‐choice items per domain.     

U.S. and Chinese Teachers' Constructing,Knowing, and Evaluating Representations to Teach Mathematics

This study examined U.S. and Chinese teachers' constructing, knowing, and evaluating representations to teach mathematics. All Chinese lesson plans are very similar, because they are all based on the Chinese national unified curriculum in mathematics. However, the U.S. lesson plans are extremely varied, even for those teachers from the same school. The Chinese teachers' lessons are very detailed; the U.S. teachers' lesson plans have exclusively adopted the "outline and worksheet" format. In the Chinese lesson plans, concrete representations are used exclusively to mediate students' understanding of the concept of average. In U.S. lessons, concrete representations are not only used to model the averaging processes to foster students' understanding of the concept, but they are also used to generate data. The U.S. teachers are much more likely than the Chinese teachers to predict drawing and guess-and-check strategies. For some problems, the Chinese teachers are much more likely than are the U.S. teachers to predict algebraic approaches. For the responses using conventional strategies, both the U.S. and Chinese teachers gave them high and almost identical scores. If a response involved a drawing or an estimate of an answer, the Chinese teachers usually gave a relatively lower score, even though the strategy is appropriate for the correct answer, because it is less generalizable. This study contributed to our understanding of the cross-national differences between U.S. and Chinese students' mathematical thinking. It also contributed to our understanding about teachers' beliefs from a cross-cultural perspective.     

Taiwanese Gifted Students' Views of Nature of Science

This study examined the conceptions of nature of science (NOS) possessed by a group of gifted seventh‐grade students from Taiwan. The students were engaged in a 1‐week science camp with emphasis on scientific inquiry and NOS. A Chinese version of a NOS questionnaire was developed, specifically addressing the context of Chinese culture, to assess students' views on the development of scientific knowledge. Pretest results indicated that the majority of participants had a basic understanding of the tentative, subjective, empirical, and socially and culturally embedded aspects of NOS. Some conflicting views and misconceptions held by the participants are discussed. There were no significant changes in students' views of NOS after instruction, possibly due to time limitations and a ceiling effect. The relationship between students' cultural values and development of NOS conceptions and the impact of NOS knowledge on students' science learning are worth further investigation.     

The Effects of a STEM Intervention on Elementary Students' Science Knowledge and Skills

The purpose of the study was to assess elementary students' science process skills, content knowledge, and concept knowledge after one year of participation in an elementary Science, Technology, Engineering, and Mathematics (STEM) program. This study documented the effects of the combination of intensive professional development and the use of inquiry‐based science instruction in the elementary classroom, including the benefits of using rigorous science curriculum with general education students. The results of the study revealed a statistically significant gain in science process skills, science concepts, and science‐content knowledge by general education students in the experimental group when compared with students in the comparison group. Moreover, teacher participation in the STEM program had a statistically significant impact on students' variability in posttest scores. These interim student performance data support the implementation of rigorous differentiated science curriculum focused on improving science concept, content knowledge, and process skills.     

Students' Concepts‐ and Theorems‐in‐Action on a Novel Task About Similarity

Similarity is a fundamental concept in the middle grades. In this study, we applied Vergnaud's theory of conceptual fields to answer the following questions: What concepts‐in‐action and theorems‐in‐action about similarity surfaced when students worked in a novel task that required them to enlarge a puzzle piece? How did students use geometric and multiplicative reasoning at the same time in order to construct similar figures? We found that students used concepts of scaling and proportional reasoning, as well as the concept of circle and theorems about similar triangles, in their work on the problem. Students relied not only on visual perception, but also on numeric reasoning. Moreover, students' use of multiplicative and proportional concepts supported their geometric constructions. Knowledge of the concepts and ideas that students have available when working on a task about similarity can inform instruction by helping to ground formal introduction of new concepts in students' informal prior experiences and knowledge.     

Three women's understandings of algebra in a precalculus course integrated with the graphing calculator

This study investigated the role of function in a precalculus classroom which incorporated the graphing calculator in the instructional process. Perspectives were taken from students, teachers, and textbooks. Emphasis was placed on choice of functional symbol system when thinking and problem solving, connections across symbol systems, the role of the instructor and the textbook in learning, affective components, and the effect of the graphing calculator.The study starts with a defination of the concept of structure as it relates to function. The account of a semester-long qualitative study on students' concept images of function and its role in problem solving follows. It was found that the students involved in the study entered the graph-intensive course with predominantly symbolic notions of algebra, in part due to prior instruction. The students also possessed highly procedural views of algebraic content. These preconceptions and expectations resulted in the students' inability to effectively coordinate graphic and symbolic notions of algebra, both in procedural and conceptual realms. Implications and curricular suggestions are provided.     

The Tabular Mode: Not Just Another Way to Represent a Function

Understanding mathematical functions as systematic processes involving the covariation of related variables is foundational in learning mathematics. In this article, findings are reported from two investigations examining students' thinking processes with functions. The first study focused on seven middle school students' explorations with a dynamic physical model. Students were videotaped during the 20‐ to 45‐minute sessions occurring two or three times per week over a period of 2 months, and students' written work was collected. The second investigation included 19 preservice elementary and middle school teachers enrolled in a course focusing on a combination of mathematical content and pedagogy. Participants' written problem‐solving work and reflective writing were collected, and participants were individually interviewed in 50‐minute videotaped sessions. Results from both investigations indicated that students often relied on a table, or some variation of a table, as a cognitive link advancing the development of their reasoning about underlying function relationships.     

Examining the Conceptual Organization of Students in an Integrated Algebra and Physical Science Class

The cross-disciplinary context of density and slope was used to compare the conceptual organization of students in an integrated algebra and physical science class (SAM9) with that of students in a discipline-specific physical science class (PSO). Analyses of students' concept maps indicated that the SAM9 students used a greater number of procedural linkages to connect mathematics and science concepts on the SAM9 students' maps than did the PSO students. The maps produced by SAM9 students also tended to show a more compartmentalized approach to thinking about the content of the two disciplines, a finding contrary to the researcher's original assertion. Traditional teaching territories and conceptual complexity were examined as possible explanations for the discrepancy between the predicted and actual outcomes.     

Teachers' ways of listening and responding to students' emerging mathematical models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students' thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed.     

The interactive constitution of mathematical meaning in one second grade classroom: an illustrative example

The manner in which a horizontal addition and subtraction number sentence activity was constituted in one second grade classroom is analyzed for the purpose of discussing and illustrating how mathematical meaning is interactively constituted in the classroom. In particular, the teacher's emphasis on different solutions contributed to students' development of increasingly sophisticated concepts of ten. In turn, students' solutions contributed to the teacher's development of an increasingly sophisticated understanding of the children's mathematical activity and their concepts of ten.     

The Winds Are Variable: Student Intuitions About Variation

This study uses the context of the weather to explore the development of students' intuitive ideas of variation from pre‐Grade 1 to Grade 9. Three aspects of understanding these intuitions associated with variation are explored in individual videotaped interviews with 73 students: explanations, suggestions of data, and graphing. The development of these three aspects across grades is explored, as well as the associations among them. Fifty‐eight of the students also answered a general question on the definitions of “variation” and “variable,” and these responses are discussed and compared with responses to the weather task. The interview protocol may prove useful for teachers, particularly with younger children, to appreciate students' developing understanding of variation and provide starting points for classroom work of a more specific nature, either with respect to weather or other contextual topics.     

Students' Understanding of the Particulate Nature of Matter

The particulate nature of matter is identified in science education standards as one of the fundamental concepts that students should understand at the middle school level. However, science education research in indicates that secondary school students have difficulties understanding the structure of matter. The purpose of the study is to describe how engaging in an extended project‐based unit developed urban middle school students' understanding of the particulate nature of matter. Multiple sources of data were collected, including pre‐ and posttests, interviews, students' drawings, and video recordings of classroom activities. One teacher and her five classes were chosen for an indepth study. Analyses of data show that after experiencing a series of learning activities the majority of students acquired substantial content knowledge. Additionally, the finding indicates that students' understanding of the particulate nature of matter improved over time and that they retained and even reinforced their understanding after applying the concept. Discussions of the design features of curriculum and the teacher's use of multiple representations might provide insights into the effectiveness of learning activities in the unit.     

Assessing Students' Mathematical Communication

Assessment of students' mathematical communication through the use of open-ended tasks and scoring procedures is addressed, as is the use of open-ended tasks to assess students' mathematical communication by providing students opportunities to display their mathematical thinking and reasoning. Also, two scoring procedures (quantitative holistic scoring procedure and qualitative analytic scoring procedure) are described for examining students' communication skills.     

The Contribution of Conceptual Change Texts Accompanied by Concept Mapping to Students' Understanding of the Human Circulatory System

The present study was conducted to investigate the contribution of conceptual change texts accompanied by concept mapping instruction to 10^th— grade students' understanding of the human circulatory system. To determine misconceptions concerning the human circulatory system, 10 eleventh-grade students were interviewed. In the light of the findings obtained from student interviews and related literature, the Human Circulatory System Concepts Test was developed. The data were obtained from 26 students in the experimental group taught with the conceptual change texts accompanied by concept mapping, and 23 students in the control group taught with the traditional instruction. Besides treatment, previous learning in biology and science process skills were other independent variables involved in this study. Multiple Regression Correlation analysis revealed that science process skill, the treatment, and previous learning in biology each made a statistically significant contribution to the variation in students' understanding of the human circulatory system. It was found that the conceptual change texts accompanied by concept mapping instruction produced a positive effect on students' understanding of concepts. The mean scores of experimental and control groups showed that students in the experimental group performed better with respect to the human circulatory system. Item analysis was carried out to determine and compare the proportion of correct responses and misconceptions of students in both groups. The average percent of correct responses of the experimental group was 59.8%, and that of the control group was 51.6% after treatment.