Student Strategies Suggesting Emergence of Mental Structures Supporting Logical and Abstract Thinking: Multiplicative Reasoning

For many students, developing mathematical reasoning can prove to be challenging. Such difficulty may be explained by a deficit in the core understanding of many arithmetical concepts taught in early school years. Multiplicative reasoning is one such concept that produces an essential foundation upon which higher‐level mathematical thinking skills are built. The purpose of this study is to recognize indicators of multiplicative reasoning among fourth‐grade students. Through cross‐case analysis, the researcher used a test instrument to observe patterns of multiplicative reasoning at varying levels in a sample of 14 math students from a low socioeconomic school. Results indicate that the participants fell into three categories: premultiplicative, emergent, and multiplier. Consequently, 12 new sublevels were developed that further describe the multiplicative thinking of these fourth graders within the categories mentioned. Rather than being provided the standard mathematical algorithms, students should be encouraged to personally develop their own unique explanations, formulas, and understanding of general number system mechanics. When instructors are aware of their students' distinctive methods of determining multiplicative reasoning strategies and multiplying schemes, they are more apt to provide the most appropriate learning environment for their students.     

Whodunit? Exploring Proportional Reasoning Through the Footprint Problem

This paper describes a proportional reasoning problem set within a real‐life context and a complete analysis of one small group discussion of this problem over the course of a 90‐minute block. The seventh‐grade students' discourse is described to provide insights into typical mathematical interpretations of this problem, as well as some generalizations for other problems of this type. The interpretations provided reveal the gradual development of proportional reasoning in a local context from additive to multiplicative understandings.     

Promoting Percent as a Proportion in Eighth‐Grade Mathematics

The literature provides many and varied suggestions for promoting conceptual understanding of percent and performing percent calculations. The diversity of ideas provides a wide selection but offers little clarity on the true nature of percent. From the premise that percent is fundamentally a proportion, this study incorporated a proportional approach for percent problem solving within an instructional program on percent. Classroom research with eighth‐grade students indicated that the method was readily adopted by students and helped them experience success in percent problem solving, with percent problem solving proficiency maintained over a delayed period. It is hypothesized that the method has the potential to promote students' conceptual knowledge of percent as a proportion and the multiplicative structure of percent, as well as to build proportional knowledge.     

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.     

Geometry‐Related Children's Literature Improves the Geometry Achievement and Attitudes of Second‐Grade Students

Use of mathematics‐related literature can engage students' interest and increase their understanding of mathematical concepts. A quasi‐experimental study of two second‐grade classrooms assessed whether daily inclusion of geometry‐related literature in the classroom improved attitudes toward geometry and achievement in geometry. Consistent with the hypothesis, only the students in the classroom with a strong emphasis on geometry‐related children's literature showed a significant improvement in their attitudes about geometry over time. While both classes improved their geometry performance over the 4 weeks of the study, the class with a strong emphasis on geometry‐related literature improved significantly more (51.2%) than the control class (33.47%). Children's literature can provide a useful and interesting context in which students can develop their understanding of geometry.     

Metaphor Clusters: Characterizing Instructor Metaphorical Reasoning on Limit Concepts in College Calculus

Novice students have difficulty with the topic of limits in calculus. We believe this is in part because of the multiple perspectives and shifting metaphors available to solve items correctly. We investigated college calculus instructors' personal concepts of limits. Based upon previous research investigating introductory calculus student metaphorical reasoning, we examined 11 college instructors' metaphorical reasoning on limit concepts. This paper focused on previous research of metaphor clusters observed among students to answer the following: (a) Do college instructors use metaphorical reasoning to conceptualize the meaning of a limit? (b) Can we characterize instructor metaphorical reasoning similar to those observed among students? (c) Will an instructor's self‐identification of metaphor clusters be consistent with our metaphor coding? We found that college instructors' perspectives vary, either graphical or algebraic, in their explanations of limit items. All the instructors used metaphors, and instructor metaphorical reasoning aligned with student metaphor clusters. Instructors tended to change their metaphors with respect to the limit item. Instructors were not aware of their use of metaphors, nor were they aware of their inconsistency in their choice of metaphor. We believe that instructor awareness of their own distinct perspectives and metaphors would assist students' understanding of limit concepts.     

Interpreting Middle School Students' Proportional Reasoning Strategies: Observations From Preservice Teachers

This paper reports the findings of an investigation of 11 preservice secondary school teachers' interpretations of the development of proportional reasoning strategies used by middle school students. The preservice teachers examined samples of solution strategies generated by middle school students in proportional reasoning situations and prepared written responses of their views concerning the developmental levels indicated in the students' work. Each preservice teacher also participated in an hour‐long interview, in which the researchers asked for elaboration and clarification of the written responses and, in some cases, challenged the preservice teachers to consider alternative interpretations for the middle school students' work. The interviews were audiotaped for later analysis by the investigators, and key aspects of both the written and audiotaped responses were entered into a spreadsheet and later tabulated into categories indicating trends in the preservice teachers' interpretations. Some implications for the preparation of preservice middle school science and mathematics teachers are included.     

Gender and Racial Differences: Development of Sixth Grade Students' Geometric Spatial Visualization within an Earth/Space Unit

This study investigated sixth‐grade middle‐level students' geometric spatial development by gender and race within and between control and experimental groups at two middle schools as they participated in an Earth/Space unit. The control group utilized a regular Earth/Space curriculum and the experimental group used a National Aeronautics and Space Administration‐based curriculum. The quantitative data sources included the Lunar Phases Concept Inventory, Geometric Spatial Assessment, and the Purdue Spatial Visualization–Rotation Test. The results indicated the experimental males and females, and the students of color and white students in the experimental group showed significant gains in their understanding of geometric spatial visualization from pre‐ to post‐implementation. However, for the control group, the significant gains were limited to the males and the white students. The findings reveal that support is needed for males, females, and all racial groups to have the opportunity to develop their spatial reasoning, which in turn, increases students' scientific understanding.     

Training Master's‐Level Graduate Students to Use Inquiry Instruction to Teach Middle‐Level and High‐School Science Concepts

Through the GK‐12 program of the National Science Foundation, graduate student fellows in a coastal marine and wetland studies program were trained to present targeted science concepts to middle‐ and high‐school classes through their own research‐based lessons. Initially, they were taught to follow the 5‐E learning cycle in lesson plan development, but a streamlined approach targeting the three attributes of science concepts—macroscopic, model, and symbolic—was found to be a better approach, while still incorporating key facets of the 5‐E model. Evaluation of the level of inquiry in the classrooms was determined using an inquiry scale from 0 to 4, differentiated by the relative number of actions that are student‐centered. The graduate fellows consistently delivered lessons at the targeted levels 2 or 3, guided inquiry. In order to assess student learning, the GK‐12 fellows were trained to develop single‐item pre‐ and post‐assessments designed to probe middle‐level and high‐school students' understanding of the macroscopic, model, and symbolic attributes of targeted science concepts. For the lessons based on the research of the fellows, about 80% of the students showed statistically and practically significant learning gains. The GK‐12 fellows positively impact the classroom and are effective science ambassadors.     

Using Student Reasoning to Inform the Development of Conceptual Learning Goals: The Case of Quadratic Functions

Despite the proliferation of mathematics standards internationally and despite general agreement on the importance of teaching for conceptual understanding, conceptual learning goals for many K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five conceptual learning goals for a complex mathematical domain, generated via a method of systematic empirical analysis of students' reasoning. Specifically, we compared the reasoning of pairs of students who performed differentially on the same task and inferred the pivotal intermediate conceptions that afforded one student deeper engagement with the task than another student. In turn, each pivotal intermediate conception framed an associated conceptual learning goal. While the empirical analysis of student reasoning is typically used to understand how students learn, we argue that such analysis should also play an important role in determining what concepts students should learn.     

Conceptual learning in a principled design problem solving environment

To what extent can instructional design be based on principles for instilling a culture of problem solving and conceptual learning? This is the main focus of the study described in this paper, in which third grade students participated in a one-year course designed to foster problem solving and mathematical reasoning. The design relied on five principles: (a) encouragement to produce multiple solutions; (b) creating collaborative situations; (c) socio-cognitive conflicts; (d) providing tools for checking hypotheses; and (e) inviting students to reflect on solutions. We describe how a problem solving task designed according to the above principles, promoted students' understanding of the area concept. We show that the design afforded the surfacing of multiple solutions and justifications in various modalities (including gestures) and initiated peer argumentation, leading to deep learning of the area concept.     

Applying Mathematical Concepts with Hands‐On,Food‐Based Science Curriculum

This article addresses the current state of the mathematics education system in the United States and provides a possible solution to the contributing issues. As a result of lower performance in primary mathematics, American students are not acquiring the necessary quantitative literacy skills to become successful adults. This study analyzed the impact of the Food, Math, and Science Teaching Enhancement Resource (FoodMASTER) Intermediate curriculum on fourth‐grade students' mathematics knowledge. The curriculum is a part of the FoodMASTER Initiative, which is a compilation of programs utilizing food, a familiar and necessary part of everyday life, as a tool to teach mathematics and science. Students exposed to the curriculum completed a 20‐item researcher‐developed mathematics knowledge exam (intervention n = 288; control n = 194). Overall, the results showed a significant increase in mathematics knowledge from pretest to posttest. These findings suggest that the food‐based science activities provided the students with the context in which to apply mathematical concepts to an everyday experience. Therefore, the FoodMASTER approach was successful at improving students' mathematics knowledge while building a foundation for becoming quantitatively literate adults.     

The fractional knowledge and algebraic reasoning of students with the first multiplicative concept

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of three different multiplicative concepts participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed.     

The Effect of Calculator‐Based Ranger Activities on Students' Graphing Ability

Middle school students can learn to communicate with graphs in the context of appropriate Calculator‐Based Ranger (CBR) activities. Three issues about CBR activities on graphing abilities were addressed in this study: (a) the effect of CBR activities on graphing abilities; (b) the extent to which prior knowledge about graphing skills affects graphing ability; (c) the influence of instructional styles on students' graphing abilities. Following the use of CBR activities, students' graphing abilities were significantly more developed in three components _ interpreting, modeling, and transforming. Prior knowledge of graphing skills on the Cartesian coordinate plane had little effect on students' understanding of graphs. Significant differences, however, were found in students' achievement, depending on instructional styles related to differentiation, which is closely connected to transforming distance‐time graphs to velocity‐time graphs. The result of this study indicates that the CBR activities are pedagogically promising for enhancing graphing ability of physical phenomena.     

Understanding the concepts of proportion and ratio among grade nine students in Malaysia

The purpose of the study was to investigate the concepts of ratio and proportion constructed by grade nine students by investigating grade nine students proportional reasoning schemes and procedures on three types of tasks: missing value, numerical comparison and qualitative reasoning. Comparisons among the different categories were made and the strategies used in solving these problems were identified. The relationship between student grades on a national examination and their knowledge of proportional reasoning was determined. The results of the quantitative analysis indicated that students performed generally well on the missing value tasks but their scores on the numerical comparison and qualitative tasks were much lower. The results indicate that only a small percentage of students who did well on the national exams were able to solve complex proportional problems and the grades obtained were not indicative of their knowledge of ratio and proportion. The difficulty experienced by the ninth graders indicated that students frequently used additive reasoning, that is a comparison of two numbers by subtraction rather than division. It appears that students cannot begin to understand the functional and scalar relationship inherent in a proportion until they first develop multiplicative reasoning.     

The Effects of Non‐CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta‐Analysis

Forty‐two studies comparing students with access to graphing calculators during instruction to students who did not have access to graphing calculators during instruction are the subject of this meta‐analysis. The results on the achievement and attitude levels of students are presented. The studies evaluated cover middle and high school mathematics courses, as well as college courses through first semester calculus. When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.     

Re‐creating a Recipe for Science Instructional Programs: Adding Learning Progressions,Scaffolding, and a Dash of Reading Variety

The purpose of this article is to focus on the development and refinement of a science instructional design program arguing for the feasibility and usability of integrated reading and science instruction as implemented in third‐ and fourth‐grade science classrooms to meet the learning needs of diverse learners. These instructional components are easily inserted into existing programs that build students' science background knowledge and abilities to apply learning through scaffolded activities focused on (1) providing structured opportunities for students to engage in hands‐on activities; (2) increasing vocabulary knowledge and understanding of concept‐laden terms, and (3) reading paired narrative and informational science texts. Extensive research shows that as students transition from third to fourth grade and beyond, they are often challenged in science by new vocabulary coupled with new concepts. Active ingredients of our reconceptualized science instructional design program are narrative informational texts, hands‐on science activities, and science textbook(s).     

Development and Field Test of the Modified Draw‐a‐Scientist Test and the Draw‐a‐Scientist Rubric

Even long before children are able to verbalize which careers may be interesting to them, they collect and store ideas about scientists. For these reasons, asking children to draw a scientist has become an accepted method to provide a glimpse into how children represent and identify with those in the science fields. Years later, these representations may translate into students' career choices. Since 1995, children's illustrations of scientists have been assessed by the Draw‐a‐Scientist Checklist (DAST‐C). The checklist was created from the common aspects or features found in illustrations from previous studies and were based initially on the scientists, broken down into “stereotypical” and “alternative” images shown in the drawings. The purpose of this article is to describe the development, field test, and reliability of the modified DAST Test and the DAST Rubric designed as an improvement of the DAST‐C to provide a more appropriate method of assessing students' drawings of scientists. The combination of the modified DAST and the DAST Rubric brings more refinement as it enables clarities to emerge and subsequently increased detail to what one could ascertain from students about their mental images of scientists.     

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.     

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.