Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics

The interpretive cross-case study focused on the examination of connections between teacher and student topic-specific knowledge of lower secondary mathematics. Two teachers were selected for the study using non-probability purposive sampling technique. Teachers completed the Teacher Content Knowledge Survey before teaching a topic on the division of fractions. The survey consisted of multiple-choice items measuring teachers’ knowledge of facts and procedures, knowledge of concepts and connections, and knowledge of models and generalizations. Teachers were also interviewed on the topic of fraction division using questions addressing their content and pedagogical content knowledge. After teaching the topic on the division of fractions, two groups of 6th-grade students of the participating teachers were tested using similar items measuring students’ topic-specific knowledge at the level of procedures, concepts, and generalizations. The cross-case examination using meaning coding and linguistic analysis revealed topic-specific connections between teacher and student knowledge of fraction division. Results of the study suggest that students’ knowledge could be associated with the teacher knowledge in the context of topic-specific teaching and learning of mathematics at the lower secondary school.     

Developing teacher content understanding by integrating pH and logarithms concepts

Logarithms are notorious for being a difficult concept to understand and teach. Research suggests that learners can be supported in understanding logarithms by making connections between mathematics and science concepts such as pH. This study investigated how an integrated chemistry and mathematics lesson impacted 29 teachers’ understanding of the logarithmic relationship and pH. Pre- and post-test data indicated 23 teachers improved their understanding of logarithms and 28 improved their understanding of pH, suggesting that teacher educators in both science and mathematics context can use this approach to foster better understanding with their teachers and ultimately school students. Our analysis also identified professional development components and teacher characteristics associated with gains in understanding of pH and logarithms, which mathematics and science teacher educators can use to strategically adapt and implement the lesson within other teacher education settings.     

The use of interactive visualizations to foster the understanding of concepts of calculus: design principles and empirical results

Calculus and functional thinking are closely related. Functional thinking includes thinking in variations and functional dependencies with a strong emphasis on the aspect of change. Calculus is a climax within school mathematics and the education to functional thinking can be seen as propaedeutics to it. Many authors describe that functions at school are mainly treated in a static way, by regarding them as pointwise relations. This often leads to the underrepresentation of the aspect of change at school. Moreover, calculus at school is mainly procedure-oriented and structural understanding is lacking. In this work, two specially designed interactive activities for the teaching and learning of concepts of calculus based on dynamic geometry software are presented. They accentuate the aspect of change and the object aspect of functions using a double stage visualization. Moreover, the activities allow the discovery and exploration of some concepts of calculus in a qualitative-structural way without knowing or using curve-sketching routines. The activities were used in a qualitative study with 10th grade students of age 15–16 in secondary schools in Berlin, Germany. Some pairs of students were videotaped while working with the activities. After transcribing, the interactions of the students were interpreted and analyzed focusing to the use of the computer. The results show how the students mobilize their knowledge about functions working on the given tasks, and using the activities to formulate important concepts of calculus in a qualitative way. Also, some important epistemological obstacles can be detected.     

Expanding context and domain: A cross-curricular activity in mathematics and physics

This article is based on my 15 years of experience as a teacher of mathematics and physics in the Danish Gymnasium (high school), and it gives an example of an interdisciplinary course between mathematics and physics. The course is centered around the concept of exponential functions. The starting point is that concepts are rooted in practice and gain their meaning through application, and the concept of a function is regarded as a tool for modelling real-world situations. It is the intention to teach a course that emphasizes factors that promote transfer of the concept and use of the various representations of the concept, to make it more practical and meaningful for the students. It is concluded that a coordinated cross-curricular activity between mathematics and physics, by offering a great variety of domain relations and context settings, has a great potential for creating a learning environment where the students, through applicational and modelling activities, are engaged actively in constructing and using knowledge.     

Promoting students’ graphical understanding of the calculus

Our purpose in this paper is to report on an observational study to show how students think about the links between the graph of a derived function and the original function from which it was formed. The participants were asked to perform the following task: they were presented with four graphs that represented derived functions and from these graphs they were asked to construct the original functions from which they were formed. The students then had to walk these graphs as if they were displacement-time graphs. Their discussions were recorded on audio tape and their walks were captured using data logging equipment and these were analysed together with their pencil and paper notes. From these three sources of data, we were able to construct a picture of the students’ graphical understanding of connections in calculus. The results confirm that at the start of the activity the students demonstrate an algebraic symbolic view of calculus and find it difficult to make connections between the graphs of a derived function and the function itself. By being able to ‘walk’ an associated displacement time graph, we propose that the students are extending their understanding of calculus concepts from symbolic representation to a graphical representation and to what we term a ‘physical feel’.     

How Students Use Physics to Reason About Calculus Tasks

The present research study investigates how undergraduate students in an integrated calculus and physics class use physics to help them solve calculus problems. Using Zandieh's (2000) framework for analyzing student understanding of derivative as a starting point, this study adds detail to her “paradigmatic physical” context and begins to address the need for a theoretical basis for investigating learning and teaching in integrated mathematics and science classrooms. A case study design was used to investigate the different ways students use physics ideas as they worked through calculus tasks. Data were gathered through four individual interviews with each of 8 ICP students, classroom participant‐observation, and triangulation of the data through student homework and exams. The main result of this study is the Physics Use Classification Scheme, a tool consisting of four categories used to characterize students' uses of physics on tasks involving average rate of change, derivative, and integral concepts. Two of the categories from the Physics Use Classification Scheme are elucidated with contrasting student cases in this paper.     

Teaching and Coherent Science: An Investigation of Teachers’ Beliefs About and Practice of Teaching Science Coherently

This article is about an investigation of six middle school science teachers’ beliefs and instructional practice about the coherence of the science they teach as articulated by National Science Education Standards ( NRC, 1996 ). Many well intentioned reform efforts focus on improving content knowledge of teachers, but many classroom teachers regularly miss opportunities to provide conceptual connections within the science ideas building the sense of coherence in science. This investigation involved a quasi experimental study to examine the efficacy of a method for collecting data about middle school science teachers’ thinking about science and to determine if they teach science coherently. The teachers were surveyed, interviewed, provided concept maps about their thinking of the science they taught, and observed to investigate whether their practice reflects their beliefs. An examination of the teachers’ beliefs, stated and unstated curriculum, the connections among topics and the nature of science revealed that one, the observation tool may have merit for identifying the content and connections among science topics, and two, that teachers ‘stated beliefs consistent with the National Science Education Standards’ vision for coherent science, did not match their demonstrated practice. The content taught could be characterized in three ways; coherent content and few connections, coherent content and connections, and not coherent content. This indicates for this group of middle school science teachers that knowing how they think about science and how those beliefs are reflected in their teaching is complex. This study can inform teacher education and professional development efforts about the need to move beyond just content enhancement to examine prior beliefs about the connections of concepts within science.     

Graphical Representations of Speed: Obstacles Preservice K‐8 Teachers Experience

This study examines the difficulties college students experience when creating and interpreting graphs in which speed is one of the variables. Nineteen students, all preservice elementary or middle school teachers, completed an upper‐level course exploring algebraic concepts. Although all of these preservice teachers had previously completed several mathematics courses, including calculus, they demonstrated widespread misconceptions about the variable speed. This study identifies four cognitive obstacles held by the students, provides excerpts of their graphical constructions and verbal interpretations, and discusses potential causes for the confusion. In particular, misconceptions arose when students interpreted the behavior and nature of speed within a graphical context, as well as in situations where they were required to construct a graph involving speed as a variable. The study concludes by offering implications for the teaching and learning of speed and its interpretation within a graphical setting.     

Understanding the integral: Students’ symbolic forms

Researchers are currently investigating how calculus students understand the basic concepts of first-year calculus, including the integral. However, much is still unknown regarding the cognitive resources (i.e., stable cognitive units that can be accessed by an individual) that students hold and draw on when thinking about the integral. This paper presents cognitive resources of the integral that a sample of experienced calculus students drew on while working on pure mathematics and applied physics problems. This research provides evidence that students hold a variety of productive cognitive resources that can be employed in problem solving, though some of the resources prove more productive than others, depending on the context. In particular, conceptualizations of the integral as an addition over many pieces seem especially useful in multivariate and physics contexts.     

Zimbabwean in-service mathematics teachers’ understanding of matrix operations

In Zimbabwe, school pupils study matrix operations, a topic that is usually covered as part of linear algebra courses taken by most mathematics undergraduate students at university. In this study we focused on Zimbabwean teachers who were studying the topic at university while also teaching the topic to their high school pupils. The purpose of the study was to explore the mental conceptions of matrix operations concepts of a sample of 116 in-service mathematics teachers. The Action Process Object Schema (APOS) theoretical framework describes the development in understanding of mathematics concepts through the hierarchical growth of mental constructions called action, process, object and schema. The results showed that many of the participants had interiorized actions on matrix operations of addition, scalar multiplication and matrix multiplication into processes. However, more than 50% of the participants struggled with scalar multiplication of a row matrix by a column matrix. In terms of notational errors, some participants could not distinguish between brackets that denote a matrix and that of a determinant, while some used the equal sign as an operator symbol and not as one denoting equivalence between two objects. It is recommended that future in-service teacher programs should try to create more structured opportunities to allow participants to engage more deeply with these concepts.     

Students’ and Teachers’ Conceptual Metaphors for Mathematical Problem Solving

Metaphors are regularly used by mathematics teachers to relate difficult or complex concepts in classrooms. A complex topic of concern in mathematics education, and most STEM‐based education classes, is problem solving. This study identified how students and teachers contextualize mathematical problem solving through their choice of metaphors. Twenty‐two high‐school student and six teacher interviews demonstrated a rich foundation for these shared experiences by identifying the conceptual metaphors. This mixed‐methods approach qualitatively identified conceptual metaphors via interpretive phenomenology and then quantitatively analyzed the frequency and popularity of the metaphors to explore whether a coherent metaphorical system exists with teachers and students. This study identified the existence of a set of metaphors that describe how multiple classrooms of geometry students and teachers make sense of mathematical problem solving. Moreover, this study determined that the most popular metaphors for problem solving were shared by both students and teachers. The existence of a coherent set of metaphors for problem solving creates a discursive space for teachers to converse with students about problem solving concretely. Moreover, the methodology provides a means to address other complex concepts in STEM education fields that revolve around experiential understanding.     

Complex calculators in the classroom: theoretical and practical reflections on teaching pre-calculus

University and older school students following scientific courses now use complex calculators with graphical, numerical and symbolic capabilities. In this context, the design of lessons for 11th grade pre-calculus students was a stimulating challenge.In the design of lessons, emphasising the role of mediation of calculators and the development of schemes of use in an 'instrumental genesis' was productive. Techniques, often discarded in teaching with technology, were viewed as a means to connect task to theories. Teaching techniques of use of a complex calculator in relation with 'traditional' techniques was considered to help students to develop instrumental and paper/pencil schemes, rich in mathematical meanings and to give sense to symbolic calculations as well as graphical and numerical approaches.The paper looks at tasks and techniques to help students to develop an appropriate instrumental genesis for algebra and functions, and to prepare for calculus. It then focuses on the potential of the calculator for connecting enactive representations and theoretical calculus. Finally, it looks at strategies to help students to experiment with symbolic concepts in calculus.This revised version was published online in September 2005 with corrections to the Cover Date.     

Implications of Informal Education Experiences for Mathematics Teachers' Ability to Make Connections Beyond Formal Classroom

The Common Core Standard for Mathematical Practice 4: Model with Mathematics specifies that mathematically proficient students are able to make connections between school mathematics and its applications to solving real‐world problems. Hence, mathematics teachers are expected to incorporate connections between mathematical concepts they teach and their applications to solving problems arising in real‐world situation. Clearly, it is assumed that the teachers themselves are able to make such connections. On the other hand, research shows that mathematics teachers find it difficult to make those connections. In this paper, we present the results of the study that investigated the ways in which exploring mathematics in informal sites, and in particular science museum, assist teachers with making connections between school mathematics and its applications in real world.     

Biomedical Engineering and Cognitive Science as the Basis for Secondary Science Curriculum Development: A Three Year Study

This study reports on a multiyear effort to create and evaluate cognitive‐based curricular materials for secondary school science classrooms. A team of secondary teachers, educational researchers, and academic biomedical engineers developed a series of curriculum units that are based in biomedical engineering for secondary level students in physics and advanced biology classes. These units made use of an instructional design based upon recent cognitive science research called the Legacy Cycle. Over a 3‐year period, comparison of student knowledge on written questions related to central concepts in physics and/or biology generally favored students who had worked with the experimental materials over students in control classrooms. In addition, experimental students were better able to solve applications type problems, as well as unit‐specific near transfer problems.     

Algorithmic contexts and learning potentiality: a case study of students’ understanding of calculus

The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.     

On the use of history of mathematics: an introduction to Galileo's study of free fall motion

In this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year university students through Galileo's experiments designed to investigate the motion of falling bodies, and his geometrical explanation of his results, via simple dynamic geometric applets designed with GeoGebra. Our goal was to enhance the students’ development of mathematical thinking. Through a scholarship of teaching and learning study design, we captured data from students before, during and after the activity. Findings suggest that the historical development presented to the students helped to show the growth and evolution of the ideas and made visible authentic ways of thinking mathematically. Importantly, the activity prompted students to question and rethink what they knew about speed and acceleration, and also to appreciate the novel concepts of instantaneous speed and acceleration at which Galileo arrived.     

Kinks in the STEM Pipeline: Tracking STEM Graduation Rates Using Science and Mathematics Performance

In an effort to maintain the global competitiveness of the United States, ensuring a strong Science, Technology, Engineering and Mathematics (STEM) workforce is essential. The purpose of this study was to identify high school courses that serve as predictors of success in college level gatekeeper courses, which in turn led to the successful completion of STEM degrees. Using a purposive sample of 893 students who had declared a STEM major between the fall of 2006 and the spring of 2008, data were collected on students' high school grades, college grades, national test scores, grade point average, gender, and ethnicity. Using analysis of variance, correlations, multiple discriminant function analysis, and multiple regression models we found that high school calculus, physics, and chemistry (respectively) were predictors of success in STEM gatekeeper college courses. Then using those courses, we constructed a predictive model of STEM degree completion. The implications of this study highlight and reinforce the importance of providing rigorous mathematics and science courses at the high school level, as well as provide some evidence of a potential mediated model of the relationship between high school performance, college performance, and graduating with a STEM degree.     

Research on calculus: what do we know and where do we need to go?

In this introductory paper we take partial stock of the current state of field on calculus research, exemplifying both the promise of research advances as well as the limitations. We identify four trends in the calculus research literature, starting with identifying misconceptions to investigations of the processes by which students learn particular concepts, evolving into classroom studies, and, more recently research on teacher knowledge, beliefs, and practices. These trends are related to a model for the cycle of research and development aimed at improving learning and teaching. We then make use of these four trends and the model for the cycle of research and development to highlight the contributions of the papers in this issue. We conclude with some reflections on the gaps in literature and what new areas of calculus research are needed.     

Cause–effect analysis: improvement of a first year engineering students’ calculus teaching model

This study focuses on the mathematics department at a South African university and in particular on teaching of calculus to first year engineering students. The paper reports on a cause–effect analysis, often used for business improvement. The cause–effect analysis indicates that there are many factors that impact on secondary school teaching of mathematics, factors that the tertiary sector has no control over. The analysis also indicates the undesirable issues that are at the root of impeding success in the calculus module. Most important is that students are not encouraged to become independent thinkers from an early age. This triggers problems in follow-up courses where students are expected to have learned to deal with the work load and understanding of certain concepts. A new model was designed to lessen the impact of these undesirable issues.