Exploring Students' Conceptual Understanding of the Averaging Algorithm

Conceptual understanding of arithmetic average includes both an understanding of the computational algorithm and the statistical aspects of the concept. This study focused on the examination of 250 sixth-grade students' understanding of the arithmetic average by assessing their understanding of the computational algorithm. The results of the study showed that the majority of the students knew the “add-them-all-up-and-divide” averaging algorithm, but only about half of the students were able to correctly apply the algorithm to solve a contextualized average problem. Students were able to use various solution strategies and representations to solve the average problem. Those who used algebraic and arithmetic representations were better problem solvers than those who used pictorial and verbal representations. This study not only suggests that the average concept is more complex than the simplicity suggested by the computational algorithm, but also indicates the need for teaching the concept of average, both as a statistical idea for describing and making sense of data sets and as a computational algorithm for solving problems.     

Intended Treatments of Arithmetic Average in U.S. and Asian School Mathematics Textbooks

This study examined how selected U.S. and Asian mathematics curricula are designed to facilitate students' understanding of the arithmetic average. There is a consistency regarding the learning goals among these curriculum series, but the focuses are different between the Asian series and the U.S. reform series. The Asian series and the U.S. commercial series focus the arithmetic average more on conceptual and procedural understanding of the concept as a computational algorithm than on understanding the concept as a representative of a data set; however, the two U.S. reform series focus the concept more on the latter. Because of the different focuses, the Asian and the U.S. curriculum series treat the concept differently. In the Asian series, the concept is first introduced in the context of “equal‐sharing” or “per‐unit‐quantity,” and the averaging formula is formally introduced at a very early stage. In the U.S. reform series, the concept is discussed as a measure of central tendency, and after students have some intuitive ideas of the statistical aspect of the concept, the averaging algorithm is briefly introduced.     

Using math magic to reinforce algebraic concepts: an exploratory study

An exploratory study was conducted to investigate the use of magic activities in a math course for prospective middle-school math teachers. This research report focuses on a lesson using two versions of math magic: (1) the 5-4-3-2-1-½ Magic involves having students choose a secret number and apply six arithmetic operations in sequence to arrive at a resultant number, and the teacher-magician can spontaneously reveal a student’s secret number from the resultant number; and (2) the Everyone-Got-9 Magic also involves choosing a secret number and applying arithmetic operations in sequence, but everyone will end up with the same resultant number of 9. These magic activities were implemented to reinforce students’ understanding of foundational algebra concepts like variables, expressions, and inverse functions. Analysis of students’ written responses revealed that (1) all students who figured out the trick in the first magic activity did not used algebra, (2) most students could apply what they learned in one trick to a similar trick but not to a different trick, and (3) many students were weak in symbolic representations and manipulations. Responses from a survey and a focus group indicate that students found the magic activities to be fun and intellectually engaging.     

An Atom is Known by the Company it Keeps: A Constructionist Learning Environment for Materials Science Using Agent-Based Modeling

This article reports on “MaterialSim”, an undergraduate-level computational materials science set of constructionist activities which we have developed and tested in classrooms. We investigate: (a) the cognition of students engaging in scientific inquiry through interacting with simulations; (b) the effects of students programming simulations as opposed to only interacting with ready-made simulations; (c) the characteristics, advantages, and trajectories of scientific content knowledge that is articulated in epistemic forms and representational infrastructures unique to computational materials science, and (d) the principles which govern the design of computational agent-based learning environments in general and for materials science in particular. Data sources for the evaluation of these studies include classroom observations, interviews with students, videotaped sessions of model-building, questionnaires, and analysis of artifacts. Results suggest that by becoming ‘model builders,’ students develop deeper understanding of core concepts in materials science, and learn how to better identify unifying principles and behaviors within the content matter.     

Thematization of derivative schema in university students: nuances in constructing relations between a function's successive derivatives

This study is part of a more extensive research project that addresses the understanding of the derivative concept in university students with prior instruction in differential calculus. In particular, we focus on the analysis of students’ responses to a sequence of tasks that require a high level of understanding of the concept, and complement this information with clinical interviews. APOS (Action-Process-Object-Schema) theory and the configuration of the derivative concept that is characterized by: mathematical elements, logical relations and the representation modes that students use to solve a task were used in the analysis of students’ responses. The results obtained suggest that thematizing the derivative schema is difficult to achieve. In addition, nuances were observed in responses given by those students who succeeded, indicating differences in the construction of relations between the successive derivatives of a function.     

Designing examples to create variation patterns in teaching

This paper uses a post-qualitative philosophical perspective to find new ways of understanding teaching and learning. The paper presents a series of examples that were used in a longitudinal study, with the aim of creating variation patterns that would make it possible for students to discern the use of the four basic arithmetic operations in different situations. The focus of this article is the potential of the examples to systematically create variation patterns that students need to perceive in order to make generalizations. The result demonstrates that well-thought-out examples help identify the correct arithmetic operation in different situations, and provide a basis from which students can discern the connection between text and the use of operation in mathematical example. The result also demonstrates that students develop rhizomatic thinking through the creation of new links between aspects of the object of learning, association and linking of different aspects to each other and the creation of a whole with unique and specific characteristics that cannot be explained by simply adding the characteristics of the individual parts.     

A modularized tablet-based approach to preparation for remedial mathematics

Basic arithmetic forms the foundation of the math courses that students will face in their undergraduate careers. It is therefore crucial that students have a solid understanding of these fundamental concepts. At an open-access university offering both two-year and four-year degrees, incoming freshmen who were identified as lacking in basic arithmetic skills were engaged in an experimental technology-enhanced workshop designed to provide them with a deeper understanding of arithmetic prior to their initial remedial coursework. Customized online content was created specifically for this experiment, and the first implementation (n = 27) yielded statistically significant improvement, not only from pre-test to post-test, but also in the subsequent remedial course. This paper also analyses the accuracy of students’ self-assessment from pre-test to post-test, as well as student attitudes about this experimental approach.     

Improving Algebra Preparation: Implications From Research on Student Misconceptions and Difficulties

Through historical and contemporary research, educators have identified widespread misconceptions and difficulties faced by students in learning algebra. Many of these universal issues stem from content addressed long before students take their first algebra course. Yet elementary and middle school teachers may not understand how the subtleties of the arithmetic content they teach can dramatically, and sometimes negatively, impact their students' ability to transition to algebra. The purpose of this article is to bring awareness of some common algebra misconceptions, and suggestions on how they can be averted, to those who are teaching students the early mathematical concepts they will build upon when learning formal algebra. Published literature discussing misconceptions will be presented for four prerequisite concepts, related to symbolic representation: bracket usage, equality, operational symbols, and letter usage. Each section will conclude with research‐based practical applications and suggestions for preventing such misconceptions. The literature discussed in this article makes a case for elementary and middle school teachers to have a deeper and more flexible understanding of the mathematics they teach, so they can recognize how the structure of algebra can and should be exposed while teaching arithmetic.     

On students’ conceptions of arithmetic average: the case of inference from a fixed total

There is more to understanding the concept of mean than simply knowing and applying the add-them-up and divide algorithm. In the following, we discuss a component of understanding the mean – inference from a fixed total – that has been largely ignored by researchers studying students understanding of mean. We add this component to the list of types of reasoning needed to understand mean and discuss student responses to tasks designed to elicit this component of reasoning. These responses reveal that inference from a fixed total reasoning is rare even in advance high school students.     

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).     

Introducing computational thinking through hands-on projects using R with applications to calculus,probability and data analysis

The goal of this paper is to promote computational thinking among mathematics, engineering, science and technology students, through hands-on computer experiments. These activities have the potential to empower students to learn, create and invent with technology, and they engage computational thinking through simulations, visualizations and data analysis. We present nine computer experiments and suggest a few more, with applications to calculus, probability and data analysis, which engage computational thinking through simulations, visualizations and data analysis. We are using the free (open-source) statistical programming language R. Our goal is to give a taste of what R offers rather than to present a comprehensive tutorial on the R language. In our experience, these kinds of interactive computer activities can be easily integrated into a smart classroom. Furthermore, these activities do tend to keep students motivated and actively engaged in the process of learning, problem solving and developing a better intuition for understanding complex mathematical concepts.     

A resource for introducing students to the integral concept

Many students do not have a deep understanding of the integral concept. This article defines what a deep understanding of the integral is in respect to integration involving one independent variable; briefly discusses factors which may inhibit such an understanding; and then describes the design of a mathematical resource for introducing students to the integral concept. The resource addresses a number of challenges when introducing the integral: (1) choosing an appropriate, intuitive context which gives meaning to the symbols in the integral expression; (2) aiding the transfer of the integral expression to different contexts via using the Riemann sum in an informal way so that students can see and interpret the rectangles which are inherent in this sum; and (3) the gradual formalizing of the Riemann sum and its linkage with the Fundamental Theorem of Calculus. The resource has been used over a number of years at this university amongst first-year undergraduate science and engineering students. Anecdotal evidence would suggest that the resource is beneficial.     

Developing Number Sense and Basic Computational Skills in Students with Special Needs

This article presents an investigation of the effectiveness of procedures undertaken to develop number sense and basic computational skills in learning disabled students. Twelve students in a K-1 classroom who had been identified as learning disabled (LD) were presented tasks which required them to subitize (i.e., recognize the number of objects in a set without actually counting them). Consistent with other such studies in special education, a qualitative research methodology was employed, involving a case study of an intact group of LD students. Also consistent with many such studies in special education, observational rather than quantitative data were collected. At the end of a four-week period, all students were consistently successful in recognizing and matching the numbers 0 through 5 and adding sums to five as determined on a teacher-administered test. Increased time-on-task and pupil independence also are reported. Suggestions for further instructional research related to other arithmetic skills are presented.     

Linear algebra students’ modes of reasoning: Geometric representations

Main goal of our research was to document differences on the types of modes linear algebra students displayed in their responses to the questions of linear independence from two different assignments. In this paper, modes from the second assignment are discussed in detail. Second assignment was administered with the support of graphical representations through an interactive web-module. Additionally, for comparison purposes, we briefly talk about the modes from the first assignment. First assignment was administered with the support of computational devices such as calculators providing the row reduced echelon form (rref) of matrices. Sierpinska’s framework on thinking modes (2000) was considered while qualitatively documenting the aspects of 45 matrix algebra students’ modes of reasoning. Our analysis revealed 17 categories of the modes of reasoning for the second assignment, and 15 categories for the first assignment. In conclusion, the findings of our analysis support the view of the geometric representations not replacing one’s arithmetic or algebraic modes but encouraging students to utilize multiple modes in their reasoning. Specifically, geometric representations in the presence of algebraic and arithmetic modes appear to help learners begin to consider the diverse representational aspects of a concept flexibly.     

Computational Estimation Performance on Whole‐Number Multiplication by Third‐ and Fifth‐Grade Chinese Students

Four hundred and three 3rd‐ and 5th‐grade Chinese students took the Multiplication Estimation Test or participated in the interview on it, designed to assess their computational estimation performance on whole‐number multiplication. Students perform better when tasks are presented visually than orally. Third graders tend to use rounding based while fifth graders tend to use written algorithm based strategies, but boys' and girls ‘performances do not differ. It is concluded that students often will not estimate simply at the request to estimate if an exact answer is within their mental computation capability, and a two‐step process is suggested for helping students decide what route to take when given arithmetic problems.     

Growth in the understanding of infinite numerical series: a glance through the Pirie and Kieren theory

In this paper, we describe the growth of mathematical understanding in university students engaged in mathematics classroom tasks regarding the concept of numerical series. Starting from the Image-Making Pirie and Kieren theory layer, students organize a mathematical concept linking different mathematical elements. In this process, there are important agents that are involved during the interactions to advance in this construction through the mechanism of folding back between different layers.     

Real-World Contexts,Multiple Representations,Student-Invented Terminology,and Y-Intercept

One classroom using two units from a Standards-based curriculum was the focus of a study designed to examine the effects of real-world contexts, delays in the introduction of formal mathematics terminology, and multiple function representations on student understanding. Students developed their own terminology for y-intercept, which was tightly connected to the meaningfulness and implicit/explicit temporality of the contexts that students investigated as part of their classroom activities. This terminology held great promise for promoting the concept of y-intercept within a multiple representation environment. However, the teacher's interpretation of different activities and his assumptions about the transparency of different representations, as well as students' past experiences left the student-generated terminology and the concept of y-intercept disconnected from one another. This resulted in student-generated terminology that had limited applicability, a fragile understanding of y-intercept within different representations, and for some students, interference between their invented terminology and the concept of y-intercept itself.     

Mathematics education graduate students’ understanding of trigonometric ratios

This study describes mathematics education graduate students’ understanding of relationships between sine and cosine of two base angles in a right triangle. To explore students’ understanding of these relationships, an elaboration of Skemp's views of instrumental and relational understanding using Tall and Vinner's concept image and concept definition was developed. Nine students volunteered to complete three paper and pencil tasks designed to elicit evidence of understanding and three students among these nine students volunteered for semi-structured interviews. As a result of fine-grained analysis of the students’ responses to the tasks, the evidence of concept image and concept definition as well as instrumental and relational understanding of trigonometric ratios was found. The unit circle and a right triangle were identified as students’ concept images, and the mnemonic was determined as their concept definition for trigonometry, specifically for trigonometric ratios. It is also suggested that students had instrumental understanding of trigonometric ratios while they were less flexible to act on trigonometric ratio tasks and had limited relational understanding. Additionally, the results indicate that graduate students’ understanding of the concept of angle mediated their understanding of trigonometry, specifically trigonometric ratios.     

'Model your genes the mathematical way'--a mathematical biology workshop for secondary school teachers

This article describes a mathematical biology workshop givento secondary school teachers of the Danville area in Virginia,USA. The goal of the workshop was to enable teams of teacherswith biology and mathematics expertise to incorporate lessonplans in mathematical modelling into the curriculum. The biologicalfocus of the activities is the lactose operon in Escherichiacoli, one of the first known intracellular regulatory networks.The modelling approach utilizes Boolean networks and tools fromdiscrete mathematics for model simulation and analysis. Theworkshop structure simulated the team science approach commonin today's practice in computational molecular biology and thusrepresents a social case study in collaborative research. Theworkshop provided all the necessary background in molecularbiology and discrete mathematics required to complete the project.The activities developed in the workshop show students the valueof mathematical modelling in understanding biochemical networkmechanisms and dynamics. The use of Boolean networks, ratherthan the more common systems of differential equations, makesthe material accessible to students with a minimal mathematicalbackground. High school students can be exposed to the excitement of mathematicalbiology from both the biological and mathematical point of view.Through the development of instructional modules, high schoolbiology and mathematics courses can be joined without havingto restructure the curriculum for either subject. The relevanceof an early introduction to mathematical biology allows studentsnot only to learn curriculum material in a innovative setting,but also creates an awareness of new educational and careeropportunities that are arising from the interconnections betweenbiological and mathematical sciences. The materials used in this workshop are available at a websitecreated by the directors: http://polymath.vbi.vt.edu/mathbio2006/.     

Abstract algebra students’ evoked concept images for functions and homomorphisms

Homomorphism is a critical variety of function in undergraduate Abstract Algebra (AA) courses and function is one of the unifying concepts across many mathematical subject areas. However, despite homomorphism’s important place in the curriculum and its existence as a particular type of function, little is known of a student’s concept image of functions at advanced levels and the role this concept image may play in a students’ homomorphism-related activities. In this paper, we share cases that explore students’ concept image and treatment of functions at the undergraduate AA level. In particular, we focus on coherence of prior function understanding and functions in AA (homomorphisms), and how this coherence may account for student activity in tasks related to homomorphism. Our results reflect that even at the AA level, students may have limited concept images of functions and their understanding of function (and coherence with homomorphism) can serve as a support or obstacle to task performance in AA. We suggest that both instructors and researchers explicitly attend to the role of function and function understanding in student activity at advanced levels.