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The standard deviation (SD) of a random sample is defined as the square-root of the sample variance, which is the "mean" squared deviation of the sample observations from the sample mean. Here, we interpret the sample SD as the square-root of twice the mean square of all pairwise half deviations between any two sample observations. This interpretation leads to a geometric visualization of the sample SD, and a more elementary explanation as to why the denominator in the sample variance is one less than the sample size.

Data collected from online learning and tutoring systems for individual students showed strong autocorrelation or dependence because of content connection, knowledge-based dependency, or persistence of learning behavior. When the response data show little dependence or negative autocorrelations for individual students, it is suspected that students are randomly guessing the answers or that they are inconsistent in learning behavior. In addition, the global and local rates of correct responses may reflect students' proficiency in the learning process. This study shows that the dependence of online data may be characterized by the fractal dimension as a summary statistic locally and globally. The rate of correct responses and the global and local fractal dimensions of individual students' responses may indicate their learning behavior in short and long learning windows. The results may shed light on when individual students are experiencing difficulties in the learning process.

This report describes the pilot conducted in the final phase of a project, Expanding Audio Access to Mathematics Expressions by Students With Visual Impairments via MathML, to provide easy-to-use tools for authoring and rendering secondary-school algebra-level math expressions in synthesized speech that is useful for students with blindness or low vision. The pilot evaluated the authoring and speech-rendition tools, including interactive navigation. Teachers participated in the portion of the study that evaluated the authoring tools. Secondary school students with blindness or low vision participated in the portion that evaluated the speech-rendition and navigation tools, comparing those tools to each student's usual method (braille or print) for working with math. The teachers received an interactive authoring tutorial and the students received an interactive navigation tutorial before using the tools studied. Prior to the pilot, feedback studies on authoring and navigation gathered information that was used to fine-tune the tutorials and the functionalities being evaluated. In the pilot we attempted to simulate a likely use-case for each group. In the students' case, the tools for reading and navigating spoken mathematics were compared directly with the methods each student typically used for reading math (braille or print in a size appropriate for the student). Teachers and content providers were instructed to author math expressions that matched those used in the pilot instruments given to the students. The studies showed that the authoring tools were easy for teachers to use as intended and that little observable difference existed between students' success in answering math and math-parsing questions when using the spoken math tools and their success in answering parallel questions posed in braille or print of appropriate size. Please see Appendix E for information on obtaining a version of this report that is fully accessible using the tools described.

The aim of the study was to determine whether pre-service elementary mathematics teachers could write and use mathematical definitions. For this reason, the question sheet containing two questions, was given to 76 pre-service teachers in a public university who was taking general mathematics course. Qualitative research method was used during data analysing. According to the answers given by them, five themes for the first and six themes for the second question were classified. In order to reach detailed results, semi-structured interviews were conducted with one pre-service teacher for each theme. Findings showed that they had difficulties in writing a formal definition assumed to be known and in using a given definition assumed to be not known of a mathematical concept.

Test score distributions of schools or demographic groups are often summarized by frequencies of students scoring in a small number of ordered proficiency categories. We show that heteroskedastic ordered probit (HETOP) models can be used to estimate means and standard deviations of multiple groups' test score distributions from such data. Because the scale of HETOP estimates is indeterminate up to a linear transformation, we develop formulas for converting the HETOP parameter estimates and their standard errors to a scale in which the population distribution of scores is standardized. We demonstrate and evaluate this novel application of the HETOP model with a simulation study and using real test score data from two sources. We find that the HETOP model produces unbiased estimates of group means and standard deviations, except when group sample sizes are small. In such cases, we demonstrate that a "partially heteroskedastic" ordered probit (PHOP) model can produce estimates with a smaller root mean squared error than the fully heteroskedastic model.

The general public consumes financial products such as loans that are administered in the realm of nominal and effective interest rates. It is debatable if most consumers really understand how these rates function. This article explores the conceptions that student teachers have about nominal and effective interest rates. The APOS theory illuminates analysis of students' levels of conception. Seventy second-year mathematics students' responses to Grade 12 tasks on effective and nominal interest rates were analysed, after which 12 students were interviewed about their mathematical thinking in solving the tasks. The findings varied. While some students could not do the tasks due to erratic use of formulae (algebra), I ascertained that some students obtained correct answers through scrupulous adherence to the external prompt of formulae. Most of those students remained stuck at the action and process stages and could not view their processes as mathematical objects. A few students had reached the object and schema stages, showing mature understanding of the relationship between nominal and effective interest rates. As most students remained at the operational stages rather than the structural, the findings accentuate that when teaching this topic, teachers ought to take their time to build learners' schema for these notions. They need to guide their learners through the necessary action-process-object loop and refrain from introducing students to formulae too soon as this stalls their advancement to the object and schema stages which are useful in making them smart consumers of financial products.

The purpose of this study is to analyze the teaching strategies of the prospective teachers of mathematics for elementary school in relevance with the geometry formulas. The participants of the study are constituted by six senior students of the program of mathematics teaching for elementary school. In this study, where the qualitative research approach is employed, the method of case study is used and the data of the study were aggregated with the techniques of semi-structured interview, observation and analysis of documentation. The data of the study were analyzed by the virtue of the technique of content analysis. With regard to the data obtained from the study, it was seen that the prospective teachers were the supporters of making students find the logic of the formulas by explaining them, instead of making them memorize, while they were teaching the mathematical formulas of which logical reason they could explain. On the contrary, it was found that the prospective teachers were the supporters of making students memorize directly the formulas of which logical explanation they could not explain. Besides, even if the prospective teachers could explain the meaning of the given formula, they could not make teaching explanations enough while they were teaching this formula.

A simulation study is presented to evaluate and compare three methods to estimate the variance of the estimates of the parameters d and "C" of the signal detection theory (SDT). Several methods have been proposed to calculate the variance of their estimators, "d'" and "c." Those methods have been mostly assessed by comparing the empirical means and variances in simulation studies with the calculations done with the parametric values of the probabilities of giving a yes response on a signal trial (hits) and on a noise trial (false alarms). In practical contexts the variance must be estimated from estimations of those probabilities (empirical rates of hits and false alarms). The three methods to estimate the variance compared in the present simulation study are based in the binomial distribution of Miller, the normal approach of Gourevitch and Galanter and the maximum likelihood method proposed by Dorfman and Alf. They are compared in terms of relative bias (accuracy) and the mean squared error (precision). The results show that the last two methods behave indistinguishably for practical purposes and provide severe over-estimation errors in a range of situations that while not the most common are perfectly credible in several practical contexts. By contrast, the method of Miller provides better results (or at least similar) in all conditions studied. It is the recommended method to obtain estimates of the variances of these statistics for practical purposes.

Solving problems is not only a goal of mathematical learning. Students acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations by learning to solve problems. In fact, there were students who had difficulty in solving problems. The students were naive problem solvers. This research aimed to describe the thinking process of naive problem solvers based on heuristic of Polya. The researcher gave two problems to students at grade XI from one of high schools in Palangka Raya, Indonesia. The research subjects were two students with problem solving scores of 0 or 1 for both problems (naive problem solvers). The score was determined by using a holistic rubric with maximum score of 4. Each subject was interviewed by the researcher separately based on the subject's solution. The results showed that the naive problem solvers read the problems for several times in order to understand them. The naive problem solvers could determine the known and the unknown if they were written in the problems. However, they faced difficulties when the information in the problems should be processed in their minds to construct a mental image. The naive problem solvers were also failed to make an appropriate plan because they did not have a problem solving schema. The schema was constructed by the understanding of the problems, conceptual and procedural knowledge of the relevant concepts, knowledge of problem solving strategies, and previous experiences in solving isomorphic problems.

In understanding upper secondary school students' interpretations of information in symbolic representations of a distance-time-relation, little attention has been paid to the analysis of the condition of the conceptual development related to utterances. Understanding this better can help improve the teaching of attribute and information in symbolic representations of different phenomena. Two theoretical perspectives have been used to conduct the analysis: Tall and Vinner's theoretical perspectives on learning and Gray's & Talls's theory of three mathematical worlds. The findings provide evidence that a detailed analyse of student's utterances show difference in quality related to student's interpretations of a distance-time relation. The qualities were related to student's concept images of functions and derivatives.

Education in society especially in Mexico, seems to be a powerful instrument of intergenerational social mobility to produce individuals with "capabilities and functions" allowing them to obtain a greater well-being. "Education as schooling," in the first instance, improves the individuals living conditions, since this is a path to a better way of living and a privileged way to achieve a higher economic and social position within a society. However, education's purpose is not only to be thought as schooling for economic well-being, but also as an element of cultural integration and general well-being within a society. Based on that, we have studied the educational inequality in two states in Mexico during the years 2008 and 2010. To this end, we have incorporated into a model of multi-dimensional logistic regression, and index of educational inequality. This index, consist of three focal variables: educational backwardness, access to technologies of information and communication, and family education spending. These variables allowed us to quantify more accurately this form of inequality. The observed results are worrisome, even more because; this derives other inequalities for the families of these societies, including reducing opportunities for better employment, and thus provoking lower standards in health, housing, nourishment, and social participation among others. Educational inequality is, therefore, another factor that originates poverty in the society.

Mathematical formulae are information objects that can be entered in a computer, visualized, and evaluated. Thus, by the majority of (mostly occasional) users it is also expected that they are transferable through the simple copy-paste procedure. This transfer is particularly interesting when users are involved in tasks that span different mathematical activities or domains. For example, when performing computations and writing a report about them, or when performing algebraic computations from geometric constructions. Essentially, using copy-paste also allows users to use a particular mathematical software for the tasks that it does best. To uncover the possible difficulties within this process, we approached students who are beginning to learn the use of mathematical tools. Through analysing their home-work where they report on the usage of various tools we observed their use of the copy-paste transfer procedure, both from their reports and from the dialogue they had with the teachers. Their attempts show a multitude of issues which we try to explain. It appears that the copy-paste procedure is often useless, especially for inexperienced users.

Financial analysts generally create static formulas for the computation of NPV. When they do so, however, it is not readily apparent how sensitive the value of NPV is to changes in multiple interdependent and interrelated variables. It is the aim of this paper to analyze this variability by employing a dynamic, visually graphic presentation using Excel. Our approach illustrates how these variables, when increased or decreased to reflect the potential range of values in a business case, change the value of NPV, and hence affect the decision about whether to proceed with the project or to reject it. Furthermore, since sales revenue is one of the least certain elements in the business case, the presentation includes a probability estimate of whether NPV will be positive or negative, assuming that sales revenue is normally distributed with a known mean and standard deviation. The business case we have chosen for illustrative purposes is a global energy project. Nevertheless, financial analysts in any industry should be able to apply our dynamic spreadsheet approach to their projects as well.

The paper investigates geometric errors students made as they tried to use their basic geometric knowledge in the solution of the Applied Calculus Optimization Problem (ACOP). Inaccuracies related to the drawing of geometric diagrams (visualization skills) and those associated with the application of basic differentiation concepts into ACOP solution were reported. A test instrument was used to collect quantitative data, while qualitative data were generated using follow- up interviews (stimulated recall). The targeted samples were freshmen students who registered for Calculus I in the department of Mathematics at a University in south eastern region of the United States of America, USA. The study indicated that students had achieved a very low success rate on the ACOP solution process, immediately after receiving/completing instruction on the optimization in their calculus I class. In general, they failed to integrate basic geometric competences required in the ACOP solution. Qualitative evidence from students' test performance indicated that failure to visualize geometric diagrams from word problems tended to preclude them getting the required formula. The overall finding of the research was that students face structural and procedural setbacks that ultimately led to a worsening of the ACOP solution process.

This study explores the understanding levels of pre-service teachers of the elementary school mathematics teacher education undergraduate program about capacitors that they learned in the compulsory Physics-2 course in the second year of the second term, and instructors' predictions about the teachers' responses. A total of 54 pre-service teachers participated in the study. The findings of the study reveal that 62.9% of the pre-service teachers did not know the function of a capacitor, 46.3% could not write any equations in connection with the capacitor and 57.4% could not explain any of the quantities in the formulas related to the capacitor. Instructors who taught Physics-2 at the undergraduate level were asked to predict the percentage rate of success of the answers the pre-service teachers gave about capacitors. The data reveal that the instructors were not sufficiently aware of pre-service teachers' knowledge of capacitors, although they taught them Physics-2 and evaluated them. [Paper presented at the World Conference on Learning, Teaching and Educational Leadership (8th, Lisbon, Portugal, Oct 26-28 2017).]

Measurement in geometry is one of the key areas of school mathematics, however, pupils make serious mistakes when solving problems involving measurement and hold misconceptions. This article focuses on the possible links between lower secondary pupils' (n = 870) success in solving non-measurement tasks and calculations tasks on area and volume and on their problems when solving measurement tasks. The study uses a mixed research design. Statistical methods are used to find correlations between the two types of tasks and a qualitative analysis is carried out to identify mistakes and misconceptions. The results show that there are indeed relatively strong links between success in non-measurement tasks and calculation tasks, and consequently, when teaching, attention must be paid to the development of both types of skills. The study identified pupils' mistakes in tasks which are within the Czech curriculum but which proved to be difficult, such as a missing link between algebraic and geometric representations, a tendency to linearize and/or to employ pseudo-analytical thinking. The study identified differences between individual classes which point to the significant role of the teacher and/or influence of the textbook used.

This article explains how explorations into the quadratic formula can offer students opportunities to learn about the structure of algebraic expressions. In this article, the author leverages the graphical interpretation of the quadratic formula and describes an activity in which students derive the quadratic formula by quantifying the symmetry of a parabola. By the end of the activity, students are equipped to use the quadratic formula to solve quadratic equations and are prepared to reason about the structure of the quadratic formula. This activity can be used with algebra 1 students to introduce the quadratic formula or more advanced algebra students to enhance their understanding. A student activity sheet accompanies this article.

Stoichiometry calculation competence tends to be particularly poor in the developing world, even among teachers. Various methods, including the unit factor method, have the potential to be effective in developing such competence. To evaluate the likelihood of such a generic proportion method, which downplays traditional formula usage, succeeding in a particular context, it is necessary first to understand teachers' existing competence with proportion and the extent to which their calculation success relies on the explicit provision of and substitution into formulae in their written solutions. This quantitative survey study of 171 South African Physical Sciences teachers' use of formulae and proportion found that most teachers failed to recognise the need to use proportion for some of the four reaction-based stoichiometry calculation questions used. Provision of and substitution into a formula in the written solution was found to be valuable in helping participants who underutilised proportion to obtain process marks, but to be largely irrelevant to obtaining the correct answer. The findings suggest that professional development interventions in similar contexts should focus on proportion within stoichiometry, particularly on recognition of its relevance to reaction-based questions. While a generic proportion method is well suited to this purpose, questions are raised concerning the likelihood that teachers would accept such a method in a context where process marks favour traditional formats of formula provision and substitution and where process mark attainment is highly valued. The findings also point to the need for teacher education to address competencies other than proportion.

The objective of this work was to evaluate the implementation of a metavisual strategy for students to revise and self-regulate concepts arising in a study of a chemical reaction between ions. For this purpose, two chemistry education undergraduate students at a Brazilian public university carried out an investigative activity, involving metavisual steps, to revise explanatory models at the submicro level. Students were given a problem, namely a reaction between ions drawn from a real-life situation and were provided with clay to construct an explanatory model of the submicro level for the initial and final stages of the reaction. The students were asked to compare their clay model with an example of a scientifically correct figure of the submicro level of the reaction generated by the researchers. At this stage students were given the option to reconstruct their model. Data were captured via photographs of the clay models and students' verbal discussions as they proceeded through the activity. The findings reveal evidence of self-regulation of mental models at the submicro level, from the interaction of prior knowledge, chemical diagrams and discussions and reflections by the pair of students. Difficulties regarding chemical formulae were also observed in relation to the symbolic level. Finally, there are implications for teaching chemistry, since teachers in training need to experience metavisual strategies for future application in their classrooms.

It is easy to dismiss the work of "teaching students to apply formula" as a low-order priority and thus trivialises the professional knowledge associated with this practice. Our encounter with an experienced teacher--through the examination of her practices and elaborations--challenges this simplistic assumption. There are layers of complexities that are as yet under-discussed in the existing literature. This paper reports a case study of her practices that reflect a complex integration of relevant theories in task design. Through examining her praxis around the theme of "recognise the form", we discuss theoretical ideas that can potentially advance principles in the sequencing of examples for the purpose of helping students develop proficiency in applying formula.

Any quadratic function has a line of symmetry going through its vertex; any cubic function has 1800 rotational symmetry around its point of inflection. However, polynomial functions of degree greater than three can be both symmetrical and asymmetrical (Goehle & Kobayashi, 2013). This work considers algebraic conversions of symmetrical quartic functions and shows the ways to express them in three equivalent formats--as is the case with all quadratic and cubic functions, shown in Miškovic (2021).

The formula for the variance of a binomial distribution is both concise and elegant. However, it is often taught without reference to the underlying reasoning. That being the case, is it important, or useful, to understand why this formula can be used to calculate the requisite result? In this article, the author demonstrates a teaching sequence that foregrounds the reasoning behind the formula. Implications for teaching are discussed, including the placement of this learning in the context of practising other valued skills.

Quality is a multidimensional and transversal concept also in the field of higher education. This and other considerations revealed some problems during application of quality principles to study programmes, emerged from the first accreditation experience in Italy. The objective of this article is to provide a simple self-assessment tool for professors who have to implement study programme quality, in order to understand their position with respect to accreditation requirements. To do this, a general procedure to create the self-evaluation questionnaire is explained. It is composed of six points: (1) studying literature or indications about the objective; (2) creating a schema of categories and sub-categories, (3) weighting the sub-categories, (4) creating a questionnaire, (5) creating a formula in order to synthesise the questionnaire in a graduation scale and finally (6) robustness analysis. A full example, based on Italian experience, is therefore developed in detail.

Continuing our critique of the classical derivation of the formula for the area of a disk, we focus on the limiting processes in geometry. Evidence suggests that intuitive approaches in arguing about infinity, when geometric configurations are involved, are inadequate, and could easily lead to erroneous conclusions. We expose weaknesses and misconceptions in arguments which attempt to articulate what the static geometric object 'in the limit', or 'at infinity', is, or what it looks like. Supported by school curricular expectations and existing research, we suggest an alternative focus and reasoning process to teaching and investigating geometric limits. Instead of discussing what happens 'at infinity', or 'at the end of a limiting process', we suggest that one should argue based on what happens as we move 'far enough', i.e. 'far toward infinity'. Thus, we suggest replacing the static situation 'at infinity' with the investigation of the corresponding dynamic geometric pattern. This way, the emphasis is placed on reasoning about approximations and estimations, which appear as objectives in school curricula of many countries. We encourage introducing and discussing geometric limits, as they raise a number of interesting questions, ask for identification of patterns, and have potentially unexpected and exciting outcomes.

The purpose of this article is to provide an explicit formula for the bounds of integration of the regular simplex centred at the origin. Furthermore, this article rigorously proves that these integration bounds recover the volume of the regular simplex. To the authors' knowledge, this is the first time that such integration bounds have been explicitly formulated in the mainstream literature. In addition, the derivation of the simplex volume using integration bounds appears to be new. As a result, this article has the potential to serve as a useful resource for a broad audience, including teachers of mathematics, engineers, and recreational geometers.

Student insight into algebraic formulas, including the ability to identify the structure of a formula and its components and to reason with and about formulas, is an issue in mathematics education. In this study, we investigated how 16- and 17-year-old pre-university students' insight into algebraic formulas can be promoted through graphing formulas by hand. In an intervention of five 90-min lessons, 21 grade 11 students were taught to graph formulas by hand. The intervention's design was based on experts' strategies in graphing formulas, that is, using a combination of recognition and qualitative reasoning, and on principles of teaching complex skills. To assess the effect of this intervention, pre-, post-, and retention tests were administered, as well as a post-intervention questionnaire. Six students were asked to think aloud during the pre- and posttests. The results show that all students improved their abilities to graph formulas by hand. The think-aloud data suggest that the students improved both on recognition and reasoning, and give a detailed picture of how students used recognition and qualitative reasoning in combination. We conclude that graphing formulas by hand, based on the interplay of recognition and qualitative reasoning, might be a means to promote students' insight into algebraic formulas.

In this paper we discuss the important Abel's summation formula, which is a very powerful tool for analysing series of real or complex numbers. We derive from it an integral test which may be useful in cases where the classical integral test may not be applied. We also discuss how this new integral test may be used when one is dealing with alternating series of real or complex numbers. The main goal of this note is to provide a supplementary material for courses dealing with infinite series of real or complex numbers.

The article shows that even a small number of iterations give a very large length of the Koch Curve. The overall purpose of this work is to show that even a small number of iterations can give a very complex fractal structure. In this case it is the very big length of Koch's curve. As examples, the distance from the Earth to the Moon and from the Earth to the Sun was compared to the length of the Koch curve.

We use Taylor's formula with Lagrange remainder to prove that functions with bounded second derivative are rectifiable in the case when polygonal paths are defined by interval subdivisions which are equally spaced. As a means for generating interesting examples of exact arc length calculations in calculus courses, we recall two large classes of functions "f" with the property that [equation omitted] has a primitive, including classical examples by Neile, van Heuraet and Fermat, as well as more recent ones induced by Pythagorean triples of functions. We also discuss potential benefits for our proposed definition of arc length in introductory calculus courses.

Mitochondrial beta-oxidation is one of the most common modes of fatty acids' oxidation in most organisms, particularly mammals. Biochemistry undergraduate curriculum often contains the description of the process, with emphasis on ATP calculations for various types of fatty acids. During our decade long teaching experience in biochemistry, we observed the difficulty faced by students in calculating energetics of several fatty acids beyond palmitic acid. We developed a canonical formula by mathematical transformations and logical derivation to aid the calculation in a much simpler manner to ease both teaching and learning experience. ATP yield of even-numbered fatty acids may be calculated using a formula [(7C - 6 - 1.5 D) - 2(D-2)], and adenosine triphosphate (ATP) yield for odd-numbered fatty acids can be calculated using [(7C - 19 - 1.5 D) - 2(D-2)], where C is the number of carbon atoms in fatty acids, D is the number of double bonds. The unbold part of the formulae is limited to polyunsaturated fatty acids. Moreover, we integrated these formulae into an HTML based web-interface for handily calculations, which is likely to augment fatty acids oxidation learning-teaching processes easier. This tool has been recently tested in our classroom programs on biochemistry and received an excellent feedback from the learners. Also, the mathematical formula is ready for being incorporated into standard biochemistry textbooks. The webtool as an opensource biochemical calculator can be effectively used in classrooms by both instructors and students while solving comprehension based questions on lipid metabolism.

One way to develop a cross-curricular lesson is to select the most common mathematical formulas used in science and carefully develop and implement tasks that allow students to make connections between the mathematical representations and theoretical/physical science concepts. The slope-intercept formula, which is used to study relationships between two sets of data, is a fundamental prerequisite concept for understanding scientific phenomenon. The temperature unit conversion formula provides an interesting opportunity to understand the mathematical concept of a linear equation, relate it to the slope-intercept form, and apply the mathematical meaning to the scientific principle. Thermodynamics, the science of heat and temperature, is a central branch of the sciences. Temperature and measurement skills are introduced in elementary school while the Fahrenheit and Celsius conversion formula, which represents the slope-intercept formula, is introduced in middle school and used through graduate-level courses. The activity described in this article is designed to help students understand the temperature unit conversion formula and its relationship to the slope-intercept equation. Skills needed for this activity include graphing calculator skills for entering data, manipulating graphs, and calculating regression equations. The activity focuses on the "Next Generation Science Standards" ("NGSS") practice Using Mathematics and Computational Thinking to develop deeper understanding of scientific formulas and follows the 5E Instructional Model: Engage, Explore, Explain, Elaborate, and Evaluate.

Background: One criterion of adaptive learning is appropriate generalization to new instances based on the original learning context and avoiding overgeneralization. Appropriate generalization requires understanding what features of a solution are applicable in a new context and whether the new context requires modifications or a new approach. In a series of three experiments, we investigate whether searching for an algebraic formalism before receiving direct instruction facilitates appropriate generalization. Results: (1) Searching buffers against negative transfer: participants who first searched for an equation were less likely to overgeneralize compared to participants who completed a tell-and-practice activity. (2) Likelihood of creating a correct new adaptation varied by performance on the searching task. (3) Asking people to sketch alleviated some of the negative effects of tell-and-practice, but sketching did not augment the effect of searching. (4) When participants received more elaborate tell-and-practice instruction, the advantages of searching were less notable. Conclusions: Searching for an algebraic formula prior to direct instruction may be a productive way to help learners connect a formula to its referent and avoid overgeneralization. Tell-and-practice instruction that only described the mathematical procedures led to the greatest levels of overgeneralization errors and worst performance. Tell-and-practice instruction that highlighted connections between the mathematical structure of the formula and the visual referent performed at similar or marginally worse levels than the search-first conditions.

In the context of an analytical geometry, this study considers the mathematical understanding and activity of seven students analyzed simultaneously through two knowledge frameworks: (1) the Van Hiele levels (Van Hiele, 1986, 1999) and register and domain knowledge (Hibert, 1988); and (2) three action frameworks: the SOLO taxonomy (Biggs, 1999; Biggs & Collis, 1982); syntactic and semantic elaborations (Kaput, 1987a, 1987b, 1989); and isomorphic, transcendent, and mixed connections (Adu-Gyamfi, Bossé, & Lynch-Davis, 2019). Along with producing a fuller analysis of student work and communication, the study found that for only the students with the lowest and highest scores regarding either their understanding or actions on the analytic geometry task might there be a predictive association between knowledge and action levels. For other students, a predictive association could not be determined. This may mean that the level of understanding a student possesses regarding a particular mathematical concept may not parallel the level of actions they use when working with an associated task.

Analysing graphs, formulating covariational relationships, and hypothesizing systems' behaviour have emerged as frequent objectives of contemporary research in physics education. As such, these studies aim to help students achieve these objectives. While a consensus has been reached on the cognitive benefits of emphasizing the structural domain of mathematical apparatus in physics instruction, the question remains open on how to accomplish this task in school practice. This study contributes to this discussion. It attempts to determine how high-school physics students perceive formulas as a means to understand covariational relationships of natural phenomena. It also explores the hypothesis that including more structural domains of mathematics makes physics content more attractive for students. A sample of 25 advanced high-school physics students was taught kinematics, dynamics, and energy sections considering functional representations of some laws and principles traditionally covered in those sections. At the concluding phase, these students were invited to contrast formulas and their covariational embodiments as a means of providing conceptual understanding. The analysis revealed that students considered formulas as algebraic tools to attain a specific value to a particular application or carry out algebraic algorithms to solve a physics problem. They found formulas converted to covariational relationships a better fit for the learning of general phenomenon behaviour and extracting new information, especially from a graphical representation of these relationships. These findings generated the conclusion that the structural domain of mathematics can aid physics understanding by alternatively presenting formulas as mathematical entities that are transferrable to functions. It was further concluded that such parallelism would benefit not only the depth of inquiry in physics but also a general students' STEM disposition.

In this paper, we present data from two iterative teaching experiments involving students' constructions of four basic counting problems. The teaching experiments were designed to leverage the generalizing activities of relating and extending to provide students with opportunities to reflect on initial combinatorial activity when constructing these formulas. We discuss three combinatorial ways of thinking that emerged from their work on this task and provide commentary on the interplay between the students' generalizing activities and engagement in the emergent ways of thinking. We address the practical goals of identifying ways that students productively reason in combinatorics, and we provide a theoretical commentary on students' reflection on activity as it occurs in the context of particular generalizing actions.

KR-21 reliability and its extension (coefficient [alpha]) gives the reliability estimate of test scores under the assumption of tau-equivalent forms. KR-21 reliability gives the reliability estimate for summed scores for dichotomous items when items are randomly sampled from an infinite pool of similar items (randomly parallel forms). The article extends KR-21 to coefficient [beta] to estimate reliability of summed scores for test with any item scoring pattern under the same assumption of KR-21. Also, the article presents coefficient G[beta] to estimate this reliability for nonlinearly scaled scores for any type of items. The article shows that coefficient [beta] is slightly different from the classical index of reliability. Results show that coefficient [beta] is not equal to the classical index of reliability for summed scores and scaled scores. Moreover, results using real data on psychological instruments reveal that different score scales yield different coefficient G[beta] reliability estimates that are less than coefficient G[alpha] by a function related to form-to-form differences in averages.

Since the content of the Theory of Interest course in an actuarial science program is tied to an external exam, instructors may be hesitant to attempt to use inquiry-based learning. In this paper, I outline how and why I did so, including descriptions of the materials that I wrote. I found that student performance on the final exam improved despite the students not being any better prepared than prior students.

One does not have to teach for very long to see students applying the wrong formula in the wrong situation (e.g., Kirshner and Awtry 2004; Tan-Sisman and Aksu 2016). Students can become overreliant on the power of the formula instead of thinking about the relationships it describes. It is not surprising that students can see formulas as a way to produce an answer as opposed to a way to understand and describe mathematical relationships. The simplicity of formulas makes them attractive to teachers and students alike as something to learn, remember, and apply. This raises the question then, where are the opportunities for students to engage in reasoning and sense making when it comes to formulas? In this article, the author offers seven techniques for supporting students' engagement of reasoning and sense making through formulas. Although the examples given are based on area formulas, many of these techniques can be applied to other less spatially intensive formulas as well.

The article discusses three empirical examples of Computer Algebra System (CAS) use in a Danish upper secondary school mathematics class that had experienced a recent change of teacher. All examples lead to didactical problems surrounding the situation and unclear expectations between teacher and students, involving loss of students' mathematical skills and confidence, loss of global mathematical perspective, and the students losing sight of the mathematical objects in question. The article is the result of collaboration between two mathematics education researchers and an upper secondary school mathematics teacher, who experienced severe difficulties when taking over a class from another teacher. CAS was experienced as a crucial part of and reason for these difficulties. As a means for investigating the potential reasons behind the difficulties, a selection of constructs from the Theory of Didactical Situations (TDS) is applied. In particular, it is observed that unclear contractual relations about the role of CAS bring with them misguided winning strategies and metacognitive shifts, eventually causing the students to "lose the game".

Admittedly, the study of Complex Analysis (CA) requires of the student considerable mental effort characterized by the mobilization of a related thought to the complex mathematical concepts. Thus, with the aid of the dynamic system Geogebra, we discuss in this paper a particular concept in CA. In fact, the notion of winding number v[f(gamma),P] = n has a profound topological and geometrical meaning. Moreover, we record the effort of some authors in order to emphasize its intuitive description (Needham, 2000). We will show some basics commands that allow too the exploration of geometric and topological ideas related to the Rouché's theorem and the Fundamental Theorem of Algebra (TFA) (Alves, 2014).

The current research around the Fibonacci's and Lucas' sequence evidences the scientific vigor of both mathematical models that continue to inspire and provide numerous specializations and generalizations, especially from the sixties. One of the current of research and investigations around the Generalized Sequence of Lucas, involves it's polinomial representations. Therefore, with the introduction of one or two variables, we begin to discuss the family of the Bivariate Lucas Polynomias (BLP) and the Bivariate Fibonacci Polynomials (BFP). On the other hand, since it's representation requires enormous employment of a large algebraic notational system, we explore some particular properties in order to convince the reader about an inductive reasoning that produces a meaning and produces an environment of scientific and historical investigation supported by the technology. Finally, throughout the work we bring several figures that represent some examples of commands and algebraic operations with the CAS Maple that allow to compare properties of the Lucas' polynomials, taking as a reference the classic of Fibonacci's model that still serves as inspiration for several current studies in Mathematics.

We explore the mathematics engagement of a group of mathematics coaches, working in k-12 mathematics education. The incenter of a triangle is used to derive an alternative formula for the area of a triangle inspired by Usiskin, Peressini, Marhisotto, and Stanley (2002).

The purpose of this article is to examine the reasoning behind the wording of the definition of the particular solution to an initial value problem. This article will be of practical importance for students taking a first year calculus course that includes the study of first order linear separable differential equations.

The development of the constraint-referenced analytics tool for monitoring algebra learning activities presented here came from the desire to firstly, take a more quantitative look at student responses in collaborative algebra activities, and secondly, to situate those activities in a more traditional introductory algebra setting focusing on procedural understanding. Procedural skill was analyzed by modeling the complexity of attempts to make equivalent transformations of algebraic expressions. The constraint-referenced analytics system uses log files of student inputs on a classroom network of handheld devices to measure success rate as students make attempts to replace one algebraic expression with another equivalent expression. The analytics engine produced psychometrically verifiable results. Moving averages of student performance revealed that when students experienced a period of struggle and persisted in attempting similar transformations, an apparent conceptual shift led to subsequent success. Students also responded to periods of struggle by switching to familiar tasks or choosing non-participation.

Aim of the study: This study examines numerical methods for solving the problems in gas dynamics, which are based on an exact or approximate solution to the problem of breakdown of an arbitrary discontinuity (the Riemann problem). Results: Comparative analysis of finite difference schemes for the Euler equations integration is conducted on the basis of an exact or approximate solution to the problem of an arbitrary discontinuity breakdown. An approach to the numerical solution of the Euler equations governing inviscid compressible gas flow is developed on the basis of the finite volume method and finite difference schemes for flow calculation of various degrees of accuracy. Calculation results show that monotonic derivative correction provides numerical solution uniformity in the breakdown neighborhood. On one hand, it prevents the formation of new extremum points, thereby providing monotonicity, but on the other hand, it causes smoothing of existing minimums and maximums and accuracy loss. Conclusions: The developed numerical calculation method makes it possible to perform high-accuracy calculations of flows with strong non-stationary shock and detonation waves and no nonphysical solution oscillations on the shock wave front.

Introduction: Digital textbooks are increasingly marketed and used, yet little research examines this medium. Within the limited research, even less investigates the role of digital textbooks in mathematics--a challenging content area for many students, but especially for students with visual impairments. Methods: Through a qualitative analysis, this study sought to understand the nature of the use of a digital algebra textbook to support secondary students with visual impairments in algebra. Results: The results suggest three main themes: (a) students' dependence on their traditional textbook medium, (b) educators' reactions to the technology, and (c) the benefits of using the digital textbook despite resistance. The three themes culminate in clear student preferences for traditional textbooks. Discussion: The general resistance to use of the technology warrants caution in terms of schools moving forward with the adoption of digital textbooks, and it suggests that additional examination of the topic is needed. Implications for Practitioners: Practitioners can consider using digital textbooks, but they will need to ensure that they and their students are both properly motivated and adequately trained to use such technology.

The purpose of the study is to determine the business need for vocational training. This article gives a detailed analysis of the problem aimed at finding optimal occupational skill structure of training, which involves all kinds of positive effects in various areas of public life--from the economy up to the spiritual sphere of human life. Moreover, the authors described relevant stages and their interconnections in the implementation process and provided a detailed description of relevant source data. The practical value is that the investigation suggests approaches to provide balance in the regional labor market, and considers the need for concerted action of regional authorities, educational and manufacturing companies in the region.

Bologna process puts in a high claim for the modern European education in terms of competency building approach. The control is conducted by the agencies which monitor learning activity level in higher education institutions. This paper presents the aspects of higher education within the conduction of competency building approach projects in Europe and Russia. We propose the technology which evaluates students' professional competence qualimetricly. The technology represents the evaluation algorithm at all levels of student training.

This paper investigates the challenges faced by secondary school teachers and pupils in the teaching and learning of algebraic linear equations. The study involved 80 grade 11 pupils and 15 teachers of mathematics, drawn from 4 selected secondary schools in Mufulira district, Zambia in Central Africa. A descriptive survey method was employed to obtain in-depth and richer information about the central issues under study. To address the three research questions, an assessment test for the pupils was administered followed by separate interviews for teachers and pupils respectively. Some identified challenges include: (i) lack of pre-requisite knowledge by pupils, (ii) lack of conceptual, procedural and strategic knowledge and skills required for solving linear equations and (iii) inappropriate approaches and methods used in the teaching of algebraic linear equations. The study also compared responses on items from teachers' and pupil's interviews and found out that there was a strong relationship between teachers' and pupils' opinions on the challenges that pupils face in solving algebraic linear equations and what needs to be done to improve the situation. It is therefore, proposed that the teaching and learning of secondary school mathematics need to focus on transforming the quality landscape that has been identified especially the transition from arithmetic to algebra. Pupils need to be exposed to formulating equations from situations with which they are familiar and that teachers get regular refresher courses to keep them up-dated with new developments in the teaching and learning of mathematics.