Aritmetiikasta algebraan : Muutoksia osaamisessa peruskoulun päättöluokalla 20 vuoden aikana

From Arithmetic to Algebra. Changes in the skills in comprehensive school over 20 years. In recent decades we have emphasized the understanding of calculation in mathematics teaching. Many studies have found that better understanding helps to apply skills in new conditions and that the ability to think on an abstract level increases the transfer to new contexts. In my research I take into consideration competence as a matrix where content is in a horizontal line and levels of thinking are in a vertical line. The know-how is intellectual and strategic flexibility and understanding. The resources and limitations of memory have their effects on learning in different ways in different phases. Therefore both flexible conceptual thinking and automatization must be considered in learning. The research questions that I examine are what kind of changes have occurred in mathematical skills in comprehensive school over the last 20 years and what kind of conceptual thinking is demonstrated by students in this decade. The study consists of two parts. The first part is a statistical analysis of the mathematical skills and their changes over the last 20 years in comprehensive school. In the test the pupils did not use calculators. The second part is a qualitative analysis of the conceptual thinking of pupils in comprehensive school in this decade. The study shows significant differences in algebra and in some parts of arithmetic. The largest differences were detected in the calculation skills of fractions. In the 1980s two out of three pupils were able to complete tasks with fractions, but in the 2000s only one out of three pupils were able to do the same tasks. Also remarkable is that out of the students who could complete the tasks with fractions, only one out of three pupils was on the conceptual level in his/her thinking. This means that about 10% of pupils are able to understand the algebraic expression, which has the same isomorphic structure as the arithmetical expression. This finding is important because the ability to think innovatively is created when learning the basic concepts. Keywords: arithmetic, algebra, competence

Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the type y″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.

Studies on Conducting Polymers and Conductive Elastomer Composites

The primary aim of this work has been to develop conductive silicone and nitrile rubbers, which are extensively used for making conductive pads in telephone sets, calculators and other applications. Another objective of the work has been to synthesise and characterize novel conducting polymers based on glyoxal and paraphenylenediamine- poly(p-phenylenediazomethine. Conducting polymer matrices were developed from polymer blends such as poly(pphenylenediazomethine), polyethylene, PVC and silica and their properties were studied.

Differentiating core and co-opted mechanisms in calculation: The neuroimaging of calculation in aphasia

The role of language in exact calculation is the subject of debate. Some behavioral and functional neuroimaging investigations of healthy participants suggest that calculation requires language resources. However, there are also reports of individuals with severe aphasic language impairment who retain calculation ability. One possibility in resolving these discordant findings is that the neural basis of calculation has undergone significant reorganization in aphasic calculators. Using fMRI, we examined brain activations associated with exact addition and subtraction in two patients with severe agrammatic aphasia and retained calculation ability. Behavior and brain activations during two-digit addition and subtraction were compared to those of a group of 11 healthy, age-matched controls. Behavioral results confirmed that both patients retained calculation ability. Imaging findings revealed individual differences in processing, but also a similar activation pattern across patients and controls in bilateral parietal cortices. Patients differed from controls in small areas of increased activation in peri-lesional regions, a shift from left fronto-temporal activation to the contralateral region, and increased activations in bilateral superior parietal regions. Our results suggest that bilateral parietal cortex represents the core of the calculation network and, while healthy controls may recruit language resources to support calculation, these mechanisms are not mandatory in adult cognition.

Customer retention management in the recession

The purpose of this article is to explore customer retention strategies and tactics implemented by firms in recession. Our investigations show just how big a challenge many organizations face in their ability to manage customer retention effectively. While leading organizations have embedded real-time customer life cycle management, developed accurate early warning systems, price elasticity models and ‘deal calculators’, the organizations we spoke to have only gone as far as calculating the value at risk and building simple predictive models.

Algebra skills and traps and diagnostic teaching for the future

Explores on some research about teaching and learning algebra and related classroom issues. Diagnostic instruments that may be used by senior secondary teachers in teaching algebra to senior classes; Strategies for remediating algebraic difficulties and misconceptions; Impact of technology on the algebra curriculum; Usefulness of copying algebraic expressions while using Computer Algebra Systems or mathematics processing software in a calculus class.

A problem-based learning project supported by the internet

This case study looks at how bank sites with interactive calculators can be used to enhance directed investigations of students in a Mathematics Studies course. In the course of these investigations, students access simulated contexts which enabled them to have a feel of how they would spend money in the real world. This case study reveals the confidence of students in carrying out searches and transferring data, learning about bank calculators and their role in real life, how hidden costs are incorporated into loans and being able to validate what is presented in these calculators with their own calculations. This case study also highlights the perceptions of the teacher regarding this strategy in teaching this topic and the areas that need improvement. This paper analyses what has happened in the teaching and learning process and endeavours to shed some light into how the Internet can be used to promote a quality mathematics education.

The mechanics of computation and human thought

Thousands of students are preparing for chemistry examinations in June. An unresolved debate is whether they should be permitted to use graphics and programmable calculators in those examinations. Some educators have not only advocated the use of graphics calculators, but have also pointed to the Danish system in which students are permitted to use computers in senior school examinations.

In some Australian jurisdictions, graphics calculators are permitted in year 12 mathematics examinations, but not in chemistry examinations. The reasoning is that information or methods of solving numerical chemical problems can be stored in the memory of graphics calculators, giving some students an unfair advantage. This means that chemistry students either have to learn how to use (and buy!) two types of calculators or, if they only have one calculator, are disadvantaged in using non-programmable calculators in mathematics examinations.

The use of technology (or its lack thereof) can limit how and what students learn. “The mechanics of computation and human thought” is an allusion to Asimov’s short story, “A Feeling of Power” in which, overuse of technology has caused people to forget how to do simple arithmetic. In our current assessment system, the insistence that students must be able to do simple chemical calculations has lead to underuse of available technology. The misperception is that the ability to do calculations is linked to understanding of concepts.

Graphics calculators, programmable calculators and computers are tools. Instead of banning or limiting technology, we should take the opportunity to rethink what is being assessed and how it is assessed. It is the proper use of technology, by combining the mechanics of computation and human thought to deepen understanding and to ask probing questions that truly leads to a feeling of power.

The effect of mathematical games on on-task behaviours in the primary classroom

A challenge for primary classroom teachers is to maintain students’ engagement with learning tasks while catering for their diverse needs, capabilities and interests. Multiple pedagogical approaches are employed to promote on-task behaviours in the mathematics classroom. There is a general assumption by educators that games ignite children’s on-task behaviours, but there is little systemically researched empirical data to support this claim. This paper compares students’ on-task behaviours during non-digital game-playing lessons compared with non-game-playing lessons. Six randomly selected grade 5 and 6 students (9–12 year olds) were observed during ten mathematics lessons. A total of 2,100 observations were recorded via an observational schedule and analysed by comparing the percentage of exhibited behaviours. The study found the children spent 93 % of the class-time exhibiting on-task engagement during the game-playing lessons compared with 72 % during the non-game-playing lessons. The game-playing lessons also promoted greater incidents of student talk related to the mathematical task (34 %) compared with the non-game playing lessons (11 %). These results support the argument that games serve to increase students’ time-on-task in mathematics lessons. Therefore, it is contended that use of games explicitly addressing the mathematical content being taught in a classroom is one way to increase engagement and, in turn, potential for learning.

Introducing linear functions: an alternative statistical approach

The introduction of linear functions is the turning point where many students decide if mathematics is useful or not. This means the role of parameters and variables in linear functions could be considered to be ‘threshold concepts’. There is recognition that linear functions can be taught in context through the exploration of linear modelling examples, but this has its limitations. Currently, statistical data is easily attainable, and graphics or computer algebra system (CAS) calculators are common in many classrooms. The use of this technology provides ease of access to different representations of linear functions as well as the ability to fit a least-squares line for real-life data. This means these calculators could support a possible alternative approach to the introduction of linear functions. This study compares the results of an end-oftopic test for two classes of Australian middle secondary students at a regional school to determine if such an alternative approach is feasible. In this study, test questions were grouped by concept and subjected to concept by concept analysis of the means of test results of the two classes. This analysis revealed that the students following the alternative approach demonstrated greater competence with non-standard questions.

The Carbon Emissions of the Boston Red Sox

The Boston Red Sox emit a great deal of carbon throughout the regular baseball season because of flights to the home fields of their opponents. Knowing that air travel is one of the biggest transportation-based contributors to global climate change, the Boston Red Sox (and all major league teams) should be encouraged to offset their carbon emissions from regular season travel. Using ArcGIS to map the flight paths along great circle routes, the distance of flights to opponents’ cities was calculated to total the number of miles traveled in the 2008 season. The price of offsetting this carbon was estimated using the calculators of carbon offset retailers, such as Native Energy, a Vermont-based retailer. This project provides the potential costs of offsetting the carbon emitted from Red Sox air travel. To take the lead in the future of the Northeast, the Red Sox should begin to consider their contribution to climate change.

PRACTICAL LAND GRADING BASED ON LEAST SQUARES.

Practical methods for land grading design of a plane surface for rectangular and irregularly shaped fields based on a least squares analysis are presented. The least squares procedure leads to a system of three linear equations with three unknowns for determination of the best-fit plane. The equations can be solved by determinants (Cramer's rule) using a procedure capable of solution by many programmable calculators. The detailed computational process for determining the equation of the plane and a simple method to find the centroid location of an irregular field are also given. An illustrative example and design instructions are included to demonstrate the application of the design procedure.

A investigação do teorema fundamental do cálculo com calculadoras gráficas

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)