Idealähtöistä koulualgebraa : IDEAA-opetusmallin kehittäminen algebran opetukseen peruskoulun 7. luokalla

This research is based on the problems in secondary school algebra I have noticed in my own work as a teacher of mathematics. Algebra does not touch the pupil, it remains knowledge that is not used or tested. Furthermore the performance level in algebra is quite low. This study presents a model for 7th grade algebra instruction in order to make algebra more natural and useful to students. I refer to the instruction model as the Idea-based Algebra (IDEAA). The basic ideas of this IDEAA model are 1) to combine children's own informal mathematics with scientific mathematics ("math math") and 2) to structure algebra content as a "map of big ideas", not as a traditional sequence of powers, polynomials, equations, and word problems. This research project is a kind of design process or design research. As such, this project has three, intertwined goals: research, design and pedagogical practice. I also assume three roles. As a researcher, I want to learn about learning and school algebra, its problems and possibilities. As a designer, I use research in the intervention to develop a shared artefact, the instruction model. In addition, I want to improve the practice through intervention and research. A design research like this is quite challenging. Its goals and means are intertwined and change in the research process. Theory emerges from the inquiry; it is not given a priori. The aim to improve instruction is normative, as one should take into account what "good" means in school algebra. An important part of my study is to work out these paradigmatic questions. The result of the study is threefold. The main result is the instruction model designed in the study. The second result is the theory that is developed of the teaching, learning and algebra. The third result is knowledge of the design process. The instruction model (IDEAA) is connected to four main features of good algebra education: 1) the situationality of learning, 2) learning as knowledge building, in which natural language and intuitive thinking work as "intermediaries", 3) the emergence and diversity of algebra, and 4) the development of high performance skills at any stage of instruction.

Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Wood procurement in the pressure of change.

Linear optimization model was used to calculate seven wood procurement scenarios for years 1990, 2000 and 2010. Productivity and cost functions for seven cutting, five terrain transport, three long distance transport and various work supervision and scaling methods were calculated from available work study reports. All method's base on Nordic cut to length system. Finland was divided in three parts for description of harvesting conditions. Twenty imaginary wood processing points and their wood procurement areas were created for these areas. The procurement systems, which consist of the harvesting conditions and work productivity functions, were described as a simulation model. In the LP-model the wood procurement system has to fulfil the volume and wood assortment requirements of processing points by minimizing the procurement cost. The model consists of 862 variables and 560 restrictions. Results show that it is economical to increase the mechanical work in harvesting. Cost increment alternatives effect only little on profitability of manual work. The areas of later thinnings and seed tree- and shelter wood cuttings increase on cost of first thinnings. In mechanized work one method, 10-tonne one grip harvester and forwarder, is gaining advantage among other methods. Working hours of forwarder are decreasing opposite to the harvester. There is only little need to increase the number of harvesters and trucks or their drivers from today's level. Quite large fluctuations in level of procurement and cost can be handled by constant number of machines, by alternating the number of season workers and by driving machines in two shifts. It is possible, if some environmental problems of large scale summer time harvesting can be solved.

An optimal local approximation algorithm for max-min linear programs

In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0 for nonnegative matrices A and C. We present a local algorithm (constant-time distributed algorithm) for approximating max-min LPs. The approximation ratio of our algorithm is the best possible for any local algorithm; there is a matching unconditional lower bound.

Local approximability of max-min and min-max linear programs

In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0. In a min-max LP, the objective is to minimise ρ subject to Ax ≤ ρ1, Cx ≥ 1, and x ≥ 0. The matrices A and C are nonnegative and sparse: each row ai of A has at most ΔI positive elements, and each row ck of C has at most ΔK positive elements. We study the approximability of max-min LPs and min-max LPs in a distributed setting; in particular, we focus on local algorithms (constant-time distributed algorithms). We show that for any ΔI ≥ 2, ΔK ≥ 2, and ε > 0 there exists a local algorithm that achieves the approximation ratio ΔI (1 − 1/ΔK) + ε. We also show that this result is the best possible: no local algorithm can achieve the approximation ratio ΔI (1 − 1/ΔK) for any ΔI ≥ 2 and ΔK ≥ 2.