Die vorliegende Arbeit widmet sich der Repräsentation und Verarbeitung arithmetischer Fakten. Dieser Bereich semantischen Wissens eignet sich unter anderem deshalb besonders gut als Forschungsgegenstand, weil nicht nur seine einzelne Bestandteile, sondern auch die Beziehungen dieser Bestandteile untereinander außergewöhnlich gut definierbar sind. Kognitive Modelle können also mit einem Grad an Präzision entwickelt werden, der in anderen Bereichen kaum je zu erreichen sein wird. Die meisten aktuellen Modelle stimmen darin überein, die Repräsentation arithmetischer Fakten als eine assoziative, netzwerkartig organisierte Struktur im deklarativen Gedächtnis zu beschreiben. Trotz dieser grundsätzlichen Übereinstimmung bleibt eine Reihe von Fragen offen. In den hier vorgestellten Untersuchungen werden solche offene Fragen in Hinsicht auf drei verschiedene Themenbereiche angegangen: 1) die neuroanatomischen Korrelate 2) Nachbarschaftskonsistenzeffekte bei der verbalen Produktion sowie 3) die automatische Aktivierung arithmetischer Fakten. In einer kombinierten fMRT- und Verhaltensstudie wurde beispielsweise der Frage nachgegangen, welche neurofunktionalen Entsprechungen es für den Erwerb arithmetischer Fakten bei Erwachsenen gibt. Den Ausgangspunkt für diese Untersuchung bildete das Triple-Code-Modell von Dehaene und Cohen, da es als einziges auch Aussagen über neuroanatomische Korrelate numerischer Leistungen macht. Das Triple-Code-Modell geht davon aus, dass zum Abruf arithmetischer Fakten eine „perisylvische“ Region der linken Hemisphäre unter Einbeziehung der Stammganglien sowie des Gyrus angularis nötig ist (Dehaene & Cohen, 1995; Dehaene & Cohen, 1997; Dehaene, Piazza, Pinel, & Cohen, 2003). In der aktuellen Studie sollten gesunde Erwachsene komplexe Multiplikationsaufgaben etwa eine Woche lang intensiv üben, so dass ihre Beantwortung immer mehr automatisiert erfolgt. Die Lösung dieser geübten Aufgaben sollte somit – im Gegensatz zu vergleichbaren ungeübten Aufgaben – immer stärker auf Faktenabruf als auf der Anwendung von Prozeduren und Strategien beruhen. Hingegen sollten ungeübte Aufgaben im Vergleich zu geübten höhere Anforderungen an exekutive Funktionen einschließlich des Arbeitsgedächtnisses stellen. Nach dem Training konnten die Teilnehmer – wie erwartet – geübte Aufgaben deutlich schneller und sicherer beantworten als ungeübte. Zusätzlich wurden sie auch im Magnetresonanztomografen untersucht. Dabei konnte zunächst bestätigt werden, dass das Lösen von Multiplikationsaufgaben allgemein von einem vorwiegend linkshemisphärischen Netzwerk frontaler und parietaler Areale unterstützt wird. Das wohl wichtigste Ergebnis ist jedoch eine Verschiebung der Hirnaktivierungen von eher frontalen Aktivierungsmustern zu einer eher parietalen Aktivierung und innerhalb des Parietallappens vom Sulcus intraparietalis zum Gyrus angularis bei den geübten im Vergleich zu den ungeübten Aufgaben. So wurde die zentrale Bedeutung von Arbeitsgedächtnis- und Planungsleistungen für komplexe ungeübte Rechenaufgaben erneut herausgestellt. Im Sinne des Triple-Code-Modells könnte die Verschiebung innerhalb des Parietallappens auf einen Wechsel von quantitätsbasierten Rechenleistungen (Sulcus intraparietalis) zu automatisiertem Faktenabruf (linker Gyrus angularis) hindeuten. Gibt es bei der verbalen Produktion arithmetischer Fakten Nachbarschaftskonsistenzeffekte ähnlich zu denen, wie sie auch in der Sprachverarbeitung beschrieben werden? Solche Effekte sind nach dem aktuellen „Dreiecksmodell“ von Verguts & Fias (2004) zur Repräsentation von Multiplikationsfakten erwartbar. Demzufolge sollten richtige Antworten leichter gegeben werden können, wenn sie Ziffern mit möglichst vielen semantisch nahen falschen Antworten gemeinsam haben. Möglicherweise sollten demnach aber auch falsche Antworten dann mit größerer Wahrscheinlichkeit produziert werden, wenn sie eine Ziffer mit der richtigen Antwort teilen. Nach dem Dreiecksmodell wäre darüber hinaus sogar der klassische Aufgabengrößeneffekt bei einfachen Multiplikationsaufgaben (Zbrodoff & Logan, 2004) auf die Konsistenzverhältnisse der richtigen Antwort mit semantisch benachbarten falschen Antworten zurückzuführen. In einer Reanalyse der Fehlerdaten von gesunden Probanden (Campbell, 1997) und einem Patienten (Domahs, Bartha, & Delazer, 2003) wurden tatsächlich Belege für das Vorhandensein von Zehnerkonsistenzeffekten beim Lösen einfacher Multiplikationsaufgaben gefunden. Die Versuchspersonen bzw. der Patient hatten solche falschen Antworten signifikant häufiger produziert, welche die gleiche Zehnerziffer wie das richtigen Ergebnisses aufwiesen, als ansonsten vergleichbare andere Fehler. Damit wird die Annahme unterstützt, dass die Zehner- und die Einerziffern zweistelliger Zahlen separate Repräsentationen aufweisen – bei der Multiplikation (Verguts & Fias, 2004) wie auch allgemein bei numerischer Verarbeitung (Nuerk, Weger, & Willmes, 2001; Nuerk & Willmes, 2005). Zusätzlich dazu wurde in einer Regressionsanalyse über die Fehlerzahlen auch erstmalig empirische Evidenz für die Hypothese vorgelegt, dass der klassische Aufgabengrößeneffekt beim Abruf von Multiplikationsfakten auf Zehnerkonsistenzeffekte zurückführbar ist: Obwohl die Aufgabengröße als erster Prädiktor in das Modell einging, wurde diese Variable wieder verworfen, sobald ein Maß für die Nachbarschaftskonsistenz der richtigen Antwort in das Modell aufgenommen wurde. Schließlich wurde in einer weiteren Studie die automatische Aktivierung von Multiplikationsfakten bei gesunden Probanden mit einer Zahlenidentifikationsaufgabe (Galfano, Rusconi, & Umilta, 2003; Lefevre, Bisanz, & Mrkonjic, 1988; Thibodeau, Lefevre, & Bisanz, 1996) untersucht. Dabei sollte erstmals die Frage beantwortet werden, wie sich die automatische Aktivierung der eigentlichen Multiplikationsergebnisse (Thibodeau et al., 1996) zur Aktivierung benachbarter falscher Antworten (Galfano et al., 2003) verhält. Ferner sollte durch die Präsentation mit verschiedenen SOAs der zeitliche Verlauf dieser Aktivierungen aufgeklärt werden. Die Ergebnisse dieser Studie können insgesamt als Evidenz für das Vorhandensein und die automatische, obligatorische Aktivierung eines Netzwerkes arithmetischer Fakten bei gesunden, gebildeten Erwachsenen gewertet werden, in dem die richtigen Produkte stärker mit den Faktoren assoziiert sind als benachbarte Produkte (Operandenfehler). Dabei führen Produkte kleiner Aufgaben zu einer stärkeren Interferenz als Produkte großer Aufgaben und Operandenfehler großer Aufgaben zu einer stärkeren Interferenz als Operandenfehler kleiner Aufgaben. Ein solches Aktivierungsmuster passt gut zu den Vorhersagen des Assoziationsverteilungsmodells von Siegler (Lemaire & Siegler, 1995; Siegler, 1988), bei dem kleine Aufgaben eine schmalgipflige Verteilung der Assoziationen um das richtige Ergebnis herum aufweisen, große Aufgaben jedoch eine breitgipflige Verteilung. Somit sollte die vorliegende Arbeit etwas mehr Licht in bislang weitgehend vernachlässigte Aspekte der Repräsentation und des Abrufs arithmetischer Fakten gebracht haben: Die neuronalen Korrelate ihres Erwerbs, die Konsequenzen ihrer Einbindung in das Stellenwertsystem mit der Basis 10 sowie die spezifischen Auswirkungen ihrer assoziativen semantischen Repräsentation auf ihre automatische Aktivierbarkeit. Literatur Campbell, J. I. (1997). On the relation between skilled performance of simple division and multiplication. Journal of Experimental Psychology: Learning, Memory, and Cognition, 23, 1140-1159. Dehaene, S. & Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83-120. Dehaene, S. & Cohen, L. (1997). Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33, 219-250. Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487-506. Domahs, F., Bartha, L., & Delazer, M. (2003). Rehabilitation of arithmetic abilities: Different intervention strategies for multiplication. Brain and Language, 87, 165-166. Galfano, G., Rusconi, E., & Umilta, C. (2003). Automatic activation of multiplication facts: evidence from the nodes adjacent to the product. Quarterly Journal of Experimental Psychology A, 56, 31-61. Lefevre, J. A., Bisanz, J., & Mrkonjic, L. (1988). Cognitive arithmetic: evidence for obligatory activation of arithmetic facts. Memory and Cognition, 16, 45-53. Lemaire, P. & Siegler, R. S. (1995). Four aspects of strategic change: contributions to children's learning of multiplication. Journal of Experimental Psychology: General, 124, 83-97. Nuerk, H. C., Weger, U., & Willmes, K. (2001). Decade breaks in the mental number line? Putting the tens and units back in different bins. Cognition, 82, B25-B33. Nuerk, H. C. & Willmes, K. (2005). On the magnitude representations of two-digit numbers. Psychology Science, 47, 52-72. Siegler, R. S. (1988). Strategy choice procedures and the development of multiplication skill. Journal of Experimental Psychology: General, 117, 258-275. Thibodeau, M. H., Lefevre, J. A., & Bisanz, J. (1996). The extension of the interference effect to multiplication. Canadian Journal of Experimental Psychology, 50, 393-396. Verguts, T. & Fias, W. (2004). Neighborhood Effects in Mental Arithmetic. Psychology Science. Zbrodoff, N. J. & Logan, G. D. (2004). What everyone finds: The problem-size effect. In J. I. D. Campbell (Hrsg.), Handbook of Mathematical Cognition (pp.331-345). New York, NY: Psychology Press. / The present thesis deals with the representation and processing of arithmetic facts. This domain of semantic knowledge has gained a substantial amount of interest as its components as well as their interrelations are well specified. Thus, cognitive models can be developed with a degree of precision, which cannot be reached in many other domains. Most recent models agree that arithmetic facts are represented in an associative, network-like structure in declarative memory. Despite this general agreement a lot of issues still remain unresolved. The open questions tackled in the present work address three different aspects of arithmetic facts: 1) their neuro-anatomical correlates, 2) neighbourhood consistency effects in their verbal production and 3) their automatic activation. In a combined behavioural and fMRI study the neurofunctional correlates of the acquisition of arithmetic facts in adults were examined. This research was based on the Triple-Code-Model of Dehaene and Cohen, the only recent model which makes explicit assumptions on neuroanatomical correlates of numerical abilities. The Triple-Code-Model assumes that a “perisylvian” region in the left hemisphere including the basal ganglia and the Angular Gyrus is involved in the retrieval of arithmetic facts (Dehaene & Cohen, 1995; Dehaene & Cohen, 1997; Dehaene, Piazza, Pinel, & Cohen, 2003). In the present study healthy adults were asked to train complex multiplication problems extensively during one week. Thus, these problems could be solved more and more automatically. It was reasoned that answering these trained problems should more and more rely on the retrieval of facts from declarative memory, whereas answering untrained problems should rely on the application of strategies and procedures, which impose high demands on executive functions including working memory. After the training was finished, participants – as expected – could solve trained problems faster and more accurately than non-trained problems. Participants were also submitted to a functional magnetic resonance imaging examination. In general, this examination added to the evidence for a mainly left hemispheric fronto-parietal network being involved in mental multiplication. Crucially, comparing trained with non-trained problems a shift of activation from frontal to more parietal regions was observed. Thus, the central role of central executive and working memory for complex calculation was highlighted. Moreover, a shift of activation from the Intraparietal Sulcus to the Angular Gyrus took place within the parietal lobe. According to the Triple-Code-Model, this shift may be interpreted to indicate a strategy change from quantity based calculation, relying on the Intraparietal Sulcus, to fact retrieval, relying on the left Angular Gyrus. Are there neighbourhood consistency effects in the verbal production of arithmetic facts similar to what has been described for language production? According to the “Triangle Model” of simple multiplication, proposed by Verguts & Fias (2004), such effects can be expected. According to this model corrects answers can be given more easily if they share digits with many semantically close wrong answers. Moreover, it can be assumed that wrong answers, too, are more likely to be produced if they share a digit with the correct result. In addition to this, the Triangle Model also states that the classical problem size effect in simple multiplication (Zbrodoff & Logan, 2004) can be drawn back to neighbourhood consistency between the correct result and semantically close wrong answers. In fact, a re-analysis of error data from a sample of healthy young adults (Campbell, 1997) and a patient with acalculia (Domahs, Bartha, & Delazer, 2003) provided evidence for the existence of decade consistency effects in the verbal production of multiplication results. Healthy participants and the patient produced significantly more wrong answers which shared the decade digit with the correct result than otherwise comparable wrong answers. This result supports the assumption of separate representations of decade and unit digits in two-digit numbers in multiplication (Verguts & Fias, 2004) and in number processing in general (Nuerk, Weger, & Willmes, 2001; Nuerk & Willmes, 2005). Moreover, an additional regression analysis on the error rates provided first empirical evidence for the hypothesis that the classical problem size effect in the retrieval of multiplication facts may be an artefact of neighbourhood consistency: Although problem size was the first variable to enter the model, it was excluded from the model once a measure for neighbourhood consistency was included. Finally, in a further study the automatic activation of multiplication facts was examined in a number matching task (Galfano, Rusconi, & Umilta, 2003; Lefevre, Bisanz, & Mrkonjic, 1988; Thibodeau, Lefevre, & Bisanz, 1996). This experiment addressed the question how the automatic activation of actual multiplication results (Thibodeau et al., 1996) relates to the activation of semantically close wrong answers (Galfano et al., 2003). Furthermore, using different SOAs the temporal properties of these activations should be disclosed. In general, the results of this study provide evidence for an obligatory and automatic activation of a network of arithmetic facts in healthy educated adults in which correct results are stronger associated with the operands than semantically related wrong answers. Crucially, products of small problems lead to stronger interference effects than products of larger problems while operand errors of large problems lead to stronger interference effects than operand errors of small problems. Such a pattern of activation is in line with predictions of Siegler’s Distribution of Associations Model (Lemaire & Siegler, 1995; Siegler, 1988) which assumes a more peaked distribution of associations between operands and potential results for small compared to large multiplication problems. In sum, the present thesis should shed some light into largely ignored aspects of arithmetic fact retrieval: The neural correlates of its acquisition, the consequences of its implementation in the base 10 place value system, as well as the specific effects of its semantic representation for automatic activation of correct multiplication facts and related results. References Campbell, J. I. (1997). On the relation between skilled performance of simple division and multiplication. Journal of Experimental Psychology: Learning, Memory, and Cognition, 23, 1140-1159. Dehaene, S. & Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83-120. Dehaene, S. & Cohen, L. (1997). Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33, 219-250. Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487-506. Domahs, F., Bartha, L., & Delazer, M. (2003). Rehabilitation of arithmetic abilities: Different intervention strategies for multiplication. Brain and Language, 87, 165-166. Galfano, G., Rusconi, E., & Umilta, C. (2003). Automatic activation of multiplication facts: evidence from the nodes adjacent to the product. Quarterly Journal of Experimental Psychology A, 56, 31-61. Lefevre, J. A., Bisanz, J., & Mrkonjic, L. (1988). Cognitive arithmetic: evidence for obligatory activation of arithmetic facts. Memory and Cognition, 16, 45-53. Lemaire, P. & Siegler, R. S. (1995). Four aspects of strategic change: contributions to children's learning of multiplication. Journal of Experimental Psychology: General, 124, 83-97. Nuerk, H. C., Weger, U., & Willmes, K. (2001). Decade breaks in the mental number line? Putting the tens and units back in different bins. Cognition, 82, B25-B33. Nuerk, H. C. & Willmes, K. (2005). On the magnitude representations of two-digit numbers. Psychology Science, 47, 52-72. Siegler, R. S. (1988). Strategy choice procedures and the development of multiplication skill. Journal of Experimental Psychology: General, 117, 258-275. Thibodeau, M. H., Lefevre, J. A., & Bisanz, J. (1996). The extension of the interference effect to multiplication. Canadian Journal of Experimental Psychology, 50, 393-396. Verguts, T. & Fias, W. (2004). Neighborhood Effects in Mental Arithmetic. Psychology Science. Zbrodoff, N. J. & Logan, G. D. (2004). What everyone finds: The problem-size effect. In J. I. D. Campbell (Ed.), Handbook of Mathematical Cognition (pp.331-345). New York, NY: Psychology Press.

The main issue in this thesis is to minimize the number of operations and the energy consumption per operation for the computation (arithmetic operation) part of DSP circuits, such as Finite Impulse Response Filters (FIR), Discrete Cosine Transform (DCT), and Discrete Fourier Transform (DFT) etc. More specific, the focus is on the elimination of most frequent common sub-expression (CSE) in binary, Canonic Sign Digit (CSD), Twos Complement or Sign Digit representation of the coefficients of non-recursive multiple input multiple output (MIMO)  FIR system , which can be realized using shift-and-add based operations only. The possibilities to reduce the complexity i.e. the chip area, and the energy consumption have been investigated. We have proposed an algorithm which finds the most common sub expression in the binary/CSD/Twos Complement/Sign Digit representation of coefficients of non-recursive MIMO multiplier less FIR systems. We have implemented the algorithm in MATLAB. Also we have proposed different tie-breakers for the selection of most frequent common sub-expression, which will affect the complexity (Area and Power consumption) of the overall system. One choice (tie breaker) is to select the pattern (if there is a tie for the most frequent pattern) which will result in minimum number of delay elements and hence the area of the overall system will be reduced. Another tie-breaker is to choose the pattern which will result in minimum adder depth (the number of cascaded adders). Minimum adder depth will result in least number of glitches which is the main factor for the power consumption in MIMO multiplier less FIR systems. Switching activity will be increased when glitches are propagated to subsequent adders (which occur if adder depth is high). As the power consumption is proportional to the switching activity (glitches) hence we will use the sub-expression which will result in lowest adder depth for the overall system.

Benchmarking of DSP kernel algorithms was conducted in the thesis on a DSP processor for teaching in the course TESA26 in the department of Electrical Engineering. It includes benchmarking on cycle count and memory usage. The goal of the thesis is to evaluate the quality of a single MAC DSP instruction set and provide suggestions for further improvement in instruction set architecture accordingly. The scope of the thesis is limited to benchmark the processor only based on assembly coding. The quality check of compiler is not included. The method of the benchmarking was proposed by BDTI, Berkeley Design Technology Incorporations, which is the general methodology used in world wide DSP industry. Proposals on assembly instruction set improvements include the enhancement of FFT and DCT. The cycle cost of the new FFT benchmark based on the proposal was XX% lower, showing that the proposal was right and qualified. Results also show that the proposal promotes the cycle cost score for matrix computing, especially matrix multiplication. The benchmark results were compared with general scores of single MAC DSP processors offered by BDTI.

Tidigare forskning visar att bråk är ett område där många elever har problem. Syftet med den här studien är att studera gymnasieelevers matematiska kunskaper i multiplikation och division av bråk. Elevernas kunskaper studerades utifrån en konstruktivistisk syn på kunskap och med procedurell och konceptuell kunskap som analysverktyg. 61 elever från kursen Matematik A har löst totalt 10 uppgifter med multiplikation och division av bråk. 7 av eleverna intervjuades dessutom för att få en bättre uppfattning om deras kunskaper. Elevernas kunskaper kategoriserades sedan utifrån procedurella- och konceptuella kvaliteter. Resultatet visar att eleverna främst använder algoritmer för att lösa uppgifterna men även andra strategier som till exempel att skriva bråken som decimaler förekommer. Elevernas kunskap i multiplikation och division av bråk är av procedurell karaktär med fokus på att komma ihåg algoritmer för att lösa uppgifterna. Elevernas konceptuella kunskaper i bråkräkning är överlag inte lika utvecklade. Det framkommer genom att eleverna visar på svårigheter att lösa uppgifter i vissa sammanhang, bristande förståelse för betydelsen av beräkningarna och för varför de olika algoritmerna fungerar. / Earlier researches show that fraction is an area where many students have problems. The aim with this essay is to study upper secondary school students’ mathematical knowledge in multiplication and division of fraction. The students’ knowledge will be studied from a constructivistic perspective of knowledge and with procedural and conceptual knowledge as an instrument for the analysis. 61 students from the course Matematik A have solved totally 10 mathematical problems with multiplication and division of fraction. 7 of the students were furthermore interviewed to get a better understanding of their knowledge. The students’ knowledge were then categorized from procedurally and conceptually qualities. The result shows that the students primarily use algorithms to solve the problems but also other strategies as example to write the fraction as decimals occur. The students’ knowledge in multiplication and division of fraction is of procedural character with focus on remembering the algorithms for the different types of problems. The students conceptually knowledge in fraction arithmetic is overall not fully developed. It comes out by the students difficulties to solve problems in certain context, deficient understanding of the meaning of the calculations and why the different algorithms work.

IDCT (Inverse Discrete Cosine Transform) is a common algorithm being used with image and sound decompression. The algorithm is a Fourier related transform which can occur in many different types like, one-dimensional, two-dimensional, three-dimensional and many more. The goal with this thesis is to create a fast and low effect version of two-dimensional IDCT algorithm, where techniques as multiple-constant multiplication and subexpression sharing plus bit-serial and bit-parallel arithmetic are used. The result is a hardware implementation with power consumption at 19,56 mW.

Computations with polynomials are at the heart of any computer algebra system and also have many applications in engineering, coding theory, and cryptography. Generally speaking, the low-level polynomial computations of interest can be classified as arithmetic operations, algebraic computations, and inverse symbolic problems. New algorithms are presented in all these areas which improve on the state of the art in both theoretical and practical performance. Traditionally, polynomials may be represented in a computer in one of two ways: as a "dense" array of all possible coefficients up to the polynomial's degree, or as a "sparse" list of coefficient-exponent tuples. In the latter case, zero terms are not explicitly written, giving a potentially more compact representation. In the area of arithmetic operations, new algorithms are presented for the multiplication of dense polynomials. These have the same asymptotic time cost of the fastest existing approaches, but reduce the intermediate storage required from linear in the size of the input to a constant amount. Two different algorithms for so-called "adaptive" multiplication are also presented which effectively provide a gradient between existing sparse and dense algorithms, giving a large improvement in many cases while never performing significantly worse than the best existing approaches. Algebraic computations on sparse polynomials are considered as well. The first known polynomial-time algorithm to detect when a sparse polynomial is a perfect power is presented, along with two different approaches to computing the perfect power factorization. Inverse symbolic problems are those for which the challenge is to compute a symbolic mathematical representation of a program or "black box". First, new algorithms are presented which improve the complexity of interpolation for sparse polynomials with coefficients in finite fields or approximate complex numbers. Second, the first polynomial-time algorithm for the more general problem of sparsest-shift interpolation is presented. The practical performance of all these algorithms is demonstrated with implementations in a high-performance library and compared to existing software and previous techniques.

Flera undersökningar har visat att svenska elevers kunskaper inom områdena taluppfattning och aritmetik har blivit sämre. I denna uppsats står därför taluppfattning, med multiplikation som utgångspunkt, i fokus. Syftet med det här arbetet har varit att analysera hur olika läromedels framställning av räknesättet multiplikation samt strukturering av inlärningen av de grundläggande multiplikationskombinationerna kan påverka elevers möjlighet att utveckla god taluppfattning. För att svara på vårt syfte har vi gjort en läromedelsanalys av fem olika läromedel i matematik, avsedda för åk 1-3. Resultatet pekar på att några av de analyserade läromedlen framställer multiplikation på ett begränsat sätt vilket kan antas ha negativ inverkan på elevers möjlighet att utveckla förståelse för räknesättet multiplikation och därmed också på taluppfattningen. Resultatet pekar också på att flera av läromedlen, genom sitt sätt att lyfta fram tankeformer och samband, strukturerar inlärningen av multiplikationskombinationerna så att elevers möjlighet att utveckla taluppfattning gynnas. / Several studies have shown that Swedish students' knowledge of number sense and arithmetic have been deteriorating. In view of this number sense, with multiplication as a basis, is the focus in this composition. The purpose of this work has been to analyze how different textbooks description of multiplication and structure of learning the basic multiplication combinations can influence students' ability to develop number sense. To answer our purpose we made a textbook analysis of five textbooks in mathematics, for grade 1-3. The results indicate that some of the analyzed textbooks describe multiplication in a limited way which one can assume have negative impact on students' ability to develop understanding of multiplication and so developing number sense. The results also indicate that several of the textbooks, by the way they emphasize mental strategies and connections between numbers, structure the learning of the basic multiplication combinations in a way that support students' opportunity to develop number sense.

An important advantage of Elliptic Curve Cryptosystem (ECC) is the shorter key length in public key cryptographic systems. It can provide adequate security when the bit length over than 160 bits. Therefore, it has become a popular system in recent years. Scalar multiplication also called point multiplication is the core operation in ECC. In this thesis, we propose the ECC architectures of two different irreducible polynomial versions that are trinomial in GF(2167) and pentanomial in GF(2163). These architectures are based on Montgomery point multiplication with projective coordinate. We use polynomial basis representation for finite field arithmetic. All adopted multiplication, square and add operations over binary field can be completed within one clock cycle, and the critical path lies on multiplication. In addition, we use Itoh-Tsujii algorithm combined with addition chain, to execute binary inversion through using iterative binary square and multiplication. Because the double and add operations in point multiplication need to run many iterations, the execution time in overall design will be decreased if we can improve this partition. We propose two ways to improve the performance of point multiplication. The first way is Minus Cycle Version. In this version, we reschedule the double and add operations according to point multiplication algorithm. When the clock cycle time (i.e., critical path) of multiplication is longer than that of add and square, this method will be useful in improving performance. The second way is Pipeline Version. It speeds up the multiplication operations by executing them in pipeline, leading to shorter clock cycle time. For the hardware implementation, TSMC 0.13um library is employed and all modules are organized in a hierarchy structure. The implementation result shows that the proposed 167-bit Minus Cycle Version requires 156.4K gates, and the execution time of point multiplication is 2.34us and the maximum speed is 591.7Mhz. Moreover, we compare the Area x Time (AT) value of proposed architectures with other relative work. The results exhibit that proposed 167-bit Minus Cycle Version is the best one and it can save up to 38% A T value than traditional one.

In digital signal processing and multimedia applications, floating-point(FP) multiplication and addition are the most commonly used operations. In addition, FP multiplication operations are frequently followed by the FP addition operations. Therefore, in order to achieve high performance and low cost, multiplication and addition are usually combined into a single unit, known as the FP Multiply-Add Fused (MAF). On the other hand, the mobile devices nowadays are rapidly developing. For this kind of devices, performance and power sustainability have to become the major trend in the research area. As a result, the mechanisms to reduce energy consumption become more important. Therefore, we propose a multi-mode FP MAF based on the concept of iterative multiplication and truncated addition, to achieve different operating modes with different errors. This MAF, with a total of seven modes, includes three modes for the FP multiply-accumulate operations, two modes for single FP multiplication operation and single FP addition operation, respectively. FP multiply-accumulate operations provide three modes to user, and this three modes have 0%, 0.328% and 1.107% of error. The 0% error is the same with the standard IEEE754 single-precision FP Multiply-Add Fused operations. For FP multiplication and FP addition operations, the proposed MAF allows users to choose two kinds of error modes, which are 0%, 0.328% error for FP multiplication and 0%, 0.781% error for FP addition. The 0% error is the same with the standard IEEE754 single-precision floating-point operations. When compared with the standard IEEE754 single-precision FP MAF, the proposed multi-mode FP MAF architecture has 4.5% less area and increase about 22% delay to achieve the effect of multi-mode. To demonstrate the power efficiency of proposed FP MAF, it is used to perform the operations of FP MAF, FP multiplication, and FP addition in the application of RGB to YUV format conversion. Experimental results show that, the proposed multi-mode FP MAF can significantly reduce power consumption when the modes with error are adopted.

Graphics processing unit (GPU) designs usually adopts various computer architecture techniques to boost the computation speed, including single-instruction multiple data (SIMD), very-long-instruction word (VLIW), multi-threading, and/or multi-core. In OpenGL ES 2.0, user programmable vertex shader (VS) hardware unit can be designed using vectored SIMD computation unit so that it can efficiently compute the matrix-vector multiplication, one of the key operations in vertex transformation. Recently, high-performance GPU, such as Telsa series from nVidia, is designed with many-core architectures with each core responsible for scalar operations. The intention is to allow for efficient execution of general-purpose computations in addition to the specialized graphics computations. In this thesis, we design a scalar-based multi-threaded GPU design that is composed of four scalar processors, one special-function unit, and can execute multi-threaded instructions. We use the example of vertex transformation to demonstrate execution efficiency of the scalar-based multi-threaded GPU. We also make comparison with the vector-based SIMD GPU.