Dissertations / Theses on the topic 'Multiplication table'

Syftet med denna studie är att belysa hur lärare beskriver sin undervisning av multiplikation i årskurs 1−3 och årskurs 4−6 när det kommer till introduktion, fortlöpande undervisning och tabellträning. Kvalitativa intervjuer med sex lärare har genomförts för att undersöka vilka mål de intervjuade lärarna har med sin multiplikationsundervisning samt hur lärarna beskriver innehållet i sin multiplikationsundervisning. Bakgrunden är att lärares uppfattning om vad multiplikation är samt vad multiplikationsundervisningen ska innehålla påverkar vilka lärandemöjligheter eleverna får. Detta innefattar val av förklaringsmodeller, arbetssätt samt lektionsinnehåll, vilket i högsta grad påverkar elevers förståelseutveckling av multiplikationsbegreppet. Att svenska lärare typiskt sett baserar sin undervisning på läromedel lyfts av forskning som en orsak till att svenska elevers taluppfattning och kunskap om aritmetik är svag. Lärare behöver därför komplettera läromedlens framställning av multiplikation i undervisningen. Studiens resultat visar att lärarnas mål med undervisningen berör områden som enligt läroplan och forskning är viktiga för elevers begreppsförståelse och procedurkunskap, men att viktiga bitar i undervisning verkar saknas. Detta berör undervisning om multiplikativa förklaringsmodeller, räknelagar och begrepp kopplade till multiplikation. Lärarnas undervisning om de grundläggande multiplikationstabellerna, där både strategier för att härleda tabellfakta samt drillövningar av dessa uppges ingå, verkar ligga i fas med vad forskning lyfter fram som viktigt för att uppnå automatisering av tabellerna.<br>The purpose of this study is to illustrate how teachers describe their multiplication teaching in grades 1−3 and 4−6 when it comes to the introduction, continuous teaching and table training. Qualitative interviews with six teachers have been conducted to examine what objectives the interviewed teachers have with their multiplication teaching and how they describe the contents of their multiplication teaching. The reason behind is that teachers’ perception of what multiplication means and their thoughts on what multiplication teaching should cover affects the learning opportunities pupils receive. This includes teachers’ choice of explanatory models, methods and lesson content which highly affects the pupils’ development of understanding regarding the concept of multiplication. The fact that Swedish teachers typically base their teaching on textbooks is indicated by research to be a contributing factor why Swedish pupils’ number sense and understanding of arithmetic is weak. Teachers therefore need to complement the presentations that textbooks contain regarding multiplication in teaching. The result of this study shows that teachers’ teaching objectives affects areas that the curriculum and research highlights as important for pupils’ conceptual understanding and procedural knowledge, but that important pieces seems to be missing in their teaching. These concerns the teaching about the multiplicative models of explanation, mathematical properties and concepts related to multiplication. However, teachers’ teaching about the basic multiplication facts, where both strategies to derive facts and drill exercises of facts is said to be included, seems to correspond largely with what research highlights as important in achieving automaticity in multiplication facts.

O presente trabalho consiste numa pesquisa qualitativa sobre o ensino e a aprendizagem de matemática, cujo objetivo foi o de propor uma metodologia, fundamentada no construtivismo piagetiano, para ensinar multiplicação nos anos iniciais do Ensino Fundamental. A observação de que o ensino desse conteúdo, em muitas escolas, privilegia a memorização da tabuada e a aplicação de algoritmos, em detrimento da compreensão, mobilizou uma investigação sobre como organizar o ensino dessa noção privilegiando o desenvolvimento da racionalidade e a aquisição de competências que transcendem o âmbito da matemática. Para isso foi realizada uma intervenção de ensino, planejada e desenvolvida pela própria pesquisadora, ao longo de um semestre (21 aulas) em uma turma de terceira série de uma escola da rede municipal de Curitiba. A multiplicação foi explorada em problemas de proporcionalidade simples (situações de correspondência um-para-muitos e de arranjo retangular). A divisão, por ser sua operação inversa, foi explorada em algumas atividades e dados relativos a essa operação foram incorporados ao estudo. As aulas foram gravadas em vídeo e transcritas em diários enriquecidos com observações e comentários da pesquisadora os quais, junto com as produções dos alunos e relatórios feitos pela professora da turma, compuseram nossa base de dados. Na análise buscamos indicativos da ocorrência de uma interação construtiva, caracterizada por progressos nos seguintes âmbitos: relações intelectuais (compreensão das operações aritméticas em questão pelas crianças); relações sociais/morais (conquistas que transcendem o conteúdo matemático); relações didáticas (efeitos sobre o próprio processo interventivo). Os resultados apontam para uma interação de qualidade construtiva uma vez que a intervenção teve efeito de aperfeiçoamento sobre os sujeitos envolvidos. Verificamos a substituição progressiva de estratégias de contagem por estratégias de cálculo, aquisição de competências aritméticas e interações entre as crianças (indícios de uma relação de cooperação). Apontamos ainda para mudanças de atitudes dos alunos no que se refere às seguintes questões: envolvimento nas atividades propostas, relação com a matemática, forma de tratar os problemas apresentados, comunicação e expressão em sala de aula. No âmbito didático destacamos o tratamento dispensado ao erro, usado como estratégia didática, o papel interventivo que a avaliação exerceu no processo de ensino e a importância da escrita para a reflexão do professor sobre a sua própria prática.<br>The present work consists of a qualitative research in education and the mathematics learning that objective was to propose a methodology based on Piaget´s constructivism, to teach multiplication in the early years of Eleentary School. The observation that shows how this content has been treated, in many schools, as a memorization and application matter, rather than a comprehension issue, directed to an investigation how about organize the teaching of this matter prioritizing the development of racionality and the acquisition of competences that go beyond matheatics. For that, a planned learning intervention was developed and made by the researcher herself over a period of one semester (21 classes) in a third grade public school from Curitiba. The multiplication was explored in simple proportionality problems (in one-tomany correspondence and rectangular array problems). The division, as it´s the inverse, was also explored in some activities and the data collected was also included in this study. The classes were recorded in video and a transcript enriched by observations and comments from the the researcher was produced. The transcript, the student´s works and daily reports done by the teacher form our database. In the analysis, we look for indications of a constructive interaction, characterized by progresses in the following areas: intellectual relationships (comprehension of the arithmetical operations used); social/moral relationships (acquisition of non mathematical aptitudes); didactic relationships (effects on the intervention process itself). The results show a constructive quality interaction as the intervention had a development effect on the subjects involved. We verified the progressive substitution of the counting strategies for calculation strategies, the acquisition of arithmetical aptitudes and interaction between the students (signs of a cooperative relationship). Also some attitudes changes were observed in the students regarding the following matters: the engagement in the activities, the relationship with mathematics, the approach on the problems, communication and expressionat the classroom. In the didactic field, we highlight the treatment regarding the errors, used as a didatic strategy, the interfered paper the avaliation exerced to the teaching process and the importance of writing for the teacher´s reflection about its practice.

The maximum subarray problem is to find the array portion that maximizes the sum of array elements in it. This problem was first introduced by Grenander and brought to computer science by Bentley in 1984. This problem has been branched out into other problems based on their characteristics. k-overlapping maximum subarray problem where the overlapping solutions are allowed, and k-disjoint maximum subarray problem where all the solutions are disjoint from each other are those. For k-overlapping maximum subarray problems, significant improvement have been made since the problem was first introduced. For k-disjoint maximum subarrsy, Ruzzo and Tompa gave an O(n) time solution for one-dimension. This solution is, however, difficult to extend to two-dimensions. While a trivial solution of O(kn^3) time is easily obtainable for two-dimensions, little study has been undertaken to better this. This paper introduces a faster algorithm for the k-disjoint maximum sub-array problem under the conventional RAM model, based on distance matrix multiplication. Also, DMM reuse technique is introduced for the maximum subarray problem based on recursion for space optimization.

De plus en plus de constructeurs proposent des accélérateurs de calculs à base de circuits reconfigurables FPGA, cette technologie présentant bien plus de souplesse que le microprocesseur. Valoriser cette flexibilité dans le domaine de l'accélération de calcul flottant en utilisant les langages de description de circuits classiques (VHDL ou Verilog) reste toutefois très difficile, voire impossible parfois. Cette thèse a contribué au développement du logiciel FloPoCo, qui offre aux utilisateurs familiers avec VHDL un cadre C++ de description d'opérateurs arithmétiques génériques adapté au calcul reconfigurable. Ce cadre distingue explicitement la fonctionnalité combinatoire d'un opérateur, et la problématique de son pipeline pour une précision, une fréquence et un FPGA cible donnés. Afin de pouvoir utiliser FloPoCo pour concevoir des opérateurs haute performance en virgule flottante, il a fallu d'abord concevoir des blocs de bases optimisés. Nous avons d'abord développé des additionneurs pipelinés autour des lignes de propagation de retenue rapides, puis, à l'aide de techniques de pavages, nous avons conçu de gros multiplieurs, possiblement tronqués, utilisant des petits multiplieurs. L'évaluation de fonctions élémentaires en flottant implique souvent l'évaluation en virgule fixe d'une fonction. Nous présentons un opérateur générique de FloPoCo qui prend en entrée l'expression de la fonction à évaluer, avec ses précisions d'entrée et de sortie, et construit un évaluateur polynomial optimisé de cette fonction. Ce bloc de base a permis de développer des opérateurs en virgule flottante pour la racine carrée et l'exponentielle qui améliorent considérablement l'état de l'art. Nous avons aussi travaillé sur des techniques de compilation avancée pour adapter l'exécution d'un code C aux pipelines flexibles de nos opérateurs. FloPoCo a pu ainsi être utilisé pour implanter sur FPGA des applications complètes.

Tidigare forskning har visat att elever har bristande kunskaper i multiplikationstabellerna. Att automatisera tabellerna ger eleverna goda förutsättningar inför övriga matematikområden. Syftet med studien var att skapa kunskap om lärares uppfattning till automatisering av multiplikationstabellerna samt att få kunskap om hur många av lärarnas elever som hade automatiserat tabellerna. Syftet var även att ta reda på vilka metoder lärare använder för att stötta eleverna i detta. Studien genomfördes med enkät som datainsamlingsinstrument och resultatet från enkäten följdes upp med fokusgruppsintervjuer. Resultatet av studien visade att många lärare upplever automatisering av multiplikationstabellerna som viktigt eftersom det underlättar för eleverna inför övriga matematikområden samt att det avlastar elevernas arbetsminne. Metoderna som flest lärare använde i sin undervisning var digitala undervisningsplattformar samt olika arbetsblad som drillträning av tabellerna.<br>Previous research had shown that students lacked knowledge in the multiplication tables. Automating the tables gives students good conditions to succeed in other areas of mathematics. The purpose of this study was to create knowledge about teacher’s perceptions of automating the multiplication tables and to examine how many of the pupils had automated the tables. The purpose was also to identify what methods teachers use to support students in multiplication table automation. The study was conducted with a survey and the results were followed up with focus group interviews. The results showed that many teachers value the automation of the multiplication tables because it prepares the students for other mathematics areas and it relieves the student’s working memory. The methods most teachers used in their teaching were digital teaching platforms and various worksheets such as drill training of the tables.

La thèse s'intéresse à l'accompagnement familial des apprentissages scolaires en mathématiques, mais aussi et surtout à l'organisation non discriminante de ses conditions. La " culture didactique " partagée dans notre société s'adapte de moins en moins aux régulations de la scolarité obligatoire. En effet, en se focalisant sur le repérage des difficultés individuelles et en encourageant les interventions précoces à l'extérieur de l'institution d'enseignement, elle transforme les aléas " ordinaires " de l'apprentissage en dysfonctionnements. Certaines tentatives d'amélioration insistent sur l'information et la communication entre école et parents. Or les discours éloignent souvent de la réalité des actions. Les " exercices à faire à la maison ", en transmettant des comportements, jouent un rôle complémentaire important. Certes, ils font rapidement surgir les divergences, parce qu'ils rendent visibles les contre-performances des élèves, et suggèrent toutes sortes de rectifications. Les devoirs sont par conséquent souvent accusés d'introduire des disparités et de pertuber les relations entre protagonistes. La thèse réexamine ce point de vue, en étudiant d'autres formes d'étude, qui s'ajusteraient mieux aux besoins des institutions didactiques. Pour simplifier la circulation des savoirs mathématiques les plus fréquemment utilisés, la société a mis en place des instruments culturels. Mais certains ont été détournés de leur fonction, ce qui a rompu des équilibres didactiques essentiels. La récitation des tables de multiplication fournit un exemple paradigmatique de la dénégation des transpositions. Les régressions métadidactiques ont en effet lentement modifié une ancienne répartition des tâches entre institutions, jusqu'à dédidactifier tout un pan de l'enseignement du calcul. La thèse éclaire la compréhension de ces phénomènes à l'aide de la Théorie des Situations Didactiques. Elle propose un nouveau concept pour une ingénierie spécifique de l'entraînement et de la familiarisation des élèves avec les connaisances les plus fondamentales : les assortiments didactiques.

碩士<br>義守大學<br>資訊工程學系<br>101<br>This thesis presents the look-up table method and Horner rule to speed up finite field multiplication in the standard basis. The polynomial can be simplified by Horner rule, and dividing into many terms. The many terms make a look-up table which is used to calculate the permutations and combinations of the results. The method can reduce the number of shift and bitwise XOR operation. Therefore, the purposes of accelerating multiplication can be applied in a larger finite field operation, such as Elliptic curve cryptography.

碩士<br>逢甲大學<br>產業研發碩士班<br>101<br>With the renovating development of the network technology, the transmission of information has become more convenient and faster; therefore, in the process of network information transmission, information encryption has to be indispensable. Moreover, because of the popularization of the smart phones, how to effectively proceed with information encryption has become a very important issue. RSA is a public key encryption and decryption algorithm which provides rather high security and great popularization, in which the RSA is the modular exponentiation. Because the computing power has become stronger and stronger, the encryption key length has also become longer and longer, which creates a very big problem, that is, the process of encryption has become very difficult and very slow. Therefore, based on the features of the RSA modular exponentiation, this paper presents a set of new hardware architecture to solve the problem of computation time. First, we choose scalable look-up table modular multiplication to serve as kernel unit and improve it. Furthermore, by means of the features of data reuse, we can effectively reduce computation times and further achieve the effect of speeding up the modular exponentiation.

Undervisning om multiplikationstabellen består främst av lärande genom repetition eller strategi. Syftet med litteraturstudien är att utifrån forskning undersöka och jämföra inlärning av multiplikationstabellen genom strategi- och repetitionsbaserad undervisning. Studien avgränsas till elever i årskurs 1 till 6. Materialet samlades in genom en iterativ sökning på åtta databaser för vetenskapliga publikationer. Det insamlade materialet bestod av sjutton vetenskapliga publikationer. Resultatet visade att både lärande genom repetition och lärande genom strategi möjliggör för elever att automatisera multiplikationstabellen. Resultatet visade också att lärande genom strategi gör att elever kan använda sina kunskaper i nya matematiska sammanhang. En nackdel med lärande genom repetition är att det är svårt att lära sig multiplikationstabellen utan att använda mönster och samband. Strategi- och repetitionsbaserad undervisning leder till olika felsvar. Felsvaren förklaras bland annat med att eleven övat för lite eller att de har en bristande taluppfattning. Forskningen presenterar en stadieteori som beskriver lärandet av multiplikationstabellen och två olika teorier som beskriver hur multiplikationskombinationer memoreras. Kombinationerna memoreras som enskilda enheter eller som system av sammankopplade erfarenheter.<br>Teaching of the multiplication table mainly consists of learning through drill or learning through strategy. The aim of this literature study is to investigate and compare the learning of the multiplication table through strategy and drill. The study is an analysis of previous research and limited to research on pupils in grade 1 to 6. The material was collected through an iterative search in eight databases of scientific literature. The retrieved material contained seventeen scientific publications. The analysis showed that both learning through drill as well as learning through strategy make pupils develop automaticity of the multiplication table. It also showed that learning based on strategies enable pupils to use acquired knowledge in new mathematical areas. A disadvantage of learning through drill is that it is difficult to learn multiplication facts without utilizing patterns and connections. Pupils make different errors depending on whether they have been taught through strategies or through drill. Errors are explained with a variety of factors, for instance a lack of practice or a lack of number sense. Previous research present three phases that pupils go through in order to master the multiplication table and two different theo-ries of how the multiplication facts are memorized. The multiplication facts are memorized either as separate entities or as a system of interrelated experiences.

Dans le premier chapitre de cette thèse, nous passons en revue les outils de la théorie analytique des nombres qui seront utiles pour la suite. Nous faisons aussi un survol des entiers y−friables, c’est-à-dire des entiers dont chaque facteur premier est plus petit ou égal à y. Au deuxième chapitre, nous présenterons des problèmes classiques de la théorie des nombres probabiliste et donnerons un bref historique d’une classe de fonctions arithmétiques sur un espace probabilisé. Le problème de Erdos sur la table de multiplication demande quel est le nombre d’entiers distincts apparaissant dans la table de multiplication N × N. L’ordre de grandeur de cette quantité a été déterminé par Kevin Ford (2008). Dans le chapitre 3 de cette thèse, nous étudions le nombre d’ensembles y−friables de la table de multiplication N × N. Plus concrètement, nous nous concentrons sur le changement du comportement de la fonction A(x, y) par rapport au domaine de y, où A(x, y) est une fonction qui compte le nombre d’entiers y− friables distincts et inférieurs à x qui peuvent être représentés comme le produit de deux entiers y− friables inférieurs à p x. Dans le quatrième chapitre, nous prouvons un théorème de Erdos-Kac modifié pour l’ensemble des entiers y− friables. Si !(n) est le nombre de facteurs premiers distincts de n, nous prouvons que la distribution de !(n) est gaussienne pour un certain domaine de y en utilisant la méthode des moments.<br>The object of the first chapter of this thesis is to review the materials and tools in analytic number theory which are used in following chapters. We also give a survey on the development concerning the number of y−smooth integers, which are integers free of prime factors greater than y. In the second chapter, we shall give a brief history about a class of arithmetical functions on a probability space and we discuss on some well-known problems in probabilistic number theory. We present two results in analytic and probabilistic number theory. The Erdos multiplication table problem asks what is the number of distinct integers appearing in the N × N multiplication table. The order of magnitude of this quantity was determined by Kevin Ford (2008). In chapter 3 of this thesis, we study the number of y−smooth entries of the N × N multiplication. More concretely, we focus on the change of behaviour of the function A(x,y) in different ranges of y, where A(x,y) is a function that counts the number of distinct y−smooth integers less than x which can be represented as the product of two y−smooth integers less than p x. In Chapter 4, we prove an Erdos-Kac type of theorem for the set of y−smooth integers. If !(n) is the number of distinct prime factors of n, we prove that the distribution of !(n) is Gaussian for a certain range of y using method of moments.

Η προσασρμοστική μάθηση (adaptive learning) είναι μια εκπαιδευτική μέθοδος η οποία χρησιμοποιεί κάποιο υπολογιστικό σύστημα ως διαδραστική συσκευή διδασκαλίας αναλαμβάνοντας να προσαρμόσει την παρουσίαση του εκπαιδευτικού υλικού σύμφωνα με τις ανάγκες των μαθητών, όπως αυτές προκύπτουν από τις απαντήσεις τους στις ερωτήσεις και τις εργασίες αλλά και γενικότερα παρατηρώντας παραμέτρους της συμπεριφοράς τους στο σύστημα. Η τεχνολογία που υποστηρίζει τη συγκεκριμένη εκπαιδευτική μέθοδο περιλαμβάνει τις πτυχές που προέρχονται από διάφορους επιστημονικούς τομείς όπως η επιστήμη των υπολογιστών, η εκπαίδευση και η ψυχολογία. Η προσαρμοστικής μάθηση έχει οδηγηθεί εν μέρει από τη συνειδητοποίηση ότι προσωπική μάθηση δεν μπορεί να επιτευχθεί σε μια μεγάλη κλίμακα χρησιμοποιώντας παραδοσιακές, μη-προσαρμοστικές προσεγγίσεις. Τα συστήματα προσαρμοστικής μάθησης στοχεύουν στο να μετατρέψουν τον μαθητή από παθητικό δέκτη πληροφοριών σε συνεργαζόμενο και ενεργό μέλος στην εκπαιδευτική διαδικασία. Ενώ μπορεί να υπάρχουν εξαιρέσεις, τα συστήματα προσαρμοστικής μάθησης είναι γενικά χτισμένα με τρία βασικά στοιχεία: ένα μοντέλο εμπειρογνώμονα ή περιεχομένου, ένα μοντέλο μαθητή, και ένα εκπαιδευτικό ή διδασκαλικό μοντέλο. Στα πλαίσια της διπλωματικής αυτής εργασίας μεταπτυχιακής ειδίκευσης αναπτύχθηκε ένα ηλεκτρονικό εκπαιδευτικό παιχνίδι για φορητή συσκευή το οποίο απευθύνεται σε μαθητές επιπέδου δημοτικού σχολείου και υιοθετεί τις αρχές της προσαρμοστικής μάθησης. Στόχος του παιχνιδιού είναι η εξάσκηση και η σταδιακή βελτίωση εκμάθησης της προπαίδειας. Το εκπαιδευτικό παιχνίδι υλοποιήθηκε σε LUA (Corona SDK) και δοκιμάστηκε σε Android συσκευές. Για να καθοριστούν οι λειτουργικές προδιαγραφές διερευνήθηκαν και καταγράφηκαν ήδη υπάρχοντα αντίστοιχα εκπαιδευτικά παιχνίδια, πραγματοποιήθηκε συνέντευξη με έμπειρη εκπαιδευτικό ώστε να προσαρμοστεί ο σχεδιασμός του σεναρίου του παιχνιδιού στην εκπαιδευτική πρακτική που εφαρμόζεται στο σχολείο, ενώ αφού ολοκληρώθηκε η υλοποίηση πραγματοποιήθηκαν δοκιμές με μαθητές α’, β’, γ’ δημοτικού ώστε να διερευνηθεί η αποτελεσματικότητα του εκπαιδευτικού παιχνιδιού.<br>Adaptive learning is an educational method that uses a computer system as an interactive teaching apparatus undertaking to adjust the presentation of educational material according to the needs of students, as reflected by their answers to the questions and work well generally observing aspects of their behavior in the system. The technology that supports this educational method includes aspects from different disciplines such as computer science, education and psychology. Adaptive learning has been driven in part by the realization that personal learning can not be achieved on a large scale using traditional non-adaptive approaches. Adaptive learning systems aim to transform the student from passive recipient of information at a partner and active member in the educational process. While there may be exceptions, adaptive learning systems are generally built with three key elements: a model expert or content, a user model, and an educational teaching model. As part of this thesis postgraduate specialization developed an online educational game for mobile device which is aimed at primary school level students and adopts the principles of adaptive learning. The objective of the game is to practice and the gradual improvement of learning the multiplication tables. The educational game implemented in LUA (Corona SDK) and tested on Android devices. To determine the functional specifications were investigated and recorded existing respective educational games, interviewed by experienced teacher to fit the design of the scenario of the game in the educational practice at school, and after the completed implementation was tested with students a, b, c grade to investigate the effectiveness of educational game.

碩士<br>國立臺南大學<br>應用數學系數學科教學碩士班<br>103<br>A Study on the Effects of Third-grade Mathematical Learning Achievements with Tablet PC as an Auxiliary Teaching --- using the Unit of Two-step Word Problem with Division and Multiplication Abstract By using the unit of two-step word problem with division and multiplication as teaching, this research was used to study the effects of third-grade students’ mathematics learning achievements solving the two-step word problems and their attitude towards learning mathematics from two modes “the use of tablet as an auxiliary teaching” and “do not use tablet as an auxiliary teaching” separately. The study was designed to analyze what the difference in problem-solving effectiveness (Answer rate) of students is by means of the two text-based tests after teaching “two-step word problem with division and multiplication” unit. Then through after two text-based tests interviews, we explore the different abilities of students in problem solving when they meet the two-step word problems. Finally by filling in “Math Attitude Scale”, in order to compare somewhat difference to their attitude towards learning mathematics.According to the above statements, we explained whether there are differences of the effects of learning achievements and learning mathematics attitudes between the mode of the use of tablet as an auxiliary teaching model (experimental group) and the mode of not using Tablet as an auxiliary teaching model (control group). The results are as follows: 1.According to the overall problem solving effectiveness (Answer rate), the experimental group had better answer rate in the two text-based tests, especially in the “before plus (minus) after ride” 、“before plus (minus) after addition” and “multiplicative” of five types of the context problems; but the control group showed better progressive situations in the answer rate of two text-based tests. 2.According to the problem solving abilities of different degree students, the experimental group, especially the high and intermediate achievers, solved more questions with a complete four-order problem-solving component of Mayer Question, but the use of problem-solving strategies did not show a little advantage. 3.According to the attitude of learning mathematics, students in two classes had no significant difference in “Math Attitude Scale”. Keywords: Tablet PC、Two-step Word Problem with Division and Multiplication、Mathematical Problem Solving、the Effect of Mathematics、Third Graders