Arithmetic cognition research was at one time concerned mostly with the representation and retrieval of arithmetic facts in memory. More recently it was found that memory retrieval does not account for all single digit arithmetic performance. For example, Canadian university students solve up to 40% of basic addition problems using procedural strategies (e.g. 5 + 3 = 5 + 1 + 1 + 1). Given that procedures are less efficient than direct memory retrieval it is important to understand why procedure use is high, even for relatively skilled adults. My dissertation, therefore, sought to expand understanding of strategy choice for adults basic arithmetic. Background on this topic and supporting knowledge germane to the topic are provided in Chapter 1.<p> Chapter 2 focused on a well-known, but unexplained, finding: A written word problem (six + seven) results in much greater reported use of procedures (e.g., counting) than the same problem in digits (6 + 7). I hypothesized that this could be the result of a metacognitive effect whereby the low surface familiarity for word problems discourages retrieval. This was tested by familiarizing participants with a subset of the written word stimuli (e.g. three + four = ?, six + nine= ?) and then testing them on unpractised problems comprised of practiced components (four + six = ?). The result was increased retrieval reported for unpractised problems with practiced components. This indicates that surface familiarity contributes to strategy choice.<p> Chapter 3 focused on another classic phenomenon in the arithmetic cognition literature, the problem size effect: Response time, error, and procedure rates increase as a function of problem size. A previous study reported a reduced problem size effect for auditory multiplication problems compared to digit problems. I hypothesized that if this reduction was due to problem encoding processes rather than an effect on calculation per se, then a similar pattern would be observed for addition. Instead, I found that the size effect for addition was larger. I concluded that the auditory format promotes procedures for addition, but promotes retrieval for multiplication.<p> Chapters 4 and 5 were concerned with a well-known methodological issue in the strategy literature, subjectivity of self-reports: Some claim self-reports are more like opinions than objective measures. Thevenot, Fanget, and Fayol (2007) ostensibly solved this problem by probing problem memory subsequent to participants providing an answer. They reasoned that after a more complex procedure, the memory for the original problem would become degraded. The result would be better memory for problems answered by retrieval instead of by procedure. I hypothesized that their interpretation of their findings was conflated with the effect of switching tasks from arithmetic to number memory. I demonstrated that their new method for measuring strategy choice was contaminated by task switching costs, which compromises its application as a measure of strategy choice (Chapter 4). In a subsequent project (Chapter 5), I tested the sensitivity of this new method to detect the effects of factors known in the literature to affect strategy choice. The results indicated that Thevenot et al.s new method was insensitive to at least one of these factors. Thus, attempts to control for the confounding effects of task switching described in Chapter 4, in order to implement this new measure, are not warranted.<p> The current dissertation expanded understanding of strategy choice in four directions by 1) demonstrating that metacognitive factors cause increases in procedure strategies, 2) by demonstrating that the process of strategy selection is affected differentially by digit and auditory-verbal input, 3) by investigating the validity of an alternative measure of strategy use in experimental paradigms, and 4) by discovering a critical failure in the sensitivity of this new measure to measure the effects of factors known to influence strategy use. General conclusions are discussed in Chapter 6.

Arithmetic cognition research was at one time concerned mostly with the representation and retrieval of arithmetic facts in memory. More recently it was found that memory retrieval does not account for all single digit arithmetic performance. For example, Canadian university students solve up to 40% of basic addition problems using procedural strategies (e.g. 5 + 3 = 5 + 1 + 1 + 1). Given that procedures are less efficient than direct memory retrieval it is important to understand why procedure use is high, even for relatively skilled adults. My dissertation, therefore, sought to expand understanding of strategy choice for adults basic arithmetic. Background on this topic and supporting knowledge germane to the topic are provided in Chapter 1.<p> Chapter 2 focused on a well-known, but unexplained, finding: A written word problem (six + seven) results in much greater reported use of procedures (e.g., counting) than the same problem in digits (6 + 7). I hypothesized that this could be the result of a metacognitive effect whereby the low surface familiarity for word problems discourages retrieval. This was tested by familiarizing participants with a subset of the written word stimuli (e.g. three + four = ?, six + nine= ?) and then testing them on unpractised problems comprised of practiced components (four + six = ?). The result was increased retrieval reported for unpractised problems with practiced components. This indicates that surface familiarity contributes to strategy choice.<p> Chapter 3 focused on another classic phenomenon in the arithmetic cognition literature, the problem size effect: Response time, error, and procedure rates increase as a function of problem size. A previous study reported a reduced problem size effect for auditory multiplication problems compared to digit problems. I hypothesized that if this reduction was due to problem encoding processes rather than an effect on calculation per se, then a similar pattern would be observed for addition. Instead, I found that the size effect for addition was larger. I concluded that the auditory format promotes procedures for addition, but promotes retrieval for multiplication.<p> Chapters 4 and 5 were concerned with a well-known methodological issue in the strategy literature, subjectivity of self-reports: Some claim self-reports are more like opinions than objective measures. Thevenot, Fanget, and Fayol (2007) ostensibly solved this problem by probing problem memory subsequent to participants providing an answer. They reasoned that after a more complex procedure, the memory for the original problem would become degraded. The result would be better memory for problems answered by retrieval instead of by procedure. I hypothesized that their interpretation of their findings was conflated with the effect of switching tasks from arithmetic to number memory. I demonstrated that their new method for measuring strategy choice was contaminated by task switching costs, which compromises its application as a measure of strategy choice (Chapter 4). In a subsequent project (Chapter 5), I tested the sensitivity of this new method to detect the effects of factors known in the literature to affect strategy choice. The results indicated that Thevenot et al.s new method was insensitive to at least one of these factors. Thus, attempts to control for the confounding effects of task switching described in Chapter 4, in order to implement this new measure, are not warranted.<p> The current dissertation expanded understanding of strategy choice in four directions by 1) demonstrating that metacognitive factors cause increases in procedure strategies, 2) by demonstrating that the process of strategy selection is affected differentially by digit and auditory-verbal input, 3) by investigating the validity of an alternative measure of strategy use in experimental paradigms, and 4) by discovering a critical failure in the sensitivity of this new measure to measure the effects of factors known to influence strategy use. General conclusions are discussed in Chapter 6.

Arithmetic cognition research was at one time concerned mostly with the representation and retrieval of arithmetic facts in memory. More recently it was found that memory retrieval does not account for all single digit arithmetic performance. For example, Canadian university students solve up to 40% of basic addition problems using procedural strategies (e.g. 5 + 3 = 5 + 1 + 1 + 1). Given that procedures are less efficient than direct memory retrieval it is important to understand why procedure use is high, even for relatively skilled adults. My dissertation, therefore, sought to expand understanding of strategy choice for adults’ basic arithmetic. Background on this topic and supporting knowledge germane to the topic are provided in Chapter 1. Chapter 2 focused on a well-known, but unexplained, finding: A written word problem (six + seven) results in much greater reported use of procedures (e.g., counting) than the same problem in digits (6 + 7). I hypothesized that this could be the result of a metacognitive effect whereby the low surface familiarity for word problems discourages retrieval. This was tested by familiarizing participants with a subset of the written word stimuli (e.g. three + four = ?, six + nine= ?) and then testing them on unpractised problems comprised of practiced components (four + six = ?). The result was increased retrieval reported for unpractised problems with practiced components. This indicates that surface familiarity contributes to strategy choice. Chapter 3 focused on another classic phenomenon in the arithmetic cognition literature, the problem size effect: Response time, error, and procedure rates increase as a function of problem size. A previous study reported a reduced problem size effect for auditory multiplication problems compared to digit problems. I hypothesized that if this reduction was due to problem encoding processes rather than an effect on calculation per se, then a similar pattern would be observed for addition. Instead, I found that the size effect for addition was larger. I concluded that the auditory format promotes procedures for addition, but promotes retrieval for multiplication. Chapters 4 and 5 were concerned with a well-known methodological issue in the strategy literature, subjectivity of self-reports: Some claim self-reports are more like opinions than objective measures. Thevenot, Fanget, and Fayol (2007) ostensibly solved this problem by probing problem memory subsequent to participants providing an answer. They reasoned that after a more complex procedure, the memory for the original problem would become degraded. The result would be better memory for problems answered by retrieval instead of by procedure. I hypothesized that their interpretation of their findings was conflated with the effect of switching tasks from arithmetic to number memory. I demonstrated that their new method for measuring strategy choice was contaminated by task switching costs, which compromises its application as a measure of strategy choice (Chapter 4). In a subsequent project (Chapter 5), I tested the sensitivity of this new method to detect the effects of factors known in the literature to affect strategy choice. The results indicated that Thevenot et al.’s new method was insensitive to at least one of these factors. Thus, attempts to control for the confounding effects of task switching described in Chapter 4, in order to implement this new measure, are not warranted. The current dissertation expanded understanding of strategy choice in four directions by 1) demonstrating that metacognitive factors cause increases in procedure strategies, 2) by demonstrating that the process of strategy selection is affected differentially by digit and auditory-verbal input, 3) by investigating the validity of an alternative measure of strategy use in experimental paradigms, and 4) by discovering a critical failure in the sensitivity of this new measure to measure the effects of factors known to influence strategy use. General conclusions are discussed in Chapter 6.

Submitted by Kamila Costa (kamilavasconceloscosta@gmail.com) on 2015-06-18T21:11:32Z No. of bitstreams: 1 Dissertação-Luiz C C Mendes.pdf: 1127626 bytes, checksum: 3867dce3c1616f07fa4984ef6374d5d5 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-07-06T18:18:27Z (GMT) No. of bitstreams: 1 Dissertação-Luiz C C Mendes.pdf: 1127626 bytes, checksum: 3867dce3c1616f07fa4984ef6374d5d5 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-07-06T18:22:43Z (GMT) No. of bitstreams: 1 Dissertação-Luiz C C Mendes.pdf: 1127626 bytes, checksum: 3867dce3c1616f07fa4984ef6374d5d5 (MD5) / Made available in DSpace on 2015-07-06T18:22:43Z (GMT). No. of bitstreams: 1 Dissertação-Luiz C C Mendes.pdf: 1127626 bytes, checksum: 3867dce3c1616f07fa4984ef6374d5d5 (MD5) Previous issue date: 2015-02-04 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This paper aims to introduce students and teachers of basic mathematical education some resolutions of problems in the field of arithmetic which can benefit the teachinglearning process. Initially, it will be addressed the divisibility with their properties and criteria. This is done after a presentation of Euclidean division and its applications in basic education. Moreover, it will be presented a brief theoretical background based on the concept and the operational properties of modular congruence with their residue classes, followed by their applications. Finally, it will be presented a brief history of the numbers in the calendars. / A presente dissertação tem como objetivo principal apresentar a alunos e professores de matemática do ensino básico algumas resoluções de problemas no campo da aritmética que pode beneficiar o processo ensino-aprendizagem. Serão abordados inicialmente a divisibilidade, com suas propriedades e seus critérios, após apresentação da divisão euclidiana e suas aplicações no ensino básico. Além disso, será apresentado um breve embasamento teórico, pautado no conceito e nas propriedades operacionais da congruência modular com suas classes residuais, seguido de suas aplicações. No final, será feito um breve histórico dos números nos calendários.