# Effect of memorization of the multiplication tables on students' performance in high school

Table of Content

List of Tables ............................................................................................................. viii List of Figures ................................................................................................................x CHAPTER 1. INTRODUCTION ....................................................................................................1 Background of the Study ...............................................................................................1 Statement of the Problem ...............................................................................................2 Purpose of the Study ......................................................................................................3 Rationale ........................................................................................................................3 Research Questions ........................................................................................................5 Nature of the Study ........................................................................................................6 Significance of the Study ...............................................................................................6 Definition of Terms........................................................................................................7 Assumptions and Limitations ......................................................................................11 Organization of the Remainder of the Study ...............................................................11 CHAPTER 2. LITERATURE REVIEW .......................................................................................13 Introduction ..................................................................................................................13 Rationale for the Research ...........................................................................................13 Theoretical Framework ................................................................................................18 Automaticity and Fluency ............................................................................................20 Methodological Justification ........................................................................................27 Methodologies and Study ............................................................................................32 Conclusion ...................................................................................................................38

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CHAPTER 3. METHODOLOGY .................................................................................................39 Research Design Strategy and Underlying Assumptions ............................................39 Theoretical Framework ................................................................................................39 Research Questions and Hypotheses ...........................................................................41 Research Questions ......................................................................................................42 Hypotheses ...................................................................................................................43 Experiment1: Performance with Memorization ...........................................................43 Experiment2: Performance with Calculators ...............................................................44 Triangulation of Data of the Two Experiments ...........................................................45 Design of the Study ......................................................................................................46 Design of the Experiments ...........................................................................................47 Site, Population and Sampling .....................................................................................48 Participants ...................................................................................................................49 Instruments and Tools ..................................................................................................49 Data Collection Procedures ..........................................................................................51 Data Analysis ...............................................................................................................52 Reporting the Findings .................................................................................................54 Validity and Reliability ................................................................................................54 Ethical Considerations .................................................................................................56 Limitations of Methodology and Strategies .................................................................56 Conclusion ...................................................................................................................57 Time-Lines for Research Activity ...............................................................................58 CHAPTER 4. DATA COLLECTION AND ANALYSIS .............................................................59

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Introduction ..................................................................................................................59 Representative Sample .................................................................................................60 Data Description ..........................................................................................................61 Multiplication without Calculator ................................................................................62 Items of the Multiplication Instrument ........................................................................63 Multiplication with Calculator .....................................................................................64 Math Assessment .........................................................................................................65 Math Assessment without Calculator ..........................................................................65 Comparison of Correct Items in Math Assessment .....................................................65 Assessment with Calculators .......................................................................................67 Grades in the Math Assessment ...................................................................................68 Questionnaire Answers ................................................................................................68 Data Analysis and Results ...........................................................................................69 Experiment 1: Performance with Memorization ..........................................................69 Triangulation of the Data of the Two Experiments .....................................................80 Testing the Null Hypothesis.........................................................................................81 Qualitative Analysis .....................................................................................................87 Performance of Students on the Math Problems..........................................................87 Results of the Qualitative Study ..................................................................................93 Summary ......................................................................................................................98 CHAPTER 5. RESULTS, CONCLUSIONS, AND RECOMMENDATIONS ...........................100 Introduction ................................................................................................................100 Findings......................................................................................................................102

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Findings from Experiment 1 ......................................................................................102 Findings from Experiment 2 ......................................................................................107 Findings from the Two Experiments .........................................................................109 Students Beliefs and Attitude towards Mathematics .................................................111 Limitations of the Study.............................................................................................114 Implications................................................................................................................115 Implications for future Research ................................................................................121 Conclusion .................................................................................................................122 REFERENCES ............................................................................................................................123 APPENDIX A. Multiplication Test .............................................................................................135 APPENDIX B. Math Comprehensive Assessment ......................................................................137 APPENDIX C. Questionnaire ......................................................................................................140 APPENDIX D. Summary of Collected Data ...............................................................................141 APPENDIX E. Experiment 1 Results of Tests Using Memory ...................................................144 APPENDIX F. Results of Experiment 2, Using Calculators .......................................................152 APPENDIX G. Triangulation of the Two Experiments (Hypothesis 3) ......................................156 APPENDIX H. Mixed Experiments Correlation .........................................................................164

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List of Tables

Table 1. Math Assessment without Calculators ...........................................................66 Table 2. Math Assessment Using Calculator ...............................................................67 Table 3. Exp 1: t-test for Groups M1 and M2 .............................................................72 Table 4. Experiment 2: T-Test Group Statistics ..........................................................78 Table 5. Group Statistics with Calculators for M1 and M2 .........................................79 Table 6. Exp.2: T-Test with calculators for M1 and M2 .............................................80 Table 7. Experiments 1 & 2: Multiple Comparisons ...................................................83 Table 8. Experiments 1 & 2: Multiple Comparisons ...................................................84 Table 9. Experiments 1 & 2: Paired Samples Statistics ...............................................85 Table 10. Experiments 1 & 2: Paired Samples Correlations .......................................86 Table E1. Experiment 1: Group Statistics.................................................................144 Table E2. Experiment 1: Independent Samples Test .................................................145 Table E3. Experiment 1: Correlation .........................................................................146 Table E4. Experiment 1: Linear Regression Correlations .........................................147 Table E5. Experiment 1: Linear Regression Model Summary ..................................147 Table E6. Experiment 1: Linear Regression ..............................................................147 TableE7. Experiment 1: Linear Regression Coefficients a .........................................148 Table E8. Experiment 1: Linear Regression Coefficients .........................................148 Table E9. Group Statistics (M1 and M2 calculators used) ........................................150 Table E10. T-Test for M1 and M2 with calculators ..................................................151 Table F1. Experiment 2: Descriptive Statistics..........................................................152

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Table F2. Experiment 2: T-Test Group Statistics ......................................................152 Table F3. Experiment 2: Independent Samples Test .................................................153 Table F4. Experiment 2: Correlations ........................................................................154 Table F5. T-Test. Comparing Means (when M1 & M2 used Calculators) ................155 Table G1. Experiment 1 & 2: Group Statistics ..........................................................156 Table G2. Experiment 1 & 2: Comparing the Means (M1-C1) .................................156 Table G3. Exp. 1 & 2: Comparing the Means of M1and C2. ....................................157 Table G4. Experiment 1 & 2: Comparing the Means (M2- C1). ...............................157 Table G5. Experiement 1 & 2: Comparing the Means (M2 -C2). .............................158 Table G6. Group Statistics M1 & M2 memory and Calculators ...............................158 Table G7. Exp. 1 & 2: T-Test For M1 & M2 memory and Calcu .............................159 Table G8. Exp. 1 & 2: Cross Samples and T-Tests (C1 -C2) ....................................160 Table G9. Experiments 1 & 2: Cross Samples and Tests (C1 -C2) ...........................161 Table G10. Multivariate Descriptive Statistics. .........................................................162 Table G11. Exp.1 & 2: Multivariate Test for Multiple Comparisons ........................163 Table H1. Exp1 & Exp2. Paired Samples Test for the paired differences .................164 Table H2. Mixed Experiments Correlations ..............................................................164

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List of Figures

Figure 1. Experiment 1: Linear Regression Curve Fit. ................................................75 Figure 2. Experiment1: Linear Regression for Math Assessment ..............................75 Figure 3. Experiment1: Linear and Normal Distribution............................................77 Figure E1. Experiment 1:. Change of means between M1 and M2 ...........................149 Figure E2. Experiment 1: Means of the Total M1 and M2 ........................................150

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CHAPTER 1. INTRODUCTION Background of the Study While students at an early age use memorization and retrieval of information for alphabets, names, numbers, vocabulary and spelling, those in the secondary and post secondary school levels use it for formulas and procedures. This fact, however, did not prevent Montessori from separating memorization from abstraction, and it did not prevent Dewey from fully devaluing its role (D. Smith, 2005). It is a commonly held belief among some scholars that students do not need to memorize math facts. Those scholars note that students should understand and practice math facts over an extended period until those facts become part and parcel of their permanent memory (Caron, 2007). Other scholars emphasize that students must acquire the basic computational knowledge and should be able to retrieve it without delay from memory in order for them to be successful in mathematics (Miller & Hudson, 2007). The knowledge of the basic math facts such as multiplication and division is acquired with rehearsal, practice, and anchors and cues. Those techniques and strategies improve the memory to control, store, and prepare the math facts for instant retrieval. The speed at which the retrieval happens can reach the level of automaticity (Marzano, 2003), or learning by heart (James, 2008), which are equivalent to rote memorization. Children start their informal and formal education by recognizing and counting items, associating numbers of items with their Arabic symbols, and then learning the basic skills (Campbell, 2005). The speed at which the retrieval happens can reach the level of automaticity (Marzano, 2003), or the learning by heart (James, 2008), which are equivalent to rote memorization. Children start their informal and formal education by recognizing and counting items, associating numbers of items with their

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Arabic symbols, and then learning the basic skills (Campbell, 2005). Although students should memorize the multiplication tables and master the basic skills by the third grade, deficiency in multiplication and solving problems are common in the high school level (Kotsopoulos, 2007; Calhoon, Emerson, Flores, & Houchins, 2007; Miller & Hudson, 2007; Watanabe, 1996). When students have deficiencies in basic math facts such as addition, subtraction and multiplication, they cannot make the connection between the new and preexisting information (Walsh, 2003). The deficiency can be in performance, skills, or both. Performance deficiency is related to behaviors more than to lack of knowledge, while skills deficiency is associated with weak foundation in the basics of mathematics (Duhon, Noell, Witt, Freeland, Dufrene, & Gilbertson, 2004). A high percentage of urban schools students struggle to pass their math classes, as well as the state-wide exams required for graduation, and many of them drop out ultimately from school (National Center for Educational Statistics, 2008; Secada, 1999; Azzam, 2007). Those who graduate are obliged in general to take math remedial courses before they start their undergraduate curriculum. This study attempts to examine the effect of memorization on students’ performance in solving math problems at the high school level.

Statement of the Problem Several theories have attempted to explain the reasons behind the deficiency in learning and performance of students in math. Some of these theories have attributed the deficiency in student performance to demographic background, teacher competency related to training and certification, instruction and delivery (Timmermans & Van Lieshout, 2003), time allocated to instruction (Rohrer & Taylor, 2006), student self-concept and motivation, behavior (Lin & Kubina, 2005), personal epistemology (Muis, 2004), and a range of other causes. Although the

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theories may explain students’ performance deficit, deficiencies in mathematics basic skills influence the acquisition of succeeding advanced skills (Shapiro, 1996; Brookhart, Andolina, Zuza, and Furman, 2004). Students solving advanced algebra problems must first become fluent in addition, and multiplication skills (Wu, 1999). Although research has dealt with memorization of math facts in primary and middle school levels (Campbell & Robert, 2008; Brasch, Williams, & McLaughlin, 2008; Flores, 2009), it has not given the same importance to the subject at the high school level (Kotsopoulos, 2007). The present study intends to provide a contribution toward understanding the role of memorization of multiplication facts in math learning. It is not known how and to what extent memorization of the multiplication tables affects students’ performance in solving math problems at the high school level.

Purpose of the Study The purpose of this study was to examine the effect memorization of multiplication tables on students' performance in math at the high school level. The study examined whether the memorization of multiplication facts constituted a main contribution toward higher achievement in math for students; it also investigated whether the absence of memorization constituted a disadvantage for students with low achievement in math. The project helped in understanding the causes behind the deficiency in math that many students had at the high school level.

Rationale Students perform at different levels in math due to various factors; among these is the student’s cultural background (Jackson & Coney, 2007). The diversity of urban schools’

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populations has prompted scholars to evaluate the practiced educational strategies compared to students’ learning background methods such as memorization (Valiente, 2008; Leung, Ginns, & Kember, 2008; Rao & Sachs, 1999). Also, students’ learning and cultural background should be taken into consideration as they may require diverse learning strategies (Valiente, 2008). If memorization is a part of the cultural learning, students should be encouraged to use it (Tsao, 2004; Wong, 2002; Zhao, 2005; Wu, 1999). The influence of culture and environment is expressed also by Taylor and Lamoreaux (2008) who suggest that the brain uses experiences from the social context for learning, and the continued learning prepares the brain to adapt to the new conditions. Neuroscientists have pointed out that the brain has a strong and well established capacity to adjust to environmental needs, a feature of the brain which is present throughout life known as plasticity (Taylor & Lamoreaux, 2008). The modification occurs as a result of strength in the connections of some neurons and the elimination or weakness in others. The degree of adjustment depends on the quality of learning, which in the long-term stimulates deeper modification (Byrne, 2009). It also depends upon the length of exposure to learning, for which infant brains undergo a high growth of new synapses. Certain periods constitute high efficiency in learning (Center for Educational Research and Innovation [CERI], 2007). Recent studies expressed the need to identify the academic skills that affect students’ learning in high school math, and differentiate between skill deficits and performance deficits (Duhon et al., 2004). Performance is based on conceptual or procedural knowledge, whereas conceptual knowledge requires a deep understanding beyond computations. The procedural knowledge is tied to algorithm use, and resides in semantic memory; it allows a person to perform computations efficiently. The conceptual knowledge enables the person to inquire and

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question whether or not the finding makes sense, and whenever needed, it initiates procedural strategies based on the existing ones stored in the permanent memory (Kotsopoulos, 2005). While performance deficits are due to various reasons, basic skills deficits have a direct influence on achievement in math (Flores, 2008; McCafferty, 2009; Brasch et al., 2008).

Research Questions The following research questions were addressed: 1. Is there a significant difference between the means of scores of the comprehensive assessment obtained by students who memorize the multiplication tables and those who do not? 2. Is there a significant difference between the means of scores of the comprehensive test of the students who memorize the multiplication tables and those who do not when all students use calculators? 3. Is there a significant difference between the mean scores of the multiplication diagnostic test when the students use memorization and when they use calculators? 4. Is there a significant correlation between the scores of the multiplication diagnostic tests and scores of the comprehensive assessments? 5. How do students who memorize the multiplication tables perform on math problems compared to those who do not memorize the multiplication tables? 6. How do students who memorize the multiplication tables feel about math? 7. How do students who do not memorize the multiplication tables feel about math? 8. How do students who memorize the multiplication tables process math calculations compared to students who do not memorize the multiplication tables? 9. How does the retention of the multiplication tables help students in solving math problems?

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Nature of the Study This study used an explanatory mixed method for triangulation of the quantitative and qualitative data. The study measured the level of memorization of the multiplication tables for students in high school, and the difference in performance between the students who memorized and those who did not memorize the multiplication tables. The use of calculators in this study made it possible to evaluate the difference in performance for calculations and problem solving with and without a calculator. The study used the data collected to identify participants who memorized the multiplication tables and those who did not. In addition, the study measured the level of proficiency in multiplication and used it to describe the quality of mathematical calculations and reasoning of participants.

Significance of the Study This study attempts to define the role of memorization in learning and helps in understanding whether memorization of the multiplication tables affects students’ performance in solving math problems. In addition, the study verifies whether memorization is helpful in problem solving and whether understanding the basic facts of math leads to higher achievements. Further, the study intends to help teachers overcome the challenges of their diverse classes. In his article “Three Hundred Years of Method” about the modern teaching at the time, Douglas (1936) states that “the memory and drill method was carried over to experience a partial surrender to those forces battling for understanding before memory. The question and answer method,

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however, had become in the early academy period almost the universal recitation procedure” (p. 137). Furthermore, scholars concerned over diversity of classrooms argue that students whose English is a second language are not given equal opportunity to learn because the methods used as active learning do not match their previous schooling experience (Valiente, 2008). In addition, students are not exposed to an evenly distributed opportunity to learn mathematics, because some of them need to use memorization in order to perform well in math (Flores, 2007). The acquisition of knowledge in the different fields occurs in various forms through anchors and complex cues (Cherney, 2008). People need to practice information a certain number of times to add it to their base knowledge (Marzano, 2003). The prior knowledge that students have about the content of a subject matter constitutes the background knowledge and represents an indicator of how they will learn new information; this prior knowledge assists students to make sense of new information (Marzano, 2004). Also, students acquire new knowledge based on their ability to process and store information, and on the number and frequency of exposures to that information (Marzano, 2001; Brasch et al., 2008; Caron, 2007).

Definition of Terms Memorization Memorization is to store, retain, and retrieve information from memory. Storing information can be done through auditory, oral, written, visual, or physical practice. Rehearsal can be done to the level of automaticity or rote memorization (Caron, 2008; Douglas, 1936).

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Involuntary Memorization Involuntary memorization happens when students are vigorously engaged in purposeful tasks and unconsciously remember the information in a non verbatim form but connect the new information with the preexisting (McCafferty, 2008). Sensory Memory It is the storage in each sensory system that keeps the information briefly while it is undergoing through the stages of processing. The visual, auditory, and other senses are considered storages for memory (Byrne, 2009). An example of the sensory memory is Braille script. Short-Term Memory It is also known as the short-term storage. It holds the information consciously which can be processed using rehearsal or repetition. The short-term memory rapidly loses the information when the person is distracted. Emotional Memory It is storage of feelings due to a certain condition of internal transformations due outside events and related to sensory responses. Working Memory The working memory is an active short term memory in which information diminishes during the transition from oral to literate form (Towse, Hitch, Hamilton, Peacock, & Hutton, 2005). It is the memory that keeps, for example, names and telephone numbers (Frith, 2007).

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Automaticity Automaticity or controlled processing is defined as the learning skills at “a level where they require little or no conscious thought” (Marzano, 2003, p.140). Knowledge Bloom’s Taxonomy places knowledge at the base (the lowest level) of the cognitive domain, because knowledge represents the foundation of comprehension, application, analysis, synthesis, and evaluation (Cherney, 2008; Walsh, 2003). Knowledge, as used in cognitive psychology, is “the information about the world stored in memory; ranging from the everyday to the formal” (E. Smith & Kosslyn, 2007, p. 151). Cognitive psychologists also distinguish between three types of knowledge: conceptual, declarative and procedural. Conceptual Knowledge Connected web of information where the linking relationships are vital to the pieces of information that are linked (Miller & Hudson, 2007). It can be considered dynamic information because it extends and changes in form and meaning. Declarative Knowledge Information based that does not require processing of information, but all information of the declarative knowledge is considered static information that does not change in form or shape, such as vocabulary, names, places, numbers, and things. Procedural Knowledge It is skills or a process based knowledge that requires the use of preexisting information to produce new items. An example of procedural knowledge is the product of two numbers in multiplication or the quotient of two numbers in long division. Efficient procedural knowledge

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requires memorization by heart with little or no conscious thought (Marzano, 2001). Although it seems a static relation, memorization has a lot to do with dynamic action as a process that continues to make production. Retrieval It is the recall of stored information. The information can be recalled (retrieved) easily when it is deeply and thoroughly processed (Cherney, 2008). Frequent repetition and increased practice make the information better remembered and ease its retrieval (Skinner, 2008; Marzano, 2003). Recent tests on human show that the medial temporal cortex of the brain is activated during retrieval compared to its resting condition (D’Esposito, 2003). However, retrieval for math learners is based on the adoption of technology advancement such as the use of computers, calculators, and other storing and displaying devices. Marzano (2001) states: Before handheld calculators, students had to learn multiplication and division to the level of automaticity. Today, this might not be the case. Which, if any, procedures should be learned to the level of automaticity or controlled processing is certainly a matter for subject matter specialist (p. 115).

The ability to recall information naturally by rote is to store information through repetition without connecting it to pre-existing knowledge. Rote memorization is “learning things by heart” (James, 1962/1899, p.64). In general, memorizing is the final process of learning to store, retain, and recall information from the natural storing memory. The information can be recalled, expressed and shared with others in its exact form through written or spoken words and or actions and gestures.

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Assumptions and Limitations The following assumptions will be considered in this study: The participants in this study use their own natural resources of solving the problems. The participants do not use any digital tool for calculating while taking the tests of the first experiment in which no calculators are allowed. 1- Students have been exposed to the table of multiplication in primary and middle school, and to the basic concepts of math and algebra in middle and high school level. 2- All participants in the study complete the tests in the given, specified and allocated time. The following limitations are present in the study: 1- The sample size is relatively small to consider a full transferability of results. 2- The participants in the study may not know the content of the questions due to excessive absences, and thus the results of the tests may not reflect the intended accuracy of the analysis and validity of the findings. 3- The results of the study are limited by the number of students who will take part in the experiments.

Organization of the Remainder of the Study The remainder of this study is divided as follows: Chapter 2 includes a review of the literature pertaining to the concepts of memorization, math basic skills, performance, performance deficiencies, and skills deficiencies. After the introduction, a review of the recent

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literature to establish a rationale for the research is provided. Also, the theoretical foundations of cognition and psychology in pedagogy are visited. In addition, the exploration of research on learning and strategies of teaching is also examined based on the three domains of Bloom. At last, the chapter further reviews the recent literature findings related to the questions and hypothesis of the present study and concludes with a summary of contributions to the field. Chapter 3 contains the methodology used in the study. The chapter starts with a theoretical framework and defines the research design strategy; it presents the sampling design, and the data collection procedures. In addition, it also describes the instruments used by the participants. The chapter also includes a description of ethical considerations and threats to internal and external validity, besides the time line and the results of the study. Chapter 4 includes the actual data collection and participants’ responses to the questionnaire. Chapter 5 presents the analysis and summary of the data and suggestions for future research; it ends with the conclusion.

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CHAPTER 2. LITERATURE REVIEW Introduction Several researchers have dealt with learning strategies of math facts in primary and secondary levels; they studied the relationships among memorization, retrieval, and fluency (Joy, Kaplan & Fein, 2003; Lin & Kubina, 2005). Other studies dealt with math facts from the perspectives of measurement of learning rates, retrieval speed, knowledge, and conceptual skills. Some others studied bidirectional associations of symbols in multiplication with the transportation and transfer of information in students' instruction (Campbell & Robert, 2008). Meanwhile, other studies claimed significant relationships between beliefs, cognition, motivation and achievement in math for students at all levels, and the existence of relationships between beliefs and learning behaviors (Muis, 2004). Most studies treating math performance and memory have different results but agree on the importance of and the need to acquire the basic facts in the early grades. The acquisition of those math basics requires that students practice and rehearse the concepts and theories to learn them to the level of automaticity (Marzano, 2001). This is seen, for example, in the lives of actors in plays, politicians, doctors, educators, students, and people of all walks of life where memorization plays a major role in their daily practices.

Rationale for the Research Piaget focuses on children learning and cognitive development through discovery and understanding of their realities to define the idea of how people construct the nature of knowing and making sense of the world around them (Piaget, 1968; Brooks & Brooks, 2007; Darling- Hammond & Bransford, 2007). Piaget argues that children construct their understanding using

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various ways and means that represent an extension of their biological growth; he states that “the logico-mathematical operations derive from actions themselves, because they are the product of an abstraction which proceeds from the coordination of actions” (Piaget, 1968, P. 81). In addition, Piaget argues that children build their cognitive instruments and modify their views about the realities based on their environment and the actions they experience. Further, a person’s learning is subjective to social and cultural influences, and working under guidance provides higher expertise. That is, children can do tomorrow by themselves what they can do today under adult guidance (Darling-Hammond & Bransford, 2005). The acquisition of computational skills necessitates practice and training. Those computational skills are evaluated by the speed and accuracy at which problems are solved, as well as the procedures used to obtain the solutions. Efficient retrieval is generally based on repeated use of effective procedures. Those procedures tend to give students variances of optional trials when an answer to a problem is not correct. The most efficient retrieval happens at the level of a long-term memory where the multiplication skills development principally employs the succession of strategies of retrieval (Siegle, 1988). When students are able to multiply single digit numbers, they can perform well on the multiplication of two or more digit numbers (Lin & Kubina, 2005). According to Lin and Kubina (2005), the mastery of basic skills of multiplication has a strong relationship with the performance requiring composite skills; their study involved 159 fifth graders who took one- minute assessments for single and multi digit multiplication problems. The results suggest that students achieved higher level of accuracy, but lower level of fluency. The study indicates that the fluent component skill may have a role in composite skill performance, and that more than one skill component may contribute to composite skill acquisition.

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In their study, Lin and Kubina (2005) used a test packet of three sections for an instrument of assessment. The first section was a pretest for students to write repeatedly on lined sheet the digits 0 to 9; the second section assessed students on 156 random multiplication problems of single digit numbers; and the third section tested students on 67 random multiplication problems of 2 or 3 digits multiplicands by 1 or 2 multipliers with and without renaming. Their finding indicated that students reached high level of fluency but lower level of accuracy, deriving the conclusion that the component of fluency skills might have a significant influence on the composite performance skills. In another context, students may or may not have a perception to visualize the abstract depending on their prior knowledge and learning strategies. Students form pictures from words and associate abstractions to concrete in the same way a word is associated with a preexisting picture in the mind. The conception of concrete and the visualization of abstraction are important factors in learning, understanding and anchoring in order for the information to be stored in the permanent memory. When students are taught in abstractions, they tend to lose the new information instantly unless they associate it with preexisting concepts or cues. If students are taught in sequence from concrete to abstract, they improve their computational skills because they associate the abstract to the concrete (Flores, 2009). The studies conducted by Flores (2009) on the effects of Concrete Representational-Abstract (CRA) sequence on elementary school students’ fluency in computing subtraction problems with regrouping, found a functional relationship between CRA instruction and subtraction with regrouping across all students. If the association of new concepts with preexisting knowledge is weak, then there is a deficiency in learning. However, the use of various learning strategies helps students to bridge

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what they already know with the newly acquired information. In a study conducted by Brasch et al. (2008), the use of flashcards appears to sustain multiplication learning for high school students with attention deficit. The study concludes that an increase in correct responses was noticed, and an intermediate ratio of mastered to unmastered was correlated for multiplication facts with trials for each participant. Further, the free recall for course content at various course levels depends on the material and strategies used for instruction to help students retain the information. A study conducted by Cherney (2008) involved undergraduate students in both introductory and advanced psychology courses showed that students were able to list concepts they learned while being engaged actively in practice and solving exercises. From another context, Caron (2008) argues that students need a superior understanding of the procedure of multiplication as well as when and how to put these facts to use. Caron stresses that rote memorization is not the solution to learning multiplication. Caron’s study presents multiplication exercises along with answers as a rescue for students to use while solving the problems. After students solve the problems they record their progress using charts. The process of solving exercises and recording their progress is practiced on a daily basis to help students memorize the multiplication tables. Although Caron claims to be against rote memorization, his only true objection is the methodology; while his own method is itself compromised by the long period it takes to have students retain the multiplication facts. A strong working memory is needed to save the information temporarily and to keep track of the arithmetic operations when successive addition is used for multiplication facts to solve problems (Mabbott & Bisanz, 2008). A weak working memory decays the information before it is tied to the one coming up, and loses the results of the previous calculations. Mabbott