THE IMPACT OF THREE MATH INTERVENTIONS ON NUMBER FACT KNOWLEDGE AMONG ELEMENTARY SCHOOL STUDENTS: EMPHASIS ON STUDENTS WITH LOWER MATH ABILITIES by ANGELA FEYTER A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN EDUCATION STUDIES, SPECIAL EDUCATION in THE FACULTY OF GRADUATE STUDIES Master of Arts in Educational Studies, Special Education We accept this thesis as conforming to the required standard …………………………………………………………….. Dr. Katrina Korb, Ph.D., Thesis Supervisor …………………………………………………………….. Dr. Kenneth Pudlas. Ed. D., Second Reader …………………………………………………………….. Dr. Lara Ragpot, D. Ed., External Examiner TRINITY WESTERN UNIVERSITY July 2018 © Angela Feyter NUMBER FACTS KNOWLEDGE ii ABSTRACT This study examined the impact of three math interventions on students with lower math abilities (LMA) in connection to their ability to gain number fact knowledge. Grade level was also used as a variable. Sixty-five students in Grades 2-6 participated in one of the following three interventions: drill-and-practice, strategy instruction and peermediated practice. At the end of 10 weeks, participants completed a number fact test that consisted of addition and multiplication statements. ANOVAs were used to analyze the results for each research question. Results demonstrated that the three interventions had no significant interaction effect on the number fact knowledge gained by the students with LMA. As well, students with LMA and without LMA benefitted equally from the interventions. It was also found that there was no significant interaction between the grade of the student and the intervention used. However, all students benefitted from all three interventions. Keywords: mathematics interventions, drill-and-practice, peer-mediation, strategy instruction, lower math abilities, number fact knowledge NUMBER FACTS KNOWLEDGE iii TABLE OF CONTENTS ABSTRACT........................................................................................................................ ii TABLE OF CONTENTS................................................................................................... iii LIST OF TABLES ............................................................................................................. vi LIST OF FIGURES .......................................................................................................... vii ACKNOWLEDGEMENTS ............................................................................................. viii CHAPTER 1: INTRODUCTION TO THE STUDY.......................................................... 1 Educational Trends ................................................................................................. 1 Local educational trends. ............................................................................ 3 Number Fact Knowledge ........................................................................................ 3 Difficulties in number fact knowledge. ...................................................... 4 Interventions ........................................................................................................... 5 Definitions............................................................................................................... 7 Purpose.................................................................................................................... 8 Theoretical Framework ........................................................................................... 9 Research Questions ............................................................................................... 10 Ethical Considerations and Researcher Bias......................................................... 11 CHAPTER 2: LITERATURE REVIEW .......................................................................... 12 Number Facts ........................................................................................................ 12 Automaticity ......................................................................................................... 14 Cognitive Skills needed for Automaticity................................................. 16 Lower Mathematical Abilities .............................................................................. 17 Grade Level ........................................................................................................... 20 NUMBER FACTS KNOWLEDGE iv Interventions ......................................................................................................... 22 Intervention Layout. .................................................................................. 23 Drill-and-practice. ..................................................................................... 24 Peer-mediation. ......................................................................................... 26 Strategy Instruction. .................................................................................. 29 Summary ............................................................................................................... 35 CHAPTER 3: RESEARCH METHOD ............................................................................ 36 Research Design.................................................................................................... 36 Participants............................................................................................................ 38 Measures ............................................................................................................... 40 Procedure for Data Collection .............................................................................. 42 Intervention 1: Drill-and-practice ............................................................. 46 Intervention 2: Strategy instruction. ......................................................... 47 Intervention 3: Peer-mediated practice. .................................................... 49 Method of Data Analysis ...................................................................................... 50 CHAPTER 4: RESULTS .................................................................................................. 53 Preliminary Analysis............................................................................................. 53 Research Question 1: Interaction Effect Between Lower Math Abilities and Intervention ........................................................................................................... 54 Research Question 2: Most Effective Intervention ............................................... 56 Effect size of interventions. ...................................................................... 56 Research Question 3: Interaction Effect Between Grade and Intervention .......... 57 CHAPTER 5: DISCUSSION............................................................................................ 59 NUMBER FACTS KNOWLEDGE v Interaction between Intervention and LMA .......................................................... 59 Most Effective Intervention .................................................................................. 61 Interaction between Grade and Intervention ......................................................... 62 Teacher Feedback ................................................................................................. 63 Summary ............................................................................................................... 65 CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS .................................... 67 Implications of Research....................................................................................... 67 Limitations ............................................................................................................ 68 Considerations for Future Research ...................................................................... 69 Conclusion ............................................................................................................ 70 REFERENCES ................................................................................................................. 71 APPENDIX A: Permission Letter .................................................................................... 78 APPENDIX B: Script for Explanation of Each Intervention............................................ 80 APPENDIX C: Post-Test .................................................................................................. 81 APPENDIX D: Teacher Checklist .................................................................................... 82 APPENDIX E: Sample Drill-and-Practice Sheets ............................................................ 83 APPENDIX F: Video Script – Lesson On Doubles.......................................................... 85 APPENDIX G: Strategy Instruction – Multiply by Four Game ....................................... 87 NUMBER FACTS KNOWLEDGE vi LIST OF TABLES Table 1. Frequency of Students in each Intervention Group ..................................................... 38 2. Grade Composition of Study Participants.................................................................... 40 3. Key Features of Number Fact Knowledge Interventions ............................................ 44 4. Mean Score and Standard Deviations of Number Fact Knowledge by Treatment Group and Math Ability .............................................................................. 55 5. Mean Score and Standard Deviations of Number Fact Knowledge by Treatment Group and Grade ......................................................................................... 58 NUMBER FACTS KNOWLEDGE vii LIST OF FIGURES Figure 1. Visual representation of the assumptions of linearity and homoscedasticity. .............. 53 2. Mean post-test score with math abilities as the independent variable. ......................... 57 NUMBER FACTS KNOWLEDGE viii ACKNOWLEDGEMENTS I would like to express my gratitude to many individuals, without whose support this project would not have been possible. I would like to thank Dr. Katrina Korb for her continual support and feedback; without her help, this project would not have made it to completion. Thank you also to Dr. Ken Pudlas for his continual encouragement to persevere and his feedback as second reader. I would also like to thank the principal and staff at the school where I currently teach. Your willingness to be involved in this project is greatly appreciated and also your continual support and ability to be flexible during this busy season of my life was a key aspect to the success of this project. Without this support, this project would never have been able to get started, much less completed. I am also grateful for the support and encouragement from my family. Their continual reminders that “You can do it” and their involvement to make sure that my work and family life were partially balanced made this journey much more successful. NUMBER FACTS KNOWLEDGE 1 CHAPTER 1: INTRODUCTION TO THE STUDY Educational Trends The American National Research Council has stated that “mathematical literacy is essential as a foundation for democracy in a technological age” (National Research Council, 1989, p. 8). Research has shown that one key component to mathematical literacy is the students’ knowledge of number facts (Fuchs et al., 2008; Stokke, 2015). The term number facts refers to the use of an operation (addition, subtraction, multiplication or division) on single digits. These are taught to students soon after they understand the ordinal concept of numbers (Ching & Nunes, 2016). As students’ progress through the grade levels, they encounter a variety of methodologies that are used to develop their number fact knowledge. Some of these methodologies stress the value of rote memorization, other methodologies place greater emphasis on the conceptual understanding of the facts and still other methodologies combine rote memorization and conceptual understanding. All methodologies have the commonality of teachers desiring their students to be successful in their number facts knowledge. The choice of what methodology to use is often reflected in the current educational trends. This is seen in the mathematical history in Canada and more specifically, as it relates to the present study, in Alberta. The present study and the author are both located in Alberta and so this particular province’s educational trends play a significant role in methodology choices. In Alberta’s recent history, the trend in math has moved from direct instruction to a guided-discovery style (Stokke, 2015). Direct instruction refers to the notion of a teacher giving explicit instructions to students without much student involvement (Klahr & Nigam, 2004). This style of instruction has shown to NUMBER FACTS KNOWLEDGE 2 have benefits when there is a limited amount of time available and when the fundamentals of a concept are being taught (Klahr, 2009). However, with a direct instruction approach, there are some concerns regarding students’ ability to recall the material at a later time (Klahr, 2009). In this study, direct instruction will be captured by both the drill-and-practice intervention and the peer-mediated intervention. Drill-andpractice involves students completing a worksheet with a designated number of number fact questions over a certain period of time. Their final score is tabulated with the goal to improve the number of facts completed in that designated time. Peer-mediated interventions involves students, in groups of two, quizzing each other on number facts using flashcards. Similar to drill-and-practice, a set time is used. A significant difference between drill-and-practice and peer-mediated practice is that peer-mediated practice involves peers working together to aid the gain in number facts knowledge. Guided-discovery is often seen as a contrasting approach to direct instruction in that the learner must find the desired information independently with minimal adult guidance (Alfieri, Brooks, & Aldrich, 2010). Guided-discovery has been shown to lead to a better recall of facts as students have discovered the information for themselves; however, the amount of time that is needed for this approach is often cited as a concern (Klahr, 2009). Due to the shift in Alberta from an emphasis on direct instruction to an emphasis on guided- discovery, the strategies to develop a student’s knowledge in number facts has shifted as well. Currently, more emphasis is placed on students independently discovering the why to the solution for the number facts, as compared to memorizing the facts using a direct instruction method (Stokke, 2015). Stokke (2015) reports a statistically significant decline in math test scores in Alberta that she believes is NUMBER FACTS KNOWLEDGE 3 connected to this shift in pedagogy. Since this is concerning, it is important to examine the various teaching methodologies to determine a partial solution to this problem. In this study, the guided-discovery method will be implemented by examining strategy instruction as a teaching intervention. Strategy instruction involves the use of various teaching methodologies such as music, visuals, manipulatives, games and worksheets to teach the students the why behind the number facts. With this understanding, the goal is to increase a student’s number fact knowledge. Local educational trends. In the author’s school, the effects of the changing trends in math are felt as well. Staff have noticed a decline both in the number facts knowledge of their students and in their general math abilities as evidenced on the students’ academic performance on tests. As a result, they have made a conscious decision to increase their students’ number facts knowledge. In order to facilitate this, the Education Committee, a committee which guides all educational decisions at the school in this study, has developed a mandate which explicitly speaks to this, “The [Math] program will include a schedule based on which students will develop mastery of basic math facts and multiplication tables” (EdComMinutes, 2016, p. 1). It is realized that if students have not mastered their number facts, they typically will not be successful at math (Fuchs et al., 2008; Kanive et al., 2014) and this lack of success is one indicator of future academic and career success (Jordan & Hanich, 2000; Stokke, 2015). Number Fact Knowledge Cardinal and ordinal principles, the basic principles behind number facts knowledge are taught to students as soon as they enter kindergarten. Once they understand these concepts of numbers, they are able to move on to counting which then NUMBER FACTS KNOWLEDGE 4 progresses to addition (Ching & Nunes, 2016). Cardinality is the concept of a number having the same value regardless of what type of objects a person is looking at and the ordinal concept of a number refers to the understanding that all numbers can be arranged in order from smallest to largest (Ching & Nunes, 2016). By the time students are in third grade, they should have reached the acceptable rate of number fact acquisition (Ball et al., 2005; Fuchs et al., 2008). If they have not reached this point, these students will most likely not be able to use number facts effectively as they get older and progress through the math curriculum (Ball et al., 2005). This then, can lead to a struggle in a variety of mathematical concepts such as algebra, probability and geometry. One of the ways in which students learn new material is through the development of their working memory. Working memory is limited in its capacity and learning becomes difficult when working memory is full (Stokke, 2015). Number facts mastery will free up working memory space (Fuchs et al., 2008) which means that the student will have more working memory available for higher level math questions and problem solving (Stokke, 2015). Besides freeing up working memory, number fact knowledge helps with problem-solving (Schoenfeld, 2004), estimation, mental math and number sense (Woodward, 2006). This results in students having an increased chance of success in higher level math concepts. Difficulties in number fact knowledge. Students who face difficulties in math are found in nearly every classroom. Some of these students have been diagnosed with mathematics difficulties (MD) while others are seen as having lower math abilities (LMA) compared to their peers. Five to 10% of students in a school setting demonstrate symptoms of a mathematics learning disability (Dennis, Sorrels, & Falcomata, 2016). NUMBER FACTS KNOWLEDGE 5 These difficulties in math may be due to a variety of factors, including cognitive difficulties, attention or executive function difficulties. However, the focus of this study is students who have difficulties in math due to a lack of number facts knowledge. If these difficulties are already evident at early years, Aunio, Mononen, Ragpot and Törmänen (2016) believe that it will be an indicator of difficulties in math into their later years. This means that these students are at risk for math difficulties in high school and even into their future careers (Axtell, McCallum, & Mee Bell, 2009). One of the underlying explanations for a student’s struggle in math is their grasp of number facts (Dennis et al., 2016). Students who struggle in math typically have weaker number fact retrieval abilities (Dowker, 2005) and have immature counting strategies which leads to retrieval and counting errors (Jordan & Hanich, 2000). As a result, these students are unable to acquire their number facts efficiently. As well, these students with MD often struggle with attentive behaviour (Fuchs et al., 2006) and having a strong working memory (Fuchs et al., 2008). In order to assist students who struggle, studies have shown that interventions lead to better success for these students (Dennis et al., 2016; Woodward, 2006). This study focused on three types of interventions: peer-mediated practice, drill-and-practice and strategy instruction. It examined which of these interventions resulted in the highest level of number fact knowledge, both for those students who struggle in mathematics and those who do not. Interventions Interventions can be offered in a variety of formats: one-on-one assistance, computer-aided and peer-directed. Regardless of the format, interventions that emphasize math fact fluency have shown to improve a student’s recall of math facts (Kanive et al., NUMBER FACTS KNOWLEDGE 6 2014; Woodward, 2006). Students who struggle in math do not acquire strategies naturally and so these interventions provide the opportunity for them to be explicitly taught the concept (Woodward, 2006). The three interventions that were used in this study are interventions that are common to classroom teachers, can easily be applied in the classroom, require minimal teacher assistance and take a limited amount of time to complete. The first intervention was drill-and-practice. In drill-and-practice, students are given worksheets that contain number facts and are required to complete as many questions as possible in a limited amount of time. Studies have shown that drill-and-practice interventions can lead to increase in retention and automaticity of number facts (Burns, 2004; Knowles, 2010). The second intervention is peer-mediated practice. In peer-mediated practice, classmates assist each other with learning their number facts through the use of flashcards. This repeated practice also involves feedback from their peers with regards to which questions are right and how to correct the equations that are wrong. Through this intervention, Arnold (2012), has demonstrated that students will reach a higher level of understanding of their math facts. The third intervention, strategy instruction, develops students understanding of the ‘why’ behind the different number facts. The focus for this intervention was the conceptual understanding of the facts. This was done through the use of videos, music, manipulatives, games and worksheets. This intervention has shown to especially benefit those who struggle in math (Dennis et al., 2016; Kaufmann, Handl, & Thöny, 2003) Based on this knowledge, this study sought to examine what intervention led to the highest recall of number facts for students who struggle in math. The students’ grade NUMBER FACTS KNOWLEDGE 7 level was also examined to determine how the connection between grade level and intervention affected number fact knowledge. In order to examine this connection, the effect of interventions on number fact knowledge was examined for students in Grades 23 and for students in Grade 4-6. Definitions In order to fully understand the discussion of this study, definitions of the prevalent terms must be presented. A difference needs to be distinguished between students who have MD and students who have LMA. This study recognizes students with MD as those students who have been formally tested and diagnosed by an educational psychologist as having mathematical difficulties, while students with LMA are those students who have difficulties mastering their addition and multiplication facts. This study focuses on students with LMA. Specifically, in this study, students with LMA are those who were ranked in the bottom 25% of their class with regards to their number facts knowledge. The 25th percentile ranking was chosen based on a similar percentage used in a similar study by Kanive and colleagues (2014). This was measured at the beginning of the study. Students who have sufficient number facts knowledge are the remaining 75%. This study compares both groups in their ability to acquire number facts knowledge. In this study, students’ number facts knowledge was the dependent variable. This knowledge will be examined by focusing on student’s automaticity. Automaticity refers to a student’s quick and effortless recall of facts without being aware that they are doing so (Axtell et al., 2009; Ching & Nunes, 2016). Automaticity often has a number value ascribed to it. Woodward (2006) states that automaticity occurs when a Grade 4 student NUMBER FACTS KNOWLEDGE 8 can complete 36 facts in two minutes while Burns (2005) believes that automaticity occurs when a Grade 3 student can complete 20 addition facts per minute. Automaticity is often connected to a student’s working memory (Ching & Nunes, 2016). Working memory is a “brain system that provides temporary storage and manipulation of the information necessary for . . . complex cognitive tasks” (Ching & Nunes, 2016, p. 481). It can only hold a small amount of novel information that will be lost after 20 seconds if it is not practiced (Stokke, 2015). If it is practiced, it is moved into long-term memory. If it is not practiced, the working memory will soon become overwhelmed which will result in the inability to add new information or new information replacing the existing information. Another term that needs to be defined is interventions. Interventions are used by the classroom teacher. Interventions are particular instructions given to students in order to make their understanding of a specific part of the math curriculum stronger (Kroesbergen & Van Luit, 2003). The duration and intensity of the intervention depends on the student’s needs. The interventions in this study are provided for the benefit of all students: students with LMA and students that have a sufficient grasp of their number facts. The focus on this study was on three types of interventions: peer-mediated practice, drill-and-practice and strategy instruction. Purpose The main purpose of this study was to consider which of the three different interventions promoted an increased rate of number fact knowledge, with a particular focus on students with LMA. Much research has been completed with regards to the development of students’ number fact knowledge through the use of interventions, but NUMBER FACTS KNOWLEDGE 9 there is a limited number of studies that have examined interventions in connection with LMA. And rarely were studies found that examined if one intervention is better than another; typically, only the benefits of one particular intervention were examined. This study provided some data towards filling this gap. As well, this study examined the possibility of a relationship between the intervention used and two different categories of grade levels. The first category was Grades 2-3 and the second category was Grades 4-6. This relationship was explored to determine if one grade level category benefitted more from a particular intervention than the other grade level category. Theoretical Framework This study is grounded in the research and theory related to positive niche construction. Positive niche construction recognizes that a student will flourish if a positive environment has been constructed for them based on their strengths (Armstrong, 2012). In order to construct a positive niche, several different components are needed. Two of these components are strength awareness and strength-based learning strategies. Strength awareness speaks to the teacher’s knowledge of their student’s strengths and in turn, developing strength-based strategies or approaches that build on a student’s strengths and abilities. Students who have teachers that instruct them based on their strengths have a higher chance of success. In her book, Weinstein (2004) argues that classrooms and schools that view children based on their strengths and abilities will likely see experience positive results in these children. This study aims to examine if there are certain interventions that are better for students who are weaker in math. This is done with the NUMBER FACTS KNOWLEDGE 10 desire that these interventions will relate to the students’ strengths in order to provide them with success in their number facts knowledge. Research Questions The following research questions were considered over the course of this study: 1) Is there an interaction effect between the intervention used and students' math abilities on number fact knowledge among students in Grades 2-6? 2) If there is no interaction effect, which intervention leads to the highest rate of number fact knowledge for students in Grades 2-6? 3) Is there an interaction effect between intervention used and the grade of the student on number fact knowledge among students in Grades 2-6? Research indicates that number fact knowledge is vital to a student’s success; however, it is not entirely conclusive about the manner in which a student can attain adequate number fact knowledge. Based on the available research, the following hypotheses have been drawn. With regards to the first research question, it is hypothesized that an intervention that develops students’ understanding of the ‘why’ behind a math concept would be the most beneficial for students with LMA. This would suggest that the strategy instruction intervention would show the most success in terms of a number fact acquisition for a student with LMA. In response to the second research question, it is hypothesized that, if there is an interaction effect, the drill-and-practice will be of most benefit to the students in Grades 2-6. Students who do not have difficulties in math often have a stronger working memory (Ching & Nunes, 2016), which will lead to the most success in drilland-practice. With regards to the third research question, Grades 4-6 should perform better on the peer-mediated interventions as they are more mature in general in their NUMBER FACTS KNOWLEDGE 11 abilities to detect their partner’s needs and respond appropriately (Van Keer & Verhaeghe, 2005). The remaining interventions are hypothesized to have no interaction effect between the intervention used and the grade of the student. Ethical Considerations and Researcher Bias Several considerations had to be kept in mind throughout the study in order to maintain high ethical standards. Prior to the study, informed consent was obtained from the parents of the participants. At the onset of the study, students and parents were informed regarding the purpose of the study and the fact that parental permission may be withdrawn at any time. Once data was collected for the study, it was kept on the researcher’s computer which was password protected. As well, in the presentation of the data, it was ensured that there was no connection between the participant’s name and the results. As this research was conducted at the school where the author of the study taught, steps were taken to ensure that researcher bias was avoided as much as possible. The author did not teach any of the interventions. NUMBER FACTS KNOWLEDGE 12 CHAPTER 2: LITERATURE REVIEW The purpose of this research was to examine the effect of three unique interventions on a child’s success in number fact knowledge. Particular focus was paid to how LMA and the intervention used interacted with a child’s ability to develop their skills in number facts. Some focus was also on how grade level played a role in a student’s ability to acquire number facts. Grade level was used as a variable in order to discover if a certain grade level had a better capacity for memorizing facts than other grade levels. Various factors are important to understand this difference and these factors are explored more fully below. The potential impact of this research is examined as well. Number Facts In the literature, authors repeatedly describe the importance of math as a predictor for future academic success. Students who were able to complete challenging math content in high-school were more likely to attend college (Borman, et al., 2017). In contrast to this, students who struggle with numeracy at the beginning of their educational career were more likely to encounter learning difficulties in math as they became older (Aunio, Riika, Ragpot, & Törmänen, 2016). Balfanz, Herzog, and Mac Iver (2007) found that if a sixth-grade student failed math, there is a high likelihood that they will not complete high school. When examining the success of students in high-school classes, it has been found that these students’ success in acquiring math knowledge in the early elementary grades was a key predictor to their math success in high school (Aunio, Riika, Ragpot, & Törmänen, 2016; Fuson, Clements, & Sarama, 2015; Stokke, 2015). In the United States in 2014, 61% of fourth graders and 66% of eighth graders were below levels of math proficiency (Kanive et al., 2014). This implies that a large NUMBER FACTS KNOWLEDGE 13 number of students will struggle in their high-school math courses. In the author’s province of Alberta, a significant decline in student math scores has been noted in the 2012 Program of International Student Assessment (Stokke, 2015). These lower levels of math proficiency are concerning, given the vital necessity of math to students’ future academic success. This leads to question as to whether there is an underlying concept or principle that is foundational to math proficiency. According to Fuchs et al. (2006), this underlying principle is number facts knowledge. Number facts are a valuable component to enabling a student to be successful in word problems and higher level math that involves more than one step. The American National Research Council (Kilpatrick, Swafford, & Findell, 2001) echoed this when they concluded that a critical component to proficiency in math is a student’s ability to attain computational fluency (or number fact fluency). Interventions that have focused on developing students’ fluency in number facts have demonstrated an increase in math fact recall and problem solving skills (Burns, 2005; Codding, Archer, & Connell, 2010). Number facts refers to the single-digit manipulation of numbers (e.g. 3+2, 5x4, 20÷4, 6-2). There are three phases to the development of number facts skills: modelling, the use of strategies and mastery (Baroody, 2006; Bay-Williams & Klink, 2014). Modelling includes using manipulatives such as blocks or fingers to derive an answer. The use of strategies includes using prior knowledge and skills to deduce the answer. Mastery refers to the stage when the student can automatically and efficiently recall the answer to the question. At this stage, they just know the answer so no deep thought is NUMBER FACTS KNOWLEDGE 14 needed. As students mature, they will progress through these stages with the goal of becoming proficient at number facts. In order to move through the three stages of modelling, using strategies and mastery, students make use of several different practices that progress in levels of efficiency. Some of these practices involve unitary counting (physically counting all of the parts), double counting (strategizing that two 5’s is 10), repeated addition and decomposition (Zhang, Ping Xin, Harris, & Dink, 2014). A student’s use of a particular strategy often indicates their maturity level in terms of math abilities (Dennis et al., 2016). When examining a student’s number fact knowledge, information about what stage they are at and what strategy they use should influence the way they are taught the math concepts and math facts. Automaticity Number fact knowledge is vital and mastery is shown when it is automatic (Baroody, 2006; Bay-Williams & Klink, 2014). Automaticity, as it relates to mathematical competence, is obtained when an individual can retrieve previous solutions quicker than they can do the mental calculation (Logan, Taylor, & Etherton, 1996). It involves the ability to recall facts quickly, unconsciously, accurately, and with minimal effort (Eiland, 2014). Automaticity is different from fluency as fluency refers to a student’s ability to respond correctly and efficiently to a given question (Axtell et al., 2009). In other words, fluency leads to automaticity. This study focuses on number fact automaticity, with the underlying knowledge that this automaticity was gained through fluency in number facts. NUMBER FACTS KNOWLEDGE 15 Deno and Mirken (1977) describe mastery of number facts occurring when thirdgrade children can complete 20 or more facts per minute. Students are measured at the instructional level when they can complete 10-19 facts per minute and are measured at a frustrational level when they complete fewer than 10 facts per minute. This is echoed by Stickney, Sharp and Kenyon (2012) who cite several studies that indicate automaticity is gained when Grade 1-3 students can complete a given answer in 1-3 seconds. In order for automaticity to occur, four different properties are important (Logan, 1997). The first property is that the student will be able to recall the fact with speed. The speed with which the student can complete a set number of questions will increase considerably during the first few practices and as the practices continue on, the speed that the student can complete those questions will level out and smaller gains in speed will be noticed. Secondly, automaticity implies that students will be able to complete the question without effort and are able to complete an additional task concurrently. The third property is that the task can be completed without intentionality and the final property is that the task is completed with lack of conscious awareness. These four properties will be evident in someone who has achieved automaticity. As students’ progress towards this goal, they may obtain these four properties at different rates and in different order, but automaticity will be seen when all four properties are evident Automaticity of number facts is essential as it increases the amount of working memory and attention available for other mathematical tasks. Individuals have a certain amount of working memory available to them and the more working memory that is available for complicated math tasks, the higher the likelihood of the student being able to perform these higher-level math tasks (Hasselbring, Lott, & Zydney, 2006; Woodward, NUMBER FACTS KNOWLEDGE 16 2006). For example, when calculating long division, students who can automatically recall their facts may be better able to focus on the algorithm required for the long division, while students who struggle with automatic recall may become frustrated as they spend more time calculating the number fact components than performing the algorithm. Automaticity in number facts is also important as number facts are a building block to many other math skills. Just as children need to be able to walk automatically in order to do advanced tasks like running and jumping or to automatically know the sounds for the words for advanced tasks like reading and comprehension, so children need to be able to automatically recall their number facts so that they can efficiently calculate answers to fraction addition, long division, sinusoidal functions and other mathematical algorithms. Cognitive Skills needed for Automaticity. In order to automatically recall number facts, a variety of skills are needed. These skills frequently complement each other and so a deficit in one skill often impacts others. One of the major contributors to these difficulties in the ability of a student to automatically recall their facts is attentive behaviour (Fuchs et al., 2008). This refers to the controlled attention that one can put to a task. If a student is able to focus on the task and not be distracted by classroom activities and noises, they will perform significantly better at their task (Fuchs et al., 2006). This is due to the amount of perseverance that they are exhibiting. In other words, a student will be more successful in their display of number facts knowledge if they are able to focus on their work. Another key contributor to an increase in number facts knowledge is working memory (Dennis et al., 2016; Fuchs et al., 2008). The student’s working memory can NUMBER FACTS KNOWLEDGE 17 only retain a certain amount of information. Once the working memory is full, the individual is unable to learn new pieces of information (Stokke, 2015). This means that a person’s ability to learn new information is limited by the amount of information already in that person’s working memory. For students who do not yet have their facts memorized but are expected to do higher-level math, their working memory will quickly become overwhelmed as it has to learn the new material, plus concentrate on the number facts that this material is relying on. They will have a higher cognitive load (Woodward, 2006). Other cognitive processes that automaticity relies on are processing speed (Fuchs et al., 2006; Fuchs et al., 2008; Jordan & Hanich, 2000), phonological decoding, and long-term memory (Fuchs et al., 2008). Processing speed refers to how efficiently an individual can perform a certain task. Phonological decoding is an individual’s ability to hold symbols in their working memory so that they will be able to establish representations in their long-term memory. Lower Mathematical Abilities In this study, students with LMA are those who struggle with number facts. Literature often refers to those who have been formally diagnosed with difficulties in math as students with MD (Dennis et al., 2016; Fuchs et al., 2008; Kroesbergen & Van Luit, 2003). This study uses the term LMA to encapsulate those students who struggle in math, but are not necessarily diagnosed using standardized testing. Students with LMA are usually weaker at number fact retrieval than any other facet of math (Woodward, 2006). This struggle will affect other areas of math as well. In one study, number facts and attentive behaviour were the only two predictors of a student’s ability to complete NUMBER FACTS KNOWLEDGE 18 math algorithms (Fuchs et al., 2006). As number facts are crucial to success in math, these students will have difficulties in math, most likely from the beginning of their education to the end (Fuson et al., 2015). One explanation for this difficulty in number fact retrieval is that students reach incorrect solutions using inefficient strategies. For example, some students do not progress beyond using their fingers until much later than their peers (Dennis et al., 2016; Hanich, Jordan, Kaplan, & Dick, 2001). In their study, Hanich et al. (2001) assessed 210 Grade 2 children by giving them seven different mathematics tasks to examine their basic calculation, estimation and problem solving skills. They found that children with MD performed worse with regards to number fact retrieval than children who did not have any difficulties. Some of this struggle was attributed to a higher reliance on their fingers when compared to their peers without MD. Another explanation for a student’s experiencing difficulty in number fact retrieval is that they are lacking the cognitive skills needed to develop automaticity of number facts. Fuchs et al. (2006) conducted a study to examine how arithmetic (the addition and subtraction of single digits) was connected with working memory, processing speed, phonological decoding, attentive behaviour and long-term memory. This was determined using a battery of tests over five testing sessions on 312 Grade 3 students. These tests included a math fact fluency test, a double-digit addition and subtraction test, a story problem test and 14 smaller tests that examined language development, working memory, processing speed and other cognitive functions. Using the results from these tests, they determined that attentive behaviour, phonological decoding and processing speed were all significant predictors for success in adding and NUMBER FACTS KNOWLEDGE 19 subtracting single digit numbers. Through this, they concluded that if a student struggled with attentive behaviour, phonological decoding or processing speed, the student would most likely struggle with the addition and subtraction of number facts. In the abovementioned study, working memory did not have a significant effect on the outcome, but other studies have shown this also as a factor in a student’s ability to automatically recall facts. If students are unable to automatically recall facts, their working memory will reach capacity much quicker and so they will experience a high cognitive load (Woodward, 2006) which will lead to learning being difficult and information being lost (Stokke, 2015). Lower mathematical abilities can be detected as early as kindergarten (Fuson et al., 2015; Woodward, 2006). A child’s math success in kindergarten is one of the key predictors of success in higher grades (Fuson et al., 2015). By the time the child who struggled in kindergarten reaches Grade 3, serious deficits will be noticed (Fuchs et al., 2008). Those students who do not develop strategies naturally often will need targeted interventions in order to experience growth (Woodward, 2006). These students progress slowly through the stages of math development or they may not even progress through these stages at all but will skip stages or stay at one stage (Baroody, 2006). Educators are not only concerned about students with LMA because these students will struggle during their school years, but because they know that these difficulties are indicative of more long-term complications. Students who fail math in Grade 6 are at a high risk of not graduating from Grade 12 (Balfanz et al., 2007). As well, students who struggle in math may be at risk for success in future careers and post-secondary education opportunities (Fuchs et al., 2008; Zhang et al., 2014). NUMBER FACTS KNOWLEDGE 20 All of these reasons together make it vital that research and intentional classroom practices be used to assist those who experience difficulties in math. Grade Level Students in schools in Alberta start learning their addition number facts in Grade 1. At this stage, they are required to use mental math strategies for facts with a sum up to 18. In Grade 3, they start working on recalling multiplication facts up to and including 5x5 and by the time they complete Grade 5, they need to be able to recall multiplication facts up to and including 9x9 (Alberta Education, 2007). Other provinces and countries have similar expectations, in that by the time a child enters Grade 5, they need to be comfortable with both their addition and multiplication facts. However, students are not always as proficient with their facts as expected. As such, it is important to know if there is an ideal grade in which to really encourage students to learn their facts. Conversely, it is important to know if there is a grade when it is impractical to teach students their number facts as it will take additional work or students may be overburdened with other mathematical content that they are required to study. When students are introduced to the concept of addition and multiplication, they should naturally progress through a series of strategies to learn these facts. One of the first stages is counting all the objects (Baroody, 2006). In this stage, to calculate an equation like 4+3, the student will lay out four items and three items and then count each item individually to determine the total. This time-consuming stage transitions to counting up in kindergarten or in the beginning stages of Grade 1 where students, when calculating 4+3, will still lay out the items (or use their fingers), but will recognize that they can start counting at either three or four and then move up from there (Bay-Williams NUMBER FACTS KNOWLEDGE 21 & Klink, 2014). If this does not happen, then this is one of the first occurrences when difficulties between a child’s ability to complete facts is noticed. This happens around eight years of age (Hasselbring et al. 2006). According to Fuchs et al. (2008), students’ ability to automatically recall their number facts should be at expected levels by Grade 3. A review of the literature though ERIC, JSTOR and Sage using search terms such as “age number facts”, “age math facts”, “grade number facts” and “grade math facts” failed to reveal further studies that gave clear indication of an optimal grade in early elementary school years by which students should be able to automatically recall their number facts. There is a paucity of research in the areas of upper elementary years as well with regards to math fact fluency. Nelson, Parker and Zaslofsky (2016) examined how Grade 4-8 students’ growth in math fact fluency predicted their achievements on state math assessments. Using regression models, they observed that statistically significant and positive effects were observed for the scores from two state assessments and the student’s average weekly growth (Nelson et al., 2016). Through their results, they were able to demonstrate that there is still value in instructing students in their math facts, even up to Grade 8. As well, it was difficult to find studies examining if there is a connection between grade level and an ideal intervention. Searches were done using key words such as ‘age and intervention’, ‘grade math intervention’, ‘grade drill-and-practice’, ‘grade strategies intervention’, etc. Although no definite conclusion was reached by the various studies regarding an optimal intervention based on age, the authors agreed both on the value of the automaticity of number facts and the need for interventions for those who were not meeting the standards. NUMBER FACTS KNOWLEDGE 22 Interventions As noted earlier, students with LMA typically struggle because of their ability to automatically recall number facts (Jordan & Hanich, 2000; Woodward, 2006). In their longitudinal investigation, Jordan, Hanich and Kaplin (2003) examined second and third grade students and their mastery of addition facts. They concluded that students who had poor fact mastery at the beginning of second grade showed very little growth in their number fact skills over the duration of Grade 2 and 3, implying that without help, the difficulties will remain. These students also performed worse on their multiplication tables when compared to their peers. The difficulties these students faced will follow them into their middle school and high school years and subsequently into their postsecondary education or careers (Axtell et al., 2009). As a result, the need for interventions has been stressed repeatedly by the various authors. This need for interventions has been demonstrated by assorted studies. Kanive and colleagues (2014) examined the effects of a computer-based intervention compared to a conceptual understanding intervention with regards to Grade 4 and 5 students’ computational fluency and problem-solving skills. Woodward (2006) developed an experimental study to determine the benefits of integrating strategy instruction versus the benefits of timed practice drills with regards to multiplication facts. Dennis et al. (2016) examined how number sense intervention and extensive practice intervention impacted the ability of a student to retrieve number facts automatically. Many other studies have had similar purposes: examine how specific interventions assist or hinder the development of number facts. In general, it was found that interventions were beneficial NUMBER FACTS KNOWLEDGE 23 to students with LMA (Woodward, 2006). The research also indicated that interventions that focused on number facts are effective (Kroesbergen & Van Luit, 2003). Intervention Layout. There are three types of math knowledge: declarative, procedural and conceptual. Math interventions typically fall under the declarative and conceptual areas. Declarative knowledge refers to the understanding about the facts of math. These facts are the building blocks to all other math concepts. Drill-and-practice and peer-mediated interventions would fall under this category. Conceptual knowledge examines the ‘why’ behind a math concept. This math knowledge connects all the pieces of knowledge that the student already has to enable them to receive a full understanding of the concept. Strategy instruction interventions fall under this category. Most interventions do not fall under the category of procedural knowledge which is that type of knowledge that refers to the rules or procedures used to complete a math algorithm (Hasselbring et al., 2006). Successful interventions have several key components. One component presented by Skinner (Cates, 2005) is that they contain an A-B-C layout. Interventions should have an antecedent (A) which involves some type of student-specific task, a behaviour (B) which would necessitate a student response and a consequence (C) in that immediate feedback is provided in connection to the student response. Careful consideration was given to this concept when developing interventions for this study. Fuchs et al. (2008) expanded on this by providing seven principles for effective intervention. These principles are: instructional explicitness, instructional design to minimize the learning challenge, strong conceptual basis, drill-and-practice, cumulative review, motivators and ongoing progress monitoring. Of the seven practices, ongoing progress monitoring is NUMBER FACTS KNOWLEDGE 24 often seen as the most important one. This is echoed in the study by Zhang et al. (2014) in that they demonstrated that feedback is one of the primary beneficial components of the various interventions that they studied. These seven principles will be used to develop and evaluate the effectiveness of three interventions in this current study: drill-andpractice, peer-mediated instruction and strategy instruction. Drill-and-practice. The first intervention studied, drill-and-practice, is an intervention that involves students completing a math worksheet on a regular basis, often for a set amount of time. This worksheet involves multiple questions of similar format. This concept is based on Thorndike’s law of frequency which states that the more two stimuli are seen together, the stronger the connection between the two stimuli (Eiland, 2014). In the number facts scenario, this is implying that the more frequent a student sees an equation such as 4+3 (the first stimuli) next to the number seven (the second stimuli), the stronger the connection will be between 4+3 and seven and soon the student will be able to recall the correct answer more efficiently. Studies have shown that the drill-andpractice interventions can lead to increase in automaticity and retention of number facts (Burns, 2005; Knowles, 2010). In her study, Knowles (2010) examined if written, timed practice drills had an impact on students’ abilities to automatically recall number facts. The dependent variable was the difference in means between the pre-test and post-test. The pre-test and post-test were similar and each contained 111 basic multiplication fact problems. The focus of this study was Grade 6 students in a single general education classroom. She included two experimental groups: students who received weekly timed practice drills and students who received daily timed practice drills, and one control group that did not receive any NUMBER FACTS KNOWLEDGE 25 practice. Using the difference in means between the pre-test and post-test, Knowles (2010) determined that written-timed tests significantly impacted the students’ rate of automaticity. In particular, daily written timed tests show significantly more improvement than weekly timed tests. Although this study only examined one classroom at one school and did not examine the value of written, timed math practice drills for students with math difficulties, it still speaks to the value of developing written, timed math facts as an intervention. Drill-and-practice is a direct instruction teaching method. Direct instruction methodology occurs when a teacher gives explicit teaching instructions and students need to follow them directly (Klahr, 2009). When contrasted with guided-instruction, direct instruction methodology has been found to be the most effective for the learning of basic math skills (Kroesbergen & Van Luit, 2003). As well, direct instruction has been found to be most successful when engaging with students who have mathematical difficulties (Zhang et al., 2014). Drill-and-practice has found to be the most beneficial when new facts are interspersed with old facts. No ideal ratio has been determined, but a ratio of new facts to known facts that has less than 50% unknown facts is ideal (Burns, 2004; Burns, 2005). In the meta-analysis conducted by Burns (2004), 13 articles were examined to find an ideal instructional ratio for drill tasks based on the strongest effect size. An instructional ratio was defined as the ratio found in drill-and-practice type tasks in which the amount of new material is compared to the amount of known material. In the consideration of 13 relevant, current articles, it was found that any ratio that consisted of less than 50% new material resulted in a strong effect size which signified that drills tasks would be more NUMBER FACTS KNOWLEDGE 26 effective at this ratio. Although a strong effect size does not always translate into significant results, in this study, the author used the strong effect size to conclude that worksheets that contain 30% new facts/70% old facts, 15% new facts/85% old facts, etc., all have a similar benefit – as long as the amount of new facts do not progress to more than 50% new facts. A concern with this meta-analysis is that it only focused on studies that had significant effect sizes which may have skewed the results. There are some concerns about drill-and-practice being used as a method to develop number fact knowledge. This intervention may not lead to transfer of knowledge nor the provision of back-up strategies for the students when they encounter questions that are slightly harder or have different values (Burns, 2004; Dennis et al., 2016). Another concern is that drill-and-practice promotes rote memorization rather than conceptual understanding. Since students have focused on memorizing the facts, they have not been taught to look for patterns or to reason through the question. This may result in a struggle for students to apply their facts in the math classroom. For many students, this method of memorization is often more difficult and causes them to feel overwhelmed. This then leads to the students giving up or losing interest in math (Baroody, 2006). One last concern with drill-and-practice is that it could reinforce student’s immature math strategies (Woodward, 2006). As drill-and-practice focuses on the memorization of facts, students are not given opportunity to develop their personal strategies and so may remain at an immature strategy (such as counting all objects) for longer than what is typical. Peer-mediation. Peer-mediation intervention involves the classmates working together to build up each other’s knowledge. This intervention, when used in connection NUMBER FACTS KNOWLEDGE 27 with the development of math facts, is used to enhance a student’s declarative knowledge of math. In order for it to be successful, it should only be used after information has been introduced to the student (Maheady, Harper, & Sacca, 1988). Extensive practice – an intervention similar to peer-mediation – has been shown to be conducive to the improvement of a student’s ability to automatically recall number facts (Dennis et al., 2016). In this particular paper, peer-mediated instruction will be evident through students quizzing each other on number fact knowledge using a specific flash-card routine. Several older studies have examined the effects of peer-mediated intervention for both the tutee and the tutor. In 1987, Maheady et al. (1988) examined how a class wide peer tutoring program affected weekly math exams. The composition of the various classes was made up of both students that were considered mildly handicapped and regular education students. The authors discovered that math scores increased by about 20 percentage points through the use of class peer tutoring program and the students who were mildly handicapped no longer received a failing grade and some obtained marks above 90%. Similarly, Fuchs, Mathes and Simmons (1997) examined the effect of peerassisted tutoring on academic achievement for students who struggle. Over the course of 15 weeks, 40 different classrooms were randomly selected to either use peer-assisted learning strategies to develop reading strategies or to serve as a control group and not use this peer-tutoring methodology. Through observation checklists, a comprehensive reading battery, and teacher and student questionnaires, it was determined that students with learning disabilities, students that were performing at lower levels and students of average achievement all made significantly greater progress using peer-assisted tutoring than the students who did not use this strategy. The results of this study are most likely NUMBER FACTS KNOWLEDGE 28 applicable today, but there are some concerns of its current relevance due to the age of the study and the changing definition of LD. In more recent history (2000-2017), the effects of peer-mediated intervention seem to not have not been studied as extensively. This author has used key words such as “peer-mediation”, “peer tutoring”, “flash cards” and “peer-mediated interventions” and only found a few studies on the value of peer-mediated interventions that did not involve other aspects. In 2005, Van Keer and Verhaeghe conducted a study that examined the effect of peer tutoring on both second and fifth grader’s reading comprehension. They found that, for second graders, same age tutoring was not conducive to increased learning. However, for the fifth graders, same age tutoring resulted in improvement in reading comprehension, regardless of the capabilities of the student (Van Keer & Verhaeghe, 2005). Although this study reflected on the value of peer-mediated interventions with regards to reading comprehension, it is applicable to this study in that the methodology was similar. When students were directly taught how to tutor their peers, significant improvement was noticed in the tutee’s math skills (Shamir, Tzuriel, & Rozen, 2006). Besides improved academic quality, peer tutoring also may positively impact students’ attitudes towards each other, even if there was a significant negative portrayal of a student beforehand (Byrd, 1990). Although it was difficult to find studies that directly examined the impact of peermediated interventions, many of the underlying principles behind this intervention have been endorsed. Interventions that provide repeated practice enable a student to apply the concept quicker (Arnold, 2012; Burns, 2004). In peer-mediation, flash-cards provide this repeated practice. As well, the procedure of combining known facts with unknown facts NUMBER FACTS KNOWLEDGE 29 has demonstrated better results for students with and without learning disabilities (Burns, 2004; Burns, 2005). This can happen naturally in flashcards in that students typically know some of the facts already and these facts are interspersed with unknown facts. Alternatively, the facilitator can deliberately set up the peer-mediated intervention in such a way that students will get a certain ratio of known to unknown cards. Another principle behind peer-mediated interventions is that students are dialoguing with each other in order to offer feedback. This method of responses (yes, you are correct; no, you are wrong) and consequences (the correct answer is ___) leads to increase academic performance (Cates, 2005). This method of feedback then results in a higher level of understanding (Zhang et al., 2014). Besides the lack of current studies providing support for peer-tutoring, a significant concern with this intervention is that peers struggle with perceiving the needs of fellow-classmates (Kroesbergen & Van Luit, 2003). Due to this, peer-mediated interventions may be more viable for students in older grades (Van Keer & Verhaeghe, 2005). Strategy Instruction. Strategy instruction intervention occurs when students’ recall of number facts is enhanced through their development of strategies that promote understanding (Dennis et al., 2016). This intervention is used to develop a student’s conceptual understanding of math. A student who has a conceptual understanding of a math concept will understand the why behind that math concept and in what context it is valuable. Students with this type of understanding are more likely to retain their math facts as they can connect these facts to the whole picture of math (Kilpatrick et al., 2001). It has been argued that students cannot fully comprehend and apply math skills unless NUMBER FACTS KNOWLEDGE 30 they develop their conceptual understanding (Kaufmann et al., 2003). The development of conceptual understanding through strategy instruction interventions also enable students to organize the facts that they do know into a systematic pattern that will help them with retaining and recalling their facts. Another benefit that is ascertained from strategy instruction is that students will perform better on tasks that require approximation skills or that expect students to do slightly more difficult problems (Woodward, 2006). Woodward studied how providing two different types of multiplication facts interventions to students with and without math difficulties would affect their automatic recall of number facts. These 58 participants were students in Grade 4 that were either a year behind in their math knowledge or had been formally diagnosed with LD. The two math interventions used an integrated strategies group and a timed practice only group. The integrated strategies group received both strategy instruction on the why behind a math problem and timed practice worksheets, as compared to the timed practice only group which received the direct instruction of the facts followed by timed practice worksheets. Using several different measures, the author examined the impact these interventions had on students’ ability to complete multiplication facts quickly and at a harder level, as well as if the interventions led to students having a superior performance on extended tasks and approximation tasks. After four weeks of instruction, the integrated strategies group did significantly better on both the extended facts test and the approximation test. An additional side benefit of using the integrated strategy intervention was that students’ attitudes towards math demonstrated improvement. However, due to a small sample size, Woodward was not able to use inferential statistics NUMBER FACTS KNOWLEDGE 31 to examine the impact the strategies had on students with math difficulties. When examining them descriptively, it can be seen that both groups improved on their knowledge of number facts, although the group with math difficulties was still below the mastery level. There are three stages through which student’s progress as they master their number facts: counting strategies, reasoning strategies and mastery (Bay-Williams & Klink, 2014). Students often try to jump from stage 1 (counting strategies) to stage 3 (mastery); however, this may not result in the student having a full understanding of the concepts. This strategy instruction intervention enables students to progress through each stage appropriately and includes a special focus on the reasoning strategies stage. This will give students the ability to determine answers to questions that they don’t automatically know, regardless of whether these questions are at the same level or slightly harder (Baroody, 2006). Strategy instruction involves both explicit instruction and students’ engagement with the various approaches needed for them to understand and memorize their facts. This can be done via games (Bay-Williams & Klink, 2014), videos, group work or teacher-led instruction (Kling, 2011). Addition and multiplication both have their own types of strategies. When developing fluency in addition, the two main strategies are doubles (e.g. 5+5, 7+7) and combinations that make 10 (e.g. 7+3, 6+4). If students can master these, they then will be able to obtain the majority of the other addition facts (Kling, 2011). Further strategies may involve commutativity and the connection between addition and subtraction (Ching & Nunes, 2017). Commutativity is the concept that 3+4 is the same equation as 4+3. In NUMBER FACTS KNOWLEDGE 32 multiplication, the strategies focus on underlying principles behind multiplication such as repeated addition (e.g. realizing 3+3+3+3 is the same as 4x3), unitary counting (e.g. when calculating 2x3, students draw tally marks and count the tally marks individually) and decomposition (breaking down an equation into simpler equations; e.g. to solve 3x6, first calculate 2x6 and then add 6 more) (Zhang et al., 2014) Other strategies are squares (a number multiplied by itself), doubles (a number multiplied by 2) or halving, then doubling (e.g. to calculate 6x4, first calculate 3x4 and then double the product) (Woodward, 2006). Pictures and mnemonics are also important when developing fluency with number facts (Wood & Frank, 2000). Studies have also focused on the benefits of strategy instruction for students with LMA. In their study, Dennis et al. (2016) examined the effects of two basic math facts interventions on students’ efficient use of counting strategies and in the students’ abilities to generalize their skills. The two interventions, number sense instruction and extensive practice, share similar characteristics to strategy instruction and drill-and-practice, respectively. The six Grade 2 students involved in this study had a mathematics learning disability for which they were on an individual educational plan. Three of these students developed skills using number sense instruction for sixteen sessions, followed by extensive practice instruction for sixteen sessions. The remaining three students were instructed in these interventions in the opposite pattern. These sessions involved a complicated, but well laid out script that was administered by graduate students. Once these sessions were complete, students were tested to determine if there was a change in their use of strategies and their ability to generalize. It was found that the number sense intervention resulted in these students having more success in generalizing their math NUMBER FACTS KNOWLEDGE 33 skills. This meant that students experienced more success when doing questions of a slightly different variation and difficulty level after they had received strategy instruction. However, due to the rigorous and intensive nature of these interventions, these results may not apply to a situation where students are in general classroom settings. In another study, it was determined that students with developmental dyscalculia exhibited clear benefits when their understanding of the math was guided from the concrete to the abstract using a strategy intervention (Kaufmann et al., 2003). In this particular study, the results of eight different mathematical sub-tests were used to examine the effectiveness of an intervention program that focused on numerical and arithmetical problem areas for children with a mathematical learning disability. The six Grade 3 students with developmental dyscalculia were provided with an intervention program three times per week for six months for 25-minute time slots. This intervention, through the use of modules, focused on skills such as memorization of numerals, memory for subtraction facts and procedural knowledge of division. Each skill included the direct teaching of the conceptual knowledge. All six children displayed positive, significant effects on the majority of the sub-tests. When the sub-tests were computed as a whole, the intervention provided a significant, positive effect for all children. Through this study, it was demonstrated that students with dyscalculia experienced success when given a strategy instruction intervention. The study did not provide an explanation for the subtests that only gave a partially significant result. In another study, Tournaki (2003) examined if strategy instruction would enable students with a learning disability (LD) to increase both their accuracy and speed on a single-digit addition sheet. These 84 urban Grade 2 students were drawn from one school NUMBER FACTS KNOWLEDGE 34 district in New York City. They were divided into groups based on having or not having a learning disability. From there, they either were taught their number facts using a drilland-practice method, a strategy instruction method or no formal number fact instruction was given. If instruction was given, it was over eight 15-minute individually administered sessions. At the conclusion of these sessions, all students wrote a 20-question addition fact test in order to calculate their accuracy and speed. It was determined that students without LD improved on both their accuracy and speed, regardless of the strategy used, while students with LD improved significantly on both accuracy and speed when strategy instruction was used. This led to the conclusion that teachers, when deciding on an instructional strategy for number facts, can base their decision on what is best for students with LD, which in this case was determined to be strategy instruction. Students without LD will benefit from any strategy. Some limitations noted in this study are the short duration of the interventions and its narrow geographic field. As well, no discussion is held regarding the diverse aspects of LD and how this could impact the results. One concern with the strategy instruction intervention is time. It takes a considerable amount of time for students to become familiar with the strategy and to adapt the strategy for their own (Klahr, 2009). Another concern is the assumption that strategy instruction will lead to automaticity. This is not always the case, particularly for students with LMA (Woodward, 2006). A third concern is that strategy instruction can lead to an overburden of the student’s working memory. Since this intervention involves a lot of information, students’ working memory can become overwhelmed and they may not master the techniques needed (Stokke, 2015). NUMBER FACTS KNOWLEDGE 35 Summary In summary, the literature has shown that there is a clear need for students to have a strong understanding of their number facts. This strong understanding should lead to automaticity. As well, the literature demonstrates that interventions lead to significant improvement in students’ ability to know their facts, regardless of their abilities in math. Much of the literature focuses on interventions that are provided individually for a small group of students. However, not much literature was found regarding the success of interventions provided in the classroom for students who struggle with math. These types of interventions would need to be a natural fit to the teaching dynamics and easy to implement and maintain in order for teachers to continue the intervention for an extended period of time. The purpose of this study is to add to the research on classroom interventions for students who have LMA. To that end, this study examines the impact of three different interventions on students who struggle whereas the majority of other studies examined the impact of one intervention on students who struggle. NUMBER FACTS KNOWLEDGE 36 CHAPTER 3: RESEARCH METHOD Research Design How research is designed depends largely on the paradigm that is driving it. This research was driven largely by a post-positivist worldview. Under this worldview, it is understood that although the researcher introduces limitations to the study, there is a certain amount of the study that can be value-free, although some bias will still be present (Mertens, 2015). This paradigm attempts to remove biases from the researcher and therefore a certain amount of objectivity can be assumed in the results (Mertens, 2015). A benefit of this paradigm is its ability to develop a stronger case of the reality being studied. Under this paradigm, this researcher chose to use quantified experimental research using experimental design. Quantified experimental research refers to the ability of the researcher to investigate data to establish if there is a difference between two or more groups (Locke, Silverman, & Spirduso, 2010). One reason for this was that the researcher wanted to determine if there was a cause-and-effect relationship between two variables: interventions, and number fact knowledge. This was with the understanding that there is no way to fully assert that one or two causes result in a certain effect. This is due to the nature of confounding variables and how it is difficult to control for them. In this research, some of the confounding variables were the teaching style of the teachers involved, the mood of the students, and the time of day in which the intervention took place. As many of these variables were controlled as possible in order to make a wellinformed, precise conclusion (Mertens, 2015). In this study, these variables were controlled through the use of a checklist completed by teachers to monitor the adherence NUMBER FACTS KNOWLEDGE 37 to the interventions as well and through the teachers choosing the best time of day to conduct the interventions based on their perceptions. The quantitative aspect of this research enabled the researcher to examine the different trends that arose and provided easily accessible, explicit information (Ryan, 2015). As a quantitative study, identifying the independent and dependent variables is important. In this study, the dependent variable was the number of answers students correctly completed on written, timed number facts tests and the independent variables were the interventions used, the grade of the student and whether or not the student had LMA. Students with LMA were determined by examining the number facts marks prior to the implementation of the study. None of the grades contained students who were formally diagnosed with MD or had an Individual Program Plan (IPP) that addressed MD. As a result, this designation was not used as a consideration for LMA. Stratified random assignment was then used to create the three groups needed for the interventions. For the first step, students with LMA (N = 17) formed one stratum and were randomly assigned to one of the three interventions. In order to randomly assign students with LMA, each name was written on an individual slip of paper. These names were then divided into three groups. These three groups were then designated as drilland-practice, peer-mediated practice or strategy instruction. The remaining students (N = 48) formed the second stratum and were then randomly assigned to one of the three interventions using the similar process. Table 1 outlines the composition of each group. This stratified random assignment controlled for the John Henry effect and effects due to maturation, testing and instrumentation (Mertens, 2015). NUMBER FACTS KNOWLEDGE 38 A post-test only group design was used for data collection. A pretest was not needed as the randomization that was used to create the groups ensured that there would be a lack of bias in the final results (Mertens, 2015). However, a screening test was given to all students at the onset of the interventions in order to calculate the internal reliability of the tests and to designate the students with LMA. Table 1 Frequency of Students in each Intervention Group Intervention Grade Peer-Mediated Drill-and-Practice Strategy Instruction 2 3 6 (3) 6 (1) 4 (0) 5 (2) 2 (0) 4 (1) 4 5 2 (0) 4 (1) 5 (2) 4 (2) 5 (1) 8 (1) 6 2 (1) 4 (0) Total 20 (6) 22 (6) Note. Number of students with LMA is in brackets 4 (2) 23 (5) Participants Research took place in a small, private K to Grade 12 school in the fall term of the 2017/2018 school year. This school is located in southern Alberta and educates 151 students with 11 teaching staff. All students in this study were drawn from the Grade 2-6 students at this particular school, which was a sample of convenience. Every student in these grades participated, with the exception of one student who did not participate due to his severe special needs (Autism Spectrum Disorder). These students (N = 65) were from five different classrooms. Grade 1, 2, 3 and 4 were all taught separately, whereas Grade 5/6 was taught as a combined class. However, they were separated for math class, so for the purpose of this study, it was seen as two separate classrooms. NUMBER FACTS KNOWLEDGE 39 Students that participated in this study shared similar characteristics. The majority of the students came from stable, two-parent households. All but two students either came from a rural setting or a small town which required them to use bus transportation to get to school. Due to the researcher being part of the school community for many years, general observations of students and their parents would suggest that, on average, the students fall within the mid-range of socio-economic status. As this is a Christian school, all students have a similar religious background and share many values and beliefs. The results of this study can be generalized to other students with these similar characteristics. Students that are coded with a special needs code (except for the individual mentioned above) were included. Grades 2, 3 and 4 each have one student with ADHD, Grade 5 has one student with ESL and Grade 6 has one student with ESL and one student with multiple medical needs. However, it was determined that these students would be able to perform well in this study and would not experience undue hardship due to excess mental fatigue, anxiety or a sense of feeling overwhelmed and therefore they were included. The percentage of students with codes (9%) in this study was similar to the number of students with codes in the entire school population (7%). Table 2 shows the distribution of the grades, as well as the breakdown with regards to gender, special needs coding and LMA. NUMBER FACTS KNOWLEDGE 40 Table 2 Grade Composition of Study Participants Grade Grade Size Male Female Students with codes Students with LMA 2 3 4 12 15 12 4 (33%) 7 (47%) 6 (50%) 8 (67%) 8 (53%) 6 (50%) 1 (8%) 1 (7%) 1 (8%) 3 (25%) 4 (26%) 3 (25%) 5 6 16 10 8 (50%) 7 (70%) 8 (50%) 3 (30%) 1 (6%) 2 (20%) 4 (25%) 3 (30%) Total 65 32 (49%) 33 (51%) 6 (9%) 17 (26%) Students in Grade 1 did not participate as they had not yet covered the concept of addition in their school curriculum. Students in Grades 7-12 did not participate due to the assumption that, at this level, students have mastered their number facts and so interventions would not have as large an impact. Measures At the onset of the study, a screening test was administered. This screening test was comprised of 50 randomly selected addition questions and 50 randomly selected multiplication questions arranged in random order. This test was built using an online website. Addition questions ranged from 1+1 to 9+9 and multiplication questions ranged from 1x1 to 10x10. This test was identical to the post-test (see Appendix C for the posttest). Students were given 5 minutes to complete this test. Grade 2 and 3 students were permitted to skip any multiplication questions that they did not know; however, all students had to complete the test in a left-to-right pattern across the rows and moving down the rows from top to bottom. If students completed one sheet, they were given as many additional sheets as needed. Once sheets were completed, the number of correct, complete answers were counted. Both questions that were answered incorrectly and NUMBER FACTS KNOWLEDGE 41 questions that were not answered were not included in this calculation. The purpose of this test was to calculate the coefficient alpha to measure the internal reliability of these tests and to determine which students would be designated as LMA. At the conclusion of the interventions, a post-test was given. The post-test was nearly identical to the screening test. It was developed using an online website that randomly placed 100 addition and multiplication facts on a sheet. There were 50 addition facts with sums up to 20 and 50 multiplication facts with factors up to nine. These facts were randomly interspersed throughout the test. Students were familiar with the concept of written, timed number facts tests; however, the blend of addition and multiplication facts was not as familiar. As such, multiplication facts were high-lighted in grey (see Appendix C for the post-test). This was to avoid students confusing a multiplication sign for an addition sign and vice versa. This highlighting of the facts is the only way the posttest was different than the screening test. In all other ways, this post-test was similar to the screening test. Students were given five minutes to complete test, were supplied with as many sheets as needed, could skip the multiplication questions if they were in Grades 2 and 3 and were required to complete the test in left-to-right order. Both addition and multiplication facts were included in this test in order to enable the researcher to use the same test for all grade levels. In order to determine the internal consistency of the post-test, the coefficient alpha was calculated using SPSS. Coefficient alpha is used to determine if individual questions on a test produce results that are dependable with the rest of the test (Field, 2013). In order to obtain the needed results, prior to the onset of the interventions, all students were given the same test that would later be used as a post-test. Students were NUMBER FACTS KNOWLEDGE 42 given five minutes to complete it. This test was then marked and all 100 questions were tabulated with a code of 1.00 indicating an incorrect answer and a code of 0.00 indicating a correct answer. Procedure for Data Collection This study was designed in the spring and summer of 2017 in consultation with faculty at Trinity Western University. On September 15, 2017, approval was received by Trinity Western’s Research Ethic’s Board. Concurrently, consent was received from the administrator of the school where this study would take place. Once approval was received, staff were informed about the study and given instruction regarding the implementation of it on a Professional Development Day. Following this, permission forms were sent home to the parents with students in Grades 2-6 to inform the parents about the study and to obtain consent (see Appendix A for consent forms). These forms requested passive permission and stated clearly that by parents not returning forms to school, they were giving permission for their child to participate in the study. This was seen as an ethical method of obtaining permission as participants in this study would be receiving instruction in the same content even if the study was not ongoing. No forms denying permission were returned to school and so the study commenced with 100% of the students in Grades 2-6. However, it was soon apparent that one student with autism would experience undue stress by completing this study and so he was not included in it (as mentioned in the Participants section). This study was started and completed in the fall of 2017. It involved a 12-week time span: 1 week to instruct the students about the interventions and to familiarize them with the routines, 10 weeks for the interventions and one week following the NUMBER FACTS KNOWLEDGE 43 interventions for executing the post-test. During the week prior to the study, students were informed briefly about the purpose of the study and emphasis was placed on the students understanding and doing the correct procedures for the three interventions. For example, for the students involved in the peer-mediated intervention, directions were given on how to mark down the number of correct and incorrect responses and what to say when the responses were correct and incorrect. During this trail week, concerns were addressed and complications were ironed out to ensure that the 10 weeks of intervention would run smooth. As well, the screening test was given during this week. At the start of the 10-week intervention period, teachers were given a script to go over with their students to ensure that all aspects of the study and intervention procedures were covered (see Appendix B for script). As time was of the essence in that the study needed to be completed by Christmas to avoid students digressing in their knowledge, the study began during the week of October 9, 2017 and was completed during the week of December 18, 2017. Once students had completed the 10 weeks of intervention, the following week they were given a 5-minute post-test. For the 10 weeks during the intervention period, the students in Grades 2-6 received one of three interventions: drill-and-practice, strategy instruction or peermediated practice. These interventions were carefully planned to ensure that they would be easy for the classroom teacher to maintain, contain realistic expectations and involve a minimal amount of classroom time. These interventions were completed three times per week for 10-minute time intervals. As these were constructed to need minimal teacher oversight, teachers were able to run all three interventions concurrently. Each group was given a designated spot in the classroom or an empty classroom to work in. In both NUMBER FACTS KNOWLEDGE 44 Grade 3 and Grade 6, an Educational Assistant was available to help with monitoring. Teachers were given their discretion to both the day and the time of day to do these interventions. All teachers chose to do it at the beginning of their time allotted for math. Each student was given their own personal folder that included any instructions or worksheets that would be needed. As well, manipulatives were readily available. Table 3 specifies some commonalties and differences among the interventions Table 3 Key Features of Number Fact Knowledge Interventions Peer-mediated Practice Drill-andPractice Strategy Instruction Features Aspect Common Features Time Allotment 10 minutes 10 minutes 10 minutes Goal Setting “How many number fact questions can you complete in 5 minutes?” Flash-cards Paper/Pencil to record results “How many number fact questions can you complete in 5 minutes?” Worksheet Paper/Pencil to graph results “How many number fact questions can you complete in 5 minutes?” iPad Manipulatives Teacher Involvement None None Minimal Student Collaboration Feedback Dyads None Entire group Partner gives oral feedback Researcher’s aide’s correct worksheet Researcher’s aide’s correct worksheet Distinct Features Materials These interventions were made to include as many possible of the seven principles of effective practice mentioned by Fuchs et al. (2008). These seven principles include instructional explicitness, instructional design to minimize the learning challenge, strong conceptual basis, drill-and-practice, cumulative review, motivators and ongoing NUMBER FACTS KNOWLEDGE 45 progress monitoring (Fuchs et al., 2008). Instructional explicitness is based on the need for some concepts to be taught directly by the teacher. The concept of instructional design to minimize learning challenges suggests that students benefit when lessons are planned deliberately, with the intent to avoid all misunderstandings and problems. The third principle, a strong conceptual basis, refers to teaching practices that emphasize the need for students to know the building blocks, or foundational principles of math. Drilland-practice techniques cannot be neglected, but rather should be used to maintain and incorporate previously learned math concepts. Opportunity for cumulative review and motivators to encourage students to persevere is also purported to assist with the provision of effective interventions. The last principle, ongoing progress monitoring is also an important component of interventions. Through this monitoring, students who struggle will be identified much quicker and can be given the help they need sooner. In this research report, these interventions will be reflected on in order to determine the quality of the intervention. Twice throughout the 10-week period, teachers were asked to complete a checklist to ensure that, as much as possible, students were receiving the same instructions and following the correct procedures for the interventions. This checklist consisted of 12 statements for which the teachers were asked to indicate if that was being done in their classroom (see Appendix D for checklist). This was given to the teachers during Week 4 and Week 9. If any inconsistencies were noticed, the researcher consulted with the teacher to determine if changes needed to be made. Data were collected after students had been participated in the intervention for 10 weeks. Teachers were provided with the post-test number facts sheet and were asked to NUMBER FACTS KNOWLEDGE 46 give it to students during an optimal period some time that week. As this was the week before Christmas holidays, the teachers chose to give it either the Monday or Tuesday of that week during regular math class. This testing was done as a class in the classroom setting. Students were given five minutes to complete this test and were expected to complete as many questions as they could. No help was supplied by the teacher. Students were accustomed to this format and should not have experienced any undue stress during this time. Six students were absent on the day of testing. Three of them were able to complete it the following day, but three students were required to complete the test following the two-week Christmas holidays. Intervention 1: Drill-and-practice. In this intervention, students were given worksheets and were expected to complete as many questions as possible in five minutes. Grades 2 and 3 were provided with a worksheet with 100 addition questions written in horizontal format. Grades 4, 5 and 6 were provided with a worksheet with 100 multiplication questions written in vertical format (see Appendix E for sample worksheets). These worksheets progressed in levels of difficulty from week to week. Grades 2 and 3 progression was: adding ones (e.g. 4+1, 6+1), adding twos, adding threes, adding fours, adding fives, adding sixes, adding sevens, adding eights, adding nines and review. Grades 4, 5 and 6 progression was: doubles (e.g. 6x2, 4x2), squares (e.g. 5x5, 6x6), multiplying by three, multiplying by four, multiplying by five, multiplying by six, multiplying by seven, multiplying by eight, multiplying by nine and review. Once students completed the five minutes on these worksheets, they would indicate on a bar graph how many questions they had accomplished. The purpose of this graph was to motivate the students to beat the previous day’s score. Students who did not NUMBER FACTS KNOWLEDGE 47 surpass their previous day’s scores were encouraged to keep trying their best, with the desire that students did not feel discouraged. Following this, the students were given the remaining time to do corrections from the previous day. Typically, they would have about four minutes to accomplish this. The teacher had marked the previous day’s work and circled any answers that were incorrect. Students would then write each incorrect question and answer over five times correctly. In total, the drill-and-practice group would spend 10 minutes on their intervention. Of the seven principles mentioned by Fuchs et al. (2008), this strategy ties in strongly with five of them. Drill-and-practice contains instructional design to minimize learning challenges as it is very routine and therefore minimizes any misunderstandings. It also contains drill-and-practice, cumulative review, motivators (the use of the bar graph) and ongoing progress monitoring (requirement of corrections). As such, it can be seen as a valid intensive intervention. Intervention 2: Strategy instruction. The purpose of this second intervention was for students to understand the ‘why’ behind the addition and multiplication equations they are required to memorize. The main medium that was used for this intervention was video; however, worksheets and games were used as well. This intervention worked on a three-day rotation. Each three-day rotation was expected to be completed within one week. On the first day, students would watch an eight-minute video if they were in Grade 2 or 3 and a 10-minute video if they were in Grade 4, 5 or 6. These videos were developed by the author of this study using an app called ShowMe. Through this app, the author was able to teach the lesson to all students in this intervention without actually being present and also was able to ensure continuity across grade levels. This video NUMBER FACTS KNOWLEDGE 48 included some narration using visuals, some music, some worksheets, and some invitations for students to use manipulatives. Students were not permitted to pause the video at any point as sufficient pauses were given for any required work (see Appendix F for a sample script). On day two of the rotation, in dyads of two, students played a math game or did a math activity. These games were simple ones that could be learned easily and completed in a 10-minute time frame. They connected directly to the previous day’s video. See Appendix G for the game the students played for their lesson on multiplying by four. On the third day, students individually completed a worksheet that had 60-100 facts from that week’s lesson. They were not timed on this worksheet, but completed as much as possible in 10 minutes. This intervention progressed through increasingly difficult skills as the weeks advanced. In Grade 2 and 3, the students progressed from adding one to skip counting to making 10 to doubles and finished off with the commutative property. This was repeated again in weeks 6-10. The skills of making 10 and the concept of commutative property are two categories of number facts that cover most of the facts that students are required to learn (Kling, 2011). The three remaining skills of adding one, skip counting and doubles, are mentioned as valuable to a student’s understanding of addition facts (Fuchs et al., 2008). Throughout the videos, reference was frequently made to the strategies of ‘counting all’ to find the answer, ‘counting up’ to find the answer, or ‘just knowing’ the answer. This was in reference to the skill progression that students make as they mature (Fuchs et al., 2008). In Grades 4, 5 and 6, the students started with doubles (e.g. 3x2, 6x2), then progressed to squares (e.g. 4x4, 6x6) and from there moved through the facts in order from the three times tables to the nine times table. The focus during week 10 was NUMBER FACTS KNOWLEDGE 49 on review. The reason for the initial focus on squares and doubles was the research done by Woodward (2006) which cites the value of these particular number facts. This intervention encompassed five of the seven principles for effective intervention (Fuchs et al., 2008): instructional explicitness, instructional design to minimize the learning challenge, strong conceptual basis, drill-and-practice and cumulative review. Strategy instruction was particularly strong in the area of conceptual basis as this intervention emphasized the reasoning behind each lesson. Intervention 3: Peer-mediated practice. The third intervention, peer-mediated practice, involved the use of two classmates working together to develop each other’s abilities in number facts. If, in a grade, there were more than four participants in this intervention, the students were randomly put into groups of two. Each group of two was provided with flashcards to quiz each on their number facts. These flashcards contained the question on the front and the answer in small print on the back. One partner would be the tutor first and would hold up the initial flashcard. If the tutee stated the correct response, the tutor would say “Yes, that is correct” and would place that card on the correct pile. If the tutee stated the incorrect response, the tutor would say “No, that is incorrect. The correct response to ________ (insert equation) is _____ (insert answer).” The card would then be placed on the incorrect pile. In this way, both students heard the correct response. At the end of four minutes, the roles would be reversed and they would do the same process for another four minutes. Once the second four minutes had elapsed, students would record in their folders the number of flashcards they responded to correctly and the number that they responded to incorrectly. NUMBER FACTS KNOWLEDGE 50 Students also progressed through this intervention in a systematic manner. In week 1, they received a limited number of cards that focused on one skill. For Grades 2 and 3, initially these cards involved adding one and for Grades 4, 5 and 6, the cards involved multiplying by two. Each week after this, they received 30% more cards than the previous week. This procedure of teaching students’ new concepts at a steady ratio of known facts to unknown facts is known as incremental rehearsal. It has shown to have better results than introducing new facts all at once (Burns, 2005). By the end of week five, students had received all the cards in their deck (N = 36) and so this process was started over for weeks 6-10. In Grades 2 and 3, once students had received all adding one cards, they moved on to adding two, adding three, etc. until the entire deck was complete. In Grades 4, 5 and 6, once the students had received all the multiplying by two cards, they received the multiplying by three cards, followed by the multiplying by four and so on, until the entire deck was complete. When compared with the seven principles of intensive interventions, this intervention incorporates instructional design to minimize the learning challenge, drilland-practice, cumulative review, motivators and ongoing progress monitoring (Fuchs et al., 2008). As such, this also can be seen as a worthwhile intervention. Method of Data Analysis The first question sought to investigate what intervention resulted in the highest amount of number facts completed for students with LMA in Grades 2-6. In order to determine this answer, a 3x2 factorial Analysis of Variance (ANOVA) was used to analyze the data. The independent, categorical variables for this ANOVA were the three interventions given and the two categories of students. The three categories for the NUMBER FACTS KNOWLEDGE 51 interventions were drill-and-practice (control group), strategy instruction and peermediated practice. The two categories of students were students with LMA and students without LMA. The dependent, continuous variable was the correct answers from the posttest results. Drill-and-practice was the control group as it is a typical routine that was already being practiced by the classroom teachers and would be used if there was no study being performed. This control group ruled out concerns due to maturation and history since the other two groups will experience the same events throughout the tenweek period as this control group. Once this information was inserted into SPSS, the interaction effect between math ability and interactions was examined for any significant difference. For the second question regarding the effective intervention for all students in Grades 2-6, the same 3x2 ANOVA was used. This time, the main effects were examined. With regards to the third question, which examined the interaction effect between the intervention used and the grade of the student, a second 3x2 factorial ANOVA was conducted. The two independent variables were grade and intervention. Grade was divided into two categories – Grades 2/3 and Grades 4/5/6 and interventions had three categories: drill-and-practice, peer-mediated practice and strategy instruction. Grade was divided in this particular fashion due to Grades 2 and 3 not being familiar with multiplication. Once again, the post-test results were used for the dependent variable. For all these question, an ANOVA was determined to be the best tool as it provides the ability to compare different means at different points in time using different entities in each category. This statistical tool determines if there is a statistically significant difference in at least two of the means. Significance level for this study was NUMBER FACTS KNOWLEDGE 52 maintained at p<.05. If necessary, significant results will be examined in more detail using a post-hoc analysis. For each ANOVA, eta squared was calculated to determine effect size. The value of eta squared indicates how much the independent variable affects the dependent variable. In order to compute this, the Treatment Sum of Squares (SSM) was divided by the Total Sum of Squares (SST). The square root of this answer was then calculated to determine Pearson’s coefficient correlation, r. A result of r = 0.1 would indicate a small effect size, r = 0.3 would indicate a medium effect size and r = 0.5 would indicate a large effect size (Field, 2013). NUMBER FACTS KNOWLEDGE 53 CHAPTER 4: RESULTS The purpose of this study was to examine if the type of intervention implemented affects students’ ability to acquire number facts knowledge. Three interventions were examined: drill-and-practice, strategy instruction and peer-mediated practice. The grade level and the mathematical ability of the student were also examined to determine if there were any additional factors that could affect a student’s success. Preliminary Analysis Preliminary analysis was completed with the data from all 65 participants. The data was initially checked for violations of assumptions. A scatterplot demonstrated the assumptions of linearity and homoscedasticity were met (see Figure 1). Figure 1. Visual representation of the assumptions of linearity and homoscedasticity. Following this, Cronbach’s alpha was used to determine the internal consistency of the post-test. The measure used to determine this was the results of the screening-test. The screening test was marked and the marks of each question for every student was NUMBER FACTS KNOWLEDGE 54 entered into SPSS, with a one indicating an incorrect answer and a zero indicating a correct answer. The results depicted a high reliability, Cronbach’s α = 0.995. As well, no question was seen to significantly impact the reliability if that particular question were to be deleted. This indicated that each item, as a group, was closely related. As a result, there is no concern regarding the internal consistency of the test. Research Question 1: Interaction Effect Between Lower Math Abilities and Intervention The initial research question examined whether there was an interaction effect between the intervention used and students’ math abilities on number fact knowledge among students in Grades 2-6. The purpose of this question was to determine if a certain intervention would be more beneficial for students with LMA. It was thought that students with LMA would benefit from a different intervention than the general population. In particular, it was hypothesized that the best intervention for students with LMA would be strategy instruction. A significant interaction effect would support this hypothesis. A 3x2 factorial ANOVA was used to determine if there was an intervention that led to more success for students with LMA. This factorial ANOVA tested the interaction between the three interventions and the two categories of math abilities. Prior to the factorial ANOVA being calculated, assumptions were first tested. Using the ShapiroWilk’s test, it was determined that the data was normally distributed. Two outliers were identified. Upon examination of these outliers, it was decided to retain them in the results for two reasons: when removed, they did not have much impact on the analysis and upon examination of those particular students, it was believed that these students were not NUMBER FACTS KNOWLEDGE 55 dealing with any extenuating circumstances that would have led to these results. As well, the data from the three students who completed the post-test after the Christmas break was examined. It was determined that it was appropriate to include their outcomes. The assumptions of linearity and homoscedasticity were met for this set of data. The assumption of homogeneity of variance was not met for this set of data (F(5,59) = 3.07, p = 0.016). Therefore, there is a violation of the assumption of homogeneity. Table 4 outlines the mean and standard deviation for this data. Results of the 3x2 ANOVA revealed that there was no significant interaction between interventions and math abilities (F(2,59) = 0.083, p = 0.920, η2 = 0.02). Due to there being no significant interaction, no post-hoc analysis was needed. These results demonstrated that all three interventions provided equal opportunities for success for students with LMA and those without LMA. Table 4 Mean Score and Standard Deviations of Number Fact Knowledge by Treatment Group and Math Ability Total Sample (n = 65) Math Ability LMA Drill-and-Practice (n = 22) Peer-mediated (n = 20) Strategy Instruction (n = 23) M SD M SD M SD M SD 34.12 14.01 32.00 16.30 38.83 8.96 31.00 17.34 NonLMA 61.58 25.39 59.38 32.80 63.29 20.54 62.22 22.42 Total 54.40 25.89 51.91 31.42 55.95 21.02 55.43 24.83 However, upon examination of the descriptive statistics (see Figure 1), it can be observed that students with LMA had a higher mean during the peer-mediated interventions than the other two. This indicates that students with LMA may experience NUMBER FACTS KNOWLEDGE 56 more success using the peer-mediated intervention; however, this rate of success was not significant in this study. This lack of significance may be due to the low number of students with LMA in this study. Although this was not a significant result, but it does have some implications for further research. Research Question 2: Most Effective Intervention The second question examined which intervention had the highest amount of number fact questions completed for students in Grades 2-6, if there was no interaction effect between the students’ math abilities and the intervention. It was hypothesized that the drill-and-practice intervention would show significantly better results than both the strategy instruction intervention and the peer-mediated practice intervention. The working memory of students would lead to the higher results in the drill-and-practice intervention. This would be shown by a significant main effect of the intervention on the scores received from the post-test. The 3x2 factorial ANOVA from question 1 was analyzed further to determine if there is a significant difference between each intervention. As assumptions were tested for question one, no further examination into this was necessary. Results from the factorial ANOVA showed that there was no significant main effect (F(2,59) = 0.248, p = 0.781, η2 = 0.03). The means are displayed in Figure 2. This indicates that all three interventions are beneficial to students and therefore enables them to learn their number facts equally well. Effect size of interventions. As part of the analyses, the effect size between the screening test and the post-test was examined. Due to the screening test being similar to the post-test, it was treated as a pre-test for this analysis. The effect size was examined NUMBER FACTS KNOWLEDGE 57 using Cohen’s d. It was determined that there was a large effect size between the screening test and the post-test (d = 0.69). This indicates that the students’ knowledge of number facts improved over the course of the interventions. Figure 2. Mean post-test score with math abilities as the independent variable. Research Question 3: Interaction Effect Between Grade and Intervention The third question examined if there was an interaction effect between the intervention used and the grade of the student based on the number fact knowledge among students in Grades 2-6. The students were categorized as either being in Grade 2 or 3 or in Grade 4, 5 or 6. This question was examined to identify if there is a certain intervention which showed more success for certain grade levels in terms of the students’ abilities to acquire number facts. It was hypothesized that students in Grades 4-6 would do better on their number facts when the peer-mediated intervention was used, due to their maturity and their ability to perceive the needs of their classmates. NUMBER FACTS KNOWLEDGE 58 Prior to the ANOVA, assumptions were first examined. Through a Shapiro-Wilks test it was determined that data was normally distributed and no outliers were evident upon examination of a stem-and-leaf plot. Levene’s test demonstrated that the variance was equal (F(5,59) = 1.82, p = 0.124). A 3x2 factorial ANOVA was used to determine the significance of the interaction effect between the intervention used and the grade of the student. It was determined that there was no significant interaction between the intervention used the grade of the student (F(2,62) = 1.097, p = 0.341, η2 = 0.007). As there was no significant result, no post-hoc analysis was performed. This lack of significant interaction indicated that all students, regardless of grade level, benefitted equally from interventions focused on number facts. Table 5 outlines the mean and standard deviation for this set of data. Table 5 Mean Score and Standard Deviations of Number Fact Knowledge by Treatment Group and Grade Total Sample (n = 65) Drill-and-Practice (n = 22) Peer-mediated (n = 20) Strategy Instruction (n = 23) Grades M SD M SD M SD M SD 2-3 4-6 55.11 53.89 21.71 28.77 58.89 47.08 29.70 32.83 50.92 63.50 19.33 22.44 57.83 54.59 11.84 28.30 Total 54.40 25.89 51.91 21.42 55.95 21.02 55.43 24.83 Upon examination of the descriptive stats, it can be noted that there is a higher mean for Grades 4-6 students in the peer-mediated intervention. Although this did not result in a significant interaction, it does suggest that there may be some benefits to using this intervention for older students. Further research would lead to more conclusive results. NUMBER FACTS KNOWLEDGE 59 CHAPTER 5: DISCUSSION This study focused on the benefits of providing number facts math interventions for students in Grades 2-6. The primary focus was on the benefit of interventions for students with LMA, but the benefits for all students was also examined. Data were also examined to determine if grade level played a role in the type of intervention that was successful in the acquisition of number facts knowledge. The three interventions that were examined were strategy instruction, drill-and-practice, and peer-mediated practice. The overall effect size for all intervention groups was examined using Cohen’s d. The resulting large effect size indicated that growth in number facts knowledge was experienced by all students regardless of the math intervention that they received. Interaction between Intervention and LMA The first question in this study examined if there was an interaction effect between the intervention used and the students’ math abilities as measured by a number facts post-test. It was hypothesized that the strategy instruction intervention would show the highest rate of success for students with LMA. However, no statistically significant interaction effect was found. This implies that a specific intervention did not interact with students with LMA to yield higher scores on a number facts tests than students without LMA. This result was congruent with a previous study (Woodward, 2006) which found that students who struggled and students who did not struggle both experienced a substantive increase in number fact knowledge when two different interventions were used. However, this was not supported by several other studies which demonstrated that strategy instruction, when compared to other types of interventions, led to improved NUMBER FACTS KNOWLEDGE 60 number fact knowledge for students who struggle with LMA (Kaufmann et al., 2003; Tournaki, 2003; Zhang et al., 2014). When examining the content of this study, there are some reasons for these opposing results. These studies all had differing characterizations for students who struggled in math. One study referred to students with mathematical difficulties defined as students who fail to achieve a certain benchmark on standardized mathematics tests (Zhang et al., 2014). Another study examined students with dyscalculia specifically and examined beneficial interventions for this particular category of students with math difficulties (Kaufmann et al., 2003). My definition for students who struggle in math is different than these definitions and this could have impacted the results. Another explanation for this difference between the reported studies and this current study is that all studies had their own definition and methodology for the implementation of the three different interventions studied in this paper. Tournaki (2003) examined both the drill-and-practice and the strategy instruction intervention. However, these interventions were administered individually and students did not progress from one skill to another until they demonstrated mastery. In another study that examined peerassisted interventions, the students were not randomly assigned as in this study, but were paired according to ability level (Fuchs et al., 1997). Although the studies were chosen because of the similarities in intervention styles, there were still differences and these differences may have led to the opposing results between this study’s results and the other reported results. However, this lack of a statistically significant interaction effect still has applications for the classroom. It indicates that, regardless of a student’s abilities in math, NUMBER FACTS KNOWLEDGE 61 they will likely do well on learning their math facts, provided they are given some type of intervention. Students with LMA will benefit equally from drill-and-practice interventions, peer-mediated practice interventions or strategy instruction interventions. This implies that when teaching students who struggle in math, the teacher does not need to be as concerned about how their personal teaching style will impact the success of their students when learning number facts. Some teachers may favour one intervention style over another due to classroom dynamics, available resources or personal preference, and this study substantiated that teachers will not detrimentally affect their students by choosing one of these three studied interventions over another. Rather their focus should be on the implantation of the intervention, rather than on the intervention itself. Upon examination of the means for the three different interventions, it was noted that, for students with LMA the peer-mediated intervention had a higher mean than the other two interventions. Although this cannot be seen as a significant result, it can be noted that there may be a slight advantage to using this intervention for students with LMA. This mean difference was most likely not significant due to the study having a smaller sample size, particularly for students with LMA. Most Effective Intervention The second question examined the intervention that led to the highest rate of number fact knowledge for students in Grades 2-6. This was if there was no interaction effect in the first question. It was hypothesized that drill-and-practice would lead to the best result for all students in Grades 2-6. Kroesbergen (2003) found that the use of drilland-practice was the most effective for students to learn their number facts. However, when the data from this particular study were analyzed in SPSS, it was discovered that NUMBER FACTS KNOWLEDGE 62 there was no significant main effect. Students did not benefit more from one intervention over another. This was somewhat surprising as the majority of studies that were examined showed a significant impact of one type of intervention over another (Kanive et al., 2014; Knowles, 2010; MacQuarrie, Tucker, Burns, & Hartman, 2002). As this question is closely connected to question one, these unexpected results may be explained by the same reason – all studies that were reviewed had their unique definitions of mathematical disability and interventions. Although the interventions in the various studies that were examined were connected to the interventions in this particular study, not one of them was identical and so this also could lead to some unexpected results. The result that no single intervention led to a higher rate of number fact knowledge is still a valuable conclusion. Teachers can feel confident that, regardless of their choice of intervention, their students will experience the same, positive results when acquiring their facts. In connection with the first question, these results may imply that teachers can use the same intervention for both the students with LMA and those without, as in both cases, one intervention was not significantly better than another intervention. Interaction between Grade and Intervention The third question examined the interaction between grade level and intervention. The purpose was to determine if there was a certain intervention that worked better for a certain grade level of students. It was hypothesized that students in Grades 4-6 would perform better when exposed to peer-mediated practice. Once again, though, the ANOVA did not show a significant interaction. NUMBER FACTS KNOWLEDGE 63 When examining the mean value of correct responses completed by students in each intervention category, there is a clear, although insignificant benefit to using peermediated practice for students in Grades 4-6. These results are echoed in several studies (Kroesbergen & Van Luit, 2003; Van Keer & Verhaeghe, 2005) who argue that, due to the maturity level of the higher-grade students, peer-mediated strategies are more successful. The fact that this result did not reach a statistical significance may be due to a smaller sample size in this study. Further research that includes a larger sample size that is more diverse may be of benefit. The lack of statistical significance in the results of this third question may also be attributed to the instrument that was used to measure number fact knowledge. The posttest was configured to include both addition and multiplication questions to enable crossgrade comparison of the intervention. The expectation was that students in Grades 2-3 could skip the multiplication questions. However, this may have led to some confusion for students, as they often mixed up the signs. As well, the students in Grades 4-6 completed fewer amount of questions due to the focus needed to ascertain if the question was a multiplication or addition question. Thus, this post-test potentially may not have measured the true ability level of the students. Teacher Feedback Throughout the interventions, teachers commented on their personal view of the value of the intervention. These comments and conversations happened informally in the hallways and at the end of the school days. Five teachers were involved in this study and all demonstrated appropriate adherence to the expectations for each intervention as demonstrated by two formal check-lists completed during weeks four and nine of the NUMBER FACTS KNOWLEDGE 64 interventions. These teachers all spoke well of the interventions and the positive growth in number facts knowledge that was seen. They felt that these interventions were easy to implement and maintain and were applicable to the course requirements set forth by the provincial program of studies. The teachers perceived that the main value of the treatment program was the increase of number facts knowledge for all students. A general comment given by most teachers was that the success of each intervention depended largely on student engagement and group dynamics. In one grade, the group dynamics were such that continual monitoring was needed as there was a recurrent lack of focus. In another grade, in a peer-mediated practice group, the two members were too far apart in ability level and so the one member became bored and restless. Thus, oftentimes the success of the student did not depend on his or her knowledge and ability, but on how well the intervention suited him or her and how well the group worked together. Teachers also provided feedback for each intervention. In general, teachers reported that the students enjoyed the strategy instruction the most – mainly due to the mode of delivery which was via video. As well, the strategy instruction provided the most opportunity for collaboration with peers and was the only intervention where teachers saw some definite carry-over of the strategies into the math lessons (this was stated with the caveat that it would be difficult to monitor the carry-over from the other interventions). A disadvantage noted by the teachers with regards to the strategy instruction intervention was that it did not provide a way to determine daily and weekly growth in number fact skills. NUMBER FACTS KNOWLEDGE 65 The drill-and-practice method worked well for students who did not struggle with focus. It was easy to implement and teachers were easily able to see progress. A graphing component was included in this intervention in order to motivate the students; however, this was only successful in Grades 4-6. It did not serve any purpose in the younger grades as they did not fully comprehend the meaning of the bar graph. The detriment of this intervention was that it was not effective for students who needed to learn some of the facts yet. The peer-mediated practice intervention was also easy to administer. The students benefitted from receiving 30% more new cards every week as this did not overwhelm them as much as receiving the entire deck at once. Nevertheless, this group was probably the group that needed the most monitoring in the different grade levels. It was easy for these groups of two to become side-tracked with their own personal conversations if they knew they were not being directly watched. Overall, the teachers could not come to a consensus on which strategy they predicted would do the best. This supports the results which showed that all strategies demonstrated equal achievement. As well, teachers felt that students who struggled were not benefitted by a specific intervention – once again this was supported by the data. Most teachers did prefer the drill-and-practice intervention as it was the intervention that they were the most familiar with. This may have led to a bias towards the drill-andpractice intervention. Summary In summary, although no significant interactions were found, valuable conclusions can still be drawn. In general, the data demonstrated that the type of NUMBER FACTS KNOWLEDGE 66 intervention was not important to the success of a student acquiring their number facts; it was only important that interventions were provided to all ages regardless of abilities. When examining the descriptive statistics, a trend is noticed in that peer-mediated practice has had higher means both for students with LMA and for students in Grades 46. This seems to suggest that there may be some advantage to this style of intervention. Teacher feedback was largely in support of the conclusions to the three questions being studied. They felt that as long as students were being given concentrated, specific time to work on number facts, there would be growth. As well, in general, teachers felt that selecting students to be in specific interventions based on student characteristics may have resulted in different, more conclusive results. NUMBER FACTS KNOWLEDGE 67 CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS This present study examined the effectiveness of three interventions on students with LMA and without LMA. The relative effectiveness was assessed by students’ score on a math number facts test. The research also examined the interaction between the grade of the student and the intervention and how this interaction impacted the overall number facts score. Three specific math interventions were looked at: drill-and-practice, peer-mediated practice, and strategy instruction. This study concluded that the type of interventions did not have an impact on the students’ number fact knowledge, regardless of the students’ math abilities or grade. Rather, it was more important that the students be provided with the number facts interventions, as seen by the large effect size observed when examining the improvement from the initial test to the post-test (d = 0.69). There was some benefit noted with the use of peer-mediated practice for students in Grades 4-6; however, no conclusive evidence could be found to support this intervention providing a significant benefit to the students. Implications of Research One of the purposes of this study was to fill the void found in research with regards to what number facts interventions work best for students who have LMA in order to increase their number facts knowledge. Many studies focused on one particular intervention and the benefit found in that intervention for students who struggle in math, but very few compared three interventions to determine if the benefits of one outweighed all the others. The conclusions drawn by this study have several implications. The ultimate implication is that all interventions are beneficial. All three interventions resulted in a NUMBER FACTS KNOWLEDGE 68 large effect size signifying a gain in students’ knowledge of number facts. This was also substantiated from the teachers’ personal comments. Interventions are needed to enable students to arrive at the point of automaticity in their facts and this study emphasizes that it is not the type of intervention that matters, only the fact that interventions are being provided. Another implication is that students with LMA do not need differing interventions than students without LMA. As both groups benefitted equally from all three interventions, classroom teachers can use the same intervention for all students. This will save teachers time in preparing their lessons and also enable them to choose an intervention style that they are the most comfortable with. Finally, since students in certain grades do not benefit from a particular intervention, teachers do not need to specifically choose an intervention based on the grade level of a student. Once again, teachers will save time and also be able to choose interventions based on their teaching preference. Limitations This study did have some limitations. The first limitation was the composition of the post-test. It was comprised of both addition and multiplication questions and when the results of this test were scored, it was evident that students often confused the mathematical signs. As a result, many questions were incorrect, but yet the student most likely knew the answer. Thus, students received a lower mark than their ability level. However, in order to determine if there was a connection between the interventions and grade level, the test needed to be administered in this way. NUMBER FACTS KNOWLEDGE 69 Another limitation is the generalizability of these results. The sample had many similar features such as SES, cultural backgrounds and family structures. As this is not characteristic of the general student population of Grades 2-6 students in Alberta, it may be difficult to apply these results to school situations that include a variety of racial, socio-economic and cultural backgrounds. As well, the sample had a relatively small size. An additional limitation is a lack of controls for cognitive, working memory or inductive reasoning. All of these factors may have had some influence on the abilities of students to develop an increased knowledge of number facts. A final limitation is that the homogeneity of variance was violated for the first research question. However, due to the robustness of the ANOVA, it is believed that the results are still valid. Considerations for Future Research There are several areas to consider for future research. Research has pointed to the positive relationship of attentive behaviour (Fuchs et al., 2008) and the student’s ability to complete their work. In order to develop this area of research further, it would be valuable to examine how ADHD impacts the success of interventions that are focused on number facts knowledge. Teachers also commented on how the students who struggled with focus seemed to have the most difficulty in the completing the requirements for their specific intervention. Certain interventions would be more difficult for children with ADHD. As such, it would be valuable to determine which type of intervention results in a significant effect for students with ADHD. If ADHD was recognized as the underlying reason for a student’s struggle, there would be a difference in how the intervention is constructed and applied. NUMBER FACTS KNOWLEDGE 70 Another area to examine for future research is if deliberately placing students in specific interventions makes a difference. This placement could be done based on student choice or based on teacher’s perception of what intervention best aligns with students’ interests. However, this would remove the random assignment of students to groups which would lead to other concerns such as generalizability and even spread of characteristics which might mitigate the results. A third area of research would be to extend this study to a larger sample. This would enhance the generalizability of the study. It could also potentially lead to findings that have more significant results. Conclusion Although this study did not completely fill the gap in research as desired, there were still valuable conclusions. This study demonstrated that the drill-and-practice intervention, the strategy instruction intervention and the peer-mediated practice intervention demonstrated the same amount of success for each student. Not one of these interventions resulted in a more significant interaction than the other. As well, there was a large effect size when examining the improvement from the initial screening test to the post-test demonstrating that students improved overall. The statements from the teachers also provide insight into the research in that they conclude that one of the main benefits of these interventions was not the interventions themselves, but the emphasis that was placed on number facts. Some teachers plan to continue using some aspects of these interventions in class, and the majority of the teachers hope to continue emphasizing a build-up in number fact acquisition. 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NUMBER FACTS KNOWLEDGE 78 APPENDIX A: PERMISSION LETTER The Hunt for Success for Children with Mathematics Difficulties in Number Facts Automaticity by Examining Strategies and Age of Child Principal Investigator: ___________________ Department of Educational Studies – Special Education (MA), Trinity Western University Supervisor: _______________________ ___ is currently a graduate student in the Masters of Educational Studies – Special Education (MAES) program at Trinity Western University in Langley, B.C. This research project is part of a thesis, which is a public document. Purpose: The purpose of this study is to examine how we can best help our students memorize their number facts. Four questions will be examined: 1. Which strategy leads to the greatest improvement of number fact acquisition for children in Grades 2-6? Three sub-questions will be answered in the context of this question. a) What strategy is most effective for students who struggle in math in Grades 26? b) At which age is the highest improvement of number fact acquisition for children seen? c) Is there an interaction effect between the intervention used and the age of the student? Procedures: In order to answer these questions, students will randomly be divided into three groups. Each group will do a different strategy. One strategy will involve the traditional drill-andpractice, another group will use flash-cards and a third group will be given strategy instruction. They will then be involved in this particular strategy for 10 minutes a day, three times a week for 10 weeks. Once the 10 weeks have passed, they will be given a test to determine their growth in number facts skills. This will be used to determine the answers to the questions above. After the answers have been determined, parents, staff and students will be informed about the results. This will be done at the spring AGM. Potential Risks and Discomforts: There is not much risk associated with this project as this will be a regular part of your child’s day and they will not notice any interruptions to the regular routine. If your child starts to feel undue stress or worry about being part of the process, they will be consulted with and a solution will be agreed on. One solution might be (if your child is extremely worried) the withdrawal of your child from the study. NUMBER FACTS KNOWLEDGE 79 Potential Benefits to Participants and/or to Society: At the end of the project, I hope to be able to have an idea of what is the best number facts strategy that works for particular types of students. Ideally, this will enable teachers to adapt their teaching style to reach more students. As well, students will have increased in their number fact knowledge and will have an understanding of what strategy they learn best with. Confidentiality: Any information obtained from this study regarding your child’s number facts abilities and growth and that can be identified to your child will remain confidential and will be disclosed only with your permission or as required by law. In my project, all documents will be identified only by a code number and will be kept in my locked office. Any information stored on the computer will be kept in my files and will be password protected. Contact for information about the study: If you have any questions or desire further information with respect to this study, you may contact ____________________. Contact for concerns about the rights of research participants: If you have any concerns about your treatment or rights as a research participant, you may contact ___________________. Consent: Your participation in this study is entirely voluntary and you may refuse to participate or withdraw from the study at any time without jeopardy to your child in any way. If you wish to withdraw your child from the study, please contact the principal investigator, ______ by December 4. Any data connected to your child will be deleted from the study. Signatures If for any reason, you do NOT wish for your child to be a part of this research process, please check the box below and fill in the required information by September 21, 2017. It is understood that if you do NOT return this form, that you are giving consent for your child to be part of this study. £ I do NOT consent for my child to be part of the thesis project undertaken by ___. Child’s name: _______________________________ Parent’s name (printed): _______________________ Parent’s signature: ___________________________ Date: ______________________________________ Please contact if you wish to withdraw your child at any time from this study. This option is available until December 4. NUMBER FACTS KNOWLEDGE 80 APPENDIX B: Script for Explanation of Each Intervention Flashcards For this activity, you will work in groups of two (or three if there is an odd number of students). Each group will receive a deck of flashcards. One person will ask questions using the flashcards for four minutes and then the second person will ask the same questions for four minutes. If you are the one asking the questions, you will wait for your classmate’s answer. • If they answer correctly, you will say “That is correct” and place that card on the correct pile. At the same time, place a tic mark in the correct column on your partner’s sheet. • If your classmate gives the wrong answer, you will say “That is incorrect” and then say the question with the correct answer. (For example, if your classmate thought 6x7 = 49, you would say “That is incorrect, 6x7 = 42”). This card will then be placed on the second pile for incorrect cards. At the same time, place a tic mark in the wrong column on your partner’s sheet. If there is still time left, go through the deck again. If the four minutes is up, switch roles. • Next week when you do this again, see if you can get more cards correct. Drill-and-Practice For these worksheets, you will receive 5 minutes to complete as many questions as you can. The goal is not to get the whole sheet done, but to complete more each time. You need to complete the questions in order, so you cannot skip all around. This means you need to start on the left-hand side and move across the page until the line is done and then start the next line. Once the five minutes are up, draw a line. Then look at the sheet that you did the time before. Choose five of the questions that you did wrong. Copy these questions out five times each on the back side of that sheet. When you copy them out, copy out the entire question and answer before moving on to the next one. If you did not receive any wrong, complete more of yesterday’s worksheet. Strategy Instruction For this activity, you will do three different activities each week. One day you will watch a video, another day you will complete an activity and the third day you will work on a worksheet. Each of these will take 10 minutes to complete. For each activity, you will need your duotang. Sometimes the video calls it a red duo-tang, but your duo-tang may be another colour. When watching the video, listen carefully and do all the work that it asks you to do. NUMBER FACTS KNOWLEDGE APPENDIX C: Post-Test 81 NUMBER FACTS KNOWLEDGE APPENDIX D: Teacher Checklist Please check off each of the following boxes if they are true for your particular grade. Drill-and-practice • Students are being timed for exactly five minutes • If students can complete more than 100 questions per five minutes, they are being given extra sheets • Students are graphing their results correctly • In the remaining time, students are correcting the previous day’s incorrect answers by copying them over five times Peer-Mediated Practice (flashcards) • Students are being timed for exactly four minutes/person • If the answer is correct, students are saying “yes, that is correct” and moving it to the correct pile • If the answer is incorrect, students are saying “No, that is incorrect. The correct answer to ______ is __” and then are moving it to the incorrect pile • Students are tallying their marks correctly Strategy Instruction (Video) • Students are not pausing the video • Students are completing all three parts of this procedure in the correct amount of time • On day three, students are taking as much time as needed for the worksheet General • If students are absent, they are caught up at the soonest possible opportunity 1. Approximately how much time does this take? 2. Do you have any general comments regarding the process? 82 NUMBER FACTS KNOWLEDGE APPENDIX E: Sample Drill-and-Practice Sheets 83 NUMBER FACTS KNOWLEDGE 84 NUMBER FACTS KNOWLEDGE 85 APPENDIX F: Video Script – Lesson On Doubles Hello students I hope you are looking forward to an exciting time to math practice. In these lessons, we are going to examine different multiplication strategies. By the end of today, you should be familiar with the double facts strategy. This is similar to adding doubles. You all have been given a red duotang that has all the worksheets for your number facts. You will need this every time you do number facts so always have it ready. But before we do that, I want to show you something neat. Pull out your multiplication chart that’s in your red duotang . . . – that looks like a lot of facts to memorize doesn’t it. Well, I’ll show you that it is not that bad. You already know the zero times tables – the answers are always zero. 0x5 = 0, 11x0 = 0, 123124x0 = 0. Since we know all those facts, colour them in. The 1’s are also easy. 1x a number is always that number. 1x6 = 6, 1x9 = 9, 1x1235 = 1235. Pretty easy – so since you know those already, you can cross those out as well. You also are really good at counting by 5’s: 5, 10, 15, 20, and 10’s: 10,20,30, right? Well, that means you can figure out those number facts pretty quickly too. You know 5x4 = 20 because you count by 5 four times 5, 10,15,20. We can also colour those in. Wow, you already know a lot of facts. We can colour a few more in. Facts such as 3x4 and 4x3 are exactly the same. That means if we know one, we also know the other. Let’s cross out one of these sets of two. Cross out 4x3, 6x3, 6x4, 7x3, 7x4, 7x6, 8x3, 8x4, 8x6, 8 x7, 9x3, 9x4, 9x6, 9x7 and 9x8. Look at that – we really only have 34 facts to memorize. Let’s get to it. In your red duotang, turn to the page called Doubles. On the doubles sheet, write out the five times tables. ( 1 minute) Here are some pictures to help you remember the double facts. For 2x2, think of a quad with two sets of wheels . . . so it has four wheels For 3x2, two rows of three pop . . . a sixpack For 4x2, think of . . . a spider with two sets of four legs . . . eight legs NUMBER FACTS KNOWLEDGE 86 For 5x2, think of . . . two hands with five fingers each . . . which gives us 10 fingers altogether For 6x2, an egg carton with two rows of six . . . a dozen or 12 eggs For 7x2, two weeks of seven days each . . . 14 days For 8x2, think of two octopi with eight legs each . . . this would give us 16 tentacles altogether And for 9x2, think of a semi with nine wheels on each side. . . . as you know, a semi has 18 wheels. . Let’s repeat this fast 2x2 = ….4 3x2 = . . . . 6 4x2 = . . .. 8 5x2 = . . . . 10 6x2 = . . . . . 12 7x2 = . . . . 14 8x2 = . . . .16 9x2 = . . . . 18 10x2 = . . . . 20 On the sheet called doubles, write out the double facts as quickly as you can. Write both the question and the answer. I’ll give you 1:00 seconds for this On your mark, get set, go. Well done – good effort. Talk to you next time. NUMBER FACTS KNOWLEDGE 87 APPENDIX G: Strategy Instruction – Multiply by Four Game Required Supplies: 10-sided die 2 different coloured markers ‘Fours’ Game Board Length of time: 8 minutes Number of Players: 2 Directions: 1. When it is your turn, roll the 10-sided die. Multiply the number that you rolled by four. Look for this answer on the board and colour in one of the circles with that answer. 2. Take turns rolling the die and colouring in circles. 3. The goal is to try to get four circles in a row (either straight or diagonally) and to try prevent your partner from getting four in a row. 4. The first player to get four in a row wins. 5. If you have enough time, play the game again. Fours Connect 4 Game Board 40 8 12 20 16 40 28 32 12 40 16 24 4 40 28 20 12 28 32 36 16 12 28 24 36 8 32 28 4 20 32 16 32 12 12