RESEARCH PAPERS INVESTIGATING PRE-SERVICE CANDIDATES' IMAGES OF MATHEMATICAL REASONING: AN IN–DEPTH ONLINE ANALYSIS OF COMMON CORE MATHEMATICS STANDARDS By C. E. DAVIS * JAMES E. OSLER ** *-** North Carolina Central University ABSTRACT This paper details the outcomes of a qualitative in–depth investigation into teacher education mathematics preparation. This research is grounded in the notion that mathematics teacher education students (as “degree seeking candidates”) need to develop strong foundations of mathematical practice as defined by the Common Core State Standards for Mathematics' (CCSSM). In this investigation mathematics Pre-Service Candidates (“PSCs”) participated in an online 15-week methods course that infused writing prompts. This research activity probed the PSCs images of mathematical reasoning. It is based on the idea that in mathematical teacher education, teacher preparation requires teaching mathematical standards. In teaching, the standards activities are required that infuse mathematical reasoning. This will aid PSCs in further infusing mathematical reasoning in their teaching both now and in the future. Keywords: Common Core, Common Core State Standards for Mathematics'(CCSSM), Mathematical Practice, Mathematical Knowledge, Mathematics, National Research Council (WRC), National Council of Teachers of Mathematics' (NCTM), Pre-Service Candidate (PSC), Standards for Mathematical Practice (SMP), State Standards, and Tri–Squared INTRODUCTION (CCSSM) Standards for Mathematical Practice (SMP) were heavily influenced by the National Research Council's Recent adoptions of the Common Core State Standards (NRC). Adding It Up (NRC, 2001) stands of mathematical across numerous states have called into question teacher proficiency; as well as the National Council of Teachers of education preparation. The Standards for Mathematical Mathematics' (NCTM) Principles and Standards for School Practice as defined on pages 6 – 8 of the Common Core Mathematics (NCTM, 2000) process standards. The strands State Standards for Mathematics reflect a need for of mathematical proficiency are defined as: Conceptual teachers to strengthen and build “processes and understanding, procedural fluency, strategic proficiencies” for their students (NGACBP & CCSSO, 2010, competence, adaptive reasoning, and productive p.6). In order to assist Pre-Service Candidates (PSC) to disposition (NRC, 2001). In the NCTMs' Principles and develop strong foundations of mathematical practice, as Standards for School Mathematics the process standards defined by the Common Core, for their future students, the are listed as: Problem solving, reasoning and proof, following research activity probed their images of communication, connections, and representation (NCTM, mathematical reasoning. This idea solicits the notion that in 2000). The Table 1 gives a brief description of the strands for teaching the standards one should seek out activities to mathematical proficiency, from NRC's Adding it Up, that infuse mathematical reasoning throughout their teaching. were used to develop the SMP. To analyze PSCs' knowledge of mathematical reasoning, writing prompts were distributed to students online during a These five strands are interdependent and interlaced 15-week methods course. together in the creation of mathematical proficiency. The Standards for Mathematical Practice (SMP) NRC states that “helping children acquire mathematical proficiency calls for instructional programs that address all The Common Core State Standards for Mathematics' i-manager’s Journal on School Educational Technology, Vol. 9 l No. 2 l September – November 2013 29 RESEARCH PAPERS its strands” (NRC, 2001, p.116). As students' progress incorporate the process standards within instruction is seen through elementary and middle school the framework as integral to creating proficient learners of mathematics provided by the five strands should empower them to The National Governors Association Center for Best contend with the mathematical challenges they face, as Practices and Council of Chief State School Officers' well as provide opportunities for their mathematical Common Core State Standards for Mathematics (CCSSM) success in future studies. It should be noted that reasoning has been adopted by most of the United States. The and reflection are essential components to developing CCSSM, in addressing best practices for teaching mathematical proficiency. mathematics, lists eight Standards for Mathematical The NCTM's book, Principles and Standards for School Practice (SMP). The SMP infuse both the processes, as Mathematics (PSSM), has served as a guide for mentioned in the NCTM's PSSM (NCTM, 2000), as well as the mathematics instruction for over twelve years. The PSSM proficiencies from the NRC's Adding it Up (NRC, 2001). The provides a recommended framework for instructional CCSSM's SMP are listed in Table 3 below. programs in mathematics through a set of six principles The SMP describe actions for teachers to incorporate within (Equity, Curriculum, Teaching, Learning, Assessment, and their classrooms to support the development of the NCTM's Technology) and ten general standards that construct mathematical processes and NRC's proficiencies. These school mathematics curriculum across several grade- types of actions, in many instances, engage students' bands (Preschool to 2, 3 to 5, 6 to 8, and 9 to 12). The ten mathematical reasoning to become mathematically standards are divided into two different types: Content and prepared and confident during their studies of process. There are five content standards, they are: NCTM’s Process Standards Number and Operation, Measurement, Geometry, Data Instructional Mathematics programs should enable all students to: Analysis and Probability and Algebra. The process 1. Build new knowledge through problem solving; Problem standards are listed in Table 2 below. 2. Solve problems that arise in various contexts; Solving 3. Incorporate a variety of strategies to solve The process standards are seen as the mathematical problems; and 4. Reflect on the process of mathematical problem processes in which students gain and use mathematical solving. Instructional Mathematics programs should enable knowledge. Similar to the strands for proficiency, the NCTM's all students to: principles and standards are seen as interrelated. They 1. Recognize and create conjectures based on observed patterns; should not be seen as separate content, principles, and Reasoning 2. Investigate conjectures and prove that all cases are and Proof true or that a counter example shows that it is not standards, but necessary components in a mathematics always true; and 3. Explain and justify solutions. curriculum. It can clearly be shown that one's ability to Instructional Mathematics programs should enable reason mathematically is an idea that threads through all students to: 1. Organize and consolidate thinking in both written each of the process standards. The principles and and verbal communication; Communication 2. Communicate thinking clearly to peers, teachers, standards are to be used to guide the methods and and others; and processes for teaching and learning mathematics. To 3. Use appropriate vocabulary to express ideas precisely. Instructional Mathematics programs should enable NRC’s Strands for Mathematical Proficiency all students to: 1. Understand that ideas are interconnected and that Conceptual Integrated comprehension of concepts, operations Connections they build and support each other; Understanding and procedures. 2. Recognize and apply connections to other contents; and Procedural Ability to perform procedures appropriately, with flexibility, 3. Solve problems that arise in various contexts with Fluency accuracy, and efficiency. mathematical connections. Instructional Mathematics programs should enable Strategic Abilities to formulate, represent, and solve problems. all students to: Competence 1. Emphasize a variety of representations to Representations Adaptive Aptitudes to think logically, reflect, explain, and justify among communicate ideas; Reasoning concepts and ideas. 2. Select, apply, and translate among representations to Productive Tendency to see content as sensible, valuable, useful, and solve problems; and Disposition worthwhile, combined with a belief that, with 3. Use representations to mod-el and interpret real life steady effort, one can effectively produce results. situations. Table 1. NRC's Strands for Mathematical Proficiency (NRC, 2001) Table 2. NCTM's Process Standards (NCTM, 2000) 30 i-manager’s Journal on School Educational Technology, Vol. 9 l No. 2 l September – November 2013 RESEARCH PAPERS CCSSM Standards for Mathematical Practices Mathematically proficient students: Make sense of 1. Explain to themselves the meaning of a problem and look for entry points to its solution; problems and persevere in solving 2. Analyze givens, constraints, relationships, and goals; them. 3. Make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt; 4. Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution; 5. Monitor and evaluate their progress and change course if necessary; 6. Explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends; 7. Check their answers to problems using “different methods”; and Reason abstractly and quantitatively. 8. Understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students: 1. Make sense of quantities and their relationships in problem situations; 2. Bring two complementary abilities to bear on problems involving quantitative relationships: the abilities to decontextualize and contextualize ; and 3. Create a coherent representation of the problem at hand, considers the units involved, attends to the meaning of quantities, Construct viable and with knowledge and flexibility uses different properties of operations and objects. arguments and critique the reasoning Mathematically proficient students: of others. 1. Understand and use stated assumptions, definitions, and previously established results in constructing arguments; 2. Make conjectures and build a logical progression of statements to explore the truth of their conjectures; 3. Analyze situations by breaking them into cases, and can recognize and use counterexamples; 4. Justify their conclusions, communicate them to others, and respond to the arguments of others; 5. Reason inductively about data, making plausible arguments that take into account the context from which the data arose; 6. Compare the effectiveness of two plausible arguments; and 7. Listen or read the arguments of others, and decide whether they make sense, and ask useful questions to clarify or improve the arguments. Model with Mathematically proficient students: mathematics. 1. Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace; 2. Apply what they know to make assumptions and approximations to simplify a complicated situation, and realize that these may need revisions later; 3. Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two - way tables, graphs, flowcharts and formulas; 4. Analyze relationships mathematically to draw conclusions; and 5. Routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense. Use appropriate tools Mathematically proficient students: strategically. 1. Consider the available tools when solving a mathematical problem; 2. Are sufficiently familiar with appropriate tools to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and the limitations. 3. Detect possible errors by strategically using estimation and other mathematical knowledge; 4. Know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data; 5. Identify relevant external mathematical resources and use them to pose or solve problems; and 6. Use technological tools to explore and deepen their understanding of concepts. Attend to precision. Mathematically proficient students: 1. Try to communicate precisely to others; 2. Try to use clear definitions in discussion with others and in their own reasoning; 3. State the meaning of the symbols they choose, including using the equal sign consistently and appropriately; 4. Are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem; and 5. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. Look for and make Mathematically proficient students: use of structure. 1. Look closely to discern a pattern or structure. 2. Recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. 3. Can step back for an overview and shift perspective; and 4. Can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. Look for and express Mathematically proficient students: regularity in repeated 1. Notice if calculations are repeated, and look both for general methods and for shortcuts; reasoning. 2. As they work to solve a problem, they maintain oversight of the process, while attending to the details; and 3. Continually evaluate the reasonableness of their intermediate results. Table 3. The CCSSM Standards for Mathematical Practices (NGACBP & CCSSO, 2010). i-manager’s Journal on School Educational Technology, Vol. 9 l No. 2 l September – November 2013 31 RESEARCH PAPERS mathematics. Teachers, as well as other educational Mathematics (1991) stated: professionals, should seek ways to bridge mathematical “Knowing mathematics includes understanding specific practices with the content during instruction. The overall concepts and procedures as well as the process of doing goal is for students to develop both procedural and mathematics. Mathematics involves the study of concepts conceptual understandings of mathematics. In order for and properties of numbers, geometric objects, functions, this to occur, mathematics classrooms need to and their uses - identifying, counting, measuring, incorporate ideas of discourse, problem-based learning, comparing, locating, describing, constructing, as well as seek out other opportunities to understand transforming, and modeling. At any level of mathematical students' mathematical reasoning. study, there are appropriate concepts and procedures to The Rationale for Mathematical Reasoning be studied (p.132).” Over the course of the last thirty years, teacher education A general definition for substantive knowledge of teaching programs took on the responsibility for much of the refers to understandings of particular topics within a research, training, and support of pre-service candidates' discipline, procedures, concepts, and their relationships to (PSC) Subject Matter Knowledge (SMK) and Pedagogical each other. Mathematical substantive knowledge can be Content Knowledge (PCK). Given the possible repercussion seen as the knowledge of mathematics and includes an that teachers' knowledge of subject matter has on their understanding of particular mathematical topics, pedagogy, many teacher educators considered ways to procedures, concepts, and the connections and incorporate discussions of subject matter into teacher organizing structures within mathematics. education programs (Grossman et al., 1989). Ma (1999) Teachers' substantive knowledge of discipline structures stated that it is during teacher education programs that has strong implications for what and how teachers choose PSC have one of three opportunities to cultivate their to teach. To learn mathematics with understanding, knowledge of school mathematics and that “their students must be exposed to relevant mathematical mathematical competence starts to be connected to a relationships and connections in their mathematics primary concern about teaching and learning school courses (Ball, 1988; NCTM, 2000). One of the implications is mathematics” (p.145). Several researchers worked with the influence teachers' substantive knowledge can have PSC to advance teacher education programs and on curricular decisions. Teachers with a strong knowledge develop courses that promote teachers' SMK and PCK (Ball, of mathematics make connections among relationships 2002, 1988; Ball & McDiarmid, 1990). within topics that promote students' conceptual Grossman, Wilson, and Shulman (1989) pointed out that understandings (Ball, 2002, 1991; Borko & Putnam, 1996; having knowledge of only the content is not sufficient. For Ma, 1999; Stigler & Hiebert, 1999). “Given the potential example, a PSC with a strong content knowledge may not impact teachers' knowledge of substantive structures may be able to make connections and illustrate relationships have on their pedagogy, teacher educators need to between mathematical topics and may teach them as if consider ways to incorporate discussions of substantive they are fragmented pieces of information. Thus, structures into programs of teacher education” (Grossman disciplinary knowledge must also include knowledge of the et al., 1989, p.29). underlying structures (substantive knowledge) and The syntactic knowledge of teaching is the evidence and knowledge of how to conduct inquiry (syntactic proof that guide inquiry within the discipline. It focuses on knowledge). Pre-service candidates' of mathematical where the discipline comes from, how it changes, and how reasoning is based on their understandings of how students truth is established within the discipline. Ball (1991) develop syntactic knowledge and substantive knowledge. emphasized that syntactic knowledge of mathematics is In support of this idea the National Council of Teachers of knowledge about mathematics. Syntactic knowledge is Mathematics' (NCTM) Professional Standards for Teaching seen as the nature of the knowledge in a field of study. 32 i-manager’s Journal on School Educational Technology, Vol. 9 l No. 2 l September – November 2013 RESEARCH PAPERS “Knowledge about mathematics also includes what it PSCs' knowledge of mathematical reasoning. Borko and means to 'know' and 'do' in mathematics, the relative Putnam (1996) suggested that learning opportunities for centrality of different ideas, as well as what is necessary or teachers be grounded in the teaching of subject matter logical, and a sense of philosophical debate within the and “provide opportunities for teachers to enhance their discipline” (Ball, 1991, p. 7). own subject matter knowledge and beliefs” (p. 702). In response to this assertion, and Schoenfeld's (1999) call for PSCs' knowledge of syntactic structures has many different theoretical research that is focused on practical and components. These components are concerned with relevant applications throughout education, the focus of establishing truth within a discipline. Truth within a discipline this research is to explore PSCs' images of mathematical comes from its preserved foundations and new evidence reasoning. It is this researcher's belief that in order for or inquiry that gives rise to debate (Lakatos, 1976). For teachers to get a better understanding of students' example, Schoenfeld (1999) asked researchers and conceptions about mathematical reasoning, they professionals within the discipline of education to themselves should evaluate their own ideas and notions “characterize fundamentally important educational are as about mathematical reasoning. for investigation, in which theoretical and practical progress can be made over the century to come” (p. 4). Research Conceptual Frameworks Since this challenge for debate and inquiry, research has The conceptual frameworks used in this research help focused and made some improvement in many of the explain the PSCs' images of mathematical reasoning when areas Schoenfeld called “sites for progress.” Grossman, examining their responses to several related questions Wilson and Shulman (1989) stressed that teachers' lack of about mathematical reasoning and proof. The syntactic knowledge can limit their abilities to learn new conceptual framework uses components of Shulman's information in their fields. Teachers with limited syntactic (1986) knowledge base in teaching and incorporates Pirie- knowledge may not be able to distinguish between more Kieren's (1994) notions of primitive knowledge and images or less legitimate claims within a discipline. Furthermore, a from their model for growth in mathematical lack of syntactic knowledge may also cause teachers to understanding that was adapted to teacher preparation misrepresent the mathematics they are teaching. by Berenson, Cavey, Clark and Staley (2001). Teachers with a limited syntactic knowledge may be Shulman (1986) described three categories of content unable to sufficiently explain relationships or engage in knowledge: subject matter content knowledge, discourse to allow their students to explore of mathematics, pedagogical content knowledge, and curricular in this case, impeding their abilities to teach as reflected in knowledge. Content knowledge refers to “the amount of the SMP and therefore impacting their students' capabilities knowledge per sec in the mind of the teacher” (p. 9). to reason mathematically. Subject matter content knowledge is seen as more than PSCs' perspectives of their discipline influence their views of the knowledge of facts and concepts of a subject, but also the roles of factual knowledge, evidence and inquiry. The includes understanding of the substantive and syntactic syntactic knowledge of teachers is instrumental in structures (Grossman et al., 1989; Shulman, 1986). PCK is a determining the classroom environment that they nurture. type of SMK that is improved and embellished by Teachers with a strong sense of mathematical syntactic knowledge of the learners, knowledge of the curriculum, structures are more likely to have classrooms that and knowledge of context pedagogy. Curriculum incorporate mathematical reasoning by including knowledge is seen as the knowledge of the range in discussions and activities aimed at developing their programs, the materials available, and characteristics of a students' awareness and understanding. curriculum, at any grade level. Lastly, Shulman described Research Focus three forms of knowledge that teachers use in their practice: propositional knowledge, case knowledge, and The focus of this research is to provide information about i-manager’s Journal on School Educational Technology, Vol. 9 l No. 2 l September – November 2013 33 RESEARCH PAPERS strategic knowledge. study was concerned with developing prospective teacher knowledge in the concepts of mathematical content and Propositional knowledge is knowledge from examples of processes. The study was done over a 15-week semester, literature that contain useful principles about teaching. involving several PSCs in two classes. One class was taught Propositional knowledge is also the wisdom of practice, on-campus and the other was taught strictly online. empirical principles, norms, values, and the ideological and philosophical principles of teaching. Case knowledge Over twenty-five pre-service candidates, who were is specific, well-documented, descriptive events of scheduled to take the course on-campus and online, propositional knowledge. Cases are specific instances in responded to the questions through Blackboard. The data practice that are detailed and complete descriptions that source included a series of responses from writing prompts exemplify theoretical claims and communicate principles regarding mathematical reasoning. Weekly, the PSCs were of practice and norms. Strategic knowledge is used when a given one reflection question, through Blackboard, in teacher cannot rely on propositional or case knowledge, which they had forty-five minutes to answer. The use of and must formulate answers when no simple solution technology allowed for the flexibility in the time for which seems possible. These strategies about teaching go PSCs could take to reflect upon the questions. By giving this beyond principles or specific experiences and are opportunity for online reflections to both the on-campus formulated using alternate approaches. The PSCs' section of the course, as well as the online section, it examples, definitions, activities, and explanations are allowed all PSCs to answer the reflections when they had examined using Shulman's three forms of knowledge to see time to reflect and did not restrict anyone to a classroom if they are using propositional, case, or strategic setting. knowledge. The questions were developed from a call for proposals for The Pirie and Kieren (1994) model for growth in the NCTM's publication Mathematics Teaching in the mathematical understanding was adapted by Berenson, Middle School. The PSCs were not privy to the questions Cavey, Clark and Staley (2001) creating a model for beforehand, unless they accessed the questions through studying prospective teachers' understanding of what and Mathematics Teaching in the Middle School. The only how to teach high school mathematics. The teacher criteria expected from the PSC were that their responses be preparation is a conceptual framework that is designed to at least a half page in length, with other formatting capture the process of learning as prospective teachers guidelines, which included font and font size, and single- come to an understanding of both what and how to teach spacing. Written statements for each of the eight questions, in high school mathematics. The “what” to teach refers to listed below, were analyzed for the PSCs' images of the school mathematics that prospective teachers will mathematical reasoning. teach after their teacher education training. The “how” to The questions used during the study were taken from a call teach is the teaching strategies that are used by the for manuscripts from the NCTM publication Mathematics prospective teachers. Both models define primitive Teaching in the Middle School. The questions that the PSC knowledge as what is known. Making/having images is were asked to respond to were: when PSCs use their primitive knowledge in new ways, and 1. What habits of mind does mathematical reasoning these images can lend insight into a PSCs' primitive entail? How can those habits be cultivated? knowledge. 2. What strategies, processes, or resources can students Synchronous and Asynchronous Methods of Inquiry use to reason mathematically? The investigation of PSC's knowledge of mathematical 3. How can teachers help students progress from reasoning was part of a study that investigated prospective concrete reasoning to symbolic or abstract reasoning? teachers' knowledge of what and how to teach concepts 4. How can teachers elicit students' inductive or in an elementary mathematics methods course. The larger 34 i-manager’s Journal on School Educational Technology, Vol. 9 l No. 2 l September – November 2013 RESEARCH PAPERS deductive reasoning? to determine if differences truly exist in the environment in which the research takes place (Osler, 2012). 5. What are teachers' beliefs and attitudes toward mathematical reasoning in learning mathematics? How Trichotomous Categorical Variables—Research Questions can these attitudes be supported, enhanced, or 1. Is Mathematical Reasoning effective as a teaching changed? method? 6. How can teachers help students develop 2. Does Mathematical Reasoning have an effect on mathematical understanding through mathematical student outcomes? reasoning, and vice versa? What challenges will teachers 3. What were teacher perceptions of Mathematical encounter? Reasoning? 7. How can mathematical reasoning be fostered by Trichotomous Outcome Variables—Responses representations, communication, curriculum, technology, 1. Yes or the learning environment? 2. No 8. How can teachers help their students develop an 3. No Opinion appreciation for the value of proofs and enhance their students' capacity to construct proofs? (NCTM, 2010). Research Hypotheses The conceptual frameworks used were instrumental in the The below hypotheses were used to assess the research analysis of the data. Analysis of the data for the research questions one and two. Each research question addresses involved coding, sorting into categories of significance, a null hypothesis with anticipation of a non–significant and establishing patterns among the categories. Data association, and an alternative hypothesis that suggests from question responses were coded and evidence of that a significant association does occur between the mathematical reasoning was extracted and examined. To variables. analyze the images of mathematical reasoning that PSCs H: There is no significant difference in the Pre-Service 0 had were coded using Shulman's (1986) three forms of Candidate (PSC) perceptions on Mathematical Reasoning teacher knowledge. They were looked at for consistency in regards to effectiveness as a teaching method, throughout the semester. effectiveness in terms of student outcomes, and Data Analysis Methodology effectiveness as a method of classroom instructional delivery. The Total Transformative Trichotomous–Squared Test (Tri–Squared) provides a methodology for the H: There is a significant difference in the Pre-Service 1 transformation of the outcomes from qualitative research Candidate (PSC) perceptions on Mathematical Reasoning into measurable quantitative values that are used to test in regards to effectiveness as a teaching method, the validity of hypotheses. The advantage of this research effectiveness in terms of student outcomes, and procedure is that it is a comprehensive holistic testing effectiveness as a method of classroom instructional methodology that is designed to be static way of holistically delivery. measuring categorical variables directly applicable to Mathematical Hypotheses educational and social behavioral environments where The Mathematical Hypotheses used in the study in terms of the established methods of pure experimental designs are the Tri–Squared Test to determine PSC candidate easily violated. The unchanging base of the Tri–Squared perspectives regarding Mathematical Reasoning are as Test is the 3 × 3 Table based on Trichotomous Categorical follows: Variables and Trichotomous Outcome Variables. The H: Tri2 =0 emphasis the three distinctive variables provide a thorough 0 H: Tri2 ¹0 rigorous robustness to the test that yields enough outcomes 1 i-manager’s Journal on School Educational Technology, Vol. 9 l No. 2 l September – November 2013 35 RESEARCH PAPERS Quantitative Research Results Reasoning in regards to effectiveness as a teaching method, effectiveness in terms of student outcomes, and Table 1 shows the Assessment of Mathematical Reasoning effectiveness as a method of classroom instructional Tri–Squared Test Qualitative Outcomes. delivery. Data Analyzed Using the Trichotomous–Squared 3x3 Table Qualitative Research Outcomes designed to analyze the research questions from an Inventive Investigative Instrument with the following In examining the PSCs' responses to the questions there is a Trichotomous Categorical Variables: a = [Mathematical need to define terminology to keep consistent when 1 Reasoning effectiveness as a teaching method?] = explaining their images. In framing their responses, the Qualitative Instrument Items: 1, 2, 4, 6, and 7; a = PSCs define a student and a teacher as we would expect 2 [Mathematical Reasoning and its effect on student them to be defined in a K-12 setting. As the PSCs draw outcomes?] = Qualitative Instrument Items: 3 and 8; and a upon their images of mathematical reasoning, the data 3 = [Teacher perceptions of Mathematical Reasoning as it has strong implications for their propositional knowledge relates to classroom instruction?] = Qualitative Instrument while examples of case knowledge and strategic Item: 5. The 3 × 3 Table has the following Trichotomous knowledge are combined throughout to exemplify either Outcome Variables: b = Yes; b = No; and b = No observations in a K-12 mathematical classroom or 1 2 3 Opinion. The Inputted Qualitative Outcomes are reported personal experiences they have had as a student in such a as follows: classroom. Tri2d.f.= [C –1][R –1] = [3 –1][3 –1] = 4 = Tri 2 In qualitatively reviewing the data, it becomes evident that [] the PSCs have strong images with regards to propositional The Tri–Square Test Formula for the Transformation of knowledge. The data reflects that the PSCs problem- Trichotomous Qualitative Outcomes into Trichotomous solving, reasoning and proof, communication, Quantitative Outcomes to Determine the Validity of the connections, and representations as processes needed to Research Hypothesis: foster mathematical reasoning. The PSCs view mathematical reasoning as an idea that can be achieved in the classroom through a multitude of strategies that can Tri 2 Critical Value Table = 0.207 (with d.f. = 4 at α = 0.975). be modeled by the teacher. As one PSC commented the For d.f. = 4, the Critical Value for p > 0.975 is 0.207. The following: calculated Tri–Square value is 17.47, thus, the null “Students must have the habit of providing a rationale as a hypothesis (H) is rejected by virtue of the hypothesis test 0 major part of every answer. They need to learn to justify which yields the following: Tri–Squared Critical Value of their ideas through logical argument. Students should be in 0.484 < 17.47 the Calculated Tri–Squared Value. Thus the the habit of logical thinking in order to decide if our answers null hypothesis is rejected and there is strong evidence that make sense and why they do so. It isn't enough to show the supports that there is a significant difference in the Pre- right answer, but a student needs to know why it is right. Service Candidate (PSC) perceptions on Mathematical These habits of mind must be cultivated by teacher example.” TRICHOTOMOUS CATEGORICAL VARIABLES Some PSCs view mathematical reasoning as student nTri = 19 a a a metacognition and self-reflection. As represented in this α = 0.975 1 2 3 comment: b 100 24 22 1 “When performing math tasks it is important to remember TRICHOTOMOUS OUTCOME b 37 30 7 to be reflective in your thinking. Always asking yourself, VARIABLES 2 “Does this make sense?” This will allow you to see that if your b3 6 4 0 answer does not make sense, or you realize you came 36 i-manager’s Journal on School Educational Technology, Vol. 9 l No. 2 l September – November 2013 RESEARCH PAPERS across the incorrect answer, that you can use this as an learning.” opportunity for learning.” Interestingly, the PSCs propositional images of The example below exemplifies that mathematical mathematical reasoning were vastly different then their reasoning is a process that is student-centered, however, it personal experiences as students. They felt as if the should go beyond the individual to the nurturing of the opportunities that they were given to reason process by the classroom teacher. As stated in this mathematically were limited. As honestly demonstrated in reflective comment: following two comments: “I think that I will start will the learning environment because I “This was one thing about learning Mathematics that think that can really foster or take away from a child's ability hindered me, as I wasn't taught in a variety of ways. I never to use mathematical reasoning. If a child feels safe in their really learned mathematical reasoning; I was taught to classroom; not just physically safe, but safe to explore their perform mathematical algorithms without an explanation thinking; then I believe they will be more likely to try new of what they represented.” things and grow in their reasoning skills.” “When I was in elementary school, we didn't a lot of the In looking at ways that a classroom teachers can nurture reasoning behind the concepts, just the algorithms. I think students' mathematical reasoning, many of the PSCs this has really hurt my mathematical reasoning skills.” focused on Polya's Problem-Solving Process or a similar While the PSCs may not have had K-12 experiences that strategy as a framework to elicit discussion. Several images allowed them to reasoning mathematically consistently, explored incorporating technology and real-life they do recognize its importance. This supports that notion applications to make mathematics more relevant to the that they would rely heavily on their propositional students' lives as stated by one PSC in the following knowledge with regards to how mathematics will be taught reflection: in their classrooms. “Mathematical reasoning can be fostered by technology Conclusion because it allows students to focus on the process of Current reform efforts such as the Core Common problem solving instead of the process of calculating Standards guiding curriculum decisions, it is imperative that numbers or amounts. Mathematical reasoning is also teacher education programs look forward to the Standards fostered by technology because it gives the opportunity for for Mathematical Practice, so that instruction can become students to get acquainted with interesting problems.” better aligned with these efforts. These practices are built The PSCs believe that mathematical reasoning must occur on established processes and proficiencies for on an ongoing basis. They believe that in order for teachers mathematics education that rely heavily on PSCs' to expect mathematical reasoning from their students that knowledge and beliefs of mathematical reasoning. Two the classroom environment must be positive and allow for important themes emerged during the qualitative image discourse, so that the students do not feel intimidated and investigation of the Pre-Service Candidates: are encouraged to explore. This is demonstrated in this PSC 1. Opportunities for students to communicate, reflect, reflection: and explore a variety of mathematical representations are “Students should also see that math is not simply seen as important for fostering mathematical reasoning. memorization; they need to become comfortable with the The PSCs saw a lack of exploration of mathematics content topics so that they are able to see that many ideas in math as a detriment to their learning. They felt as if they were are interrelated and they can use one concept to help taught in ways that required understanding through rote them solve another. Overall, I would say that the best habit memorization without opportunities for exploration of the of mind to have when learning math concepts is content. In looking back on their own limited experiences persistence, because seeing math as something that you with reflection and communication, some PSCs felt as if are determined to figure out is the best way to want to keep i-manager’s Journal on School Educational Technology, Vol. 9 l No. 2 l September – November 2013 37 RESEARCH PAPERS they could have been more familiar and self-assured in for Research on Teacher Education (pp.437-449), New their learning of mathematics content if they had York: Macmillan. additional occasions in which to discuss and internally [5]. Berenson, S., Cavey, L., Clark, M., & Staley, K. (2001). process their learning. Adapting Pirie and Kieren's model of mathematical 2. Mathematical learning environments must nurture a understanding to teacher preparation. Proceedings of the climate of mathematical reasoning. Twenty-Fifth Annual Meeting of the International Group for the Psychology of Mathematics Education. Utrecht, NL: One of the most promising results of the study was the PSC's Fruedenthal Institute, 2. 137-144. image of the importance of mathematical reasoning and the role in which the classroom teacher plays in facilitating [6]. Borko, H., & Putnam, R.T. (1996). Learning to teach. In this process. They knew that teachers needed to afford D.C. Berliner & R.C. Calfee (Eds.), Handbook of Educational students multiple chances to reflect and communicate Psychology (pp. 673-708). New York: Simon and Schuster their mathematical understanding. They saw a teacher's Macmillan. role as going beyond “just telling” to establishing a [7]. Grossman, P.L., Wilson, S.M., & Shulman, L.S. (1989). classroom environment where explanations are processed Teachers of substance: Subject matter knowledge for and explored in ways that incorporate technology or other teaching. In M.C. Reynolds (Ed.), Knowledge base for the tools for learning. beginning teacher (pp. 23-36). New York: Pergamon. 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