Running Head:  DIRECT VERSUS INDUCTIVE APPROACHES

 

 

 

 

 

 

 

 

 

Direct Versus Inductive Approaches to Mathematics Instruction

Pamela R. Hudson Bailey

George Mason University

May 24, 2009

EDCI 855: Mathematics Education Research on Teaching and Learning

 

 

 

 

 

 

 

 

 

 

Direct Versus Inductive Approaches to Mathematics Instruction

            Debate of methodologies that promote student learning, retention and application of concepts is ongoing. The National Council of Teachers of Mathematics (NCTM) has been on the forefront and an opponent of the reform movement and debate. Misconceptions of inductive approaches, along with personal beliefs and biases, have led to the war of mathematics regarding student learning and successful methodologies. Prince and Felder (2006) posit that different studies on the same subject will produce different results since they are dealing with people and their interpretations of a situation. Problem solving, social interactions, knowledge retention, motivation, student-centered concerns, self-explanation and sense-making are all characteristics brought forth in the research on inductive approaches and direct instruction. Teacher and student risk taking is also a concern when dealing with individuals and the experiences they bring with them into the mathematics classroom. The reform movement controversy is not a new issue.  

Mathematics War: Why Controversy

In the mid 1980’s computers were becoming more common place in the workplace, schools and homes, the failure of traditional mathematics instruction had been documented and scientific studies on teaching and learning mathematics led to the Reform Movement (Battista, 1999). The scientific research results revealed that mathematics knowledge needs to be constructed and make sense to the learner. Battista stated that mathematics instruction had been dictated for more than forty years by behavioral methods. Opponents to the mathematics reform, as with any new development, wanted a back to the basics approach that was founded on non-research gathered information. Mathematics educators were on one side of the debate with the opposing side, the mathematicians, on the other side (Mervis, 2006). The opposing force to the reform movement, a back to the basics approach, information was built on the fear of the unknown, lack of knowledge and understanding and the risks that teachers and students would be taking. Back to the basic meant direct instruction, rote drill, procedures and memorization. The opposing dichotomies were that the reform movement was bad and the basics were good. Direct instruction versus learning by discovery was the argument between the opposing sides. The reform movement meant teaching students “fuzzy math” (Martinez & Martinez, 1991).

            By the late 90’s more students were taking upper level mathematics as a result of the reform movement (Martinez & Martinez, 1991). The movement was a call to go beyond rote drill without meaning with students needing to understand and apply mathematics to real-world situations. Guidance for mathematics educators was found in the Principles and Standards for School Mathematics (NCTM, 2000). NCTM’s teaching principle and the learning principle state that students should be able to learn mathematics with understanding from competent teachers. Actively building conceptual knowledge from their experiences and previous knowledge will allow students to solve authentic problems that will encounter in their futures (NCTM, 2000). Along with the principles, NCTM presents process standards that guide each level of instruction and includes making connections, reasoning and proof, communication, representations and problem solving. Documents published by NCTM were a guiding force for the reform movement but the math war continued. The Common Ground Initiative, consisting of six members, helped the opposing sides to agree on many issues (Mervis, 2006). One of the results of the initiative led NCTM to develop a list of topics. Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (2007) gives three major ideas for each grade level from kindergarten through eighth grade thereby answering a concern stated by Battista (1999) regarding a need to go deeper instead of wider by studying fewer topics. The Common Ground Initiative negotiated principles that affected K-12 education (Mervis). Principles included a recall of basic facts, reasoning skill importance, mastery of significant algorithms, correct usage of calculators and real-world focus for problems.

The process standards are needed in a global society in which technology is ever changing and affects everything one does (Battista, 1999). The inductive approaches that had risen and fallen over the years gained strength with the NCTM documents and the reform movement. Concerns that existed, and still exist, include a teacher’s lack of knowledge about content and how students learn mathematics (Battista). Hence a concern and debate over teaching methodologies that increase student knowledge. In order to compare direct and inductive approaches one must first recognize the characteristics and concerns for each type of methodology.

Definitions of Methodologies

Inductive Approaches

Inductive approach is an umbrella term for many methodologies.

            Since the mid 1950’s inductive approaches to teaching mathematics have been on a roller coaster ride (Kirschner, Sweller & Clark, 2006). Inductive approaches are considered to be the opposite of school learning as they surface, dip, and then resurface with slight changes being the new curve on the inductive roller coaster (Illeris, 2007). The ride began with discovery learning which led to experiential learning, problem-based learning, inquiry-based learning, and constructivist teaching (Kirschner, Sweller & Clark). Prince and Felder (2006) included project-based and just-in-time teaching in their list of inductive approaches to teaching and learning. Inductive approaches may also be called student-centered or learner-centered. All of the approaches make connections to learning through collaboration, discussions, and active learning (Prince & Felder).         

Discovery approach.

            Mayer (2005) discusses two types of discovery learning, pure and guided. For either approach, students are given a problem, question or an observation and complete the work individually, deciding what, when and how to obtain the solution (Prince & Felder, 2006). Pure discovery involves students receiving no guidance however the teacher may provide some feedback (Prince & Felder). The teacher will provide guidance throughout the learning process with guided discovery. Studies conducted on the discovery approach have shown that the method promotes better transfer and conceptual understanding for students (Rittle-Johnson, 2006; Star & Rittle-Johnson, 2008). Star and Rittle-Johnson’s (2008) study required students to solve equations more than one way by discovering their own procedures to meet the needs of the specific problem. As students solved problems multiple ways, knowledge and usage of multiple strategies did increase. Even though student usage of strategies increased their ability to determine when to use and apply a strategy was lacking. Prompts given by teachers using the guided discovery approach was found to increase student knowledge. Presenting new material using guided discovery was better than employing a pure discovery approach.   

            Students are required to explain the processes they employ and justify and defend their statements throughout the discovery process (Muthukrishna & Borkowski, 1995). Reflecting on their processes and solutions also play a large part. This approach resulted in better usage of processing strategies by students which was greater than direct instruction or a combination of both. Star and Rittle-Johnson (2008) state that the guided discovery process should be considered a complement to direct instruction. Student engagement plays an important role in the discovery process as well as in experiential learning.

            Experiential approach.

            Learning by doing is the foundation for experiential learning according to Illeris (2007) who traces its beginning to Kolb and Dewey. Kolb felt that all learning is gained through one’s experiences (Illeris, 2007; Warwick, 2008; Hartshorn & Boren, 1990; Kirschner, Sweller & Clark 2006). Dewey took it one step further stating that one needed to be concerned with the quality of the experience (Illeris, 2007). One’s experiences need to be connected to his or her past as it will affect future actions. Interactions between individuals, the individual and the environment or the individual and the experience are all controlled by the learner. Warwick (2008) also refers to Kolb model of experiential learning (abstract conceptualization, active experimentation, concrete experience, and reflective observation) while relating the stages to a situational leadership model (directing, coaching, supporting and delegating) developed by Hershey and Blanchard. Beginning with new material, the teacher employs the abstract conceptual model along with direct leadership. This is a very traditional stage with the flow of information going one way, from the teacher to the student. Active experimentation follows with a coaching model of leadership applied by the teacher. Students try problems on their own with input from the teacher, two way communication. The third stage involves concrete experiences with a style of leadership that delegates control of learning to the student. Amount and type of support needed is determined by the student. Lastly, supportive leadership style is combined with reflective observation as students look back on the experience with discussion guided by the teacher.

            Illeris (2007) looks at experiential learning as having three sides, content, incentive and the social side. Content is related to cognitive learning, incentive to one’s emotions and social to the interactions he or she has while learning. All three play a part in gaining knowledge; forming schemas as one recalls and applies new situations to similar learning and assimilates new material to existing schemas. Learning takes place as one creates a change or rebuilds mathematical understanding and/or attitude toward learning mathematics, altering schemas. According to Illeris, experiential learning is determined by the amount of student engagement in the learning process and the intensity of the type of engagement. Hartshorn and Boren (1990) posit that the active involvement in experiential learning leads to greater student achievement and that using manipulatives increases student understanding of concepts. Manipulative use helps students move from concrete thinking to abstract thought. The longer that a manipulative is used by students leads to increased effectiveness.

            Students carry out an action in experiential learning but one needs to acknowledge the effect of the actions for the given situation (Kirschner, Sweller & Clark, 2006). The acknowledgement is important so that when one encounters similar circumstances he or she will be able to apply the action. An individual’s learning style also plays a part in the experience and the approach one might take in the actions applied (Kirschner, Sweller & Clark). The situation becomes more vague in the problem-based learning approach.

            Problem-based or project-based instruction.

            Open-ended, authentic and/or an ill structured problem that is analyzed by teams of students is considered problem-based learning whereas teams in project-based learning may be given several problems that form the overall project (Prince & Felder, 2006). Problem-based is usually established on previously learned material with students focusing on acquiring new knowledge as they develop their own solutions. Focus for project-based learning is on applying the knowledge gained throughout the project. Problem and project based learning improves conceptual knowledge, problem solving skills and attitudes toward mathematics. Trouble transferring learned concepts to future project and problems is viewed as a concern by Prince and Felder. Inquiry-based learning is very similar to problem-based, project-based and discovery learning.  

            Inquiry-based instruction.

            Inquiry-based instruction shares the spotlight with discovery learning as the umbrella of several inductive methods (Prince & Felder, 2006). Students are given a problem, question or an observation, same as discovery, with a focus on students learning to develop good questions, recognize type of data needed and method for collecting the data, report results and articulate a conclusion while evaluating its affect on society. The answer and its justification and affect are the focus of inquiry-based learning. Prince and Felder mention five types of inquiry learning: structured, guided, open, teacher and learner. The types range from students being given the problem to solve to determining what the problem might be given a scenario. For best results using the inquiry-based approach, questions should be such that students can answer them through the investigation while making a connection to something familiar. Being actively involved in the learning process in order to build knowledge is also the vision of constructivism (Mayer, 2004).

            Constructivist approach.

            Knowledge should be attained through active learning and making sense of the process and material in order for understanding to be coherent (Mayer, 2004). Handal (2003) places emphases on the personal and social interactions involved as students build knowledge which he terms socio-constructivism. Students need to explain and justify their thinking and activities which reinforces the importance of social interactions. Knowledge should be built based on physical and mental activities with students reflecting on their actions (Handal) while making connections to prior knowledge and experiences (Prince & Felder, 2006). Cycles of inconsistencies and uncertainty will evolve as students go through the process of problem solving and exploring, working toward a solution that makes sense (Handal). These periods are when learning occurs (Hills, 2007). Active learning in most situations implies physical activity (Mayer, 2004; Kroesbergen, Van Luit & Maas, 2004). Contribution to learning is based on what the student does however the active learning motivates students to want to participate (Kroesbergen, Van Luit & Maas). The teacher does not instruct students on how and when to use a strategy therefore if the student does not discover the strategy they will never get it (Kroesbergen, Van Luit & Maas). Teachers do guide discussions through questioning. Constructivists feel that if you do the mathematics then you will know the mathematics (Handal). A thematic unit approach enables students to make connections between many concepts as they do the mathematics.

Pure constructivist learning of new materials have shown to employ multiple strategies (Hills, 2007) as well as groups receive more guidance compared to students learning via direct instruction (Timmermans, Van Lieshout & Verhoeven, 2007). When new material makes connections and integrates into the existing schema then learning is said to have taken place (Prince & Felder, 2006) with students employing multiple strategies (Hills). Assessment of students results in the opposite being true. Students use strategies of which they are comfortable and will not complete the challenging problems when completing assessments (Hills).

Boys and girls also look at constructivism differently (Timmermans, Van Lieshout & Verhoeven, 2007). Girl’s performance in a constructivist activity is greater than girls involved in direct instruction however boys in both groups show no difference. Boys used more strategies than girls on a pre-test however the opposite was true on the post-test with boys using one strategy, girls multiple strategies. One must always remember that what an individual brings to the situation will affect instruction that includes past experiences, acquired knowledge (Kirschner, Sweller & Clark, 2006), fears, prejudices, beliefs, preconceptions and misconceptions (Prince & Felder, 2006). In general, low performing students in regular schools learning using a guided constructivist approach have not been successful; more specifically low performing girls did show improved performance.(Timmermans, Van Lieshout & Verhoeven).

Students receiving instruction based on the constructivist approach need enough leeway so that they might be active cognitively to insure sense-making of the learning and the solution (Mayer, 2004). Guided constructivism is preferable to pure constructivist approaches. Students need guidance, according to Mayer, so they might construct useful knowledge and make application to the given situation. It is not how much activity students are involved with in the learning process, it is the meaningful cognitive activity that truly creates learning. 

            Cased-based instruction.

            Hypothetical, historical or authentic cases are given to students with the document being well structured and detailed, enabling learning to be applied (Prince & Felder, 2006). The case may involve more than one problem or a complex scenario. Decisions or solutions to the case may be omitted so students will determine their own solution without influence. Cases may be approached individually or in groups with multiple approaches to obtaining a solution encouraged. Just-in-time teaching is an individual activity or one approached electronically. 

            Just-in-Time instruction.

            Students answer questions about assigned readings during face to face class time or discussion groups online, prior to formal discussions (Prince & Felder, 2006). The approach is mainly employed with web based classes. Searching for the answers to the questions and for a thorough understanding of material is the responsibility of the student. During class time, whether face to face or web based, sessions are collaborative between students and teacher.

 Direct Instruction

Inductive approaches are student-centered, direct instruction is teacher-centered (Kroesbergen, Van Luit & Maas, 2004). The teacher will explicitly state how, when, and what students need to do when learning new material with very little student input. Modeling problem solving, understandable explanations, remedial feedback, and praise for accurate answers are considered characteristics of direct instruction (Mayfield & Glenn, 2008). These back to the basics ideas correlates with the non-reform movement (Battista, 1999). Battista revealed that studies indicate that the traditional method of direct instruction affect student growth in mathematics, problem solving and reasoning. Kroesbergen, Van Luit and Maas’ study of 265 low achieving students in the Netherlands confirmed Battista’s findings revealing that the scores of those participating in direct instruction were greater in problem solving than those in the constructivist group. Mayfield and Glenn agree with Kroesbergen, Van Luit and Maas’ regarding low achieving students with the added comment that cumulative practice of processes and concepts is the key that leads to better problem solving skills. Problem solving interventions for low achieving students using direct instruction include cumulative practice, tiered feedback, solution sequence instruction, and review practice of at least one of the targeted skills (Mayfield & Glenn). The tiered feedback did not include additional strategies or improve problem solving however solution sequence instruction did improve problem solving due to students receiving feedback plus additional interventions.

Direct instruction was shown to not be needed all the time for low achieving students and was not more motivating (Kroesbergen, Van Luit & Maas, 2004). Being a “sit and get” type of learning, students accept learning without questioning (Illeris, 2007). Students were shown to forget procedures when taught using direct instruction (Battista, 1999). Transfer is gained by continuously working examples. Repeated procedural practice helps with retention that leads to conceptual knowledge (Rittle-Johnson, 2006). Teaching a method/procedure may show the way for students to create additional procedures (Rittle-Johnson). Direct instruction, combined with self-explanation, leads to improved conceptual understanding and greater procedural learning and understanding (Rittle-Johnson). Students are more apt to resort to using methods learned through direct instruction on similar problems (Star & Rittle-Johnson, 2008). Star and Rittle-Johnson’s (2008) findings show that direct instruction strategy demonstrations do not support multiple strategy usage but do increase flexibility in the strategies that are used. All the different methodologies are useless unless students are motivated or have the motivation to learn.                 

Characteristics for Comparison

Student Motivation

            Motivation to learn is based on how a person perceives that they need to know the material (Prince & Felder, 2006). Along with a desire to know more, the instructional method chosen will influence a student’s motivation to learn concepts (White-Clark, DiCarlo & Gilchriest, 2008). The constructivist approach to learning may bring out various types of motivation (Muthukrishna & Borkowski, 1995). Prince and Felder includes all inductive approaches that use real-world problems, case studies and data as ways to increase student motivation but explicitly states that it is because the methods generate a need or desire to know. The time students devote to learning is an indicator of the level of motivation they possess (Prince & Felder). This goes along with students enjoying learning when they are more challenged and therefore, more motivated (Kroesbergen, Van Luit & Maas, 2004). When motivated, students are more task focused and have a better attitude toward mathematics. Problem solving skills improve as well as automaticity in mathematics when students are motivated to learn (Kroesbergen, Van Luit & Maas).

Muthukrishna and Borkowski’s (1995) study found that the discovery method, as well as the combined method of discovery and direct instruction, will result in improving student motivation and understanding of concepts. They go on to also state that learning environments that put emphasis on comprehension and deep processing may lead to a transformation in student motivation. Timmermans, Van Lieshout and Verhoeven (2007), focusing on gender, found that girls experienced positive emotions when involved in guided instruction while boys experienced positive emotions when involved in direct instruction. Combining this remark to Kroesbergen, Van Luit and Maas’ (2004) statement that motivated students have better attitudes would imply that girls are more motivated when taught using guided instruction and boys with direct instruction. Applying the same thought, Kirschner, Sweller and Clark (2006) posit that lower academic students taught using inductive methods like mathematics more, therefore are more motivated than those taught using direct methods.

Adding it up: Helping children learn mathematics, by the National Research Council (2001), lists five strands needed for mathematical proficiency. One of the strands, productive disposition, refers to a student’s view of mathematics and seeing the usefulness of the concepts and a belief in oneself. Again, one’s disposition, likeness and efficacy seen in the subject, correlates with the motivation one might have towards mathematics. 

Communication and Social Interactions

            Being able to communicate mathematically is emphasized by NCTM (2000) and the National Research Council (2001). Another strand of mathematical proficiency, adaptive reasoning, represents a student’s ability to apply logic, reflect on their actions or statements and explain in writing or orally justifying the procedure or process (National Research Council). Communication, a process standard, states that students should be able to use mathematical language as they coordinate and consolidate their thoughts so that others, fellow students and teachers, might understand the reasoning and process (NCTM).

            Students involved in the guided discovery method are able to communicate mathematically better than students taught using direct instruction (Muthukrishna & Borkowski, 1995). Inductive methods emphasize a student’s ability to communicate his or her understanding of problems and concepts (Muthukrishna & Borkowski). Reflecting on a solution or a process, collaboration and verbal interactions are positive end results for those using a discovery process (Muthukrishna & Borkowski). Teamwork is a characteristic of many of the inductive methods including problem-based and project-based learning (Prince & Felder, 2006). Mevarech’s (1991) study combined mastery learning and cooperative learning into cooperative mastery learning (CML). The combined method resulted in more communication between participants, verbally and in writing, than either of the two separately. CML provides an opportunity for students to receive feedback and correct mistakes. Discussing misunderstanding and errors is also a form a verbal metacognition for those participating in CML. As students verbalize procedures and justifications they are thinking and analyzing their own assessments.    

Retention of Concepts

            Students need to know and recognize how they learn which leads to transfer of knowledge (Prince & Felder, 2006). Once transfer is made it is considered to be in long term memory and readily available to the individual for recall and usage. A study involving instruction for low-achieving students in the Netherlands using explicit instruction and constructivism showed that after three months, including the summer break, retention was the same or just a little lower than the post-test for both approaches (Kroesbergen, Van Luit & Maas, 2004). Direct instruction and discovery methodologies, combined with self explanations led to retention of knowledge two weeks after instruction on the topic ended (Rittle-Johnson, 2006). Hartshorn and Boren (1990) looked at long term use of manipulatives finding that their usage increased student retention and ability to transfer to abstract thinking. Understanding what has been learned leads to retention, will elevate fluency, and aid in making future connections (National Research Council, 2001). The aptitude to transfer knowledge is helpful when students are trying to solve problems.   

Problem Solving

            One might say that transfer has been successful when a student makes connections between concepts, within a concept or between one or more similar problems or situations (Prince & Felder, 2006). Once concepts are transferable and connections are made, flexibility in using multiple strategies will be readily available for the student to use when problem solving (Star & Rittle-Johnson, 2008). A lack of flexibility may be directly linked to low academic achievement since flexibility implies an efficient use of strategies. NCTM’s (2000) problem solving process standard states that students should be able to use an appropriate strategy as they develop new concepts and solve problems. Being able to scrutinize ones process and reflect on ones actions is necessary as problem solving skills are transferred to solve a variety of contexts. Mayfield and Glenn (2008) posit that explicit instruction may improve problem solving performances of learning disabled students. Explicit instruction practices that were used in the study include review practices, feedback, prompts, and specifically stated strategies. Combining Star and Rittle-Johnson’s findings with Mayfield and Glenn’s would imply that low performing students are unable to be flexible in strategy and procedural usage. It also might mean that explicit instruction promotes flexibility. Whether employing direct instruction or an inductive approach, without guidance, as in pure discovery, students may lose focus of the problem and/or procedures needed to solve the situation (Mayer, 2005).

Sense-Making

            Sense making is one of the key principles of the Common Ground Initiative (Mervis, 2006). It is not enough to “plug and chug” when using algorithms. One must know why and how the algorithms work in order to apply them successfully (Mervis). Using open ended questions encourages students to make sense of the problem, process, procedure and solution by justifying statements, making sure they are relevant for the given situation (Mervis). Students need to make sense of mathematics so that he or she might retain material longer and communicate their knowledge to others in order to justify statements (Battista, 1999). This means that one must be given enough independence and guidance to be functionally cognitive so that the end result is useful to the learner and the situation (Mayer, 2004). Utilizing inductive methods, teachers need to establish a climate of sense making by rethinking class norms (Muthukrishna & Borkowski 1995). Class norms that are conducive to student-centered classrooms enable students to work effectively in groups and to express views with confidence.   

Self-Explanation

            As expressed earlier, self-explanation implemented along with discovery or direct instruction has resulted in retention of concepts (Rittle-Johnson, 2006). The benefits to discovery learning are when the students engage in manipulating, connecting and evaluating information through explanation (Rittle-Johnson). Direct instruction and self-explanation were shown in the study by Rittle-Johnson of 85 students in the third through fifth grade to improve conceptual understanding. Engaging in actively processing the information, regardless of whether it is combined with direct instruction or discovery, lead to higher academic achievement. Students who did not explain reverted back to incorrect procedures as explanations are a form of actively processing information. Those involved in self-explanation are metacognitively acknowledging how knowledge was attained which will in turn promote learning, retention and transfer (Rittle-Johnson). 

Concerns Of Student-Centered Learning

            Learning using a student-centered approach increases the likelihood of being taught using an inductive method or constructivist approach however concerns have risen (Zaslavsky, 2005). When a student-centered approach is employed a concern is that outcomes are not readily available. Students lack the confidence to justify and analyze the process and results or methods may not even be known that could used to verify results (Zaslavsky). One must realize that worthwhile learning is linked to uncertainty (Zaslavsky). There are competing claims of statements contradicting past learning or student understanding (Zaslavsky). A learners belief’s about a concept may even be in contradiction to the outcome. Teachers employing a cyclic approach of conjectures and conclusions are allowing their students to work through conflicts and contradictions that may arise by repeating the process of creating a new conjecture based on the most current conclusion. Teachers also need to be aware of how their students learn mathematics (Zaslavsky). Reflecting on activities to develop meaningful situations, pedagogically and mathematically, will enhance student and teacher relationships, student attitude toward the subject, learner retention and even motivation (Zaslavsky). Besides all of the concerns, Zaslavsky states that students experiencing uncertainty will have a desire to prove their processes \and mathematics. Teachers should acknowledge a need to create student doubt and uncertainty.

Teacher Risks And Concerns

            How one is taught is usually how one perceives teaching (White-Clark, DiCarlo, & Gilchriest, 2008). This includes the beliefs one brings with him or her as they will play a part in how material is taught. Teachers need to recognize their own beliefs and also what students bring with them to class (Hills, 2007; Wood, Cobb & Yackel, 1991). Beginning teachers usually begin their teaching careers viewing instruction as rule based and procedural (Wood, Cobb & Yackel). Change is uncomfortable. Hills (2007) posits that every person has a different view of social risk and will respond differently to situations of discomfort. Adding to the discomfort are the evaluations, formal and informal, by peers, administration, parents and students. There must be a need or a desire to try the unknown and have support from peers and administration in order to step out into an uncomfortable situation (Hills). Teachers want to feel competent with a desire for impressions to be positive and avoid negative perceptions (Hills). One’s competence is threatened as the instruction becomes more student-centered with discussions and activities that have the possibility of bringing forth some unknown.

Wood, Cobb and Yackel (1991) conducted a study of one teacher as she changed from traditional methodologies to constructivism. The teacher being studied encountered many conflicting situations in her growth. A need to find ways for students to work in groups productively as well as communicate effectively was the first point of disequilibrium as the social norms in the classroom needed to change. The norm of having students tell the teacher what he or she wanted them to learn was changing as students expressed their own thoughts about a topic. As student discourse evolves, teachers should not make evaluative remarks. Another point of disequilibrium for the teacher was when the students gave wrong answers and she could not jump in to correct. Good questioning skills were learned and applied to steer students toward the correct answer and procedures. Muthukrishna and Borkowski (1995) discuss the need to let students be responsible for their own learning. Students change as they focus on ideas versus procedures and solutions. Understanding procedural methods may be a conduit to creating and using their own procedures (Wood, Cobb & Yackel). As the teacher in the study grew, she realized that constructively teaching did not result in anything goes, she was still responsible for student learning and guiding student’s experiences.

Teachers need to know how much discovery and guidance to use and how much direct instruction (Mayer, 2005). Clear learning objectives should be developed that specify student outcomes (Prince & Felder, 2006) as there might be a concern about engaging student in activities where the content, method and product are not specifically defined (Hills, 2007). Determining the methodology that a teacher can employ and still maintain a learning environment is a personal decision that takes considerable thought. Teachers are unsure and uncomfortable with losing or having a lack of control, never knowing what students might ask (Hills, 2007; White-Clark, DiCarlo, & Gilchriest, 2008). Content uncertainty, unknown answers and being unable to respond with the answer are fears of teachers as they try new approaches (Hills, 2007). Experience with alternate methods of teaching mathematics that expose students to authentic problems that are student-centered will aid in transitioning to student-centered learning (White-Clark, DiCarlo, & Gilchriest, 2008). Traditional methods are safe with one answer problems that keep feelings of risk and anxiety levels low (Hills). As teacher anxiety levels rise, views on constructivism fall. In order to encourage others to change, one must know more about the individuals concerns to see how it affects his or her willingness to change. Determining the support teachers need may be difficult as those who need the encouragement and assistance will not likely let it be known (Hills). This is the same for students. Those who need assistance will likely not let their concerns be known. Lastly, teachers are concerned regarding content being learned (Wood, Cobb & Yackel, 1991). Mathematical procedures have been developed over the years to ease mathematics computations (Wood, Cobb & Yackel). These traditional procedures may not be developed or applied by students who are creating their own methods leading to a concern for students taking future mathematics courses and their ability to recall and apply procedures with ease.     

The amount of instruction is an ongoing decision when employing a method presented by Koichu (2008) called “If not, what yes?” Cycles of conjecturing and justifying statements facilitate conceptual understanding by having participants prove that they state is true. With each cycle the new conjecture is refined and more specific. Teacher questioning skills play an important role in the process as he or she leads students to developing concepts with a feeling of ownership. Preparedness for each anticipated conjecture permits the teacher to plan and guide students to develop the end result. Flexibility and knowing when to step in and when to question further are concerns for teachers employing this method as well as the fear of the unknown.

Another cyclic approach that focuses on procedural and conceptual learning is discussed by Rittle-Johnson, Siegler and Alibali (2001). Their study of fifth grade students showed that using correct procedures help students to recognize the important characteristics of a problem and may lead to a better understanding of concepts. Conceptual understanding reinforces procedural learning. Conducting a pre-test assessing conceptual knowledge is a good predictor of procedural learning. Likewise, assessing procedural learning is a good predictor of additional conceptual learning. Neither conceptual nor procedural learning is fully developed when learning new material and one does not always come before the other. A person’s prior knowledge and experiences in a specific domain will affect which type of learning will come first. Rittle-Johnson, Siegler and Alibali posit that competence is a combination of procedural plus conceptual knowledge which they have facilitated in their study using a constructivist approach.

Teacher education programs could assist pre-service teachers by creating an awareness of how to develop tasks that increase student uncertainty, like the Koichu (2008) approach, thereby developing learning situations (Zaslavsky, 2005). Tasks from various sources can be modified to create the uncertainty so that students are curious and motivation is heightened (Zaslavsky). Modified textbook problems are one such source along with a task that acknowledges student misconceptions and mathematical difficulties and authentic problems that elicit student debates and discussions. Education should include discussions about changing social norms which Wood, Cobb and Yackel (1991) mentioned. Zaslavsky continues the discussion on norms stating that the classroom climate and setting should be conducive to accepting creativeness and wrong answers. Teacher training should also include education on how to implement manipulatives effectively so that students gain understanding (Hartshorn & Boren, 1990). On the secondary level there is a lack of manipulative use so an awareness needs to be created early with pre-service and in-service education.

In addition to education there is a concern about leadership in the student-centered classroom (Warwick, 2008). Very little research has been conducted on teacher leadership with instead an emphasis on classroom management (Warwick). Traditional teacher actions include planning lessons and student activities, organizing the classroom before and during the day and controlling student actions. Leadership skills on the other hand include motivating and inspiring students (Warwick). Positive student attitudes lead to having the confidence to take risks and explore a question.

Forming support groups is another method to aid in communication between groups to promote school reform (Weidemann & Humphrey, 2002). The support group would be a network of individuals with a common goal of changing teaching techniques, include problem solving, technology and manipulatives into learning situations, methods of cooperative learning, increasing mathematical discourse, and creating alternative assessments. For a productive group, the network should be involved in a series of projects, form a common vision, and practice/present model lessons. Interactions and sharing between group members and organizers is very important for communication and at the same time enabling the coordinators to determine and provide for member needs and requests. The long range effect of the network is a change in teaching due to support and knowledge gained in addition to knowing that one is not alone in his or her endeavors.

Inductive Approaches: Positive and Negative

In summary, inductive approaches have many positive and negative aspects. Kirschner, Sweller and Clark (2006) state that inductive methodologies are not based on brain research or research on long and short term memory retention. Everything we see, say and do is based on what we have stored in long-term memory. Instruction should therefore be based on altering one’s existing long term memory. If long term memory is not altered then learning did not take place. One’s working memory is limited in capacity and duration. When one is learning new material, working memory is where information is initially stored. Information in working memory needs to be transferred to long term memory where one might recall and apply to new situations. Guided discovery, an inductive approach, is said by Kirschner, Sweller and Clark to ignore the limitations in working memory as if it were nonexistent.

Teachers instructing using an inductive approach will end up giving a great deal of guidance to students, resorting to direct instruction according to Kirschner, Sweller & Clark (2006) and Kroesbergen, Van Luit & Maas (2004). Solving problems without some guidance leaves students with no focus or direction (Mayer, 2005). This lack of direction results in students unable to select or recognize information or concepts when needed. Kirschner, Sweller and Clark state that those using a pure discovery approach will often lead to student frustration, confusion and misconceptions. They also felt that lower ability students will not retain material taught inductively. The study by Kirschner, Sweller and Clark showed that lower ability students scores on post-tests were lower than the pre-test scores which was attributed to student confusion. Applications of learned material may also be limited (de Haan & de Ridder, 2006). Limitations would include teacher decisions and depth of knowledge to make connections and applications.

Illeris (2007) mentions barrier to learning intended concepts. One is not concerned with what students have learned but instead with the non-learned blocked material (Illeris). Teachers need to determine why the concepts were blocked in order to counteract and overcome the barrier. Lack of participation, prior knowledge or communication may be reasons of blockages. In addition, determining whether the barriers were conscious or unconscious will make a difference in the approaches teachers would need to undertake for learning to be the result (Illeris).

On a more positive note, inductive methods were considered to challenge students to solve authentic problems where knowledge was obtained through experiences (Kirschner, Sweller & Clark, 2006). Students learned to reflect on their thinking processes and procedures, making connections to the real-world situations (de Haan & de Ridder, 2006). Slow thinking was encouraged so that students would have the opportunity to make connections and be able to justify their answers. De Haan and de Ridder’s study, even though it was non-mathematical, showed that while engaged in an inductive lesson participants rated learning high. Topcu and Ubuz’s (2008) study on web-based instruction showed that students involved in the indirect approach retained the material longer due to their participation and involvement in the lessons. Midterm exam results showed that those involved in direct instruction had better scores than indirect. The opposite was true for the final exam with indirect achievement gains greater than the direct achievement. Explicit guidance about how to manipulate working memory, indirect learning to long term memory is needed for learning to take place (Kirschner, Sweller & Clark). Students created personal connections with each other while constructing their knowledge internally, creating their own way of understanding the material (Topcu & Ubuz, 2008). Mayer (2005) posits that learning is best when active learning is combined with guidance.  

Shull (2008) posits that a hands-on approach keep students actively engaged, reduces anxiety and provides students with a heightened awareness as they build concepts. Beginning instruction with familiar concepts gave students confidence to proceed to complex situations (Shull). This is followed by the importance of pacing instruction as the teacher moves from the familiar to a new concept or situation of uncertainty. Students might lose focus and become distracted if familiar concepts are dwelled for too long a period of time. A combination of traditional and inductive methods need to be employed to sustain student engagement while maintaining pace (Shull).       

Combined Methodologies

            Direct instruction and discovery learning are not mutually exclusive (Star & Rittle-Johnson, 2008). Both will lead to a gain in knowledge by using the approaches together to increase efficient manipulation of strategies and multiple strategy use. Muthukrishna and Borkowski (1995) found that there was no difference between direct instruction and guided discovery approaches on transfer of knowledge. The discovery approach group performed better on the transfer problems on the long term assessment than the direct instruction group or the combined direct instruction-discovery group. Short term retention resulted in the combined and direct instruction group performing better than the discovery group.

            Teachers need to address inductive and direct instruction approaches. Cooperative mastery learning (CML) is one method that does combine a concern for learning conceptually and procedurally while employing inductive and direct instruction approaches (Mevarech, 1991). Students, working in groups, give and receive help using a CML approach. Assessment of student knowledge occurs frequently with the opportunity to receive feedback and corrections. Interactions between students are encouraged through the cooperative learning process. The result of the combined processes of mastery learning with cooperative learning showed higher mathematics achievement. Large amount of the time on task for the CML group was actually less than the time spent in traditional instruction. Student discourse was the largest benefit of CML. 

Conclusion

            Many teachers, parents, and administrators believe that some students have mathematical abilities while others lack the intellectual ability to do mathematics (Battista, 1999). As shown above, many studies have been conducted with low achieving students however upper level students may be left out in the debate over inductive reform movement approaches and traditional direct instruction. Higher achieving students may be hurt due to lack of attention to reasoning and problem solving skills in the mathematics classroom. Teachers need to meet all student needs. The National Research Council (2001) states that teachers can only teach what they know. Inductive approaches and the reform movement affect teacher confidence. Content knowledge, methodologies and teacher confidence need to be addressed through educating pre-service teachers and through in-service or professional development for practicing teachers. Martinez and Martinez (1998) quote Secretary Riley’s statement about believing in the educational system and teachers: “We need to have faith in our teachers who, given the proper resources and training, will teach to the highest standards (p. 3)”. In addition to faith in teachers, one needs to have faith in the reform movement and research (Martinez and Martinez).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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