Literature Review
Importance of Mathematics and Problem Solving Strategies
Our students are living in a world where education is changing and evolving rapidly. With the rise of Common Core Standards in the classroom, mathematics is an area with which more and more teachers are finding a need to reevaluate their pedagogy. Math can be taught in numerous ways and through multiple venues. Nevertheless, too often are teachers become stuck in the pattern of simply using computational practice and story problems as one of the main methods of math instruction. According to Inoue (2005), it is common to find that, “…the most popular activity is repetitious exercise of algorithmic procedures, rather than making sense of mathematical ideas in terms of their everyday experience” (pg. 70). A worrisome effect of this type of instruction and learning is that students can merely solve straight computation but cannot apply their computational ability into solving real-life mathematical problems.
Our students are living in a world where education is changing and evolving rapidly. With the rise of Common Core Standards in the classroom, mathematics is an area with which more and more teachers are finding a need to reevaluate their pedagogy. Math can be taught in numerous ways and through multiple venues. Nevertheless, too often are teachers become stuck in the pattern of simply using computational practice and story problems as one of the main methods of math instruction. According to Inoue (2005), it is common to find that, “…the most popular activity is repetitious exercise of algorithmic procedures, rather than making sense of mathematical ideas in terms of their everyday experience” (pg. 70). A worrisome effect of this type of instruction and learning is that students can merely solve straight computation but cannot apply their computational ability into solving real-life mathematical problems.
The National Council of Teachers of Mathematics (NCTM, 2013) suggests that learning mathematics is most effective through a deep conceptual of understanding of math. Students become most effective when they are, “…aligning factual knowledge and procedural proficiency with conceptual understanding” (NCTM, 2013, pg. 2). Rittle-Johnson and Siegler (2001) suggest that students need to develop and link their conceptual understanding of different principles within mathematics with their procedural ability in solving finite problems. In this model, students that advance their conceptual and procedural knowledge develop a higher competence in the mathematical domain (Rittle-Johnson and Siegler, 2001). The NCTM (2013) identifies certain Process Standards that enable students to attain and utilize their learned procedural knowledge and conceptual understanding in math. Students need to develop problem solving strategies, the ability to communicate, and the capacity to make connections amongst different mathematical ideas in order to broaden their thinking capacity and view Mathematics as a coherent whole (NCTM, 2013). Kazemi and Stipek (2001) argue that it is important to not just take the standards at surface value, but to actually stimulate students’ conceptual understanding of mathematics by providing complex problems to solve and pushing them beyond what comes easily for them. Furthermore, this type of pedagogy in mathematics allows students to create develop a deeper and more meaningful understanding of mathematics.
Why Students have to Think and Reason
Problem solving (which does not equate to simply solving word problems) helps students comprehend the computational and procedural side of mathematics (NCTM, 2013). Inoue (2005) argues that when faced with rote mathematics that do not provide meaning or substance, “…students mindlessly execute arithmetic operations without evaluating their actions in reference to our common sense understanding of real life practices” (pg. 70). Through this method the mathematics becomes mechanical, and the learning that occurs is not considered deep learning. Students are unable to use the mathematic procedures they have learned to implement them into real life situations. Schoenfeld (1992) suggests that, “Instruction should be aimed at conceptual understanding rather than at mere mechanical skills, and at developing in students the ability to apply the subject matter they have studied with flexibility and resourcefulness” (pg. 32). The level of comprehension will reflect their understanding of how to approach problems. To achieve this type of learning and practice within the classroom from the students, the instruction of mathematics and the environment in which the math is taught has to reflect the problem solving mentality (Schoenfeld, 1992).
Using problem solving as a means for students to think and reason helps students put together mathematical concepts in an applicable and intelligible way. “When students solve problems, they apply their knowledge to some real world situations and do not merely perform a set of abstract exercises that can be solved in an algorithmic manner. Problem solving is seen as a decisive test of genuine skill and understanding” (Wyndham and Säljö, 1997, pg. 361). Students take their procedural knowledge of computations and utilize them in a learning environment that assesses their ability. According to Schoenfeld’s (1992) framework on how to think and reason within mathematics, “Learning to think mathematically means (a) developing a mathematical point of view — valuing the processes of mathematization and abstraction and having the predilection to apply them, and (b) developing competence with the tools of the trade, and using those tools in the service of the goal of understanding structure — mathematical sense-making” (pgs. 3-4). Learning deepens once students begin to develop a sense of understanding for the concepts and procedures within mathematics.
My goal is to focus on how to apply their conceptual and procedural abilities in math. Therefore, my question to investigate is:
· How can I use both procedural and conceptual instruction to enhance my 5th grade students learning in math?
Additionally, I plan to help manage and focus my research with these sub-questions:
· How can I ensure I am improving their ability to execute a multi-step mathematical inquiry?
· How can I make sure that students are problem solving instead of simply working on word problems?
I will be utilizing a real world problem to guide their mathematics instruction during Phase 1. I believe that by addressing the way the students approach math in the classroom, I will be able to stress the importance of problem solving and approaching Mathematics on a conceptual level. They will be working in small groups and interacting as they work through a math inquiry project that they will then present to the class. My hope is that through this research I can help students be more involved in their problem solving mindset, become more cognizant of their procedural and conceptual knowledge, as well as be primed to be successful in today’s competitive generation.
Why Students have to Think and Reason
Problem solving (which does not equate to simply solving word problems) helps students comprehend the computational and procedural side of mathematics (NCTM, 2013). Inoue (2005) argues that when faced with rote mathematics that do not provide meaning or substance, “…students mindlessly execute arithmetic operations without evaluating their actions in reference to our common sense understanding of real life practices” (pg. 70). Through this method the mathematics becomes mechanical, and the learning that occurs is not considered deep learning. Students are unable to use the mathematic procedures they have learned to implement them into real life situations. Schoenfeld (1992) suggests that, “Instruction should be aimed at conceptual understanding rather than at mere mechanical skills, and at developing in students the ability to apply the subject matter they have studied with flexibility and resourcefulness” (pg. 32). The level of comprehension will reflect their understanding of how to approach problems. To achieve this type of learning and practice within the classroom from the students, the instruction of mathematics and the environment in which the math is taught has to reflect the problem solving mentality (Schoenfeld, 1992).
Using problem solving as a means for students to think and reason helps students put together mathematical concepts in an applicable and intelligible way. “When students solve problems, they apply their knowledge to some real world situations and do not merely perform a set of abstract exercises that can be solved in an algorithmic manner. Problem solving is seen as a decisive test of genuine skill and understanding” (Wyndham and Säljö, 1997, pg. 361). Students take their procedural knowledge of computations and utilize them in a learning environment that assesses their ability. According to Schoenfeld’s (1992) framework on how to think and reason within mathematics, “Learning to think mathematically means (a) developing a mathematical point of view — valuing the processes of mathematization and abstraction and having the predilection to apply them, and (b) developing competence with the tools of the trade, and using those tools in the service of the goal of understanding structure — mathematical sense-making” (pgs. 3-4). Learning deepens once students begin to develop a sense of understanding for the concepts and procedures within mathematics.
My goal is to focus on how to apply their conceptual and procedural abilities in math. Therefore, my question to investigate is:
· How can I use both procedural and conceptual instruction to enhance my 5th grade students learning in math?
Additionally, I plan to help manage and focus my research with these sub-questions:
· How can I ensure I am improving their ability to execute a multi-step mathematical inquiry?
· How can I make sure that students are problem solving instead of simply working on word problems?
I will be utilizing a real world problem to guide their mathematics instruction during Phase 1. I believe that by addressing the way the students approach math in the classroom, I will be able to stress the importance of problem solving and approaching Mathematics on a conceptual level. They will be working in small groups and interacting as they work through a math inquiry project that they will then present to the class. My hope is that through this research I can help students be more involved in their problem solving mindset, become more cognizant of their procedural and conceptual knowledge, as well as be primed to be successful in today’s competitive generation.