Literature Review

Introduction:

All across the country, higher educational programs are being appropriated for the education of intermediate children (Gestwicki, 2011). Many families are seeing that the work load brought home is much greater than what they remember from their own school experiences. Our society is very fast paced and data driven and therefore strives for the most out of our student’s education. The new common core standards are pushing students to strive to be college ready with ripple effects that go all the way down the academic chain. “The Common Core State Standards focus on core conceptual understandings and procedures starting in the early grades, thus enabling teachers to take the time needed to teach core concepts and procedures well—and to give students the opportunity to master them (Common Core).” They give guidelines that will develop young minds into adapting mathematics into their everyday activities.

In my experiences, students in grade five struggle very much with fractions and their operations. This concept is one that consistently creates struggle among students. It is my wonder whether or not this struggle has to do with the concept itself, being taught before the adolescent mind is ready, or if the methods that are being used are insufficient for this concept.

Curriculum:

A curriculum is a guideline for teachers developed to help make decisions about what will be taught in the classroom as well as how the instruction will be implemented. For years now, it has seemed that most teachers would teach through lecture. However, today’s standards have our students working harder than ever. In the concept of fractions, students need to have rich connections between symbols, models, pictures, and context (Cramer, Post, delMas, 2002). Teachers’ curriculums need to be developed around the students learning styles and abilities, and at the same time, be able to adapt to the diverse needs of a classroom. Teachers should enhance the students’ previous knowledge and build off of it. If teachers have a knowledge of the children’s readiness level at the beginning of each academic year, an emphasis on developing the misunderstandings and new concepts should be presented ensuring that each student is expanding their mathematical skills (Bjonerud, p. 349)

Different theories suggest a multitude of ways in which educating on fractions can be handled. Heaton and all (1992) suggest that the current mathematical world develops teachers to educate in one of two ways. The first entails active construction of knowledge through what students already know. This methods suggests that teachers, instead of modeling or explaining, should allow students to construct concepts on their own, based off of the knowledge already acquired. The second is that teaching is telling and should reinforce the computational procedures that must be memorized. This method, contrary to the first, teaches students to master the correct answer rather than creative problem solving (Heaton, Putman, Prawat, Remillard, 1992).

In my experience, after viewing both lecture type classrooms and creative development classrooms, I have noticed a much more solid and capable understanding of concepts within classrooms that promote creative learning, group learning, cooperative activities, and project based learning. Fractions cause many conceptual problems for students.  It has been found that many students have trouble conceptualizing the relationship between the numerator and the denominator, connecting them to the natural numbers. It is thought that through tangible measurement activities, students will be able to further develop the concept of a fractional unit (Stafylidou, Vosniadou, 2004).  By using concrete materials, you are allowing students to act with something, manipulating it in a multitude of ways, and you are enabling a grounded conversation within the class (Thompson, 1994).  These materials can promote the conceptualization that the students may need. Along with that, allowing students the chance to think critically for themselves develops skills that can be put towards tasks that were not directly instructed to them by that teacher. In a study done by Cramer and all, it was found that the students who were given the opportunity to “investigate” math problems had a stronger conceptual understanding of fractions and were better able to relay their knowledge of fractions to further tasks that were not directly taught (Cramer, Post, delMas, 2002).  Therefore, it seems clear to me that the style of lecturing students has proved to be less respectable than that of discovery learning.

As the curriculum unfolds in the classroom, having “simple but powerful” ideas, reinforcing them, and revisiting them often, demonstrates the understanding of the fundamentals that a teacher must have. Being knowledgeable about the previous year’s achievements as well as the following year’s goals is also beneficial to reviewing as well as preparing for the following year. Teaching for understanding is key. Students must develop the concepts and be capable of applying them to a multitude of problems.

Struggling Students:

As a teacher, when only one student is struggling, it becomes difficult to hold back the class. The most common thing that I have seen is focused around basic concepts; not knowing which part is which, or not knowing previous math facts that should have already been mastered. According to Small (2012), teachers can differentiate their lessons in order to accommodate every students and perhaps prevent the struggling student to arise. By asking students “open questions” you are inviting meaningful responses that can be from any level. You are also allowing yourself to see the levels of knowledge present. If this method does not work, a response to intervention may be necessary. By providing intervention, students will be able to receive more in depth instruction. According to Gersten and all (2009), explicit and systematic instruction will allow students positive gains. By providing students with a verbalization of the thought process , guided practice, and opportunity for reviewing, they will be receiving the information in a multitude of supportive instructional methods therefore enhancing their knowledge. In order for students to be successful in their later years of mathematics, they need to achieve complete mastery level of the basics.

Conclusion:

The advancement in our society will only increase overtime. While it may seem that students may not have the academic ability or perhaps the development to understand every concept, it is our duty as teachers to help them as best as we can. Through the use of developmentally appropriate curriculum and planning, students will be better able to understand and grow off of their knowledge (Gestwicki, 2011).  Through the use of differentiated learning, curriculum maps that develop the learner, and memorable activities, teachers should be able to achieve a mastery level of fractions and operations with fractions within students. Developing a basis of such a crucial topic is key to allowing students to succeed throughout their academic career. It is crucial to their academic achievement that students have a well develop and mastered understanding of the basic principles.

References:

 Askey, R. (1999). Knowing and teaching elementary mathematics. American Educator, 23, 6-13.

Ball, D. L. (1990). Halves, pieces, and twoths: Constructing representational contexts in teaching fractions. National Center for

Research on Teacher Education, Michigan State University.

Bjonerud, C. E. (1980.). Arithmetic concepts possessed by the preschool child. The arithmetic

          teacher, 7(7), 347-350. Retrieved from http://www.jstor.org/stable/10.2307/41184342

Bright, G. W., Behr, M. J., Post, T. R., & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for research in

          mathematics education, 19(3), 215-232.

Cramer, K. A., Post, T. R., & delMas, R. C. (2002). Initial fraction learning by fourth and fifth grade students: A comparison of the

effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for research in

          mathematical education, 33(2), 11-144.

Gersten, Russell, Sybilla Beckmann, Benjamin Clarke, Anne Foegen, Laurel Marsh, Jon R. Star, and Bradley Witzel. 2009. Assisting

students struggling with mathematics: Response to intervention (RtI) for elementary and middle schools (NCEE 2009-4060).

Washington, DC: National Center for Education Evaluation and Regional Services, Institute of Education Sciences, U.S.

Department of Education.

Gestwicki, C. (2011). Developmentally appropriate practice: Curriculum and develop in early education. (4 ed.). Wadsworth, CA:

Cengage Learning.

Heaton, R.M., Putnam, R. T., Prawat, R. S., & Remillard, J. (1992). Teaching mathematics for understanding: Discussing case studies of

four fifth grade teachers. The elementary school journal, 93(2), 213-226.

Small, M. (2012). Good questions: Great ways to differentiate mathematics instruction. (2 ed.). New York, NY: Teacher’s College

Press.

Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and

          Instruction, 14(5), 503-518.

The Common Core State Standards. Common Core Standards for Mathematics. Retrieved January 28, 2013, from

          http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Thompson, P.W., Concrete materials and teaching for mathematical understanding. Arithmetic teacher, 41(9), 556-558

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