Learning Centers- Orange Pp

“Learning science and math is child’s play. More than that, a child’s brain is actually hard-wired for learning and equipped by evolution with programs that enable, even require, it to explore and discover.” Popular Science

It is an educational paradox that, given children’s innate curiosity and propensity for investigation and exploration, many struggle in the subject of mathematics. Mathematics is a science, specifically “the science of pattern and order,” (Van de Walle 1998) a science with practical application and a subject we use often in our daily lives. A common area of confusion is the concept of fractions. “Adults who enter continuing education programs, for instance in the armed services, often specify fractions as an area of mathematics that has always confused them. Kindergarten teachers, however, observe young children in everyday situations demonstrating a beginning understanding of fraction concepts. What happens between kindergarten and the army?” (Riddle 2000) Indeed, as adults we rely upon our knowledge of fractions frequently, in cooking, sewing, time management, travel, carpentry, decorating, and budgeting. Why, as educators and researchers Margaret Riddle and Bette Rozwell asked, is there this disparity between school fraction work and real life fraction applications? Recent studies concerning students’ understanding of fraction concepts provide the answer to this important question. “Traditional teaching somehow interferes with children’s natural inclinations to make mathematical sense of the world.” (Riddle 2000) In other words, there is a great deal of difference between knowing math and doing math. (Van de Walle 1998) School mathematics has for years placed emphasis on procedural knowledge of mathematics. (Aksu 2000) This is “doing math,” and refers to rules and procedures one follows to perform specific mathematical tasks. Providing students with experiences that develop their conceptual knowledge, on the other hand, has been neglected. Conceptual knowledge “consists of relationships constructed internally and connected to already existing ideas.” (Van de Walle 1998) This type of knowledge is what Jean Piaget referred to as “logico- mathematical knowledge” (Copeland 1974) and refers to the specific cognitive process of taking experiences and experimentation in mathematics and understanding the broader implications. Simply, it is applying mathematical tasks to real-life situations, a unique, independent, and personal process that occurs in the learner’s mind through experience, interaction, and then reflection. The shift in teaching style from the traditional pedantic authoritarian approach to a more partner-oriented approach has been a slow process in the teaching of mathematics. (Aksu 2000) However, this shift, which actually entails de-emphasizing procedural knowledge and stressing conceptual knowledge, is essential. As teachers we are daily asked, “Why do I have to know this stuff?” and should be daily able to justify the content we teach. Further, as teachers we need to trust our learners and their understanding of their own thinking. “Our students give us blueprints for instruction. In teaching fractions, we must match our instruction to the way in which children think about parts and wholes. We can be mathematicians together- each with a role to play in the educational process.” (Riddle 2000) The New York State Standards for Mathematics, Science, and Technology and the National Council of Teachers of Mathematics curriculum for school mathematics (Troutman 1995) have restructured the approach to teaching fractions. There is more emphasis on conceptual knowledge throughout the students’ thirteen years of mathematics education. Traditionally, the curriculum focused lightly on fraction concepts through grade three, and then students experienced “an explosion of procedural knowledge (symbolic rules) at about grade five that was not supported by strong conceptual knowledge of fraction meanings because the curriculum simply did not provide time for the complex development that fraction concepts require.” (Van de Walle 1998) Hands-on and visual fractional representations that students can see and touch help them to better understand the idea of fractions. “Models must be used at all grade levels to develop fraction concepts adequately.” (Van de Walle 1998) Region models are the most common models for fractional representation. These models include pattern blocks, paper folding, geoboards, dot paper and graph paper, and circular pie pieces. However, to make presentation of the concepts of fractions most effective and most representative of real-life applications, measurement and set models are also important. Measurement models include rulers, number lines, Cuisinere rods and line segment drawings. These models are important in helping students transfer fraction use to measurement, and percents and decimals. Set models are fractions represented as by a whole that actually is a collection of objects. “It is the idea of referring to sets of counters as single entities that contributes to making set models difficult for primary school children.” (Van de Walle 1998) Activities that incorporate cereal, gummi bears, m and ms and the like with graphing and sharing tasks can help ease students into thinking of sets as fractional parts and wholes. Another conceptual difficulty concerning fractions is that “fractions are the first place in children’s experiences where a number represents something other than count.” (Van de Walle 1998) Size of fractions violates a child’s intuitive understanding of numbers; knowing that seven is bigger than four implies that sevenths would be bigger than, not smaller than, fourths. “Children need experience with models that convince them that as the denominator becomes greater, the value of the fraction becomes smaller…and they also need sound instruction in place value.” (Troutman 1995) “Number sense with fractions demands ‘more.’ In particular, number sense with fractions requires that students have some intuitive feel for fractions.”

2 (Van de Walle 1998) One aspect of fractional number sense that children bring with them to the classroom is the notion of sharing and partitioning. Relating fractions to sharing and dividing helps students to attach this new concept to prior experience and knowledge. “Constructing knowledge requires reflective thought, actively thinking about or mentally working on an idea. Networks of ideas that presently exist in the learner’s mind determine how an idea might be constructed. These integrated networks, frequently referred to as cognitive schemas, are both the product of constructing knowledge and the tools with which new knowledge is constructed. The more connections with the existing network of ideas, the better the new ideas are understood.” (Van de Walle 1998) Effective teaching of fraction concepts begins with this fundamental understanding of what fractions are, being sensitive to common misconceptions students have concerning number sense versus fractional number sense, and having a working knowledge of how students think. The next step for an educator is to have an understanding of how students learn in order to implement sound strategies to enhance classroom learning. “The National Council of Teachers of Mathematics (1989) has stated that ‘learning to solve problems is the principle reason for studying mathematics.’ Therefore, in relation to developing concepts, Carpenter and Moser (1983) proposed that teachers use word problems as a basis for developing …concepts, rather than teaching computational skills first and then applying them to solve problems.” (Aksu 2000) The seven New York State Standards for Math, Science, and Technology each clearly emphasize the skill of problem solving, particularly in relation to real- life problems and decision-making. Research concerning effective ways to teach students to become good problem solvers is consistent. “The useful pursuit of knowledge is a social endeavor,” and social skills are a vital part of the education process because, “professions are social organizations whose goals are the maintenance of and transmission of knowledge and skills.” (Hashway 1990) The idea that two (or more) heads are better than one, particularly in relation to mathematical problem solving, was appreciated by the earliest mathematicians. In 387 B.C., Plato’s Academy in Athens was an intellectual center devoted to group work that focused on solving mathematical problems and using cooperative learning and deductive reasoning to formulate general statements and principles concerning mathematical concepts and geometric figures. (Dunham 1990) The modern mathematics class more closely resembles this ancient learning model than the classrooms of the late 1960s and early 1970s where my own mathematical learning took place. Conceptual knowledge and problem solving skills are developed most effectively through exploration and interaction. “No better way exists for wrestling with an idea than an attempt to articulate it to others.” (Van de Walle 1998)

3 Understanding new concepts and working toward solutions “requires the ability to use the transactional power of language. In fact, the learning of mathematics is sometimes regarded as the acquisition of a special, formalized language.” (Ausubel 1969) Learning language, and learning the language of math, require expression. “It is a great mistake,” according to Jean Piaget, “to suppose that a child acquires the notion of number and other mathematical concepts just from teaching.” (Anderson 1965) Inventing, proving, taking risks, and discovery through interactive problem solving are key components to students’ effective grasp of the necessary concepts of mathematics in general and fractions in particular. Most rules or algorithms for fractional calculations “have no real intuitive connection with the concept.” (Van de Walle 1998) Therefore, providing students with experiences that allow them to uncover algorithms for themselves is more effective than rote teaching of procedures for fractional computations. “The student must be an active learner, manipulating objects and cooperating with other students. According to Piaget, this was how someone learned, with hands-on activity and from peers. The role of the educator was to provide the learner with the necessary materials and to structure the situation so that the student could spontaneously construct the generalization on his/her own. According to Piaget, this tendency to learn and understand the world is natural to a human being.” (Hashway 1990) This Piagetian concept of experiential learning is echoed through many modern teaching practices and instructional strategies. Psychologist Lev Vygotsky believed that social interaction was an essential element in a child’s intellectual development and a key factor in the eventual emergence of the “inner speech that enables humans to plan and regulate their activity,” (Wertsch 1985) or, in other words, verbal interactions “eventually become internalized in the form of private speech.” (Borich 2000) With today’s educational high-stakes testing, this development is important, and this transition from social learning to independent thinking through inner speech is a vital cognitive milestone. Solving problems with others, and expressing themselves mathematically in speech and writing, prepare students for the independent assessments they will face throughout their school years. The more problem solving strategies students have access to, the better their chances for success. The implementation of Multiple Intelligences, Learning Styles, and Cooperative Learning philosophies in a classroom is an effective way to teach students mathematical concepts in a richly exploratory and interactive setting that emphasizes student strengths and sets the stage for optimal student learning. Multiple Intelligence theory (Gardner 1987) is a way of thinking about human intelligence that “that broadens the scope beyond the confines of the IQ score.” (Armstrong 2000) “Gardner seriously questioned the validity of determining an individual’s intelligence through the practice of taking a person out of his natural

4 learning environment and asking him to do tasks he’d never done before- and probably would never choose to do again. Gardner suggested that intelligence has more to do with the capacity for solving problems and fashioning products in a context-rich and naturalistic setting.” (Armstrong 2000) Linguistic, Logical-Mathematical, Interpersonal, Intrapersonal, Musical, Spatial, Bodily-Kinesthetic, and Naturalist intelligences are possessed by all learners to varying degrees, and all eight intelligences can be developed in all learners. (Gardner 1987) Professor Rita Dunn’s Learning Styles theory complements Gardner’s ideas and provides educators with insights and strategies on student learning. (Dunn 1991) Dr. Dunn’s Learning Styles course gives teachers myriad ideas for matching content and concept presentation with students’ individual learning styles and offers many options for the creation of classroom set-ups and educational aids conducive to a variety of student learning styles. “Learning style is the way in which each learner begins to concentrate on, process, and retain new and difficult information. That process occurs differently for everyone.” (Dunn 1991) Learning Styles analysis includes identifying students’ left-brain and right- brain processing preferences, their perceptual strengths (oral, visual, or tactile modes of presentation), responses to the physical environment, and whether students think and learn holistically and globally, or sequentially and deductively. These preferences are best catered to in classrooms that offer learners choices in how they work and learn, and student involvement in deciding what is learned. (Dunn 1993) Cooperative Learning (Kagan 1990) emphasizes peer interaction and collaboration to engage students in educational discovery. Interactive learning experiences are authentic situations, because the ability to work with others cooperatively is a fact of life. Every day, on the job, at home, and in one’s community, there are virtually dozens of occasions where cooperation and interpersonal exchanges take place. Providing enriching experiences that involve communication and collaborative efforts give students vital real-life social skills. “All learners form their attitudes and values from social interaction.” (Borich 2000) A classroom that incorporates activities that involve thinking, working, and learning together along with independent tasks is an excellent model of the world of work and the life of a family, and one that instills sound principles of citizenship. (Kagan 1990) (Borich 2000) Educators equipped with the knowledge of Multiple Intelligences, Learning Styles, and Cooperative Learning techniques can effectively present difficult mathematical concepts like fractions. “Creating a learning environment in which teachers engage students in meaningful mathematical tasks and encourage them to actively explore and discuss mathematics is essential in developing the inquisitive minds of young mathematicians. (NCTM 1989, 1998)” (Drier 2000)

5 A teaching strategy that uses all these philosophies in concert with NCTM and New York State standards and what we know about student learning is the learning center. Learning centers have a variety of purposes and myriad organizational options. Learning centers are “activities which organize individualized instruction in ways that encourage students to assume major responsibility for their own learning. This means that any independent activity in which the students provide major direction for their own learning is considered a learning center.” (Maxim 1977) Specifically in the teaching of fraction concepts, learning centers can be used at the start of the unit to offer “review and practice for those who may lack task-relevant prior knowledge or skills.” (Borich 2000) They may be used at the end of a fraction concept unit for enrichment and assessment. “When a learning center can contain media, supplemental resources, and/ or practice exercises directly related to applying lesson content, it becomes an integral part of your lesson plan.” (Borich 2000) In The Complete Guide to Learning Centers, educator Susan Petreshene (1978) discusses her transition from the traditional teacher-centered classroom, to the student-centered model which focused on the implementation of exploratory and enrichment learning centers. “I was convinced that the child who did not learn to get along with his or her peers was not a happy child, and quite possibly, would not be a happy adult. Peer interaction would be necessary so students could discover the give and take of human relationships.” (Petreshene 1978) When this give and take in the mathematical classroom is centered around problem solving, students begin to behave “like young mathematicians.” (Drier 2000) In an authentic setting, the science of mathematics is studied as any other science would be, through experimentation and investigation, discussion and comparisons of trials and results. The learning center is an authentic setting that fosters mathematical thinking. “In problem solving as elsewhere, nothing succeeds like success. Once a student ‘gets a sense of’ a strategy and successfully applies it to some new example, a dramatic change can often be observed in his performance.” (Ausubel 1969) This change, fostered through teamwork in a cooperative setting, is often evidenced as self-confidence (Ausubel 1969) and this leads to the beginnings of independent problem solving. The synthesis of concepts that allows students to apply knowledge to new situations that they analyze together is deep thinking on the highest levels of Bloom’s taxonomy. (Borich 2000) This higher level thinking is difficult to assess through pencil and paper testing, but can be accurately evaluated through direct observation in the group setting. (Maxim 1977) This is one of the many benefits of the learning center. “Learning centers provide students with the opportunity to engage in ‘active’ learning. They serve as oases in the desert for many students

6 who are thirsting for something other than dry worksheets and individual work at their desks.” (Armstrong 2000) Performance task assessments are effective measures of learning center behaviors. Performance assessments are either “observations of a process while skills are being demonstrated, or evaluation of products created.” (Stiggins 1997) Rubrics can be designed to specifically evaluate certain components in the problem solving process, or creative steps in the development of a solution, as well as cooperative behaviors in the group setting. Students’ comprehension and application of standards-specific concepts can be assessed as well. In addition, performance task assessments in learning centers that offer students choices “provide excellent information about students’ proclivities in the eight intelligences.” (Armstrong 2000) Because students tend to select activities that they feel most competent to handle, learning centers that offer selections in as many of the eight areas as possible can be doubly effective. Teachers can make observations and discoveries about student strengths by noting what students elect to do, and also foster development in all intelligence areas by requiring students to try activities in other intelligence areas to “activate a wide range of learning potentials in students.” (Armstrong 2000) The goal of the use of learning centers as teaching strategies is to insure academic success and social development. Learning centers help to individualize instruction to meet the needs of all students; they can at once enrich and remediate. Because they are cooperative endeavors, the activities in learning centers help to increase a sense of classroom community and a spirit of caring. When choices are included, students develop ownership and responsibility for their own learning. Learning centers let students “explore many mathematical concepts and skills in the context of solving appropriate and engaging tasks by using a variety of exploratory tools.” (Drier 2000) Further, learning centers “create an environment that accepts, respects, and emphasizes the ideas of students rather than their skills. If skills are stressed the consequences of their creative attempts might be failures.” (Five 1992) Instead, learning centers encourage the free exchange of ideas, risk-taking, and a spirit of “try and try again.” They bridge the gap between “doing math” and “knowing math” by encouraging students to cooperatively problem solve and think and express their ideas mathematically.

7 As a future educator, I will certainly employ the use of learning centers in my mathematics classroom. Realizing that fraction concepts are introduced in kindergarten and more deeply developed each year throughout a child’s school experience, there is no question that I will be teaching the concept. Authentic, practical applications of fraction concepts can be developed from the early Egyptians’ Rhind papyrus. This text, which dates from 2000 B.C., contains about 80 mathematical problems that developed from real-life situations. The concept of fractions and mathematical operations with fractions are perfectly illustrated in this ancient record. One tablet contains an ancient form of a spreadsheet, which outlines portions of goods that were dispensed as payment. The complex distribution of goods in payment, with different workers receiving different fractional parts of the whole stock of goods, illustrates better than many modern word problem examples some practical situations where fractions are multiplied and divided. (Bunt 1976) This authentic illustration of fractions in action makes mathematics meaningful and the history of mathematics comes alive. The Rhind papyrus problems would make an excellent learning center activity with many enrichment and elaboration options, as well as practice in computational skills in a collaborative setting. When it comes to fractional learning center activities, those that involve edible items are quite useful. Children explore concepts of dividing, sorting, and sharing, and enjoy a treat when the tasks are completed. The following books contain good beginning lessons and ideas for learning center projects:

Boegehold, L. (1997). The gummi bear counting book. New York: Scholastic, Inc.

Mc Court, L. (1999). Candy counting: Delicious ways to add and subtract. US: Bridgewater Paperback.

Pallotta, J. (1999). The Hershey’s fractions book. New York: Scholastic.

Pallotta, J. (2000). Reese’s Pieces count by fives. New York: Scholastic.

Investigation of the use of learning centers as a strategy for teaching the concept of fractions has opened many doors to resources I had never before explored. Web sites such as: www.teachnet.com/leson.html, www.teacherlink.usu.edu/links.html www.kent.wednet.edu/curriculum/math/edmath/teachers.html www.thejournal.com/

8 www.edupuppy.com/ www.microsoft.com are all high quality education sites that offer many ideas for lessons and learning center extensions in many areas, including the teaching of fractions and mathematics. Books devoted to learning center development and assessment that are extremely well written and organized are:

Alton, E., Gersting, J., & Kuczkowski, J. (1987). Bulletin board learning centers for math and science. Indianapolis: Indiana University.

Armstrong, T. (2000). Multiple intelligences in the classroom. Alexandria: Association for Supervision and Curriculum Development.

Davidson, T., Fountain, P., Grogan, R., Short, V., & Steely, J. (1976). The learning center book: an integrated approach. Santa Monica: Goodyear Publishing Company.

Forte, I., Pangle, M., & Tupa, R. (1982). Center stuff for nooks, crannies, and corners. Nashville: Incentive Publications.

Maxim, G. (1977). Learning centers for young children: Ideas and techniques for the elementary school classroom. New York: Hart Publishing Company.

Petreshene, S. (1978). The complete guide to learning centers. Palo Alto: Pendragon House.

Thomas, J. (1975). Learning centers: Opening up the classroom. Boston: Holbrook Press.