Goals of Optimizing Students Learning - Essay Example

Unit Overview: Surface Area of Cones, Pyramids and Spheres Duration: 10 hour sessions Stage       5         Year 10   Syllabus Outcome(s) Students learn about Surface Area of Pyramids, Right Cones and Spheres • identifying the perpendicular and slant height of pyramids and right cones • using Pythagoras. theorem to find slant height, base length or perpendicular height of pyramids and right cones • devising and using methods to calculate the surface area of pyramids • developing and using the formula to calculate the surface area of cones Curved surface area of a cone = πrl where r is the length of the radius and l is the slant height • using the formula to calculate the surface area of spheres Surface area of a sphere = 4πr 2 where r is the length of the radius • finding the dimensions of solids given their surface area by substitution into a formula to generate an equation Key Ideas Apply formulae for the surface area of pyramids, right cones and spheres Student Prior Knowledge: Substitution of correct values in the application of formulae. 3-dimensional solid shapes Knowledge of measurement Notes: Lesson plan shows a balance of teacher-directed and learner-centred activities. It takes into account the specific abilities of the students when they are provided worksheets and may be grouped in accordance to their ability levels. Apart from working individually, students have opportunities to work in pairs and in small groups and as a whole class. Activities try to balance theoretical and practical knowledge, as concepts are likewise applied to real life situations. Students learn about: 1 Basic concepts of the unit, definition of terms, formulae for the surface area of cones, pyramids and spheres. 2. Surface Area of Pyramids 3. Various forms of Pyramids 4. Surface Area of Pyramids 5 Surface Area of Cones 6. Surface Area of Pyramids and Cones 7 Surface Area of Cones 8. Surface Area of Spheres 9 Proving Formula for the Surface Area of a Sphere 10. Surface Area of Pyramids, Cones and Spheres. Students learn to: Identify the solid shapes of cones, pyramids and spheres and the formulae of their surface areas. Compute for the surface area of pyramids given some values to substitute to the formula. Create various pyramid forms using given values and 3 dimensional materials. Mastery of computation for the surface area of pyramids using the formula Compute for the surface area of cones given some values to substitute to the formula. Compare cones and pyramids and differentiate the computation of their surface areas. Create various cones using given values out of paper and glue. Compute for the surface area of spheres given some values to substitute to the formula. Using 3-dimensional objects students get to prove that the formula for finding surface area of sphere is accurate. Appreciate the value of the concept of surface area in the real world. Mastery of computation of surface areas of pyramids, cones and spheres. Assessment Opportunities Matching the solid shapes to their corresponding formula. Substituting values and finding missing values using the formula. Exercises on finding surface area of pyramids Using formula of surface area of pyramids and knowledge of various forms of pyramids to create 3-dimensional pyramids out of sticks. More exercises in the seatwork on finding surface area of pyramids. Exercises on finding surface area of cones Group work on finding surface area of pyramids and cones Using formula of surface area of cones to create 3-dimensional cone out of paper. Exercises on finding surface area of cones Orange Peel Activity to prove the formula for surface area of spheres is accurate. Group exercises on mixed review Long individual unit quiz Resources: Wooden representations and actual objects with conical, pyramidal and spherical shapes. Worksheet on surface area of pyramids. Various lengths of sticks and wood glue. Worksheets. Worksheets. Object representations of cones and pyramids Worksheets paper, scissors, string, ruler, pencil, glue and some dimensional values. Worksheets Oranges, String, Knife, Paper, Pencil (or pen), ruler, calculator, Worksheets, Long Quiz Lesson Plan on the Unit of Surface Areas of Pyramids, Cones & Spheres Lesson Number 1 The Key Idea(s) for this lesson are: Introduction to basic concepts of the unit Time Guide 10 mins 35 mins 15 mins Learning Experiences Introduction: Review 2-D shapes and how they measured it in terms of perimeter, area and volume. Body: Introduce 3-D shapes of sphere, cone and pyramid and discuss basic definitions of unit concepts and formulae Conclusion: Establish the importance of the concept of surface area in practical living/ Teaching Strategies Show them wooden representations of each and let them touch it and pass around to the others. Discuss the definitions of basic - for the unit: Surface Area - A measure of the number of square units needed to cover the outside of a figure. Pyramid- a solid figure that has a polygon for a base and triangles for sides, or lateral faces. Pyramids have just one base. The lateral faces intersect at a point called the vertex. Formula is: 1/2 × Perimeter × [Side Length] + [Base Area] Circular cone- another solid figure and is shaped like some ice cream cones. Circular cones have a circle for their base. Formula is: r: is the radius of the base h: is the height s: is the length of the side s can be calculated by using the Pythagorean theorem: Base area is: The surface area of a cone is therefore given by: Sphere – A sphere is a body bounded by a surface whose every point is equidistant (i.e. the same distance) from a fixed point, called the centre. Formula of SA of Sphere is 4 pi r 2 Class Organisation Whole class discussion. Once in a while, teacher throws questions at them and they can discuss in small groups using the wooden shape blocks as their guide. Assessment Children being able to answer trick questions of teacher and explaining their answers clearly. Matching shapes to formulae Lesson Number 2 The Key Idea(s) for this lesson is the concept of the pyramid and how to compute for its Surface Area Time Guide 15 mins 35 mins 10 mins Learning Experiences Introduction: Discuss the grand pyramids of Egypt and some trivia about how they were built. Ask students to think about how the early Egyptians computed for the dimensions and what kind of measuring units they used. Body: Discuss the computation of the surface area of pyramids and let students break into small groups to compute surface areas for various dimensions Conclusion: Students come together again to compare their results and calibrate it to the correct application of the formula Teaching Strategies Teacher engages the students’ interest by showing them pictures of Egyptian pyramids. Discuss how they came to be and how ancient people built them without the sophisticated tools of today. Review previous lesson on the formula of getting the surface area of pyramids. Explain that the surface area has two parts: the area of the sides (the lateral area) and the area of the base (the base area). The Base Area depends on the shape, there are different formulas for triangle, square, etc. But the Lateral Area is surprisingly simple. Just multiply the perimeter by the side length and divide by 2. This is because the sides are always triangles and the triangle formula is base times height divided by 2 Using the formula: 1/2 × Perimeter × [Side Length] + [Base Area], give the students different problems and let them solve these in small groups. After everyone is done, come together as a class and compare answers. Class Organisation Initially, the whole class discusses together the concept of pyramids and how their surface areas are computed. Later on, students are grouped according to their abilities, given various number problems to solve for the surface areas of various pyramids. At the end of the session, everyone comes together to compare their answers and how they came up with their answers Assessment Correct computation of surface area following the formula for surface area of pyramids. Lesson Number 3 The Key Idea(s) for this lesson are the various forms of pyramids Time Guide 15 mins. 40 mins. 5 mins. Learning Experiences Introduction: Discussion of the various pyramid forms Body: Students create various pyramid forms out of various lengths of sticks and attaching them with wood glue. Conclusion: students label and show their pyramid forms to the rest of the class. Teaching Strategies Pyramids can come in various forms. Parts of a Pyramid A pyramid is made by connecting a base to an apex Types of Pyramids There are many types of Pyramids, and they are named after the shape of their base.   Pyramid Base   Triangular Pyramid: Details >> Square Pyramid: Details >> Pentagonal Pyramid: Details >> Right vs Oblique Pyramid This tells you where the top (apex) of the pyramid is. If the apex is directly above the center of the base, then it is a Right Pyramid, otherwise it is an Oblique Pyramid. Right Pyramid Oblique Pyramid Irregular Pyramid: This tells us about the shape of the base. If the base is a regular polygon, then it is a Regular Pyramid, otherwise it is an Irregular Pyramid. Regular Pyramid Irregular Pyramid Base is Regular Base is Irregular Computation for the surface areas of pyramids follows the same formula, whatever shape the pyramid is. Class Organisation Class Discussion on various pyramid forms and types. Individual and Group work on making various pyramid forms given various lengths of sticks and glue. At the end of the session, the students will show their creations to the class and name the form that they came up with given the sticks they had. Assessment Creation of various forms of pyramids out of sticks Labeling the pyramids according to its forms. Estimation and measurement of sticks to come up with the intended pyramid form. Lesson Number 4 The Key Idea(s) for this lesson is solving for the surface area of pyramids Time Guide 5 mins. 30 mins. 25 mins. Learning Experiences Introduction: Review of formula for SA of pyramids Body: Teacher demonstration of solution and application of this knowledge on individual seatwork. Conclusion: Proving answers by showing it on the board. Teaching Strategies Review past lesson on solving surface area of pyramids. Demonstrate: Formula is: 1/2 × Perimeter × [Side Length] + [Base Area] Sample problem: a square pyramid with a base that is 20 m on each side and a slant height of 40 m. Find the surface area of the base and the lateral faces. Base: A=s(s) or 20(20) A=400 Each triangular side: A=1/2 bxh or ½ (20)(40) A=400 area of the base plus area of the four lateral sides SA=400 +4(400) SA= 2000 square meters. Give worksheets appropriate to the level of the students on solving for SA of pyramids. Check answers after by allowing students to demonstrate how they came up with the answer on the board. Class Organisation Class Demonstration of Problem Solving of SA Individual Seat Work Proving of Answers by Boardwork Assessment Seatwork on Finding the surface area of Pyramids given appropriate to ability level of students. Lesson Number 5 The Key Idea(s) for this lesson is the concept of the cone and how to compute for its Surface Area Time Guide 5 mins. 30 mins. 25 mins. Learning Experiences Introduction: Review formula of surface area of cones. Body: Demonstrate how to derive the surface area of cones, then apply this knowledge onto a seatwork of problems on surface area of cones Conclusion: Prove answers by showing it on the board. Teaching Strategies Review: r: is the radius of the base h: is the height s: is the length of the side s can be calculated by using the Pythagorean theorem: Base area is: The surface area of a cone is therefore given by: 1. Identify the radius of the cones base circle. If the diameter is given, cut it in half to get the radius. If you have the slant height and perpendicular height, use the Pythagorean theorem. 2. Write the radius somewhere off to the side, where its labelled and easy to find, because it will be needed several times in several different calculations. 3. Find the area of the base circle by squaring the radius and multiplying by pi (3.14). If the instructions say anything like "exact value", it means that you write the Greek letter for pi and leave it. So a radius of 3 gives an area of 9pi. Otherwise, use 3.14 or the calculators pi button to finish the multiplication and get a decimal version for the area. Values can be rounded off, but keep at least 3 digits after the decimal point for now. 4. Write that answer off to one side, somewhere where it is labelled "base area" and easy to find. 5. Identify the slant height of the cone. This refers to the height along the slanted side of the cone, not the height from the tip of the cone to the center of the circle. The radius, the perpendicular height (from tip to center), and the slant height are related by the Pythagorean theorem. 6. Multiply the slant height times the radius times pi. Again, "exact value" means write pi as pi; otherwise, use 3.14 to get the decimal approximation. 7. Write that answer off to one side, somewhere where it is labelled "lateral area" and easy to find. 8. Add the "base area" from step 4 with the "lateral area" from step 7. 9. Round, as needed. This is your final answer. Apply this formula to various problems in search for the surface area of cones. Check answers after by allowing students to demonstrate how they came up with the answer on the board. Class Organisation Class Demonstration of Problem Solving of SA Individual Seat Work Proving of Answers by Boardwork Assessment Seatwork on Finding the surface area of cones. Lesson Number 6 The Key Idea(s) for this lesson is finding the surface area of pyramids and cones Time Guide 5 mins. 35 mins. 20 mins. Learning Experiences Introduction: Discussion of the comparison between cones and pyramids. Body: Showing the difference in computation of surface areas of cones and pyramids. Small group work to apply knowledge gained from discussion. Conclusion: Proving answers by showing solution on the board. Teaching Strategies Compare and contrast cones and pyramids. Cones have a circular base while pyramids have angular base. They have different formulae to solve for its surface area. Formula for computing Surface Area of Cones is: Formula for computing Surface Area of pyramids is: 1/2 × Perimeter × [Side Length] + [Base Area] Do the following exercise in small groups. Come back together as a class and discuss the solutions Class Organisation Class discussion on the Comparison between cones and pyramids. Comparison of the formulae for deriving surface area of cones and pyramids Small Group activity of solving simple problems in getting surface area of cones and pyramids Proving answers by showing solutions to class. Assessment Active participation and engagement of each member of the group in contributing their ideas to the solutions of the problems. Getting to apply the correct formulae to the problems posed. Lesson Number 7 The Key Idea(s) for this lesson is the concept of creating cones given some of its values Time Guide 10 mins. 40 mins. 10 mins. Learning Experiences Introduction: Discussion on conical shapes and identification of objects in real life with this shape. Body: Showing how cones look in 2 dimensional form, and identifying parts and probable values of the cones both as a full cone and as one flattened out. Students pair up to create their own cones with the materials and values given by teacher. Conclusion: Measurement of the cones to check if they have the correct values. Teaching Strategies Review the concept of cones. Discuss which real-life objects are conical in shape. How do they think they can create cones give some of its dimensions? Challenge the students to create various cones given a large sheet of paper, scissors, string, ruler, pencil, glue and some dimensional values. Children work in pairs. Class Organisation Brief discussion on cones. Teacher shows 2-d pictures of cones on the board and challenges students to translate this into a 3-D figure out of paper. Students pair up and get materials and assigned values for the cones they will create. (Some cones will have missing values that students are supposed to compute for). Cones are measured to check if they correspond to the assigned values. Assessment Measuring instruments (ruler, tape measure, etc.) to check if the 3-D figures are correspond to its prescribed measurement. Lesson Number 8 The Key Idea(s) for this lesson is finding the surface area of spheres Time Guide 5 mins. 40 mins. 15 mins. Learning Experiences Introduction: Discuss concept of sphere. Show various examples. Body: Review formula for computing surface area of spheres. Teacher demonstration of application of formula, or she can call on a student to do it. Individual seat work. Conclusion: Proving of answers by showing the solutions. Teaching Strategies Review concept of sphere by showing examples and remembering the formula of its Surface Area. One-half of a sphere is called a hemisphere. Formula of SA of sphere: 4 pi r 2 Demonstrate computation of Surface Area: Solution:   Do seatwork on finding the surface area of spheres individually. Prove answers by showing solutions on the board. Class Organisation Reviewing of formula for surface area of sphere. Teacher demo on computation of solution. Individual seatwork on surface area of spheres. Showing solutions derived to seatwork problems. Assessment Seatwork on finding surface area of spheres given in accordance to ability level of students. Lesson Number 9 The Key Idea(s) for this lesson is proving Formula for the Surface Area of a Sphere Time Guide 5 mins. 40 mins. 15 mins. Learning Experiences Introduction: Review concept of sphere and its surface area. Teacher gives directions for activity. Body: Students collect materials for the activity of proving the formula for the surface area of a sphere. Students do the Orange Peel Activity. Conclusion: Discussion of activity, check for correctness. Teaching Strategies Prepare the materials: Oranges, String, Knife, Paper, Pencil (or pen), ruler, calculator, Estimate the surface area of your orange and explain your reasoning. With your string measure around the orange and record the data. Using a knife ¨C cut the orange in half. Place half of your orange upside down on the paper, and trace the orange 6 times ¨C without overlapping. In the first circle measure the diameter. On the chart next to the students’ name have him write in the circumference, the diameter, and calculate the last column ¨C Circumference ¡Â Diameter. Combine the class data. Now peel the orange and fit the peelings into the circles. ** Don’t lose any Record your findings. Discuss results. Class Organisation Students can either work in small groups or individually, depending on teacher’s preference or availability of materials for distribution to all. Assessment The students see or not see the value of solving for surface area applied in the real world The students achieve a better understanding of the concept of spheres for every step of the lesson and hopefully apply such knowledge on paper. Lesson Number 10 The Key Idea(s) for this lesson review of all concepts in the unit, basically, looking for the surface areas o pyramids, cones and spheres. Time Guide 10 mins. 40 mins. 10 mins. Learning Experiences Introduction: General review of the unit on surface areas of cones, pyramids and spheres. Body: Group solving of problems on the board. Smaller group seatwork Checking of answers. Long Unit Quiz. Conclusion: Checking of answers to long quiz for students to know how much they have understood from the unit. Teaching Strategies General review of the unit on surface areas of cones, pyramids and spheres. Matching formulae to solutions. Exercises on solving for SA with missing values. Students break into small groups to solve various problems on finding surface areas of cones, pyramids and spheres. Check for correctness Long Unit quiz Class Organisation Short review of the unit concepts. Group seatwork Unit quiz. Assessment Find the surface area of the following: 1.   a  =      10 mm b  =      14 mm c  =      14 mm 2.   a  =      5.1 in b  =      17 in c  =      17 in 3.   a  =      3.9 cm b  =      15.8 cm 5.   a  =      4 in b  =      9 in 6.   a  =      2 mm b  =      11.7 mm 7.   a  =      10 mm 8.   a  =      6 cm 9.   a  =      3 in b  =      17 in 10.   a  =      3.28 cm 11.   a  =      3.4 mm b  =      8 mm 12.   a  =      16 in b  =      6 in c  =      6 in   Part B: Rationale Education today has undergone various transformations from days of old. Schools at present may have similar goals of optimizing students’ learning and maximizing their potentials but may have differing philosophies, approaches and educational strategies in fulfilling these goals. “Predictably, the traditional teacher-centered model in which knowledge is “transmitted” from teacher to learner is rapidly being replaced by alternative models of instruction (e.g., learner-centered, constructivist, and sociocultural ideas) in which the emphasis is on guiding and supporting students as they learn to construct their understanding of the culture and communities of which they are a part (Brown et al., 1993; Brown, Collins, & Duguid, 1989; Cobb, 1994; Collins, 1990; Duffy & Cunningham, 1996; Pea, 1993). In the process of shifting our attention to the constructive activity of the learner, we recognize the need to anchor learning in real-world or authentic contexts that make learning meaningful and purposeful. “ (Bonk & Cunningham, 1998, p.27). The lesson plans prepared above were meant to combine old and new methods of teaching and learning. It has the elements of both teacher-directed as well as learner-centred learning activities. Math concepts and problem solving still need teacher demonstrations, but it also needs practical application for the students to be able to appreciate it. The theories of Piaget and Vygotsky were based on their predecessors’. “Piaget believed that children create knowledge through interactions with the environment. Children are not passive receivers of knowledge; rather, they actively work at organizing their experience into more and more complex mental structures.” (Brewer, 2001, p.6). He insists that children need to use all their cognitive functions. These theories were designed to form minds which can be critical, can verify, and not accept everything they are offered. Such beliefs reflect his respect for children’s thinking. This gives the students more power in the acquisition of learning. Using prior knowledge, they are encouraged to invent their own solutions and try out their own ideas and hypotheses with the able support of their teachers. This way, they can indulge in concrete experiences that focus on their interests. The process of searching for information, analysing data and reaching conclusions is considered more important than learning facts. Ability Grouping It is a known fact that each person is unique. One has his own set of natural or acquired strengths and weaknesses. In a group of people, sometimes these abilities (or lack of it) complement and at other times, clash, especially when close interactions between the members of the group are heightened. This is especially true in a classroom setting. In a class, children of different personalities, temperaments, social and cultural backgrounds, skill levels and learning styles come together to learn. They not only learn from the teacher but from each other as well. However, in cases when children do not achieve success in their learning as measured by the set learning outcomes, any teacher would be alarmed. The word ‘ability’ can be used to describe the quality of being able to perform, a quality that permits or facilitates achievement or accomplishment, or the possession of the qualities required to do something. In education ‘ability’ is can be described in its most simple term to illustrate a person’s aptitude and competence in the learning process. In academic circles, such ability refers to intelligence. Evolving research shows that the concept of ability – its components and whether or not it is a fixed entity in individuals, is hugely diverse and complex and is much deeper than simply saying a child is more or less able than another. It is with this dilemma that the strategy of grouping children based on their abilities or skill level is introduced. Teachers believe that it is an effective way to meet children where they are at and customize their lessons to the groupings accordingly. Children working in groups achieve success when targets are set and time limits are given at each phase of the activity and that they are given enough time to share their own ideas and to explain these to one another. The teacher ideally intervenes, or at some point becomes a contributing member of the group modelling appropriate listening and collaborative behaviours (Williamson, 2006). Williamson (2006) concluded in her study with grouping children that individual work may spur pupils’ feelings of isolation, with no one to bounce ideas off. They feel more confident and motivated when working in a group and reporting back to the class due to the elevated confidence in the work they were part of producing as a group. They felt less frustrated when a problem seems unsolvable because in working as a group, they are more willing to spend time discussing with others alternatives to tackling problems. Vygotsky (1978) believed that children’s intellectual development is influenced more by social context than by individual experiences. His theory places a great deal of emphasis on effective social interaction. Collaboration in learning brings about more knowledge acquisition than what would be generated through independent, individual learning. “Ability grouping refers to the process of teaching students in groups that are stratified by achievement, skill, or ability levels.” (McCoach, O’connell & Levitt, 2006, p. 339) It is used as a pedagogical instrument to promote collaborative learning, active engagement with material, critical thinking and communication as a strategy. Indeed, as outlined by Allebone (2006); “group work is used as a tool to facilitate the organisation and management of the class and as an approach to support differentiation for the range of attainment within the class.”. From this quote it could be suggested that ability grouping facilitates focused teaching which increases pupil achievement by reducing the range of ability within a particular group. The idea of grouping children according to their ability has many advantages. Many teachers may argue that it is a more effective way of teaching. Children who have had a period of absence for example can benefit from peer guidance in group situations: Where there is an opportunity for children to communicate, the teacher need not repeat instructions thus avoiding a continual line of pupils asking questions. It could also be suggested ability grouping makes better use of time. If children of similar ability were grouped together for example, it would be easier for a teacher to talk and give instructions directly to that group. Paying particular attention to time restrictions, it is worth noting of Moyles’ (1993) suggestion that tasks would be “more efficient and less tedious” for children if they were working collaboratively with children of similar ability. In addition, working in similar ability groups enables the teacher to assess each groups needs more effectively and allows children on the same level to have “the opportunity to share ideas with each other” and support each other in their learning. Ability groups are often viewed to make teaching a class with a wide range of skill levels more convenient for the teaching because she is able to aim their lessons at the middle range of students without being concerned about lower and higher abilities since the specific groups will process the lessons in their own pace. Ability groups are more effective with short-term instruction designed to help students who are experiencing difficulties or who require more extensive instruction in a particular area. Scholz (2004) The literature provides a lot of studies on the use of ability grouping in Mathematics. Harlen (1999) argues that it serves more relevance in the area of Math than other learning areas due to the huge range of abilities that are often exhibited within the same class. What makes it more complex is the fact that Mathematics is a hierarchical discipline where concepts build on previous concepts and more often than not, need full understanding before proceeding to the next, more complicated concept. (Ruthven, 1987). Hence, Math teachers are more likely to support homogeneous grouping than their colleagues in other subject areas. Cahan and Linchevski (1996) report from the UK where 80% of Mathematics teachers believe the inappropriateness of mixed ability teaching groups whereas, in contrast, only 16% of science teachers and 3% of English teachers believed the same. According to Ruthven (1987), similar trends are prevalent in primary schools where grouping was predominant in Math but not so in areas such as Science and Social Studies. He also believes that although ability grouping is consistent in Math, it diminishes in other subject areas as students move through the year levels.. Ability groups encourage collaborative learning because the students of equal status work together in small groups toward a common goal. It fosters much social interaction, which may be the key to understanding others better (Saleh, Lazonder & De Jong, 2005). As in most interactions, communication is essential as both parties get to have a better grasp of concepts talked about. The party receiving explanations gain more understanding while the party doing the explanation benefits from the “cognitive restructuring” involved in peer tutoring which may trigger the realization and repair of misconceptions and knowledge gaps (Webb & Palinscar, 1996). Knowledge becomes co-constructed by group members as they build on each other’s ideas and considering and discussing the significance of personal beliefs until mutual agreement is reached (Damon & Phelps, 1989; Slavin, 1995). This agrees with Vygotsky’s (1978) concept of intersubjectivity. This is when two people are engaged in a task and begin from different understandings but with interaction, comes to an agreed, shared understanding. This is usually manifested when students initially debate opposite arguments but upon more understanding of the concept because of listening to each other’s opinions, will both end up seeing the concept in one direction. The whole unit planned has high hopes for optimizing student learning and collaboration. It aims to translate theoretical paper & pencil concepts and tasks to real-world understanding of mathematical concepts. It also addresses individual needs and abilities of students so each and every one gets to be an active learner in a subject that is usually feared by many students. References Bonk, C.J. & Cunningham, D.J. (1998) “Searching for Learner-Centered, Constructivist, and Sociocultural Components of Collaborative Educational Learning Tools” in Electronic Collaborators. Retrieved on September 5, 2009 from: www.publicationshare.com/docs/Bon02.pdf Brewer, J.A. (2001) Introduction to Early Childhood Education.Boston: Allyn and Bacon Brown, A. L., Ash, D., Rutherford, M., Nakagawa, K., Gordon, A., & Campione, J. C. (1993). “Distributed expertise in the classroom.” In G. Salomon (Ed.), Distributed cognitions: Psychological and educational considerations. New York: Cambridge University Press. Brown, J. S., Collins, A., & Duguid, P. (1989). “Situated cognition and the culture of learning”. Educational Researcher, 18(1), 32–41. Cahan, S. & Linchevski. (1996). The cumulative effect of ability grouping in mathematical achievement: A longitudinal study. Studies in Educational Evaluation, 22(1), 29–40. Cobb, P. (1994). “Where is mind? Constructivist and sociocultural perspectives on mathematical development.” Educational Researcher, 23(7), 13–20. Collins, A. (1990). “Cognitive apprenticeship and instructional technology”. In L. Idol & B. F. Jones (Eds.), Educational values and cognitive instruction: Implications for reform Hillsdale, NJ: Lawrence Erlbaum Associates. Damon, W. & Phelps, E. (1989). Strategic users of peer learning in children’s education. In T. Berndt & G. Ladd, eds, Peer Relationships in Child Development, pp. 13–157. New York: Wiley. Duffy, T. M., & Cunningham, D. J. (1996). “Constructivism: Implications for the design and delivery of instruction”. In D. H. Jonassen (Ed.), Handbook of research on educational communicationsand technology. New York: Scholastic. Harlen, W. (1999). Making Sense of the Research on Ability Grouping. Accessed September 8, 2009 from http://www.scre.ac.uk/rie/ nl60/nl60harlen.html. McCoach, D.B., O’Connell, A.A., Levitt, H. (2006) “Ability Grouping Across Kindergarten Using an Early Childhood Longitudinal Study”, The Journal of Educational Research, July/August [Vol. 99(No. 6)] Pea, R. D. (1993b). “Practices of distributed intelligence and designs for education”. In G. Salomon (Ed.), Distributed cognitions: Psychological and educational considerations. New York: Cambridge University Press. Ruthven, K. (1987). Ability Stereotyping in Mathematics. Educational Studies in Mathematics 18, 243-253. Saleh, M., Lazonder, A. W. & De Jong, T. (2005) “Effects of within-class ability grouping on social interaction, achievement, and motivation”, Instructional Science 33: 105–119 Scholz, S. (2004) “Ability Groups Ineffective or Ineffectively Used?”, Australian Primary Mathematics Classroom, 9 (2) Slavin, R. E. (1987). Ability grouping and student achievement in elementary schools: A best evidence synthesis. Review of Educational Research, 57, 293–336. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press. Webb, N.M. & Palinscar, A.S. (1996). Group processes in the classroom. In D.C. Berliner & R.C. Calfee, eds, Handbook of Educational Psychology, pp. 841–873. NewYork: MacMillan ). Williamson, V. (2006) “Group And Individual Work” , Mathematics Teaching Incorporating Micromath.,March 2006 Resource Websites: http://teachers.net/lessons/posts/4013.html http://www.edhelper.com/math/volume106.htm http://www.mathsisfun.com/geometry/pyramids.html http://www.mathsteacher.com.au/year10/ch14_measurement/19_sphere/20sphere.htm http://www.wikihow.com/Find-the-Surface-Area-of-Cones Read More