Manipulative as Supplements in Teaching MathematicsUsing manipulatives and technology as supplement to mathematics instruction is gaining momentum, popularity and much debate. Foundations of these concepts and value judgements on their positive or negative effects must consider the basic premise of mathematics, its learning and teaching strategies.1) The first computer was built during World War II, to help the British intelligence service crack military codes.
After the war there came a growing need to perform other complex calculations much more rapidly than could be done by paper and pencil or adding machine.
To fulfill these needs technicians designed electronic calculators. Although slow and cumbersome by today’s standards, these devices represented a huge leap forward of information processing power. (Russel, 2006)2)Manipulative use in today’s mathematics classrooms is taken for granted.
Yet using manipulatives effectively to engage students in meaningful experiences that promote mathematical understanding is a function of the tasks for which a teacher conceives them being used (Moyer, 2001). Manipulatives are designed to represent explicitly and concretely abstract mathematical ideas.
Research on their use has shown that students who use manipulatives during learning outperform those who do not (Driscoll, 1983; Greabell, 1978; Raphael ; Wahlstrom, 1989; Sowell, 1989; Suydam, 1985,1986). Some studies show that student achievement levels are related to teachers’ experience in using the manipulatives (Raphael & Wahlstrom, 1989; Sowell, 1989).
Although much of this research documents manipulative use in prescribed teaching and learning interactions (Parham, 1983), few reports tell us how teachers use the manipulatives (Moyer, 2001 ), nor do they describe how students use manipulatives when they are given a choice to do so. (Moyer, 2004)First Math is a language. Math is the universal language of science. If science is life, then math is the universal language of life.
Of the many languages in the world, mathematics is the most accurate.3) When I think of the development of Mathematics over the last 2500 years, I am less surprised that early mathematicians left lasting results than that, given the tools they possessed, they achieved anything at all that could have lived through centuries. Just think of it. Zero gained widespread use only in the last millennium.
Systematic introduction of modern algebraic notations began only in the sixteenth century and is most often associated with the French mathematician François Viète (1540-1603). René Descartes (1596-1650) was first to use letters at the end of the alphabet for unknowns. He also introduced the power notations: x2, x3. The sign of equality (two equal parallel strokes) has been invented by Robert Recorde (c.
1510-1558) in his The whetstone of witte (London, 1557 (:To help you appreciate the expressive power of the modern mathematical language, and as a tribute to those who achieved so much without it, I collected a few samples of (original but translated) formulation of theorems and their equivalents in modern math language. (a + b)2 = a2 + b2 + 2ab. If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. (Euclid, Elements, II.
4, 300B.C.) (Bogomolny, 2006)Secondly, Math is both process and product of intelligent learning. Since learning is not confined within the four walls of the classroom, it is imperative to incorporate life and the immediate environments around the student in teaching math.
This is where manipulative are effective because they bring three dimensionalities to concepts taught and learned.4) Once a problem can be solved with manipulatives, students draw a picture and then write a number sentence to represent what was happening with the manipulatives as they solved the problem. This gradual 6 progression from concrete to pictorial to abstract representation provides a solid foundation of understanding upon which the students can build. Every method or algorithm can be understood, and even reinvented, with manipulatives, thus replacing rote learning of algorithms with understanding of concepts and ways to efficiently apply them.
Once the concepts have been firmly established and students understand how the algorithms work, they move away from using concrete manipulatives. However, manipulatives can be revisited at any time to remediate or extend a concept as needed. (Madden, 200.) Thirdly, the use of manipulates though treading in paths laden with positive and negative implications further develop the cognitive and affective responses in students.
As different learning theories are currently being studied and research, much of constructivist and Piagean educational advocates lobby use of manipulative in teaching math. Students of the modern world are grown amidst the background of technology. Manipulatives are part of their life and therefore incorporating it to the learning field is imperative. The way that teachers use these manipulatives in teaching mathematics will spell success of failure in the students’ progress.
5) Three types of choice are fundamental, those relative to the choice of content to be taught, the planning of interactions between learners and the knowledge to be learnt; the interventions and role of the teacher in a class situation. The choice of teaching content and its organization is based on epistemological-type hypotheses and learning hypotheses. In the practice of teaching there is generally a process of contextualization of the knowledge taught – that is to say, the organization of a context which situates this knowledge, in which the activity of the pupils can operate. The interactions between knowledge and pupils operate through the context, the milieu (Brousseau, 1986).
(Wilder, Sue. 2004). Theories such as the Piagetian theory which is knowledge base finds the manipulative usage as effective for mathematics6) According to this emerging theory, students need to construct their own understanding of each mathematical concept. Hence, we believe that the primary role of teaching is not to lecture, explain, or otherwise attempt to “transfer” mathematical knowledge, but to create situations for students that will foster their making the necessary mental constructions.
A critical aspect of our approach is a decomposition of each mathematical concept into developmental steps following a Piagetian theory of knowledge based on observation of, and interviews with, students as they attempt to learn a concept. (Dubinsky, 2006)Lastly, research must be done to further realize insights in using manipulatives in teaching mathematics not only as supplement but as prime tool in at least showing how the concepts work. If science and life need to be understood, mathematics will be able to effectively explain the many concepts of science.7) In recent years, mathematics textbooks have changed to incorporate calculator-based lessons, and the College Board has begun to permit students to use graphing calculators while taking the SATs.
Teachers are gradually learning how to modify both the mathematical content of their curriculum and their approach to teaching to take advantage of these calculators. Some schools have equipment that links calculators to sensors for collecting data such as temperature, pH, movement, and light. The calculators can plot these data as a function of time and graph the results. They can also be connected to equipment that projects the work from any one calculator to a display so that everyone in the class can see it.
In these ways, graphing calculators can support inquiry, thinking, and dialogue about mathematics and scientific data, rather than the “plug and chug” routines that have characterized U.S. mathematics classrooms. (Wiske, 2006.
)Non advocates are afraid that the consequence of not memorizing the multiplication table is devastating. A person who relies on the calculators won’t rely anymore on his own mental multiplication skills. Whether a person who’s calculator conks out buys another set of batteries or mentally calculates the mathematical problem at hand is rooted out in values that have been taught in early life. Appreciation of mathematics that is the responsibility of the parents and creativity of the teachers is first step in helping a person to like math.
As it is, dogma and traditions have scared the wits of children towards math. Manipulatives help children realize that Math can be fun afterall.8) Most students benefit from visual representations of concepts. While the research is scarce on mathematics achievement resulting from using virtual manipulatives in the classroom, Kelly Reimer’s and Patricia Moyer’s action research study (2005), Third-Graders Learn About Fractions Using Virtual Manipulatives: A Classroom Study, provides a look into the potential benefits of using these tools for learning.
Interviews with learners revealed that virtual manipulatives were helping them to learn about fractions, students liked the immediate feedback they received from the applets, the virtual manipulatives were easier and faster to use than paper-and-pencil, and they provided enjoyment for learning mathematics. Their use enabled all students, from those with lesser ability to those of greatest ability, to remain engaged with the content, thus providing for differentiated instruction. But did the manipulatives lead to achievement gains? The authors do admit to a problem with generalizability of results because the study was conducted with only one classroom, took place only during a two-week unit, and there was bias going into the study. However, results from their pretest/posttest design indicated a statistically significant improvement in students’ posttest scores on a test of conceptual knowledge, and a significant relationship between students’ scores on the posttests of conceptual knowledge and procedural knowledge.
(….., 2006)9) So why are children still taught mathematics as a pencil and paper exercise which is usually somewhat solitary? For most of us mathematics, like music, needs to be expressed in physical actions and human interactions before its symbols can evoke the silent patterns of mathematical ideas (like musical notes), simultaneous relationships (like harmonies) and expositions or proofs (like melodies). Regretfully I hold Mathematicians (with a capital M) largely to blame for this.
They are so good at making silent mathematics on paper for themselves and each other that they have put this about as what mathematics is supposed to be like for everyone. We are all the losers. Music is something which nearly everyone enjoys hearing at a pop, middle-brow or classical level. Those who feel they would like to learn to perform it are not frightened to have a go, and those who perform it well in any of these varieties are sure of appreciative audiences.
But Mathematicians have only minority audiences, consisting mostly or perhaps entirely of other Mathematicians. The majority have been turned off it in childhood. For these, the music of mathematics will always be altogether silent.(Skemp, 2006) REFERENCES Bogomolny, Alexander.
2006. Mathematics as a Language. http://www.cut-the-knot.
org/language/index.shtml Dubinsky, Ed. 2006. Calculus, Concepts, Computer and Cooperative Learning.
http://www.pnc.edu/Faculty/kschwing/C4L.html Moyer, Patricia.
2004. Controlling Choice: Teachers, Students, and Manipulatives in Mathematics Classrooms. School Science and Mathematics, http://www.findarticles.
com/p/articles/mi_qa3667/is_200401/ai_n9348825 Madden, Nancy. 2001. MATHWINGS Effects on Student Mathematics Performance Robert E. Slavin, Johns Hopkins University Kathleen Simons, Success for All Foundation Report No.
39 (revised) July, 2001 Published by the Center for Research on the Education of Students Placed At Risk (CRESPAR), supported as a national research and development center by funds from the Office of Educational Research and Improvement (OERI), U.S. Department of Education (R-117-D40005). This work was also supported by the Charles A.
Dana Foundation, New American Schools, and the Carnegie Corporation of New York. However, any opinions expressed are those of the authors and do not necessarily reflect the policies or positions of our funders. Russel, Peter. 2006.
The Global Brain. http://www.peterussell.com/GB/Chap8.
htmlSkemp, Richard. 2006. The Silent Music of Mathematics. www.
tallfamily.co.uk/david/skemp/pdfs/silent-music.pdfSkemp, Richard R.
1987. The Psychology of Learning Mathematics. Lawrence Erlbaum Associates.Wilder, Sue.
2004. Fundamental Consructs in Mathematics Education. Routledge Falmer Wiske, Stone. 2006.
A New Culture of Teaching for the 21st Century http://www.edletter.org/dc/wiske.htm ….
, 2006. Math Manipulatives. http://www.ct4me.net/math_manipulatives.htm
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