Mathematics manipulatives I discovered are very useful and important in a mathematics classroom at any level. When I started my Mathematics Education 530 at the University of Tennessee - Knoxville, I knew what a mathematics manipulative was but had no idea there were so many different ones or how to use any of them. So when selecting my articles, I selected two articles which showed how to use certain manipulatives in the classroom and I also chose one article which discusses the possible problems with using manipulative in the classroom. The articles that showed how to use manipulatives in the classroom used technology manipulatives such as a calculator and Microsoft Word. I selected these because most of the schools I know of have calculators for every student in each class and rentable computers for each student of each class.
This is an article on Mathematical Manipulatives and their uses in the classroom. I selected two articles that showed how to use manipulatives in the classroom and I also chose one article that discusses the possible problems.
In the article "Using Microsoft Word to Teach Area," Duncan Symons (2011) starts off by saying he is always looking for ways to incorporate technology into his classroom because he believes that is motivates students and enhances their learning experiences . He believes that "technology is far more easily and naturally incorporated" (Symons, 2011, p. 20). He quotes Way and Webb in his article with technology, it creates a shift to constructivism, student-centered learning activities, global resources, and an increase in the complexity of tasks. Of course he refers to technology as Microsoft Word which can be used as a manipulative in his mathematics classroom. The reason he uses Microsoft Word as a math manipulative and the reason I found this article so helpful and educational was that Microsoft Word is available on almost every computer that is available in a mathematics classroom.
Duncan Symons (2011) starts the lesson by using "tuning in" questions, and the first one being the definition of area. He then moves on the asking questions about how to find the area of an irregular shape. He states "the fact that class members were volunteering their ideas and having them evaluated by other class members, with my intervention, was evidence that the lesson had gone from a teacher-centered activity to a student-centered learning activity" (Symons, 2011, p. 21). He then suggested putting a grid and noticed that none of the students had thought about it this way.
Duncan Symons then started the activity. Each student had access to their own computer while he solved the problem on his interactive board. The directions of the activity were find a picture you would like to find the area of and open it in Microsoft Word. Using the table function in Microsoft Word create a grid to put over your picture. Finally print it out. He then asked the question, "How can we work out the area of these irregular shapes?" (Symons, 2011 23). The students then had many ideas about how to solve the problem. In his conclusions Duncan Symons (2011) states, "the lesson did allow for the students to construct their won knowledge of the content", and through that "they were able to develop a more conceptual understanding of what are actually is." He then reinforced his ideas on why he chose Microsoft Word. The reasons again were it being easy and available for everyone in the world and technology increases the level of interest in the subject of mathematics (Symons, 2011, p. 24).
The next article I looked at was "Pedagogical instruction with calculators" by Karen K. Lucas and Jo Ann Cady. In the background section of their article they agree with the idea that Mathematics Educators need to take a balanced approach to teaching computation. They believe that if used correctly calculators increase a students conceptual understanding but if used only as a procedural tool then it decreased student conceptual understanding. They quote Ellington saying "students received the most benefit when calculators had a pedagogical role in the classroom and were not just available for drill and practice or checking work" (Lucas & Cady, 2012). Karen Lucas and Jo Cady state in their article that it is the teachers job to help the students understand when to use a calculator, pen and paper, or their head. In "Pedagogical instruction with calculators" calculators are used to develop number sense by working with concepts of place value, relationships between operations, and estimation.
The first "game" Lucas and Cady created was called wipe out. In wipe out a student was expected to be given a number and then given a number in that number make that place value zero. Lucas and Cady even suggest the question of "why did the nine disappear completely when the nine was wiped out of the number 9256, but the seven turned into a zero when wiped out of 3271" (2012)? They provide an extension of this activity which is the same game involving decimals. Through this game students can begin the understand the need of zeros as placeholders.
The next activity of Lucas and Cady they call the the "related operations activity (2012). In this activity a student uses a calculator and starts with a number. Then the teacher gives the student another number. The student then subtracts this number from the first number until they get zero. Then the student should be able to determine how many groups of the second number can be made. Lucas and Cady believe this activity "develops students' comprehension about the interconnectedness of mathematical operations" (2012). Throughout this activity they stress the importance of the teacher asking the students to articulate their own thoughts. The next "game" Lucas and Cady call the Range game and it deals with estimation. "The game begins with the teacher stating a start number, an operation, and a target range" (Lucas & Cady, 2012). A student then estimates to reach a value within the target range. If it correct then that student gets a point but if not correct the other student gets to guess. A great point made about this game is the fact that it allows students when using division in a calculator to see the division symbol written in two different ways. In the conclusion of this article it is pointed out that the use of calculators is inevitable but they can be used "for activities like searching for patterns, developing concepts, promoting number sense, encouraging creativity and exploration, or solving problems involving computations beyond the students ability (Lucas & Cady, 2012).
The final article is "Concrete Material and Teaching for Mathematical Understanding" and it discusses how to and not to use concrete manipulatives in the classroom. The first point the article makes is the assumption that the use of concrete material helps student understanding. It has a beneficial influence on all learners, but using them does not guarantee accomplishment ("Concrete Materials," 1994). The next point in the article is manipulatives do not automatically have mathematical meaning to students. "The material may be concrete, but the idea that the students are intended to see is not in the material ("Concrete Materials," 1994). Along with this idea is the point that it should be the teacher's goal for a student to make all interpretations of a problem. "They are empowered when they recognize the multiplicity of viewpoints from which valid interpretations can be made" ("Concrete Materials," 1994). This is not possible if the teacher before the lesson has not thought of all of the possible interpretations his student might have.
In "Concrete Material and Teaching for Mathematical Understanding" much is made of manipulatives when trying to learn fractions. It makes the point that factions cannot be taught so manipulatives cannot be used to make fractions so many out of so many. This is because how then are you supposed to take six things out of five. Next when using manipulatives to teach fractions "it only makes sense to combine amounts measured as fractions when both are measured in a common unit ("Concrete Materials," 1994).
In the conclusion of "Concrete Material and Teaching for Mathematical Understanding" it states that it is easy to use mathematical manipulatives but it is really difficult to use them well. A difficulty with this is when a teacher uses a manipulative to model a symbolic procedure. "Our primary question should be 'What do I want my students to understand?' instead of 'What do I want my students to do?' So therefore it is "problematic when teachers have a prescribed activity in mind - a standard algorithm - and unthinkingly reject creative problem solving when it doesn't conform to convention or prescription" ("Concrete Materials," 1994).
Mathematical manipulatives according to the article "Concrete Material and Teaching for Mathematical Understanding" are used to allow the students and teacher to discuss something concrete and manipulatives allow students to participate in the classroom. "Effectiveness is contingent on what one is trying to achieve ("Concrete Materials," 1994).
Both the articles "Using Microsoft Word to Teach Area" and "Pedagogical instruction with calculators" provided ways to use virtual manipulatives in the classroom. They used tools which are easily available teachers and students from which the students can use to increase their knowledge. In all of the activities these articles suggested each one started with questions to start the students thinking. In "Concrete Material and Teaching for Mathematical Understanding" suggested that using concrete material does not always lead to success. Both of the other articles mentioned agreed with this statement and stressed the importance of the teacher in the activities they suggested. Again in "Concrete Material and Teaching for Mathematical Understanding" it stated the issue of looking for just one answer to a certain question. Symons agreed with this idea because he allows his students to created their own pictures and which lead to their own interpretations. With using a calculator in activities it stresses a single correct answer or estimation. There is not much interpretation with using a calculator so this was a difference in the articles that I found.
The final criteria in "Concrete Material and Teaching for Mathematical Understanding" was understanding is more important than doing. Again, Symons agrees with this statement because in his activity he shows why grids are used in area. Lucas and Cady concentrate on the doing in their activities and not so much on the understanding. This is not to say I do not think Lucas and Cady came up with some great activities they are just not addressed in their article. I believe the activities in the first two articles are really close to what "Concrete Material and Teaching for Mathematical Understanding" wants in their activities when using manipulatives. I believe if a manipulative activity follows the guidelines provided in "Concrete Material and Teaching for Mathematical Understanding" then it will be very successful in helping the students in a classroom learn.
I found in these articles math manipulatives when used correctly can increase a student's participation and enjoyment of mathematics. Then when technology is used as the manipulative this participation and enjoyment of mathematics is increased even more. Also, the great thing about using technology as manipulatives allows for each student to get their own and can be very inexpensive or in most cases free.
References
Lucas, K., & Cady, J. (2012, February). Pedagogical instruction with calculators. Teaching Children Mathematics, 1, 384-389.
Symons, D. (2011, September 22). Using Microsoft Word to Teach Area. Australian Primary Mathematics Classroom, 1, 20-24.
Thompson, P. (1994). Concrete Materials and Teaching for Mathematical Understanding. The Arithmetic Teacher, 9(41), 556-558.
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