NỘI DUNG NHƯ SAU Ạ - AI CÓ TIME THÌ DỊCH GIÚP EM VỚI, EM NGU TIẾNG ANH QUÁ! HUHU LEARNING AND TEACHING MATHEMATICS : A DEVELOPMENTAL VIEW Decoding begins with a symbolic expres – sion or a symbolic, the meaning of which is then demonstrated with the use of models (Post, 1980; Van de Walle, 1983) Examples of Encoding Figure 2.12 show a simple concept activity for addition. Nine counters are put on a two-part mat. One student separates the nine counters , placing some in one section of the mat and the rest in the other. Before doing any writing , two students working together the combination aloud: “ Two and seven is nine”. Then this combination is encoded as they write an addition equation. The activity: continues by rearranging the counters to show different combinations with a new equation for each. In Figure 2.13 children represent each fraction with pie pieces and then find an equivalent way to show both fractions using all the same size pieces. When the equivalent models are found, a new equation is written down (encoded) along with the sum. The focus of the otherwise symbolic exercise is kept on the connection between the materials and the corresponding symbolism. Similar activities could be assigned as homework, with the students being required to draw pic- tures of the pie pieces and/or write short explanations of what they did to arrive at the symbolic results. Notice that encoding is recording something that is being done conceptually, usually with models. Simply writing down answers is not the same thing. Examples of Decoding Suppose that students in the second grade have used base ten materials (one, tens, and hundreds models) to do addition problem with pencil and paper. They then take turns using their familiar base ten materials to “teach” each other how the problem was worked. The same type of decoding can be done in a larger group, with one student providing an explanation with models for an exercise presented by the teacher. It is fun to purposely make an error in a computation and have students use models to explain where you made the mistake. In the upper grades there is the advantage of students being more adept with simple drawing and writing brief verbal explanations. For example, students can be asked to select the larger of two decimals or to put three decimals in order from least to most. Having done this, they can be told to draw representations of the decimals by shading in a 10 x 10 grid or by sketching a number line as in Figure 2.14. they might also be asked to write a sentence explaining what they did. AVOID PREMATURE SYMBOLISM As a rough guideline, the first 50 to 60 percent of the total time spent on a topic should be devoted to concept development and making connections with procedural knowl-edge. This is a unit or chapter perspective, not a lesson-by-lesson guideline. This large proportion of time given to conceptual and connecting activity is highly unusual in most classrooms. Most basal textbooks are written with both symbolic and conceptual material in nearly every lesson. However, the development of a new concept almost always is a matter of days or even weeks. That means that the developmental teacher should make modifications in the way concepts are presented in books. The good activities and especially in the teacher’s guide should be used and, where appropriate, additional activities added. However , just because there are symbolic exercises in the text lesson does not mean they must be done that day. To use a textbook while teaching developmentally means to view the text as a resource rather than as a series of lesson plans. The following lines from Roach Van Allen provide food for thought: What I can do, I can think about. What I can think about I can talk about. What I can say, I can write. What I can write, I can read. The words remind me of what I did, thought, and said. I can read what I can write and what other people can write for me to read. (Cited in Labinowicz, 1980, p.176) Although written about a language-experience approach to reading, these thoughts make compelling common sense when applied to mathematics. Consider what it would mean if a child trying to do symbolic mathematics could not “do” it conceptually. Would he be able to think about it? Would he be able to write about it (use mathematical symbols and rules)? Children should only be asked to use symbolism for ideas they have explored, reflected on, and discussed-what they have done, thought about, and talked about. Hallmarks of Teaching Developmentally In this chapter we have explored the nature of the mathemat-ics we want children to learn (relational understanding) and have examined a theory of how children develop this knowl-edge (a cognitive-constructivist theory). These ideas provide a framework and a guiding philosophy for how to teach mathe-matics developmentally. Principal characteristics of this ap-proach are summarized here. Teaching mathematics developmentally- Is child-oriented rather than content-oriented. It is listening to children to gain their perspective. It is realizing that children, not teachers or books give meaning to ideas and procedures. Is based on a cognitive-constructivist view of learning. It is recognizing the role and value of manipulative models in helping children form conceptual relationships. It is understanding that existing ideas give meaning to new ones. It is encouraging children to talk about concepts and rela-tionships. It is emphasizing the interconnections among concepts. Emphasizes meaningful connections of concepts with symbols and procedures.