Memo 1

Hiebert & Carpenter (1992) Learning and Teaching Mathematics with Understanding (MEMO #1)

           Heibert and Carpenter (1992) start off talking about external representation and internal representation as an introduction for students’ understanding. The authors believe that external representation can be constructed or represented by objects, language, written symbols, and etc. On the other hand, internal representation can be represented what the students already know or what the students already have in their mind of the content. Furthermore, the authors also talk about that understanding can bring a lot of pros in learning, such as promoting remembering, enhancing transfer knowledge, and etc. Most importantly, the authors discuss the importance of conceptual knowledge and procedural knowledge and how they can be linked together in teaching a classroom.

            The authors define skill efficiency as somewhat similar to procedural fluency or knowledge. There need not to be understanding in the content because it is more of a procedure format. An example would be adding and subtracting numbers because only computation will be involved throughout these problems. There will not be much of a conceptual understanding going on. However, procedural knowledge is required for mathematics because procedures allow mathematical tasks to be completed efficiently.

            The authors define conceptual knowledge as knowledge that is rich in relationship. In other word, conceptual knowledge is not stored as information, but it is part of a network. Conceptual understanding in mathematics can also mean that students can make connections with the contents. Furthermore, conceptual knowledge is necessary for mathematical learning because it helps student’s gain conceptual understanding from the content.

            Classroom supports conceptual understanding when the teacher helps the students connect their prior knowledge of mathematics and acquire real-world context. For example, when the students are subtracting two fraction numbers like ½ - ¼, the students can use a real-life problem that is similar to the problem ½ - ¼ to help them visualize the problem. For instance, if I have ½ of a pizza, and I gave ¼ of the pizza away, how much pizza do I still have? The students acquire conceptual understand if the students can have these kind of questions in their mind while computing the difference of the fraction problem. Thus, the students have to make connection to the problems while computing them in order to gain conceptual understanding and procedural fluency at the same time.

            Teaching in US classrooms sometimes involves conceptual understanding and procedural fluency. However, students tend to experience more of procedural activities in mathematics. Heibert and Carpenter’s claimed, “Because students in mathematics classes often are asked to memorize procedures and rules for manipulating symbols as individual pieces information, it is not surprising that many students believe that mathematics is mainly a matter of follow rules, that is consists mostly of symbols on paper, and that the symbols and rules are disconnected from other things they know about mathematics” (p. 77). Students are often asked to use computation and memorize formulas in a math classroom, but they do not have the opportunity to study more about the content in depth, so that they can gain conceptual understanding.

            In some cases, even if the teaching format is very procedural, conceptual understanding can still be gained through the lesson. For example, adding a three digit number to another three digit number can be really procedural, but in order to compute the problem, students have to understand about how to carry and line up place values in order to solve the problem efficiently and accurately. This may sound like procedural fluency, but students can gain conceptual understanding at the same time working on computation problems.