Hiebert, J. and
Carpenter, T. (1992). Learning and teaching with understanding.
In summary, Heibert and Carpenter's Learning and Teaching With Understanding addresses how students can successfully learn mathematics through internal and external connections. Internal connections refers to the understanding the occurs while learning. External connections refer to symbols, objects, and pictures the construct understanding. External mathematical representations influence internal mathematical representations. Heibert and Carpenter explore how understanding mathematics can be beneficial for learning more mathematics later on. One of the key ideas that are addressed is how thinking and talking about similarities, differences, and patterns can help students construct relationships. They also address issues related to understanding mathematics such as conceptual understanding and procedural understanding, prior knowledge and current learning, and math being taught mechanically.
The authors define conceptual understanding in several ways. Understanding occurs through thinking and talking about similarities, difference, patterns, etc. Conceptual understanding can also result as different representations for a concept are connected. If students have constructed strong connections, or networks, for a given concept, then conceptual understanding is likely and can influence the understanding of new relationships. For example, if students can see the connection between money, blocks, and decimals have to place value, they may demonstrate a conceptual understanding.
We can recognize that a student understands a concept through a variety of ways. A student understandings when he/she can connect prior knowledge to new knowledge. If they are able to represent something internally, understand patterns and relationships, and explain a concept, this may be evidence of understanding. For example, students have conceptual understanding of operation symbols such as +, -, x, and / if they understand each operation. If students know to add, subtract, multiply, or divide when given a certain symbol, they have conceptual understanding of what the symbols mean. They should be able to work out the operation on paper, explain what it means, and be able to represent it using manipulatives.
Understanding leads to many beneficial consequences. It can lead to the ability to more easily construct meaningful connections, promote retention, reduce the amount needed to be understood at a time, and enhance the transfer of knowledge to new situations. One example of a consequence of understanding is the connections between addition and multiplication. If a student is familiar with what it means to add, the student will more easily be able to grasp the multiplication means to add a certain number of times.
The growth of understanding occurs an understood concept is connected to new concepts. As connections are internalized, new connections can be constructed. For example, if the a student understands the idea of the addition of natural numbers, this can be extended integers and multiplication. An understanding of addition and multiplication can be extended to subtraction and division. These concepts can be built on by incorporating variables in algebra. As a result, we can see that understanding grows through connections to new concepts.
In summary, Heibert and Carpenter's Learning and Teaching With Understanding addresses how students can successfully learn mathematics through internal and external connections. Internal connections refers to the understanding the occurs while learning. External connections refer to symbols, objects, and pictures the construct understanding. External mathematical representations influence internal mathematical representations. Heibert and Carpenter explore how understanding mathematics can be beneficial for learning more mathematics later on. One of the key ideas that are addressed is how thinking and talking about similarities, differences, and patterns can help students construct relationships. They also address issues related to understanding mathematics such as conceptual understanding and procedural understanding, prior knowledge and current learning, and math being taught mechanically.
The authors define conceptual understanding in several ways. Understanding occurs through thinking and talking about similarities, difference, patterns, etc. Conceptual understanding can also result as different representations for a concept are connected. If students have constructed strong connections, or networks, for a given concept, then conceptual understanding is likely and can influence the understanding of new relationships. For example, if students can see the connection between money, blocks, and decimals have to place value, they may demonstrate a conceptual understanding.
We can recognize that a student understands a concept through a variety of ways. A student understandings when he/she can connect prior knowledge to new knowledge. If they are able to represent something internally, understand patterns and relationships, and explain a concept, this may be evidence of understanding. For example, students have conceptual understanding of operation symbols such as +, -, x, and / if they understand each operation. If students know to add, subtract, multiply, or divide when given a certain symbol, they have conceptual understanding of what the symbols mean. They should be able to work out the operation on paper, explain what it means, and be able to represent it using manipulatives.
Understanding leads to many beneficial consequences. It can lead to the ability to more easily construct meaningful connections, promote retention, reduce the amount needed to be understood at a time, and enhance the transfer of knowledge to new situations. One example of a consequence of understanding is the connections between addition and multiplication. If a student is familiar with what it means to add, the student will more easily be able to grasp the multiplication means to add a certain number of times.
The growth of understanding occurs an understood concept is connected to new concepts. As connections are internalized, new connections can be constructed. For example, if the a student understands the idea of the addition of natural numbers, this can be extended integers and multiplication. An understanding of addition and multiplication can be extended to subtraction and division. These concepts can be built on by incorporating variables in algebra. As a result, we can see that understanding grows through connections to new concepts.
You did an amazing job of summarizing the article. I found Hiebert and Carpenter's article to be very dense and wordy but the way in which you clearly lay out their points is extremely easy to understand. I also appreciated how you included specific examples of what they were talking about. This definitely exemplified that you had an understanding by telling the reader what you thought they meant by their discussions.
ReplyDeleteCera, I really enjoyed how you discussed the different ways a student can show understanding of a particular concept. One of the ways we can show a student understands is if they are able to connect their prior knowledge to the new topic or problem at hand. For example, when students need to solve 3x + 2x=5 they need to use their prior knowledge of combining like terms to solve this equation. I agree that understanding can lead to many beneficial things. One that stands out to me is students being able to connect and relate the concepts and knowledge they learned in past weeks to the new ideas they are currently learning. This creates a more cohesive curriculum the student can build off of.
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