Kieran, C. (1992). The
learning and teaching of school algebra
In summary, Kieran's article addresses some of the issues involved the content, teaching, and learning of algebra. One of the key points Keiran points out is that many student see algebra as memorizing sets of rules and procedures. This is a result of the combination of both content difficulty and the way in which algebra is taught in schools. Before students are taught algebra, they are learning procedures which involve arithmetic operations and numerical results. Once students begin learning algebra, students need to transition from procedures to structures, which involve different operations on algebraic expressions. Unfortunately, algebra is taught in such a way that students are not fully understanding how algebra works structurally, and the symbolism behind symbols and letters. When students are introduced to algebraic word problems, they work backwards using arithmetic backwards strategies. After a certain point, student may no longer depend on the backwards strategies and must work forward using algebraic structures. Many algebra teachers felt that their focus was on classroom management and covering all the material required by the school, and guided by the text book. Many teachers were also unaware that much of what they were teaching were not actually being learned by students. The procedural process of learning algebra seems to be more accessible to students than the structural process. Student difficulties in algebra included attempts to convert expressions to equations. Many students did not have a real sense of the structural aspects in algebra. As a result, more explicit attention needs to be given to the transition between procedural thinking and structural thinking from arithmetic to algebra.
In summary, Kieran's article addresses some of the issues involved the content, teaching, and learning of algebra. One of the key points Keiran points out is that many student see algebra as memorizing sets of rules and procedures. This is a result of the combination of both content difficulty and the way in which algebra is taught in schools. Before students are taught algebra, they are learning procedures which involve arithmetic operations and numerical results. Once students begin learning algebra, students need to transition from procedures to structures, which involve different operations on algebraic expressions. Unfortunately, algebra is taught in such a way that students are not fully understanding how algebra works structurally, and the symbolism behind symbols and letters. When students are introduced to algebraic word problems, they work backwards using arithmetic backwards strategies. After a certain point, student may no longer depend on the backwards strategies and must work forward using algebraic structures. Many algebra teachers felt that their focus was on classroom management and covering all the material required by the school, and guided by the text book. Many teachers were also unaware that much of what they were teaching were not actually being learned by students. The procedural process of learning algebra seems to be more accessible to students than the structural process. Student difficulties in algebra included attempts to convert expressions to equations. Many students did not have a real sense of the structural aspects in algebra. As a result, more explicit attention needs to be given to the transition between procedural thinking and structural thinking from arithmetic to algebra.
1.
What is algebraic thinking?
Algebra itself is defined as the branch of mathematics dealing with symbolizing mathematical structures and numerical relationships. Algebraic thinking varies depending on which feature of algebra we are addressing. This includes terms, expressions, equations, word problems, functions, and graphs. It can mean recognizing the symbol representations and the structural aspects. Symbol representations include the idea of a variable representing an unknown number, or the equal sign meaning symmetry. Structural aspects might mean understanding operations can only be carried out onto like-terms.
Algebra itself is defined as the branch of mathematics dealing with symbolizing mathematical structures and numerical relationships. Algebraic thinking varies depending on which feature of algebra we are addressing. This includes terms, expressions, equations, word problems, functions, and graphs. It can mean recognizing the symbol representations and the structural aspects. Symbol representations include the idea of a variable representing an unknown number, or the equal sign meaning symmetry. Structural aspects might mean understanding operations can only be carried out onto like-terms.
2.
What
are the central concepts, connections,
and habits of mind for teaching algebraic thinking?
One key concept, connection, and habit of mind I want students to understand is the equal sign. Before algebra, the equal sign has been represented as something the generates an answer, but in algebra, it represents symmetry. In other words, it means the the left and right side must be equal to each other. To successfully understand the structure of algebra, understanding the equal sign is a highly important concept. Students need to get in the habit of mind that if something is done to the left side of the equal sign, the same must be done to the right in order to keep the equation symmetric. I also want students to be able to represent word problems algebraically. It can be incredibly helpful if student are able to translate from mathematical phrases to algebraic expressions. Some problems are to complex to work through by using guess and check methods, so understanding how to represent ideas algebraically can help simplify work. This can be done by showing what phrases can represent a variable and what phrases can represent operations.
One key concept, connection, and habit of mind I want students to understand is the equal sign. Before algebra, the equal sign has been represented as something the generates an answer, but in algebra, it represents symmetry. In other words, it means the the left and right side must be equal to each other. To successfully understand the structure of algebra, understanding the equal sign is a highly important concept. Students need to get in the habit of mind that if something is done to the left side of the equal sign, the same must be done to the right in order to keep the equation symmetric. I also want students to be able to represent word problems algebraically. It can be incredibly helpful if student are able to translate from mathematical phrases to algebraic expressions. Some problems are to complex to work through by using guess and check methods, so understanding how to represent ideas algebraically can help simplify work. This can be done by showing what phrases can represent a variable and what phrases can represent operations.
- What are recommendations for teaching algebra for understanding?
Algebra can also be taught with the use of objects, such as algebra tiles. My cooperating teacher uses algebra tiles to help them work with equations visually. This helps them see how an equation needs to be kept balanced. If they add squares to one side, they need to add it to the other. Having a visual model can help many students see the connection.
To address student difficulties, I think it is helpful for students to see common errors and try to reason through why it is incorrect. If students can see common misconceptions, I they are less likely make common errors, and also recognize when they are doing something incorrectly.
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