Fostering Algebraic Thinking Chapter 7
1. Summary/ What is Algebraic Thinking?
Algebraic thinking, as mentioned in last week's memo, involves connecting the function and structure of a system and being able to manipulate it using a variety of tools. In Chapter 7 the focus is linking multiple representations and finding connections within them. For teachers, the toughest part in helping students link representations is deciding when to give the necessary help. Understanding how each representation is linked together will help the student grow in their algebraic thinking, so this is something that they should really do on their own.
2. What are the central concepts, connections and habits of mind for teaching algebraic thinking?
The whole chapter emphasizes the importance of students understanding how and why different representations connect together. One connection, out of the many that Chapter 7 said students lacked, involves interpreting graphs at a global perspective and not point-wise. I think this connection is important, because it brings students from viewing points as a single entity (5th grade) to viewing graphs, its data and the function as a symbiotic team (higher level thinking).
By knowing how and why representations connect, students will be able to develop their habit of mind involving doing-undoing. This habit of mind will become easier to establish and do when students know how each representation connects to another; being able to manipulate one representation and see how this manipulation will affect the other representations.
3. What are recommendations for teaching algebra for understanding?
Many recommendations were given to teachers when teaching with multiple representations. One of the most emphasized recommendation was to not give students the answers; building connections must be done on their part because it will have the most meaning when they struggle to find answers on their own (not when they are given them). Because this chapter was on teaching algebra with multiple representations, there are plenty of examples showing the use of different representations in teaching the velocity equation (for example). In the excerpt describing a conversation between the teacher and students, the teacher can be seen scaffolding the material. He may have given them suggestions to approach the problem a specific way, but he never told the students what they should find or the answers. The teacher was also addressing the students' difficulties by asking specific questions that would determine where their misunderstandings lay.
Not giving students the answers, i think, is one of the most important things for teachers to keep in mind. Just giving students the answer benefits no one. The student doesn't learn anything, and you as a teacher have to spend more time re-teaching the information. It's a lose-lose situation. I liked the idea in the book of instead of giving them an answer, asking the student to explain where they are having trouble, and then guiding them through their representations to find where they might be able to fix their problem.I think that tying in their question to the representations is a great way for the students to make more connections through their representations to the original problem.
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