Week 6; Memo #5

Week 6; Memo #5

1.      What is algebraic thinking?

Algebraic thinking is reaching “a deeper understanding of the underlying mathematics, it is important that students see functional relations in both [globalizing and extending] ways and connect the two ways of seeing patterns to each other” (page 96).

2.      What are the central concepts, connections, and habits of mind for teaching algebraic thinking?

a) Concepts: What concepts do you want students to understand?

There are two concepts presented in the article that I want students to understand. The two concepts are generalization and extension. Generalization is “making the leap to what is “always true” for a functional relation.” Extension refers to “wondering about and pursuing the mathematical directions that an algebraic result may suggest, beyond the result itself. These two concepts are fundamental in understanding functional relations.

b) Connections: What connections (between representations, between procedures and concepts, etc.) do you want students to develop?

As a teacher teaching functional relations, it is important to make connections to what students know and where they need to be. In Chapter 5, recognizing patterns is not a big problem because they know how to do so from elementary years. The problem lies with describing the process of how they recognize the pattern and what it leads to. The teacher must help students understand functional relationships by asking questions that involve solving the problem from a different standpoint. It is important for students to show how they arrive at their answer.

A second thing teachers must do is make the connection between small scale pattern recognition and general case recognition. It is important for students to switch from a table and charts way of thinking to an algebraic way of thinking about functional relations.

c) Habits of mind: What habits of mind do you want students to develop?

I want students to be able to ask open ended questions that generate more understanding from one problem. Questions such as “Why?,” “What do we get if we go into higher dimensions?,” and “Can a large pyramid have x number of toothpicks?” Open ended questions promote higher levels of thinking because students have to critically think about how to proceed with solving the problem.

In addition to critically engaging students, I want students to understand that they often can get stuck with a problem. For this reason, I encourage students to work on their social skills. In a classroom, I would do so by allowing group work. The way I see it, group work benefits students in such a way that allows them to acquire higher level thinking by talking about different approaches peers have on tackling the same problem. Exposing students to a variety of methods adds on to their internal representations that can be used to solve future functional relations.

3. What are recommendations for teaching algebra for understanding?
1. What should I emphasize when teaching algebra?

Along with pattern recognition, I want students to learn how to explain various forms of solving algebraic patterns and apply it to real-life questions. Students should be able to formulate a general expression and be able to justify it and apply it to various real-life questions. The goal is to have students think outside the box, in addition to exercising their generalization and extension skills.

2. How can I teach algebra using objects, pictures, and word problems?

As a teacher, I provide word problems that require students to make follow up questions for the word problem. The activity involves them creating follow up questions for word problems and having their classmate answer it. Creating and finding solutions for follow up questions works on students’ extending abilities and recursive word problems work on students’ generalizing abilities.

I can use pictures to show students the patterns. It gives them concrete objects to focus on that way the student can better relate to what is being asked. Pictures give a visual of the algebraic problems presented in Chapter 5. Also, pictures are used to describe patterns which are easier for students to do than represent patterns symbolically.

3. How can instruction address common difficulties?

Giving tools for students to use when solving complex patterns lowers the intimidation factor of the complexity of the pattern they need to find. Tools such as pictures, tables, charts, predicting, and explaining with words and written words help break down the challenging problem. Breaking it down allows students to focus on solving one aspect of the problem rather than losing focus and getting overwhelmed with the problem as a whole. I can use instruction to let students know the strategies they have “chunking” pattern problems and how it can be helpful to understanding how to find a general formula.