Week 5: Memo #4

Memo #4: Handbook of Research in Mathematics Teaching and Learning

1.      What is algebraic thinking?

Algebraic thinking refers to “the capacity to represent quantities situations so that relations among variables become apparent” (Driscoll 1). It includes “being able to think about functions and how they work, and to think about the impact that a system’s structure has on calculations” (Driscoll 1). In other words, algebraic thinking is important because it helps students work on recognizing patterns and finding a shortcut to the system. Essentially, students are playing around with numbers (undo and do mathematics or going forward and backwards and decomposition) and making discoveries (trying new methods and building new rules) that play a role in how they think about algebra.

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?

I want students to recognize structural arithmetic. Structural arithmetic can include a + b – c = a – c + b, but is not limited to 1/2*(a) = a/2. Having a strong foundation in structural arithmetic will allow teachers to build off a strong foundation to learn complex abstract concepts such as the quadratic formula or graphing.

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

Even though I am a big fan of teaching for conceptual understanding, I believe procedural understanding is just as important. To emphasize procedures in a coherent way, I want students to develop the ability to learn how to apply an algorithm to a problem. Instead of teaching students all these different algorithms to solve different things, as a teacher, I have to show the differences and similarities among the algorithms so that students do not see mathematics as a bunch of disconnected algorithms.

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

I want students to learn how to be able to transfer concepts from one body of knowledge to the next. For example, students who know how to simplify expressions may have a hard time simplifying radical expressions or polynomials.

What are recommendations for teaching algebra for understanding?

a.       What should I emphasize when teaching algebra? Can be seen on page 15.

As a teacher, I should emphasize three things. First is the property of reversibility. I want my students to be able to know the ins and outs about the processes they use. Using a process to reach a goal is only half the battle. The other half is to start at the end and work backwards which would expand their understanding of the process. Second is to build rules to represent functions. Being able to recognize patterns and organize data is crucial to a student’s algebraic learning because it works on their information processing skills. Linking ideas to patterns helps students grab a better picture of the concept. The third, is abstracting from computations. I want students to be able to get as much information as they can by thinking about computations independently rather than dependently. It helps fixate the abstractness of algebra into something more comprehendible.

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

I can teach algebraic thinking by using word problems to show that there are multiple ways of approaching one problem. In addition to this, as a teacher, I have to show a variety of ways that are range from one end of the spectrum to the other end. The two ends I am talking about are doing direct and indirect methods for solving word problems. Solving the word problem using forward operation and backward operations is an alternate way of representing and solving word problems.

Most of the time algebra is represented in the form of numbers, variables, and symbols. All these are abstract things students have a hard time grasping. To solidify the important concepts algebra brings, pictures and objects are needed to bridge the connection between abstract thinking and practical thinking. This connection must be made so that students do not fall in a deep hole where they lose understanding of algebra and eventually lose interest in learning the subject. For examples, drawing graphs lets students physically see how changes in functions work.

c.       How can instruction address common difficulties?

Having clear instructions let students know what is expected of them to do. They know that they have to get from point A to point B provided that the teacher assists them.

Another way instruction addresses common difficulty is by incorporating English into math lessons. Instead of having students procedurally solve problems and use verbal explanations to justify their method, I will have students write out their reasoning and use sentence structures to formulate defending arguments. Having to write out their explanations instead of verbally communicating them accesses a different part of their mind so that they become use to the written math literacy which will help them in higher level math such as calculus.