Memo #4

Summary
In the FAT readings we learned about developing algebraic habits of the mind. Algebraic thinking involves thinking about the function and structure of a system and having the abilities to do and undo computational/ mathematical processes, creating specific rules for specific situations and abstracting information from a given computation. Overall, the FAT reading dove deeper into these topics, furthering the discussion in common misconceptions/ hardships students deal with in developing their algebraic habits by asking engaging and thought provoking thoughts. Kieran's article further discusses algebraic thinking; represented in two different types. Student misconceptions/ hardships with thinking algebraically is also considered.

1. What is algebraic thinking?
Kieran addresses two types of ways students look at algebraic representations. Students either approach algebraic representations in a procedural or structural mindset. Kieran uses the term "procedural" to illustrate students thinking about algebraic representations in an arithmetic way. For instance, in 2x+5=11, students would substitute a number in for x until the left side of the equal sign was equivalent to 11. Thus a procedural thought process is strictly computational. Another mindset that students may have when approaching an algebraic representation is the structural mindset. Kieran uses the term "structural" to denote students thinking about algebra in terms of the algebraic expression; students operate on the expressions themselves, not the numerical aspects. In referring back to the example of 2x+5=11, students with a structural mindset would subtract 5 from both sides of the equal sign than divide both sides by 2. Algebraic thinking refers to the structural mindset, students having an algebraic thinking are able to see "mathematical entity as an object means being capable of referring to it as if it was a real thing-- a static structure, existing somewhere in space and times. It also means being able to recognize the idea "at a glance" and to manipulate it as a whole, without going into details." (Kieran 253) Students with an algebraic thinking are able to manipulate an expression and treat equations and such as a real thing, without stumbling over the fact that most of what they're working with is unknown.

2. What are the central concepts, connections, and habits of mind for teaching algebraic thinking?
One concept that I really want my students to understand is the relationship between division and multiplication. For instance, many students don't realize that when they are dividing 2x by 2, they are actually multiplying 2x by one half. This connection between division and multiplying by a reciprocal is very important and through this, students would be able to realize why when dividing fractions, you invert and multiply by the second fraction. This concept is a "magical rule" that students think come out of no where so I really want to emphasize that there is a reason we do everything a certain way in mathematics. Many math classes today focus on memorizing a formula, plug and chug computations and getting the correct answer. One habit of mind that I want to instill in my students is the importance of understanding why they are doing something. I don't want my classroom and students to be focused on only getting the answer; a mindset like this only hinders learning and reinforces that only the end product is important. Understanding why they do something will help students develop a structural mindset of algebraic thinking and through this they will be able to fully understand their computations; being able to do and undo their thought processes.

3. What are recommendations for teaching algebra for understanding?
When teaching algebra I should emphasize the importance of knowing why something is being done. I want students to make the connections between non-mathematical ways of solving a problem, and the mathematical ways of solving them. In word problems, I want my students to understand that there are multiple ways to solving them--sometimes using common sense or daily problem-solving skills will get you through a problem. I also want students to understand that there is no correct way to solve a word problem, there is most likely an algebraic way, but anyway that will get them through the question can suffice. Although students look at word problems and proceed to do them with an arithmetic view, I want them to know that this is also correct, however a much easier way would be to set up an algebraic expression. Showing students multiple ways of solving a problem will allow them to make connections through the work they are doing. Another difficulty students have is thinking of the equal sign as something other than "do something." In my placement I've seen a few exercises to help students realize that the equal sign is a relational symbol that says both sides are balanced. Perhaps working with activities that addresses this problem before beginning one-step equations will help students view the equal symbol differently.