Why Algebraic Reasoning Matters

By, Anne Marie Burdick

The role of the TEKS mathematical process standards is to guide students to think about mathematics so that students are able to develop and establish a conceptual understanding of mathematical content. Specifically, when teachers create algebraic thinking tasks for their students, students are given the opportunity to, “analyze mathematical relationships to connect and communicate mathematical ideas.”  

            

Through the extension of numerical patterns, teachers should be able to transform their classroom textbook problems into algebraic tasks. When teachers give their students opportunities to engage in algebraic tasks, students are able to practice algebraic thinking and recognize mathematical relationships while also participating in activities that “can provide large amounts of computational practice in a context that intrigues students and avoids the mindlessness of numerical worksheets” (Blanton & Kaput, 2003, p. 73). When algebraic tasks are structured carefully, students are given the opportunity to complete work that has multiple entry points and multiple solutions. In addition students’ algebraic literacy will develop as they are actively engaged with mathematics content in meaningful ways. For example, this problem that was presented in a Pre-Algebra textbook:

At a frozen yogurt shop, frozen yogurt costs 45 cents for each ounce; a waffle cone to hold the yogurt is $1. How much will the sale cost if a customer buys 10 ounces of frozen yogurt?

This task can be modified into an algebraic task by extending the number of ounces of frozen yogurt purchased. When students solve for greater quantities of frozen yogurt, there will be a need for a more generalized relationship. Students would be looking for how much Z ounces of yogurt costs. Therefore, this textbook problem could be rewritten as an algebraic task as follows:  

At a frozen yogurt shop, frozen yogurt costs 45 cents for each ounce; a waffle cone to hold the yogurt is $1. How much will the sale cost if a customer buys 10 ounces of frozen yogurt? 25 ounces? 100 ounces? How much will the sale cost if a customer buys Z ounces of yogurt?           

By adding different amounts of frozen yogurt to this problem students will have to come up with more than one answer and a general equation to describe the relationship between the number of ounces purchased and the total cost.  When students are finding their solutions for 10 ounces they have to add 45 cents to 1 dollar ten times (1 + .45 + .45 + .45 + .45 + .45 + .45 + .45 + .45 + .45 + .45), they will soon discover that they will need to find a new method for when they are looking for a solution with 100 ounces. This will encourage students to formulate a more general equation so that they can find the cost of frozen yogurt no matter how many ounces there are. By doing this students will come up with an equation y = .45Z + 1 where y is the total cost and Z is the ounces of frozen yogurt. By extending the number of ounces purchased and creating a need for a generalization and by making modifications to the simple textbook problem, students analyze mathematical relationships and develop their algebraic thinking skills.  

Resources are available for teachers interested in developing algebraic thinking tasks for their students. A good resource for teachers if they want to show their students a visual representation of the equation developed in the frozen yogurt problem is Math Flyer. In this application students could type y = .45Z + 1 in the application and see what their equation looks like. When students view the graph they will see how steep the slope is, which will tell them how the price is affected depending on how many ounces of frozen yogurt is bought. 

Retrieved from: http://download.cnet.com/Math-Flyer/3004-20415_4-12069669-1.html

Another good resource for educations who are looking for ways to integrate algebraic thinking in their curriculum is the Center for Algebraic Thinking. This website provides technology tips, sample modules, and assessments for teachers who need extra resources.  For example, the video explains why algebra is so important to our students’ success and their future plans, and provides an overview of the mission of the center. It notes the vast amount of research that has been completed on how students think about algebra and complete algebra to validate their stance on why algebra is a principal subject.

Therefore, in order to help our students be successful problem solvers and mathematical thinkers, we need to be actively incorporating the TEKS Mathematical process standards, such as “analyze mathematical relationships to connect and communicate mathematical ideas” into our daily mathematics lessons. Algebraic thinking tasks “extends learners’ understanding of arithmetic and enables them to express arithmetical understandings as generalizations using variable notion” (Ketterlin-Geller, Jungjohann, Chard & Baker, 2007, p.68). Algebrafying our problems in the classroom promotes students to generalize and think algebraically instead of having our students simply solve word problems with one solution.

Blanton, M. L., & Kaput, J. J. (2003). Developing elementary teachers’ “Algebra eyes and ears”.
     Teaching Children Mathematics, 10(2), 70-77.

Ketterlin-Geller, R. L., Jungjohann, K., Chard, J. D., & Baker, S. (2007). From arithmetic to

     algebra. Educational Leadership, 65(3), 68.