Why Should We "Algebrafy" Problems?

By Megan Hancock

The TEKS Mathematical Process Standards describe ways in which students should interact with mathematics.  These methods encourage students to apply mathematics, communicate mathematical ideas, analyze mathematical relationships, and justify mathematical ideas and arguments.  Specifically, students are to “analyze mathematical relationships to connect and communicate mathematical ideas.” As teachers, we often find ourselves using worksheet generators or textbook questions as practice problems for our students.  By creating the Process Standards, the state of Texas is encouraging teachers to go beyond the basic textbook problems.  From an early age, students should begin developing algebraic thinking in order to prepare them for their future encounters with algebra.   

In “Developing Elementary Teacher’s ‘Algebra Eyes and Ears’”, Blanton and Kaput (2003) encouraged teachers to “algebrafy” problems.  They stated, “Our ‘algebrafication’ strategy involves three types of teacher-based classroom change: algebrafying instructional materials, finding and supporting students’ algebraic thinking, and creating classroom culture and teaching practices that promote algebraic thinking” (p. 70 – 71).  Teachers should begin with a simple algebra problem and then extend it.  This problem was presented in a Pre-Algebra textbook:

A pebble falls into a pond.  From the surface, it descends at a rate of 2 feet per second.  Where is the pebble in relation to the surface of the pond after 5 seconds?

This problem can be changed from a basic word problem to an algebraic task by modifying one of the variables.  When students are required to think about the placement of the pebble in relation to the surface at multiple times, they begin to recognize patterns in the data.  This is also referred to as learning to generalize.  Barbara Kinach (2014) states, “…describe the transition to working with symbols as a process of continually refining one’s representations of a problem situation…Attention shifts from interacting with the actual problem while problem solving to interacting with successive representations of it” (p. 434). 

Our new, algebrafied problem would look like:

A pebble falls into a pond.  From the surface, it descends at a rate of 2 feet per second.  Where is the pebble in relation to the surface of the pond after 5 seconds? 10 seconds? 100 seconds?

By adding more time options to the end of this problem, we are encouraging students to think about more than just one answer choice.  Students could use the original method of solving the problem to determine the pebble’s placement after 10 seconds, but they would need to find another method of solving to determine its placement after 100 seconds.  This encourages students to begin thinking of the problem in more general terms.  The students would come up with the equation y = 2x where y is the number of feet below the surface and x is the number of seconds.  They would use this equation to determine the location of the pebble after 100 seconds. 

In elementary school, students should have experience with three types of algebraic thinking: generalizing, equality, and unknown quantities.  In their article, “ThreeComponents of Algebraic Thinking: Generalization, Equality, and Unknown Quantities”, EdTech Leaders Online discusses the importance of these types of algebraic thinking in late elementary school.  Before students reach Algebra 1, it is important that they understand the importance of these three types of algebraic thinking.  The NationalCenter for Improving Student Learning & Achievement in Mathematics &Science provides even more examples for how to algebrafy student’s tasks to prepare them for Algebra I and Algebra II in high school. 

Retrieved from http://ncisla.wceruw.org/

By algebrafying our students word problems, we encourage algebraic thinking and generalization.  The TEKS Mathematical Process Standards describe how students should be able to analyze mathematical relationships and communicate mathematical ideas.  By algebrafying tasks and encouraging students to generalize their thinking, we are encouraging them to learn how to identify mathematical relationships instead of just solving basic problems.   

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

Kinach, B. M. (2014).  Generalizing: The core of algebraic thinking.  The Mathematics Teacher, 107(6), 432-439.