By: Ashlyn Smith
Algebra is a subject for many K-5 educators that invokes childhood memories of confusion, frustration, and complex concepts. This uncertainty leads educators to question whether they are adequately prepared to teach algebra to their students. Furthermore, research says that implementing algebra in the early years of elementary school lays the groundwork for later mathematical success. Thus, algebra in the early years is a hot topic in schools today. (Brizuela, Martinez, & Cayton-Hodges, 2013) Because many elementary educators are unfamiliar with early algebraic ideas K-5 educators are looking for support and resources to effectively and meaningfully incorporate algebraic tasks into their curriculum.
In Texas the Texas Essential Knowledge and Skills (TEKS) for mathematics includes algebra concepts throughout the elementary grades. In addition to content expectations, educators are asked to engage students in the TEKS Texas Mathematical process standards while teaching algebraic topics. The TEKS mathematical process standards support students in acquiring a deep understanding of mathematics and critical thinking abilities. The challenge for K-5 educators is how to create engaging learning experiences to prepare students for the expectations of mastery in formal algebra. Blanton and Kaput (2003), describe a strategy called “Algebrafying” that “aims to help teachers learn to identify and create opportunities for algebraic thinking as part of their normal instruction and to use their own resources such as textbooks and supplementary materials" (p.73). Algebraification is a tool used by teachers to focus on helping students generalizes their mathematical thinking; promote mathematical expression and justification of their generalizations. An “algebrafied” approach focuses on solving meaningful related problems for which the answers typically are student-generated generalizations. When students are making generalizations they are finding common characteristics in problems, strategies or methods to solve a problem, and relying on past knowledge. As an example let’s take, the following third-grade basic arithmetic force and motion problem from StemScopes and turn it into an algebrafied problem. (Math Connections. (n.d.). Retrieved from http://www.acceleratelearning.com/)
Jane is on a playground swing and asks her friend Mary to push her. She looks at her watch and notices that it takes 3 seconds from the force of one push to the next. What would the total time for 6 pushes be?
To algebrafy the Force and Motion problem, one extension would be to vary the number of seconds per push, e.g., "She looks at her watch and notices that it takes 5 seconds...", "10 seconds", "20 seconds". As the number of seconds per push increases, students can then generate a set of data, analyze the mathematical relationship and generalize their observations using a verbal description or corresponding equation.
According to Earnest and Balti ("Instructional Strategies for Teaching Algebra in Elementary School" 2008) there are three strategies that can be used to modify arithmetic problems to foster algebraic thinking (a) unexecuted number expression (b) large numbers, and (c) representational context. Additionally, students can benefit from utilizing pictorial, graphic, and verbal description as these elements bring forth prior knowledge, better retainment of new material and promotes critical thinking. First, unexecuted number or strand of numbers refers to finding multiple expressions that are equal, (e.g. 3 + 3 x 2 or 3 x 2 +2.). "Unexecuted strand of numbers provides a way to record information from a problem and encourages students to discern emerging patterns" (Blanton & Kaput, 2005). Large numbers provide students a reason to consider the relationship between the input number and the output of numbers. Plus, it provides a stepping-stone in making the problem more abstract. Lastly, representational context refers to the "interactions and discourse co-constructed by the teacher and students that allow students to explore, test, reason, and argue about mathematical ideas" (Ball, 1993, p.160). According to Blanton and Kaput (2005), this step is where variables can be introduced, discussed, which in turn pushes students to reason algebraically within the lesson. The following is an example of adding these strategies crate a new algebrafied problem:
Jane is on a playground swing and asks her friend Mary to push her. She looks at her watch and notices that it takes 3 seconds from the force of one push to the next. If 6 pushes takes N seconds how many pushes happen in 30 seconds? 60 Seconds?
We could continue to alter this problem for students to analyze by varying the numbers to add larger more complex numbers. Modifying the StemScope problem to include larger numbers such as the number of seconds between each push, or the time spent on the swing, creates an algebraic learning opportunity. With larger numbers, students cannot easily compute the answer so must rely on their understanding of emerging patterns and the relationship between them to solve the problem. Representational context can be utilized throughout this problem through students exploring, prediction, building relationships, and reasoning with one another. Applying other variables such as the number of seconds for the force of the swing to come down, by the number of pushes, P, can be found by multiplying "3 x P", or alternatively this information can be presented in an input, output table. Generating a table or pictorial allows students to visually see the number pattern that represents the relationship between the two values in the algebrafied problem.
By generalizing the situation into a numerical expression students can discuss the relationship more generally. They are actively communicating these mathematical ideas and relationships. Altering an existing problem and designing it to go from a single numerical answer to a progression of problems that form multiple number sentences provides students with an advantage to analyze a sequence of events and engage in the process standards TEKS (c) “Students will use mathematical relationships to generate solutions and make connections and predictions and will effectively analyze mathematical relationships to connect and communicate mathematical ideas."
To algebrafy the Force and Motion problem, one extension would be to vary the number of seconds per push, e.g., "She looks at her watch and notices that it takes 5 seconds...", "10 seconds", "20 seconds". As the number of seconds per push increases, students can then generate a set of data, analyze the mathematical relationship and generalize their observations using a verbal description or corresponding equation.
According to Earnest and Balti ("Instructional Strategies for Teaching Algebra in Elementary School" 2008) there are three strategies that can be used to modify arithmetic problems to foster algebraic thinking (a) unexecuted number expression (b) large numbers, and (c) representational context. Additionally, students can benefit from utilizing pictorial, graphic, and verbal description as these elements bring forth prior knowledge, better retainment of new material and promotes critical thinking. First, unexecuted number or strand of numbers refers to finding multiple expressions that are equal, (e.g. 3 + 3 x 2 or 3 x 2 +2.). "Unexecuted strand of numbers provides a way to record information from a problem and encourages students to discern emerging patterns" (Blanton & Kaput, 2005). Large numbers provide students a reason to consider the relationship between the input number and the output of numbers. Plus, it provides a stepping-stone in making the problem more abstract. Lastly, representational context refers to the "interactions and discourse co-constructed by the teacher and students that allow students to explore, test, reason, and argue about mathematical ideas" (Ball, 1993, p.160). According to Blanton and Kaput (2005), this step is where variables can be introduced, discussed, which in turn pushes students to reason algebraically within the lesson. The following is an example of adding these strategies crate a new algebrafied problem:
Jane is on a playground swing and asks her friend Mary to push her. She looks at her watch and notices that it takes 3 seconds from the force of one push to the next. If 6 pushes takes N seconds how many pushes happen in 30 seconds? 60 Seconds?
We could continue to alter this problem for students to analyze by varying the numbers to add larger more complex numbers. Modifying the StemScope problem to include larger numbers such as the number of seconds between each push, or the time spent on the swing, creates an algebraic learning opportunity. With larger numbers, students cannot easily compute the answer so must rely on their understanding of emerging patterns and the relationship between them to solve the problem. Representational context can be utilized throughout this problem through students exploring, prediction, building relationships, and reasoning with one another. Applying other variables such as the number of seconds for the force of the swing to come down, by the number of pushes, P, can be found by multiplying "3 x P", or alternatively this information can be presented in an input, output table. Generating a table or pictorial allows students to visually see the number pattern that represents the relationship between the two values in the algebrafied problem.
Blanton, M., & Kaput, J. L., (2003). Developing Elementary Teachers’ “Algebra Eyes and Ears.” Teaching Children Mathematics, Vol 10 (No. 2). www.jstor.org/stable/41198085
Brizuela, B. M., Martinez, M. V., & Cayton-Hodges, G. A. (2013). The Impact of Early Algebra: Results from a Longitudinal Intervention. Journal of Research in Mathematics Education, 2(2), 209-241. Doi: 10.4471/redimat.2013.28
Earnest, B., & Balti, A. A., (2008). Instructional Strategies for Teaching Algebra in Elementary School: Findings from a Research-Practice Collaboration. National Science Foundation. http://sdcounts.tie.wikispaces.net/file/view/strategiesforteachingalgelem.pdf
Picture Retrieved: http://ncisla.wceruw.org/
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