Developing from Arithmetic to Algebraic Understanding: Strategies to Create Algebraic Mastery

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." 

The separation of early mathematics and algebra has been around for decades, but this separation is having an impact on later student success in grades 8^th on up. When algebra is introduced in the elementary grades, the foundation of algebraic reasoning is established therefore strengthening the understanding of arithmetic and paving the way for more advanced learning. There are many resources out there for elementary teachers to implement into their math curriculum to enhance basic computations into “Algebrafied’ problems such as SEDL, National Council of Teachers of  Mathematics, and National Center for Improving Student Learning & Achievement in Mathematics.  As educators' it is important to adequately prepare ourselves to integrate algebraic thinking into our classrooms so that we provide opportunities for our students to build their analytical and algebraic understanding.

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/

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. 

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.  

Beyond Textbook Problems

By Cassandra Hatfield 

More often than not teachers rely on curricular materials at face value for quality instructional tasks. After all, the state goes through a lengthy process of accepting instructional materials. Many lessons within textbooks are tagged with Mathematical Content Standards and integration of the Mathematical Process Standards is also shown. This can be misleading, therefore, elementary mathematics teachers need to be provided with strategies that will build better opportunities for their students that have more depth of understanding.  Particularly, I would describe depth by the quantity and quality of knowledge that students can gain. This can be achieved when students' experience an “algebrified” problem instead of a worksheet with fifteen of the same type of problem on it.

Blanton and Kaput (2003) describe how teachers can “algebrify” a problem in their article "Algebra Eyes and Ears." Algebrification is a modification of “problems with a single numerical answer to opportunities for pattern building, conjecturing, generalizing and justifying mathematical facts and relationships.” Algebrified problems can be integrated into lessons instead of as a "challenge" problem or the problem at the end of a sequence of computational problems so students can experience the cohesion of the incorporation of the mathematical process standards. Specifically, when students interact with these problems they have the opportunity to “display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.” (TEKS, 2012)

Below is an example of a traditional task (original) and an example of how the task can be algebrified. The original task was found at the end of a lesson in a fourth grade textbook. The algebrified task is also appropriate for fourth grade students.

The algebrified task asks students to describe the relationship between weight on the Moon and weight on Earth. Since the “key to abstract reasoning is using algebraic expressions to describe problems,” (Ketterlin-Geller, Jungjohann, Chard, & Baker, 2007) this task provides an opportunity to see a range of responses from students and facilitate moving students beyond their current understanding of a relationships to a more abstract representation (or generalization) during a whole class discussion.

Thus, the class discussion is a key component of algebrifying a task. Rather than engaging students in a show and tell of their answers, teachers should use five key practices that will support students’ productive engagement in a discussion of their solutions. These key practices are outlined in Figure 1 and described in more detail in the next paragraph. (Stein, M.K., Engle, R., Smith, M., Hughes, E., 2008)

Figure 1

First, the teacher should anticipate student responses. In the modified task provided, some students are likely to write out the relationship in words while others will use an expression. It is possible that some students will only identify the numerical relationship between 3 ounces and 1 pound 2 ounces rather than recognize a generalized relationship between the two weights. The anticipation of how students will respond to the task is helpful in determining the appropriate types of questions to ask students or supports students will need when working through the task. Next, a teacher should monitor how students are interpreting and solving the problem while they are working. This observation support the teacher in selecting which students should share during the discussion. Additionally, the teacher should consider the sequence in which the students will share their responses so that connections can naturally be made and student understanding can be moved to a new level. (Stein, M.K., Engle, R., Smith, M., Hughes, E., 2008)

The algebrification of problems leads to an incorporation of the mathematical process standards and it also provides a rich content for a whole class discussion beyond show and tell.  

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

Ketterlin-Geller, R., Jungjohann, K., Chard, D., and Baker, S. (2007). “From arithmetic to algebra”. Association for Supervision and Curriculum Development, 66-71.

Stein, M.K., Engle, R., Smith, M., Hughes, E.,  “Orchestrating productive mathematical conversations: five practices for helping teachers move beyond show and tell”. Mathematical Thinking and Learning, ) 10(4), 313-340.