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.