Problem Solving and the Common Core

Content

Background

In the Commonwealth of Virginia, students are expected to solve single-step and multi-step addition and subtraction problems with whole numbers by the fourth grade. I have been teaching this content for two years in the same school and I have found it helpful to spend a long time on this standard, as word problems are a particularly difficult concept for my students to grasp. Typically, I spend approximately a week on each aspect of the standard. For example, during the first week of the unit I attempt to teach addition word problems in isolation. During the next phase of the unit, I would spend another week concerned with subtraction word problems. Through my research, I have concluded that it would be more conducive to student understanding of addition and subtraction to discuss addition and subtraction together, as they are linked by an inverse relationship. It is very important for students to recognize this connection.

This unit draws heavily on an understanding of the taxonomy of several types of word problems. More information on taxonomy of the specific problems I have chosen will be discussed under the following sub-headings, under Content: One-Step Problems and Two-Step Problems. It is also important to note that this unit is being taught with the assumption that students are already competent in multi-digit addition and subtraction computations. If students are also struggling with basic mechanical errors in their work, this unit may be overwhelming.

Also included is a set of word problems that will target a specific issue: misleading ‘key words’ in written math problems. These problem sets can be found in the appendix. To help students become more analytical, I have constructed a set of word problems that intentionally include misleading “key words” that may guide students to perform the wrong computation. For example, a problem will use the words “greater than” or “in all”, which students typically associate with addition, when in fact the problem requires subtraction. Another example would be when the words “less than” appear but it is necessary to add to find the correct answer, rather than subtract. These issues often appear in comparison problems, but ‘inconsistent’ key words can also be found in other types. What I hope students will conclude after closely studying these problems is that the “key word” approach that they often depend on becomes very unaccommodating as they encounter more rigorous problems of a wide variety, and beyond that, that a careful reading and analysis can lead to their desired results.   

My wish is that, from a close examination of several types of problems, my students will gain an understanding of the relationships among variables in a problem. I would like them to focus more on analyzing problems rather than looking for specific words. What I mean by that is that instead of asking students to focus on “key words”, I would like them to focus on what relationships can be identified in each problem. For example, I want students to start asking them selves, “What part is missing in this problem?” or, “Am I comparing anything in this problem? If so, which part is greater, which is smaller, what is unidentified?” I do not think it is necessary to explicitly teach students the taxonomy of word problems. I believe it is more beneficial for students to create their own meanings and types by solving many different word problems and sorting them by their own definitions and categories. This will be elaborated on more in the “Activities” section. Below is a figure that discusses the taxonomy of one-step addition and subtraction problems. It provides an example of each type of problem. The scenarios that are discussed in my unit and presented in the appendix have been constructed to reflect the types of problems my students may encounter on standardized tests.  I will discuss a few types of problems that appear in the chart in detail and give a brief explanation of the relationships I want students to identify. I have designed sets of problems with fixed variables and set scenarios in order for students to easily see the relationship among each variable in a problem.

General Taxonomy of One-Step Addition and Subtraction Word Problems

Figure 1

In each type (shown as a row), any one of the three quantities in the situation can be unknown, leading to the subtypes shown in each cell of the table. The table also shows some important language variants, which, while mathematically the same, require separate attention. Other descriptions of the situations may use somewhat different names. This chart is adapted from CCSS, p. 88, which is based on Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, National Research Council, 2009, pp. 32–33⁵