From Arithmetic to Algebra: Variables, Word Problems, Fractions and the Rules

Introduction and Rationale

Next year I will teach third grade. This is a change from teaching fourth grade, where I have spent most of my teaching career up to now. Since I am switching grade levels, I recently reviewed Virginia’s standards for third grade mathematics. I was not surprised to see that the importance of problem solving is emphasized, considering that problem solving is always the most difficult aspect of the math curriculum to teach. I believe that many factors combine to create the difficulty: the nuances of each problem, the reading comprehension component, the application of skills, and the need for deeper understanding of computational skills. These aspects correspond roughly to Polya’s steps 1, 2, and 3.¹ In addition to the skills needed to understand and solve a problem, students need to have perseverance to bring it all together.

The Mathematics Standards of Learning for Virginia Public Schools² state: “Problem solving has been integrated throughout the six content strands. The development of problem solving skills should be a major goal of the mathematics program at every grade level. Instruction in the process of problem solving will need to be integrated early and continuously into each student’s mathematics education. Students must be helped to develop a wide range of skills and strategies for solving a variety of problem types.”

However, despite the emphasis placed on problem solving, there is little guidance or training on how to do that. In fact, as I read through the specific standards, one of the few utterances about problem solving that I found says the students will, “create and solve problems that involve multiplication of two whole numbers, one factor 99 or less and the second factor 5 or less.” The standards do not mention any strategies or methods to use to solve problems.

When I go through the pacing guide provided by the school district, it skips around the textbook and provides random websites, some of which no longer exist, for additional practice. If a lesson is not directly tied to the standard, it is excluded. Fortunately for my students, I choose to follow the book more sequentially and intertwine Singapore math strategies, along with knowledge I have gained from working with Roger Howe through the Yale National Initiative.

Another issue that is promoted widely throughout resources and my school district is the strategy of key words to solve problems. By focusing on key words, children tend to spot the key word, then they choose what they deem to be the appropriate operation based solely on a word or two in the problem. Then they perform the “matching” operation on the numbers given in the problem. The key word strategy is limited, since a word problem with the word more may require addition or subtraction or, later in the curriculum, multiplication or division; and a word problem with the word less may require addition or subtraction depending on the given scenario. In order for our students to become more comfortable with problem solving, they need to comprehend the action and relationships described in the problem instead of relying on the key word strategy. To reiterate, the standards say, “The development of problem solving skills should be a major goal of the mathematics program at every grade level.” For this reason, I believe our students deserve better. Teaching the key word strategy to children is a disservice, especially given the importance of problem solving as stated in the Virginia State standards.

In order to address the issues listed above, I am creating a curriculum unit that focuses on problem solving techniques. I wrote a curriculum unit for fourth grade in 2007, titled Dr. Word Problem: Solving Word Problems with the Four Basic Operations of Mathematics Using Singapore Bar Models.³ I do not want to duplicate that unit. This unit will address some of what I consider to be the best problem-solving strategies. I will briefly touch on Singapore bar models, but if you are interested in learning more about them, my 2007 unit is listed in the resources, and includes its own list of resources.

The crux of this unit is understanding the Common Core taxonomy of problems,⁴ while infusing Polya’s steps,⁵ and aspects of the Singapore model method.⁶ I include a collection of word problems that identify the 14 types of one-step addition and subtraction problems and the nine one-step multiplication and division situations. The curriculum unit will include Polya’s four-steps as he is still considered the “guru” of problem solving for his work from 1945.⁷ It will also include some steps from Singapore that are likely based on Polya’s four steps, as they seem to work hand in hand.