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

Content and Objectives

I will use the taxonomies of one-step addition/subtraction problems and one-step multiplication-division problems, included in the CCSSM as corestandards.org /Math /Content/mathematics-glossary/Table-1/² to guide my writing and discussion of word problems. I will begin with the one-step types. I would allow time for student to become comfortable with the different types of one-step problems, and then another period of time dealing with two-step problems and figuring out how to decompose them into one-step problems before going into the three-step problems and more. Students need to get to the point where they can deal with multi-step problems fairly readily. The key is getting them used to reading the problem carefully and figuring out what it is saying. My overall goal is that my students will practice a mixture of varied problems. All problems will have a strong cultural aspect, and will especially involve the hooghan. In this way, my students will begin to see how home and family are valued at school, and how mathematics fits into these aspects of life, and helps to understand math problems and culture.

As students learn the process of math problem solving, they need to understand that there is no single process that works all the time. There are various strategies and skills they can learn to use to be able to read various problems and to understand how each problem is slightly different. The most important practice to develop is the habit of reading the problem carefully and thinking about how the different parts are related. The practice of problem solving is crucial because when problems become difficult they require specific analysis and breakdown in certain situations. As the problems are solved, students will need to understand how they solved the math problems; this is when thinking about thinking comes into play. The cognition, to be able to explain orally or in writing how the word problem was solved, and to ask if it is correct, are skills I want to emphasize. This is when everyday life and home activities make the connections between cognition and context when using culture and language.

After my students, have gained some familiarity and comfort with word problems, I hope to introduce some of the more formal aspects of arithmetic, to strengthen their abilities to work with symbolic expressions, and to prepare them for dealing with algebra. Students learn to substitute variables while writing equations and this increases the rigor and the pedagogy of the mathematical thinking. So, for students to be able to work successfully with math word problems, we need to be strategic in teaching appropriate classroom techniques to assist and build on these linkages. By emphasizing the home and school connections for students in my problem sets, I want to make students feel successful and take ownership in deciphering the mathematics. We will build bridges when the instructional objectives are for children are useful for them, and will promote their enduring understanding.

The challenge of solving math word problems for elementary teachers is the need to be pedagogically sound when teaching the arithmetic and algebraic solutions. I hope that using the table created above will assist teachers in thinking about the appropriate problems and working through the process of solving the steps. The practices of problem solving in various ways will become a smooth process for elementary teachers to get comfortable and confident in math.

Below are word problems with arithmetic solution, then the algebraic solution. This is how Roger Howe’s seminar, “From Arithmetic to Algebra” modeled the applications.

Problems:  The arithmetic solution, then the algebraic solution. First is Roger Howe’s example problem,³ then addition problems are shown as examples, like One-Step Equation, Additive with result unknown, change unknown, and then start unknown.