Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra

Rationale

This curriculum unit focuses on fostering students’ conceptual understanding of the role of variables and their units in linear equations- specifically in writing linear equations from word problems, justifying these equations, solving these equations, and evaluating their answers. In the past, I have taught translating and solving linear word problems after teaching solving, evaluating, and graphing linear equations; I noticed that, rather than gaining a comprehensive understanding of linear equations, this sequence resulted in students’ overreliance on rote procedures and did not culminate in the “Aha!” moment I was hoping they would reach.

Research out of UC Santa Barbara found that “unsuccessful problem solvers base their solution plan on numbers and keywords that they select from the problem (the direct translation strategy), whereas successful problem solvers construct a model of the situation described in the problem and base their solution plan on this model (the problem-model strategy).”¹ Many U.S. textbooks teach word problems in conjunction with a very specific type of mathematical concept, thereby promoting the direct translation strategy. For instance, after learning about velocity, students will read word problems about velocity, identify the necessary values in the problem to plug in (distance and time), and determine where these values go in the velocity equation. However, this rote proceduralism does not foster meaningful learning and, instead, results in students’ inability to solve the same kinds of problems outside of the classroom context.

According to the California Assessment of Student Performance and Progress results from 2021-22, more than two thirds of all 11^th grade students at Overfelt High School did not meet state standards in math. Over two out of every five students scored “below standard” in “Problem Solving and Modeling & Data Analysis,” meaning that they “[do] not yet demonstrate the ability to solve a variety of mathematics problems by applying… knowledge of problem-solving skills and strategies. [They do] not yet demonstrate the ability to analyze real-world problems, or build and use mathematical models to interpret and solve problems;” nearly a third of students scored “below standard” in “Communicating Reasoning,” meaning that they “[do] not yet demonstrate the ability to put together valid arguments to support [their] own mathematical thinking or to critique the reasoning of others.” Finally, nearly three out of every five students scored “below standard” in Concepts & Procedures, meaning that they “[do] not yet demonstrate the ability to explain and apply mathematical concepts or the ability to interpret and carry out mathematical procedures with ease and accuracy.” Other categories of scoring include “near standard,” in which some ability is perceived, and “above standard,” in which a thorough ability is perceived.²

My Math I students reflect these same struggles in their comprehension of linear equations. They lack the conceptual understanding that equations, graphs, and tables can all represent the same real-world scenario or problem despite presenting very differently. My students also lack the confidence as well as the mathematical vocabulary to justify their use of numbers, symbols, and variables in place of words. My curriculum unit will try to bridge this gap of understanding by introducing students to linear equations through word problems, so that they can truly comprehend the meaning (or a possible meaning) behind each equation as it is translated, justified, and then solved. My hope is that this unit will not only will this facilitate student success in working with linear equations, both as word problems and as algebraic expressions, but this will further prepare students when asked to work with linear equations as tables and graphs after the planned unit is taught.

Students’ use of the critical thinking that is needed when creating algebraic expressions and equations from English sentences will likely bolster their success in Problem Solving and Modeling & Data Analysis, as it will facilitate their “ability to analyze real-world problems, or build and use mathematical models to interpret and solve problems.”³ Additionally, the explanations they provide around why they wrote the expression or equation as they did will likely improve achievement in Communicating Reasoning, as it will encourage their “ability to put together valid arguments to support [their] own mathematical thinking…”⁴ Furthermore, having students evaluate and simplify expressions and solve equations will improve student performance in Concepts & Procedures, as they will be more apt at “interpret[ing] and carry[ing] out mathematical procedures with ease and accuracy.”⁵