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

Rationale

In the United States today, teaching students the skills to truly understand Algebra is one of the most important ways to improve their economic status. Bob Moses, civil rights activist and the founder of Algebra Project, states, “In today’s world, economic access and full citizenship depends crucially on math and science literacy” (1). Algebra is a “gate keeper” because it is a requirement for students to enroll in higher math classes and science classes. Therefore, if students are considering careers in the fields of science and technology, they need to pass Algebra sooner rather than later. Unfortunately according to David Foster, executive director of Silicon Valley Math Initiative (SVMI), the state of California is experiencing an “Algebra Crisis”. One piece of evidence for this claim is that three out of four students scored below proficiency in the 2008 Algebra 1 California Standards Test (CST) (2). Even more troubling, according to the California Dropout Research Project 2008 of Los Angeles Unified School District, the high school dropout rate is approximately double for students who do not pass Algebra 1 by 9^th grade (70% versus 35%) (3) Consequently it has been said that “Algebra is the #1 trigger of dropouts in high school” (4). Thus we can see some ways that Algebra performance can affect opportunities for all students regardless of career choices.

In response to the “Algebra Crisis” and student underperformance on international math assessments, many states, including California, adopted Common Core Standards. My district, Berkeley Unified School District (BUSD), implemented Common Core Math five years ago. Unfortunately too many of our students, especially students of color, have had limited success in math from early school years. In fact the achievement gap has become wider and wider in higher grades. In BUSD on the 2016 Smarter Balance Assessment for Math, Black/African American students scoring at or above proficient decreased significantly from 26% in 6^th grade to 11% in 8th grade. Similarly, Latino students also decreased from 43% in 6^th grade to 34% in 8^th grade. This is in stark contrast to the situation for White students of whom 82% were proficient in 6^th grade, and also 82% in 8^th grade, no change at all. I predict that this downward trend for students of color and the achievement gap between white peers will only get worse in Algebra 1 and in other high school math classes.

Knowing the importance of passing Algebra 1 in our students’ career choices and their future economic standings, many elementary and middle school math teachers are preoccupied with the question of how to make Algebra concepts more accessible to all of our students. Like my peers, I have tried many different instructional strategies and pedagogies in my 8^th grade classroom. However, a much lower percentage of my students are proficient in Algebra strands than in Geometry strands. On 2016 District Assessments, 90% of my 8^th grade students were proficient in Geometry (Congruency and Similarity) strands, but only 71% were proficient in Algebra (Expressions and Linear Equations and word Problem) strands. Even worse, only 38% percent of students of color and English learners were proficient, which means nearly two thirds are not mastering these essential concepts. I find this troubling, because these students will fall farther and farther behind in high school math classes, which can close doors to more lucrative career choices.

In every one of my fourteen years of teaching, there have been a number of students who think the value of x is 1 because x represents an unknown value in the equation and they hear that “x is a shortcut for 1x.” When solving 4x = 10, some students subtract 4 from both sides of the equation instead of dividing by 4. Every time I notice such misconceptions as above, I ask the following questions: What does 4x mean? What are we looking for? What operation do we need to do to find the value of one x? Through this process of questioning, I can get students to understand what to do with the equation we are discussion. Unfortunately these “aha” moments do not stick. One of the goals of this unit is for students to see concretely the meanings of equations such as 4x = 10 and ¼ - x = 10 by referring the symbols to the situations described in the word problems.

Although my own student work analysis and research show that misconceptions around variables is one of the key reasons for mistakes in solving equations, most of my 8^th graders would say that they know all they need to know about variables. In reality most students understand variables only in the form of labels representing objects: t for table, c for centimeters, and x for a number. Phillips (5) discusses the many different roles the variables can take on (labels, constants, unknowns, generalized numbers, varying quantities, parameters, abstract symbols) depending on the situation. Studies (6) show students of all mathematical level, even in college, have misconceptions regarding how to think of variables due to the multiple roles they play. By intentionally drawing connections between the context in the word problem and the given equation, I hope that my students should arrive at the broad conception that a variable is a place holder for any number in some set of numbers, but that it can seem to play the various roles mentioned above, depending on text.