Many students going through public schools in the United States often utter the question, “Why do I need to learn this stuff?” This is an interesting question that many teachers or parents answer by saying, “you will use it when you are older” or “you need it to pass the quiz/test/homework.” But these answers fall short of getting to the heart of why certain topics are important for students to not only comprehend, but to understand. One such subject that falls into this category is algebra. Many high school students sit in class just doing or remembering enough to get through the class without actually internalizing or understanding what was happening in the classroom. Too often, the teachers, who are expected to be the experts in the area, themselves, do not fully grasp the importance of understanding algebra.
Usiskin (1995) attempted to give reason for why algebra is important to learn by stating that:
Without a knowledge of algebra, you are kept from doing many jobs or even entering programs that will get you a job; you lose control over parts of your life and must rely on others to do things for you; you are more likely to make unwise decisions, financial and otherwise; and you will not be able to understand many ideas discussed in chemistry, physics, the earth sciences, economics, business, psychology, and many other areas (p. 31).
This answer seemed to be just a more detailed explanation of the one given by teachers and parents for years. Although Usiskin (1995) did go on to explain the necessities of algebra more in depth and attempted to connect it to the students who might be in an algebra class. For example, he matches the concept of linear equations to understanding and calculating a phone bill, slope to how fast a car can change speeds, exponents to credit card bills and other issues in finance, quadratics to physics formulas, and logarithms to the Earth sciences. Usiskin also describes in detail how algebra can be useful in deductive arguments to establish truths. Algebra could be used to determine if something is always true, not just relying on many examples, which does not prove a statement to be true.
Usiskin (1995) goes on to state that many people use algebra because it can make things easier to understand and can bring enjoyment through discovery, and proving results. He also adds that many adults attempt to avoid algebra at all costs, by using computations that can take much longer and is comparative to “people who go to a foreign country but do not know enough of the language to converse with native speakers in that country” (p. 37). These statements seem to hold true for most adults using algebra or other forms of mathematics, but it still seemed like better connections should be made for the students in grades kindergarten through twelfth deserve better explanations and more connections to their “world”.
But before connecting it to the world of the student, why is algebra so important in today’s society and in the realm of education. Spielhagen (2011) noted that algebra is the cornerstone to mathematics literacy, but went on to say that the instruction for students should be authentic algebra instruction. This authentic algebra instruction should not be one that weeds students out by only allowing a select few to enter the world of algebraic thinking. Instead, algebra should be used to increase equity to bring forth increased excellence in the performance of students. Algebra is a vehicle that must be used to get more students accepted to four-year colleges (Spielhagen).
So how did some school districts attempt to have more students take algebra? One way was to mandate all students in the eighth grade to take algebra. This is what occurred in Chicago, but it failed to achieve the goal of raising test scores. Even though this was viewed as a failure, many other school districts began implementing the same format with the same outcomes and also creating another problem. Many students who failed algebra in the eighth grade were assigned to repeat the course in high school, where many students failed again and some even did worse the second time around (Spielhagen, 2011).
This problem of where algebra fits into the American mathematics curriculum is still creating confusion and arguments in the field. There is no national curriculum for all to follow, but even with some standards set in place, Spielhagen (2011) identified the states’ rights argument for sabotaging the “algebra for all initiative” (p. 5). This is a strong sentiment that is taken head on by Cueves and Yeatts (2001), in the Navigating through Algebra series. This series attempts to show how algebra permeates through all levels of mathematical learning and is very important for students to use and learn about from a very young age.
According to Cueves and Yeatts (2001), “algebra is dynamic and a necessary vehicle for describing a changing world” and is “a natural extension of arithmetical thinking” (p. 1). But much must be done to how students are introduced to algebra and how this learning leads to higher level understandings about the power of algebra. The algebra curriculum must be coordinated from grades K-12 and must be well articulated for educators to understand, so that each stage is developmental and coherent for students moving through the grades (Cueves & Yeatts). The students, who go through each grade level building upon previous knowledge, must be continually challenged to learn and apply algebraic thinking in new situations. These situations should be anchored to the lives of the students, including use for algebra in the schools, home, and other life settings. Instead of more concepts for students to memorize, a few “big ideas” should be selected as the focus of study, and these ideas should transcend the grade levels.
Beginning in the elementary grades, teachers need to expose students to a variety of algebraic concepts in the classroom. These concepts should be presented in a fashion that they make sense and connect to what students already understand or at least know about; some of these concepts and methods are described by Schwartz (2008). Some of the basics of algebra that Schwartz described were equations and equality, inequality, variables, integers, and graphing on a coordinate plane. The way that teachers could present these algebraic concepts to young students include using items that students are familiar with such as balances or number lines.
For the areas of equations and inequalities, Schwartz (2008) describes using a balance to demonstrate each item. The equal sign is placed above the balancing point or fulcrum of the balance to represent that each end of the equation must be the same for it to balance perfectly. This opens up the idea of having more them one term on both sides so that students do not get the misconception that the equal sign is a symbol that instructs students to perform an operation. This would lead the students, with some instruction, to the understanding that there could be infinite many correct responses to equal a certain value. For example, 5+4=3+1+5 while at the same time 9=3+1+5, etc.
Moving from equations to inequalities, students could use the same format of a balance to understand the concepts of greater than or less than. Students could manipulate both sides of the balance to determine if quantities are greater than or less than another quantity by viewing which way the balance tilts. Students have many misconceptions when it comes to inequalities, because they assume that they can be manipulated in the exact same manner as equalities (Prestege & Perks, 2005). These misconceptions have been linked to students not understanding the meaning of what inequalities represent and how they are affected by multiplication and division of negative numbers (Blanco & Garrote, 2007). This deeper understanding is what educators must look for and aim for. This understanding should begin in the younger grades for future success in higher-level algebra courses.
Educators can then focus on the concept of representing a quantity with a variable. Utilizing the concept of the balance, students could begin to realize the usefulness of a variable or come to appreciate “a variable as a place holder” (Cueves & Yeatts, 2001, p. 3). Students can begin by representing unknowns with an empty box that needs to be filled by some quantity to make the equality or expression true. In algebra, it is important for learners to understand that there could be multiple answers that could make a mathematical statement true. For example, a student might be asked what number is less than 19, or visually as >19. A student might respond with 18 and they would be correct, but educators “eventually want learners to understand that any number that is less than the stated number can fulfill the requirements” (Schwartz, 2011). The empty box could then be replaced by a letter variable, and students could make connections that they represent the same idea.
Students could also begin working with algebra by developing an understanding of patterns and relationships. Educators should begin with young children by incorporating concrete manipulatives that the students can see and touch. One example comes from Hatfield, Bitter, Edwards, and Morrow (2005) that explains a type of function machine that the teacher can use to spark interest in the concept of pattern. The teacher begins with a shape that is a specific color. The teacher drops the shape into the function machine (a box with other shapes inside) and pulls out a different object that has either changed in shape, size, or color. The students watch as other examples are put into the function machine with their specific outcome. Students then work to explain what the function machine is doing to change the original shapes all in the same way. This then leads to students creating their own function machines, but instead of shapes, numbers are put into the machine with different outcomes. Students can then attempt to describe what is happening to change each input value.
These relationships and patterns can then be organized so that they are easier to understand or view. “By using tables, charts, physical objects, and symbols, students make and explain generalizations about patterns and use relationships in patterns to make predictions” (Cueves & Yeatts, 2001, p. 2). From these initial tables, students could then begin using coordinate planes to graph coordinate pairs to discover the relationship between the ordered pair. This graphic representation is important because it “allows us to find an output number that is associated with any input number without having to actually do the computation for that pair of numbers” (Schwartz, 2008, p. 257). This could then lay the foundation for older students when the concept of rate of change is introduced.
When students are beginning to move into using integers, educators can involve using a number line to help students visualize about what is really happening when integers are combined (Schwartz, 2008). Another method for allowing students to work with integers is to introduce a game involving a letter carrier (Hatfield et al., 2005). This game allows one student to be a letter carrier and delivers either checks (positive integers) or bills (negative integers). The other students start with a certain amount of money (usually a positive amount) and then begin to receive checks or bills from the letter carrier. The students then must calculate how much money is left when they receive each type of delivery. The concepts of adding a positive integer (checks) or adding a negative integer (bills) could be explored and discussed. To further the exploration, the letter carrier could be prone to mistakes and must retrieve letters that were delivered to the wrong places. The letter carrier could then come back and take away checks or bills to allow students to subtract using positive and negative integers.
The same game, utilizing the letter carrier, could be developed for multiplication of integers also. The letter carrier could deliver multiple checks or bills at the same time, and the students would then need to calculate the total loss (negative) or gain (positive) received at the time of delivery (Hatfield et al., 2005). To incorporate division with integers, Schwartz (2008) suggests taking the idea of bills and checks a step further. The example given describes a situation where a person owes $36 dollars to four different people. This idea could be represented as -36÷4. This allows students to work with dividing a negative integer by a positive integer. Schwartz then describes how students could then take the $36 dollars that they owe and view it as 36 individual IOU’s. Next the students would be asked to make piles that would represent negative three dollars. This could be represented by -36÷-3 to show that there are in fact a number of piles greater than zero, or a positive integer.
As for dividing a positive integer by a negative integer, Schwartz (2008) admits that, “there may be some highly unusual situation that occurs that would be represented by a positive number divided by a negative number, but we do not see such situations in everyday life” (p. 255). Thus the best way to explore this situation would be to have students make connections between multiplication and division to determine if the quotient would be positive or negative.
All these situations bring to light what the outcome of learning algebra should be for students, “being able to represent generalized relationships” (Schwartz, 2008, p. 261). Algebra education needs to reflect this ideal by incorporating methods that best teach algebraic concepts in context, focus on learning, focus on the mathematics in elementary grades leading to algebra, create a substructure that assists all students leading to algebra, and investigate what is happening in each classroom to determine what the teacher is doing to help students understand algebra (Hatfield et al., 2005; Schwartz, 2008). Educators and policy makers must remember that algebra is easier to learn at a younger age and practically anyone can learn it (Hatfield et al., 2005).
So if algebra can be learned by anyone, why have so many students failed in courses offered in schools? Robert M. Capraro (2011) made some insights into why this is happening, especially in the state of Texas. Capraro argues that to improve mathematics education, a regular tetrahedron should be used as a model. A regular tetrahedron is a shape where “all sides are the same and no matter how it falls, it has a stable base or foundation on which to build” (Capraro, 2011). Each face of the tetrahedron represents a facet that builds quality mathematics instruction. The first face represents a rigorous curriculum that was developed as a scope and sequence for educating students in mathematics. Each previous concept builds upon the next so that full understanding is the goal to be achieved.
The second face of the tetrahedron is that assessments should be developed around what the content being taught. This just makes sense, due to the fact that how would an assessment be deemed valid, if it were testing information the students have not come in contact with? Capraro’s (2011) tetrahedron’s third face is representative of quality teachers in each and every mathematics classroom. He explains that, “teachers are the linchpin or keystone to mathematical success for children” (Capraro). Educators need the appropriate knowledge and understanding of mathematics to support high levels of education in the classroom with different levels of students.
Finally, the fourth face of the tetrahedron represents models of education that “offer evidence for improving mathematical learning” (Capraro, 2011). Models should be useful to teachers but at the same time build a professional learning community within the school district. The model should build upon the teachers’ strengths to develop higher levels of expertise and mathematical knowledge to support mathematical education in the classroom. This face also incorporates the partnerships between school districts and colleges of higher learning so that the gap can be bridged between high school and the university level.
Looking into the future of algebra education in the United States, it seems that a couple of items come to the forefront. First, teachers should be introducing algebra to younger students, even beginning in the lower grades of elementary schools. Focus should be on a couple of “big issues” that encompass algebra at all different levels. Students should also work with algebra in context of their world. Connect lessons and content to what students already are familiar with and have the students uncover how algebra is useful in their lives. Last, educators and policy makers need to understand what it takes to create a system for mathematical learning to occur in the classroom. This system cannot be viewed as one-dimensional, but a tetrahedron with multiple faces that allows focus to shift from one to the other, working in harmony on one ultimate goal, giving every student the best opportunity to understand and embrace algebraic thinking.
Blanco, L.J., & Garrote, M. (2007). Difficulties in learning inequalities in students of the first year of pre-university education in Spain. Eurasia Journal of Mathematics, Science & Technology Education, 3, 221-229.
Capraro, R.M. (2011, September 15). Testimony given to the State Board of Education, Austin, Texas.
Cuevas, P., & Yeatts, K. (2001). Navigating through algebra in grades 3–5. Reston, VA: National Council of Teachers of Mathematics.
Prestege, S., & Perks, P. (2005). Inequalities and paper hats. Mathematics Teaching, 193, 31-34.
Schwartz, J.E. (2008). Elementary mathematics pedagogical content knowledge: Powerful ideas for teachers. Boston: Allyn & Bacon.
Spielhagen, F.R. (2011). The algebra solution to the mathematics reform: Completing the equation. New York: Teachers College Press.
Usiskin, Z. (1995). Why is algebra important to learn? American Educator, 19, 30-37.
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