Students' understandings and misconceptions of algebraic inequalities.

The National Council of Teachers of Mathematics [NCTM] requires students in grades nine through twelve to be able to explain inequalities using mathematical relational symbols and be able to understand the meaning of inequalities and their solutions (NCTM, 2000). Studies have shown that not only middle and high school students have difficulties with inequalities, but those of college age as well, particularly in the following four areas: a) regarding inequalities as equations (Blanco & Garrote 2007; Vaiyavutjamai & Clements, 2006), b) a limited understanding of the terms "more" and "less" and of the corresponding relational symbols (Warren, 2006), c) difficulties relating and using different solving techniques (Tsamir & Almog, 2001; Blanco & Garrote, 2007), and d) interpreting solutions (Tsamir & Bazzini, 2004).

Vaiyavutjamai and Clements (2006) conducted a study using 231 ninth-grade students. They found that the students often treated inequalities as equalities and demonstrated confusion of the meaning of solutions to inequalities. These researchers claimed that in lessons on inequalities, instruction was brief, minimal links between algebraic inequalities and their use in daily life were made, and while symbol manipulation was taught, less time was spent discussing the meanings of the symbols being used and of the manipulations.

A similar study by Blanco and Garrote (2007), in which 91 students in their first year of university studies participated after having received instruction on algebra and inequalities, the researchers found that "Many students understood the greater than and less than signs to be a nexus between two algebraic expressions. They then carried this nexus through the various steps in solving an inequality without attaching any meaning to it, even to the point of simply substituting and equals sign" (p. 224). The students' treatment of inequalities as equalities led many to fail to understand the solution they found; they had trouble understanding which values made the inequality true and which ones did not (Blanco & Garrote). Pre-service teachers at the end of their university studies may struggle with these same difficulties, passing their own misunderstandings to their students. Shaughnessy (1992) emphasizes that "we have to first deal with teachers' misconceptions before we can expect them to be competent at helping their students to overcome misconceptions" (p. 484).

According to Usiskin (1996), "If a student does not know how to read mathematics out loud, it is difficult [for him or her] to register the mathematics" (p. 236). Much of students' confusion about inequalities can be attributed to their lack of a complete understanding about the terminology used to describe inequalities-words like "less than," "greater than," "at least," "at most," "no more than," etc. Teachers often hurriedly brush over the terms "more" and "less" as used in a mathematical context (Warren, 2006). Warren conducted a longitudinal study in Queensland, Australia over a period of three years to determine how students were being taught the mathematical language associated with algebraic inequalities. During this time, she analyzed the mathematics textbooks used in the first five grades of primary school and discovered that only five percent of material aimed to develop a mastery and understanding of the concepts of the terms "more" and "less." She developed a framework that outlined students' progression in understanding the terms. Warren claimed that students fail to operate at the highest level of understanding the terms because most classroom instruction focused on arithmetic procedures that involved solving number or word problems with addition and subtraction. These arithmetic lessons with "more" and "less" were not necessarily transferable to working with the same terms and concepts in algebra.

Students also have difficulties with using various representational methods to solve or represent inequalities. Tsamir and Almog (2001) found that students use algebraic manipulations most often when solving inequalities; however, they found that this technique led to the highest rate of incorrect solutions compared to other forms of representation. The study suggested that the choices students made when determining which form of representation to use was based on the classroom techniques the teacher used most often; one method of representation was often used exclusively when solving one particular type of algebraic inequality rather than using various methods (Tsamir & Almog). The authors further suggested that teachers are not representing inequalities to the students in a visual way as often as they might should, especially with the growing availability of graphing calculators. Blanco and Garrote (2007) agreed that students' learning was enhanced when they were introduced to various methods to solve inequalities. The different strategies also helped a greater number of students understand the concept of inequalities. Students may have avoided many of the mistakes made using the algebraic strategy if they were able to see a visual representation of the solution and evaluate its reasonableness. To be able to accurately understand and correct these misconceptions about inequalities held by students, teachers must know how to represent inequalities in various ways.

Teachers also should be aware of the intuitive misconceptions students have about inequalities (Tsamir & Bazzini, 2004). Tsamir and Bazzini conducted a study of 148 high-school students in high-level mathematics courses. All students were given a questionnaire and 21 were selected to interview privately regarding their answers. The researchers concluded that the participants held two intuitive beliefs: the solving process for inequalities and equations is the same and solutions to inequalities cannot be equations but must also be an inequality. To help overcome these wrong intuitions, the Tsamir and Bazzini suggest that teachers be constantly aware of the misconceptions students have and promote student awareness of their misconceptions. However, teachers themselves must be free from misconceptions about inequalities and have a deep understanding of mathematical knowledge or they will lack the insight needed to identify, interpret, and respond to students' misconceptions (Ball, Lubienski & Mewborn, 2001).

References

Ball, D. L., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers' mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed.) (pp. 433-456). New York: Macmillan.

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.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and directions. In D. A. Grouws (Ed.). Handbook of research on mathematics teaching and learning (pp. 465-494). New York: Macmillan.

Tsamir, P., & Almog, N. (2001). Students' strategies and difficulties: The case of algebraic inequalities. International Journal of Mathematical Education in Science and Technology, 32, 513-524.

Tsamir, P., & Bazzini, L. (2004). Consistencies and inconsistencies in students' solution to algebraic 'single-value' inequalities. International Journal of Mathematical Education in Science and Technology, 35, 793-812.

Usiskin, Z. (1996). Mathematics as a language. In P. C. Elliott & M. J. Kenney (Eds.), Communication in mathematics, K-12 and beyond (pp. 231-243). 1996 Yearbook of the National Council of Teachers of Mathematics (NCTM), Reston, VA: NCTM.

Vaiyavutjamai, P., & Clements, M. A. (2006). Effects of classroom instruction on student performance on, understanding of, linear equations and linear inequalities. Mathematical Thinking and Learning, 8, 113-147.

Warren, E. (2006). Comparative mathematical language in the elementary school: A longitudinal study. Educational Studies in Mathematics, 62, 169-189.

Rebecca V. Rowntree

Texas A & M University