# Term Paper: Algebra Lesson Plans and Curriculum

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[. . .] (p. 38).

A large category of algebra misconceptions has been documented in Algebra: Some Common Misconceptions under the category of "The Meaning of Letters." These relate to students' difficulty with the meaning of letter variables. They may completely ignore letters and consider them irrelevant. For example, add 3 to x + 4 and the answer is 7. Or they may think that there are: 1) "rules used to determine which number a letter stands for," 2) specific letters for specific values, 3) a letter can only have one value, 4) a letter has to be a natural number, and 5) forgetting the signs for the variables. These represent a small group of misconceptions surrounding the use of variables in algebraic expressions. None are explicitly addressed in the lessons but all must be understood if correct answers are to be obtained while completing the lessons.

Funmaths.com web site also contains a fifteen item list of common misconceptions experienced in solving algebraic equations entitled Homework Humdingers! Some mirror those listed above and others are unique. But all must be implicitly understood to correctly complete the three lessons under consideration here. Should students show through their completed work (or while the teacher is walking around the room during the assignment period) that they are not falling for these misconceptions. If the instructor notices mistakes being made, this would indicate that a separate lesson specifically dealing with relevant misconceptions is in order. The more important misconceptions which are either implicitly or explicitly covered in the lesson plans are: A) "if k = 4 then 3k + 5 = 39, B) if n = 1 and m = 3 then n squared = 2 and m squared = 6, C) if p = 3 then 2 p squared = 36, D) if g = 5 then 4 + 3 g = 35, E) if 6 x + 9 = 21 then 6 x = 30 and x = 5, F) 2 x - x = 2, G) 2 a + 3 b = 5 a b.

A research paper by Aziz, Pain and Brna discusses two particular misconceptions that are directly addressed in lesson one (Solving Equations) and lesson three (Translating Words into Equations). The focus of the paper is "i) the domain of translating simple relational algebra word problems to algebraic equations [lesson three]; ii) misconceptions related to the reversal error [lesson one]." Unit 16 Algebra: Linear Equations Teaching Notes then echoes some of the misconceptions mentioned above. "Misconceptions abound in algebra as pupils do not stick to the rules; for example, when solving equations, common misconceptions are: x/2 = 4 so 2 = 2, x + 3 = 5 so x = 5 + 3 = 8, x - (-4) = 0 so x = 4.

As a final comment, The American Association for the Advancement of Science (Algebra for All) evaluated twelve algebra textbooks for their Project 2061. The conclusion was that seven were adequate but none of them got a high rating. One of the main failures as it relates to the present discussion revolves around misconceptions. The finding was that "No textbook does a satisfactory job of building on student's existing ideas about algebra of helping them overcome their misconceptions or missing prerequisite knowledge."

Addressing the Transition from Arithmetic to Algebra in the Lesson Plans

The issue of making a transition from arithmetic to algebra is a real one. The web site edhelper.com refers specifically to ideas drawn from the NCTM "that many students have difficulty making the transition from elementary school mathematics to middle school algebra." The site then provides numerous worksheets to aid in this transition. The lesson plans take this transition problem into account by always reserving the possibility that should students need remedial work, it will be made available to them. After all, one of the benefits of the computer/technological approach is the ability to create individualized, self-paced programs to aid those students who need extra help.

One interesting misconception listed on the Mathforum.org website deals with the concept of the negative sign when used with a variable (i.e. -x). The individual posting this item, Dave Marain, correctly points out that this is more of a failure to appreciate an algebraic concept. Students making the transition to algebra from arithmetic could understandably run into this type of situation. Negative numbers as used in arithmetic can be grasped after some practice. But then students have to make a leap to the concept of a negative variable. Lesson one concerning Solving Equations uses this negative variable in the highlighted problem.

David Tall from the University of Warwick states quite well the recognized connection between arithmetic and algebra when he states that: "Algebra is often seen as 'generalized arithmetic' and approached through number patterns...the initial algebra attainment targets [are] a search for pattern before letters are introduced to stand for numbers." His approach also takes into account the realization that there are certain "cognitive difficulties... [in]... The nature of the thinking that the child may bring to algebra from arithmetic." The lesson plans under consideration here do not specifically address these potential "cognitive difficulties." However they are set up in a fashion that these difficulties will be easy to diagnose based both on the worksheets handed in at the end of the lesson as well as an observation by the instructor while the lesson is progressing at each individual computer station. The instructor can use this written and observed feedback to recommend additional steps that would need to be taken to correct these perceived difficulties in translating arithmetic to algebra.

Lesson three addresses one particular aspect of the translation from arithmetic to algebra. According to Tall, "arithmetical thinking shows a spectrum of interpretations of arithmetic symbols." The simple expression 2 + 2 can be described either as adding 2 to 2 (which is a process) or as the sum of two and two (the "concept produced by that process") (Tall, p. 1). This ambiguity may seem like a bad thing but studies show that those who understand the ambiguity and can move back and forth from one to the other are much more successful. Lesson three which deals with the understanding of a how words can be translated into a mathematical expression and then solved involve this ambiguous way of thinking.

Why is this ability so important? Again, turning to Tall: "Children with different attitudes to notation are likely to approach algebra in radically different ways." Students who only think in a procedural way (process) cannot easily translate an expression with a variable because the variable is an unknown quantity. But students who can move in between the ambiguity of expression can see the algebraic situation as a "potential process, which could be carried out when x is known, but is also more likely to be conceived as an object that can be manipulated mentally." (Tall, p. 1). The practice afforded by lesson three will (at the very least) introduce the process by which words can be translated into algebraic expressions which can then be manipulated to arrive at a final numerical answer.

All too often the middle school classroom is run in a way that is counter to that specified in the NCTM Principles and Standards. In the rush to complete paperwork and attend to the other (often) mundane chores which comprise the teaching profession, novel ways of inspiring interest in mathematics through interesting and challenging lesson plans are left behind. The three lesson plans discussed above clearly defy this all-too-common pattern. They not only uphold the NCTM approach but they also address head-on the two fundamental issues of algebra misconception and how thinking should change as students progress from an arithmetic to an algebraic way of thinking.

References

Algebra for All - Not with Today's Textbooks, Says AAAS. (2000). Retrived April 1, 2003, at http://www.prject2061.org/newsinfo/press/r1000426.htm.

Algebra: Some Common Misconceptions. (n.d.). Retrieved April 1, 2003 at http://www.quesnrecit.qc.ca/mst/mapco/pdf/algemisc.pdf.

Aziz, N., Pain, H.G., Brna, P. (1995). Modelling and Mending Students' Misconceptions in Translating Algebra Word PRoblems Using a Belief Revision System in Taps (Abstract).

Presented In the proceedings of the 7th World Conference on Artificial Intelligence in Education, AI-ED 95, Virginia, AACE. Retrieved April 1, 2003 from DAI Database.

Homework Humdingers. (n.d.). Retrieved April 1, 2003 at http://www.funmaths.com / worksheets/downloads/view.htm?ws0100_1.gif.

Making the Transition from Arithmetic to Algebra. (n.d). Retrieved April 2, 2003, at http://www.edhelper.com/arithmetic_to_algebra.htm.

Marain, Dave (2002). Reply to "misconceptions in algebra?

need input from teachers."

Retrieved April 1, 2003 at http://mathforum.org/epigone/math-teach/twirkahpald/

Positive Exponents. Retrieved March 29, 2003, from MTSU WEB SITE:

htp:/ / www.mtsu.edu/~jan2h/Lesson2.htm.

Principles for School Mathematics. (n.d.). Principles and Standards:National Council of Teachers of Mathamatics. Retrieved April 1, 2003 from ttp://standards.nctm.org / document/

Solving Equations. Retrieved March 29, 2003, from MTSU WEB SITE:

htp:/ / www.mtsu.edu/~jan2h/Lesson1.htm.

Tall, David (1992). The Transition from Arithmetic to Algebra:… [END OF PREVIEW]

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