Course Guide - USAID Teacher Education Project

Unit 2 AlgebraWeek 1, Session 2: Algebra as Generalized Arithmetic1. What are the important concepts?a) Algebra is a symbolic way to express what students already know from their workwith arithmetic, number, and operations.b) "x" is not only an "unknown" in equations but also a variable in expressions.c) Equivalent formulas can be expressed in different formats.d) Adults' past experience learning algebra affects their definition of algebra and howit is taught. Addressing these experiences may require "unlearning" images ofalgebra, replacing them with images of broader algebraic thinking rather than justsymbolic notation and algebraic formulas.2. How do children think about these concepts?a) From the article “What Do Students Struggle with When First Introduced toAlgebra Symbols?”: "It was found that only a small percentage of students were ableto consider algebraic letters as generalized numbers or as variables, with the majorityinterpreting letters as specific unknowns."Youngster's formal introduction to algebra usually consists of finding x as theunknown, a particular "answer" for a given equation. However, x is not only anunknown. It can also represent a variable where, for example, there is an infinitenumber of answers for y = x + 1. This distinction between the unknown and thevariable is crucial for pre-service teachers to understand.b) Youngsters think the terms x and y are somehow both mysterious andunchangeable. (I had a school principal once ask, "Why x and y?" To which Iresponded that x and y could be any letters (sometimes n) and that they were purely amathematical convention. I also noted that x usually represents the independentvariable (the input) whereas y usually represents the dependent variable (the output).c) Youngsters usually are not clear about the relationship among equivalentrelationships such as these descriptions for the area of a square. Youngsters tend tosee them as totally different.• Measurement from a numerical formula (the area of a square with a sidelength of 3 can be expressed as 3 x 3 = 9)• A symbolic formula (A = s x s)• The algebraic formula for the area of a square expressed in x and ynotation, y = x 2d) Note the difference between the visual orientation for notation in algebra andarithmetic. In most arithmetic equations, the answer comes "to the right." Instead,most algebraic formulas have the dependent variable y (“the answer”) on the left.