Educators have been increasingly interested in the transition from arithmetic to algebra, accompanied by an interest in algebraic thinking in the elementary grades. This interest has been amplified because so many students seem to have trouble learning algebra. It is not uncommon to hear from adults that they did fine in mathematics until it became about the last three letters of the alphabet. While some people define algebra as “generalized arithmetic,” it in fact is a very different way of thinking than merely numerical or computational arithmetic. It is a system of logical reasoning. It is a representational system involving manipulation of symbols, not numbers, and a subject of study in mathematics. It is about structures and relationships.
For example, if a student is given the equation 2x + 5 = 15, a good arithmetic student can solve this mentally, not by finding an equivalent equation that isolates the unknown. Perhaps a better strategy would be to ask the student to write the equation that represents the question, “5 more than double a number is 15, what is the number?” This question focuses on the representation of a relationship using a variable, and not on the particular numbers.
The transition from arithmetic to algebra is challenging, but if students are introduced to algebra concepts throughout the elementary grades, that transition is made easier. This can be done first by making sure students have sufficient understanding and fluency with critical topics such as whole number computation, fractions, ratios, and proportionality, and second by explicitly teaching critical algebraic ideas and reasoning. Some of the topics explicitly taught include generalizations, recognizing structures, properties, equivalence, use of variables, and creating visual models to solve algebraic problems.
Mathematics is the study of generalizations—of finding a relationship that applies across a set of objects. It requires looking across objects and cases for similarities, differences, and definable commonalities. It is leading to abstraction, to a stripping of context.
For example, students in the beginning of third grade are asked to consider the effects of adding an even number to an even number, an odd number to an odd number, and an even number to an odd number. After trying several versions of each of these, students are asked if that relationship is always true. Students often will provide additional examples as proof that an even sum results when the two addends are even or when both are odd. The teacher is then suggested to ask why this relationship holds, which forces students to move away from additional examples and instead focus on the meaning of even and odd numbers. What do all even numbers have in common and what do all odd numbers have in common?
The use of manipulatives encourages students to see even numbers as pairs and odd numbers as pairs plus 1. In later grades this will become 2n and 2n + 1. Students will discuss if 0 is an even number, then later in the year, students will look at the multiplication table in the same way and make generalizations about the products of even and odd numbers. They will answer the questions of whether there are more even or odd products on the table and why.
When the multiplication facts are taught as related facts and students learn facts by using these relationships, they are introduced to structures and properties.