Teaching Structure in Algebra

In the observation of August Grossman’ journal —in 1924, teacher from St. Louis, describes that pupils who study algebra,  from the first year in high school up to college with almost comical regularity mistakes in the subject cancelation of fraction. Pupils are apt to cancel any two numbers or expressions that are same, no matter in what context. Such as,

In Ethan M. Merlin teaching, he have developed that address the missed “essence of the matter” of algebraic transformation. He use “glue” in expressions, “trees” in drawing expressions, and describing “subexpresssion addresses”.

Which Operation is The Glue?

This operation want to show an algebraic expression containing multiple operations and to name that operation that glues the expression together. Glue is the student’s way to see how order of operation creates the very architecture of an algebraic expression. For example, x + 3y , the addition in this operation is the glue.

Diagramming Expressions As Trees

Tree diagrams are powerful tool for helping students see the multileveled structure of algebraic expressions. The first step, we identify the gluing operation and write it an circle. Next, we draw branches from gluing operation, one branch for each piece that is glued together.

Where does A Subexpression Live?

Students need vocabulary to name parts of an expression according how they are put together, especially factor and term. We can use the analogy of street address. When we addressing a letter, we start with the most specific details of recipient’s location and expand  outward from street, to city, to state. The same outward progression applies when describing the structural location of a particular subexpression.

Tackling “Strange Mistakes” In simplifying Fractions

Many student can better handle the classic symbol manipulation work of algebra when equipped with tools for analyzing expression structure. And to determine whether or not an expressionis a factor of each, a student must be able to see whether one thing is a factor of another in general. Students who can already identify gluing operations, create expression trees, and name the address of a subexpression are better equipped to answer the question, Is x a factor of y?

Teaching The Essence Of The Matter

Algebra is powerful because it permits us to work directly on general expressions, a skill that is primarily gained through direct practice. Students must eventually be able to transform algebraic expressions structurally according to the rules of the game. A deep conceptual understanding on students can’t be genuine fluency in memeorization approach. In any case, once students have mastered these skills, they are delighted to have become technically proficient enough to truly understand the structure of the algebraic expressions with which they are working.