Students need to build procifiency between tables, graphs, symbolic rules and verbal rules. Understanding the connections between these representations is essential for building competency and flixbility with algebraic expressions.
IN this art of the border problem experiement, they intentionally skipped the stage of using a t-table to show the growth pattern.
The objective they are trying to get to is to represent the pattern symboliclly.
The most common misconception about variables is that letters are initials or labels. In other words, choosing to use b as the variable to represent the number of squares in the border made too strong a connection between the word border and the variable. This lead to the conclusion that it was important to emphasize that b represented the number of squares in the border but it could have just as easily been x or alpha. Another common misconception is that different letters in an expression cannot represent the same number or that changing a letter somehow changes the solution.
Students have the perception that the equals sign is "Get the answer" symbol but when working in algebra, finding the equation is often the answer that is sought.
Verbal representations of the pattern or relationships is critical for understanding. This step comes before writing the generalized, algebraic expression.
It's encouraging that even Jo Boaler has students tuning out sometimes when she's going deep into extracting exactly what she wants from their understanding.
There can be a tendency for students in their early understanding of algebra to replace all the numbers with variables rather than just the unknowns.
It's important for the students to construct their own learning of the algebraic relationships. Relationships can not be taught. Relationships are constructed, or made, be each individual.
Phrasing the prompting as "what does joe need to know to figure out the number of unit squares in the border". In this problem, "let x represent the number of unit squares on one side of the equation.
Prompt with the question of "what is staying the same in the equation and what is changing?".