Prior to starting a lesson on algebraic expressions, I administered a pre-assessment as a means of assessing background knowledge. Using the document camera and the LCD projector, I wrote five different algebraic expressions on the board. Two of the expressions involved addition, “n + 5” and “4 + p,” two more involved subtraction, “10 – m” and “x – 12,” and the last expression was a multi-step expression involving both addition and subtraction “(x + 10) – 5.” I instructed the students to copy all five expressions down on a piece of paper and write as many verbal phrases as they could to represent the expressions.
On the first two expressions involving addition, all the students used the word “plus” correctly and grammatically. For the expression, “n + 5,” they wrote “n plus 5” as opposed to “5 plus n,” maintaining the syntactic order of the factors. Most of the other phrases given were generally correct, but they were not grammatical. For example, one student wrote, “n added to 5,” for “n + 5.” Because mathematics is read from left to right, the factor “5” would have to be added to the factor “n.” Thus, the student should have written “5 added to n.” Another phrase that was written was, “5 increased by n.” Even though “increased by” is a correct phrase to use to represent addition, it should have been written, “n increased by 5.” Again, the syntactic roles of the factors were switched. Interestingly, the students that used “added to” and “increased by,” wrote “added to” first before writing “increased by.” Based on the syntactic order of both phrases, “increased by” follows the same order as “plus” (i.e. “n plus 5” and “n increased by 5”), while “added to” reverses the order of the factors. This could have caused the students to reconsider the order of the factors when using the phrase, “increased by.” Their confusion could have also originated with the use of prepositions. The phrases “added to” and “increased by” both possess prepositions, and are thereby classified as verbal phrases. If the students made this connection, they would have reversed the syntactic roles of the factors using the phrase, “increased by.” It could also be that the commutative nature of addition implies a sense of bi-directionality in the syntactic order of the phrase given. Thus, if “n + 5” is equal to “5 + n,” then “n increased by 5” should be equal to “5 increased by n.”
The students’ phrases for “10 – m” and “x – 12” were much the same as those involving addition. All the students used the word “minus” correctly and grammatically. About half of the students wrote the phrases “10 subtracted by m” and “10 decreased by m” correctly and grammatically. The rest of the students wrote “10 subtracted from m” and “10 take away m.” All four phrases have maintained the order of the factors, “10” and “m”; however, this order does not maintain the grammaticality of the expression, “10 – m.” This strict adherence to the order of the factors probably stems from the lack of commutativity of subtraction. Still, a phrase like “subtracted from” requires the factors to be used in reverse order. For example, “10 – m” should have been written as “m subtracted from 10.” Even though the phrases “subtracted by” and “subtracted from” include the same verb “subtract,” their prepositions require different syntax. The preposition “by” implies agency. Thus, the factor being subtracted (i.e. “m”) should follow the preposition “by” (i.e. “by m”). On the contrary, the preposition “from” implies movement, with the origin of that movement (i.e. “10”) following the preposition (i.e. “from 10”), and the agent of that movement (i.e. “m”) prefixing the verb (i.e. “m subtracted”), resulting in the phrase, “m subtracted from 10.” Without a strong background in syntax and the use of certain prepositions, it seems that the students are ordering the interaction between conflicting constraints. First, they maintain the lack of commutativity of subtraction by upholding the order of the factors used in the algebraic expression. Second, they correctly write verbal phrases using transitive verbs with an expressed subject and direct object (i.e. “10 minus m”). Finally, if a preposition follows a verb, they uphold the first two constraints at the risk of forming ungrammatical phrases.
The results from this pre-assessment revealed a disjunction between the students’ understanding of algebraic expressions and the linguistic structure of phrases that represent these expressions. If anything, the rubric designed to evaluate the students responses failed to include the grammaticality of the phrases provided by students. Instead, it focused on the students’ ability to represent algebraic expressions using correct verbal phrases without any consideration for the syntactic order of the phrases. Based on this, I would revise the rubric to include grammaticality (see revised rubric below).