Processes of Thought: Math and Latin

The phrase "puer vocatus est" means "the boy was called" in English. This is the common or simple translation of the sentence. "Vocatus est" is called the predicate, but is actually the combination of the perfect passive participle and a form of the verb "esse", "to be". For students who have reached this level in Latin, it is likely the first time that they have encountered a predicate made up of two or more words. Unlike English, most Latin predicates, with the exception of these perfect passives, are single words, verbs. Thus, the verb ambulabam signifies any of the following English expressions: I was walking, I did walk, I walked, I used to walk, and I kept walking. Where English adds words to make verb phrases, Latin changes endings and verb forms.

"I was walking" is truly a sentence, composed of a personal pronoun, a form of the verb "to be", and the present participle of the verb "walk". We grasp the meaning being expressed in English, if it is our native tongue. But the beginning Latin student, who has spent several months unlearning this way of thinking, has great difficulty understanding the function of "vocatus est".

A way to clarify this concept is to introduce the linear versus FOIL methods of multiplying binomials. A similar act of cognition is occurring. In the FOIL method, a shortcut, which eliminates the need to write the equation in a vertical fashion, lets the student achieve a quick and immediate answer, yet they miss "the stuff in the middle", according to a former student. For example:

(x+3) (x+2) using foil produces: (x²+2x+3x+6), which equals (x²+5x+6). Though this works, and a little math is required to combine 2x and 3x, an automaton could be taught to do this ad infinitum. When the equation is reconsidered, it helps the student to make connections, and have a grounded understanding of the function:

(x+2)

X (x+3)

3x+6

x²+2x

x²+5X+6

Similarly, a hyper-literal translation of the perfect passive participle helps ground the student in the concept of how Latin uses the phrase to produce the passive voice. "Vocatus" can be taken to mean "in the state of having been called" or "having been called", and "est" is the verb "is", so "the boy is in the state of having been called" or "the boy is having been called". Neither is a normal English translation, but the student is aware of the precise mode of thinking required in the target language. Common usage produces "the boy was called", which collapses the "est" and the "vocatus" into the English past perfect: "is" + "having been" = "was". The confusion of "est" in this phrase with the simple present tense of this verb is always eliminated with the hyper-literal translation. When the individual student becomes comfortable with this concept, he or she is then free to skip to common usage.

In the same way, FOIL, which represents common usage, should be taught only after the vertical method. Problems which become more complicated in the FOIL method are rendered simply by the linear. Students should become grounded in the way that shows the "stuff in the middle".