I am of the generation that took algebra in my freshman year at high school. Everybody took algebra. It was the only math class for freshmen at my consolidated learning place. About half the class of ’77 were in one hour, the other half in class the other. Small school.

The teacher was Mrs Allen. She was at least 102 years old. The school board had been trying to get her to retire for over forty years and had probably given up on the idea at least ten before I sat in her classroom. Having class with Mrs Allen was like having class with your mean-spirited great-great-grandma. While she looked sweet on the outside, her smile was always ironic and her attitude towards teaching was strict and old-school. No fooling around in Mrs Allen’s class.

Once we all got the basic “rules” down during the first semester, the second semester was basically solving problems. Mrs Allen would spend the first five or so minutes of class chalking problems onto the lower half of the black board (she couldn’t reach the upper half) and then instruct us to solve them. She would then sit at her desk and doze off. Yes, she actually went to sleep.

We, of course, took advantage of this and “worked together” with our neighbor, or the person behind us, to make sure that we understood and solved the problem correctly. No one wanted to suffer Mrs Allen’s silent, ironic smile when she pointed out just where we had gone wrong with the figures.

After a quarter of an hour or so, she would suddenly open her eyes, look confusedly about the class, figure out where she was, and call on different students to come up to the board and show his or her figures and the final answer to each of the problems. She would either subtly praise us or ironically curl her lips.

The last trimester of the year, we had a student teacher, can’t remember his name, let’s call him Mr B. Mrs Allen silently observed from her desk as Mr B tried to convince us that we could understand “higher algebra”. One particular problem, I remember, he called it a “booger”, language Mrs Allen considered inappropriate both for a class setting as well as for talking about algebra.

One day Mrs Allen spent the class period in the teacher’s lounge, leaving us alone with young Mr B. He put a complex problem on the board and then told us to work on it with our neighbors. When one student mildly protested that Mr Allen always expected us to figure out the problems on our own, he let us in on a little secret that Mrs Allen had shared with him, a secret that had even deepened his own basic understanding of mathematics.

Mrs Allen never really dozed off, she was just pretending. She came from a generation of math teachers who in general believed that individuals should be able to learn and use math without the help from others. Each student should be evaluated upon their individual merits. Mrs Allen, though, believed that math was a unique language that communicated thoughts in a way that spoken language could not. She felt it important that her students orally practiced this language with one another, learning to use the language to communicate beyond the basic rules that old-school teaching had always insisted were more important.

This anecdote came back to me years later, as I was conversing with an adult student in Barcelona. Cristina was a working mathematician and we were discussing the similarities and differences between math and linguistics. I mentioned that one of the reasons I enjoyed linguistics so much was because I often got emotional when looking at language.

Here, Cristina pulled out one of her workbooks from work and showed me a page that was half-covered with numbers and symbols. She asked me if I knew what the graphic images meant. I, of course, did not, though I could identify some of the basic traits that I had learned years before with Mrs Allen. Cristina explained that she could not translate into English what the formula meant, that’s why it was written in a formula. As she ran her fingers through the lines, she suddenly stopped at a particular symbol and told me that right here, on reading this line, she became choked up with emotion.

The emotion came from two sides. One was the emotion she felt because she knew that she understood what the symbols were communicating. She understood the language on that page. The other was the emotion she felt because she not only understood the language, she understood what was being communicated, the concept behind and within the language. Though she could not put it into words, she could put it into symbols.

The “lesson” here, then, is the difference between learning the rules of any language and the actual use of that language to communicate. We can teach only so much grammar and structure to our students. On the other hand, we should try to get our students emotionally worked up over their understanding of those rules and how that understanding leads them to understanding how a speaker of the language uses those words to communicate thought, and how that thought can be understood. That is an important emotional connection that is sometimes missing from our language classes.