In the beginning of this school year, I wrote a post called: In Math Class: To Write is Right. It discusses some thoughts on how we use writing to express ourselves, and how we use the written word in today’s world. I also promised to follow-up with some strategies that I have tried, and want to try, in order to help my students practice writing. Here is that follow-up:
In class, I have incorporated writing in a few small activities. In the beginning of the school year, I asked each student to write down goals for math class for the upcoming year.
Some responses included:
“To get better at math.”
“I want to not fail and work harder.”
“Have fun and enjoy math! 🙂 ”
Responses tended to be shorter, and more phrase-like, especially from my Freshmen classes. Students wrote these responses on Post-It notes, and we displayed them on large posters in the room. This way, students are always able to refer back to their original goals. It holds a certain accountability when they can see their own words written in their own handwriting.
Phrases, however, aren’t enough. I want students to gain a level of sophistication in how they express themselves in math class (appropriate for each individual, of course). So, I have begun to ask more thought-provoking questions about our work throughout the year, asking students to write down their answers using complete sentences. Furthermore, I have tried to make my questions more specific, rather than vague. I have come to realize that specific questions elicit more specific answers, while vaguer questions elicit more vague answers.
Most recently, I asked my Algebra students: “If you check your solution to a system of linear equations, why does the solution work in both equations?” Students know to substitute their coordinate point in each equation to see if it “works,” but I really wanted them to understand why. It also helps bridge the gap between the graphical representation and the algebraic representation of a system of linear equations.
Here are some responses:
“The x value and the y value work in both equations because they are the same in both equations.”
“Because there are three different methods of doing this type of question, and you need to check your work to see if you get this equation/problem right.”
“The solution is the point of intersection of the two lines.”
“It is the same equation written differently.”
“It’s linear, so they have the same slope.”
“Both equations share the same x value and y value.”
“They both have a point in which they cross.”
While some responses are a bit more reasonable than others, I was pleased to see students using decent vocabulary. They were also writing slightly more complex sentences than their phrases from the beginning of the year. It’s clear they have some sense of what the answer means to my question, even if their answers need more fine-tuning.
In my Algebra 2 class, I asked the following question: “Which should always have a higher value: sine or cosecant of the same angle? Why?”
Here are some responses:
“Cosecant, because the hypotenuse is the longest side of the triangle and for cosecant the hypotenuse is on the top.”
“The cosecant should be higher because it is the flipped of sine.”
“Cosecant has to be higher, because the hypotenuse is the longest side of the triangle.”
As we can see, they were trying to express the concept that cosecant is the reciprocal of sine, which brings the hypotenuse, the longest side of the right triangle, to the numerator. They used some good vocabulary, while other vocabulary could be expanded upon (i.e. use “reciprocal” instead of “flipped,” or “numerator” instead of “the top”), but they were definitely trying to describe the differences between sine and cosecant.
As the Math Research director at our school, I advise students throughout the year in researching a topic in advanced mathematics. These students also write a 10-15 page research paper and prepare a presentation for a local competition involving dozens of school districts and over 400 students. This is no easy task. These students struggle to begin their papers, because they have found that they have never written an essay about math before. It also takes some practice interweaving equations and mathematical concepts throughout the paper, without sounding like a textbook. I have worked, and continue to work, to help my students use detailed and clear explanations, while using proper vocabulary to describe the mathematics that they are researching.
The trend between my classes and my math research students is that students feel more comfortable, and are more adept at, writing about math in a straight-forward, fact generating way, rather than truly analyzing the concepts. We could all use a little more practice in this area. Students can definitely learn to write more analytically about mathematics, and teachers can also learn to assign specific writing prompts in order to elicit detailed, more complex responses. I hope to continue to practice helping students become better, more analytical writers in the field of mathematics for years to come.