Convince Yourself, Convince a Friend, Convince a Skeptic

I have recently learned a new strategy for students to use as they work in groups called "Convince yourself, convince a friend, convince an enemy" described in Jo Boaler and Cathy Humphreys' book Connecting Mathematical Ideas. However, in order to keep this strategy from developing a negative connotation, Humphreys changed the last portion to "convince a skeptic" instead of using "enemy."

This strategy focuses on encouraging students to reason through problems and justify their arguments. As students work through a particular problem and eventually find a solution they think solves the problem, students must follow three steps. First, the student must convince herself that the solution she obtained is indeed correct. Next, the student must convince her partner (or "friend") of her answer by explaining her reasoning and showing evidence that the answer is correct. Finally, once her partner is convinced, the student must justify her reasoning to a "skeptic." This skeptic is one of her group mates that begins by being skeptical of the answer (whether this student actually is or not). The skeptic's job is to look at the evidence presented for the solution and determine whether this evidence is sufficient in supporting the solution.

By using this in the classroom, less emphasis is on whether the solution is correct, and students can focus on determining whether an explanation of the solution is convincing. I want to test out this strategy one day in my own classroom in order to get students thinking about the reasonableness of an answer and practice delivering and analyzing arguments.

Developing Conceptual Understanding of Fractions

In fall 2013, I began my senior year at Grand Valley State University teacher assisting in a sixth grade classroom. As the semester progressed, I, along with a fellow GVSU student, was given the task of teaching sixth graders how to multiply and divide fractions. This particular mathematical topic tends to be one of the most challenging for students to grasp. I had found that one of the reasons for this was due to the fact that students had a limited conceptual understanding of what fractions actually represented. Thus, as I taught my unit on multiplying and dividing fractions, I found myself constantly reviewing the basic concept of fractions ( which should be learned in earlier grades) before I could even discuss multiplying and dividing them.

This semester at GVSU, I am enrolled in a Teaching Middle Grades Mathematics class. The current topic of discussion is, not surprisingly, multiplying and dividing fractions. While learning strategies to use for teaching multiplying fractions (such as the area model or number line method), we have also been discussing why students seem to have such a difficult time developing this idea of multiplying and dividing fractions. Through many readings and discussions, we have come to the conclusion that one of the largest reasons is that students are usually introduced to the procedures performed on fractions before they gain a conceptual understanding of what fractions are and what is actually happening when these numbers are being added, subtracted, multiplied, and divided. In fact, I have seen this issue of being taught procedurally before conceptually (or solely procedurally) present when teachers begin instruction on many mathematical topics.

Looking back on my teacher assisting experience, I can see this clearly based on how my students approached multiplying and dividing fractions. When I would give students real-world contextual problems, they would automatically apply a procedure (of multiplying or dividing) that they partially remembered from fifth grade before even fully understanding what the problem what asking. Thus, they had no contextual understanding of what they were calculating in regards to the problem, and, because they depended so much on their knowledge of the procedures, when they would obtain an answer that made no sense contextually, never did they once question the soundness of that answer.

This reminds me of a quote by W.W. Sawyer, a mathematician and math educator of the 1900s. He said, "The depressing thing about arithmetic badly taught is that it destroys a child's intellect, and, to some extent, his integrity. Before they are taught arithmetic, children will not give their assent to utter nonsense; afterwards, they will."

I see this time and time again as I work with math students. Before procedures are drilled into their brains, they tend to think and question mathematically. The more procedures become the sole emphasis in the classroom, the less they reason through problems, justify their solutions with sound reasoning, and consider the sensibleness of their answers.

 Nancy Mack, currently a professor at GVSU, wrote an article in 1990 discussing how to develop an understanding of fractions with students. Mia Ong Wenbourne and Andrea Hall, from Drexel University summarize Mack's article, and this summary can be found by clicking on the following hyperlink: http://mathforum.org/sarah/Discussion.Sessions/Mack.html

Ultimately, while the concept of fractions (especially multiplying and dividing them) may be more challenging than other topics, perhaps we are making it more difficult that it has to be. As I continue my education and begin a career as a mathematics educator, I must keep myself from falling into the easy habit of only teaching procedures and thus building a superficial understanding of the content. Whether I am tasked with teaching fractions or not, I can make sure I teach conceptual understanding, the importance of justifying thinking, and checking the reasonableness of answers along with the procedures.