When today’s elementary school students learn about fractions, they are sometimes asked to explain how they reason, for example, that one fraction is greater than another. By constructing their own arguments to explain how they came to a particular mathematical conclusion, they take on more agency in their own learning.

“We know from research that when students have opportunities to devise their own approaches [in math], they tend to come to view math as more of a creative, open discipline, where they have some voice,” Michael Jarry-Shore, assistant professor of elementary education at North Carolina State University and McGill graduate, said in an interview with The Tribune.

Of course, with this approach to learning, students are bound to sometimes create false arguments. Teachers, then, must be able to identify and refute such arguments. One way to accomplish this is by using a counterexample, as it can quickly expose where a student’s reasoning has gone wrong and clarify what a more general rule might be.

In a recent study published in The Journal of Mathematical Behavior, Jarry-Shore and his collaborators assessed what kind of knowledge this style of learning would demand of prospective elementary school teachers. They were interested in how prospective teachers would go about employing counterexamples to refute students’ false arguments in the context of fractions. 

Jarry-Shore worked with participants enrolled in a teaching program at a Canadian university, having them assess five arguments from imaginary students as they explained their reasoning when working through fraction problem sets. Two of the five arguments were false. For example, imaginary Student A stated, “When you compare them, 7/8 is greater than 6/9 because 7 is greater than 6.” While it is, of course, true that seven is greater than six, this fact is not sufficient to determine which of the two fractions is larger, as the values of both the numerator and the denominator must be taken into account.

The researchers found that participants easily identified which students made false arguments, a skill that relies on a participant’s subject matter knowledge—that is, how well the prospective teachers understand the mathematical concepts of fractions. However, participants struggled with pedagogical knowledge, which refers to the ability to explain concepts effectively to children. This was evidenced by the quality of their counterexamples.

“[It was pretty rare] to present, for example, a more generic counterexample that didn’t just consist of one pair of fractions that successfully refute [the student’s argument], but that maybe suggested a mechanism to follow in developing more and more counter examples, and thus a wealth of kind of disconfirming evidence [….] We didn’t see [any general] counterexamples,” Jarry-Shore explained.

Furthermore, while participants often produced counterexamples that successfully refuted an imaginary student’s false argument, they would occasionally mismatch the mathematical complexity of the counterexample with the student’s original fractions. For example, in response to imaginary Student A’s argument, one participant brought forth 6/8 and 5/6 as a counterexample, a case in which the fraction with the larger numerator is, in fact, the smaller of the two fractions. However, determining which of the two is larger in this scenario requires a number of additional calculations.

“It could kind of diminish the power of the counterexample because [the student might think], ‘Look, this is supposed to help me see that this argument is false, but we just went down this whole other path of equivalent fractions and inverse relationships,’ and so on and so forth,” Jarry-Shore said. “So it can kind of, you know, take away from the point of the counterexample.”

After observing these difficulties, the researchers developed a framework that would allow prospective elementary school teachers to produce counterexamples that were not excessively complex—among other metrics—but remained convincing to students. The framework would guide prospective teachers to evaluate the quality of their counterexample based on a number of factors, ultimately revealing how well the counterexample is likely to resonate with a student. Jarry-Shore noted that the effectiveness of these counterexamples would need to be empirically validated with actual elementary school students.

 
Ultimately, this study reminds us that teacher education must work to foster the development of both content knowledge and pedagogical knowledge, which requires collaboration and research among students, teachers, and teacher-educators alike.