“It won’t work every time”: Prospective elementary teachers’ counterexamples for students’ false arguments about fractions

In today’s elementary mathematics classroom, students are urged to construct arguments. If this is to enhance students’ learning, teachers must be able to identify and refute students’ false arguments. This requires substantial knowledge, yet little research has examined the nature of this knowledge with prospective elementary teachers. We asked 17 prospective teachers to assess the validity of students’ arguments regarding the comparison of fractions and to refute those that were false using counterexamples. Teachers did well with the mathematical aspects of this task, successfully identifying false arguments and refuting them with correct counterexamples. The pedagogical aspects of the task were more challenging, as only one counterexample explained why an argument was false and counterexamples were hampered at times by distractors. We propose that teacher educators emphasize pedagogical considerations in preparing prospective elementary teachers for such work. However, which considerations to emphasize requires additional research examining elementary students’ reactions to counterexamples. • Prospective elementary teachers assessed students’ arguments about fractions. • Teachers successfully identified and refuted false arguments with counterexamples. • Few counterexamples explained why an argument was false. • At times, counterexamples were hampered by potential distractors or “noise”. • Which pedagogical considerations matter most for elementary students is unknown.