Children begin to reason from their own experiences and create arguments, even mathematical ones, before they enter the classroom as students. As students begin a more formal of study of mathematics, teachers should seek to build on the understandings that students bring to the classroom and move students toward commonly employed techniques for mathematical arguments and more formal language.
Teachers should seek to develop and maintain a classroom culture that supports and nurtures students as they develop their logical thinking and reasoning skills. Students’ early mathematical reasoning often relies on pattern recognition, generalization from examples, and classification; therefore, teachers should provide students with ample opportunities for students to use these skills as they make conjectures, search for evidence, and explain and justify ideas. For example, the teacher may ask, “If we skip count by twos, will we ever get a number that ends in 5?” Students can offer conjectures about the answer and study the pattern of numbers created when skip counting by twos. When they reach conclusions, the teacher can ask them to explain their thinking and critique the explanations of their classmates. Students need such opportunities on a regular basis in order to develop the ability to create and examine mathematical arguments.
The arguments that students make should move from a basis in perception, through empirical grounds, and finally toward more abstract foundations. For example, a teacher could provide students with a drawing of a rectangle and a square of different dimensions and ask students to compare their sizes. Initially, students may decide by only looking at the shapes. A more sophisticated empirical approach would be to cut out the shapes and physically compare sizes. Ultimately, students could be encouraged to reason more abstractly by using the dimensions of the shapes and methods for calculating the areas of the shapes.
Young students are capable of learning about acceptable and inappropriate techniques in mathematical arguments. Through opportunities to discuss rich problems, students can learn that a single example does not provide a general justification, but a single counterexample demonstrates that an argument is incorrect. They also need to understand that mathematical arguments are not based solely on an outside authority, for example, accepting a claim as true because the teacher said it was, even if they do not understand it. They should eventually be guided toward using mathematical properties and relationships as the bases for their arguments.
As students have opportunities to develop understanding of acceptable mathematical arguments, the community of learners also needs to learn about the importance of appropriate vocabulary and build a shared understanding of terms to use in their arguments. For example, if some students do not understand what odd numbers are, then they will not be able to successfully understand, create, or critique arguments about odd numbers. While mathematical words need to be described accurately, they should be described in ways that are accessible to all students in the classroom in order to support shared understandings of the words. Teachers should model precise language use and support students as they learn appropriate terms and their place in mathematical argument.