Mathematics is constructed on a foundation of logical reasoning, and the National Council of the Teachers of Mathematics (NCTM) has been calling for an elevation of reasoning and argumentation in math education since at least 2000. Formal logic and the mathematical proof share an origin story, and the most influential figure in argument studies over the past 60 years, Stephen Toulmin (creator of the ‘Claim – Data/Evidence – Warrant/Reasoning’ argument model), had as his primary objective to expand the role and influence of informal logic.
Mathematical argument – ‘a line of reasoning that intends to show or explain why a mathematical result is true,’ according to the Encyclopedia of Mathematics Education (London: Springer Media, 2014) – is now recognized as an essential component of rigorous, college-directed 6th – 12th grade mathematics. Of the five NCTM process standards, two are closely connected to argument: ‘reasoning and proof’ and ‘communications.’ As part of mathematical ‘reasoning and proof,’ students must be able to:
- ‘Develop and evaluate mathematical arguments and proofs’
- ‘Select and use various types of reasoning and proofs’
As part of mathematical ‘communications,’ students must be able to:
- ‘Analyze and evaluate the mathematical thinking and strategies of others’
- ‘Explanations should include mathematical arguments and rationales, not just procedural descriptions or summaries’
And in its Standards of Math Practice 3 – ‘Construct viable arguments and critique the reasoning of others’ – the Common Core codified this argumentative move. Elaboration of this standard makes clear the deep commitment to mathematical argument that leading-edge education thinkers, theorists, and administrators have.
Mathematically proficient students . . . justify their conclusions, communicate them to others, and respond to the arguments of others. Mathematically proficient students are able to compare the effectiveness of two plausible arguments, distinguish correct reasoning from that which is flawed, and – if there is a flaw in an argument explain what it is.
Moving far beyond mastering a particular calculation technique or building competence with formulae and functions, to be proficient in math, 21st century middle and high school students must be able to ‘generate arguments’ and ‘evaluate arguments.’ Students will use empirical, pre-formal, and formal reasoning – all common in a wide range of argument production and practice.
Even states that have rejected the Common Core have retained the prominent place of argument in their new math standards. Indiana, for example, has in its new math standards ‘Construct viable arguments,’ in which students are required to ‘read the arguments of others’ and ‘improve their arguments’ by critiquing their reasoning.
High-performing schools and leading math educators recognize that mathematics argumentation is borne of the need to make generalizations, establish definitions and properties, identify relationships, and create proofs. The top math teachers have students critiquing (i.e., refuting) flawed reasoning, and evaluating arguments is given significant emphasis. Students must analyze the implications of a claim, recognize argumentative method, and determine the validity of warrants or reasons for solutions, approaches, techniques, thinking.
Argument-Centered Education works with the teachers of mathematics to bring these deeply challenging but at the same time professionally exciting and expanding trends and directions in pedagogy into their classrooms and into their own professional skill sets and capacities.