Mathematics and Argument

Mathematics education has over the past 15 years increasingly elevated argumentation. The National Council of the Teachers of Mathematics established five ‘process standards’ for mathematical proficiency, and two of them (40%) 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’

The Common Core Standards took 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 (Springer Media, 2014) – a step further when in their Standards of Math Practice 3 they require that students demonstrate a proficient ability to ‘construct viable arguments and critique the reasoning of others.’

Deductive vs. Inductive Arguments

The algebraic number sequences and series unit gives us an opportunity to investigate differences in the basic ‘types of reasoning’ recognized by authorities in argument education. One fundamental dichotomy in the field separates deductive arguments and inductive arguments.

As part of a periodic series in The Debatifier on argument and math, today’s post backs up a step from Conor Cameron’s previous post on ‘Sometimes, Always, Never’ questioning in Algebra as a form of argument-making, and examines the broad and now well-established move in K-12 mathematics education toward thinking and reasoning skills at the base of mathematical formulas, procedures, and algorithms. The clear implication of this pedagogical direction is that students should be regularly engaging in argumentation in the classroom, as the articulation and performance of this thinking and reasoning.

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.

By Conor Cameron

The Common Core State Standards demand that students of mathematics, “Construct viable arguments and critique the reasoning of others.” In a vacuum, that requirement probably seems more descriptive of what might occur in an English or social studies classroom. Many math teachers probably think to themselves, “Students may do that sort of work in their Geometry classes, not so much in mine.”

By Yanelly Villegas

Northern Illinois University’s College of Business and Department of Communication were brought together several years ago over the common goal of developing student’s critical thinking skills. The connection nurtured a mutually beneficial relationship for both NIU departments, and all of the educators and students taking part. My involvement in the collaboration has grown throughout its development. It began with providing example debates as a senior undergraduate member of NIU’S debate team; then I provided more extensive support as a graduate intern; and now I am integrally involved with the project as its professional consultant.

One of the signature features of the services model developed and employed by Argument-Centered Education is its embeddedness. Not only is its teacher coaching embedded within schools and active classrooms, so that teachers get observation-based feedback and targeted modeling support, but its curriculum design and adaptation works from curriculum that its partner schools and teachers are currently working with and to which they are committed. Instead of importing argument-based curriculum from outside, we work to build argumentation from the inside of an individual teacher’s, or department’s, or school’s, or district’s in-place instructional content and methodology.

External curricular components can often feel like diversions from the trajectory of a course. They can be and often are tried a couple times then quietly dropped. They can generate understandable even unspoken resistance from educators who entered the profession in part because they have an intellectual passion for certain fields of learning, things they know, and have dedicated their professional lives to sharing with the next generations. And they can impair the effectiveness of an on-going and embedded professional development strategy because they restrict demonstration of the use of argumentation to an external curriculum.