In this chapter, you will learn
- What argumentation is and what it is not
- How to use a four-part model of argumentation: generating cases, conjecturing, justifying, and concluding
- About argumentation as a social process
- Why teaching is disciplined improvisation and how improvisation supports argumentation, norm setting, and student engagement
- Steps for introducing argumentation in your mathematics classroom
- About argumentation in lessons and argumentation lessons
- How to share new ideas for teaching mathematical argumentation in working together with your colleagues
Argumentation Is Important!
There are things that we need to communicate in everyday life, especially in the society in which we find ourselves now with all kinds of complexities. If we could just step back and think critically about it, then we should be able to come up with some kind of solution to the problems we face. Thatâs why I just think that math argumentation is so great, not only for education, but so that you will be able to function as a human being and a citizen in this society.
âSeventh-grade mathematics teacher
Thatâs what a middle school mathematics teacher, one of our workshop participants, had to say about the potential for mathematical argumentation to make an impact outside of the classroom. What if students can use the same kind of reasoning to solve problems in their lives as they do to, say, establish that the sum of two odd numbers is an even number? As we consider our studentsâ futures, the kind of careful reasoning that they do together in classroom mathematical argumentation is an important 21st century workplace and life skill. Making logical connections among abstract ideas and interacting with others to clarify their ideas are both deemed necessary in an increasing number of good jobs (Partnership for 21st Century Skills, 2008).
And then there are current mathematics standards. You probably know about the emphasis on mathematical practices or processes in most current state standards, including the practices that students âconstruct viable arguments and critique the reasoning of othersâ (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010) and âjustify mathematical ideas and arguments using precise mathematical language in written or oral communicationâ (Texas Education Agency, 2012). The ability to make sense of the story mathematics tells, construct a viable argument about that story, justify oneâs reasoning, and critique the reasoning of others are essential skills in almost every line of work and in citizen participation.
In addition to these practical considerations, we believe that the practice of mathematical argumentation is the most important of the mathematical practices because it is the fundamental way in which mathematicians communicate with each other. The search for mathematical truth is ongoing, as mathematicians create new ideas and justify them and as students reason together in a classroom.
Furthermore, access to mathematical argumentation is an equity issue. Every student should have access to this high-level disciplinary practice. Providing this access in elementary and middle school puts students on a path to higher level mathematics in high school and college. Current research indicates that about a third of the difference in mathematics achievement between students of color and white students, and between students from low- and high-income families, is attributable to the opportunity to learn high-level mathematics that they are given in class (Schmidt, Burroughs, Zoido, & Houang, 2015). The techniques in this book provide practices that support equitable access to high-level mathematics.
Although argumentation is serious business, itâs also true that engaging in argumentation can make your mathematics classroom more joyful. Students get to play with mathematical ideas and take ownership of them in a way that often delights them. Youâll most likely feel a boost yourself. One participant in our early workshops proclaimed that every Friday was argumentation day, and her class eagerly looked forward to it. While we advocate including argumentation most days, not just Fridays, we appreciated the spirit of her designation and found it a positive step in her own professional development.
The approach, techniques, and activities in this book were developed while working with teachers in a variety of settings. We have worked, in particular, with teachers in urban schools with high proportions of youth of color and students from low-income families. We have also worked in schools in more affluent communities. Teachers in all of these settings have used these methods to bring argumentation to their classrooms. Additionally, we have worked with teachers of students who receive special education services and students who are English Language Learners, and they have found that their students, too, can participate in mathematical argumentation. The vignettes and examples we present are informed by what we learned from these teachers but do not represent any one teacher.
What Argumentation Isâand Is Not
To understand what mathematical argumentation is, it is important to understand what it is not. We can contrast it with other mathematical practices and processes. Take, for example, a graph of distance as a function of time, as shown in Figure 1.1. It can be the starting place for lessons on problem solving, modeling, or argumentation, depending on the prompt that goes with the graph. A problem-solving prompt, for example, is âCreate a trip with three segments that ends at 200 feet.â It calls for a solution, carefully reasoned but not necessarily an argument. On the other hand, a prompt that calls for argumentation goes like this: âRaj says that if one line is steeper than another, then it represents a faster motion. Is this always true?â Notice the question, âIs this always true?â In this book, we help you develop a repertoire of ways to use that simple question, among others, to engage your students in building arguments throughout the school year.
Figure 1.1 Time Versus Position Graph
A second question asks for argumentation: âHow do we know it is true?â In this question, the focus is on a public demonstration of why a statement is true or false. The onus is on students to come up with an argument that is convincing to others. This press for truth is key in fostering argumentation that takes place among students so that students not only construct arguments but also critique each otherâs reasoning.
This approach to argumentation positions it as a social practiceâwhat we engage in to find out the truth together (Thurston, 1998). For example, you can tell students that the area of a parallelogram is calculated by multiplying the lengths of the base and height. What if students multiply the base, height, and the other side length to find the area? You could simply tell them this is wrong. But it is more powerful for students to explain to each other why multiplying these three numbers together does not make sense, calling on the concept of area as a measure of two-dimensional space.
Students will likely also need help understanding what argumentation is and what it is not. They may bring their own notions of what an argument isâfor example, a fightâand it will take some work to help them develop a new way of thinking about argumentation as a mathematical practice, as a way of reasoning together about the truth. Weâll have more to say about classroom norms for argumentation in the following chapters, but this may be the most fundamental norm of all: We are finding out the truth together.
A Four-Part Model of Argumentation
Our model of argumentation is based on what mathematicians do and what philosophers and educators have posited as parts of argumentation. We distilled the expertsâ (e.g., Harel & Sowder, 1998, 2007; Krummheuer, 1995; Lakatos, 1976) views on argumentation into a structure that works for teachers and students just beginning with argumentation as well as for those with more experience. The model has four parts:
- Generating casesâcreating something to argue about
- Conjecturingâmaking bold claims
- Justifyingâbuilding a...
