Mathematical Argumentation - How it can help in class?
Students need to have a variety of experiences with a topic to have the best opportunity to fully understand that specific topic. Developing activities that are structured around the eight mathematical practices will ensure students will have a variety of experiences to develop their understanding of a topic. Teachers who develop and use activities that incorporate mathematical argumentation will infuse Mathematical Practice 3 (construct viable arguments and critique the reasoning of others) into the structure of their classrooms.
Math argumentation was a topic in the November/December 2018 NCTM's Mathematics Teacher issue. The authors Graham and Lesseig talked about multiple activities that require students to justify, defend, and discuss math at a deeper level. I am going to just focus on one of these activities for you to use in your classroom.
Always, Sometimes, or Never True questions will provide students with an opportunity to think deeper, discuss their reasoning, and defend their thinking. During this type of activity, students will be required to provide justification to support their answer.
Here are some examples of Always, Sometimes, and Never True questions:
Adding two even numbers will result in an even number.
Adding an even number and an odd number will result in an odd number.
Multiplying two numbers together will produce a larger number.
Dividing a number in half will result in a smaller number.
The graph of y=mx+b is a line that will pass through three quadrants.
The graph of a parabola will pass through three quadrants.
The square root of a number is half of the original number.
Adding two negative integers will result in a negative integer.
Subtracting two negative integers will result in a negative integer.
a+b result in a positive integer.
As students determine if the statement is always, sometimes, or never true, they are required to support their reasoning. Students can work in small groups and/or individually on these tasks.
If you use small groups, individual students need to have private work time to think about their answer and justification of their thinking about the statement prior to discussing it with their group. Next, students need to discuss their justification with others. This is a great time for the students to listen and utilize Accountable Talk stems to question or confirm a student's justification and reasoning. Through the discussion of the group a common justification should emerge, and the group needs to create a display on a poster, large sheet of paper, or whiteboard to present their work publicly. The use of a gallery walk provides time for students to read and view other's thoughts about the statement. At this time students can use post-it notes to make comments, ask clarifying questions, and provide additional information to each group in the class. After the gallery walk, the groups need time to edit their reasoning based on the comments and questions from the class. Lastly, the students will present their justification to the class.
During class multiple discussions will occur and these discussions will provide opportunities for students to question or affirm their own reasoning over the topic. Journal writing about the process of how their thoughts changed or became solidified during the process will be an important aspect of a student's understanding of their own learning.
If you use this type of question as an individual activity, a test question, or a formative assessment, you will need to provide detailed feedback to students to correct misconceptions or affirm their reasoning. Students can edit their justifications and hand their corrections to you so you can provide additional feedback. Shaping student's thinking through the editing process will allow for a deeper understanding of the topic. This type of individual activity can be differentiated for any level of student. Discussions and feedback lead to higher expectations for all levels of math students and a deeper understanding of math concepts.
In this student example, the student did get the correct answer, but they did not demonstrate a full understanding of the concept. After providing feedback, the student was able to adjust their justification to demonstrate their full understanding of the concept of adding a negative number. A teacher's expectations of a the product of a student can be raised through the use of feedback. Through this activity other students demonstrated their understanding by drawing counters or positive and negative signs. This type of open-ended question gives students multiple pathways to demonstrate their understanding of the problem or statement.
We spend so much time in math grading math problems that are correct or incorrect; this will provide a time for you to develop a student's thought process about math in a deeper way.
