Too often students come into math class ready to abandon all writing and communication skills. For many years, students were able to succeed in school by simply computing fluently and memorizing algorithms. Now, with the Common Core curriculum, students and teachers alike are finding that this is no longer sufficient. How then, do we encourage students to see beyond algorithms and focus on the mathematics behind them so that they can flourish as mathematicians, as well as students?
Pushing students to discuss their work with their peers and build justification for their strategies (as well as their solutions) is key to this success. Not only does communicating their mathematical ideas deepen students’ understanding of the content, but it also develops listening and speaking skills. By encouraging students to partake in group work, we are simultaneously encouraging them to share their ideas as well as listen and respond to the ideas of their peers. Hearing strategies and justification of classmates allows students to think about what makes sense and what makes a strong argument.
From these discussions, students are more able to think through their own ideas thoroughly and provide written justification for their work. This helps students move toward writing formal proofs. At first, there is often pushback from students about having to write about mathematics. We have to help students past convincing themselves that they are right (something most students do immediately) and towards convincing others. By having students read through the written justification of others, they can determine whether or not they find the argument convincing. The phrase “convince yourself, convince a friend, convince a skeptic” can become a tool for students to measure how strong an argument is. Students who are struggling to understand can also “play the skeptic” without fear of being judged by other students.
Unfortunately, students cannot jump into justification without these skills being taught, modeled, and practiced. There are four ways to introduce these concepts to students and help them become comfortable with justification.
- Require them to use specific language. When a student shared an idea, have them state “My conjecture is ________________, and my justification is __________________”. Using this language consistently will get students in the habit of always following up their ideas with their reasoning.
- Use logic puzzles. Logic is one of the foundations of mathematics, but students don’t always see that. Working on logic puzzles increases students’ problem-solving skills and organically forces them to discuss their reasoning.
- Use a listening game called “He/She Said, I Say.” In this game, students respond to a prompt in their notebooks. The response should be about a sentence. For example, “the best thing I did over vacation was go swimming with my cousins”. Starting this game with something not related to math increases student comfort. Once students have completed the prompt, the first person stands up and says, “I say, the best thing I did over vacation was go swimming with my cousins”. This student then chooses a student to go next. The next student stands and says “He says the best thing I did over vacation was go swimming with my cousins, and I say, the best thing I did over vacation was ride a roller coaster at the amusement park”. The game continues until all students have shared.
- A prompt called “Always, Sometimes, Never”. In this, students are given a statement and asked to decide if it is true always, sometimes or never. Students can use examples, diagrams, etc. to justify their response. An example of a statement would be “When you cut off a portion of a shape, you decrease the area and the perimeter.” Again, it may be helpful for students to start these prompts with something less mathematical until they are more comfortable with the task.
Using these four tasks in the classroom consistently will help students build their confidence in sharing ideas as well as crafting powerful arguments.
Conjecture + Justification = Argument.
Elizabeth Masalsky is a 6th, 7th, and 8th grade mathematics teacher at Battery Park City School in Manhattan. She has a post-baccalaureate in mathematics from Brandeis University and her master’s in secondary math education from Bard College. Elizabeth is a Math for America Master Teacher and continues her professional development through workshops with Math for America, Metamorphosis and Math in the City.