## 3. Construct viable arguments and critique the reasoning of others.

How are **second** and **third **grade teachers helping their students construct viable arguments and critique the reasoning of others?

How can students be helped to:

- understand and use stated assumptions, definitions, and previously established results in constructing arguments
- make conjectures and build a logical progression of statements to explore the truth of their conjectures
- analyze situations by breaking them into cases
- recognize and use counterexamples
- justify their conclusions, communicate them to others, and respond to the arguments of others
- reason inductively about data
- make plausible arguments that take into account the context from which the data arose
- compare the effectiveness of two plausible arguments
- distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is

*The CCSS states:*

Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?

Our class in Vermont talked about the need to provide students in grades 3- 5 (the focus of this class) with some helpful language that could guide them in presenting their “argument” and providing one another with feedback. Phrases such as “I propose…” “I found that (this) is (that) because…” can help a student present their thinking. The important part comes in giving and receiving feedback. Language such as “I agree” or “I disagree” along with “Could you clarify?” “Can you demonstrate?” or even “Can you show us proof?” can all help these elementary level students to learn how to have respectful conversations about mathematics.

I believe that one of the most important components to being able to construct a viable argument and critique the reasoning of others is time. In order for students to be able to articulate their argument, they need to be able to share their argument with the class. Likewise, in order to critique the reasoning of others, students need to be able to actively listen to their peers and have the opportunity to question and argue their reasoning. Currently, many of us are locked into pacing guides or math programs that do not allow for any lengthy student discussion, let alone discussion that is not directly related to the math topic that is currently being taught. If students are to develop proficiency in this Practice, they need the time and scaffolding to do it and teachers need to be able to allow time for the rich discussion to happen.

The other benefit of time, in this Practice, is for students to be able to think about the math they are being taught and apply their prior knowledge and real world connections to be able to analyze and reason new learning. It is only when we, as teachers, are not rushed will our students be able to have the rich math discussions and arguments that are inherent in Practice 3.

The key to this is that kids need to be comfortable about having their work critiqued by their peers. That it’s okay to make mistakes and that the rest of the class can still learn from those mistakes. This “trust” that nobody is going to make fun or tease needs to start being built from day 1. Of the examples I’ve seen this week, the most effective lessons are those in which the teacher has backed away from the discussion and let the students defend their reasoning. In one example, a third grader almost had me convinced that 6 was an even number! Letting kids reason, justify and critique their own work and that of their peers creates a much richer lesson than having the teacher just give directions and feedback. It also holds the kids responsible for their opinions and ideas and makes those solutions and ideas relevant and important enough to defend.

Classroom climate is the key to this standard. Students should be comfortable with sharing their ideas and thinking, not afraid of being wrong or having others critique their work. This starts at the beginning of the year, and modeling, practice, practice, guidance, and support should be ongoing throughout the year! In addition, being patient, allowing wait time, being flexible, and allowing discussions to develop rather than obsessively sticking to a planned schedule increases the depth and complexity of the discourse. This also helps to meet the standards of speaking and listening (ELA).

How do you promote a classroom environment where all students can be a participant in discussions?

There will always be one or two students in a classroom that do not have the confidence to share their thinking, even if you tell them their answer is correct.

To help increase confidence, have children share in small groups might not be so overwhelming. As these shy students get more comfortable, confident, sharing groups can get larger.

Having these students begin sharing something with the class that their an expert at may help build confidence.