Section 1 - Construct Viable Arguments & Critique the Reasoning of Others

Part I And Part II: What is the difference?  

Part I: Facilitator Information

Describe and explain the background and rationale for the math practice of constructing viable arguments and critiquing the reasoning of others. This section breaks down the techniques needed for students to be successful at justifying and critiquing and teaches the skills they could employ to do this in the classroom. Some activities are included in this section for use with students. They include both non-math and math-specific activities. These activities will help the students and facilitator develop their skills with this Math Practice. 

Part I has two sections:

Section I explains Math Practice 3 and provides you with resources to understand Math Practice 3 in more depth. 

Section II helps the facilitator teach the skills of constructing viable arguments and critiquing the reasons of others in more depth. Teaching techniques are taught and students have non-math and some straightforward math examples to practice these skills based on their grade level.

Part II: Student Practice

Provides math-based activities to practice reasoning and critiquing skills at the different grade levels. Students will have learned and practiced this skill in Part I and can now formalize their knowledge in Part II with specific math practice. (Part II) 

Materials Needed

All the materials needed in this toolkit are listed within the individual activities or lessons. Depending on your background knowledge, you may or may not need to reference all of the activities provided.  Additionally, you may want to use these questioning prompts or sentence stems to foster dialogue and structured conversation among students and yourself. 

As you engage with the activities and lessons provided,  you might find that some students prefer to explore the skills with manipulatives and hands-on materials.  They might also enjoy exploring these online, interactive Math tools.

Please note, for all resources and materials provided that do not belong to CDE: Non-Endorsement Statement: Site/s contains information and content supplied by third parties. The information provided here does not constitute endorsement or recommendation by the Colorado Department of Education (CDE) and CDE and its employees are not liable for any improper or incorrect use of third-party information. All information is provided for informative purposes only and does not constitute a legal contract or other covenant or agreement of any kind between CDE and any person or entity unless otherwise expressly specified.

The teaching of mathematics involves not only the teaching of mathematical content (ie: how to add or factor) but also how to best teach our students using different techniques and strategies. Math content is the “what” we teach; math practices are the “how” of what we teach.

This toolkit will focus on the Math Practice Standard of constructing viable arguments and critiquing the reasoning of others. Section I will give you knowledge about the math practice itself. It will explain the practice with videos and readings. Then the toolkit will give you strategies for teaching students how to think about “how to critique with math” and how to “justify with math”. Section II will have some math activities that are designed to practice these evaluative and justifying skills.

The toolkit is set up by the various grade levels. The skills and math concepts chosen to include for the K-12th grade practice activities are meant to not provide overly challenging content so that the facilitator and student can easily flow through a practice activity and focus on the Math Practice Standard.  The first section of the background section is for everyone. The math activities in the second section are meant to focus on MP3.  As a facilitator, if your students are finding the math content too challenging, feel free to choose activities from another grade band.

Background Knowledge For Facilitator

Below we explain the rationale behind this practice and offer techniques and tips to help your students develop this math practice. 

• First, we will explain the rationale behind the practice. 
• Then we will explain the questioning prompts and phrases you can use with students to develop these skills. 
• Lastly, we’ll provide a unique lesson that will practice these skills at each grade band.

The Math Practice Standard 3 has two main parts. 

• The first part is to construct a viable way to discuss and defend problem-solving. Constructing a viable discussion and defense of one’s work is like a plea in a courtroom. The lawyer says their client is innocent or guilty. They make a claim (guilty or innocent) and then present information as to why they are making this claim. Another analogy is an essay. In an essay you write a thesis statement, you claim something and then you justify your claim with evidence. Being able to make a claim and then justify it is math practice 3, constructing a viable argument.
• The second part of Math Practice 3 is to critique the reasoning of others. If we continue with the courtroom example, when you cross-examine the witnesses for the other side, a lawyer is debating and trying to explain why what they said was wrong. This is an example of critiquing the reasoning of others.

If you are interested in reading in more depth about MP 3: 

• This Elevated Group Article, Teach Your Students to Construct Viable Arguments and Critique the Reasoning of Others explains how one might teach the practice of justifying thinking and critiquing the thinking of others to students at specific grade levels. 
• The article, Digging Deeper into SMP 3 – Construct Viable Arguments and Critique the Reasoning of Others is another resource that helps explain how one might craft good arguments. 
• A useful website link, Standard 3: Construct Viable Arguments & Critique the Reasoning of Others, from the Charles A. Dana Center, includes information and videos of what this Math Practice Standard might look like in the classroom.
• Standards for Mathematical Practice is another website that more thoroughly explains math practices and was created by the organization that wrote the math practices, the Common Core. 

All of these articles and websites may provide a greater understanding of math practices, why they are a beneficial and important part of thinking and understanding in math, and how math practices can be used in the classroom.

Student I-Can Statements

This Math Practice Standard is to be taught at all grade levels, but it will look different depending on the age and development of the child. 

Here is the practice explained using kid-friendly vocabulary with “I can…” statements so that you can understand what each piece of the practice standard will look like at each grade band. These are statements that students in this grade should be able to access, understand, and use. 

The techniques listed below are appropriate for use across all grade bands. The depth and complexity with which they are used will depend on the level of math understanding and development of the student. 

• Math Discourse: You can use this Question prompts handout for ideas to use with your students to justify their solutions and responses.
• Model Positive Reasoning Practices: Equally question correct and incorrect solutions shared by students verbally or in writing.
  • Students may interpret being asked to explain their thinking as an indication they’ve made an error. Students must learn they’re expected to share their reasoning always. 
• Connect this Standard to Other Subjects: Use persuasive/argumentative writing models that are used in other subject areas (reading, writing, science, art) in a grade level or school. 
  • Claim-Evidence-Reasoning (CER) writing model can be adapted for math writing. A student’s Claim is their solution to the math problem, their Evidence is any calculations, drawing, or diagram they use to find their solution, and their Reasoning is the thinking work - definitions and strategies - that helps them set up their Evidence.
  • RACE: Restate, Answer, Cite evidence, Explain, and Summarize. A student can restate the problem being asked. A is their answer to the problem. Students citing the evidence are their worked-out examples. The summary is their logical thinking to justify the answer.
• Building Math Identity & Confidence: Students may be resistant to explaining their solutions or thinking when doing math activities.  If so, use the chart below to help you respond to students to foster a safe environment for sharing all ideas, encourage diverse thinking, and collaborate on ideas:

The techniques listed below are appropriate for use across all grade bands. The depth and complexity with which they are used will depend on the level of math understanding and development of the student.

• Activities with Multiple Answers: Introduce and practice this skill with students using low-risk activities where multiple responses or choices can be justified. In math, some problems have multiple right answers and some wrong answers provide opportunities for deeper discussion and learning. Even problems that have a definite right answer can be solved using different strategies. The goal of this practice is not to find the right answer but to practice justifying and explaining thought processes.
• Non-Math Activities: Allow students to practice with non-math activities so they can focus on their critiques. Here are some examples: 
  • Art: Critiquing the composition of a painting or drawing
  • Science: Evaluating the accuracy of conclusions drawn from scientific observations.
  • Sports: Discussing the decision-making process in a particular play or move.
  • Music: Critiquing a musical performance, focusing on aspects like timing, expression, and technique.
• Math Discourse: Provide sentence stems or question prompts to help students talk about math with each other.