If you look at nearly any set of math practice standards, you will find something related to the need for solid mathematical communication and reasoning. When the Common Core outlined the Standards for Mathematical Practice in 2010, they articulated this in Mathematical Practice #3: Construct viable arguments and critique the reasoning of others.
This is the fourth post in our Mathematical Practices series. To see the previous three posts, check out the links below.
While our past posts have focused on standards that are hard to interpret or just not well understood, here we’ll look at one of the most popular (and important!) standards of them all. Let’s dig in.
Unpacking MP3
MP3 reminds us that math is not about answer getting but about reasoning and communicating that reasoning. Jo Boaler (2019) defines reasoning as “talking about why methods are chosen, how they work, and how they link to one another, describing the logical connections between them.” Reasoning and constructing arguments is at the heart of mathematics. When a student justifies their thinking process or gives a rationale for their strategy, they are practicing MP3.
When engaging in MP3, students are
Explaining their strategies
Justifying conclusions
Exploring the validity of their conjectures
Using a logical progression of statements to build an argument
Asking clarifying questions to understand and improve someone else’s argument
Deciding whether an argument makes sense
Identifying a flaw in an argument
Providing a counterexample
Comparing the effectiveness of two arguments
When students construct an argument, they are essentially answering the question “Why is this true?” or “Why does this work?”. It is not about finding the answer, it is about being able to explain why that answer is valid and sensical, based on diagrams, pictures, verbal descriptions, algebraic or numeric methods.
You don’t need special tasks to help students practice constructing arguments and evaluating the claims of others. Even a calculation problem like 42x15 can turn into a reasoning problem by asking students to prove that 42x15=630 in at least three different ways.
At the high school level, this might sound like:
Determine if the two events are independent and explain your reasoning.
Prove that ∆ABC is congruent to ∆DEF.
What is the smallest number of imaginary zeros this polynomial can have? Give a reason for your answer.
Find the number of dots in the n th figure and prove that your rule works for all n.
You may have noted that there is nothing inherently special about these prompts. They simply ask students to explain and reason, rather than just recall or calculate.
Arguments range in formality
Not only does the mathematics range in formality, but the types of arguments students should be expected to make are equally varied. Explaining one’s reasoning and making arguments should be the bread and butter of any math class. But when students are exploring topics for the first time, their reasoning will be more informal and intuitive. They don’t yet have the tools for rigorous arguments—they are barely convinced of the new concept themselves! This is the stage of conjecturing and exploring and should be matched with friendly, open questions that are not intimidating to fragile ideas. As the body of concepts and skills that students have around a topic grows, we can help students move toward more sophisticated arguments. While we often focus on formal proofs like 2-column proofs or flowchart proofs in Geometry, students must know that this is not the only way to be convincing. A “proof without words” or a clever counterexample can be just as persuasive.
Here are some prompts you can use to help students construct arguments, from very informal to very formal:
How do you know?
Explain.
Give a reason for your answer.
Why does that work?
Provide a rationale.
Convince a friend that…
Convince the class that…
Convince a skeptic that…
Show that your rule always works.
Justify your answer.
Prove that you are correct.
Write a (paragraph, flowchart, 2-column, inductive, etc.) proof to show that…
Refining arguments
Making arguments is inherent to almost all good mathematics problems. This is because good tasks require problem solving, which incorporates reasoning and strategic thinking. Modifying tasks to include more reasoning and argument building can be as simple as adding “Explain” to the end of the prompt. But it doesn’t end there. Students need opportunities to practice building, sharing, revising, and refining their arguments. Just because a student is asked to explain does not ensure that their argument is viable. This is where the second part of MP3 comes in.
In order for students to build solid, viable arguments, they need a chance to share an initial argument and get feedback on it. This doesn’t have to be anything formal. Students sharing their ideas in a small group will inevitably lead to some level of feedback from others in the group. “I have no idea what you just said” is feedback that the argument is not clear or organized enough to be accessible to others. “But what about…?” is feedback that the argument doesn’t work for all cases, and needs to be refined. “That won’t work” is feedback that either the problem itself was misunderstood or the proposed solution does not meet all criteria posed by the problem.
Building a culture conducive to sharing and critiquing ideas
How (and if) students talk to each other about their ideas is dependent on the classroom culture you have in place. If math class is all about getting answers, being fast, or earning all the points, then that competitive and socially risky environment will impede students from sharing their ideas, let alone accepting feedback from others on their ideas. If, on the other hand, math class is a place of exploration and trying out ideas, then students will feel comfortable offering a strategy even if they are not yet convinced for themselves it will work. If the expectation is that we build on and revise each other’s ideas, then when a peer says they disagree or point out a flaw in the argument, the student’s self-esteem isn’t crushed. Unless all students are convinced that math class is a place to workshop ideas, they will not share their contributions.
Discussing the ideas of others should be modeled in a whole-class context where the teacher facilitates the conversation. That way the teacher can ensure that interactions are respectful, focused on the mathematics, not the person, and honoring of diverse contributions. A skillful teacher is able to select approaches worth discussing as a class, identify the important mathematics in each approach, and help students develop their thinking through careful questioning.
Before asking students to evaluate the argument of a classmate, they must first understand it. This is why I will always remember the teaching move of a master teacher I got to observe during my undergraduate years.
“Without saying if you agree or disagree, who can restate _____’s idea?”
Ensuring that students understand a classmate’s idea (by paraphrasing, restating, or summarizing it) before evaluating its strengths and weaknesses is a pivotal piece of MP3. Clarifying questions can be used to allow others to follow the logical flow of a student’s approach and also to ensure that the student’s idea is represented as intended, not how it might be interpreted by others, or “cleaned up” by the teacher.
Once the student’s idea or approach is understood, it can be assessed and evaluated. This can be particularly effective if multiple approaches are being compared.
Here are some questions you can use when discussing multiple student approaches side-by-side:
What is the same in both approaches? What is different in the approaches?
What are the strengths of this approach? Are there any gaps or flaws in this approach?
Will this approach always work?
Which approach is easier to explain? Which approach is easier to use?
Is one approach more efficient than the other?
Is one approach more sophisticated than the other?
Which approach makes more sense to you? Why?
5 Quick Tips
Are you looking to make MP3 a more integral part of your math classroom? Here are a few practical ideas to help you get started.
Provide sentence stems to help students analyze each others’ arguments (ex: “What did you mean by…?” “Something I found convincing in your argument is…” “Something I did not find convincing is…” “One way your argument could be improved is by…” )
Instead of asking students to find an answer to a problem, state the prompt as “Convince me that…” and then give both the problem and the solution. For some sample prompts, click here.
Have students practice an argument or line of reasoning with a partner before sharing with the whole class.
Use Cathy Humphrey’s framework of “Convince yourself”, “Convince a friend”, and “Convince a skeptic” to help students understand various levels of justification
Use the group role of “skeptic” to ensure that students are regularly practicing giving and critiquing arguments even in everyday math tasks. The skeptic asks questions like “Why?” “Can you prove it?” and “How do you know?”.