Instead of "Show Your Thinking..." Try This!

As educators, our intentions are always rooted in fostering understanding and depth in our students' learning experiences. Often, we prompt children to "show their thinking" or "show their work," hoping to gain insight into their problem-solving processes. Despite our efforts to scaffold this task by encouraging the use of pictures, numbers, and words, it can be disheartening when these prompts fall short, resulting in mere compliance rather than genuine engagement with mathematical thinking.

This realization prompted me to question the purpose behind asking children to show their work. Reflecting on this, several key points emerged:

• Understanding Thought Processes: One of the primary reasons for asking students to show their work is to gain insight into how they approach problem-solving. It serves as a window into their reasoning, thus allowing us to better support their development.
• Authentic Audience: Who is the intended audience for students' mathematical thinking? Is it solely me, the teacher, or can students also consider their peers as their audience? If we aim to cultivate an authentic mathematical community in the classroom, students should view their classmates as fellow mathematicians. After all, mathematicians collaborate to prove and refine new ideas and theories.

With this deeper understanding of the why behind the concept of "showing your work," my understanding has evolved into a notion of providing proof. As Catherine Fosnot aptly puts it, "Questioning, defending, justifying, and proving are all processes characteristic of human activity." (Conferring with Young Mathematicians at Work Making Moments Matter) Children naturally gravitate towards establishing certainty and convincing others of their ideas, which not only boosts their confidence and math identity, it also nurtures a deeper understanding of mathematics. Incorporating Jo Boaler's insights, particularly her emphasis on convincing oneself, a friend, and even a skeptic, further solidified this approach. Proof, in this context, becomes a meticulously constructed argument supported by sound reasoning and evidence.

Here is how I have moved from the idea of ‘Show your work’ to the idea of proof:

• Setting the Stage: Introduce the concept of "proof" to students, eliciting their understanding and ideas. Use Adrienne Gear’s One Word task: Proof. Start with the word Proof written in the middle of the board. Ask the children “what is proof?” and record their thinking.
• Gallery Walk: Display the problem-solving papers from a previous day's task and encourage students to look at them for proof, using sticky notes to mark examples of what they think or see as strong proof.
• Shared Reflection: Gather students to discuss their observations and insights from the gallery walk. What did they notice about proof? Encourage them to refine their understanding of proof based on the examples they've encountered. Add some new ideas of proof to the original thoughts in your Setting the Stage discussion.
• Class Discussion: Select a few examples to analyze as a class, highlighting why they demonstrate strong proof.
• Consistent Reinforcement: Continuously reinforce the importance of proof over several days, encouraging students to identify and share examples of proof with one another. You can also consider using feedback and compliments to scaffold this reinforcement. 

Throughout the school year, it's crucial to consider the audience for students' mathematical work and provide opportunities for authentic engagement, such as through gallery walks, peer feedback sessions and sharing time.

In the words of Jo Boaler,

"Mathematicians prove conjectures by reasoning—talking about why methods are chosen, how they work, and how they link to one another."

By fostering a culture of reasoning and collaboration, we empower students to become confident mathematicians who can effectively communicate and defend their ideas.

Some guiding questions to help facilitate this idea include:

• How do you know that?
• Can you prove that to me/us? 
• Why did you solve it that way?
• I'm not convinced, yet. Tell me more?

Through these practices, we not only deepen students' mathematical understanding but also cultivate a community where mathematical thinking is valued, shared, and celebrated.