Using Problem-Solving to Build Community From Day 1

“Find a 10-digit number whose first digit represents the number of zeros in the number, the second digit represents the number of ones, and so on, until the 10th digit represents the number of nines.”

If this problem is scary, start with an easier version: “Find a two-digit number where the first number represents the number of zeros and the second digit represents the number of ones.” With fewer options, guess-and-check feels doable. Try the same problem for a three-digit number. Look for patterns. Can you solve the original puzzle now?

This is one of several problems that I give my high school math students on the very first day of school. While it has nothing to do with the calculus or statistics content that we’ll explore together, it sets the stage for the classroom community that we will cocreate throughout the year.

The First Day

The vibe of the first day sets the tone for the year. Starting with a review of the syllabus, going over classroom rules, and practicing routines and procedures might send a message like “This is an orderly space where compliance is valued.” Diving right into content might signal that “in this class the teacher dictates what is important to know.” 

I strive for a classroom filled with curiosity and collaboration. When students trust me and trust each other, they are far more likely to allow themselves to be vulnerable, ask questions, and take on difficult challenges, all keys to success in math.

The goals for my first day are: building community, getting to know each other, inciting excitement about math, and setting the desired tone for the year. When we kick things off by solving problems together, we are set on a path toward achieving these ideals.

The Activity

I carefully curate the puzzles beforehand. None of them require specific mathematical skills beyond addition and subtraction, which levels the playing field and gives more students the opportunity to feel successful.

Next, I prepare manipulatives and materials that help the learners visualize the puzzles and tackle each one from a variety of modalities. For example, in the problem at the top, I would be prepared to offer the same hints that I provided to you. I would also have little slips of paper with at least 10 copies of each digit between 0 and 9 so that the students could move them around, facilitating guess and check. I’ve also provided things like small cups, cotton balls, string, paper, and scissors.

In class, I use a random number generator to create groups of three. The students start by introducing themselves to their teammates, usually by sharing three numbers that represent something important to their identity. Afterward, I give each group a different logic puzzle. As they work, I provide scaffolding, hints, and manipulatives as needed.

After solving a puzzle, they get a new one to try. We dedicate time at the end of class to discuss and appreciate all the different methods used to solve each problem.

Why It Works

There are so many reasons that I keep coming back to this activity. It achieves all of the goals that I strive for on the first day.

Because the students are in small groups and the problems are challenging enough that they can’t be done alone, they start building collaborative relationships with each other right off the bat. When we come back together as a large group to share the wide range of solutions, the students feel validated and appreciated. All of this helps us build community.

The activity helps me get to know the students as learners. As I observe, I get a glimpse into how they think, respond to challenges, and interact with each other. Some students spend a chunk of time quietly thinking before they write or say anything. Others start to scribble immediately or jibber-jabber nonstop until some fruitful idea emerges. Learning these patterns helps me tailor my support to each student throughout the year.

Problem-solving also helps incite excitement about math. The manipulatives and scaffolding provide many pathways to success, so that the students actually have fun puzzling. Afterward, they are excited to come to class the next day.

The activity sets the tone for our year together. It gives my students a chance to start using their critical thinking skills right away and start feeling comfortable tackling challenging tasks. Because the puzzles don’t require much prior math knowledge, the activity eases math anxiety.

The students leave class knowing that this is a place where they have support from me and their classmates, where everybody’s thoughts and questions are valued, where experimentation and creativity are more important to mathematical success than computation, and where they can overcome difficult challenges.