Building Problem-Solving Habits With Desmos and Fermi Problems

One thing that puzzled me for many years was how my students would sometimes arrive at a solution to a math problem that just couldn’t possibly make sense in context, but they would fail to notice this inconsistency that could have helped them identify an error in their work.

The root of these illogical solutions was more than just a lack of attention to detail or not checking over work. It started with whether students could come up with a reasonable estimate of the “neighborhood” of values that would make sense for the solution, followed by an interpretation of their answer in context.

Fostering Real-World Problem-Solving Habits

There’s often a noticeable gap between the problem-solving skills used in the math classroom and the skills that are needed to solve complex problems in the professional and scientific world. Real-world problems don’t just demand recall of an algorithm or procedure but also require a range of problem-solving habits.

When students are first gaining confidence with a mathematical concept or procedure, it’s typical to allow them to practice the skill multiple times in isolation before adding in the complexity of application or other content. I realized that my students needed this same type of streamlined practice with problem-solving skills. Below are the habits I identified that would be most important to focus on.

• Estimation
• Determining relevant information
• Interpreting the reasonableness of a solution

An idea for how to build these specific habits clicked for me one day when I encountered the Desmos Activity Builder Estimation starter screen within the Classroom Data Collection. I realized that this screen could play a powerful role in engaging my students in Fermi problems. A Fermi problem asks students to estimate the answer to a question without knowing all the data or specific information needed to precisely calculate it. In some cases, it may not even be possible to know the true solution. I was excited to create a routine in my classroom for students to attempt these problems because they are naturally engaging and have access points for all learners.

The simplest answer to a Fermi problem could be just guessing a number that seems like it’s not too low or too high. Students can then evaluate the range of all possible answers in the classroom to determine which of those numbers are reasonable and which ones are not, potentially enabling them to seek out information to make a better estimate. This discussion leads to students refining their thinking and reevaluating whether their own solution is reasonable.

More Tips for Using Desmos

In my classroom, the Fermi problem routine is part of the warm-up and follows the below steps.

1. Students enter a response to the Fermi problem in the Classroom Data Collection screen for Estimation.
2. Other student answers populate in the form of a box plot on their screen, and students can adjust their response as needed for one to two minutes.
3. There is an answer box for students to explain their thinking to encourage them to reflect on their estimation process.
4. Once the Desmos activity is paused, students engage in partner discussion about which responses they believe are reasonable.
5. Student discussion leads to consensus about which responses are reasonable and what information or calculations are useful in determining an estimate. This may include students asking for or looking up relevant information to refine the estimate.
6. Students reflect on their original estimate and whether they would change it now given the additional information from the discussion. 

The Desmos Classroom Data Collection Screen for Estimation is a key part of this routine because it makes it easy to collect student responses and for students to see the range of different answers within the classroom. It displays student response values in a box plot, and as students repeat this routine throughout the year, they gain familiarity with reading the graph as well.

One additional edit I make to the question is to include a response box for students to explain the thought process for how they came up with their estimate. The goal is for students to improve at this routine throughout the year, so students who begin the year by typing something like “I guessed” might move on to describing information they used or possibly a calculation they made in later attempts. 

The contexts of Fermi problems can vary widely. One of my favorite Fermi problems is how many ping-pong balls can fit inside our classroom. It usually doesn’t take long for students to ask for the dimensions of a ping-pong ball. Others involve more realistic scenarios, like estimating how many cups of coffee Starbucks sells in Seattle in a single day, which might bring up statistical ideas involving averages and variation or rates. 

The criteria I use when creating these problems are below.

• There is not one right answer, but there is a range of reasonable answers.
•  The information needed to develop an estimate is either likely to be known to students or easily estimated based on their experiences.
• The context is relatively familiar to students. 
• The context is somewhat interesting, surprising, or useful—this is subjective, but we all know our students and what might inspire or interest them!
• It is not easy to look up the answer.* 

*It’s important to note that Google and particularly AI will certainly attempt to come up with answers to these problems, and those answers are just as interesting to evaluate because of the assumptions that they depend on, but part of the purpose of this routine is to illuminate the many different estimates that you can arrive at, depending on which information you choose to use. 

After we used this routine regularly, the improvement in my students’ ability to approach complex problem-solving tasks was noticeable. Students built stronger habits around checking the reasonableness of a solution and interpreting the solution to a problem in context.

Perhaps the biggest impact, though, was that the routine made problem-solving fun and interesting. The open-ended nature of the Fermi problems, as well as the ability to add fun and unique contexts to learning, make these problems one of my students’ favorite parts of class.