# Using Creative Math Prompts

What is a Creative Math Prompt?

A Creative Math Prompt is an image that contains the seeds of a deep, open-ended math investigation. Students and teachers use the prompts to generate their own observations and questions to explore. Creative Math Prompts develop mathematical creativity along with problem-solving and reasoning skills. They lead to questions around specific mathematical content goals while simultaneously addressing broader goals of mathematical inquiry. The investigations often have no clear endpoint, because there are always more questions to ask! A single prompt may lead to investigations that last days, weeks, or even months.

Why don’t Creative Math Prompts have directions?

Most Creative Math Prompts have few, if any, words! The prompts are designed to help children (and adults!) learn how to ask their own mathematical questions. Some questions may help to clarify the meaning(s) of an image. Others may relate to learning goals that underlie a prompt.

Expect the unexpected! When students create questions that you did not anticipate, follow up on their ideas.

What if my students have trouble thinking of things to notice and wonder?

Be patient. Students will gradually become more comfortable sharing once they realize that the focus is not on right or wrong responses and that you really want to hear their ideas. Accept their initial responses without judgment, either positive or negative. Show a genuine interest in the ideas themselves—all of the ideas! (Some teachers like to leave a copy of a prompt posted for a couple of days in front of the classroom so that students have plenty of time to look at it and think about it before sharing their ideas.)

What are some different ways to use Creative Math Prompts?

Here are three possibilities. You may think of more!

1. Just notice and wonder. The most basic way to use a prompt is simply to show it to your students and ask them to notice and wonder as much as they can. Even if they do nothing more, they will develop skill in observing and questioning, and you will gain valuable insight into their thinking. Even more importantly, students will begin to relax and open up about their thinking as they come to realize that their ideas matter!

2. Build a lesson. You may also use the prompts to help you design lessons. Some Creative Math Prompts would work well in a traditional teacher-focused lesson. The difference lies in how you use it! Instead of trying to explain it to students, begin the lesson by showing students the prompt and asking them to generate their own ideas. The observations and questions that they share make excellent input for formative assessment.

3. Create an investigation. The full power of Creative Math Prompts comes from using them to inspire long-term mathematical investigations. This requires some practice (and maybe a little courage) on your part, but when you begin to see the growth in your students’ (and your own) thinking, you will see that it is well worth the effort!

What might a Creative Math Prompt investigation look like?

A typical investigation will have five parts. All five parts involve noticing and wondering!

1. Begin. Become familiar with the prompt. Observe as much as you can. Ask clarifying questions. Think of one or more first questions to investigate.

2. Explore. Calculate; estimate; gather, organize and visualize information; find, analyze, and extend patterns; make and test predictions, etc.—all with the goal of answering your own question(s). Record your ideas as you work!

3. Create. As you explore, create new diagrams, equations, drawings, designs, patterns, lists, ideas, predictions, stories, tables, graphs, strategies, problems, questions, etc.

4. Reflect. Think back on both the big picture and the details of what you have learned from your investigations.

5. Extend. Think of still more questions to ask (even if you don’t plan to answer them right away)!

The process will usually go in roughly the order above. The Explore and Create parts happen simultaneously, and students may cycle through parts 2 through 5 many times!

How do I plan for such open-ended lessons and investigations?

Begin by selecting a prompt that fits the learning goals you have in mind. Once you find one, become a student for a while; try noticing and wondering. If possible, work with a colleague! As you produce ideas, try to imagine how your students will respond to the prompt, and how you will respond in turn. Plan some ways to gently steer the conversation toward the learning goals. (Remember that there is no need to rush this process). If the prompt you have chosen has supporting notes and comments (which I will complete for all of them eventually), read them and use what you learn to further inform your planning*.

Think about how you want to structure the investigation in your classroom. How will you introduce the prompt? When will students work on the investigation? Who will work together? Will groups stay fixed or will they be fluid depending on changing needs? What will your behavioral and academic expectations for students look like? What will students’ final products look like? How will you assess their work?

*Note: Keep in mind that most of what I write in the notes and comments is geared toward learning goals that I have in mind for the prompt. Your students’ ideas will probably be much more varied and creative than any sample responses that you and I may come up with!

How can I help my students create good mathematical questions?

Your students are likely to do a lot more noticing than wondering at first. Thinking of their own math questions is likely to be new experience for them. Consider giving them a Noticing and Wondering T-chart to record their thoughts.

If the left side fills up faster than the right, ask them to focus on the right for a while.

Realize that some of your students’ questions are likely to be about clarifying the meaning(s) of the prompts. Others may reflect doubts or feelings, such as

Why are we doing …?
What does … have to do with math?
What am I supposed to do?
etc.

These are important questions. You need not always respond immediately, but don’t forget about them! Some questions may resolve themselves as students work.

At some point, students will share mathematical questions that lend themselves to investigation. They may need help with this, especially if Creative Math Prompts are new to them. Mathematicians like to ask questions such as

• Why (or why not)?

• What if...?

• Is that always true?

• Is that ever true?

• How do I know?

• Is there another answer?

• Is there another way to think about it?

• Can I find a pattern?

• Does that make sense?

Share this list with your students, and suggest that they try to come up with a few questions like these. And if they ask these kinds of questions on their own, be sure to point it out!

How can I help my students keep track of their work and their ideas over days or weeks?

I like to have them use something that I call thinking paper. Thinking paper is an alternative to scratch paper. Students use it whenever they have problems that take a long time to solve. Thinking paper is not turned in. It is a safe place for students to record and test ideas, make notes, make mistakes, look for patterns, etc. It only has to be neat and organized enough for them to read. Students may save their thinking paper in a folder and take it out each day to remind themselves of what they have done so far. They may also use it to help them put together their final product when the investigation nears its end, choosing the most important ideas and examples and leaving the rest out.

How can I keep my students focused and engaged throughout a long investigation?

One of the goals of Creative Math Prompts is to help students learn to stay engaged for a long time! They are likely to be excited initially, because they are exploring their own questions rather than the questions from a book. However, they may want to stop working sooner than you would hope. If so, try to figure out why they are losing steam, and respond accordingly.

One thing that may make them want to stop is getting stuck. Because many bright students are accustomed to finding answers quickly and moving on, long math investigations will be a new experience for them. You can head this problem off by making it clear from the very beginning that getting stuck is normal and expected and that you will ask them to push through it when it happens. (For more ideas on this, see the next question.)

Sometimes, students may simply want more variety. It’s always a good idea to have alternate activities handy that they can turn to in times like these. Even then, however, I encourage them to return to the investigation after a short break. Worthwhile mathematical problems take a long time to solve, and I find that students often get a second wind.

Students will also lose focus if they lack clear objectives and expectations. This can be a special challenge when you are dealing with very open-ended tasks. Once you and your students decide on a question or two to investigate, get it down in writing and make the expectations clear. Students are also much more likely to remain engaged over the long haul when you ask them to produce some sort of product that summarizes and explains what they have learned. Typical possibilities include reports, posters, presentations, or videos.

How do I respond when my students get stuck?

Getting stuck or off-track is a normal, even necessary, part of problem-solving. Resist the urge to rescue your students. Teach them to rescue themselves by asking them what they already know, what they still need to know, what they are thinking, what they have tried, what specific questions they have, and what they can learn from any false starts.

Help students find coping strategies such as putting a problem aside and coming back later. Remind them that they are doing something very challenging and that sticking with hard problems is the best way to strengthen their brains!

How can I tell when it is time for students to stop working on an investigation?

My general rule of thumb is to push students beyond what they think they can or want to do—a little at first and more as time goes on. They are almost certainly capable of much more than they think!

Consider how long and hard they have been trying, and assess their chances for making further progress. Struggle should be productive, not pointlessly frustrating. If they have been trying for a long time and are being held up by something small, consider dropping a hint. When you decide that it’s time to stop, acknowledge their progress and effort, and reinforce what they have learned

As the school year goes on, students’ stamina generally increases, and they are able to stay with investigations for a longer time.

How do I respond when I don’t know the answers to students’ questions?

Open-ended problems can be uncomfortable for teachers, because there will always be questions you can’t answer! James Tanton of the Global Math Project has the perfect reply: “be your honest true self in front of students. Wonder about math; be honest if you don’t know the answer and then help students try to find it.”

One of the most powerful things about Creative Math Prompts is that you and your students can learn together. Every time you use a prompt, you gain a deeper understanding of the mathematics behind it by hearing and reflecting on your students’ ideas—professional development at its finest, and it’s built right in to your instructional time!

How do I assess students’ learning on Creative Math Prompts?

Written comments related to students’ thinking are the key. You don’t need to write a lot, but you need to let students know that you are interested in their ideas and that you have thought about them. If you need a numerical score, consider using a rubric like the Assessment Tool on this page for scoring concept-based problems. Ask students to self-assess as well.