By Chris Bolognese

The Scottish Book

In the early 1930s, local mathematicians gathered in a small café in a town in Poland to converse about their latest mathematical conjectures. Now renowned for their contributions to analysis and topology, Ulam, Mazur, and Banach would meet at this hangout regularly, known as the Scottish Café. Since chalkboards were not available, they would use the marble table tops to scrawl mathematical formulas, theorems, and conjectures, only for their work to be erased each night. Banach’s wife suggested they use a large ledger book as a record, not only of their discoveries, but most notably, their unsolved problems. Soon other mathematicians would come visit the café with the aspiration of interacting with The Scottish Book. Perhaps they could solve one of the problems thereby receiving a prize from the original poser or even add their own problem that they could not solve. After years of use, The Scottish Book was filled with close to 200 problems by many different authors all in the pursuit of posing and solving problems as a community. This collective knowledge of research-level mathematics was so exalted that the Book was buried deep in the ground to protect it from being destroyed as the threat of World War II reared. These mathematicians developed a way to collaborate and communicate by helping one another solve conjectures of those in their community. The goal was not a grade or one of the offered prizes (such as brandy or even a live goose), but rather, to have a space to wonder about and contribute generatively to mathematics. What parallels can we make between this story and the teaching and learning of mathematics now?

Motivated by this story, I purchased a book of blank pages from Amazon for my students to write their questions. I eagerly told my high school Geometry students about The Scottish Book. Although it took some time, students started to contribute their own mathematical wonderings. Some wondered about connections between Geometry and music, others about different types of transformations or alternative definitions. Students perused these queries and responded either with an answer or new questions of their own. For a number of years, students in different classes have been using our Scottish Book to record their own mathematical conjectures. One student went on to write up an article about graphing shapes with equations, which was published in a journal.

The Practice of Problem-Posing

The introduction of the Scottish Book in my classes five years ago has encouraged me to continue to reflect on how we can better position students not just as problem solvers but as problem posers. Our students can go through their entire compulsory education without having the opportunity to pose their own problems. Instead, the problems they are asked to solve are almost always provided to them by some other authority, such as their teacher, a worksheet, or their textbook (Silver, 1994). Jeremy Kilpatrick (1987) states, “[T]he experience of discovering and creating one’s own mathematics problems ought to be part of every student’s education” (p. 123). The purpose of this blog post is to share how we can all empower our students to engage in the practice of generating their own mathematical questions and solving them as a classroom community. In engaging in this work, not only do students feel empowered to create and negotiate their own mathematics, but they share in the authority and authorship of the classroom.

Problem Posing Examples

Asking Questions

How do you get started with having students pose their own problems? A good first step is just to practice posing mathematical questions without the goal of answering them. Providing students with an interesting context allows them to make observations and pose queries. For example, I recently showed Calculus students a video of me filling up my water bottle and asked them to pose what they noticed and wondered. Some screenshots from the video and some student-generated questions are below. The questions highlight key mathematical ideas including rates of change, units, and accumulation, all paramount to the study of Calculus. What picture, video, or context could you provide to encourage students to practice posing questions?

Student questions about a water bottle task

Formative assessment

A few times during each unit, my students are asked to pose their own problem and show how to solve it. Often I don’t provide any restrictions or constraints, but you might impose some, such as:

• Write and solve your own system of equations that has a solution in the third quadrant.

• Write a problem about vectors whose answer is a vector with magnitude between 3 and 4.

• Write a problem where one strategy is more advantageous than another. Why is that strategy better?

I have found that something as simple as asking students to create their own problems related to our current unit of study is constructive for many reasons. Consider the three student-generated problems below, all from a Calculus course.

Omar’s Problem

Viewership for the third season of the show Twin Peaks decreased continuously after every episode at a rate directly proportional to the number of viewers present. The first episode of the season brought in 620,000 viewers while the 8th episode brought in just 250,000. Write a differential equation modeling this scenario, its general solution as well as its particular solution.

These problems showcase so many things. For one, problems like Omar’s and Andrea’s allow mathematics to bridge a conversation to better know my students’ personal interests. Such problems also give students the creativity to personalize mathematics to be meaningful to their own life experiences. Omar’s problem struck up a conversation about our favorite episodes and characters. Andrea’s problem got us talking about our pets and their idiosyncrasies. In what ways can having your students author their own problems allow you to know them not just as mathematics students but as human beings?

Problems like Cole’s showcase students’ prior knowledge (such as log and exponential properties from a prior class) as well as extensions by trying new riffs on problems already solved. Each student-generated problem initiates an opportunity for students to conceptualize mathematics in their own terms. Moreover, it sparks a dialogue for feedback, revisions, and extensions, whereby my students are encouraged to treat each problem-posing assignment as a rough draft (Jansen, 2020). They get to measure their own understanding and get actionable feedback as to how to revise their problem or solution moving their thinking forward. These problems can be used as daily warm-up problems or lesson closers with the student author as the facilitator.

Summative assessment

Student-generated problems can also be leveraged for an end of unit test or project. Each of our tests is a compilation of problems that students have written with attribution to the author before each problem. Students light up with excitement when they see one of their own problems featured. I have also created a portfolio-style assessment where students can choose which problems they solve from each category; however, each problem selected must be by a different person and they cannot solve their own problems. An example is included below.

Using problem-posing as a vehicle for alternative assessments builds classroom community because students can see the mathematical interests and thinking of others. Unlike traditional tests, the portfolio example also provides students choice and autonomy to select problems that they know they can solve.

Conclusion

Problem-posing is essential to exploring and creating mathematics. Having students write their own problems is far from a new pedagogical idea. Brown and Walter’s (1990) “what if not” approach to posing new questions or Polya’s (1945) method of looking back to create a new problem from one solved provide a canon of which to build upon. However, problem-posing still is a rare experience for many K-12 and post-secondary students. It is my hope that this post gives you the confidence to support and position your students as problem-posers. Please add and tag me @eulersnephew on Twitter if this post resonates with you.

To close, my student, Gurby, said it best: “I really enjoy writing my own problems! It gives me an opportunity to look over the problems we reviewed in class and imagine the process of finding the solution. Sometimes I think we, as students, prioritize finding the answer, which limits us from considering why problems are written the way they are and what they’re trying to get us to do, if that makes sense. I feel like problem-posing opens a new window.”

References

Brown, S. I., & Walter, M. I. (1990). The art of problem posing (Second). Lawrence Erlbaum Associates, Inc., Publishers.

Jansen, A. (2020). Rough draft math: Revising to learn. Stenhouse Publishers.

Kilpatrick, J. (1987). Problem formulating: Where do good problems come from. Cognitive science and mathematics education, 123-147.

Polya, G. (1945). How to solve it. Princeton University Press.

Silver, E. (1994). On mathematical problem posing. For the Learning of Mathematics. 14. 19-28.

https://web.archive.org/web/20180619155008/http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka1pol.pdf (Accessed February 20, 2022.)

Chris Bolognese is the George S. McElroy honorary chair of mathematics at The Columbus Academy in Ohio where he teaches upper school mathematics and computer science. Chris is a Presidential Award for Excellence in Mathematics and Science Teaching state finalist and enjoys facilitating math teachers’ circles.