Many years ago I taught a course for non-math majors (MAT102) at SUNY Oswego, a required course for majors other than math, science, business or computer science. One of the topics was Euler paths and circuits. For a youtube© link on Euler paths and circuits go to htttps://www.youtube.com/watch?v=REfC1-igKHQ. The Euler Circuit requires an even number of vertices. So what? I think this topic can be taught to Kindergarten to Grade 16 students without very much additional content preparation.

The content lends itself to real-world examples, such as a mail delivery route or creating a public transportation system, where students could use mathematical language and thinking. Students could discuss the number of paths to get from one location to another, and in the discussion use mathematical language and reasoning.

For example, I spent almost every day, in Budapest, using the public transportation system. With Budapest’s system, you can purchase a 7-day pass for 4950 forints (approximately $17.00) and travel anywhere on Budapest’s public transportation system. One day, to figure out how to get around Margaret Island, I kept on taking the bus back and forth on the Island. What a great way to make your very own tourist style HOP ON, HOP OFF. For a while, I kept on seeing the same bus driver following his route. He never said anything each time I showed the pass.

I used the public transportation system for all the school’s visits. Each of the journeys included some combination of walking, biking, trams, buses, or subway. (I never took a taxi in Budapest.) By far, my favorite mode of transit was the tram. Tram’s con, next to the electricity and run even during rainstorms. Often when I used my GPS on my mobile phone, multiple routes were provided. My technology allowed me to remember Euler.

Using a “problem-solving approach” to math teaching is difficult (as mentioned in a previous post), but the benefits of using real-world problems or theoretical problems that ignite creative thinking outweigh the challenges. Providing students with engaging problems, where students employ a democratic process for thinking, engaging with one another, and solving a problem together is at the heart of real math learning. I posit that engaging students in mathematical problem solving require a particular protocol of teacher behaviors (Sawada, 2002). Teacher behaviors should support the following student actions. Students should want to solve the problem; The solution is not obvious; There is more than one way to arrive at the answer; The problem is interesting enough that students want to find a solution; The student is unable to proceed directly to the solution; The solution requires the use of mathematical ideas.

Teaching for democracy, therefore, creates a space for citizens (our students) who have problem-solving abilities in other areas. At stake is whether we continue to teach in a way that covers the material, to students who see mathematics as a place for only negative experiences, or do we work to help students view mathematics as an opportunity to build democratic processes. Students who have problem-based experiences in math may be more confident problem solvers in general.

One of the teachers I met (I will call her Maria) is about to release her first group of students she has been working with for the last 3-4 years. (In many Hungarian schools, a group (their class) of students is assigned a single math teacher, four years. So students keep the same math teacher for grade 9, 10, 11, and 12. These students have the benefit of having a teacher who knows their mathematics strengths and weaknesses. The teacher can scaffold the learning differently, and in many cases provide the differentiation needed for each student to build their math potential.

I witnessed these particular problems, play out in two of my observations, grade 2, and grade 7 classes.

** Grade 2 problem in a Roman Numerals lesson**:

*How do you change one stick in the equation to make the statement true?*

X I + V = V

V + II = V

V – II = VII

V – I = IX

(At the end of one of class a 2^{nd} grade girl went up to the teacher with one of the solutions. She had been given time with her peer to find a solution. Students also completed this mathematical exercise using sticks and the teacher’s role was completely different.)

** Grade 7 problems related to an Exponents lesson**:

*Problem 1:*

For homework students were asked to create the largest possible number using five 2’s.

*Which of the following numbers (brought by the students) is the largest?*

Problem 2:

a) Which number is larger, and why?

b) What is the __last digit__ of the number. Explain?

The students in each of these classes (grade 2 and grade 7) worked in different modes, alone, collaboratively, and as a full class. The teachers did not have organized systems for the grouping expect for grouped seating. I observed student facilitation of learning as teaching.

These theoretical problems, respond to teacher objections about not having relevant enough applied problems to try the problem posing/solving approach to teaching. These theoretical problems are not contrived and are not necessarily connected to the current curriculum being taught. Theoretical problems can move student thinking forward and allow access to mathematical knowledge and processes. I witnessed democracy in action in my observation.

A ride on public transportation informs my thinking about applied problems and theoretical problems. Thank you BKK (Budapest Public Transportation system).

Sawada, D., Piburn, M. D., Judson, E., Turley, J., Falconer, K., Benford, R., & Bloom, I. (2002). Measuring reform practices in science and mathematics classrooms: The reformed teaching observation protocol. School science and mathematics, 102(6), 245-253.