The first phase of my work in Budapest is almost complete. I will return from my hiatus in a week. Yesterday I had an opportunity to observe a grade 12 mathematics class at an alternative school established right after the 1989 revolution. (I am keeping the details out of this post to protect the people.) The translation was completed by my colleague from BSME from Hungarian to English. My interpretations, therefore, are based on the accounts from my partner in the observation

The organization of the classroom was non-traditional. Five tables were pulled together, conference style, so that students could all work together. There were several late students, but ten students eventually participated in the classroom with the teacher. There were six boys and four girls. This number of students is the maximum number of students allowed in classrooms at this Foundation school.

The teacher, (Let’s call him Harold.) engaged the students using a problem-solving approach. From my perspective, if a teacher is using a mathematics problem-solving constructivist approach then the teacher may use some of the following behaviors.

- Use cognitive terms, such as classify, analyze, predict and create
- Encourage student inquiry, such as asking meaningful questions or using meaningful contexts
- Support student’s ability to ask questions, such as seeking elaboration of the student’s initial response, or engage students in experiences that enhance understanding.
- Allow wait time after posing questions
- Provide time for students to construct relationships, such as encourage and support student autonomy and initiative, use raw data and primary data, or use manipulatives, interactive and physical materials
- Allow student responses to drive lesson, such as shift instructional strategies, or alter content accordingly
- Inquire about student’s understanding of concepts before sharing yours.

Harold created a constructivist-approached lesson with a diverse group of students at a “Foundation” school. Why is this an important statement? I have spoken about the POSA method used with, so-called, “talented students” selected by their teachers to participate in weekend camps. The POSA method as discussed earlier, assumed that children were capable and mathematically gifted as recommended by their home school teachers. While the researchers are working to bring POSA to general (typical) students, there are questions about how this may be possible. I witnessed, not POSA, but a problem-solving approach with students at this school.

These are the problems we worked on solving problems like these:

2〖 ∙ 3〗^(x+1)=3^3-9^x

9^x-2∙ 3^x-3 = 0

4∙ 3^x +3 = 20

There was a back and forth with the teacher. The teacher asked questions about what to do and why, and the students responded. Harold, the teacher, took them down a path to why Logs worked and the approximation for why we used Logarithms. The process was beautiful and painful, and some students got it, and others did not. There was an incredible patience from the teacher. At one point a student was brave enough to say without frustration, but I don’t understand why that works, and Harold took the time to explain in 3 different ways. He also used the whiteboard, the computer, their calculators, and their inquisitive nature to teach his lesson. There was some direct instruction, but it was iterative and required student engagement.

Problem Solving approaches to teaching are possible with all kids. My theory about building a democracy around the way we teach grew arms and legs through this observation. Thank you, Harold, for showing me an example of problem-solving, constructivist mathematics teaching.