Well, here I am ready for another school visit. This vocational school in another part of town with another set of circumstances and public transportation puzzles. I say taking public transportation, transports one to trying to figure out the challenge of figuring out how to get to the right place at the right time without spending the entire day. Because I am worried about these things, I leave as if it will take an hour even Google says it will take 22 minutes. This adventure or journey each I go to a school visit reminds me of the problem-solving approach to teaching. You know the constructivist approach. It is clear that if we are serious about this constructivist approach to teaching, then we have to define the parameters for success.

The journey, fun or not, to figure out public transportation in a big city like Budapest, maybe another good metaphor for the problem-solving process. Do you have a map? Do you have experience in a big city? Do you know the language? Can you read? Can you walk? Is it the rush hour in the morning? Is it a safe city? How important is the destination objective? Do you have to get there on time? Are there consequences for getting there late using public transportation? Do you have an infinite amount of money to ditch the public transportation and take a car or a taxi? All of these questions are the same in the constructivist approach to teaching mathematics? These are very similar parameters when teaching using the problem-solving approach or learning using the problem-solving approach.

Yesterday I witnessed an excellent example of productive struggle I have ever witnessed. I left thinking, how patient the teacher was for allowing this, and I wanted to see more of this productive struggle. What is the productive struggle in the context of this public transportation metaphor and learning mathematics, or even teaching mathematics? Students were confident at the board, and their peers listened intently. Students did not interrupt, and the teacher allowed the student in the explanation to explain their thinking so that the student, in the end, found his flaws, explained his work to the class and appeared overjoyed by the success. It was not like watching someone peel an onion. It was like watching someone peel an orange. The slow process did not follow a particular model. You were not sure where the student was going to go. You did not know if the student would give up, but in the end, the process produced an orange you could eat. It produced a student who knew what he was talking about mathematically. It produced a student who earned a point for the day, but also he earned what could be referred to as “street cred.” He risked, and he was rewarded for taking a chance on the black/green board.

What did the teacher do here? The teacher set up a three-year system of collaboration and struggle. The teacher set up a place was struggle brings forth anticipation for success. This is an example how the teacher is expecting students to struggle without just telling them the answer. How can this teacher be this patient in a room full of teenagers? This teacher is engaged in a project with the University to use the POSA method with all kids, not just those deemed talented. I saw the fruits of this labor yesterday.

Let’s talk about shoes. In a big city like Budapest, the men’s shoes are diverse. What do I mean? It would be easier to talk about women’s shoes, but it is more interesting to talk about men’s shoes. There are a lot of pointy leather and leather looking shoes. Men are wearing the shoes, with straight-legged pants, formal dress pants, and leather jackets. The shoes seem to go with the clothing, in that no one appears unkempt. I just saw a man drive by on a big with shoes that I might see any American person wear with a suit. All the shoes in my observation of 10 minutes were clean and well kept. In fact, the last man in my 10-minute observation walked by with suede shoes and a very nice matching suede jacket. I am not sure if all of these people walking by are students, faculty or workers. It is not known, but I am writing about the diversity of shoes to broach the cultural diversity question in mathematics. These are the observations of an American on a Hungarian college campus. All I see is the shoes, and I begin to make thoughts in my head about who these people are and how the shoes make a difference. I am looking at the shoes and figuring out how I might interact with the variety of shoe wearing men. The cultural diversity question issues can unpack through the men’s shoe observations. Keep in mind that this process is about how to help teachers understand their perceptions and expectations translate into a set of behaviors that affect student learning.

Who are the students in the mathematics classroom, and how do the perceptions of the students translate into expectations for learning mathematics. What are my perceptions of the shoes that are on men’s feet, and how does one form complex interpretations of the observations? How do teachers form complex interpretations so that they can be more effective teachers, especially during the problem-solving teaching process?

How can we figure out what it means to teach all students?