Lénárt’s Spheres


Four professors from different US universities visiting BSME
BSME Distinguished Visitor Program
Isvan Lenart
Lenart, Istvan, demonstrating Lenart’s Spheres at BSME workshop

Professor Lénárt, István is the inventor of the spheres from Eötvös Loránd University in Budapest, Hungary. His passion is infectious. He talks about learning mathematics for everyone. I shared an office with him, spoke to him on multiple occasions, attended one of his workshops and attended one of his university classes where he was teaching future kindergarten teachers how to integrate geometry into the classroom (this was in Hungarian).

I watched him demonstrate how to do geometry constructions using the sphere. I watched him ask questions and make sure that everyone in the class could answer with confidence. I watched him discuss formally abstract concepts of geometry using a simple sphere and questioning. Every interaction with my new colleague István (Steven) was both intellectually and emotionally enjoyable. He knows that students can do mathematics and he expects them to learn.

When I am discouraged and wonder why I believe in mathematics for all, I will remember my interactions with István.

My Visit to Margaret Island, 10/15/17

Hungarian Flag without the communist symbol
Hungarian Flag

My visit to Margaret Island, Budapest, Hungary reminded me about breathing and that everything is about context. Margaret Island is an Island in Budapest where the Hungarians and tourists alike enjoy the outdoors and nature. It is a simple train ride onto the island, but you feel transported to another time and space, where the worries of life are no more. I am still trying to figure out how the people walking by with American accents have dogs. People around me are walking, jogging, carrying babies, strolling, playing, and people watching.

A Hungarian bank holiday commemorating the 1956 Revolution is on Monday, October 25. University students further ignited the revolt; these college students sacrificed their education and their lives for democracy. University students are an essential factor when it comes to making societal changes. The students in 1956 risked their lives and their future livelihoods to build a democracy. I like to think about the Hungarians who decided that the way things were did not work. They stepped into the unknown fight the then current communists. I do not expect to give my life to make changes to our mathematics educational system, but I feel like the urgency around a revolt is needed to make changes in the way we teach mathematics. I believe that teaching any subject can be about building democracy. Teachers can work to ensure that every student has access to rigorous math teaching and learning. Mathematics learning can contribute to the democratic process.

From my experience Mathematics teaching is conducted in basically the same way in the places I have visited (Hungary, England, France, Benin, Brazil, and the United States) all over the world. In these western locations most recently in Budapest, Hungary, there are pockets of teachers, who are experimenting with a democratic, constructivist, problem-solving approach. I endeavor to describe this because, with the description, we have an opportunity to understand how this teaching, more difficult may create an atmosphere of learning for all students.

What does constructivist teaching look like, and why is a democratic process? When you give human beings opportunities to ask questions and engage with difficult topics (even in math), then this is how democracy is promoted. Asking questions and getting answers is why democracy is so difficult. I think learning math in a problem-solving way can only help us engage in what it means to be democratic.

In a previous post, I talked about approaching the teaching and learning of mathematics using a Polya connection to helping students understand and learn mathematics in their context. Using a problem-solving approach requires planning, patience, priorities, and perseverance,

Planning (by the teacher and the student) for a constructivist-focused lesson may require more planning time than of a direct instruction focused lesson. In a constructivist lesson, students need to prepare problems as assigned by the teacher in advance (you know the whole flipped classroom thing) and the teacher must develop a series of problem-based questions for engaging the students in the classroom. These problems cannot be trivial. The teacher must prepare in content and process of learning to succeed in using the constructivist approach. It might take 4 hours to prepare for a single one-hour problem-solving class.

The problem-solving approach requires patience. The Teacher may have to wait while students process their learning and their errors. Patience is a learned behavior, and I believe that integrating technology as part of the constructivist process may be of use to compliment the teacher-student student-student interactions.

Learning about prioritizing covering material or slowing down so that everyone understands the details of the math. Is it possible for the teacher to decide to reinforce understanding via cooperative learning, or should she just keep going just in case she is accused of not covering the material? There are standards (common core), end of the year examinations (in Hungary, typically, at the end of grade 8 and the end of grade 12). If the 8th grader scores poorly then students sorted and selected for the less competitive high schools. And if you do not score well on the end of high school examinations, then you wait a year, or you cannot move forward. There are high stakes examinations in both the US culture and the Hungarian culture. These high stakes exams and processes give students the idea that if something goes wrong, one’s life might be ruined. The ideas about using the problem-solving approach in the classroom cannot ignore the system that our teachers and students exist within. A friend told me that she used to say to her kids, “it is just high school.” Knowing high school is not the end of the world is the correct way of neutralizing the perceptions, but this works I a setting with love and support and a safety net. What if you don’t have a safety net, then high school, or an exam, might be the only avenue to a different life. The pressure to succeed in the system sometimes ignore real learning so that students can meet particular checkpoints. How can we help teachers and students better prioritize?

How can mathematics learning include time for patience and priorities, when the system lets us know that it is not about learning, but about meeting the next milestone in life and learning.

Finally, when I think about my teacher observations, I worry about how we can help new in-service teachers persevere. This morning, I witnessed a teacher allow her students to learn. Right at the beginning of class, when the teacher presented three challenging trigonometry problems, one student said: “we have not done problems like this before.” The teacher said, “that’s right, but we are going to do it together.” The teacher allowed the students to unpack the problems in various groupings, small groups, individually, and on the board alone. Not every student was writing working, but every student was engaged in the process on a continuum. Some students were in shock by the level of rigor, but this was not a direct instruction lesson. Different students took the lead to share their thinking with their classmates. I witnessed this teacher allow her students to learn, no matter how painful it was to watch. This observation is one class, and my call is for more classes to happen this way. I am overwhelmed by how daunting this is. Can I persevere?

Uniforms and Slippers


I had an interesting visit to a school that has a novel approach to uniforms. I love the idea of identifying a particular class with its own color. For example, when the 9th grader enters, they are given a red or some other colored jacket with open sleeves. This jacket( at least the color) stays with the student as they enter, and then this sort of letterman’s jacket leaves with you 4 years later. There is a sense of camaraderie with others from your same class. You can identify those from a particular graduating class by their color. Students still keep their identities with their clothing, but also build an identity with their individual class.

At this school the students also remove their shoes and put on slippers or indoor sandals. A student told me that the slippers are an effort to keep the school clean. I believe this very specific practice of putting on your academic clothing and putting on your shoes, has a psychological effect. The jacket and slippers together send an internal and an external message that the individual is ready to learn. Simple and novel.But, what if the student fails and is ,,not part of the graduating class? This seems like a recipe for ostracizing the student.

This school as with every other school I have visited, the teachers care about their students and their learning, but the classes are heavily tracked.

The expectations of the student’s ability to do math at some if these highly selective schools, is mapped to the student’s test scores and the student’s perceived interest in learning math. More than one teacher has said, ”these students only care about humanities”. The interpretation is that the test scores may not be inevitable-lower because the student is not motivated in the math way.

I think a little self fulfilling prophecy is going on here. If the teacher perceives a lack of interest, then the teacher behaviors might be different than if the student is perceived as interested. The teacher expectations for rigor might be different. Maybe the potential for that student may be stifled because of the teacher perception and teacher expectation is lower.

Slippers and jackets go a long way to build capacity at a school that has very high achieving competition and test scores already? What might these slippers and jackets mean for a high school student who was not selected for a top school?



The solution is easy to find, but not so easy to check

“The solution is easy to find, but not so easy to check.”

While visiting one of the BSME (mathematics education) university courses on Wednesday 10/11/17, two students agreed to demonstrate their completed geometry homework solutions on the board. Their explanations were both conceptually and procedurally well done.   Upon completing some additional problems from the homework, the professor said: “the solution is easy to find, but not so easy to check.”

These words resonate around my discussions on equity, access, and actions for all students in the process of learning mathematics. No one seems to argue about the value of learning math, in fact, in New York State, the high school Advanced Regents Diploma requires Algebra, Geometry, and a second Algebra course. While the argument for more math classes is that students have more career opportunities, we have to take into account that some students may become more alienated through two more years of negative experiences (insert research). So we want learners to understand the breadth of mathematics through our curriculum choices, but the system is not always interested in the quality of the average student’s experiences. So back to the quote.

The quote is an admonishment of my loud proclamation that we have to do a better job of teaching mathematics for all students, but the solution may not be so easy to implement.

Is it about resources? Yes.
Is it about teacher expectations? Yes.
Is it about the lack of technology? Yes.
Is it about the stratification of who gets good math teaching and who does not? Yes.

George Polya (Heuristics around problem-solving in mathematics–see notes below.) might say, to solve any problem the first step is to “Define the problem.” I am not really past the first step yet. My journey has been about making sense of the problem through my interactions with colleagues in another part of the world. I will not step back (or step off as young people might say) without a plan for equity, access, and actions around mathematics learning. What is different,

Geometry Spheres Workshop
Lenart Istavan Workshop on Spheres

is that I realize that I cannot just ram the access mantra down your throat, but I have to figure out how to add you to the conceptual framework of equity in mathematics.

Maybe I have defined the problem.

Polya’s four-step approach to (math) problem solving

  1. Preparation: Understand the problem.
  2. Thinking Time: Devise a plan.
  3. Insight: Carry out the plan.
  4. Verify: Look back.

A Constructivist Approach: An Example of the Journey

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.

  1. Use cognitive terms, such as classify, analyze, predict and create
  2. Encourage student inquiry, such as asking meaningful questions or using meaningful contexts
  3. 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.
  4. Allow wait time after posing questions
  5. 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
  6. Allow student responses to drive lesson, such as shift instructional strategies, or alter content accordingly
  7. 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:

The Journey is possible.
Polya Garden in Balatonfured, Hungary

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.

Reality Check on Mathematics and Constructivism

I had a conversation with a new colleague at the university. The new math colleague (we will call David) has been teaching high school for more than ten years and is one of the first students admitted into the new Mathematics Education Ph.D. program. David teaches in a high school 1/2 time with head teacher/homeroom responsibilities and is a university student the other 1/2 time. He is fully funded to do his work in both places. David’s arrangement reminds me of the Master’s Degree I earned from Syracuse University. They had a program where I completed a master’s degree in Education and was hired to teach in the Syracuse City Schools as my assistantship. I learned about teaching mathematics, but teaching math, but Syracuse University paid for 30/36 credits of my master’s degree. I wish I had taken better advantage of that opportunity.

My conversation with David was about the research study to bring Posa to regular kids. Currently, the Posa program requests that teachers recommend the talented ( mostly boys) to the weekend math camps. David is involved in a research project to bring Posa to schools in general. I said, “there is a lot of risk around doing this problem-solving, constructivist approach to teaching, math(s) to regular kids.” I said this because I wonder how hard it is to teach percieved talented mathematics to kids who would probably be successful with or without the math camp.

His response was refreshing. We are not very successful in teaching regular kids math, so why not try this problem-solving way? We sometimes lose many students to the love of mathematics, by instructing in the same mundane procedural only, memorization process, and with lower level repetitive thinking by doing many problems without conceptual understanding. Why not use the POSA method to teach critical thinking through mathematics to all secondary school children. There are jobs, not yet invented that might require problem-solving, so why not teach in a problem-solving way to produce the citizens of the future? By teaching kids how to learn in a constructivist way, we might have a jumping off point to contributing to our democracy.

It is about using the teaching of mathematics as a democratic act. By teaching kids how to learn in this constructivist way we might have a jumping off point to other larger democratic processes.

My Goodness. It was a wonderful to hear David, this seasoned teacher, talk about why he teaches. He mentioned that he approaches his teaching to build the fabric of mathematics for the next generation. He talked about how success in mathematics is a process that he enjoys witnessing in children. His explanation feels like the journey metaphor I am pushing. He seems to love the process of learning and helping the next generation of kids learn mathematics in a way that is it dynamic, creative and rewarding.

He made me smile. I realized that my work around helping teachers make a difference in the mathematical lives of all students is important, democratic, academic, emotional, and maybe even memorable. Why do I say memorable?

After visiting the Hungarian Parliament today, as an American tourist, I learned that the Hungarians had re-written their histories. The rewriting may have been with the Austrians, the Turks, the Germans, the Russians, and now the Europeans. Each time the Hungarians recover from some war or occupation, internal or the external occupying forces, they work to rewrite the version of history created for them by the occupiers. Some might say that the story (history) always goes to the victors. Some might say that the interpretations of history are therefore important and relevant for us to engage.

The interpretations about ensuring the access and success in mathematics for all is an ongoing journey with factors and people who have different visions for the future. Teacher expectations make a difference in student learning and helping teachers unpack their expectations might just be the work. Dr. Dolores Grayson, here you are again with the Generating Expectations for Student Achievement (GESA).

David, a talented (as perceived by his teachers) alumna of the POSA method is now teaching using the POSA method to his students in his school. He mentioned many of teachers who are teaching using POSA were taught POSA as children. The concern is that without Posa experiences a child, how difficult will it be to transform a teacher’s conceptual framework around teaching and learning mathematics this way. It might be better to concentrate on new teacher training. The research project is hoping to bring the POSA method to teachers who may not have been POSA students.

David uses the Posa method but not all the time. He mentioned that the time it takes to prepare a 1-hour class using the POSA method is 3 -4 hours. We agreed that this could be a resource issue. We talked about how to work with teachers to use this method when we know that there may not be enough time, or resources to accomplish this rigorous way of teaching for all. He said. “I use POSA sometimes and then the typical didactic methods other times. He made me smile because he was so honest about the POSA method. Another colleague (let’s call her Joanne) is working on a research project for creating sets of problems that will support a system of problem-solving. This other faculty member is investigating how these problem sets might scaffold problem-solving differently. How do we put math problems together so that students can use the discovery process? It has been wonderful to find a set of mathematics people interested in delving into this type of thinking.

I am an insignificant want to be mathematics educator person interested in ensuring that more students have access to good math teaching of the constructivist problem-solving type. These math people, in another country, have similar interests. What a privilege it is to talk with them. I could not have asked for more on my time away.


Defining the Problem

September is almost over, and I think I am over the, “I want to go home jitters.” Thank goodness, since I was looking to find a way home for cheap. I listened to a story on the BBC this morning about the Silent Forest. It is a story about the over-harvesting of birds in East Asia. The author gave a dynamic story about the intricacies of the rainforest, the marketers, the poachers, and most importantly the external and internal factors that make it difficult to find a solution to making the forest silent. Of course, my brain is thinking about mathematics for diverse learners. From my perspective, it is about helping teachers and administrators in K-12 settings understands that we are cutting off possible solutions to our world problems by limiting the best and the brightest teaching to only the top 20%. The top 20% will say that they will be held back if they have to wait for the rest of the class. The “average” student will say they will be left back if the special education students are part of a more inclusive classroom. The middle-class student will say that the working class student does not have the vocabulary to keep up with the rigorous nature of mathematics. My kids don’t know how to interact with those children from the other side of the tracks, so we have to spend valuable teaching time helping them understand one another. The college preparatory student will say that they should be able to expand their learning without the vocational student. The teacher will say “how do I teach all the students in the classroom when they are all in a different place on their standardized test scores, or they are all different in their capacity for learning?” These statements and questions are not new in education, or they are not new to teaching mathematics with an inclusive approach.Talk about the problem.

Talk about the problem.

I have been witness to learning about the POSA method. The Hungarian POSA method as described earlier. The inventor has used this problem-solving, constructivist approach with gifted children. The gifted children, mostly boys, are selected by their teachers to attend these weekend camps, where they learn with their peers to engage in mathematics using the approach. This “constructivist teaching” mostly by mathematicians, is an attempt to bring the most gifted into mathematics, for any purpose. The reason for this approach I assume was to make sure that the next generation of math people could be encouraged and POSA was encouraged. But now the mathematics researchers are interested in bringing this “POSA” into general-typical mathematics classrooms. The teachers are taught using the method, at the university as part of their regular professional development, and then the teacher works with her administration to build this in each of their school settings.

I met one such teacher, who is experimenting with the POSA method. In the school at which she teaches, and this is a common practice in Budapest schools, to take a group of students organized from highest to lowest, and place them in two different classes. Typically the students know if they are in the top group or the lower ranked group. There might some self-fulfilling proficiency( about who will succeed) going on here when someone is placed (based on their scores) at the bottom group based on their end of the year examination in primary school. The students assigned to the top or the bottom group based on the high school entrance test taken at the end of the primary level education. (Primary schools are broken down into two four-year sections: 1, 2, 3, 4 and then 5, 6, 7, and 8. Students take an examination around age 14 which will begin the determination of their fate for the rest of their lives. High schools are highly selective, based on the end of the high school level examinations. If students score well enough, then they are supported via scholarship to attend university. If you do not score well enough, then you are allowed to go to University, but at your own expense. This system has many characteristics similar to the United States system in which the author is familiar. This teacher has agreed to take a mixed group of kids, ones from the top of the high school exam ranks and those from the bottom of the secondary school examination group and teach them as a heterogeneous group. Parents and the head teacher of the school granted this seasoned teacher permission to place these students in one class. This heterogeneous grouping is a new thing, and I commend this teacher for undergoing this democratic experiment. These students will be together with the same math teacher for the next four years. The research will help us understand how these methods for teaching mathematics are valuable for all students. The teacher is already amazing, and I believe her expectations for success in this setting will be the marker for the program’s success.

In the United States, and maybe in New York, if you live in a nice neighborhood and you are white, then you have access to the better public schools. If you are poor and of color, then how do you have access to the school in your neighborhood. In New York City the selection of schools appears to be very close to my understanding of the Budapest/Hungarian system. The context of ensuring that children get the most rigorous education around mathematics is a system problem. If one is to engage the system, then each must be willing to act as a part of that system. While there are activism responsibilities, one must fight the problem of ensuring that all students get what they need is an individual responsibility to take it on for the betterment of the system. Maybe this is the democracy talk. Maybe this is where we have to look at ourselves in this democracy and decide what is going to be best for the democracy.

So what about Meritocracy? A meritocracy only exists in a system that is fair. One’s birth position (i.e., first middle or last) or one’s birth station (poor or rich, white or black, smart or smarter, abled or disabled) should not determine one’s station in life. This discussion about mathematics is a discussion about how we want our democracy to survive. Social class, and race, and religion, and ability class should not be the factors which determine the course of our lives. Teachers may play a vital role to play a role in interrupting a system with low expectations for a particular set of students. With how they interact with their students in each class. GESA (generating expectations for student achievement), Grayson____ . GESA is an about how we use our interactions with students to make a difference in their learning. It is about how our expectations make a difference in their education, and how we monitor our expectations to better teach the students in front of us. Our expectations do affect student learning. Expectations are often explicitly spoken, but the unspoken, implicit expectations often have more effect on student learning.

Something else to think about: Check out the App Photomath and let me know what this means in the teaching of high school algebra.