Scared and Ready

My last post was on Nov. 17.  This means two months without a post.  After returning from Budapest, I wondered how my learning about math teaching in Hungary could benefit my university and the interactions I have with students at my institution.  I have been asked to supervise Mathematics student teachers. What a wonderful opportunity to discuss the creative process of learning and teaching mathematics. I will have an opportunity to help the next generation of teachers find their teaching selves for the purpose of ensuring that all students can learn math well.

I attended the JMM ( Joint Mathematics Meeting) January 10-13, 2018.  This is a joint conference of Mathematics Association of America ( MAA) and American Mathematics Society (AMS).  Every type of (as the Europeans say) maths was presented.  I attended many talks on teaching and learned about math teachers who teach University and K-12 students how to be mathematicians.  In fact, Francis Edward Su has amazing writing where he shows his vulnerability around doing mathematics and teaching mathematics.  I had no idea that others, certainly more high powered that I, had insecurities about their abilities, but also had the attitude of paying it forward.  The paying it forward idea came through loud and clear at each presentation. The important detail is that we do not know everything there is to know about how to teach mathematics, but our passions, interest, curiosity, intentions, and attention to learning will help us move forward from where we are in our teaching.

I was asked by a researcher if I did mathematics for fun. At the time I said, “I have not done the math for a long time.”  I decided to do some math because I felt like I was out of the loop. The process of doing math for oneself is important.  One should consider doing math, whether it is a puzzle or a theoretical problem or a game because the process of doing mathematics keeps your brain engaged in the process of learning.  If you are teaching or teaching teachers to be mathematical, it is difficult to maintain one’s credibility without actually doing math.

I tried a calendar problem from NCTM as my new entry to doing mathematics.  I spent time on the problem and found it challenging, but in the end, it helped me understand how to approach my teaching and supervision responsibilities. Teaching is an interactive process, especially in math.  I think that without ongoing engagement with mathematics, it is difficult to ask students to do math that is unknown, challenging, and brain expanding. The struggle of the process of doing math is part of why we should continue to do math.

I look forward to working with a group of new student teachers.  I look forward to helping them see themselves as mathematicians because math teachers are created, made, not born. Each of the students has decided to become a teacher for different reasons.  I am interested in helping them grow because I do math because it is challenging and I never know until the end if the answer is correct. The process and the product are equally a part of doing the math. I hope that I can help this cohort of student teachers to grow to the next level.

I am scared and excited. Making mistakes is part of the process. I hope that our learning together will be productive.

 

Public Transportation–Applied and Theoretical Problems

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?

manipulative
Using sticks

X I + V = V

V + II = V

V – II = VII

V – I = IX

(At the end of one of class a 2nd 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?Screen Shot 2017-11-04 at 11.45.20 AM

Problem 2:
a) Which number is larger, and why? Screen Shot 2017-11-04 at 11.47.17 AM

b) What is the last digit of the number. Explain?

Screen Shot 2017-11-04 at 11.54.53 AM

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.

 

The Market: Kilograms and Forints

IMG_9416
Pazmany Peter Setany 1c, University Math Building
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Auditorium Chair at Eotvos Jozsef Gimnazium

I have less than three days left and what do I do? I go to the market. The market is dangerous when you have a short time left because there are vegetables and meats and cheeses to buy, but you are alone and who is going to eat it? I have created a monster for my last few days. I bought popcorn and oranges and tangerines, and sausage and bacon and Brussel sprouts and onions and tomatoes. All of these things are great, but as I said, who is going to eat it? I wanted to buy eggs and cheese, but seriously, why. I also purchased some favorite desserts to which I have become accustomed. I have paid anywhere between 3200 to 1000 forints for this dessert, but today, at the market, the favorite dessert–Somloi Galuska was only 320 forints. I bought two.

Every interaction with people in the market is affirmative and engaging. I do not know Hungarian, and for the most part, they do not speak English, but we figure it out. I learned after only two months in Hungary that when it says Kilograms and you are trying to figure out if the price is right, that 1 pound is approximately 2.2 kilograms. Now I think I knew this before, but when you are in another culture, and you don’t speak the language, you forget about your school math and work double time to figure out forints to dollars and pounds to Kilograms all at the same time. The merchants could have been swindling me, but they did not/ When I give a merchant 5000 forints for something that cost 230 forints, which means I should have given the person the 500 forint note, they could just take my money and move on. They could take advantage of me, but they do not. I spent another 10 minutes trying to figure out that I did not want a blueberry extract for 6000 forints, but before I knew it there were four different sets of people trying to help me communicate that. I learned a little more about forints and kilograms today and nice Hungarian people in a market off the 4 – 6 tram and the number 19 bus.

I will remember, with affection, the experiences in the community, probably more than I remember the school visits. We have many possibilities for which we can assist our youngest and oldest students to connect better with math. I am in awe of the resilience exhibited by the Hungarian people over the last 100 year, war, fascism, communism, and democracy. They are doing yeoman’s work to figure out democracy, and so are we in the United States.

The market is where I go to learn about the culture and myself.

Teacher Talk

A Math Lesson (with minimal) Teacher Talk

What is needed to learn mathematics? I don’t know. When students struggle with math, learning occurs? How does math students struggle productively? Here is my problem. I witnessed two students in a 11th-grade course preparing for the end o the year leaving examination. The teacher decided to make the students use their English for my benefit. One student agreed to go to the board to find the difference in area between an inscribed circle and a circumscribed circle with a radius of 5 cm. The student went to the board, in English, (he was Hungarian speaking using English to practice his English in a math class) walked the entire class through his thinking. There was a lot of trust in that classroom. Maybe he was one of the best students. I don’t know, but his struggle showed me that creating a classroom atmosphere of trust one another and make mistakes together is a sign that this is a community of learners ready for anything.

Many years ago I studied social science education at Syracuse University with Dr. Jack Mallan. I had no idea what he was talking about, but now I do. His book, titled “No GODs-No Givers of Directions in the classroom” makes sense now, after 37 years.   Dr. Mallan wanted his teacher candidates to perceive ourselves as facilitators of learning. My interpretation of Mallan’s work is that teachers can serve as facilitators of learning if we think about how to help students construct knowledge. If we give students a chance to work, they can demonstrate their learning. With our guidance, then learning will occur. Some teacher talk or some direct instruction is required, but the real learning happens when students are demonstrating, creatively how they understand what we want them to learn. The reformed teacher practice process requires that the teachers provide a safe environment for learning that is interactive, sustainable, and measurable. This National Science Foundation-funded project created a reformed teacher observation protocol (Piburn, 2000) for documenting reformed math teacher behaviors and student learning. The lesson I witnessed, did not conform to a typical teacher lesson with lots of teacher talk, but using the “reformed teacher protocol” the outcomes were met.

Teachers need to talk, but often, teacher talk (add a reference here later) interferes with student learning. Dr. Mallan had a mantra for convincing pre-service teachers to understand their role as facilitators of learning. Creating a space for students to demonstrate their learning is a complicated process. I witnessed the creativity of the students and the patience of the teacher during the lesson. For patience, teachers need time to prepare excellent lessons, so that students can experience the creativity that mathematics has to offer. It was an honor to have witnessed this lesson.

Mallan, J. T., & Hersh, R. H. (1972). No Gods in the Classroom: Inquiry and elementary social studies (Vol. 2). Saunders.

Piburn, M., Sawada, D., & Arizona State Univ., T. T. (2000). Reformed Teaching Observation Protocol (RTOP) Reference Manual. Technical Report.

 

 

 

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?

I Have Something To Say

I have something to say. Access to mathematics is a diversity issue. After re-watching a video interview with a Dr. Geneva Gay on culturally responsive teaching, I realized that my vocabulary around access to mathematics and Gatekeeping has to be strengthened.  There is a history of stratification around who gets to learn mathematics, and after reviewing several New York Times articles from the past 100 years, the stratification about who and what is taught in classrooms is also part of cultural divide.  When I talk about how to open the GATE, to mathematics learning, I am requesting that teachers, students, parents and the rest of society rethink their role in maintaining the system of math minds and non-math minds. Many theorists in mathematics education would say that everyone can do mathematics, but we are all controlled by the expectations and perceptions we hold in the system. My conceptual framework (My belief) is that everyone can do and be successful in mathematics, but in our American society, in general, only certain populations are allowed to do more than the typical arithmetic.  This stratification means that only certain populations have access to all there is to offer in being a confident math person.  My approach to Mathematics for All is to use our perceptions and expectations to unpack our histories around what it means to transform who we are around mathematics. I have something to say to our pre-service teachers about their role in the stratification process. Lets see what my time in Hungary does to my conceptual framework.

 

Organizing My Time

I spent the day figuring out all the promises I made to visit classrooms in Hungary and as part of the process  I realized that a Mathematics Education Department Chair, Odon Vancso agreed to help me find additional schools to visit.  He is at Eotvos Lorand University and his colleague Andras Ambrus is allowing me to use his office for a time while in Budapest.

I also realized that after the first scheduled conference in Baltonfured, September 10-15, my arrangements to stay at an airbnb in Budapest, almost three hours away, was going to result in me sleeping in the streets while attending the Polya conference. Luckily, now I have arrangements to stay at the same Hotel Annabella for two more nights. One less move, even if it is pretty expensive.

I leave for “The Mathematics Education for Future Project, Fourteenth International Conference, Challenges in Mathematics Education for the Next Decade” on September 8 and the first conference ends on September 14.

The second conference, specifically about Polya is in the same town, Balatonfured, Hungary, but three hours from Budapest, so I will at least stay in the same community , instead of traveling for 3 hours back and forth to Budapest for two days.  I am glad I figured that out before it was too late.

Once the second conference is over, I will be working with Agnes Tuska and Reka Szasz to shadow the Budapest Semester in Education (BSME) students in their classes, and in the school visits between September 18 and September 29.  I hope to do a little bit of touring during this interval too.  My Friend Banna Rubinow just came back from Budapest, so will have some recommendations for me.

In the time interval between October 4 to 10 I will visit Hailey Ihlow, Brad Wray and Lori Nash in Denmark.  That trip might be the extent of my tourist behaviors. When I return I will be attending a Distinguished Teacher math conference at the BSME for two days. This is another opportunity to meet other faculty from the United States interested in problem solving and improving mathematics education.  I have high expectations for continued work with this group of faculty.

Between October 15 and October 31 Odon Vancso has planned some school visits for me too.  I plan to take some time off to write and plan for a semester abroad  in Budapest for students in the future. My work in Budapest is important as I revamp my modules in the Mathematics for Diverse Learners Course I taught back in 2006.