I read one of the National Council of Teachers of Mathematics (NCTM) Math Teacher
articles. I was looking for some motivation to move forward with my agenda on diversity and Mathematics for all. In the past, I have recommended that math teachers engage in personal problem-solving as part of their ongoing professional development. The premise is that by continuously placing ourselves in math problem-solving modes, we may be better equipped to support our students as they engage in the same process. These personal problem-solving activities such as, reading mathematics education articles, experimenting with math problems or puzzles, making sense of the history of mathematics, exploring art and mathematics, or investigating realistic, applied math problems.
I chose the November 23, 2017, monthly calendar problem from the Math Teacher to pursue my professional development around experimenting with math problems.
The number puzzle is from an 1852 algebra textbook: “A certain number consists of three digits, which are in arithmetical progression; and the number divided by the sum of the digits is equal to 26, but if 198 be added to it, the digits will be inverted. What is the number?”
After feeling scared that I might not be able to solve the problem, I wrote the problem down and tried to understand the problem. I then made sense of the problem by writing some examples of my understandings. I thought, since this is from an algebra textbook, I decided to try to use some algebra. After about 5 minutes of setting up some equations, I decided that it would be easier for me to follow my instincts and just try out some example cases of solutions to the problem. I then gave up a little bit and wrote some arithmetic progressions using the digits 1, through 9. I worked for about 30 minutes until I was successful.
Finding the solution was empowering. The selected problem has very little to do with an applied problem, often recommended for a performance task in math and often recommended so that student witness the value of a real-world problem. The answer is not obvious; there is more than one way to arrive at the answer or perhaps the problem has more than one answer; there is a perplexing situation that is understood; the solver is interested in finding a solution; one is unable to proceed directly to a solution; and the solution requires the use of mathematical ideas. (Find the reference.)
I felt stress and anxiety. I wondered why I was doing this. I believe it is essential for pre-service and in-service teachers to engage in problem-solving because it puts you in the shoes of the student. I had forgotten what it felt like to not know exactly how to approach any math problem. I had forgotten about the tension involved in solving a problem. If we expect students to do this, solve problems, then we have to engage the process ourselves, and students must be taught how to engage in this process.
The process of solving the problem may mirror the process mathematicians engage when discovering a solution to a theoretical question. I only felt a sense of courage and confidence about solving the problem once I solved the problem myself. The steps, the feelings, the success, the stress, and finding the answer, all might be transferable to problem-solving in general. If we expect citizens in our democracy to make decisions and solve essential life problems, then how do we use the curriculum to engage students in problems that provide them with the experiences we want them to have for the classroom and life.
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 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?
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.
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.
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, 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 went for a hike today in preparation for my journey to mathematics. While I have hiked 3 of the 46 Adirondack New York High Peaks, the last time I hiked, was a year ago at Tremont State Park in Ithaca, New York. I am out of shape, but why is hiking important to learning mathematics? Today, on my hike to Labrador Hollow Unique Area. I wanted to stop after only 5 minutes. My lungs hurt. My feet hurt, and I wondered why I was out in the woods. My body said stop, but my brain said “you came all this way from Oswego to hike with friend, so forget the
pain and hike”. What does this have to do with learning mathematics? Learning mathematics often requires the type of struggle experienced while hiking. Just as hiking up a mountain requires some physical and mental struggle, learning mathematics can be the same. I believe that if I had understood that struggle was part of the learning process in mathematics, I may have been able to go further in my learning. In my early days of learning mathematics, I did not know that struggle was part of the learning process. I thought that everyone knew more than I did, and I th
ink I stopped learning as much as I could have learned because I though that struggle was a sign of weakness. An important quote from Albert Einstein, “Do not worry about your difficulties in mathematics; I assure you that mine are greater”.
So, if Albert Einstein is quoted as accepting the struggle, my job as a mathematics educator is to help others see that struggle is part of the process. I am proud of my hike today. I had a chance to think about nature and how my work with learning mathematics is a universal struggle. Think about anything that is worth learning. If it comes easily, is it as satisfying as if you have had to struggle? I am not sure. My journey begins as I am reminded about why this work is important.