Use of the online graphing calculator enables students to use tools to analyze and represent data in various ways to facilitate problem-solving. Read The Unreasonable Man

In 1982 a graduate student DM supervised my student teaching at a middle school in Central New York. I had some time to plan lessons, and I was scared out of my wits. DM was gentle and direct. She gave me little things I could work on to improve my practice. I did not learn how to teach during student teaching, but I did learn that nothing is perfect and you have to put yourself into a different space to help students learn. I learned how difficult teaching was. It was not until a few years later; I read “Freedom to Learn” by Carl Rogers. This book allowed me to understand the power I had to assist in student learning, but why didn’t I get that then. Why didn’t I get that teaching was about facilitating learning. Jack Mallan’s “No Giver’s of Direction–No Gods In the Classroom” did not make sense to me when it was required, but Carl did, once I started to teach. The power to teach is in the trust you have in yourself and the trust you build in your students to learn. Teaching math is a subject that screams trust your students because we all are born mathematicians. We all can engage with one another as we build our capacity to learn math. It is a subject that requires interaction, iteration, experimentation, communication, and creativity. These processes are we learn and do mathematics.

In hindsight, it was Dr. HJ who convinced me that Mathematics Education was the path for me, and I learned about teaching EOP students as part of a summer program back in 1983. I was trusted by Dr. HJ to teach mathematics to new students at a good university. Somehow I was trusted as a new teacher to figure out what to do. Why Dr. HJ. trusted me, I have no idea. I taught exactly how I was taught. Students learned, but I also learned things that are never in a textbook. I learned how to ensure that students’ learning, but not by accident. Everything I learned in school, taught me how to teach, but only after I was teaching.

I learned about “mathematics anxiety while working for the EOP program at SUNY Oswego. I will never forget the mentoring done by PP and others. They believed in a process for helping students get past their bad experiences in mathematics to a new experience of passing a math class to meet the graduation requirement. In that short almost four year experience, I learned that at any point in someone’s lives they could change. They can change their attitudes towards mathematics and build success for themselves. I learned patience, and I gained gratitude for all the teachers who had to deal with me over the years. The EOP program, and the courses I taught helped me realize that I did believe that everyone could do the math and that it was my responsibility to ensure that my practices included both emotional support, cognitive supports, and hard work reminders. Without the EOP experience, I might not have understood that there is another world out there preparing people to investigate mathematics and their place in it.

What a gift it is to be supervising student teachers. I see all of my faults in their behaviors. I see myself 30 years ago trying to figure out how to talk to the students, teach content, watch the clock and figure out why students don’t stay in their seats. There is no need to figure all of this out. The most important thing with this math thing is to trust in the students who have to learn to learn. We cannot assume they are going to make mistakes, but we can assume that when they make mistakes, that there is someone there to help learn from those mistakes. Yes, yes, we learn more from our mistakes than we learn from not pushing ourselves to the limit. I love math because it challenges my thinking, my interactions with others, and my deepest desires to know be successful.

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.

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 2^{nd} 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?

Problem 2:
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.

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.

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 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?