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.

What are the parameters for success for teaching using the problem solving approach in mathematics?

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

The journey, fun or not, to figure out public transportation in a big city like Budapest, maybe another good metaphor for the problem-solving process. Do you have a map? Do you have experience in a big city? Do you know the language? Can you read? Can you walk? Is it the rush hour in the morning? Is it a safe city? How important is the destination objective? Do you have to get there on time? Are there consequences for getting there late using public transportation? Do you have an infinite amount of money to ditch the public transportation and take a car or a taxi? All of these questions are the same in the constructivist approach to teaching mathematics? These are very similar parameters when teaching using the problem-solving approach or learning using the problem-solving approach.Hero's Square, Budapest, 2017

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

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

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

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

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

 

Posa Method at BSME

Today I had a chance to observe the Posa ( pronounced posha) method.  A mathematics colleague that I met at the Polya conference delivered the BSME class, using the Posa method.  There is a Hungarian way of teaching the gifted, and that is the Posa method, but my colleague  Dr. Joseph Lewis (this is a pseudonym) is involved in a research project where he is teaching using this method to research subjects in a high school.

The Posa method follows the interactive approaches to mathematics teaching we believe is the best way to teach mathematics.  My colleague kept on saying “do you understand” and I did not completely understand even though I have had number theory.  I wonder if the American students in the class, taking it for a grade, understood?  I should have understood, but for the first time in my life I did not feel upset because I did not understand.  I want to understand, but I would have never said I did not understand. (What does that say about me?)I already went to the internet to find proofs that I remember doing years ago, but remember, I am not sitting in this class for a grade.  I am sitting in the class to better understand the POSA method. This is how I understand the POSA method.

Keep in mind, I fully respect the problem solving approach to teaching mathematics and I think this is the way we should be teaching all mathematics classes.

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1. Problems are given to students to work on before the formal lecture.  There is no textbook. There are carefully selected problems for the purpose of  doing the problem solving approach and in this case, the problems are what I call fun problems and proofs. Fun problems are the ones where you can via trial and error work out the answer. Usually these problems have the feedforward and feed-backward.  You know if the answer it correct without the telling you.  Sometimes the process of working on the problem is the full mathematical experience.

2. Step 2, the students arrive to class with some version of the completed work.  Some students have completed the work, and know what the teacher is talking about and confirms the teacher’s work or engaged explanation of the work.

3. Step 3. The teaching of the lesson is a series of questions, with responses to lead students through the process. Here is an example of the questions I documented:
a. “Who has a nice example?”
b. “What would you do now?”
c. “What do you think?”
d. “What if it is nonlinear but bounded?”
e. “How many of them can be counted?”
f. “Who has the construction so that the product  that does not equal 72?”
g. “Who can extend the set , so that we have enough elements and the product is divisible by 72?”
h. “If you understand, what is the next construction?”
i. “What if you don’t have any two’s?”
j. “Can you extend the this subset so that the product is not…?”
k. “If you don’t like 3’s then what?”
m. “How do you this is the maximum?”
n. “Should I give you a hint?”
o. “Is it clear?”
(I am going to compare these questions to the type of questions asked by the high school teachers. )

4. Step 4, the inclass interplay is with the students who understand the problems completed the class before.

5. Step 5, introduce a hands on problem that requires manipulating some sort of manipulative.  This can be done individually or in small groups. This provides some concrete -non abstract examples of the work discussed abstractly.

6. Students engage in some small in class writing and then the homework for the next time is assigned. 

What a wonderful day I had with BSME.  There is a lot of support for our American students to learn about the problem solving approach to teaching mathematics. I expect to return to the classroom to see more next week.

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Match Boxes Can you arrange the boxes so that they have exactly 2 faces touching?

 

Istvan’s (Steve’s office)

The Eotvos Lorand University Mathematics Department Chair, Odon Vancso, has given me a key to share an office with a retired, but still teaching mathematics professor Lenart Istvan. They say the last name first here, and then the first name. I knew that I was going to be okay when Istvan stopped what he was doing with a student, and greeted me with kindness.  He shook my hand, and showed me how to use the phone and the lights and gave me a space in the office that is larger than any office I have ever had except as dept. Chair. The poster in the office was the clincher for comfort though. You may have seen this quote before, but this is a full sized poster that reinforces what I have been saying these past weeks about struggle and mathematics. We are all mathematical. We can all do mathematics.  Remember, Einstein was a genius, and he struggled with mathematics as the poster says.  Let’s keep working on making it okay to struggle in mathematics. According to Einstein, hard work, with struggle is okay.IMG_8340

15 minutes a Day to Publication

You know what happens when you return from a conference? Within a few days you realize you don’t have enough time to transition from the wonderful conversations and learning to the next steps.  I have no excuses, because I have the time and energy to implement, at least via publication my thoughts on math(s) education. I very nice, long term teacher, turned into Assistant Professor gave me the same advice given to me by Dr. Alok Kumar. The advice given by Dr. Kumar was write 1 hour a day.  I don’t even make time to exercise an hour each day, so an hour of writing seems impossible. I can make a commitment to write for 15 minutes a day.  I have not make it to 15 minutes a day yet, but today is the first day of the rest of my life, of 15 minutes a day.

Get ready, and thanks for reading.

Happy Trails.

Now What? So What? Reflections from the International Mathematics Conference–Balatonfured, Hungary

Now what? So What? On our “three hour tour” on Lake Balaton, I had a conversation with colleagues from the conference. We had a conversation about what our next steps on this journey towards improving mathematics education might be.  The conversation began quite depressingly with “how is this conference any different than any other conference, where we talk about exactly the same content and continue to do exactly the same thing in our own settings with no real changes.

Two days later, at the plenary, Marjorie A. Henningsen, Grey Matters Education, Beirut, Lebanon. engaged us by creating a list questions.  The questions helped us all re-visit the paper presentations, and our individual responsibilities to improving math(s) education.

Questions relevant to the future of Math(s) Teacher Education inspired by the last few days (sessions and conversations):

  1. To what extent do our carefully structured and sequenced curricula build on children’s real life understanding and math competency or simply ignore it?
  2. How can we help teachers to be able to pose and frame good questions and problems in mathematics using diverse real REAL WORLD contexts? (not fake real world contexts)
  3. Gaps in content knowledge are well-documented by now—How much of a gap is OK to start teaching?   How can teachers become aware of their own gaps in knowledge and learn to take steps to address that? Do we have as good a sense of what teachers do know as we have about what they don’t know?
  4. What kind of knowledge is more important in maths teacher development: ways of thinking and dispositions toward knowing, or particular bits of knowledge?
  5. How can we interrupt the systems that produced us so that we can make change happen? Are we too immersed to see how—or even to see the problems?
  6. How do our “research-based” and presumed ideas of good teaching practice hold up in situations where children have real agency—when learning is more student-led than teacher-led? (e., How much of our work depends on the existing traditional classroom and school structures?)
  7. Some say multidisciplinary and/or transdisciplinary, mission-driven education is the future—what implications might that have for our work as maths educators and maths education researchers?
  8. How can we foster more fluid coherence among different phases of teachers’ developing professional identity as mathematics teachers (from preservice to induction to inservice experiences) so that beliefs and practice are more aligned?
  9. How can we foster a fuller understanding of mathematics as a human endeavor—a cultural production as important as other cultural artifacts?
  10. What is our image of the child in general and in relation to mathematics learning? Do we really believe in the agency of the learners in their own 
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    Taken at Polya Conference 9/15/17

    learning processes? What does that image imply for teaching? (This question can be also asked about the image of the teacher held by teacher educators, school leaders and others.)

  11. Over the past thirty years a consensus has been forming around what needs to happen to make teacher preparation better—so why aren’t we all doin
    g it? How do we take all this common wisdom from our minds into our bodies, into our practice, into our hearts?
  12. What is the impact of the testing and textbook industries on maths education and maths education research? Does it have to be that way?
  13. Does every always have to be doing/learning the same thing at the same time all the time?

These questions were introduced and everyone chose and formed groups to discuss them “world café” style for 17 minutes. Then participants were asked to shift their discussion to sharing specific or concrete steps each person might ta

ke to make a change or to address the issues in these questions. In the final five minutes, four participan

ts shared their action plans.   45 minutes would have been a better amount of time for this to allow for more sharing. Participants seemed to enjoy it and found several of the questions provocative and resulting in a good small group discussion. Marj Henningsen

 

Presentation Script on Gatekeeping Revealed, September 11, 2017.

The short version (r1_powerpoint_revealed_burrell_16revised) of my presentation and a long version (powerpoint_revealed_burrell) of my presentation can be found or read the script.

Hello my name is Marcia Burrell. I am from SUNY Oswego in Northern NY on Lake Ontario. I come to this conference because there is support for my ideas about teaching mathematics to all populations. This talk has a U.S. context, but I realize that our concerns in the US are similar to those of yours across the world.

There is nothing new here, but one of my jobs is to help educators who teach mathematics realize that we are in control of who can and cannot do mathematics. This is a large responsibility.

Before I forget, please consider following my Blog. https://wordpress.com/posts/ajourneytomathematics.blog

I will be posting my thoughts about my observations over the next several weeks, while observing in classrooms in Budapest. My interest is in how teachers approach their teaching in this Hungarian system.

I am collaborating with BSME (Reka) and with two other maths teachers Emese Gyorgy and Ödön Vancsó

For today, the small study is about how we prepare pre-service and inservice teachers who teach math to all learners. I hope to reveal their thinking, after what I deem to be my interventions through my class teaching. The paper is about what teachers say about clearly established approaches to teaching and learning mathematics. The revelations are the interesting part of this discussion, but I want to set up so you understand my approach to ensuring that those who teach mathematics have an understanding of the perceptions, expectations and behaviors that often reinforce the good teaching in the classroom and sometimes the not so good things that happen in the classroom.

There are very good data about how people learn, and more importantly how people learn mathematics. We are not confused about what is good teaching. The constructivists have it. There is a Hungarian way of teaching, and they have the corner on the market. There is research on math anxiety, but I think this may be about disaffection as quoted by Lewis, 2013. Is it really about how we socialize certain audiences to love, struggle, hate, or have some other passion for the subject? We have many research studies, which help us understand about procedural and concept attainment.

Using the history of Mathematics (as echoed by Douglas Butler this morning), as a means to humanizing mathematicians is also important. The personal stories about who we are as mathematicians is an important part of helping students learn the difficult concepts.

Donavan documented the research about learning mathematics through 3 basic principles. There are specific things that teachers can do to help students learn. I have to give Hilary Povey credit for her presentation this morning, because her list clearly explicated the list of things that can be done. Thank you Hilary.

But there are hidden things that reveal themselves (called dispositions by Hovey) around culture and stratification and misunderstandings about who can and should learn mathematics.

While in Benin West Africa, as part of a study abroad program for students at SUNY Oswego I had the honor of observing teachers, with almost no resources, except chalk, explain difficult mathematical concepts to eager students. Now I am not naive about the fact the mostly boys in that classroom had the personal, financial and cultural resources to succeed. The Girls I saw in these classrooms were amazing, and no one was saying that they did not have the capacity. They demonstrated the capacity and the teacher expected students to succeed. Yes I know these students know how to suffer through the standardized exams, so already are considered the top of their populations. I get it. I witnessed no disaffection for mathematics at this level. In the US as soon as students begin to struggle with mathematics, we separate students into distinct categories. Again, as Butler said, “there are things that all students should learn. There should not be a divide between those we believe have the capacity and those we believe to not have the capacity.

It appears, at least in the research that outside of the US that struggle is a part of learning. Struggle appears to be a part of any learning process.

Our Master’s degree students in mathematics education were required to take a diversity course. I spent the summer creating a course, “Math for Diverse Learners”. I gathered articles on Math learning and proficiency. Many of the articles are references, but this is a limited list. One of the ongoing assignments for the course required students to read about mathematics pedagogy. Part of the assignment was for them to write brief reaction papers with the following headings, cognitive, affective and conative. I then mapped their writings to Perceptions, expectations and Behaviors. I pulled the comments from their papers. I was hoping that students could reveal to me and to themselves the kinds of behaviors that assist students (k-12) in their learning.

I set up an elaborate scheme to summarize how good teaching could create a new generation of mathematics learners. I hoped to unpack the perceptions, expectations and behaviors through their writing.

These writings are from a small group of students writing for an assignment. I wondered, are they writing what they think I want to hear, or are they giving real accounts of their understandings.

I want this group of pre and in-service teachers to reveal to themselves their ability to be the gatekeepers for learning. I want the course to provide a window to access, through their writings. It Is important for my teachers to see themselves as gatekeepers. Take a look at gates here as barriers, but there are many gates such as the “Door to no return”, the Brandenburg Gate, the Golden Arch, which is a gate to the west. All openings. The mathematics writings I had students produce were an effort for them to see themselves as gatekeepers, not the barrier gates.

So let’s divert our thoughts a bit to contradict my intentions…

Play audio.

So I do believe that we have a way of looking at things that reveal what we have to do, but our hidden expectations have to be engaged in order to see changes in our mathematics teaching systems.

Marcia

 

 

Thank you.