I am coming to believe that, much like the concept of substitution, developing a deep understanding of the idea of betweenness is a huge part of the psychological and conceptual work of Algebra 1.
I have been dissatisfied for years now with the fact that we tell students about the geometric interpretation of absolute value, but we don't really get them to live it. And yet that idea of the "distance from zero" on the real number line is not something we give students time to really marinate in.
And you know what? That feels dumb to me.
So this week and part of the next, my students and I are really wallowing in that idea.
I've taken a page from my studies as a young piano student. In the study of the piano, there are certain studies of technique that really force you to slow down and take apart the finger movements. There are specific figures that you have to practice over and over and over so that they become part of your finger memory. How they feel in your fingers is how you come to relate to them.
This is not just about developing automaticity, although that is a side benefit. This is about learning to feel these foundational figures in your bones. In your body. They become so fundamental that as you learn and grow as a musician, you come to feel them when you see them coming up in a new score you are studying.
The technique does not replace musicianship. The technique supports the musicianship.
I've been noticing lately how my own experience of absolute value is about noticing boundary points at the periphery of my mathematical perception. I see them out of the corner of my mathematical mind's eye. And how an inequality is said to relate to them defines how I relate to those boundary points.
So I am taking the risk of sharing this mathematical experience with my students.
As with young piano students, we take this slowly. One figure at a time. Right now we are only dealing with the case of an absolute value being less than a nonnegative quantity. We are dealing with situations of betweenness, where an inequality presents us with a figural situation that is going to wind up with a quantity being between two boundary points.
That is all. And that is enough.
I see the effort in their faces and in their fingers as they rewrite, revise, calculate, solve, and sketch graphs. I see them noticing and wondering whether they need to use a closed dot or and open dot.
And I hear them developing the confidence that comes from experience in developing a relationship with these quantities.
They are not following rules. They are listening to their own deeper wisdom. Everybody knows something about the situation of being "between" other things. Betweenness is one of the most elemental human ideas.
They are making friends with mathematics.
cheesemonkey wonders
Showing posts with label Zen. Show all posts
Showing posts with label Zen. Show all posts
Wednesday, January 13, 2016
Sunday, January 3, 2016
Seating charts for equity
I can't imagine traveling to new lands and not wanting to try their cuisine. But there really are people who bring their own food with them. One of the best things about traveling in my opinion is being educated in the sense of the Latin root word — being led out of my own ignorance.
The same is true for me about attending a large, great school. It always has been. From the moment I arrive in a great new school, I feel excited and open to meeting and learning with all different kinds of people from different cultures and backgrounds. I want to expand my own limited world view.
But it seems inevitable that, without outside intervention, I often end up knowing and hanging out with the other Buddhists and Jews in any room. Cultural affinity is a force that possesses a tractor beam all its own. Fortunately, I am not the first to have noticed noticed this.
Our amazing counseling department and our Peer Resources program noticed this phenomenon too, and when they did their most recent student survey of our very large, urban, diverse student body, they put in some questions about this in their student well-being section. And the results were very moving to me.
Students overwhelmingly reported that when they first arrived at our school, they felt enormous pressure to connect with their cultural affinity groups. And for this reason, they reported, they deeply appreciate seating charts in classes that take this pressure away. This practice overwhelmingly helped them to feel that they fit in here and that those who are different from them in some ways are more like them in other ways than they are inclined to believe. It also created a zone of psychological and emotional safety to explore social connections with others not as "Others" but as fellow explorers in a safe space.
These findings touched my heart. Our kids' deeper wisdom never fail to blow me away.
So I sit here on the Sunday before the first day of Spring term making up seating charts, making sure that everybody arrives in my classes in the same boat as everybody else, and with the same opportunity to experience connection with others in as safe a space as I can create.
I will also pre-make Seating Charts #2, #3, and #4 so that it's convenient for me to change the seating without having to think. Sometimes "don't think" is the best rule.
I don't have any scintillating conclusions to draw here. I just wanted to document for myself what I am doing and why so that when I forget, I can more easily remember.
The same is true for me about attending a large, great school. It always has been. From the moment I arrive in a great new school, I feel excited and open to meeting and learning with all different kinds of people from different cultures and backgrounds. I want to expand my own limited world view.
But it seems inevitable that, without outside intervention, I often end up knowing and hanging out with the other Buddhists and Jews in any room. Cultural affinity is a force that possesses a tractor beam all its own. Fortunately, I am not the first to have noticed noticed this.
Our amazing counseling department and our Peer Resources program noticed this phenomenon too, and when they did their most recent student survey of our very large, urban, diverse student body, they put in some questions about this in their student well-being section. And the results were very moving to me.
Students overwhelmingly reported that when they first arrived at our school, they felt enormous pressure to connect with their cultural affinity groups. And for this reason, they reported, they deeply appreciate seating charts in classes that take this pressure away. This practice overwhelmingly helped them to feel that they fit in here and that those who are different from them in some ways are more like them in other ways than they are inclined to believe. It also created a zone of psychological and emotional safety to explore social connections with others not as "Others" but as fellow explorers in a safe space.
These findings touched my heart. Our kids' deeper wisdom never fail to blow me away.
So I sit here on the Sunday before the first day of Spring term making up seating charts, making sure that everybody arrives in my classes in the same boat as everybody else, and with the same opportunity to experience connection with others in as safe a space as I can create.
I will also pre-make Seating Charts #2, #3, and #4 so that it's convenient for me to change the seating without having to think. Sometimes "don't think" is the best rule.
I don't have any scintillating conclusions to draw here. I just wanted to document for myself what I am doing and why so that when I forget, I can more easily remember.
Wednesday, July 30, 2014
"The organism moves towards health" — reflections on TMC14
Everybody is writing blog posts about feeling like a fraud after an amazing experience at Twitter Math Camp 2014. Impostor syndrome. I feel like a fraud too, at least, most of the time, but I am trying to practice refraining from my conditioned habits of reacting automatically and giving in in response to that defense mechanism. I am practicing not-reacting. I am trying to notice the positive energy that is there and to just allow it. I am trying to allow myself to experience myself as a competent, good-enough teacher I have respect for and want to continue to be.
What worked in the Group Work Working Group session was setting up a structure to sustain that positive energy of presence. Having learned how to do that is a huge gift I have given myself over the past 25 years of dharma practice. It's a "gift" that comes from working very hard at being present and practicing every day, rain or shine, whether I feel like it or not, whether I do it well or do it badly. I follow the three teachings my teacher Natalie Goldberg learned from her teacher Katagiri Roshi: Continue under all circumstances. Don't be tossed away. Make positive effort for the good. I have done that in my practice every single day for over 25 years. It's the one thing I know in my feet that I am good at.
So I decided to bring THAT to Twitter Math Camp this year.
The structure works because it is a structure for teaching and sustaining presence — learning to be present with an open heart. I have dharma sisters and brothers all over the world, but when we practice together online, we practice asynchronously — each of us on our own, in our own lives, in our own homes. When we come together using the asynchronous forum of online communication, we maintain that same structure of presence. Write when you write. Read when you read. Listen when you listen.
No comment.
"No comment" is the most important part of the structure, and the hardest part to implement online in a forum like Twitter, which is designed to support comments. "No comment" is about allow there to be space for everyone. It is about all being present together authentically and about staying present with whatever arises. THAT is the thing that most adolescents don't get exposed to in their lives, and it is the thing that can make the greatest possible difference in the quality of their experience — both in the math classroom and everywhere else in their lives.
Most people in our culture don't have a lot of experience in being present and staying present. It takes an enormous amount of energy to learn how to stay present and not flinch. But do that with anything you love and you will have a magical experience. Do that with math, and you'll unlock the treasures of your amazing human mind. But being present with others in a big open space is hard. At first it can be scary. It's very naked. That is why the structure of "no comment" is so important. It helps to create a shared space of emotional safety. It gets everybody focused on their own stuff and supports dropping the "act" you bring to most in-person interactions. That's why it's good to do so right from the start. It's about reframing our conditioned habits of personality.
Very quickly, the timed structure and the practice of "no comment" makes the practice of presence very freeing. You begin to relax into that big open space. You become curious. Your defenses soften. You begin to notice the interesting patterns of your own mind. Best self and worst self. Curious self and bored self. Zen mind and monkey mind. Defense mechanisms, such as snark.
The practice of "no comment" creates a space in which the authentic thoughts of your own amazing human mind can arise and step forward. And we honor that process by persisting in not-commenting as we continue.
Natalie describes this process as stepping forward with your own mind.
Once you get a taste for being present, you'll naturally begin to crave it more. That is something I count on in my classroom management practice. Fred always said, "The organism moves towards health." That is one of his greatest teachings for me. "The organism moves towards health" means that, in the process of growing up, we all fall away from the naturally sane and healthy patterns of our organism. "Fight or flight" is a falling away from the natural discharge cycle of "rest and digest" we experienced as infants. When you're hungry, you eat. When you're tired, you sleep. Fred said there is a deeper wisdom inside us that is always available for us to tap back into. It's like an underground stream that is part of our psychological and emotional water table. When we practice being present through structures like Talking Points or meditation or writing practice, it feels like a homecoming — a homecoming to a natural state that is healthy and inquisitive and curious to see what will happen next. It is a natural reconnection with our own inner growth mindset that is our birthright — not some artificial fantasy state we impose on students from without by telling them to have one.
A growth mindset is just the psyche's way of attuning to the fundamental idea that our organism moves naturally in the direction of health if we will let it — if we can get out of its way and allow it to unfold as it needs to. Allowing means learning to refrain from interfering with that natural movement, and so we use structures that make it manageable for ordinary human beings like us to access the extraordinary ocean of intellectual and creative possibility that is mathematics.
Ten minutes at a time is about what I can muster, I have learned over the years.
In my experience teaching meditation and writing practice and other structures that cultivate presence, I have found it is about what most people can handle. Ten minutes of Talking Points, no comment — GO. Ten minutes of writing practice — keep your hand moving, no comment, GO. Ten minutes of mathematical conversation, no comment, GO. Learning how to be present with the big, scary openness of not-knowing is no small thing. That is why we zone out, check our phones a hundred times an hour, play video games, watch TV, assault-eat, numb out, zone out, distract ourselves. We all crave the real stuff, but connecting with it feels like sticking a butter knife into the electrical socket. So we break it into more manageable chunks. We set a limit for ourselves and dive in for a limited period. We practice being present for ten minutes at a time. And then we give ourselves and our students a break. It helps us build our tolerance for the intensity of presence and it builds our courage to come back and try it again the next time.
Natalie says that monkey mind is the guardian at the gate, protecting the treasures of our heart and strengthening us for the challenge of opening ourselves to presence and to Big Mind. The structure of "no comment" makes it feel safe for us to touch in to that fire at the center of our being. It helps us to close the gap between what we THINK we've been doing and what we have ACTUALLY done. For me, it's about strengthening students' courage to open their hearts to contact with their amazing mathematical minds — with what my friend Max Ray of The Math Forum at Drexel calls their "mathematical imaginations." We math teachers know the secret that everyone has this mathematical imagination. Our greatest challenge is to get students to trust that they have it too and can access it safely and reliably.
________________________________________
TRY THIS
Read Taming Your Gremlin by Rick Carson.
 |
| me practicing accepting myself as a competent, good-enough teacher, seen here with supportive tweeps & a giant margarita |
What worked in the Group Work Working Group session was setting up a structure to sustain that positive energy of presence. Having learned how to do that is a huge gift I have given myself over the past 25 years of dharma practice. It's a "gift" that comes from working very hard at being present and practicing every day, rain or shine, whether I feel like it or not, whether I do it well or do it badly. I follow the three teachings my teacher Natalie Goldberg learned from her teacher Katagiri Roshi: Continue under all circumstances. Don't be tossed away. Make positive effort for the good. I have done that in my practice every single day for over 25 years. It's the one thing I know in my feet that I am good at.
 |
| starting with restorative classroom circles |
So I decided to bring THAT to Twitter Math Camp this year.
The structure works because it is a structure for teaching and sustaining presence — learning to be present with an open heart. I have dharma sisters and brothers all over the world, but when we practice together online, we practice asynchronously — each of us on our own, in our own lives, in our own homes. When we come together using the asynchronous forum of online communication, we maintain that same structure of presence. Write when you write. Read when you read. Listen when you listen.
No comment.
 |
| using the Talking Points structure; no comment |
Most people in our culture don't have a lot of experience in being present and staying present. It takes an enormous amount of energy to learn how to stay present and not flinch. But do that with anything you love and you will have a magical experience. Do that with math, and you'll unlock the treasures of your amazing human mind. But being present with others in a big open space is hard. At first it can be scary. It's very naked. That is why the structure of "no comment" is so important. It helps to create a shared space of emotional safety. It gets everybody focused on their own stuff and supports dropping the "act" you bring to most in-person interactions. That's why it's good to do so right from the start. It's about reframing our conditioned habits of personality.
Very quickly, the timed structure and the practice of "no comment" makes the practice of presence very freeing. You begin to relax into that big open space. You become curious. Your defenses soften. You begin to notice the interesting patterns of your own mind. Best self and worst self. Curious self and bored self. Zen mind and monkey mind. Defense mechanisms, such as snark.
 | |
|
Natalie describes this process as stepping forward with your own mind.
Once you get a taste for being present, you'll naturally begin to crave it more. That is something I count on in my classroom management practice. Fred always said, "The organism moves towards health." That is one of his greatest teachings for me. "The organism moves towards health" means that, in the process of growing up, we all fall away from the naturally sane and healthy patterns of our organism. "Fight or flight" is a falling away from the natural discharge cycle of "rest and digest" we experienced as infants. When you're hungry, you eat. When you're tired, you sleep. Fred said there is a deeper wisdom inside us that is always available for us to tap back into. It's like an underground stream that is part of our psychological and emotional water table. When we practice being present through structures like Talking Points or meditation or writing practice, it feels like a homecoming — a homecoming to a natural state that is healthy and inquisitive and curious to see what will happen next. It is a natural reconnection with our own inner growth mindset that is our birthright — not some artificial fantasy state we impose on students from without by telling them to have one.
 | |
|
Ten minutes at a time is about what I can muster, I have learned over the years.
 | |
|
Natalie says that monkey mind is the guardian at the gate, protecting the treasures of our heart and strengthening us for the challenge of opening ourselves to presence and to Big Mind. The structure of "no comment" makes it feel safe for us to touch in to that fire at the center of our being. It helps us to close the gap between what we THINK we've been doing and what we have ACTUALLY done. For me, it's about strengthening students' courage to open their hearts to contact with their amazing mathematical minds — with what my friend Max Ray of The Math Forum at Drexel calls their "mathematical imaginations." We math teachers know the secret that everyone has this mathematical imagination. Our greatest challenge is to get students to trust that they have it too and can access it safely and reliably.
________________________________________
TRY THIS
Read Taming Your Gremlin by Rick Carson.
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