cheesemonkey wonders

cheesemonkey wonders

Sunday, May 30, 2021

Piercing the Enchantment

Having survived more than a full year of online learning, I am readier than ever to return to in-person learning. 

I've been thinking a lot about what I've learned over the past year and what I will carry forward with me into the classroom as we return to in-person learning.

As a person with a lifetime of getting hopelessly lost, I have learned that once you notice that you are lost, the first thing to do is to just stop. Stop and get reoriented. This is one of the things I have learned through the power of meditation. When we meditate, we sit down and stop paying conscious attention to the crazy stories our panicked monkey minds are trying to tell us. We anchor our mind in our breath and just stay there. When the river of thoughts delivers another raging dumpster fire of crazy thoughts, we notice it, label it, and disidentify from it. Huh. That’s interesting. Another dumpster fire worth of crazy thoughts. And each time, we quietly return to anchoring ourselves in our breath. 

The more we do this, the less power these storylines have over us. As Suzuki Roshi says, we start to understand that what we refer to as "I" is really only a swinging door. Breath flows in, and breath flows out. When we anchor our panicking minds in our breath, we return to the safety and goodness of the present moment.

When I teach, I anchor each lesson in an Essential Question. Under ordinary circumstances, my Essential Question is always some variation on the meta-question, Why do I believe this is worth your time and attention today? But these have not been ordinary times. Instead of the usual four-plus hours a week of math class I have always had, under distance learning, we have had no more than an hour and a half of teaching and learning time together each week – for the whole week. Instead of synchronous time, we've had to make do with asynchronous learning experiences, which can be isolating, discouraging, and frustrating.

This has forced me to rethink my entire concept of the Essential Question for my classes. It's hard to keep the momentum going when you lose that day-to-day in-person connection. Many students reported feeling so alone without the daily contact of in-person schooling.

And so to help them – and to stay grounded in my efforts to help them – I changed the focus of my classes. 

I began to think of my class as an Essential Anchoring Place for students first and foremost. An anchoring place where we anchor our minds using math.

My Essential Question for each day turned into something like this:

How can I guide your attention to some things that can help you when you feel absolutely and utterly lost?

When you feel lost, the first thing to do is to stop. Just stop. Stop moving, stop striving, stop efforting. 

Just sit the heck down.

Reconnect with the body, with the breath. Return to the ground.

In meditation, we sit down and anchor our minds in our breath.

In Classical mathematics, I realized, we anchor ourselves in definitions. Definitions are mathematical bedrock. We bolt ourselves to them, making a conscious decision to take them to be true and move on from there.

We use definitions to orient ourselves. We do this not because they were handed down on stone tablets from Mount Zion or Mount Olympus but because for thousands of years, human thinkers have decided – as a thought experiment – to take these to be true and explore what can happen next.

These are our assumptions, and we acknowledge them as such.

And just as a map enables us to construct a working mental model of our journey, a mathematical definition gives us a working mental model of mathematical reality. A map is a tool – a good-enough humanly constructed tool that encodes our best, most reasonable, working understanding of how the world fits together. The journey we are on is a relay, and the maps have been handed down across generations for thousands of years. We are only responsible for our portion of the journey, though we inherit both the tools and the biases they encode.

Maps are cultural artifacts – texts which are products of the terrible racist systems in which they were constructed. They may contain some of the best thinking people were capable of, but they are also encoded with some of the worst, most wrong-headed, and most biased thinking of the dominant cultures in which they were developed. 

This is why we take them only as heuristics. They are imperfect pointers to a truth, not the truth themselves. As the Buddha often said, “My teaching is like a finger pointing to the moon. Do not mistake the finger for the moon.” 

So just as a map guides our thinking about how we journey in the outer world, a mathematical definition helps us to take mathematical journeys in the inner world of our thoughts and minds. 

Students learned that our foundational mathematics are built on definitions. We do not prove these – we take them to be true. 

Definitions provide an on-ramp for a crucial way of thinking in mathematics: the foundation of thinking in conditions. What is necessary and sufficient for a figure to be considered a circle? We start with its Classical definition: a figure is a circle if and only if it is the set of all points in the plane equidistant from a given point, the center. Where does it start? With a center. Does every circle have a center? Yes it does – by definition. What else does this definition tell us a circle has? Students fasten on the idea of a fixed and equal distance. Is the circle a set of points? How many points? Does the circle as a mathematical figure include the points inside the figure? How do you know?

The definition becomes my students’ friend. It contains a set of tests. What if one point of the figure were discovered to NOT lie in the plane? Would it still be a circle? Why or why not? How would you know?

The idea of a set of true-false tests becomes a foundation in which students can ground their thinking. They always know something, and if worse comes to worst, they can go all the way back to the ground of the definition. In this way, mathematics becomes a tool for getting yourself oriented. It becomes a culture and a community of belonging. We have to look beyond ego, beyond personalities, beyond individual likes and dislikes, to uncover what is true, enduring, unshakable. Definitions open a door to ways of thinking that have proven themselves to be durable and useful over time – over years, centuries, millennia. 

We really go all Platonic in our search for definitions, seeking out the perfect and ideal mental forms. We developed a crazy love affair with thinking in conditions. What are the necessary and sufficient conditions for an object to qualify as a member of this category? We start with real numbers and the real number line. We unpack the definitions of positive, negative, and zero. We go a little Aristotelian for a moment. Every real number has a fixed address on the real number line, and all addresses on the real number line fall into one of three categories: positive, negative, or zero, which is defined as being neither positive nor negative – the perfect inflection point. What does it MEAN for a number to be positive? to be negative? to be zero? How do you know? Slowly we construct an answer – these are by definition. They are what we are choosing to take as being true. 

This leads us to another mathematical love affair – the habit of thinking in cases. What is the set of all possible cases here? Is this a possible case? Why or why not? How do you know?

Students do a lot of casting votes in the chat window. “OK, does this figure meet all of the conditions required to qualify as a circle – yes or no? Don’t hit return until I count down.” I give them a moment. “Three, two, one, zero – hit return.” 30 votes pop up in the chat window. Yes, yes, yes, no?, yes.

I run an anonymous survey to see how things are going. Students tell me, “I like this class.” “This class feels the most normal.” I’m surprised. My Zoom policy/default is cameras-off. Many students were too self-conscious about their living situations. 

I worked with that. Every day starts just like my class would start in person. Slide with instructions, homework, and a countdown timer running for two minutes. Hawaii Five-O theme music playing loudly. I mostly used my iPad and Apple Pencil to do what I would have done on the smart board or document camera. More direct instruction than I would like because we have only 30 minutes together three times a week. Camera transitions and breakout room logistics eat up too many precious minutes. I give up after the first two weeks.

How do we reorient ourselves when we feel hopelessly lost?

We stop, sit down, and think about what a question is actually asking us. We allow ourselves to wonder what tools we have that could help us find the good-enough appropriate next step.

And then we do it again.


  1. I'll definitely have my preservice teachers read this. But it seems like it would be directly applicable to math learners, too. Do you share with them via your experience, like this, or ... or something more Cheesemonkeyish?

    1. I share with them as we go along, and there's a lot of repetition over time.

      When we do notes together, definitions are ALWAYS in blue ink, so when they have to look back for something, I always hear them saying to each other, "Definitions are blue." :)

      When we are lost as a class, I ask them, "What does the definition say?" Flip flip flip flip... "definitions are blue"... I hope that we are cementing in some habits.

      We active USE the definitions all the time, so my hope is that we are building in these habits.

      Thanks for your comment, John!

      - Elizabeth

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