Mistakes and Erasers

Today I was looking at my child's online math learning platform, and I stopped on some very pointed instructions made in a video lesson, "If you made a mistake, cross out the mistake, and write the correct answer next to it. Don't erase."
I've long struggled with finding ways to address my child's fear of imperfection, and my students', and my own. I have a policy in my classes that you should do your math homework in pencil and erase your mistakes. When I bring it up, I always reiterate, "Remember you will make mistakes. Not because of anything about you, just because you're human. Be prepared." I have the "write in pencil and erase errors" instructions on all exams. I'm wondering how much impact it has to make a switch to encouraging people to keep their mistakes visible rather than erase them.

I looked for some research specifically on the question of whether having students erase mistakes is helpful or not, and I came up with basically nothing. Lots of research about the value of de-shaming mistakes. Here is one particular article that sort of summarizes some of the research well, as well as a video of one 7th grade teacher's approach to helping her students learn from their mistakes.

One of the things that I've sort of quietly struggled with in the past, and I've not ever taken the time to make a decision, is how much I treat in-class quizzes and exams as finished pieces of work. I've always cringed at disorganized work and big messes on exam pages, and frankly have punished students for it. I have a concept category of "Organization" that I judge for every piece of work that is turned in. It's totally wrong to do this. When you're performing a type of math that you already know, then you can have the cognitive capacity to attend to the presentation of the work. But if you're working out your understanding, then it'll be messy, and there will be mistakes along the way. And I want to give credit to students who are working out their understanding on exams and quizzes. The neatness and organization shouldn't be assessed until 2nd-draft presentations. So, maybe what it should be is that students get two shots at an exam: an original, and a make-up. The make-up can be done on the students' own time, and they are asked to be very tidy and organized for that one.

Back on pencils and erasing: I did find this blog post, which contains the anecdote of a person receiving the gift of a pencil from their mentor, with the following wisdom of an eraser.

1. Erasers allow us to correct mistakes, and mistakes are correctable. Everyone makes mistakes. You have to erase them, and let go of the mistake.
2. Erasers give us the power to adjust, and our best work requires adjustments and updates. Your plans absolutely must be written down, and they need to be written in pencil. The world and your situations will change. The eraser will help you be more flexible and adaptable.
3. Erasers give us a safety net, and remind us that the pressure isn’t as great as we might think. Think about it, if you do a math assignment in pen, the pressure is on to get the answers right the first time. This is not an excuse for doing poor work, but having an eraser does take off some of the pressure.
4. Erasers invite us to edit, and results of a second (or third) draft are almost always better. If you don’t succeed (or succeed as completely as you hoped), the eraser lets you make edits and try again.
5. Erasers erase the past, and help us focus on what’s next. Yes, you must learn from the past, but you must be forward focused. The beauty of a great eraser, like the one on the pencil I’ve given you, is that it erases well enough that you CAN focus on what’s next.

While I appreciate the sentiment of this advice, I have to poke at some of its lessons. The biggest one is just generally the concept of erasing mistakes. I have made sooo many mistakes in my life, and I am a better person for them. I understand so much more because of them. If you ask me whether I regret having made the decision that put me into a really bad situation, almost every time I will answer with a no. I regret having had to live through certain situations, in the sense that I remember the pain of living through them, and often still feel the pain in the remembering; but I do not regret in the sense where I would choose something different if given a time machine. I was married at a very young age, and I got divorced at a similarly young age. The whole process of my divorce was incredibly painful for everyone involved. But there is no way in the world that I would give up everything that I had during that 10-year relationship for the sake of "erasing" that mistake. Actually, truth be told, I don't necessarily think it was a mistake for us to marry; but it would have been nice if we could have gone into it with more of an agreement that either or both of us might choose something different later (here's an example of where we might benefit from normalizing error). 

I love the notion of erasers giving you an invitation to edit, but I don't think that erasing is a great metaphor. For one thing, second and third and fifteenth drafts are never done on the same sheet as the first draft, so the eraser is irrelevant in this analogy! Most editing of physical documents happens on computers these days, and in fact it's helpful to have the original draft with all of the editing marks all over it, so you have the opportunity to puzzle through the edits and the comments.

I actually really like the metaphor of editing. When we edit, we take the time to look at the previous draft and make improvements. In the end, we throw away the old draft and only keep the new. But we don't forget that the old draft existed. When we have a final product, we remember how many drafts we went through, and we will remember a few of the specific edits we made. This feels right to me, in terms of how I edit my personal life and self. I have the me that is present, and I remember lots of what went into the making of me, including some of the specific changes that I've made along the way. But none of those past drafts of me are hanging around for others to see. They exist in my memory, and in the memories of others, but those old memories are of previous drafts. If anyone asks who I am now, the best way for them to find out is to get to know the me of now.

I think the points about having a safety net for experimentation and a tool for focusing on the future rather than dwelling on mistakes are fine, but I don't love erasing as a metaphor. Sometimes there is joy in all of the weird things that you did along the way of experimentation (Post-It notes, anyone? Silly Putty?). And again, focusing on the future does not require forgetting mistakes of the past; in fact it usually requires remembering mistakes of the past.

Going through this metaphor of erasers of life, I think I've decided that I'm not strongly in a camp of encouraging students to erase their mistakes or to keep them, but I do think we need to have better normalizing of the making of mistakes. We need to have a culture that values experimentation as a way of progression, or even just getting things wrong sometimes and being willing to correct errors. For my teaching, I think this means I'm going to change the way that I grade in-class work on neatness; maybe I'll only grade on neatness for less-frequent projects that are done at home.

Lessons From Learning French: Brain Growing Pains

As a teacher, I've been happy to share information about brain plasticity with my students. I talk to them about the fact that adults in fact can learn, and I love to share the stories of the London taxi drivers and the observed physical change that occurs in their brains when they study for and pass the test, ominously called The Knowledge, of all 25,000 London streets. I also talk to my students about ways that knowing about brain growth and plasticity can help us be more kind to ourselves. We talk about analogies of the brain being like a muscle, where you have to build up strength and endurance by practice, and how stopping practice causes atrophy. Through this understanding, we can be patient with ourselves as we struggle through new concepts, changing "this makes no sense to me; I must be because math isn't my thing" to "this is completely new to me, so it'll take some time for my brain to adapt to it." Also through this analogy of the brain to the body, we talk about fatigue associated with mental effort. So I ask my students to be patient with themselves if they are extra tired in the first few weeks of a school year, letting them know that it is an expected side effect of increased mental effort. [Side note: I just got distracted by this podcast about connections between mental and physical endurance... It's fascinating!]

Because of this awareness of brain plasticity and brain energy consumption, I was not particularly surprised when I experienced some of these side effects on my path to learn French. However, I was surprised by some of the ways in which they manifested (and the strength with which they manifested).

A little background: I started practicing French in June 2018, entirely through Duolingo. I don't remember exactly, but I believe I was practicing daily, with very little failure, up until the point where I started working with tutors in mid-September. I started with 3 tutors, meeting one of them twice per week and the others once per week. Needless to say, this completely changed the way I was learning. Rather than just following the routine provided by one app, I was introduced to new language learning supports, such as spaced repetition software, grammar support sites, and pronunciation support sites. So in the matter of days, my initial practice of spending about 30 minutes per day following prescribed routines turned into 3-4 and sometimes 8 hours per day practicing all sorts of new things, including flashcard studying, podcast listening, grammar researching, pronunciation practice, and also having live conversations in French.

I looked back on my timeline, and I can see that my first tutoring session was on 13 September. I remember around this time that I was tired. I was so tired, all of the time. I would wake up, have a morning tutoring session with my tutors in Columbia and France, then often I'd have a nap. I was certain it was from the effort of learning French. Then on 01 October, I emailed one of my tutors about what I later started to refer to as my "French headache."

It was the type of headache that stopped me from being able to function. In my memory it was less about pain, but like a balloon being blown up inside my perception field, squeezing up all of the space between the functional me and the tools I needed to access language. I was always closing my left eye, in response to the sensation. That Monday morning I had a tutoring session in which I couldn't grasp any French words or phrases, even the ones I knew. I thought it was just a regular ol' headache. However, my experience throughout the day told me otherwise. After I hung up the morning tutoring session, left eye closed, I took some aspirin and did some things in the kitchen. I had decided that I was going to take it easy and maybe watch a movie until the pain went away, but within minutes I was fine. Eager to get back to work, I opened up my computer and opened my flashcard software. Just a few flashcards in, my headache came back full force and I was once again one-eyed. I thought, hmm, that's odd. So I put the computer down and did something different. The headache went away again quickly. Still, I decided to give myself a rest, so I didn't practice any more that day. And I didn't have any more headache, until later I picked up a crossword puzzle. As I was searching my brain for words that matched clues, again I felt the pressure of that balloon squeezing my brain.

It's been a long time since I've had the same intensity of learning and practice in French, but occasionally I will still get my French headache, especially if I consume lots of French after several days (or weeks) of slacking off. In my mind, I equate it to the muscle fatigue of doing bicep curls to the point of exhaustion, and then not being able to lift a single book afterwards. Another thing that happened, all very related and I believe in the same time period, was that I got actual muscle fatigue in my mouth. The uptick of intensity of practice wore out all of the muscles in my mouth, as I was exercising those muscles differently for the French pronunciation. I remember my lips and my tongue being tired and feeling like useless lumps when I tried to speak. It was astounding! I think that if I didn't have the background mindset of thinking about brain growth and the related fatigue, then between the constant fatigue, headaches, blurred vision, loss of vocabulary and slurred speech I would have worried I was having symptoms of a serious health problem.


I don't expect that my students will experience this level of content-specific brain fatigue, mostly because I don't think they'll be putting as many hours into a single subject (I had the privilege of being on sabbatical). But I do think it's an experience worth sharing, to give students an awareness of some of the possible side effects of learning new things. For me, it was helpful to not worry myself about the symptoms I was feeling; but also it really excited me because it was evidence that my body was physically responding to the work that I was putting in. It was almost like I could feel my brain growing!

Lessons from Learning French: Being Corrected

I've been learning French since September 2018, with varying degrees of intensity at different time periods, and it has been super interesting to be in this position of focused learning, which I've not experienced for a long time. As such, I'm constantly reflecting on my experiences and trying to put them into a greater context of understanding how learning works, and of course translating this into directives for my own teaching.

In this particular post, I'm going to focus on the concept of error correction, specifically when other people correct my errors for me, and the role it has played in my French learning.

When I decided to learn French, I read lots of stuff and watched lots of inspirational videos that convinced me I needed to be speaking to other real human beings from Day 1, and I needed to embrace the fact that I would definitely make errors. I did not speak to other humans from Day 1; instead I used the paid version of Duolingo for daily practice. When I finally got an online tutor three months later, it was clear that my learning progress took on a completely different pace. (I tell people that I've been learning French since September because I really think that's when I started learning, despite having practiced almost-daily with Duolingo since June.) I got on well with my tutors and made lots of progress quickly, much because I was willing to try things and make mistakes. They would correct me sometimes (often, actually), and I would say, "Merci," and I would repeat the corrected form that they gave me in order to continue with what I was saying. Sometimes this led to some discussion of "oh, I thought that I was supposed to say it this way; please explain the difference," which would derail the conversation at hand in favor of learning something about the language. In general, I tried to embrace the corrections, thanking my tutor and moving on without any negative feelings about being corrected.

Everything felt very different when I spoke with friends or colleagues in French (it turns out that I have quite a few friends who are fluent). If they corrected me, I still tried to embrace the corrections and thank my companion, but inside it felt very different from how it felt when I was corrected by my tutor. In social context, being corrected was painful, and often offensive. I actually stopped trying to speak French with my friends because of this. It just doesn't feel safe. In fact, when I saw a friend who told me she was learning French in school, and I shared some experiences of my learning, I was actually surprised to hear myself say, "Your French teachers will tell you that fluent speakers will embrace you and be supportive when you try to converse in French, but that's actually not true in my experience." Now, the reason that I said this at the time was not so much about being corrected, but about the needs in the conversation. Which is to say, whenever I spoke French with friends, it took less than 5 minutes before they became impatient with my pace of speech and reverted to English, because they wanted to actually communicate, and I don't yet have the skills to do that in French. To me, this is different from, but related to, the emotions felt around being corrected when trying to speak.

While I no longer speak French with friends, I still speak French with my child, largely because he's still learning too (he started French-English bilingual school in September), and he is happy to play with the language, sometimes intentionally making mistakes because he thinks it's funny. We often communicate in a playful form of "franglais." It's the best.

The frustrations with being corrected came to a head a few weeks ago when I went to a French language meet-up. I put on my brave hat and invited my friend (who, I gather, hasn't spoken any French outside of classes). It was clear when I walked into the meet-up that we were the least fluent people there. This was my first time speaking French in a group like this, and I panicked a bit when people came up to me rattling off a bunch of French, so I led my conversations with deer-in-the-headlights looks and disclaimers of the form, "je suis une débutante; je suis lent" (I'm a beginner; I'm slow). After that, people were generally very friendly, and they seemed to slow down and simplify their conversation when speaking with me. It was mostly fine, and I actually remained in French for over 2 hours. That's a solid 4-6 times as long as I had ever stayed fully in French up to that point. C'est super, non?

The difficulty came with one person in particular, D, who kept correcting me. These corrections had such an impact, that when I remember the meet-up, my "general impression" memories revolve around resentment over those corrections, rather than joy and pride for having made it out and participated with other French speakers for a record amount of time. To be clear, D was being super supportive and kind. She was patient as I slogged through finding words, and frankly I felt like she was patient just in talking to me, since there were many other people there who were much easier to converse with. However, the corrections were not helpful, and here's why. When I'm conversing in French, it takes every bit of my attention to stay with the conversation. I'm focusing acutely on the sound and appearance of someone who is speaking, to get every bit of context I can get to help me understand what they're saying. Similarly, when I am speaking, I'm working extremely hard to connect my ideas to words that I know, constantly flipping through my own inner French dictionary, and often referring to my inner English thesaurus to find a word that I actually know how to translate. All of this focus means that my field of attention is extremely narrow. I've found that if I don't understand at all what someone is saying, and I ask them to repeat a sentence very slowly, then about the 8th word or so, I've already forgotten how the sentence began. This means that having context is critical for me to be able to follow a conversation, so I have a framework to place sentences into. And if I have that framework, then oftentimes I can leave some individual words unknown without interrupting, because I can generally get the gist from the context. It doesn't make for great nuanced conversation, but I find that it allows for fuller communication than if I stop to fill in all of the gaps, because it maximizes the flow of ideas and the building of the framework of context. This is what I was battling with the corrections. I was engaging in conversation, using all of my brainpower for mapping my thoughts into French words, and then listening to the French words that came back to me and attempting to map them back into thoughts. I really needed context for that mapping to work. Here's one example where I got left out on a limb: I had tried to tell a story about how my first non-English language was American Sign Language, and when I learned Russian in college, any time I didn't know a word in Russian I would revert to ASL. Conveying this was extremely difficult, and I was helped a lot by hearing other people express the same phenomenon with their language learning. Apparently it's a common thing that when you're learning a third language, the second language comes up to fill in gaps in vocabulary. When I tried to share my own experience, I didn't know how to say sign language in French, so I asked, "comment tu dis 'sign language'?" (how do you say 'sign language'?) When nobody knew, I quickly moved on and continued my story, inserting the English phrase 'sign language' wherever it was needed. My goal was to convey an idea. I wasn't sure of my grammar, and I knew that I was throwing in an English word, so I was intensely checking in on other people's responses to see if I had said what I meant to say. D responded to what I said, and her response went on and on, and I was having troubles mapping any of her words to anything meaningful. I struggled and struggled to attach meaning to her words, trying to fit them into the framework of the conversation, and I was only able to grab a few isolated words per sentence. After what felt like a solid 3 minutes of her talking nonstop, I finally realized that she wasn't responding to me at all, but she was instructing me on what to do when I didn't know a word in French, rather than insert the English word. When I noticed this, it completely deflated my momentum and made me question whether I should engage at all. It really felt like, if people are going to correct the way that I speak rather than respond to the content of what I'm saying, then there's no point in me even trying to convey my ideas because there's no way for me to do it without making mistakes, and I want to talk about my ideas and not about my mistakes. It also felt like a clear communication that I didn't belong there. Other people at the meet-up were allowed to have real conversations about whatever they liked, but I was limited to having conversations about my use of the language. It felt pretty awful.

So what do I do with this understanding? How can I use my understanding about the experience to more forward and progress more comfortably? And how does it figure in to my teaching of mathematics?

The first thing that I am going to do is to practice some things that I can say to people when they correct me as I'm speaking French. Here are some things that I would say to my friends in English, as a way of setting some guidelines on correction. I'll work with my tutor to construct phrases that convey similar meaning in French, and to practice conversations where I can insert them.

• Please don't correct my French.
• I have a tutor who can help me with my grammar. Right now, I'm trying to have a conversation.
• If you are correcting my grammar, then you are not listening to what I'm saying.
• If you are correcting my French rather than responding to what I'm saying, that conveys a clear message that I'm not welcome to speak with you in French until I can do it without mistakes.
• If my French is so bad that you don't understand what I'm saying, then please ask me questions to help me clarify. For example, "Do you mean to say..." will help me find the right words to convey my meaning, while staying in the context of the ideas that I'm trying to communicate.

Next, I'm thinking dually about my French learning and my teaching of math, and I'm wondering what the guidelines should be on corrections. It seems clear to me that in social contexts I want pretty strict adherence to the above guidelines. But what should be the guidelines when I'm working with my tutor? I'm certainly open to having conversations that are entirely about language construction, but I also want to have the opportunity to practice just speaking, without being corrected. I spoke with my tutor about this, and we agreed that in each 45-minute session, we'd speak only in French for 30 minutes, and then we'd have the freedom to speak in English for the last 15 minutes. Furthermore, for at least the first 15 minutes of the French-only portion, we will follow the social guidelines outlined above, whereas the second 15 minutes he could correct me if he thought it appropriate. I think part of the goal here is just for me to have a safe space in which to practice this way, and see if those really are the guidelines I want. The reason for allowing the last 15 minutes to have English is to get across some of the concepts without laboring through the language. For example, it may be helpful to dissect the grammar and say, "the reason that this word is in this form is because it's the direct object of the phrase," and I'd rather just hear that than have to slog through French in order to get that understanding.

I'm also not sure how this translates to practices I should use as a teacher of math. One of the distinctions I make with the French conversations is being able to say, "Please don't correct me or teach me when we're communicating socially; I have a tutor that covers that." This suggests that I'm only open to being "taught" when I'm in a learning-specific environment. On the face, this could be translated into an understanding that it's okay to correct students in class, because that's their learning environment, and I just won't correct them when we're mathing socially. But that seems incredibly wrong, as I know from experience that I need to sometimes back off of correcting in the classroom, so I'm trying to figure out the nuances here. Back on French, it's actually not true that I'll always be offended if I'm corrected in a social situation, and I'm also not okay with being overly corrected in my tutoring sessions. I know that as a teacher, I have been tuning my sensitivity in mathematical conversations, and I'm very selective about where I correct my students. I used to correct students a lot, and now I correct them much less. I can't say that I know exactly where I draw the line on whether to correct a mistake or not, but I think that these are my rough guidelines:

1. Respond to the content of what the person is saying, rather than the words that they are using.
2. Don't interrupt what someone is saying in order to correct a mistake in what they said.
3. Don't correct someone within the first ~10 seconds after their mistake.
4. Allow silence, to give space for the person to reflect on their work and maybe notice their mistake independently.
5. Assess: Is that mistake something that is critical to their understanding of the overall concept being discussed? If not, then maybe let it slide without comment.
6. Assess: How fluent is the person with the overall content? How much cognitive focus are they using to be able to even converse on the content? If they have general fluency but have made some mistakes, then they may be able to hear a correction, assimilate it, and move on. If they are struggling with basic skills, then the correction could feel like a roadblock that completely diverts the conversation.

When I'm working with students (or with my child), I'm tuning in to what they are saying and how they are saying it, and picking up on lots of cues to see whether and how they are open to receiving feedback. Every mistake I witness, I do a fast assessment of whether there is a net benefit of correcting it. I think this is a non-trivial skill, and I'm quite proud of my progress in developing it, because I think it's one of my more valuable assets as an educator and a parent. Ironically, though, I don't know what the parameters are of this skill. I don't know exactly what cues I'm picking up on and what decisions I'm making. This is why I'm trying to dissect it here, so I can make sense of it and develop it more intentionally, both in spaces where I'm the teacher and in spaces where I'm learning. I also want to have some way of communicating these guidelines to my students. I think they should be aware of some guidelines so that they may also advocate for themselves. They should be able to say, "I know that my language is imperfect, but please respond to the content of what I'm saying."

When I think about the effects of correcting on learning, I always mentally review a particular interaction I had with my child and his father. When my child was still at the beginning stages of learning to read and still super timid about writing, there was one evening when he was on my lap and was writing out a word. He was clearly laboring over every single letter that he wrote. His entire body was tense as he wrote, and he was holding his breath for each stroke then exhaling and taking in a new breath for the next stroke. When he wrote one of his letters backwards, I started my internal assessment of whether I should correct it, and decided that the first approach would be to just wait and see if he caught his own mistake. But before he'd had time to review his writing, his dad spoke up and pointed out the mistake. I gave Dad a sort of "back-off" glare, which he did, and then we talked about it later when the child wasn't around. From my perspective, I was coming from from having developed some intuition about when to correct, based on years of interactions with students. I see when my corrections cause a student to freeze up in shame, and when they don't. I think I'm quite good at assessing when and how is a good way to correct mistakes, but I didn't necessarily have any facts to back up my intuition. My child's father was sitting on a logical argument of, "I don't see the benefit of allowing him to make that mistake, and then have it engrained in how he writes until it gets corrected later." (I believe this is referred to as fossilizing, which seems to only be a term used in the context of language learning, but it seems like a thing that can happen in any field.) I can definitely see the appeal of that position, but intuitively, from my thousands of interactions with real learners, I know that it is fundamentally flawed, and that sometimes it's better to let a mistake go in favor of giving the learner space for what they need. It's really hard to get a meeting point when one person is standing on their logic without having contextual experience, and the other person is standing on intuition that hasn't been logically validated, and so it went for that conversation. We did not agree about whether the child should have been corrected at that juncture. However, I again reviewed this situation when I was struggling with being corrected with my French learning, and this prompted me to see what research there is on the effects of error correction on learning. I found one metastudy: The effect of error correction on learners' ability to write accurately. I did not read the whole paper, but the Abstract says that they found, overall, that error correction doesn't seem to be helpful, and it can be a little bit harmful. Here's an interesting quote about one study they reviewed:

Sheppard (1992) compared an ESL group that received extensive correction over a 10-week period to one that had identical instruction but received only content-oriented comments, including marginal statements saying when a portion of the writing was difficult to understand. Students in each group had individual conferences with the instructor, in which they talked entirely about their errors (the correction group) or entirely about meaning (the content group). In the final results, the content group had significantly higher scores in marking sentence boundaries, yielding a large effect size (d) of .939. On accuracy of verb forms, the content group obtained nonsignificantly higher scores (d = .478). Thus, correction in this study was not only ineffective but also probably harmful to students’ learning, relative to providing feedback only on meaning. 

This resonated with me on my experience in learning French, as well as the intuition I've developed from working with learners. The message is,  interact with learners on a content level, allowing for requests for clarification if needed, but don't focus on the correctness of the actual delivery.

What Colors Are On Children's Books?

Last week I led a data collection activity in my child's Kindergarten class.

We began with a little game of Which One Is Different (adapted title of Which One Doesn't Belong; I'm sensitive to language around not "belonging" as related to being different... plus, the acronym WOID is pronounceable).

I had hoped to introduce this game using Christopher Danielson's WODB book a few days beforehand, so they were familiar with the game before my lesson. But it didn't happen, so I took a little risk in just jumping in with books. My child wanted to introduce the rules, which he did, and the kids jumped right in with noticing differences. It was magical. They noticed all sorts of things, including that The Book of Mistakes was the only one where the author's name was written in green, and Du Iz Tak is the only one where the title is written inside a cartouche (and we learned the word cartouche).

This was a primer for an activity of gathering data about the colors found on the books in the classroom. For this, the students were split into pairs, and we gave each pair one data collection chart and four books: two written in English and two written in French (it's a French-English bilingual school). I first modeled what they would do. For each book, they should put a mark in the row of each color that appears on the cover of the book. The chart looks like this:

I modeled with two books written in English, asked if there were any questions, and then asked for some predictions about what we might find. There were lots of blank faces, so I asked what color they thought would come up the most. Two kids  we then set up the pairs of kids.

Afterwards each pair came to me, and I recorded their data on a larger chart that I made ahead of time. I had built the chart with the intention of using post-it page flags, but when it came time to do it, I found that the page flags were a little fussy, so I just put check-marks in the boxes. They're not removable like the flags, but I decided that the efficiency of marker won over removability of flags.

Here are all of the individual data charts that the children made (with my demo chart on the far right):

Here is our compiled data:

We noticed that blue, black, and bleu were tied for the most books. One child thought that he saw equality between blue and yellow, and there was some discussion about that. Even with lines drawn on the charts, he was convinced that they were the same. There was a little noticing that the English books seemed to have more check marks in general. And we noticed that violet was used very little on the French books, which sparked a conversation among the teachers about how they don't see purple being used much in French culture, or in clothing.

I finished off by introducing the terms data and data scientist, in the context of what we did. I asked if they had any questions that might be answered by collecting data. Their response was crickets and blank stares.



Overall, it was super fun to come in and just do stuff with the kids. They really got into their tasks on the project. I think I had hoped for a little more participation on the noticing and wondering, and I was definitely hoping that they would have some fun ideas about what else they could collect data about, but I'm not all that surprised. Maybe the concept was too new for them to be comfortable with asking questions of their own, and having more subsequent activities would spark some ideas. Alas, the year is coming to an end, so I won't be able to do more activities. I still consider it to have been a delightful and worthwhile experience on its own.

Oh... one more thing: I realized about 40 minutes before the lesson that I had forgotten to learn all of the students' names. I quickly made an Anki deck from a pdf we got at the beginning of the year, and reviewed for about 10 minutes. I was able to call on each child by name, which felt really great. I haven't reviewed the deck again since that day, but today at the Kindergarten year-end performance, I was able to name all but 2 kids (in my head) when they came up on stage. I'm so using it to learn the names of students in my own classes in the fall.

Early Morning Contemplations

If you cut a square out of a piece of paper, do you have a square?

No. It's a piece of a cube, because the paper has height, or else it wouldn't be there.

So, does this mean that squares don't actually exist?

No, because you can make one on your computer.

Lines are 2-dimensional, because they have length and width (and if you put it on paper, it would also have depth). So maybe there's no such thing as a real line, even on a computer.

(moooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo)(moooooooõooooooooooooooooooooooooooooooooooooooooooooooooopoo)

Shifting the Paradigm

NOTE: I found this sitting in my drafts and published it as it was, which is kind of unfinished.

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I am a community college math instructor. I was educated in traditional (lecture/practice/test) classrooms, and I mostly succeeded in this environment, and I didn't really see much need for anything different. When I dropped out of graduate school, I managed to get a job working for Key Curriculum Press, a now-extinct company that published inquiry-based math curriculum. I didn't see this move as meaningful for me, career-wise, partly because I didn't really have any goals at the time (a topic for another blog post), and partly because, if I'm being honest, I saw it as being beneath me. I was, after all, trained in much higher level mathematical thinking. This attitude of mine led to a couple of gentle, and one or two not-so-gentle, nudges by some of my colleagues that I might consider trying out a new perspective (and a new approach for how I spoke with others). Well I worked for Key and KCP Technologies for nearly 4 years, and I met some amazing educators and learned a lot about inquiry-based instruction, plus of course was exposed to some wonderful materials and tools (toys!) for use in education. This primed me well for deciding that I wanted to get back into teaching. By fortunate coincidence, Key's downsizing and my layoff were exactly aligned with my deciding to get a teaching job and spring application season. A couple bumps down the road, I got my current position at City College of San Francisco.

When I started teaching at City College, I was totally game for guiding inquiry-based and collaborative learning. But, it turns out, having materials was not enough. Every time I tried to get students to be curious, I was met with resistance. Every time I tried to get students to work in groups, I was met with resistance. I had no training into how to make students more active and engaged. In the end, I settled on a style of giving clear lectures, regular homework, and regular quizzes and tests. I would try group work occasionally, because I had read (and kept reading) that collaborative work was somehow more effective, and I knew that I needed to get students talking math and explaining it to each other. But every single time I tried group work, it felt uncomfortable for me and for the students. Students were reluctant to move into groups, and most groups had at least one (if not four) students who was quiet, I assumed because they didn't feel they were smart enough to participate. I knew, I KNEW, that my teaching wasn't effective. In my very unscientific estimation, considering they types of engagement I got from my classes, I think I was providing meaningful instruction to about 20% of my students. But I had no clue how to change it. Every change I attempted to make was painful.

Jump to about 3 years ago, when I read Amanda Palmer's book, The Art of Asking. I do recommend it, if you're curious. What struck me most was Amanda's connection and community with her fans. She had that community from the beginning of her time performing as half of The Dresden Dolls, and she kept it up throughout her career. She is the one who introduced me to the notion that Twitter is a place for community. I had previously thought it was a place for individuals to express opinions and announce accomplishments to the world in 140 characters, and I simply wasn't interested in that. But through listening to her explanation of Twitter, I got on and found basically two or three people I knew of in the math ed world, followed them, and then proceeded very cautiously to view their networks and very selectively follow a few more people. Within one hour of getting onto Twitter, I managed to find an open invitation to a local meet-up of real humans, which was going to meet (omg omg omg) at Desmos HQ. So uh... yeah. This was the beginning of me finding some community.

But the online math ed community wasn't what I was writing about here. It was about transforming my teaching. So through this newly discovered community, I learned generally about the concept of student-centered learning. It took a loooong time for me to understand what this means. I think, no matter what your first introduction is to this concept, you won't get it until you try it. These days, my teaching is FAR more student-centered than it was three years ago, but it still has a long way to go. I have gone through a whirlwind of changing my teaching in the past three years, and I'm about to go on sabbatical because I need it. I am exhausted, from all of the changing and trying and failing and learning and adjusting and failing and succeeding and failing and trying and trying and trying. It has been an exhilarating journey, and I'm a better teacher for it, but right now I need (a) a rest, and (b) some time to step back and assimilate all of the things that I've been learning and trying.

So. My teaching today looks very different than how it looked when I started. My instruction is far more student-centered, but it still has a long way to go. I'm not satisfied with how I teach, but I am far more satisfied with my understanding of my goals as an educator. I know from experience that the process of changing from teacher-centered to student-centered teaching is extremely difficult. It's difficult to change habits; it's difficult to embrace new methods. It's also difficult to understand what it means to support flexible thinking without implicitly giving students the message that the correct answer is the most important part. I have been in meetings with colleagues, where we have talked about the importance of allowing for failure, and acknowledging that getting things wrong is a valid and even necessary part of the process of learning. Yet nobody in the room could give an idea of what to do when a student gives an incorrect answer to a question, other than to point out where they are wrong (and thereby give students the message that their contribution to the discussion was not the contribution that we were looking for. Now I have a much more extensive toolset for supporting flexible student thinking, and for helping students to find their own ways of correcting their errors and moving to a more correct understanding. I am much better than I used to be at listening to students, like really listening to them, understanding what they are thinking, validating what they are thinking, and connecting to them on the level of how they are thinking about a thing, rather than how I am thinking about it. This is not a skill that you can read about and perform, and it is not a thing that you can do in the context of a regular lecture; it is a thing that takes lots of practice, in the context of a non-lecture forum. It can sort of happen as a small piece of a lecture, but lectures are not really a place where flexible student thinking is well supported.

What's the one change that I made to practice this skill?
There is one thing that I can point to for transforming my approach, and for giving me practice in listening to my students, and that is incorporating number talks. I don't call them number talks in class, because that title sounds elementary to me, and I'm very cautious about my adult students feeling condescended to. I call them mental math practice. If you don't know about number talks, you should. Here is one explanation of how to do a number talk. There are many more if you are interested, and many videos to watch. The main thing is to (a) get students thinking mentally, (b) give students time to think, (c) get the answer(s) out of the way quickly, without validation of "this one is right and these ones are wrong", and (d) focus on the process. It is best done if you incorporate visuals into the explanations.

Intentional Classroom Culture: a cry for help.

Every semester and every class, I wonder what the classroom culture is going to be. I often have very positive classroom cultures, and I often attribute that largely to my approach to teaching [pats own back]. However, every once in a while I have a class where everything falls apart, and the class feels like poison to walk into. The class is grumpy and doesn't want to work or think, and I turn from prompting participation to requesting it, or even begging. Usually I am unable to turn such a class around to a positive and supportive culture. I know that my students hate it every bit as much as I do, and we all just wish the semester would end. This might manifest as a class that is just non-participatory and sluggish, or worse as a class that is actively vocalizing bad attitudes. When the latter happens, I can usually point to 1 or 2 students who were vocal about their bad attitudes in just the right way to infect the entire class.

Wait.

This is also true of the great classes. The classes that I love to walk into, where everyone has become friends, and they are texting on weekends to talk about math, and they all go "wow" whenever we settle another concept. Those classes, too, I can usually point to 1 or 2 students who were vocal about their positive attitudes in just the right way to infect the entire class. So maybe it's not about my approach to teaching [hold off on that self-back-pat].

I mean, I know that I do some good things as a teacher, and I firmly believe that I generally influence my classroom cultures toward the positive side of the spectrum. But I want to find ways to be far more intentional about the specific classroom culture I engineer, so that it is less dependent on the particular group of students that make up the class.

This semester, I have to observe a colleague in a class that has turned to poison. He's looking for help, and I honestly don't know what to say to help him. So I'm turning to the internet: what resources can you suggest, say reading materials or specific tools to try, to help us recover from unproductive/uncomfortable classroom culture? And what resources do you have to help us get better at engineering a more intentional classroom culture from the beginning?

I feel like it's relevant to mention that I teach at a community college, where students are adults who are in our classrooms for 150–250 minutes per week, and the classrooms are almost all built to cram exactly 45 students sitting side-by-side and facing the front of the room. None of these outright barriers, but definitely large hurdles for a lot of the group-work goals I aspire to. Also, I still think I'd have this issue even if I had better group-work habits in the classroom.