I aim to blog about my interactions with math, logic, life, teaching, and the web they spin together.
My favorite divisibility rule is that of three.
I think this should be "for the topics you are likely to be teaching, including those up to three years beyond what you'll be teaching" and then we can discuss the range you have here and what to do about it in professional development, PLCs, etc.If an elementary teacher has a math anxiety about algebra 2 topics, I don't really care. That means that there's no pressure on that person to become fully competent and comfortable, so no one needs to stress over it ... and the teacher will be best off in a low-pressure, exploratory environment, or picking up neat things from co-workers in a peer-review and classroom-visit environment.If it's an elementary teacher's anxiety about fractions, then it's a major concern. This anxiety has been shown over and over to be damaging to the students' development and can be seen throughout their education if it isn't corrected/dealt with in the succeeding couple of years. This teacher needs to allow someone else to teach this topic until the anxiety has been overcome. A more complete program is needed exploring all around the topic in questions, finding and learning about all those extensions and digressions that make up good teaching. The University of Vermont has a program (Vermont Math Initiative) for just this purpose. If a formal VMI education isn't right, this teacher needs to hit workshops, learn more about the topic, practice, learn some more, trade stories and develop enough math confidence to alleviate the anxiety.I used fractions as an example, but it's really anything that comes under "I never did well in math", including fractions or appropriate level number theory, or new and better methods for basic operations such as division or multiplication. Likewise, a middle school teacher should be comfortable with up to geometry or algebra 2, etc.The "math cocky" types are always going to be that way if you simply present a problem - solving such is right up their alley - instead of presenting the same problem with the instructions, "How should we approach this problem in the classroom? What instructions, organization, scaffolding, etc." If we make it about solving, then that's what you get. If you make it about teaching, then that's what you get.Okay, there's your starter, people.Talk amongst yourselves.
Curmudgeon, I always appreciate your perspective and insight. You have raised some wonderful points. You have great suggestions for teachers where VMI isn't feasible and I will save that list for teachers who are ready to hear it. You said, "If we make it about solving, then that's what you get. If you make it about teaching, then that's what you get." and I needed to hear this, especially so I can think deeper about it. Ironically, the two are connected. That said, it feels like the solving will suffer greatly if we don't prioritize the teaching.
"If we make it about solving, then that's what you get. If you make it about teaching, then that's what you get."Nice!
I loved this question. I love questions, and this is a good one!That was my first reaction. My second reaction was a bunch of follow-up questions. What does "support each other mean"? What does "foster environments with each other" mean? Are we talking about on twitter? In person? With colleagues? With teachers we're coaching?
Thanks Michael. I can always count on your for follow-up questions."support each other" can mean many things, right? For me, I fall back on systems. What explicit systems (or maybe social norms) are in place (or lacking) with our colleagues that allow us teachers to communicate with each other when talking about unit planning, lesson planning, assessments, problem-solving, etc. etc.?I think "foster environments with each other" also falls back on systems. As to who these apply to, there's a various range. For me, I where many hats and the support and environment looks different in all cases. I left the question general because of this fact: we all where different hats and communicate with other math colleagues through different mediums and environments. What works for me, might not work for you, right?For me, I'm perplexed when I might do some consulting with school districts and see teachers shut down because their colleagues do the mic drop on them. Ultimately, that worries me for the sake of their students. As for my daily job, I'm perplexed when math departments don't know what each other are doing. I'm the first to say I wasn't perfect at collaboration, especially as a new teacher. However, the last half of my classroom teaching greatly benefited from collaboration with my colleagues.What hat or environment do you see this being relevant to you?
Thanks for your response. I appreciated your focus on systems. You could've asked about individual characteristics (like "openness" or something) but I'm glad you're focusing on systems. This is a system-level problem, it doesn't need a solution at the level of individuals.Things feel simpler for me than for you. I wear one hat. My math department is great, and I think it's a great environment.One thing that I think we do well is that we LOVE math. We're all excited to learn new math. The high confidence teachers in our department show this by getting excited about new problems that anybody shows them. It's safe to learn math in our department not because everyone is high confidence but because learning math is something we all do.No department is perfect, and I'm sure we have our flaws. But our department is a great place to be a math teacher, and I think this is part of what makes our environment great.
I am still working my way through to chapter 12, but shared math experiences (doing rich tasks together) help people build confidence that they can contribute ideas and also learn that there may be other ways of seeing/more to learn about a math problem or concept. So if it feels like you have people on the edges of the spectrum the first time the group comes together to do math, will they have a chance to come together again? If not, have you maybe reshaped each person a little bit for the next time they do math in a group? I had an interesting experience taking a class with struggling MS students and teachers as co-learners. At the beginning of the 9 days almost every MS student lacked confidence and intellectual autonomy it was tough to get them to share or even explore their own solution pathways. I felt the status issues erode a little more each day as we did math together. The teachers realized they could learn from the students and the students gained confidence that their thinking could contribute to the discourse and learning. The dynamics in the group were very different day 1 to day 9. Doing math together (number talks and menu tasks every day), having ample time (more than one math task) to build community as a group, and facilitators that could highlight important mathematics, connect representations, and mitigate status in the classroom over time seemed to be important factors. I don't know if/how these students or teachers are engaging differently around mathematics now that they have returned to their regular math settings. It would be interesting to find out, though.
SKIP ahead Molly!hahahaThanks for sharing your 9 day experience. I'm hearing that it takes time and doing math together to build an environment that is supportive of each other and eliminating status issues. It's interesting how you point out that you're unsure if the students or teachers are engaging differently. There's a section in Ch. 12 that talks about 98% of teachers making shifts on vertical non-permanent surfaces and visibly random groups. SKIP AHEAD!
This isn't original, but for me a lot of the answer in the classroom is the shift to thinking away from answers. I am a little interested in what you got, but I am very interested in how you got it, what questions you have and how else you might think about it. If you can convince students (or fellow teachers) that thinking is what you're after, now some of the people who have not had the chance to share now have the most to share.
You're not the only one, and that's perfectly great with me, John. Many here agree with you that shifting the focus away from answers. Thanks for your encouragement. You've helped me consider the value in explicitly stating the importance of thinking and not answer-getting by saying,"I am a little interested in what you got, but I am very interested in how you got it, what questions you have and how else you might think about it."Thanks!
"How should we approach this problem in the classroom? What instructions, organization, scaffolding, etc." If we make it about solving, then that's what you get. If you make it about teaching, then that's what you get." this is an awesome approach by Curmudgeon. I also like using/getting student work. You do the problem first, now look at the student work, what comments, questions, feedback do you have. How do you nudge the student in the correct direction?
Thanks Amy for checking in.Your last question "How do you nudge the student in the correct direction?" concerns me only if some steps haven't been taken first.For example, a teacher might assume they now need to get the student to the correct answer by only going the one "teacher-correct" direction. At some point prior to this question, I would need to establish what's important to this teacher: student thinking, multiple solutions, one-way solutions, algorithms, etc. If the teacher is cool with multiple solution strategies and student thinking, then your question is spot on. If the teacher is not okay with multiple solution strategies and student thinking, then there's more work to be done. I'm wondering what questions might help elicit the information from above. I'd be interested in asking the teacher what questions they could come up with to better understand what the student is thinking.Thoughts?
That is where multiple approaches can be shown with student work. Still let the teacher do the work BEFORE showing student work. This is an amazing place for "mind blown" moments. When the teacher table groups then see the student work, maybe show multiple correct paths for teachers to examine, THEN show and examine several "off the ramp" approaches. Teachers do NOT have to show each other their work, but make it about student understanding. From this approach, perhaps the teacher comments on student work can help the less confident mathematician lead the the cocky teacher to understand the student misconception, and the cocky teacher can see a valid other approach. Wow, check me out, in both cases the less confident is teaching the cocky. Hmmmmmm...
I'll admit to having been on both ends of that math confidence spectrum. Going through school, I was surrounded by incredibly intelligent mathematicians who regularly left me in the dust when solving problems. They were never rude nor impatient when waiting for me to catch up, but I internalized my slowness as not being good enough at the math we were doing and it made me incredibly anxious. I think this experience is what made me turn away from teaching high school and "settling" on a more comfortable middle school career (although I no longer feel like I settled...I love middle schoolers!)My first job was at a small private school where I taught 6-8 math and things flipped...I was now THE math person at the school. At all of our math in-services I thrived on being the one getting the answers first and quickly sharing my answers. I'd turned in to those same mathematicians that were so intimidating in college.After a few years I matured a bit and got a job in a larger district school. I joined a math liaison program that brought teachers from every school in the district together to learn best practices. Every time we met (about once a month) they started by going over our group norms. This was the first time I had experienced expectations for how to communicate with another person while solving problems. One of those expectations was to give everyone a chance to work independently and to ask permission before beginning to discuss any problem. Ok, so yes...NOW that seems obvious...but to my 24 year old self it was a revelation. "I'm not supposed to just start telling people how to solve a problem right away? I'm supposed to give them time to think for themselves and then make sure they're ready to discuss the answer or ask for help?" It was this moment I realized the people in my group were most likely feeling the way I was in college. Just having that set of norms laid out in every situation you're doing math with others is important. I've been through enough experiences that I'm now confident enough to tell people I'm not ready talk or to ask questions when I'm stumped...but it took me a while to get there. I still work with people who lack confidence in math and people who mic drop their pencils when they've solved the problem but I've found that the facilitators who focus on the process, alternative methods to solving problems and making connections between our solution paths seem to bring those with less confidence out of their shells while reminding the cockiest that there is still a lot left to learn.
Thanks Elizabeth for sharing your experiences. I absolutely love that you're touching on something that I think this dilemma absolutely needs: norms.You said, "One of those expectations was to give everyone a chance to work independently and to ask permission before beginning to discuss any problem."This sounds so obvious, but necessary to say. You've got me thinking more about being explicitly clear about norms like this.My concern is that norms can sometimes be assumed instead stated. Addressing this concern is absolutely necessary. You've also got me thinking about the systems in place for teachers realizing that norms /need/ to be set and understanding why norms need to be set.
Can't wait to delve into this dilemma! Just downloaded the ebook and started reading chapter 12 - only a couple of pages in, but I wanted to brain dump my 'first thoughts' (Which is a skill I teach my Algebra students when working on a meaty task) This discussion is exactly what I need right now as I question my pedagogy ....I also like the twist on how that applies to teachers as we work in our PLCs. The comment in the book that I'm ruminating on right now is:"If we want students to make more progress together than they can alone—like Robinson and Matijasevich—we need to give them good, mathematical reasons to work together, and we need to teach them how."Specifically - the 'teach them how' part. I often think that we just expect kids and coworkers to have the skill set to collaborate - and it becomes very clear that is not the case. What I see happening is the 'mike droppers' take over the discussion, and those quiet types - perhaps less secure, become more quiet and less secure.I'm excited about unpacking the rest of the chapter, but I wanted to get my initial thoughts - after only reading and pondering a bit - on paper. I'm curious to see how my thought pattern changes throughout this give and take.Carrie, Middle School Algebra 1 teacher.
Thanks for sharing your first thoughts, Carrie.I also like how we need to provide students with good, mathematical reasons to work together. I feel this way with teachers too.I like how Michael Pershan has pushed me to focus on the system and not the individuals. The mic dropping is a systemic issue, not an individual issue. Curmudgeon and others have helped me focus that system on teaching and not answer getting.What do you think? Is that a fair assessment so far? If not, what am i missing?Thanks for sharing your initial thoughts. I hope to hear more.
Andrew, thanks so much for starting this discussion. It's a great question, and I'm so curious about how you see the relationship between your question and CH12. (BTW, CH12 is my favorite. I had the most fun writing it. I sometimes wonder if that should have been its own book, and I could have kept going with those ideas.) As for your question, it's a meaty one. I offered some thoughts on working across grade levels (and confidence, and anxiety) last year at TMC. The video is linked up here: http://tjzager.com/2016/07/19/twitter-math-camp-keynote/ I hope it might be helpful. I think it pertains a lot.More generally, I'd say my approach is the same as it is with my students. We all have areas of strength and we all have areas to work on. Often those math-cocky teachers have solid content, but have not yet learned how to listen to students and their math thinking. They don't yet understand how to help students who struggle without just telling them what to do. Or they may only know one method and are aces at it, but lack flexibility or creativity. On the flip side, sometimes those teachers who are anxious about their content knowledge have learned how to leverage their own experiences in math to make it more accessible for their students. I'm thinking specifically about Patricia's comment, here: http://tjzager.com/forums/topic/math-autobiographies/#post-4189 She has anxiety, but because of it, she has learned to give students more TIME to think and imagine mathematically than most teachers do. So her anxiety has helped her pedagogy, and math-cocky teachers would do well to listen to her!I think the questions about norms are on point. It's up to the facilitator to set and defend the tone throughout the work together, not just at the beginning. I've learned a lot from Elham on this. I watch her position herself ALONGSIDE the rest of the participants. She talks about the things she has to learn, and how she can't wait to learn from everyone else. She'll be specific, saying something like, "I hope there are kindergarten teachers here because I don't know anything about teaching kindergarten! I have so much to learn from you!" She shares her respect freely because it's genuine humility and honesty and eagerness to learn. People who were holding their breath begin to exhale. One thing I feel painfully in my work is that it's much harder for us coaches to come together and talk about these problems of practice than it is for teachers to talk about their work with students. These are our colleagues, and blogging about what happened in our workshop or what we're struggling with is a violation of the relationship. So we keep mum. But we /do/ have common problems of practice and we /do/ need our own colleagues with whom we can learn together. I don't know how to solve that problem. I mean, NCSM is great. But the all-year-round awesomeness of the #MTBoS isn't the same for coaching as it is for teaching. I wonder sometimes if Kristin and you and I and some other coaches should have our own password-protected site--a slack channel or something--where we could seek each other's advice and help get better at adult education. It makes me miss BTR, where I worked briefly. We would do things like video ourselves debriefing with a teacher and then come together with other coaches and use a protocol to watch that video together. The coach would get coached. learned a ton. I miss that.Thanks again for kicking it off. I'm staying tuned.
Hi Tracy,You have added a lot to this conversation and I appreciate it. I'll revisit your keynote later this week. Thanks for sharing.Thanks for referencing Elham and her facilitation skills. Having been in a couple of her sessions, I remember her sincere desire to learn from others in the room. It was very contagious and I hope to emulate that skill.I think you raise a valid concern for us coaches in a supportive role. I would never dream of blogging about my meetings with my fellows (teachers I support). However, there needs to be some way for coaches to communicate with each other in order to better our own practice. The group of DLCs in my district get together once a week to discuss tools, lesson ideas, and problems of practice. It's is so valuable to our practice as coaches. I will bring the video idea to our group and see if that's something feasible.Back to norms... I'm reassured that this is more of a systemic challenge instead of individual challenge. I need to remind myself to establish explicitly clear norms in workshops with teachers and encourage them to do the same in their PLCs (or other forms of department meetings).I also love how both Michael Pershan and Carrie Hardy raise valuable points. Michael is fortunate to work with colleagues who LOVE math. Carrie loved the part in your book on providing students with good, mathematical reasons to work together. In turn, teachers needs to be provided with (or find) good mathematical reasons to work together. How can we develop a system so this is happening and what role do I play as a coach, colleague, teacher, consultant, etc.?My concern for Michael is if someone new to his department comes in and doesn't LOVE math. What system does his department have in place so this new person's feelings toward math don't initially hinder the productivity of the department? Does that make sense?
"One thing I feel painfully in my work is that it's much harder for us coaches to come together and talk about these problems of practice than it is for teachers to talk about their work with students. These are our colleagues, and blogging about what happened in our workshop or what we're struggling with is a violation of the relationship. So we keep mum." YES Tracy! I'm personally struggling with this since I'm one of very few coaches (it's not even allowed to be in my title) and I have very few colleagues to share my dilemmas with in hopes of growing as a coach. I'm starting to explore the idea of blogging but I'm afraid my experiences with teachers and classrooms would be hard to keep out of my writing. I would never want to betray my colleagues' trust but sometimes I need a coach to help me work through the things I see and hear!If you do ever begin a closed coaches group, count me in!
Andrew: What system does his department have in place so this new person's feelings toward math don't initially hinder the productivity of the department?A system? I don't think we have a system.I think we have a culture, and it's a culture of amateurism. We ask stupid questions in public. People get to watch other people learn things. People talk about wanting to know things, and about things that they wish they understood. People share rough drafts. People ask questions they don't know the answers to.Part of what supports this, I think, is that we are mathy enough that we put ourselves regularly in contact with higher-level mathematical books and ideas that we don't understand completely. It also helps that we have multiple math PhDs hanging around and that they also are publicly learning math. We all know some things and don't know others.Amateurism is about striving for something you don't have, and maybe can never get. I think increased mathematical exposure supports this in our department.
I still haven't had a chance to read Tracy's book, but based on everything I've been seeing before and after publication, I can't wait.This question you bring up is an important one. One approach I've taken in my classes is to present a problem and the answer, and then challenge students to come up with as many ways to get the answer as possible. Last week, my geometry students generated about a dozen ways to find the area of a trapezoid. Not all of them were valid and there were some that were essentially repeats, but everyone was engaged and excited to work together. I wonder if a similar approach would help in a PD setting to make all teachers feel like they had incentive to contribute something. I know that I've always appreciated hearing how many ways different teachers in a room will do mental math, and how much variation there can be based on a teacher's years of experience and what level they teach. With more intimidating problems, where the approach is what we want our audience to explore, maybe giving the answer to start with is an approach to consider?
Thanks Ethan. The idea of giving the answer is interesting to me and I want to think more about that. What I understand from your post is that everyone will have something to contribute and we have to honor that. It seems like that would be a norm that needs to be established. I have mistakenly made the assumption that most teachers see your idea. Learning from my past, it sounds like this would also need to be established as a norm with teachers and colleagues...which in my mind, would pay dividends throughout the school year...everyone has something to contribute.
What a great question. Math teachers are often expected to know everything, as perceived by parents, students, and administrators. Therefore, teachers may be less likely to admit they have math anxiety due to fear of losing their job or the respect of others. How does a fellow math teacher support teachers who they suspect feel this fear? How does one approach the subject?Additionally, I'd like to propose a 2-axis model that would couple with the spectrum you suggest. The y-axis would reflect teaching flexibility (+y) and teaching fixedness (-y). How do we support teachers who fall into Quadrant 3 (math anxiety and teaching fixedness)? "I'll always teach this way because it's what I've always done, and it's what the book says to do." In short, how do we address a weakness, if no one will admit they have a weakness?
Yes, yes, yes Claire! This is my biggest hurdle! Coaching is easy when teachers reach out and say, "Can you help me?" But those are usually not the teachers in your quadrant 3...unfortunately I see teachers in that quadrant do the most 'damage' in the classroom. :(
I think this also applies to our students. Often times our students have weaknesses they do not want to show and need our help but will not ask for it. Which begs the question, how do we find those students and help them?
Ashley, I agree. I think that we can use data to find the students, but how do we approach them in a way that helps them build the confidence to ask questions when they get stuck? Same with teachers!