Quick post here as I’m preparing for lots of other things:
Watch this clip from the movie ‘Indie Game’ – it’s game designer Edmund McMillen talking about how he designs the first few levels of a video game:
Now can we talk about how amazing this clip is if you replace the phrase ‘level design’ with ‘task design’ or ‘activity design’ or ‘worksheet design’?
Related: Dan Meyer & his whole ‘Open Middle’ analogy between good game design and good task design
I’m getting worse at keeping my blog updated… I’ve been wanting to add something to this for a while because I haven’t liked that the first post people see when they come here starts with ‘Shameless Promotion!’. That’s just tacky.
So – in an effort to move that from the top of my front page, I want to write about Twitter. In particular, I want to write about one person on twitter: Alexis Huicochea. This person is 90% of the reason I still use twitter today.
Alexis isn’t a teacher or educator. She’s not someone that I follow for professional development – that’s the other 10% of why I use twitter. I’ve never met her in person, nor have we ever had a conversation on twitter. Alexis works for the local newspaper in my city – she writes primarily about local education. I like reading her articles because they keep me informed about local educational news – things happening with districts and stuff. Most cities probably have this – that’s not a big deal.
Here’s the big deal: Alexis Live-Tweets Every School Board Meeting.
And they are fascinating. They are enlightening. They catch things that don’t make it into the newspapers. They are, at times, hilarious. Seriously – check out this gem right here.
There’s a lot going on these days in education policy – changes to statewide assessments, funding issues, unions & tenure – all sorts of stuff. The more I know about what’s happening in my district, the better. And I can’t think of a better way to get that info unfiltered and raw, which is what I want. I don’t want the boiled-down bullet points for the general public – I want to know what’s up for discussion and where people are leaning. I want to know what might be coming down the pipeline because these people affect me directly. I want to be informed damnit.
So – if you’re a teacher on twitter, I encourage you to find someone in your area who does this same thing. Maybe this is common and I’m late to the game. Or maybe this is the start of a trend that I hope starts to pick up – live-tweeting school board meetings so we all know what’s going on.
Actually – here’s a question for you, reader of this post: are there any journalists in your area that go to board meetings and tweet the results? Do any school districts in your area do this voluntarily? If not, why not? Why don’t more people do this? Can we get more people to do this?
Lastly, for the record, I live in Tucson, AZ and I follow Alexis because she live-tweets the Tucson Unified School Board (TUSD) meetings, which is one of the largest districts in the state. So if you live here too, you should give her a follow. Ironically, I don’t teach in this district anymore and no one live-tweets the school board meetings from my district. I wish someone would.
Thought I’d share: Adrian Pumphrey has started a neat little podcast called MathEd Out where he’s been interviewing lots of awesome people in Math Education. I’m on his most recent episode talking about all sorts of things – first-year teaching, Standards Based Grading, what my classroom looks like, Standards of Mathematical Practice, PARCC implementation, an amazing Math joke in the last few minutes, etc. It’s 40 minutes! Holy crap – how did that happen?
Anyway – if you’re interested, you can listen to that here: Mathy McMatherson on MathEd Out.
And, if you’re really interested, I talk a lot about first-year teaching and preservice teaching and feedback and SBG and lots of other things in the first episode of Infinite Tangents, a podcast by Ashli Black that started strong but has since petered out (which I think is because Ashli is now a big deal facilitator for Common Core Professional Development as part of the Illustrative Mathematics team). This was recorded in the Spring of my second year teaching as I was just beginning to seriously reflect on my implementation of SBG, leading to a whole series of posts that people seem to find useful.
Anyway – if you’re interested, you can listen to that here: Mathy McMatherson on Infinite Tangents.
Cheers – Mathy
See Previously: Some Thoughts on Interventions & Answer-Getting
So here’s another thing I did a few times in my class this year with dramatic effect: Mastery Quizzes
A Mastery Quiz Is: A collection of skill-based questions designed to be completed in 10-15 minutes. The types of problems on the quiz should be very similar to the ones they’ve done in class – no surprises. For full effect, some problems should be procedurally difficult in the sense that they are multi-step and require you to be careful with all the little details of the problem. Think along the lines of “Perseveres in Solving Problems” for the remedial student. It is not multiple choice.
A Mastery Quiz Works Well With: Skills that have some kind of concrete or procedural foundation that, if students just relied on this foundation, they would get it right. Examples include:
- Integers for students who know how to use the number line but try to take shortcuts and make little mistakes
- Multiplication for students who can use the box model but get stuck with the standard algorithm and make little mistakes
- Exponent Rules for students who understand the individual rules but try to do too many steps at once and make little mistakes
- (Procedural Skill) for students who understand (the foundation grounded in a process or scaffold) but (do this bad thing) and make little mistakes
Here’s my Mastery Quiz for Integers:
Here comes the important one:
You Grade a Mastery Quiz By: Everyone Either Gets a 0% or 100%
See previously: Some Thoughts on Interventions & Answer Getting
So here’s something I started doing in an effort to keep students more engaged in problems and less focused on the most direct route to getting to the answer: I started having them analyze their own mistakes. This isn’t new – Kelly O’Shea’s idea for The Mistake Game has been around for a long time and it was a definite inspiration for my explicit focus on mistakes. But, I didn’t think I could jump right into having my students create their own mistakes – here’s why:
- Generating mistakes requires you to be confident enough that you could solve the problem without a mistake in the first place. My students don’t start out having this confidence – I needed to work on building it first.
- Generating mistakes requires you to care about how you got your answer, not what the final answer ends up being. Its possible to generate two mistakes which, as the problem progresses, cancel each other out and give the final answer. As a teacher, I see this as an incorrect problem even though the final answer is correct. Students with the Answer-Getting Mindset will see this as correct because the final answer is correct – learning to see the problem as an entire body of work is something that I need to train them to do before purposefully making mistakes has any meaning to them.
- Generating mistakes requires me to ask a “how/why” question rather than a “what” question. Answer-Getting is all about what: “What is the answer?”, “What is the next step?”. How and Why questions focus on process: “How did you get from this step to this step?”, “Why are you allowed to do this?”, “Why can’t I do this instead?”, “How would you explain this process in words rather than numbers or symbols?”. In general, I want to be asking more how/why questions rather than what questions.
At the start of the year, all of my students took a pretest that covered basic arithmetic, solving basic equations, and basic graphing skills. As the year progressed and I had begun to explicitly teach these skills and build confidence, I would eventually come to a lesson where we revisited these tests. To do this, I took lots and lots of pictures of these initial tests (without any student names showing):
Students would walk into class and see a single picture on the board. They wonder whose mistake it is, which gives some social buy-in. Each person wants to be the first person to find the mistake, which gives some competitive buy-in. As class starts, I ask them: Where is the mistake? At first, students tell me the answer is wrong (which is true). Then they try to work it out themselves and give me the right answer – but, since we’re still in the middle of mastering these skills, these answers still aren’t very reliable. Usually there are multiple answers, so I let them debate for a while. Then I interrupt:
“You’re not answering my question. I asked ‘Where is the Mistake?’, not ‘What is the Answer?’. I don’t care what the answer is. I care where they made a mistake”
I let that sink in for a minute because, for someone with an Answer-Getting mindset, this stops them cold. A teacher just told them that they don’t care about an answer. This either makes this class new and intriguing or new and terrifying. Either way, we’re gonna work through this.
“I want to get to the point where I can circle the part of the problem where this student made a mistake. This means I need to look at each line of their work and ask myself ‘Do I understand what they did?’ and then ‘Was it the right step? Did they get the right number?'”
And we’re off. I guide them through looking at each step of the work. I ask them to describe that this person did from one line to the next, ask if this was an okay thing to do, then verify that they got the right numbers. We talk about how not showing our work can make it harder to find mistakes. I show them how to circle the individual step of the problem that is incorrect. We look at the next mistake, which is the same starting problem but with a different mistake somewhere in there. Isn’t it interesting how one problem can have so many different mistakes? Is this a mistake you might have made? Are there multiple mistakes?
For students who’ve built up walls around these problems in the form of dismissal and anger and solve-as-fast-as-I-can, they tend to give these problems a second-chance because they’re not actually being asked to solve the problem. This slight change in what I’m asking them to do is enough for them to engage with the work even if working out the problem itself would have been a motivational challenge. For students who’ve built up walls around these problems in the form of apathy and not-trying and fear of failure, I found that they start to quietly participate in these discussions because this type of question is new and doesn’t have any past stigmas of failure associated with it. Either way, I have students genuinely engaging with a set of problems and, hopefully, feeling successful as they do it.
Full Disclosure: I’m leaving out all the important work that happens in the classroom as we discuss these – the graceful handling of student pushback; the subtle encouraging and guiding of students thoughts; the questioning strategies and No Opt-Out mentality that I have; etc etc. There’s lots of other explicit and implicit things that need to happen, but they’re the things that differ from teacher to teacher.
After doing this as a class, I would give students a problem set in two-columns. The left-column had a problem and a hand-written solution that intentionally had a mistake. The right-column had the same problem and a space for work. Students needed to find the mistake and circle it in the left column, then solve the problem correctly in the right column.
Push-Back Against Answer-Getting
- Part of the Answer-Getting mindset is avoiding mistakes because they’re bad. Explicitly analyzing mistakes and getting exciting about them starts to remove a lot of that stigma and fear. This is all over Math & Science Education research – the idea of normalizing mistakes and internalizing it as growth versus verification – and this is how I managed to do that in my classroom.
- Students get practice explaining their steps and discussing how to get to answers versus what is the answer. There are lots of students who make little mistakes leading up to answers, but are actually very capable of explaining their thinking out loud. For some students, this oral explanation needs to become part of their problem-solving toolkit. For these students, they start to realize that all their steps may be correct even if their final answer isn’t, which is an easier problem to fix and helps to build confidence.
- When students get a problem incorrect, it starts to become normal for me to ask them to find their mistake. In doing this, students begin to see value in showing their work. And not just because the teacher said so – because it makes it easier to retrace their thought process and identify their mistakes. This is one of the first times in the class that I say to students “I think you should do this because it’s helpful to you, not because I’m being mean and want you to do extra work”, and they begin to believe me.
- This is one of the first times where I’ve made it explicit that there’s more to math than just getting the right answer. Which, in terms of motivation and attitude towards math, may be something my students need to hear.
A Few Other Things
Here is Michael Pershan’s Math Mistakes website, which has tons of other mistakes to peruse.
The same day I published these thoughts, Michael Pershan publishes his thoughts on How He Gives Back Quizzes. Embedded in this post is his method of displaying mistakes as a way to generate discussions as part of quiz feedback – if you enjoyed reading this, try reading his ideas too.
Even with all of these resources, I highly recommend finding a way to naturally generate your own student mistakes to analyze. The buy-in this creates from students – “Who’s is that? Is it mine? Is it _______ from 4th period?” is important, and the fact that we’re normalizing our own mistakes is important.
A long while ago, Andrew Stadel posted a call for ideas on Intervention Strategies and the only person I could see who valiantly answered was Michael Pershan. Then, more recently, this topic showed up again here and here. If there’s one thing I’m good at, its noticing trends in the blogosphere.
In looking at these posts, I found myself wanting to write something about interventions because I live in the world of intervention. For the last year, I’ve only been teaching an intervention class called ‘Math Lab’ to a group of sophomores who have (literally) failed math most of their life. I’m part of a brand new school-wide math intervention process designed to increase math fluency and confidence. I’ve been helping to develop the curriculum and identify effective strategies for my particular demographic. My students have built up mental walls and self-handicapping strategies and a slew of negative coping mechanisms to deal with their distaste and distrust of mathematics. Their issues aren’t only their skill deficits and cognitive issues – it’s their underlying behavior and mindset that causes the most issues. These are my students.
I want to contribute somehow to this discussion of intervention, but this post has been in the ‘draft’ status for months because I’m not sure who my intended audience is. Teaching in a class designated as strictly intervention with a 15-student cap is not the typical classroom setting, which makes some of my best strategies less feasible to the typical classroom teacher (which may be you). At the same time, in talking to teachers in town and across the twitterverse, some kind of school-wide intervention model seems to be the new attractive thing for schools and districts looking to respond to low test scores and the incoming PARCC assessments, which might make these thoughts attractive to someone who may be facing an intervention class next year (which may also be you).
In trying to find the overlap between these two audiences, I realized what I’d like to do is share my experience combating something I’ve started calling the Answer-Getting mindset. Hopefully this is something the typical classroom-teacher can relate to, and it’s definitely something that an intervention teacher will face head-first in their own classroom. In thinking of all the things I consider intervention strategies – from affective techniques to teaching strategies to grading systems – the underlying theme is how all of these are designed to defeat this Answer-Getting mindset that is developed in the students with the most need.
We’re doing a mini-unit on Probability in my Math Lab class this week. Probability is a tricky thing because it’s hard to predict how much a student has seen in their previous math classes and it’s hard to predict how ‘intuitive’ some students will find it. Today was our first day, so I wanted to take it slow: I wanted to explore how to find the probability of an event by examining the probability space. This breaks down into two separate sub-topics: situations where you already know all possible combinations, and situations where you first have to generate all possible combinations.
I designed the lesson so that we started by performing an experiment – I had every student write down the following on an index card: their name, their age, and their favorite number between 1 and 10. I collected everything, then asked questions around “If I pick a card at random, what’s the probability that the person I pick is…. a boy; a girl; is under 21; wears glasses”. These are all things students can answer from observation. Then I asked “Whats the probability that the card has a favorite number of 5?”, which they can’t answer by observation alone: I have to actually go through the cards and say what everyone wrote down. As I do this, students make a frequency chart – using the chart, we answer the question, then talk about what it means for something to be ‘more probable’ vs ‘least probable’. The whole time, I’m emphasizing the context behind the numerator and denominator of a probability fraction – what we care about / total possibilities (sidenote: I still haven’t found a good catch-all for how to think about the numerator. I shuffle between “What we care about”, “the event happening”, “number of ways to win”, and a few other things depending on the context).
After a bunch of explicit full-group and small-group practice based around this experiment, I give them this problem:
These answers looked familiar – they were fractions just like what we had been doing the last 10-15 minutes as a class. They followed the ‘What we care about/total possibilities’ model. Most worked through this with ease. If there was a mistake, most students forgot to also change the total number of students in problems 14, 15, and 16 (more concretely: before the new student, there are 11 students in the class, so the denominator for #s 1-13 should be 11. After the new student, there are 12 students in the class, so the denominator for #s 14-16 should be 12. Most students left it at 11, even though they correctly changed the numerator)
After we checked this problem, I had them flip their paper over, where they saw this problem:
The goal of this problem is to introduce a situation where students must first generate the probability space (ie: find all possible combinations), and only then can they start thinking about probability. I wanted this problem to be the catalyst that forced them to think about systematic ways of listing things. My plan was to model a few possibilities, give them some ambiguous clues as to how to think systematically, let them explore independently and privately acknowledge the students who seemed to have found a system for counting, and then have those students share their strategies with the class. I didn’t tell them how many total possibilities there were, but I did tell them there were more than 16 and less than 30.
Here’s where I had this strange reflective moment that inspired this post
As I found myself describing this second task – listing all possible combinations – I found myself also clarifying my expectations for the purpose of their work. I found myself saying “I’m not expecting you to know the answer right away or to be able to see this immediately. I expect you to start listing possibilities in whatever way you can, but then I hope you find yourself looking for patterns or trying to organize things so that you’re sure you don’t miss any and you’re sure you don’t repeat any”. By the time I was teaching this lesson for the third time today, this speech evolved into something like this: “This task isn’t like the questions on the other side of your sheet where we had practiced and I was expecting you to know the answers right away. This isn’t something we’ve done before, and I’m purposefully letting you explore a little bit before I offer some more guidance. I’m not expecting you to see the answer right away because I want you to give this a try and see if you can find patterns or some clever way to organize your work. But again – the goal is to try something so we can talk about it, not to get the definite right answer right away”
Explicit practice is something we do a lot of in class. Problems for the purpose of exploration and discussion are also something we do a lot of in class. This isn’t the first time I’ve layered both in the same lesson: problems for practice and clarification, then problems for investigation and discussion. BUT – this is the first time I can remember being aware of the difference as class was happening and this is the first time I can remember explicitly describing this difference in expectation to my students. And, as I think about it now, this is also the first time I’ve realized how much of an impact this could have on how my students approach this task. In the past, I’ve probably glided seamlessly from one type of problem to another – “Alright, you guys feel confident about those ones? Let’s try these” without even realizing that my purpose behind the two sets of problems may be fundamentally different.
Being aware of my own expectations and communicating them to students is something I’ve been doing to help with classroom management and class culture, but I think today’s the first time I was aware enough to realize that I also have expectations of how students will react to problems or questions (“They should be able to do this on their own”, “They should do fine up to this step, then we’ll have to regroup”, “We’ll use these to frame our class discussion”), that those expectations can change rapidly mid-lesson, and that these expectations are important for the mindset my students have as they approach these problems. I wonder how many lessons I’ve given where I’ve had this transition and never mentioned the shift in expectation to my students. I wonder if it made a difference in the past – if students struggled with these investigative questions purely because I didn’t make clear that I was no longer expecting an immediate answer. If they shut down quicker because they weren’t sure how to solve it – or, even worse, if I was more impatient because I transitioned from practice questions to investigation questions without even realizing it myself.
Anyway – in this particular lesson, students took risks and started to write down possibilities even if they weren’t sure that they were on the right track; they were receptive when I had a fellow student explain one of their methods for counting; and they participated in the ‘how…? why…?’ discussion that happened afterwards. I wonder if the same thing would have happened if I had just given them the task without clarifying that this was a different type of problem with a different expected outcome. I wonder if students would have begun the task thinking it was just more practice and more familiar, only to become frustrated and shut down more easily. I wonder how many times I’ve accidentally done this in the past too.
Update 1/22: Added another Problem Generator link to the bottom of the post. Original post is below:
Full Version of this Post:
So here’s something I found annoying as a geometry teacher: it’s a pain in the butt to create my own geometry problems that incorporate solving an algebra problem. Some examples:
The difficulty in generating these problems is:
- Any algebra expression you create usually has to involve positive variables or a positive evaluated result (since it wouldn’t make sense for x = 7, but when you plug it in, the length of AB = -4)
- The algebra expressions depend on each other (for example: if you have two parallel lines and one of the angles is 47, then any algebra problem students need to solve must end up being equal to 47)
- There usually aren’t enough of these types of problems in textbooks to make them worth your time, but they’re really valuable for reinforcing algebra skills while also teaching geometry.
Right now I teach an intervention class for sophomores who are in geometry, so I would love it if I could find a ton of these problems because they let me talk about both geometry and algebra at the same time (which is great!)
So – if you’re interested in how I do this, I made a video and put it on Youtube. All you need is a copy of Geogebra and to be somewhat familiar with how Geogebra works. You don’t really need to know any programming, if you want to make more complicated problems, you’ll want to play around with what I have below.
I’ve been in something of a blogging rut lately, so I thought I’d try to find something small to share and hope it sparks more posts. So here we go:
I’ve been trying to find more ways to create positive reinforcement in my classroom. The response to the Wall of Champions in terms of attitude and motivation was more than I could have imagined, so I’ve been experimenting with other ways for students to receive positive feedback for their behaviors. This is also a manifestation of one of my core classroom beliefs: my students will care about the things that I care about. If I show them that certain types of behaviors are important, they will also think that these types of behaviors are important. So, I decided at the beginning of the year to create little ‘award’ cards to give to students when I see them doing something I like. Here they are:
These awards are meant to reward students who are doing things that I value in a student without the student necessarily realizing it or intentionally behaving this way. They’re not always the students who are incredibly eager to volunteer themselves in front of the whole class, which means its hard to find those moments when they open themselves up for positive reinforcement. I could find time to compliment them individually, but sometimes I forget or the moment has passed. These are the hard-working students who tend to fade into the background. And these awards are my quiet ways of saying “Hey – I notice you. And you’re doing a good job. Keep it up”.
Here’s how they work: I printed them on colored cardstock, cut them out, and carry a few in my back pocket throughout class. When I see a student doing something that fits in these categories, I find a moment to write their name on the award, and then quietly slip it in front of them. I try not to make a big deal about it and purposely ignore them when they ask “What’s this?” – they need to read it first. I don’t make a big deal about it, but my nonchalant attitude is sometimes more enticing and mysterious than the most elaborate performance I could create. Once a few of these start circulating, students pick up on it and start noticing it. The students who receive them feel validated that their hard work is noticed – that they’re doing something right – and the students who don’t receive them now know that these are the behaviors I’m looking for. And, hopefully, they’ll start imitating them.
A Fun Anecdote: These cards and this presentation is partly inspired by a story that a former colleague told me. Different organizations at my school sell candy bars in the hallways as a fundraising opportunity. My friend would always buy a Snickers in the morning, then put the candy bar on top of his smartboard. Throughout the day, he’d be on the lookout for a reason to give the Snickers away – something positive, worthwhile, and non-academic that one of his students would do. It was usually something different every day, in a different class period, to a different student – but he always gave it away.
One day, during a break in his lesson, one of his top students – without any prompting or explicit motivation – got up, changed seats so he was sitting next to a struggling student, and started helping him with the problem they were working on. After a few minutes, my friend grabbed the Snickers and handed it to the top student. Another top student saw this and asked the teacher “Wait, what just happened? Why did he get the Snickers?”. My friend responded, “I don’t know. He decided to help someone else out. It was a good thing to do”.
The next day, during a break in the lesson – without any prompting or explicit motivation – both students got up and went around and tried to help struggling students.
I’m doing something new this year: I’m teaching four sections of a math intervention course. I have all sophomores who’ve had a history of doing poorly in their past math classes. I’m loving it.
One thing I did on the first day with these students was give a survey. The purpose of this was for students to reflect on their past experiences and how their time in this class would work.
Their responses were honest and heartbreaking and sobering and important. Its amazing for me to see and interact with these students in my classroom, but then read about the experiences and self-perceptions that brought them here.
I’ve curated some choice quotes from their responses and I want to present them with a minimal amount of comment. I’m putting them here because they definitely made me reflect on some of my assumptions about this demographic of student, so maybe they’ll do the same to you.
Prompt 1: What is it that makes math classes hard? Is it the way that its taught? Did you have a bad class when you were really young? Is there something about Math itself that you just don’t like?