The Signal from Tölva is a wonderfully atmospheric exercise in tactical archaelogical drone combat, with a hint of existential horror. Or whatever “existential horror” means when you sort of don’t exist.
Anyway. It’s good. Read on for details. (Mild spoilers by some standards)
The all-in approach usually plays out something like this (in my experience):
1. Form groups (via cards or whatever). 1-5 min
2. Call students around, tell them the story of today’s problem, tell them to get at it. 3-5 min
3. Watch students find their groups, give them a few minutes to talk to each other, try not to interfere; physically hang back in the middle of the room. Let them ask each other clarifying questions; don’t jump in to help them just yet. 5ish min
4. Problem solving time! This is the hard part and length of time will vary based on how many extensions you’ve got to your problem and how tough it is. The good culture-building challenging problems from Peter’s site can easily eat up an hour of class time if you want it to.
Here’s what you’re doing as you circulate around the groups::
answer questions that clarify the problem’s requirements and constraints.
DON’T answer any “stop thinking” questions. eg. “how do I do this?” “what do I do first?” Deflect / defer to group thinking and peer knowledge.
if it’s some other kind of question, just ask yourself, would answering this be spoilers for someone who is genuinely trying to figure this out as a puzzle? if it’s not spoilers, answer away; if it would be spoilers, either defer or just give a hint. (don’t hint immediately!)
watch for students who are dropping out, letting the other two group members (groups of 3 are optimal) do all the thinking. walk over, verbal encouragement, take the group’s marker and hand it to that person as needed.
when a group has solved the initial problem, hand them an extension / continuation problem to keep them going
when a group asks you IF they’ve solved the problem, ask them! “How would you know if you’re right?”
When you hit your stop-goal – usually something like “every group has solved the core problem(s)” – then this phase ends and we go to…
5. The Recap – 5-10 min
Call everyone around where you are; kind of like setting the stage for your story, at the whiteboards where students were working. (They don’t have to physically move far, but you’re shifting the tone and focus of the room back to you now.)
Now work your way through the story of what students just solved. This is actually the hard part, I lied before. You want to start from the start (“level from the bottom” as Peter says) and essentially recap the entire problem. Use student work as part of your narrative, but don’t feel that any particular group’s entire process needs to be explained in depth. Your goal is to connect the threads of what everyone’s been doing so that no one’s missed out. When you’ve told the story to the end of the core problem, you can also give a high-level view description of what happened in the more advanced extensions that some-but-not-all of your groups will have made it to.
6. Notes for your future forgetful self – 5-10 min (might be optional, depending on if this was a curricular content problem or not)
Give students time to take notes for themselves. There’s a lot of variety in how this can be done – totally unstructured, partially structured, as long as it’s not just them copying down what you tell them to. (Except that *someone* will probably beg for that or refuse to write anything otherwise, so fine, put some premade notes on the class website or let them see your copy later or something.)
Final journal entry for my current math course. Lots to think about looking back.
Comment on our discussion of point gathering vs data gathering.
Short version: every teacher (in Peter’s words) is either doing one of these two things, or trying to mush them together. Point gathering is what it sounds like: a gradebook where everything gets points and you just add the points up at the end to get a grade. Data gathering is where you collect data that you can look over holistically and read a narrative from that tells you whether someone’s learned. That data narrative can then turn into a grade based on professional judgment and/or by translating it into scores to add up / average, but then you’re scoring based on learning, not based on specific assessment events.
This is more-or-less the idea behind Standards-Based Grading (SBG) which the math blog world already got on board with heavily around the time I was starting teaching. It’s also shown up in various other names, with different twists and spins on how it’s organized, from other assessment gurus and researchers. Whatever the specifics of the system, though, it’s fundamentally different than point-adding because we’re using assessment events as data to generate a measure of learning, rather than letting specific events (eg. old, early quizzes back when someone was just forming new ideas) dictate the grade.
So, yeah, I’m on board with this, no doubt.
Re-respond to the following questions…
What is mathematics?
Mathematics is the practice of finding logical patterns and exploring them logically. It is the community of mathematics practitioners and their shared wealth of understanding of the world’s mathematical patterns. It is the language in which we communicate those ideas to each other.
How does someone best learn mathematics?
They best learn math by being given resources and an environment in which to do mathematics that are on the edge of their current understanding. Those resources can include peers, technological tools; the environment ought to be social, supportive, safe and guided.
How does someone best teach mathematics?
By providing an effective, efficient environment in which to think mathematically while also giving well-designed challenges to the learner that afford mathematically ‘flow’ in the direction of new ideas and understandings.
Read your entire journal and comment on what has changed the most for you?
Mostly, I’ve gotten what I wanted to get, which was to be supported in implementing the Thinking Classroom approach to math class. I’m more ready to just dive in and do it; I’ve got a better theoretical and data-driven background in why these techniques work and what they’re meant to improve. I’ve got a much greater appreciation for the nuance and skills that this style of teaching requires, and know I’m both on the right track and also have a long road ahead to work on improving this for myself.
I also have more questions about designing good tasks than I had when I first started. I feel like there’s a lot of room to research and investigate how to design a good task, rather than stumble across one, and how to pass those design strategies on to teachers for their own work.
So during this class, we discussed homework. The research on who actually does it (not the kids who need it), the effect it has on inequality in education (makes it way worse), and generally why it should be knocked off its high pedestal.
Now, honestly, a lot of us teachers kind-of-sort-of know this but then assign “homework” anyway, but give time in class to complete it. Does this fix the problem? Well … there’s the matter of what students actually *do* when you assign homework.
If homework is “marked” in any way – including a simple check-for-completion! – you get the graph on the left (based on Peter’s research observations in multiple classrooms; image grabbed from a ppt from the course). ‘Getting Help’ means that someone like a tutor or a parent did the work ‘with’ them – and they probably can’t do it on their own afterwards.
If you remove homework marks completely, the cheating almost disappears and is split roughly evenly between Did It and Didn’t Do It. So it’s an improvement … that’s still only helping around 1/3 of your students. That’s, uh, not great numbers.
So what’s the fix? For starters, recognize that there really is enough time in-class for students to learn things. You might need to recover some of that time from things that aren’t students-doing-the-thinking, but it’s possible.
The next step is to replace homework with “check your understanding” opportunities … wait, no, bear with me, yes I know your textbook calls its question sets “check your understanding” and that hasn’t helped, just hold on before you leave. It’s not about replacing the questions or just what we call them – it’s about replacing the mentality and the motivation for doing them.
So the example we discussed was, a set of CYU questions are put out for students to work on together, labelled as to what topic they’re checking and/or what difficulty. Students are not required to complete them all. They’re there solely as a way for students to check whether they can do them. They’re there to give some indication of where the student is going and where they’re currently at.
Now, a weird effect that’s been observed is that even though textbooks have stuff you can use for this, you almost definitely are better off not giving students something in a textbook format – like not even a photocopy of a textbook page. Textbooks have too strong a “complete this busywork” association tied to them already, such that it poisons the well.
The where they are and where they are going guideposts are the most important part to any approach to this. Homework as ‘practice’ more-or-less doesn’t work: those who need it, don’t get it (see above), and unlike physical practice where you actually can get even better at something like weightlifting by continuing to lift over and over and over, math skills are not that transferable. Once you know how to factor quadratics, you reeeeeeally don’t need to do 50 more. Like, okay, maybe a few. But not 50.
But if someone isn’t yet there, like maybe they can factor the easy ones but don’t know how to handle the harder ones, they need some way to know that. Solving 50 easy ones won’t actually build up mental “muscles” that suddenly enable them to solve hard ones! That’s nonsense. They need to know that there’s more to learn and roughly how to get there.
Comment on our discussion on numeracy in general and our discussion on the relationship between numeracy and mathematics in particular. (and etc on numeracy tasks)
Okay, so Prof. Liljedahl has a very particular idea of what numeracy is and what it isn’t. I’m convinced, but the challenge will be whether the rest of the world chooses to understand the term this way.
What numeracy isn’t: being able to add, subtract, multiply and divide. knowing your multiplication tables.
What numeracy is: stepping up with whatever mathematical tools you’ve got and getting the job done.
Pithy, but requires unpacking.
The ‘job’ in this case would be, any kind of messy situation where math may be useful. Like are you willing and able to take this weird scheduling situation and apply some mathematical reasoning to it? Juggle the numbers around and make sense of it? Even when there isn’t an easy “right” answer?
The obstacle helps rein in the huge degrees of freedom by giving a creative constraint. The ambiguity and the degrees of freedom will, in some problems, draw out questions of personal choice and value judgments into the solutions. For example, we worked on one problem to do with how fundraising dollars were split among students who had done uneven amounts of fundraising, and whose ski trips would cost varying amounts. Questions of individual responsibility, offsetting the expenses for the underprivileged or those new to the sport, and generally trying to make sure no one would be left feeling taken advantage of were all themes that came up in our solutions. Just by asking us to sort out how to split some numbers fairly!
So, okay. Back to the definition of “numeracy”. This isn’t how everyone uses the word. Often ‘innumeracy’ is associated with poor estimation skills, inability to mentally process differences in large numbers, or simply being unable to multiply. Peter’s take on those cases is that they’re not about numeracy, they’re about “number-acy”. Which, okay, also probably not a word that everyone else will buy into, but it gets the point across. Being numerate is more than arithmetic in the same way that being literate is about more than spelling or grammar quizzes. You have to be willing to engage with the world through the medium of language to be literate. Likewise, you have to be willing to engage with the world through the medium of, and with the toolset of, mathematics to be numerate. If you aren’t willing and able to grapple with life using the mathematical tools you’ve got, that’s akin to being unable to handle situations in life that require reading.
One of the bigger misconceptions about the Thinking Classroom approach (that I’ve already seen come up in discussion at my school) is that students are never shown a glimpse of how the teacher thinks about the subject. Peter refers to this as “leveling to the bottom” (I think he means leveling from the bottom), or consolidating at the end of an activity. In class, I’ve sort of defaulted to calling it a “recap”.
The way this tends to work: wait until all groups have reached a “stop goal”, which is usually to have found the solution to the main problem (but not all of the extension questions). Then tell everyone to “gather ’round” (or something like that – usually I cheese it up a bit) until you’ve got them at least loosely clustered around you at the whiteboard(s).
Then the show begins.
Frankly, at this point you could argue this is lecture. The difference is, it’s lecture about something students already know. You’re telling the story of what they’ve just done, from the beginning, using student work as illustration sometimes, reworking something yourself sometimes. The retelling lets you weave in mathematical language, helps students who were falling behind see the whole picture and perhaps even catch up, and provides a chance to highlight what you want to highlight as the Main Ideas of the day.
Here’s what I’ve noticed:
Students, for the most part, are listening. It’s been a tougher slog lately with the more abstract curricular questions, but with the better problems it’s been simple to maintain students’ attention. (Possibly the challenge has been that on the clunkier material, my summary is clunkier as well…)
The “gather round, kids” mentality creates a different space for the students even if they’re hardly moving at all. In my classroom, there isn’t room for students to all circle around the front, so when I do this move there’s really only a minimum of actual physical movement. Some are still seated at front-row desks, a handful are still at the back as they got annoyed with window glare and worked on paper … but they don’t behave the same way as when they’re told to just “sit down”.
The recap helps reify (yeah it’s a great word) the thinking that’s been done, to an extent, but it needs follow-up: the mindful notes, the outlines so they can put it into a larger context of what’s been learned in a span of days or weeks. (This is not news to anyone doing Thinking Classroom stuff, just worth noting for myself.)
There’s a lot of nuance possible in how you approach these recaps which, fortunately, isn’t required to simply give it a go. If you like telling stories you’re probably well set to just give it a try and think about how to improve your technique later, like after a few weeks of just winging it. (Or, I don’t know, maybe future me would be super embarrassed at how clumsy these recaps were. Good thing they weren’t recorded.)
It’s Spring Break, I’ve made it this far. Now for the unfortunately-large task of figuring out which journal responses I’ve yet to catch up on … you know this whole blogging idea seemed great a few weeks ago …
Going to make a big post of things I missed blogging about earlier, write until it looks awfully large, then start another one and schedule them to post a bit later.
One thing we discussed in class is the really, really abysmal evidence of the usefulness of teacher-dictated notes. Peter’s phrase here is to replace them with “mindful notes”, which is fairly loosely defined (I think) as “notes based on student thinking instead of teacher thinking”.
Journal-writing time for my class, and by that I mean, “hmm I’m three weeks behind now … wait, 1, 1, 2, 3 … dang it I’m turning into a Fibonacci sequence, this is bad!”
First, an update on how my math classes are going.
I’ve been sticking with (what I know of) Thinking Classroom strategies pretty consistently since the last update. I’m less overwhelmed than 1st semester, but still a lot of last-minute decision making as class is about to start. (I keep reminding myself that someday my kids will be old enough to, you know, just go to bed without requiring two hours of policing in the evenings and maybe I’ll be able to actually prep everything in advance …)
Students are functioning pretty well with group work at the whiteboards & windows, and I’m doing an okay job adapting the textbook concepts into Decent Problems. (Not “Good Problems”, but they’re getting there.) I’m still wrapping my mind around how to identify a good *extensible* problem – something that you can add more interest to for groups who get to the goalpost sooner. Right now this feels like A Problem To Work On: what are the strategies we can use to extend a mediocre problem into something with more to think about? What are the requirements? Which starting points should we just throw out immediately (if any)?
However, since this is going sort-of-okay, I’m not putting my full attention on that this week. On Thursday I wrote down a four-day plan for myself that would give me some structure for getting course assessments in gear. It went something like this:
Day 1: Work on a Good Problem, talk about <mathematical competency>, get students to share good vs bad examples.
Day 2: Work on a Good Problem, then have groups self-assess with rubric made from their examples.
Day 3: (because I need to get content assessment going as well) Group quiz on <content assessment topic>.
Day 4: Individual quiz on <content assessment topic>, then work on something else (either intro to next unit, or just something for fun).
I’m currently just done Day 2 of said plan. I forgot to do the “walk around with a clipboard and assess three groups myself” step, so they didn’t drastically improve, but I decided to assess on Reasoning & Analyzing (a heading in our “competencies” doc which amounts to stuff you to do start working on a new problem) and that wasn’t something I needed to see drastic improvement on anyway.
Hi intrepid readers, so here’s me homeworking again.
First, the classroom action:
My first week of semester 2 classes has been pretty great. Both math classes have had a full 5 days of beginner-Thinking-Classroom stuff: verbally delivered problems that have interesting patterns / solutions, room for extensions, and working in visibly random groups on whiteboards & windows.
There’s been majority buy-in from the students so far, and the rest are pretty firmly in the “I hate math class because math class has always hated me” camp from what I can tell. No one’s had a meltdown over not working with their favorite friends. I have had a few questions along the lines of, “So when are we going to have a normal class?”, but I’ve just kind of shrugged that off. “Maybe never, we’ll see!”
Content-wise, it’s all been a selection of Good Problems taken from either what I’ve worked on in class myself or from Peter’s site. I’ve been using the same ones in both my Foundations 11 and my Math 10 classes … which just backfired today as the afternoon grade 10 class had talked with someone from the morning 11 class and been given “the answer”. (They still didn’t know *why* it was the answer and they kind of spilled the beans in asking me, but it did mean the time spent on the problem was significantly shorter.) Oh well, was going to move to curriculum-specific problems tomorrow anyway!
I have a few coursework-specific questions to answer that I’ll put after the break.
Playing some honestly-I’m-behind catchup on my course journaling here, I’m going to sum up what I’ve got in mind so far and maybe add more later to actually properly finish the assignment.
It’s 3:09pm after the first day of a new semester. Today went *great* but I feel like my tension levels are running too high. I need to figure out how to make this more manageable for myself without compromising on what matters – student engagement and building a culture of expecting-to-think.
Quick summary of the last couple weeks: four of my five courses wrapped up, with a mix of ‘Thinking Classroom activities coupled with straight-up commonplace review packages. Of those four, one entire class ended the year with exit interviews instead of an exam; one class was a mix of exams, projects, or interviews; the other two were mostly-exam with a couple of exceptions.
Exit interviews were great, but they would NOT scale well, at least not like this. For interviews, I went through each content unit and asked questions; I had intended to ask some high-level, open-ended questions and let the student guide me through what they learned, but quickly found that neither of us were quite sure how to make that work, so I fell back on specifics. My goal was to be able to gauge their understanding in each area on a 4-point scale; not trying to cover every kind of question an exam would have, but hitting on some concrete example of the main ideas.
Foundations 12 interviews took about 20-25 min; the few PreCalc 12 interviews I did ended up 45-60 min each. This would not have worked out well for me if they weren’t such small classes (under 10 each; they were a combined class in my timetable).
I had a couple of summative projects; they were all right, but again my initial criteria were high-level questions, and the students didn’t have enough experience working at that level in a large-scale way. Not a bad idea but needs to be built up towards next time if I want to repeat it, I think.
So, today. Two new math courses, all started off with students at the whiteboards and windows on some good problems (Pirates and split-25). The classes ran well, but at some point I noticed my tension levels rising. I think(?) what’s going on is I’m just nervous watching them all without either sitting down or talking! I don’t want to sit because I need to circulate and hint/extend – I don’t need to talk a lot unless a group is stuck, and they weren’t often (yay!).
Gonna keep at it for now – I’m committing all my powers of stubbornness to see this through properly and not let fear pull me down …
I have a lot of homework to catch up on for this course, and other questions to discuss in this journal entry, but now off to class.