There seems to be some misunderstanding, though, about exactly what HPL proposes an effective learning cycle ought to look like. Since in HPL, there is a place for everything, here is my 30,000-foot understanding and implementation of the four-stage process it advocates. I don't claim to be the definitive voice in any of this. I'm just taking this opportunity to document and share my practices in using their model because I believe that understanding this framework can go a long way toward helping teachers make good instructional decisions that can help their students to learn and thrive.
Specifically, HPL advocates:
STAGE 1 - a hands-on introductory task designed to uncover & organize prior knowledge. In this stage, collaborative activity provides an occasion for exploratory talk so that students can uncover and begin to organize their existing knowledge;
STAGE 2 - initial provision of a new expert model, with scaffolding & metacognitive practices woven together. The goal here is to help students bring their new ideas and knowledge into clearer focus so that they can reach the next level. Here again, collaborative activity can provide a setting in which to externalize mental processes and to negotiate understanding, although often, this can be a good place to offer some direct instruction;
STAGE 3 - what HPL refers to as "'deliberate practice' with metacognitive self-monitoring." Here the idea is to use cooperative learning structures to create a place of practice in which learners can work within a clearly defined structure in which they can advance through the 3 stages of fluency (effortful -> relatively effortless -> automatic)
STAGE 4 - working through a transfer task (or tasks) to apply and extend their new knowledge in new and non-routine contexts.As with all good models, there is a lot of fluidity and variation in each stage, depending on how the teacher "reads" the learners in her classroom. Here are some of my notes on each of the stages and how I have learned to look at each stage realistically and pragmatically:
A good discovery activity can be a powerful catalyst for learning in Stage 1. But unfortunately, sometimes there just really isn't a great discovery activity that leads students captivatingly but inexorably to a blinding insight that will transform their learning forever.
Sometimes the best you've got is a mediocre discovery activity from a textbook that kinda sorta leads students in the general direction — but not without a lot of heavy-handed guidance. Or perhaps there is some other deficiency in what is available to you.
Like Gattegno, I believe that all learners have an energy "budget," and that means I have to make savvy and strategic decisions about how I'm going to ask my students to apply theirs. A boring or mediocre discovery activity requires just as much energy as a great one, but without the payoff of leaving students energized.
So sometimes I've learned I have to ask myself, is a discovery activity the best choice I can make here at Stage 1? Or do I have some other kind of introductory task I could use — such as a simulation, a story, a funny or interesting deleted scene, or some other kind of analogy — that will get my class into the learning episode faster and free up more of their energies to developing the necessary fluency that a rich and interesting transfer task may require?
To me, the most important thing that can happen in Stage 1 of a learning episode is that students come sharply to appreciate the Burning Question of this segment. Whenever possible, I really like for my students to arrive at a Burning Question through a collaborative discovery activity that they own because when they own it, they buy into it.
But realistically, this is simply not always possible with every single topic in the curriculum. So I have a range of strategies for Stage 1 that can get my students to a Burning Question even though there may be a gap in my pedagogical arsenal.
If the purpose of Stage 1 is to motivate students to ask a Burning Question, then the purpose of Stage 2 is to provisionally "pay off" the Burning Question — and to whet their appetite for knowing more. I say that my purpose here is to provisionally pay off the Burning Question because I believe a huge part of growing up as a learner is developing your own internal capacity for identifying questions and finding ways to pay them off and extend them.
So for me, this is where I "earn" the right to give my student a little bit of lecture, although when I work with them, I always call it "doing some notes" or "organizing our ideas" or "investigating ways in which others before us have thought about this problem." I say this not because I'm trying not to admit that I am lecturing (I am lecturing here) but I am also modeling note-taking and annotating practices that they will need when they arrive at a class where there is no other learning mode than lecture. No matter who you are and no matter where you study, at some point, somebody is going to lecture at you. If you are lucky (like I was at Princeton), those who lecture at you will consider it a high art form and will put great thought and care into their storytelling and argumentation modes.
Realistically, though, a lot of the lecture we encounter in our lives is not thrilling. But you need a certain degree of note-keeping and annotating skills that will enable you to survive those instructors and their inanimate lecturing practices so you can take what you need from their teaching and move on in your life.
So I use Stage 2 to also teach my students these note-taking/note-keeping/annotation survival skills as well as some metacognitive practices that will help them to get the greatest possible "bang" for their note-taking "buck."
As HPL clearly says, Stage 2 is about the "initial provision of an expert model." This is the place where we are sharing what students cannot find or develop on their own — or at least, what they cannot find or develop very efficiently given the time constraints of teaching and learning.
So please don't tell me there's no place for a transmission model in the HPL learning cycle. It's there, we all do it, and we all need to do it from time to time. Enough said. Let's move on.
With some new knowledge or ideas in hand, and having borrowed a more expert model from me as a tradeoff for accelerating the learning cycle, students need time to practice thinking these new thoughts, using the new model, and discovering what happens when they take it out for a spin. Deliberate practice with metacognitive self-monitoring is not the same thing as drill-and-kill. It's a form of experiential learning, like what a young child develops as they are integrating new vocabulary words. I've heard that a toddler needs to hear a new word used appropriately in context between 10 and 20 times before s/he can try it out for herself or himself. Mathematical ideas are no different. Students need to try and stumble, try and wobble, try and fall over, dust themselves off and try again until something takes hold in their unconscious. Nobody really knows what this secret crossover point is for every learner in every subject and every topic. So we provide a range of experiences for our students to help them find this crossover point for themselves.
Once students achieve some degree of "relatively effortless" fluency, they can dive into a transfer task.To me, an inspiring transfer task is more important than all of the mediocre discovery tasks in the world combined. An inspiring transfer task takes a learner seriously as a professional, and offers him or her an engaging, in-context opportunity to apply their new learning with all its glorious, messy, gravity-driven moving parts. One lightbulb moment from a transfer task — say, as Barbie is launched over a balcony railing, held aloft only by a series of looped rubber bands in answer to the question "How do we balance 'thrilling' and 'dangerous' to give her the greatest possible bungee jump that does not split her little plastic skull open?" — can last a lifetime. Being tasked to figure out experimentally and quantitatively whether or not Double Stuf Oreos do indeed contain double the "Stuf" as regular Oreos... or whether they are another marketing fraud being perpetrated on the Oreo-eating public can easily push a student over the fence into losing themselves in doing mathematics.
And frankly, to me, that is the whole point.
You know when you read a blog post and it makes you want to stand up and applaud? Yeah, that's right now. You're making me miss teaching.ReplyDelete
Thanks for this day-brightener, Kate. It's nice to know that I've gotten something right. ;)Delete
- Elizabeth (@cheesemonkeysf)
Great cover of the HPL ideas! You make me want to go pull out my free electronic copy and read it - again. In fact, I think I will....ReplyDelete
Very nice. Two thoughts come to mind. First is what math concepts are appropriate for this model. That is to say, what are the "big" math ideas to design HS curriculum/lessons around vs. what mathematical ideas are inappropriate for lesson design of this sort. It is a mostly open question for me; I think I have a few ideas.ReplyDelete
Second is to note that maybe a first transfer task is a response to an unsettled question that was created in Stage 1. For example, if a linear programming question was posed at Stage 1, where students explored and made reasonable guesses--but maybe lacked a sophisticated argument as to what might me maximum. Then after further development, the same question was returned to. In other words, I like the framework you've outlined from HPL as a way to view Problem-Based Learning. Maybe a small caveat would be a large cycle with iterations of the cycle inside while smaller concepts were developed--to get back to resolving the big problem set in the first Stage 1. (This big problem should/could be a fairly big mathematical idea.)
OK - a third thought: I cringe when the term "discovery" is used. It suggests to me that the teacher has a certain way of knowing or thinking in mind for the student to use. This seems to me to be a coercive way to interact with people. C. Kamii uses "re-invent" which seems better to me (don't know why). I most prefer invent--maybe because it is my way to remind myself to allow students to make the meaning that they do, it does not need to be mine.
Brian, Thanks for these thoughts. I use a lot of Problem-Based Learning and I apply this structure of learning to topics large and small. Sometimes a "small idea" is actually a big idea! And sometimes a big idea is better broken up into more manageable chunks. Adolescents don't exactly have the longest time-planning horizon, so I find it can be useful to think globally but plan lessons or units locally. Variety and an element of surprise and sometimes a sense of humor are valuable in keeping students' attention fresh. Hope this is helpful.Delete
As for the term discovery, I can understand your perspective. I don't believe in playing "guess what I'm thinking" because I find it psychologically and emotionally insulting. Instead, when I hear the word discovery, I always think about it as DIS-covery, or as UN-covering some dusty old idea or an abandoned sense of wonder or surprise. For me, DIS-covery is synonymous with being the catalyst for students to activate their own prior knowledge. As one of the great modern yogis has said, "We already know. We just have to improve our knowing."
That's also why I like using a dis-covery activity as the motivation to raise a Burning Question. Most of the time, students have had a "Wait, what...?" moment about the topic, but life is speeding by too fast for them to stop and notice everything.
Finally, I agree with you about the importance of avoiding coercive models of interactions. It's one of my main beefs with so-called Non-Violent Communication, which I find to be highly coercive and violent. Maybe a better way of thinking about dis-covery activities is as an equivalent to saying, "Hey—look over here!" :)
Thanks for commenting.
- Elizabeth (@cheesemonkeysf)
I love the ideas, but what I really need to know is: what does this look like day to day? Would love to see some video of teaching done this way, just so I can get a feel for it (although I know it will play out differently in every class).ReplyDelete
Kara, I don't think it looks like anything "special" from day to day. What it does mean to *me* is that there is always this throughline going through my head (one-two-three-four), but it is a longer-term throughline.Delete
One thing you would see is that when I'm scouring blogs for great ideas or activities, I am thinking very deliberately about where in the learning cycle they would fit. Speed dating, for example (see Kate Nowak, FunctionOfTime.blogspot.com ) is an example of a superb "deliberate practice" activity (Stage 3). Right now, my advanced Geometry students are working on mastering the structure of elementary proofs, so for the last couple of days, we've had a segment in which we are "speed dating" through tiny proofs. Each student has a proof that s/he is the expert on, and we rotate partners, trade proofs to solve, and help each other out. It's like scales and arpeggios for proof.
Another thing you would see is some explicit lecture and summary (Stage 2) after a Stage 1 activity.
A transfer task is a biggish project, like Barbie Bungee.
I know this isn't exactly what you had hoped for, but I can't envision hiring a documentary crew to follow me around (plus it would be boring).
Thanks for commenting,
- Elizabeth (@cheesemonkeysf)
Thank you, E, this post is an outrageously useful document. A few things that are occurring to me...ReplyDelete
I rarely ever felt like I had time for the 4th part of this. (Not never, but not enough.) I'm also reminded of how Pershan is frequently going on about how transfer is impossible, and that Willingham book that talked about research showing just how very concrete people are when they're learning and thinking about new ideas, and how very very difficult abstract thinking is and how difficult it is to apply new learning to novel situations. Despite all that, I absolutely think this is a thing we should be asking students to do in math class, I'm just wondering about judicious ways to approach what and how you ask kids to do in 4-type activities. (You don't have to answer that. It's just a thing that's on my mind.)
You say that when you're scouring blogs, you're thinking about where in the learning cycle they would fit in. I've had a ton of frustration over the years (as have many people, I know) over translating "really awesome activity" to "where and how do I use this." The various virtual filing cabinets do a great job of sorting ideas into MS and HS courses and even broadly defined units within courses, but I think we lack a common language like your 1,2,3,4 for signifying the purpose of the things we share. I'm wondering if maybe some kind of loose taxonomy might be a productive project for folks interested in that sort of thing. Like wouldn't it be cool if a virtual filing cabinet also included a dimension describing where in a learning cycle that resource might be deployed.
Last thing, also from Willingham's book (I just hearted it so), he talks about how "discovery" activities and demonstrations often turn out to be a poor use of time, because students don't have the background knowledge to fit what they are seeing into any sort of schema. And that things we teachers use as launch activities might be better saved for a little later, after the kids know some stuff. I don't know if I agree with him, because I feel like I was always very deliberate about structuring those experiences so that they connect with something the kids already know and lead to a Burning Question that they all understand, as you describe, even if they are left unresolved. But, also something I've been thinking about.
Kate, I am slowly coming to the conclusion that transfer tasks are way better "bang for the buck" than discovery activities. But I'm also coming to see that the set-up/motivation for a transfer task is the most critical piece. If students aren't bought into earning the pay-off, they won't invest in the learning. So I'm taking the time to create a project-based framework for these tasks. For example, I *LOVE* Andrew Stadel's idea of giving students a "budget" for a project and requiring them to get a signed permit for the activity by explaining their mathematics to a patient third party (like the principal).Delete
I agree with you that there's no point in structuring some experience unless you can hook it to something they already know. That's why I've broadened my definition of a "discovery" task. Sometimes it's a script or "deleted scene" in which they get to eavesdrop on others discussing or learning this stuff. It just has to create some engagement.
I must know what this Willingham book is. Can you please share the title?
Thanks for thinking with me.
Oh yes sorry, it's Why Don't Students Like School? He's a cognitive psychologist, I believe, at UVA. (I am enough of a fangirl that I might go over there and say hi.) The book is meant to frame relevant research about how people learn to inform classroom teaching.Delete
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Great post. Excited to dive into HPL. It is a dense read. Could you point to where in the book it outlines the four stages? How do you think this relates to the workshop model?ReplyDelete
Hi Sean, My advice is keep this outline at hand as you read HPL — it took me about a week to condense it down to my outline! Don't worry about trying to take every single thing in on your first read. It is, as you say, extremely condensed and a lot gets consolidated. The first half of the book is where they outline the four stages.Delete
What is the "workshop model?"
- Elizabeth (@cheesemonkeysf)
I will certainly dive into the text. Workshop is an instructional design where you apprentice learners alongside an expert. Reading and writing workshops have been around a long time in literacy. There is a great book Minds on Mathematics by Wendy Hoffer that describes a workshop. The main structure is Open, Minilesson, Practice, Reflection. If you google the book and author, the first chapter is available from the publisher. Great read!Delete
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