23 comments on “A Problem Based Learning Starter Kit”

A Problem Based Learning Starter Kit

You’ve seen the tasks. You’ve read the research. You’re basically bought in. But how do you begin? More importantly, how do you introduce students to inquiry driven learning?

Or maybe you’re not convinced. Perhaps you maintain that the teacher is the primary knowledge constructor. Perhaps you’ve been burned in the past by inquiry driven instruction. You tried it and didn’t see kids learning much and you feel like you wasted some amount of class time when you could have been actually teaching. I can speak from experience: if I wasn’t part of a cohesive team (all subjects, as part of an entire school effort) I quite possibly would have tossed inquiry, Problem Based Learning, groupwork and everything else in the trash after my first miserable experience with it.

Or maybe your students are burned out and beaten down on math. They’ve been labeled “remedial” and by golly, they’re living up to that stamp that your district has placed on them. To them, math is an arbitrary bunch of rules to follow and steps to regurgitate. Their test scores stink and they have difficulty applying math in new and novel situations. Applying math in new and novel situations is probably an entirely foreign concept. Up until now they’ve had example problems or math instructional software to guide them through their problem packet.

It’s always tough to be the first. In many cases, you might be the first teacher to actually ask students to solve complex math problems without pre-instruction. Students might look at you cross-eyed the first time you ask them to work in groups collaboratively on a problem that may not look like the stuff they see in their textbook. There isn’t an example problem for them to look at. Yes, you are the first line on the shores of Normandy.

Not all problems are created equally and some may be more easily acquired and delved-into by students. If you’re not careful with your first exposure of kids into a new way of mathematical task-posing, you and the students could easily frustrated with the process (if you even have one yet). As Dan states perfectly in one of my favorite posts this year on first-steps toward inquiry, “The Unengagables“, “you’ll be hearing from their attorney.” Dan poses three quick methods of introducing kids to mathematical inquisitiveness, be sure to check those out, and follow the comments. I’ll follow with a few tasks here that I think make for good first-foray’s into Problem Based Learning (PrBL).

I like these tasks as first-forays for a few reasons, pointing two directions.

For the teacher:

  • The problems kind of “implement themselves.” That is, there isn’t a whole lot to do to massage the task to make it implementable. While I don’t necessarily advocate a plug-n-play curriculum, it’s ready to toss in the oven.
  • It doesn’t take too long. Maybe a day, maybe two at most. I’m not sure any first-foray into PrBL should last more than a couple days.
  • The task includes facilitation notes and/or other supporting resources.
  • The task naturally fosters student and peer-to-peer dialogue. Obviously any good task should do just that, but these tasks especially do that with minimal teacher-prompting.

For the student:

  • It’s naturally engaging or intuitively interesting. Real-world is nice, mathematically perplexing is better. 
  • The problem allows for multiple ways of being mathematically smart. Hopefully some of these tasks will spur the conversation about being smart in math in multiple ways. Habits of a Mathematician type stuff.
  • The task at hand is clear. And gets to the point.

Here are a few problems that I’d consider starting with. Or, if you’ve been burned or you’re skeptical, problems to try and experiment with.

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Why it’s a good starter problem:

It ties together a visual and number sense. There are several ways to prove or demonstrate a solution. It gets to the point.

Why it’s a good starter problem:

The task allows for guess and check. The task is intuitive and understandable. The scaffolding task involves analysis of samples of student work, a non-threatening way of fostering dialogue.

Why it’s a good starter problem:

The scaffolding involves manipulatives. The math naturally folds into multiple representations and modeling.

Why it’s a good starter problem:

The task prompts students to ask the question. There is an “either-or” possibility for initial guessing and estimating. The task allows for easy differentiated instruction (don’t know how to find the diagonal of a right triangle? how ’bout a workshop on Pythagorean’s Theorem?).

Why these are good starter problems:

You probably have a file cabinet in your room.You probably have a door through which students enter your room. Students have seen and interacted with post-it notes. Students have seen and interacted with styrofoam cups. And with a phone, you could recreate this exact Act 1 video. The task may incorporate multiple ways toward mathematical smartness. Kinesthetic learners might engage via experimentation with post-it-ing the file cabinet themselves.

Dialogue is an inevitability with Always/Sometimes/Never. It can be tailored to your specific classroom. The notion of finding counter-examples is one of the most mathematical ways of thinking I can come up with, and one that kids intuitively understand (it’s a shame we rarely bridge that). If you incorporate some Geometry-type Always/Sometimes/Never cards (like these), kids will be begging you for scratch paper.

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Why it’s a good starter problem:

This task, like all of Yummymath’s, include well thought out worksheets with questions that allow for deep conceptual understanding. If you’re not comfortable with driving the car, let the questions that Bryan provides steer for a while.

Why it’s a good starter problem:

For teachers, I think this models nicely how to modify a textbook problem to something more interesting. For students, they have specific math-like things to do. It gives them exposure to modeling from an authentic scenario.

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Why they’re good starter problems:

It’s got the presentation – usually with video – ready to roll. The Mathalicious team is adept at both humor and conceptual understanding. Like Yummymath, they can steer the ship for a while until you’re more comfortable with less lesson plan structure and organizing groupwork. The lessons are all aligned to CCSS.

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So, there are a few problems to experiment with.

“But they don’t address my particular standards.” you say (For the record, all of these tasks do address specific content standards, see?, but possibly not yours). To that I’d say don’t worry about “coverage” for a day or two. Teachers lose teaching days all the time due to pep-rally schedules, fire-drills, or whatever. And (this is a whole other post waiting to happen but) coverage is overrated. If you can get kids buying into math – possibly for the first time ever – that’ll go a lot farther than coverage.

What are some of your favorite “starter problems”? What other advice would you give teachers that are starting out? Or maybe, better yet: what was your first experience with any kind of inquiry-based instruction like? What was it like for your students? Feel free to share in the comments.

Update 11/22/2013: Andrew, as always, doing great work. I feel like weekly POPs might be another way to dip you and your students’ toes into non-routine problem solving.

19 comments on “A non-linear approach to curriculum mapping”

A non-linear approach to curriculum mapping

I often hear teachers and parents talk about how math skills build on each other in a way that other subjects do not: you have to know how to add before you can subtract, you have to know how to multiply before you use exponents. This is certainly true to an extent, however, I’m wondering if we’re reinforcing these modes by our overly linear curriculum maps (*ahem*). In an inquiry based approach of mathematics, we often preach about “multiple solutions or solution paths” or “multiple entry points.” If we believe what we’re selling, doesn’t that fly in the face of a laddered approach to curriculum mapping? Are we just paying lip service to the whole “multiple solution paths” bit because we know the real way to solve the problem?

linear curriculum map
A linear curriculum map

I was talking with Kelly Renier (@krenier), director at Viking New Tech, and we began discussing the concept of “power standards” or “enduring understandings” or “What are the Five Things you want your students to know when they leave your class?” then build out from there. However, we didn’t discuss building those Five (or whatever number) Things out into linearly progressing units, but rather concentric circles.

concentric circles

Tasks and/or concepts may go in some ring of each of these concentric circles.

ccircles_tasks

Think of it as an outward moving spiral.

However, standalone, this still operates somewhat linearly: you start with the middle stuff (which is allegedly easier or essential) and progress outward, just like you would at the start of a unit, progressing to more complex concepts. But we make an entire curriculum of concentric circles and rotate from concentric circles cluster to concentric circle cluster every few days, or even in a week, potentially moving outward from the center of each set of concentric circles along the way.

ccircles_course
A [circular? iterative? vortex? Archimedean Spiral?] curriculum map.
There are two Moving Parts here, which probably should be addressed individually, but I’ve mashed together, either like a fluid Girl Talk album or Frankenstein’s Monster, take your pick.

  • Moving Part 1: Constructing units as concentric circles
  • Moving Part 2: Rotating through and revisiting topics

That said, I’m not sure you could do Moving Part 2 without doing Moving Part 1. We probably need a name for this type of Scope and Sequence / Curriculum Map: Circular Curriculum Mapping? Iterative Curriculum MappingArchimedean Spiral Curriculum Mapping?

This is getting a little mad-scientisty, I realize. Still, this may have a few potential benefits.

1) Students get to revisit a general topic every few weeks, rather than a one-and-done shot at learning a concept.

2) Students have time to “forget” algorithms and processes and when they see a scenario they have to fight their way through it accessing prior or inventing new knowledge, rather than relying on teacher led examples. Yes, I consider this a benefit.

3) Teachers may formatively assess more adeptly.

4) Students may see math as a more connected experience, rather than a bunch of arbitrary recipes to follow.

5) It probably better reflects the learning process, which happens in fits and starts, and frankly, cannot be counted upon to be contained within a specified time frame.

Personally, I find this framework compelling to a point. I think it better exemplifies recent research and advocacy toward math education. It certainly is messier than a linear approach to curriculum mapping. Your syllabus could potentially look elegant and beautiful or ugly and convoluted. Your administrator might back you, she might not. I’m guessing if you were forced to follow a district scope and sequence, or your math department wanted to be teaching the same things at the same time this would be a non-starter.

So this is just a sort of framework I’ve been playing around with, mostly in my head and I thought I’d throw it out there. I haven’t really developed anything useful. I’d be interested to hear your thoughts. How would you feel about a framework such as this? Do you think it adheres to best practices around mathematics instruction? Would this just work to create more confusion within students? Just how impossible would this be to develop in a public school? Maybe some math departments or curricula are already doing this or something like this? And if it does adhere to best practices and it isn’t implementable due to external constraints, then there may be additional implications for a teacher, school and district. For now, we’re just trying things on. And possibly tearing things apart and starting from scratch. Again.

5 comments on “Kicking things off: How do I start the facilitation of a problem?”

Kicking things off: How do I start the facilitation of a problem?

So you’ve decided to undertake inquiry-based learning. That’s great. I’m really glad you see the inherent value in having students swim through a challenging problem on their own a bit before the teacher jumps in with instruction. I’m also glad you’ve been creative at creating new mathematical tasks with cool entry videos, perplexing pictures, and solid scenarios. Looks like you’ve got your curriculum mapped out, all ready to go for the 2013-2014 school year. Really, you’ve done incredible work this summer as you’ve restructured your curriculum with the help of awesome, engaging tasks from the MathTwitterBlogosphere. It’s fantastic. You’ve come a long way. You’ve shown tremendous agency.

Now what? We’ve got all these nifty tasks tied to standards, but what do we actually, you know, do with them? Sadly, even though we’re all rowing the same direction with regards to inquiry based learning and complex mathematical task driven learning, your students are (probably) not at the place where you can just say “GO!” and they’ll spring in to action. Facilitation needs to happen. And while it’s great to have a protocol like the Know/Need-to-Know process (below) handy, if you’re doing 3-5 tasks per unit, any single protocol, no matter how effective, can get pretty boring after a couple rounds of it. While I do believe in giving students the power in common language, it needn’t be that common.

Here are a few ways facilitate the transition from the entry event (the artifact or problem scenario that launches the task) to the student work time.

1. The Know/Need-to-Know Process (NTK).

I’ve blogged a bit about the NTK process before. It’s certainly my go-to protocol. It works well when deconstructing longer (or wordy) problem scenarios. It’s got its problems though. If you’re not adept at facilitating the protocol or just leave it to the students to fill in some blanks, you’ll get some pretty crappy Need-to-Knows, heavy on logistics (when is it due?) or worthless Next-Steps (teach us how to do the math in this here problem). The point of the NTK process isn’t to establish how many words are in a written task, it’s to aggregate prior knowledge and begin brainstorming solution strategies.

2. #anyqs

The good old “Do you have any questions?” “protocol”. Certainly one of the more fun ones. I’d suggest having students jot down their questions before aggregating them as a class. Dan Meyer does a nice job of this by adding “+1’s” when there’s a repeat question. Ideally you’ll have an overwhelming majority of students asking the same question.

2 1/2. Related: Jeff  de Varona (@devaron3) does a nice bit about “what do you think I’m going to ask you?” after producing the problem scenario. I’ve never done that but it seems on point to me. Here’s an entirely stolen-and-published-without-permission of one of Jeff’s worksheets that has that little nugget in there.

3. Visible Thinking Routine: See, Think, Wonder (STW)

Also similar to #anyqs, but slightly more structured, STW was developed as a way of interpreting and discussing works of art, which, if you’ll allow me to opine, ought not to be so different from math problems. It also has the added bonus of adding a layer of evidence-based things the students notice about the picture or video that #anyqs sometimes lacks. Students observe an artifact and discuss what they see, what they think about what they see, and what it makes them wonder. This protocol also works well when having students peer-evaluate each other’s work.

bridgeSTW

4. Estimations

I think many of us know the power of having students put some estimations up before launching in to the problem. See, they know how to do it over here.

5. Have students develop a concept map before they begin working on a solution.

004

This is a nice way of having students recall previous lessons and mathematical knowledge. It also helps to bin those logistical need-to-knows that often muck of the NTK process (above).

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So there are 5 (and a half) quick ways to move from that awesome, engaging entry event of yours into actual mathematical work. What are some additional protocols or structures you have in your classroom to elicit mathematical strategery?

7 comments on ““Isn’t Problem Based Learning easier than Project Based Learning?” and 10 other myths about PrBL. (“Real or not real”)”

“Isn’t Problem Based Learning easier than Project Based Learning?” and 10 other myths about PrBL. (“Real or not real”)

About a year ago, I started advocating and pushing towards a Problem Based approach in mathematics, as opposed to a solely Project Based approach, which many/most of my peers currently employ. But before we go any further, let’s better parse the differences between Project- (PBL) and Problem-Based Learning (PrBL). I realize that different people define and implement Problem and Project Based learning wildly differently. Some things I would define as problems, others would define as projects, and vice versa. So here’s what I mean by Problem and Project Based Learning:

Pick your favorite diagram.

Figure 1.

Figure 2.

Figure 3 (sorry for the word- and acronym-slaw in this one).

The primary differences are size, scope, and end product. And then also a particular mathematical concept may require several problem scenarios instead on just one project scenario. Here are some untruths and half-truths I’ve heard about moving toward a Problem Based approach. I’d say the two hallmarks of PrBL (in my head at least) are as follows: 1) Each problem is about 1-4 days long and 2) the problem scenario comes first, the instruction and scaffolding succeeds the scenario.

Here are some misconceptions about Problem Based Learning. Let’s do this “Real or not real” style, since the Hunger Games is so popular with you kids.

“Problem Based Learning represents a step back in giving students authentic mathematical experiences.”

Not real. Maybe we need to define what is an “authentic mathematical experience.” A mathematical experience to me is something that promotes mathematical habits of mind. I’m totally going to steal from Bryan here (@doingmath) and point you towards his thinking through mathematical habits of mind. I’d argue that a problem can better facilitate the many of the mathematical habits of mind.

“Problem Based Learning is easier to facilitate than PBL.”

Not real. Due to the tighter time frame, a PrBL Math teacher always has to be formatively assessing students, differentiating instruction and generating good problems. You don’t have weeks to create a workshop, you have a day or so. So you either have to be well prepared for any and all student knowledge gaps and/or quick to respond to Need-to-Knows from students. To me, the work of PrBL is more challenging.

“Problem Based Learning allows teachers to be more ‘traditional’ in their facilitation.”

Not real. Like in PBL, the teacher acts more as a coach or facilitator, rather than a primary source/gatekeeper of knowledge. Students are still given a fair amount of autonomy in their problem solving strategies. Also, lectures still suck.

“Problem Based Learning is more about the math content, whereas PBL is more about 21st Century Skills.”

Sort of real. While I do believe PrBL can mesh nicely with 21st Century Skills, I am acknowledging of the fact that it does promote math content knowledge more than, say, writing a position article for a local newspaper. The fact is that by having the end product be a Socratic dialog among students rather than, say, a powerpoint presentation, may promote math content more than 21C Skills.

“Problem Based Learning is better for students who require more math remediation.”

Not real. In fact, I’d suggest that PrBL is exceptionally appropriate for advanced math courses such as Calculus. PrBL does tend to strip away some of the “psuedo-context” that often makes the math hidden within the weeds of a scenario.

“Problem Based Learning is better for novice teachers, or teachers new to an inquiry-based approach.”

Not real. As I mentioned earlier, PrBL is more work, more difficult, and requires higher order mathematical thinking and content knowledge and skills. It also requires teachers to always be on the lookout for a good math problem. It enhances your mathematical “spidey-sense”.

“If I’m teaching a mathematical concept, I have to make the choice to use either PBL or PrBL; it’s one or the other.”

Not real. There’s no reason a mixture of problems and projects, where each are appropriate, may be used. I would just caution to really think about your Projects: is the time spent in product refinement really enhancing mathematical understanding? Or is it just time spend doing “cool stuff?”

“Problem Based Learning doesn’t allow for authentic real world connections.”

Not real. The main differences are in the size and scope of the undertaking. Here are some or my problem ideas that take 1-4 days that have real world applications:

For more, I’d highly recommend Brian Marks’ awesome YummyMath.

However, I will concede that PrBL does allow for the use of non-real-world connections, and promote pure mathematical conceptual understanding, whereas a Project really sort of needs to be couched in some real-world product. My follow-up question: is it really such a bad thing if the math has nowhere to hide in a problem?

“In Problem Based Learning, students are just given a problem and expected to work through it on their own.'”

Not real. This might be a misconception about PBL in general: that students are just supposed to “figure it out” or that the teacher isn’t expected to “teach” as much. This is flat out incorrect or improper implementation of PBL and PrBL. Every problem ought to have some form of scaffolding along with it. Maybe it’s a lecture, maybe it’s one of these tasks, maybe it’s students sharing their solution route ideas and then going forward as a class. While students are expected to begin to attempt a solution without initial handholding, the facilitator is still expected to address Need-to-Knows with some sort of scaffolding. And the teacher is also responsible for getting students off on the right foot by facilitating some sort of strategizing and brainstorming process.

“You don’t really assess problems in the same way as you do projects.”

Sort of real. I suppose this is more personal preference, but there’s no reason you can’t assess the a problem in the same way you can a project. The main difference is that it might be impractical to develop, say, a full, zillion rowed and columned rubric every couple days. However, the practice of using a rubric to assess across multiple skills and proficiencies is still an excellent practice (despite what others have suggested) . So, a suggestion:

  • Develop one rubric per unit, or batch of problems. Focus on a couple or one particular key learning outcome aside from Math content. For example, written communication: have students have some sort of writing component in every problem for a couple weeks and use the same rubric, or the same written communication component of that rubric.

One other note about assessment for now: I’d suggest it’s easier to assess something like Critical Thinking through the use of a Problem rather than a Project. In a Project, you summatively assess the final, polished product. In a problem, all that scratchwork, brainstorming, and multiple solution attempts are right there on the paper. I’m saying it’s any easier to assign a numerical score to Critical Thinking, just that it can be more evident in Problem Based Learning.

“Non-math teachers can’t help me refine my problems.”

Not real. If anything, I’d suggest non-math teachers are particularly adept at helping you refine your problems. Often, math teachers become so insular and like-minded we can see what the problem is trying to get at. We’re particularly adept at sifting through “psuedo-context” to find out what problems are really asking, while other teachers (and, ultimately, our students) may not be. So by enlisting their support, either by Critical Friend-ing or by actually facilitating the Need-to-Know process, you’ll probably get a better idea of what the problem will look like in an actual real-live class with real-live students instead of a bunch of math teachers.

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What misconceptions have you head about Problem Based Learning that I/we can address in the comments?

3 comments on “Problem Based Learning, start-to-finish, in Ten Minutes”

Problem Based Learning, start-to-finish, in Ten Minutes

Want to learn more about Problem Based Learning but don’t have time to read several posts with graphics? Want to see what a student-centered math unit looks like from start to finish, but would prefer to see it visually and hear it in a nasally voice? Well, look no further, my friends! I recorded a little video in which I discuss five-ish steps to a problem, start-to-finish. It’s about 10 minutes and you can hear me doing terrible impressions of students all while I had a cold. Also, it was made with a crappy movie making program (I won’t reveal what program it was, but let’s just say it comes free with Windows 7 and it rhymes with “Shmoovie Shmaker”). So apologies for that. Hopefully what it lacks in design it makes up for in usefulness.

Here are five stages to Problem implementation discussed in the video, start-to-finish.

  1. Posing of the Problem
  2. Work on the Problem
  3. Intervention as questions arise
  4. Students apply scaffolded instruction
  5. A solution is reached

Although, as I look at it now, it probably shouldn’t appear so linear and step-by-step. It should probably be something like this:

1. The Problem is Posed

2-???. Work ↔ Intervention ↔ Apply scaffolding

???+1. A solution is reached.

Anyway, for more on Problem (or Inquiry) Based Learning and more in-depth discussions, here is some linkage.

Also, I zipped up all the files used for the above presentation, including slides, audio, and the transcript. So if you want to, like, re-record it in your own less annoying/nasally voice, have at it. Or I supposed you could isolate the audio and listen to it as a podcast for some ungodly reason. Anyway, here are the files (note: the slides are in PNG images format and not in proper order; anything else?).

Zip_files_PrBL.zip

Standard disclaimer: I would also like to formally declare that I don’t have all the answers. Frankly, I’m not sure I have very many answers at all. I do have a lot of questions though.

6 comments on “Inquiry-based mathematics: the posing of a problem is only the beginning of the problem-posing process.”

Inquiry-based mathematics: the posing of a problem is only the beginning of the problem-posing process.

We’ve been exploring some of the steps to an inquiry-based lesson in mathematics recently. In the last post, I tossed out a few .png images and laid out a few general steps in preparation for actually getting into the meat of inquiry-based mathematics instruction. Which we’ll do so starting today.

Step 1: Posing a problem

To oversimplify (more-so than I already do), the primary difference between an inquiry-based classroom and a traditional classroom is the placement of the problem statement: it comes before the instruction, thereby giving rise to the need for instruction. The problem is front-loaded, not the instruction.

For a run-down of what makes a quality problem, there are lots of places to go. In short, they A) are interesting to students (either in the scenario itself or in the manner in which it’s posed), and B) have multiple entry points. It’s also nice if there are multiple pathways to a solution (or even has multiple solutions) but that’s not a dealbreaker.

Now that you’ve got the problem, it has to be posed to the students. This can be done several different ways. Maybe it’s a video, a letter, a demonstration or a student simulation, but something delivers the problem to the student. New Tech calls this an “entry event”. Dan Meyer calls it “Act 1.” Whatever it’s called, it is intended to ignite student curiosity about the problem.

However, the problem posing does not end after the problem is posed. The posing of the problem is only the beginning of the problem-posing process. For most classrooms, along with an “entry event” there needs to be a strategizing process in place for students.

One way of doing this is the “Know/Need-to-Know/Next-Steps” process. Here, students begin brainstorming everything they know about the problem and everything they need to know about the problem in order to solve it. This can be done as a class or in groups, jigsaw style. Google docs is a nice tool. As is scrumblr and wall wisher. Of course, a white board works just fine.

But even this neat sounding process is fraught with peril. Here’s an example of Knows/Need-to-Knows I saw in a class the other day, captured in a google doc.

I love this list. Not because these are good knows/need-to-knows/next-steps but because of precisely the opposite: it encapsulates what can go wrong with the K/NTK/NS process.

As you can see, the knows, need-to-knows, and next-steps are not very mathematically rich. And frankly, if your need-to-knows simply restate the problem or are focused on things like “when’s it due?” then there’s really not much of a point to it.

So how do we get from moribund knows/need-to-knows to mathematically rich (and useful) ones? It’s a difficult quandary. How do we get students to say things like “how do I find the zeroes on a graph?” when they haven’t been instructed on finding the zeroes on a graph? It’s a tough quandary, and frankly, you may not get students reciting the exact content lesson you desire. But here’s one way to prepare for this.

Develop a list of anticipated knows/need-to-knows/next-steps and a list of desired K/NTK/NS. Then start filling in the questions you could ask to get from one to the other. Like this.

Try to anticipate the strategies students will come up with and generate questions that will steer them into a more mathematically and problem-solving-esque mode. Or better yet: try running the K/NTK/NS process with non-math teachers at your school, see what they come up with: I’m betting mathematically it’ll serve as a pretty good approximation of of student brainstorming.

Perhaps the most critical act that can sink an inquiry based lesson is when a problem is posed and students are simply set loose without any guidance into their work time. I’ve seen it (and been a part of it many times): teacher poses a problem, students brainstorm, teacher sets students loose, students are unable to engage in the problem as the brainstorming process yielded very little in terms of actionable items. In general, I wouldn’t set the students off to work until they have a list of two or three potential next-steps along with two or three good need-to-knows. But then, that may be just me.

Often teachers ask “yeah, but when do I actually teach?” when transitioning to a inquiry-based environment. The answer is: the nano-second after the problem is posed. Your instruction begins with helping students craft strategies and access prior knowledge that pertains to the problem. That’s as much teaching as anything else you could provide for students.

3 comments on “An oversimplified model of an inquiry-based lesson, with visual aids”

An oversimplified model of an inquiry-based lesson, with visual aids

Last week, I mentioned that, having begun to attempt to slay one of the two giants of inquiry-based math instruction, I’d be steering into a potentially trickier aspect of inquiry based instruction: namely that of instruction and facilitation.

Most of us learned math like this.

We have decades of evidence suggesting that this method of instruction is not only ineffective, but damaging – both to students’ confidence and love of the subject (see Jo Boaler’s awesome “What’s Math Got To Do With It” for more). But honestly, I think that battle has essentially been won. Most of us (I think) are in agreement that this isn’t the ideal way to teach math, or any other subject. But the question is “how?”

Even in more-or-less traditional high schools nowadays, you’ll see something more like this.

Students need to be actively and collaboratively involved in the problem solving process, but what does that really look like? How do we give students both the freedom of solving a problem collaboratively and in novel ways, while still providing enough support to help students along the way?

That’s where the real art of teaching lies. One has to be nimble to adjust instruction based on student need, but prepared enough to be able to anticipate and address the need.

So let’s start with the “ideal” inquiry based lesson, start-to-finish, then in future posts we’ll go back and analyze the process further.

A model of inquiry-based instruction.

Step 1: The problem is posed.

Usually the problem is introduced along with some sort of class or group discussion facilitated by the teacher where students identify key components of the problem and begin strategizing.

Step 2: Students begin work on the problem.

As the students work together toward a solution, the teacher checks in with each group and each student, probing for understanding and answering any clarifying questions.

Step 3: Questions begin popping up from the students.

As the students are working through the problem, questions related to the intended content begin to crop up. As students begin to struggle with the problem a “critical mass” (or “tipping point) of perplexity occurs.

Step 4: Appropriate scaffolding and/or instruction is provided by the teacher.

Based on the students questions, the teacher provides instruction in some format. Maybe it’s a lecture, maybe it’s groups sharing out, maybe it’s analyzing samples of student work, maybe it’s a research resource, maybe it’s an investigation, activity or lab.

Step 5: Students work on the problem some more, after being provided instruction.

Having acquired the requisite content knowledge from the instruction, students proceed to work on the problem.

Step 6: Students solve the problem.

Students finalize their solutions. Usually some sort of informal sharing out or presentation is accompanying. The teacher asks probing questions to get students to make generalizations about their work and promote sense-making.

Fin.

Now, this is clearly an over-simplified model of what a classroom actually looks like. Every step along the way is fraught with different challenges and obstacles to understanding which need to be addressed. I hope that the simplicity of this model does not imply that inquiry-based instruction is simple: far from it! For example, in our fantastical little classroom above students appeared to be all having the exact same question at the exact same time. Obviously that doesn’t ever happen in classrooms. (also, it looks like we lost a couple students from the first image of this post to the next)

I would also like to formally declare that I don’t have all the answers. Frankly, I’m not sure I have very many answers at all. I do have a lot of questions though.

In order to address the monumental challenges, we’ll be looking at each step in depth over the next few weeks, discuss particular challenges, differentiation strategies, etc. My preference would be to get your input and suggestions, since I’m far from an expert.

But before I do, what do we think of this little utopian situation? Did I miss anything? Would you swap out one of the steps for something else? In my desire for simplicity I may have glossed over something or left something out entirely. Please chime in in the comments.