## Ten Design Principles For Engaging Math Tasks

My work with the Pearson Foundation has changed. They still include some of my three-act tasks (all of which are available for your non-commercial use at this page) but more often lately I review units for engagement. “Dan thinks like a child,” said one of the authors, which I chose to take as a compliment. The bottom line is that engagement is incredibly tricky to nail to the wall. At one point I was asked to draft a document outlining some guiding principles for designing engaging math tasks. I’ll reproduce that document below.

1. Perplexity is the goal of engagement. We can go ten rounds debating eggs, broccoli, or candy bars. [references a debate, long since settled – dm] What matters most is the question, “Is the student perplexed?” Our goal is to induce in the student a perplexed, curious state, a question in her head that math can help answer.
2. Concise questions are more engaging than lengthy ones, all other things being equal. Engaging movies perplex and interest you in their first ten minutes. No movie on this list took more than twenty minutes to set up its context, characters, and conflict. The same is true of engaging math problems, either pure or applied. Use a short sentence or simple visual to “hook” the student into the space of the problem. Use later sentences to expand on it. This order is often inverted in problems that fail to engage students.
3. Pure math can be engaging. Applied math can be boring. The engagement riddle isn’t solved by taking pure math problems and shoehorning them into contexts that don’t want them. It’s hard to argue that two trains traveling in opposite directions from Philadelphia at different speeds is more engaging than “How many ways can you think to turn 20 into 10?”
4. Use photos and video to establish context, rather than words, whenever possible. Rather than describing the world’s largest coffee cup in words, show a photo or a video of it. Not only because our words fail to capture what’s so engaging about the coffee cup but because we should find ways to lower the language demand of our math problems whenever possible.
5. Use stock photography and stock illustrations sparingly. The world of stock art is glossy, well-lit, and hyper-saturated and looks nothing like the world our students live in. It is hard to feel engaged in or perplexed by a world that looks like a distortion of your own.
6. Set a low floor for entry, a high ceiling for exit. Write problems that require a simple first step but which stretch for miles. Consider asking students to evaluate a model for a simple case before generalizing. Once they’ve generalized, considered reversing the question and answer of the problem.
7. Use progressive disclosure to lower the extraneous load of your tasks. This is one of the greatest affordances of our digital platform: you don’t have to write everything at once on the same page. While students work on one part of a problem, there’s no need to distract them by including every other part of the problem in the same visual space. Once they answer the first part of the problem, progressively disclose the next. This technique has far-reaching applications.
8. Ask for guesses. People like to guess, speculate, and hypothesize. Guessing is engaging. Before disclosing all the abstractions of parabolic motion on the basketball court, just show a video and ask the question, “Do you think the ball will go in?” Once they’ve answered, continue the rest of your unit, lesson, or problem, now with more engaged learners. They’ll want to know if they’re right or not so be sure to pay off on that engagement later by showing them.)
9. Make math social. More engaging than having a student guess whether or not the ball goes in is showing her how all of her classmates guessed also. Summarize the class’ aggregate responses with a bar chart. Students will enjoy seeing each others’ short answers and opinions but we can also use the same social interactions to engage them in pure math. Have your students a) select three x-y pairs and b) check if they’re solutions of x + y < 5. If everyone in the class sees the results of everyone else’s investigation, a visualization of linear inequalities will emerge on the class’ composite graph.
10. Highlight the limits of a student’s existing skills and knowledge. New mathematical tools are often developed to account for the limitations of the old ones. You can’t model the path of a basketball with linear equations – we need quadratics. You can’t model the growth of bacteria with a quadratic equations – we need exponentials. Offer students a challenge for which their old skills look useful but turn out to be ineffective. That moment of cognitive conflict can engage students in a discussion of new tools and counter the perception that math is a disjointed set of rules and procedures, each bearing no relationship to the one preceding it.

What would you add? What would you subtract?

• When possible, reveal information only when requested. Current word problems will have 3 numbers given and they will all be used and nothing more is necessary. Knowing what is necessary to solve a problem and what is possible to measure is key to real-world application problems. [CalcDave]
• Once the problem has been completed, explain the cultural and historical context of this problem, if it exists. [David Wees]
• Go crazy. You know how high 5 cups would be? What about 5,000? You can factor this trinomial? Try this octnomial. What would happen if we composed these functions 100 times? 200? Asking these sorts of questions empowers students by making them aware of just how robust the abstractions they’ve earned are. At the same time, they humble students who think that they deserve a cookie for directly measuring the height of 5 cups. [MBP]

2012 May 19: Here’s a predecessor of this document that I totally forgot I wrote.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### CalcDave

April 17, 2012 - 9:35 am -

What about something like:

When possible, reveal information only when requested. Current word problems will have 3 numbers given and they will all be used and nothing more is necessary. Knowing what is necessary to solve a problem and what is possible to measure is key to real-world application problems.

2. #### David Wees

April 17, 2012 - 11:08 am -

Not that it will be used by Pearson, but here’s an important idea.

Once the problem has been completed, explain the cultural and historical context of this problem, if it exists.

Too often I think students think of mathematics as something which just exists, rather than something that was discovered and created over an extremely lengthy period of time.

I’d also like to see more open-ended problems which cannot be easily solved in a single class period. We need to de-sitcom mathematics education. Textbooks are among the only sources where we can provide this more rich explorations for teachers and students to use, unfortunately.

3. #### Steve J. Moore

April 17, 2012 - 11:25 am -

Dan, I love this post. I’m printing it out for discussion with my literacy/comm arts team of teachers in our middle schools. Your “math” ideas represent so much of what I view as effective generalist teaching with a PBL kind of slant. Good stuff which I hope will spark more conversation and writing! Thanks for sharing this!

4. #### Mike T (@gfrblxt)

April 17, 2012 - 11:27 am -

This might be orthogonal to the question asked, or maybe it’s not, but I’d love to have engaging problems developed around this idea:

Encourage students to think on scales other than “human” ones. Problems that are “people-sized” are more likely to be immediately engaging (for example, things about cars, pools, bicycles, basketballs), but developing a sense of scale beyond that is important as well.

Is there a way to engage students with nucleus-sized (cell OR atomic) problems? How about planet-, or star-, or galaxy-sized?

5. #### David Cox

April 17, 2012 - 11:53 am -

I’m surprised by the absence of 8b. Ask for a range. (ie. too high/too low) unless, of course, I read too quickly and just missed it.

You could probably combine #4 and #5.

I’m glad you included #3. I think it’s often assumed that the “application” problem is the only way to have students apply what they know.

6. #### MBP

April 17, 2012 - 12:02 pm -

Go crazy.

You know how high 5 cups would be? What about 5,000? You can factor this trinomial? Try this octnomial. What would happen if we composed these functions 100 times? 200?

Asking these sorts of questions empowers students by making them aware of just how robust the abstractions they’ve earned are. At the same time, they humble students who think that they deserve a cookie for directly measuring the height of 5 cups.

7. #### Kevin D

April 17, 2012 - 12:04 pm -

Love design principle #9. I would consider adding another deliberate step. After students make a written guess, the teacher facilitates a brief “elbow partner” conversation to allow students the opportunity to share their guess with a classmate. Would help students to clarify what the concise question is asking, and begin to dabble in and deepen their own state of perplexity. Then, transition towards the whole-class aggregate bar chart.

8. #### MBP

April 17, 2012 - 12:04 pm -

Just to clarify: Just because something is empowering or humbling doesn’t make it engaging. The distinction between what’s good in some ways and what’s engaging is important.

But I think that in this case, people like flexing their math muscles. That’s fun.

April 17, 2012 - 3:14 pm -

I am excited about the “make math social” bullet, especially with regard to technology. I see lots of room to push the envelope here as you have repeatedly done. I am also happy to see the shout out to “pure math” — thank goodness we have not given up on the sheer beauty of algebra and number theory.

To the extent that anything is missing, I think it is this: Implementation matters. An effective teacher knows where to look for the Easter eggs and so can transform even a dull series of computations into an exploration. Conversely, an ineffective teacher may transform even your three-act tasks into the equivalent of a famous “stupid joke worksheet.” Writers should always keep asking, “What are *teachers* going to do with this?”

10. #### mr bombastic

April 17, 2012 - 4:29 pm -

Nice list. I agree that it is crucial that the essence of the problem is conveyed quickly and easily, but it also seems to help if there is a little ambiguity in the posing of the question. For example if you ask what is the biggest triangle you could fit in the classroom the students can ask about whether it has to be a right triangle, or has to go on a surface of the room (wall/ceiling/floor), or has to fit through the door, or whether big means area or maximum side length, etc. Clarifying questions are another relatively low risk opportunity for students to engage in the problem. I have better results if there are some non-numerical aspects to talk about as well — like describing where you envision your triangle going in the room.

11. #### Nathan Kraft

April 17, 2012 - 6:56 pm -

Don’t let the learning stop.

I recently had my students work on my rubberband ball video

. (And yes, I see now how I gave too much information in the beginning. I will have to remove the text from the video soon.) And students had all sorts of questions ranging from how many rubberbands were in the ball to how much time did I waste making it to how could I have made it cheaper to how to make it double in size). In reality, I don’t have enough time for the students to answer all of these questions in class. I think that resolving those questions can be a great homework or extra-credit assignment.

I had a girl come up to me the other day because she took a problem home to solve and got her dad to help and was so proud of the answer they came up with together. This girl forced her dad to explain some mathematical concepts to her, not because it was some compulsory math assignment with 50 problems that would be worth 50 points, but because she just had to know the answer.

12. #### Shawn Cornally

April 17, 2012 - 7:28 pm -

I miss your musk, and by that I mean fully-implementable posts like this. You changed my teaching with a glaring lambaste of an algebra problem during a PD session. Thanks!

I have the hardest time with #6. How do you make that ceiling explicit without ruining the first or second act?

13. #### Roy Wright

April 17, 2012 - 7:39 pm -

Comment #3: “Once the problem has been completed, explain the cultural and historical context of this problem, if it exists.”

A very important suggestion, in my opinion, except why wait until the problem is completed?

It’s a real shame that the historical background of so many ideas and problems in mathematics has been scrubbed from the curriculum. If we don’t let students in on the reason that someone originally became interested in a concept, or show them an honest reason that people are interested in the concept today, it’s pretty silly to expect the students to muster enough interest to learn the concept themselves.

14. #### LP

April 18, 2012 - 4:39 am -

Dan,

This is great advice. I must say, if you don’t show the students the relevance of the context they always ask “When am I ever going to use this?” I find my students really are more engaged when I give them real scenarios to work out.

Liz

15. #### Bruce Ferrington

April 18, 2012 - 4:40 am -

Not sure about the 3rd point. Is the emphasis on “can”? Also not sure that the train leaving the station problem really qualifies as a real life inquiry. It’s just as abstract as the “pure math” problem.

16. #### Janet Abercrombie

April 18, 2012 - 5:03 am -

Maybe I confuse “pure math” with “applied math” being an upper elementary teacher. But, I have to say that the applied math regarding the floor covered in pennies seems like applied math – and I can’t tell you how engaging it has been for my 5th graders.

I read your blog thinking, “How can I make this relevant in an Upper Primary classroom?” The applied math seems most relevant to them (especially if it has to do with food or money).

17. #### Dani Quinn

April 18, 2012 - 5:19 am -

Good list, and thought-provoking. Two quite different questions:
– What role do you see for more abstract but investigative maths, like that developed by nrich (e.g. http://nrich.maths.org/5469)? I find that *once* students are on track, they enjoy it and the abstract nature makes it ‘cleaner’ for them to explore things, but engaging them in the first place can be tough.

– What do use to (a) capture and (b) edit your videos? I think the editing in particular looks really smart, and would love to know what you use, and how long it took to have basic shooting/editing skills.

Thanks!

18. #### Michael Paul Goldenberg

April 18, 2012 - 6:52 am -

Great stuff here, Dan, as usual. Thanks for the post.

On making mistakes: I can’t resist this quotation from the NOVA episode on Wiles’ proof of FLT:

GORO SHIMURA: That was when I became very close to Taniyama. Taniyama was not a very careful person as a mathematician. He made a lot of mistakes, but he made mistakes in a good direction, and so eventually, he got right answers, and I tried to imitate him, but I found out that it is very difficult to make good mistakes.

19. #### Dave Foster

April 18, 2012 - 8:09 am -

If mathematics is a hierarchical “tree” of connected concepts (per the the French mathematicians whose work underlies ALEKS math software), wouldn’t it be great if we had an easily navigated library of teachable lesson plans associated with each of that tree’s topics–lessons which meet the design criteria you and others have specified?

I’m confidant that the collective intelligence of all of us would exceed that of any one of us. Must each of us design the majority of his/her own collection from scratch? My mantra on this is: imitate the best, then create the rest! If such a wikipedia-like tree existed, the lesson plans submitted from around the world, clustered by topic, would be a real boon to those of us who are brand new math teachers and would like a fast-start to engaging students. Beats boring them while we painstakingly rise up our own learning curve of “lesson creation capability”.

20. #### Dan Meyer

April 18, 2012 - 8:38 am -

I’ve added several of your contributions to the post proper. I like MBP’s quite a lot:

Go crazy. You know how high 5 cups would be? What about 5,000? You can factor this trinomial? Try this octnomial. What would happen if we composed these functions 100 times? 200? Asking these sorts of questions empowers students by making them aware of just how robust the abstractions they’ve earned are. At the same time, they humble students who think that they deserve a cookie for directly measuring the height of 5 cups.

Right. Check out this task. Decide where it goes wrong by MBP’s maxim and decide how to make it right.

mr bombastic fills in some color on my advice to reveal the task quickly and concisely:

I agree that it is crucial that the essence of the problem is conveyed quickly and easily, but it also seems to help if there is a little ambiguity in the posing of the question. For example if you ask what is the biggest triangle you could fit in the classroom the students can ask about whether it has to be a right triangle, or has to go on a surface of the room (wall/ceiling/floor), or has to fit through the door, or whether big means area or maximum side length, etc. Clarifying questions are another relatively low risk opportunity for students to engage in the problem. I have better results if there are some non-numerical aspects to talk about as well — like describing where you envision your triangle going in the room.

Shawn Cornally:

I miss your musk, and by that I mean fully-implementable posts like this.

You and me both. Afraid they’re in shorter supply since I left the classroom. That’s real.

I have the hardest time with #6. How do you make that ceiling explicit without ruining the first or second act?

The bummer case is to write down all the extension problems on a piece of paper and hand it out to students. They know it’s coming. They know they’re going to have to do a lot more work once they finish their early stuff. My favorite is to pose the extension to a group as they finish and to frame it as though it’s something I’m seriously wondering.

“So this guy’s made this enormous pyramid of pennies, which you’ve now found out the value of. His new task he’s set up for himself –Â and I think this is crazy, by the way – is to make a pyramid of one billion pennies. And I can’t help but wonder, should he start building here, in this room, or will it blast through the walls or the ceiling. What kind of room should he build in?”

Something like that, I guess.

@Bruce, Dani, Janet, my only point is there’s a billion ways to make pure math problems engaging and a billion ways to make applied math problems boring.

Dave:

If such a wikipedia-like tree existed, the lesson plans submitted from around the world, clustered by topic, would be a real boon to those of us who are brand new math teachers and would like a fast-start to engaging students.

Dave, meet BetterLesson. BetterLesson, meet Dave. Or 101questions.

21. #### Belinda Thompson

April 18, 2012 - 8:58 am -

Full disclosure: I worked for Pearson for a few years on a PD project.

I like your list, and your position on the goal of engagement. To tweak your list, it seems like some points are about the spirit of the task and some are about the implementation of the task. Is it possible to make this distinction explicit? I see #1, 3, and 9 as more global. #2, 6, 8, and 10 seem critical to the initial planning stage (does the task pass muster as far as these are concerned; 2: will students understand the task with minimal lead in? 6: is there an entry point for everyone? 8: does it make sense to make guesses? 10: does it push students to need to learn something new?). Then #4, 5, and 7 when it comes to planning the nitty-gritty. Would these be concentric or sequential? I’m not sure.

Of course I would add at the global level: “There is a clear goal for what students should understand as a result of engaging in this task.” We tend to get seduced by a cool problem without explicitly stating what students should come to understand.

As an extraneous comment on “engagement”: Lately I’ve been leaning toward the goal of “investment”. To me this is a richer descriptor of what Dan is shooting for.

Belinda

22. #### Matt

April 18, 2012 - 9:13 am -

“As an extraneous comment on “engagement”: Lately I’ve been leaning toward the goal of “investment”. To me this is a richer descriptor of what Dan is shooting for.”

I love this. School and learning will never be as engaging as fun activity X. I think too many teachers sacrifice learning on the alter of engagement. But investment is a worthy pursuit. It also implies something deeper and richer than fun and excitement. You can get students invested because an idea, concept or problem is important, or valuable or relevant not just engaging.

23. #### Matt Vaudrey

April 18, 2012 - 10:43 am -

I agree with Brendan, welcoming wrong answers and discussing them is a part of the solution.

Welcome wrong answers, because they aren’t wrong, they’re just not totally correct yet.

“Is this answer right? Are you sure? How do you know? Do you disagree with her answer? Why? What do you think we’re missing?”

24. #### David Hughs

April 18, 2012 - 1:09 pm -

1. Initially allow brute force solutions then ask if it can be done quicker, thus motivating the formal math

2. Every now and then, let the student “play” with the math

25. #### Bruce Ferrington

April 18, 2012 - 1:26 pm -

Thanks Dan. I like the provocation to reconsider my preconceptions.

Pure math can be interesting? As an elementary teacher I am inherently suspicious of the whole idea and prefer to live in the realm of “real” math. But now you’ve got me thinking…

26. #### mr bombastic

April 18, 2012 - 4:14 pm -

@Belinda,
I disagree that there should necessarily be a clear goal for what students should understand after completing a task. The ease in which I could give an administrator friendly goal for a task is inversely related to the amount of mathematical thinking required for that task. Many of the activities I see don’t seem to involve a lot of mathematical thinking. But that is a different issue. I don’t think we are in any danger of having too many lessons that require good mathematical thinking, but don’t fit super neatly into a unit with a clearly defined goal.

27. #### Belinda Thompson

April 18, 2012 - 5:28 pm -

@mr bombastic

I completely disagree with the inverse relationship you describe. I don’t think that mathematical thinking need be diminished when the teacher has an understanding goal in mind. We may have a different definition of an understanding goal. Mine is this: An understanding goal describes what students should be able to say about the mathematics at work in a task. This is different than a goal about students regurgitating the steps or formulas used to solve a problem.

28. #### BrianM707

April 19, 2012 - 5:27 am -

Adding an element of competition will increase student engagement in an assignment. Setup your game – put a couple extra points at the end for the winner – and you will get their interest. It’s not intrinsic, but it’s fun.

29. #### Sean

April 19, 2012 - 7:27 am -

Just an awesome piece of work. (Your crusade against stock photography is both hilarious and necessary.) Also, for memo-design enthusiasts: note the headers, pacing, and parsimony.

A broader question. 101qs and a lot of the excellent work on this site is focused on classroom tasks. Because these tasks are by design open-ended, I would imagine the path to authentic assessment is a challenging one.

How would you formally assess the ticket roll problem, for example? Giving the students a quiz on Friday with a picture of another ticket roll seems to both defeat the purpose and kill the soul of the project. Put differently, when your goal is to develop habits of mind, how do you assess that students are developing habits of mind?

I can imagine a parent walking into and immediately loving a math class with that gumball picture. Students making guesses and then using math to make precise what intuition couldn’t. I can also imagine the same parent irritated after seeing the gumball photo on an assessment.

30. #### Gwen Jenkins

April 19, 2012 - 11:45 am -

“It’s hard to argue that two trains traveling in opposite directions from Philadelphia at different speeds is more engaging than “How many ways can you think to turn 20 into 10?”

Typical mathematician bias. Discovering the number of ways to turn 20 into 10? Yawn. I’d rather read a book. The stories implicit in word problems always made them much more interesting. The only way I could get through math homework was to make up my own stories to illustrate the pure math exercises.

Comment #18 seems to assume that the trains will run on the same track and collide. (Yes, unlikely.) Real world scenario: Train A and Train B run from different directions on separate tracks at different speeds. What point do both trains reach at the same time? (An efficient place to locate a branch line, or the spot most likely to become a meeting place…) Word problems can also hint at the cultural and historical context.

“When possible, reveal information only when requested.” Or include extraneous information requiring the student to identify what is relevant.

Comment #32 (assessment): Use a different problem that requires the same habits of mind to solve. Or ask for the the formula used for solving the ticket role problem. Or provide the formula and ask them to provide another practical illustration of its use.

31. #### Sean

April 19, 2012 - 2:22 pm -

Gwen:
“Use a different problem that requires the same habits of mind to solve.”

Fair enough. But if the habit of mind underlying the Ticket Roll is- to name one- visualizing proportionality, I think it gets tricky. Can you think of a series of homogenous tasks that would reliably measure that? And do you even want to? I think it undermines the spirit of it a bit.

“Or ask for the the formula used for solving the ticket role problem.”

This assesses a different goal that could be delivered more efficiently- and arguably with more clarity- than the ticket roll problem.

“Or provide the formula and ask them to provide another practical illustration of its use.”

I think this conflates an excellent question with a valid assessment.

32. #### Brian

April 19, 2012 - 4:58 pm -

Should we also think about the role of students’ mathematical ideas? Does your engaging math task have a mathematical objective in mind? A content standard or small set set of standards you seek to engage students in? If so, consider how we might use student mathematical ideas to drive the lesson and the lesson closure. Consider allowing students to communicate their reasoning and critique the reasoning of others (MP.3) through posters or the showing of work through a document camera. You don’t need to have every student or group in the class share out. Did students have different ways of solving the problem? Are there varying levels of sophistication? Look for different methods and discuss how each method connects to another. Your lesson goals, examples and follow up “notes” can come from the mathematical ideas of your students. Students that get to share will better articulate their thinking. Other students may get more out of the task/lesson by hearing from students and listening to your questioning.

33. #### Brendan Murphy

April 19, 2012 - 6:01 pm -

Thinking assessment while creating the task I think might be the one big point that Dan is missing here.

In my district we use a textbook that I am very happy to use. That is in part because I read the implementing guide and follow the intent of the authors.

What some teachers see as questions to be answered I see as discussion starters.

What some see as homework and practice I see as review only for those who think they need it.

Most importantly when some finish the questions in the book they finish the lesson while I spend twenty minutes asking my students what does it all mean?

I mean really who cares how far away an exploding bridge is until someone slaps their forehead and says, “Oh that’s what they meant by d=r*t”

34. #### mr bombastic

April 20, 2012 - 3:15 pm -

Seems like this task (how many pennies fit in different size circles) has a lot of room for improvement.

Put up a large circle and ask for guesses on how many of the circular objects would fit at the start. That is my clearly defined goal – administrators & CCSS be damned.

If you want to use regression, give students a mixed bag of slightly different sized objects (washers, pennies, etc). It is silly to use regression (which conveniently avoids the need for students to have an understanding of area) if the objects are all the same size. This also makes it a little less scripted as they will need to think about a way to “fairly” choose which objects to use to fill the circle.

Have students combine their data with a neighbor and run a regression on the combined data. There is now a reason for regression (due to the variation caused by the different sized objects), and the reason is more apparent when you have a couple of data points for each circle.

Find a way to efficiently check your model’s guess for the number of objects — everyone counts out 100 objects and puts them in the circle, etc.

This activity in its original state follows an all too familiar template of going from easy explicit instructions at the beginning, and without any transition, asking for very difficult explanations at the end: explain why you would expect the number of pennies that fit inside a circle to be a quadratic function of its diameter. Please!

35. #### Don Breedwell

April 20, 2012 - 5:47 pm -

This is really inspiring. I am a Kindergarten SPED teacher and trying to get my students engaged in math is one of the hardest subjects to get them involved. These steps and the subsequent comments have inspired me to help me redefine my methodology to introduce math to my students. If they can start learning math concepts in a more interactive and “social” way, then it can become a subject they can possibly find as one they will want to learn more and more about. I’ve always said the “math is everywhere” and this just reinforces that belief.

Thanks for the great ideas and for all of the comments as well.

Don

36. #### Johanna Langill

April 20, 2012 - 8:41 pm -

I’ve been thinking recently about how perplexity is absolutely essential, but there’s no such thing as a teacher-proof curriculum. I think I’ve got the idea behind perplexing in the beginning, and drawing students in with guessing and intuition. But I would love to see how more experienced teachers map out path(s!) for moving through Act 2. I think a lot of potential perplexity and engagement can be lost in mathematizing these situations. As a brand new teacher, I really have to accept the imperfection. But I don’t want students to have a low threshold but then a cliff face before they can start climbing the mountain. I know this isn’t particularly your focus right now, but I’d love to see your/others’ take on planning out *how* to use these.

37. #### Phil Smith (@liketeaching)

April 21, 2012 - 2:50 am -

Loving this post! I think I’m gonna turn in it into a checklist to pass my activities through for approval.

One thing that I find really helpful for engagement are activities which have instantaneous feedback mechanisms: things like Kate Nowak’s Row games ( http://function-of-time.blogspot.co.uk/2009/12/row-games-galore.html ) or this app ( http://www.fi.uu.nl/toepassingen/02018/toepassing_wisweb.en.html ). Especially for those pupils who still find maths a terrifying topic.

38. #### Dan Meyer

April 21, 2012 - 10:21 am -

Sean:

How would you formally assess the ticket roll problem, for example? Giving the students a quiz on Friday with a picture of another ticket roll seems to both defeat the purpose and kill the soul of the project. Put differently, when your goal is to develop habits of mind, how do you assess that students are developing habits of mind?

Generally speaking, the solutions of these tasks require a particular skill, like finding the area of an annulus in the case of the ticket roll. My assessment, then, will be a naked number problem involving the area of an annulus. I won’t assess a kid on the water tank, for another example. I’ll assess her ability to find the volume of a prism.

Basically, I can offer a lot of formative feedback on the habits of mind inherent to three-act problems (and other problem-types) but I set the bar a lot lower for the summative kind of assessment that will require a student to retake a math class.

@Gina, the product on which I’m consulting isn’t yet on the market.

Johanna:

But I would love to see how more experienced teachers map out path(s!) for moving through Act 2. I think a lot of potential perplexity and engagement can be lost in mathematizing these situations.

The middle act of these problems is the one that most defies my efforts to script it. While the beginning and (to a much lesser extent) the end are tightly scripted, the tools, information, and resources necessary for moving from one to the other vary wildly. I suppose I’d say that the more tools that exist for solving a particular problem, the more likely I am to back off and see what students do with the task. If only one tool can crack the problem open, I’m more inclined to explain that tool’s use on a separate, generic problem and send them back into the original problem.

I’ll offer Pyramid of Pennies as an example of the former and Falling Rocks as an example of the latter.

39. #### Matt Wilson

April 21, 2012 - 5:57 pm -

Great post and discussion. I just wrote an article that relates to this very closely; it is a draft of one chapter of a book I am writing and I am very interested to get some feedback from other educators. I apologize for the self-promotion, but I was reading your blog and it just fit in so well I had to share – think of it more as a response too long to fit in a comment window.

Thanks

http://everythingreform.wordpress.com/2012/04/20/you-want-school-reform-brace-yourself/

40. #### Marshall Thompson

May 11, 2012 - 5:54 pm -

It’s funny, I’m not going to say that engagement is easy, but it’s certainly easier than perplexity.

And the ultimate challenge is funneling that engagement and perplexity into the standard.

The disconnect between the way kids best learn and the way kids are assessed is significant. And I’m not sure I know how to resolve that conflict yet. Maybe that’s where the money is!

41. #### Eileen

July 7, 2012 - 6:36 am -

My most engaging lessons come when the students ask the questions, not me. Your videos and pictures have had a positive impact on these activities. I encourage all math teachers to read, Make Just One Change, a book that helps teachers develop the questioning ability/skill in all their students. The questions that my students have created mirror the comment Go Crazy.