Big Problems vs. Problem Sets

UPDATE (6/9/2020): My thinking around all of this has developed quite a bit, and actually blossomed into the design for a whole new math elective, which you can read about here.]

There are two big structures for "Doing Math" that I really value in math class, and want to understand better. This post is basically a reflection and narration of me working through the relationship between those two structures. They are "Big Problems" and "Problem Sets." The kind of math class I'm trying to learn how to lead is centered around these two big structures, and I need to do some reflection to figure out what they even are.

• Big Problems
• I consider a "Big Problem" (BP) to be one task that is so deep and rich that it could take 2-4 days to complete. Consider my BP-interpretation of this classic problem, given as is:
• The best BPs are problems where the "Big Question" kind of asks itself as you start to play around with it. They tend to start out with an image or description of a context, and then I may or may not follow up with some scaffolding questions. Here are a few examples like the one above. My biggest inspiration for BP Design is Play With Your Math. One of the leaders, Joey Kelly (@joeykelly89) reflects on the design process for some of the problems in his blog, if you wanna learn more.
• I might give kids just this for a day or two, and then slowly give them some follow up problems to guide them towards whichever of the million content objectives I could have with this problem. The students' product is some recording of their work in their notebooks, whatever whiteboard work they did. I track their work with some observation notes and asses their products.
• Problem Sets
• I consider a "Problem Set" (PSet) to be a collection of related problems that all get. Here are a few examples I've used in the past. The header on each PSet give the big idea:
• "These problems are meant to help you if you get stuck, or would like some help organizing and thinking about your work. Do some of them, all of them, or none of them. The order isn’t SUPER important. The problems do tend to build on each other as you go."
• My PSet design is based on what I've experienced from a limited exposure to the problem sets I've experienced at PCMI and PROMYS. The PSet is broken up into three different sections:
• "Definitely Related to the Problem" (A Big Problem At Heart)
• These PSets were designed as follow-ups, after students had already spent a day with a related Big Problem-looking task. The questions in this section are directly designed to help them along the way with the main part of the task.
• "PROBABLY Not Related to the Problem" (Different Surface, Same Depth)
• These questions are where I will engage students in problems that have a "Different Surface, Same Depth." That is, problems that aren't obviously related to the main problem, but share some deep structure at the deeper level. This design feature is related to, but converse to Craig Barton's (@mrbartonmaths) work on "Same Surface, Different Depth" problems, which you can learn more about here.
• "Finding More Spice in the Problem" (Building Rigor & Adding Advanced Math Tools)
• These questions are designed partially as extensions and partially as differentiation. These questions often require more specific background knowledge, like alternative bases, multiplying and polynomials, or modular arithmetic. This is in contrast to the previous two sections, where I design so that pretty much any high schooler can start. They are also designed to help increase the mathematical rigor with which students discuss the main problems.

• Big Problems vs. Problem Sets
• Question: What is the practical difference between a BP and a PSet?
• I'm not totally sure! I know there is a meaningful difference of scope. That is:
• I think that BPs are more focused on specific nuggets of knowledge. For example, I did a BP this year where students discovered and articulated the Handshake Theorem (which upon research just now, is only a lemma--oof). That's pretty targeted content.
• As designed in the PSets I shared above, the first part of a PSet is the breadth of the problems, all of which point towards a pretty big representation or idea (Pascal's Triangle, Fibonacci, the solvability and equivalence of all impartial combinatorial games, GCD and LCM, etc.)
• I think it's possible to experience the same narrative of a PSet, without actually doing a PSet. We all start and finish a BP together, then in each of the following days we tackle a a different "Probably Not Related" problem, or "Finding the Spice" problem. We're basically all just walking through a PSet together.

• Why not just give them a PSet? The PSet is a structure that really gives students a chance to experience independence and ownership of the process, which can be an enormous feature of differentiation.
• The PSet is an "idealization" of mathematical exploration--the mathematical narrative. You got some good math problems, you attack it, connect it to other problems, and use your mathematical machinery to rigorously analyze and articulate the problem and its underlying mathematical structure. These courses then are basically a sequence of scaffolded PSet--scaffolded by our moving together as a class. As such, the structure of the class can be represented below:

• The course is built around a sequence of BPs, with smaller supplemental lessons between them. Then a big PSet towards the middle and at the end. Sometimes the green "regular lessons" between BPs can follow the same narrative of a PSet, just without the explicit structure. Then over the course of the semester, I just giving them the PSet (which we do before each exhibition, at the middle and end of the semester). Having experienced that same "PSet narrative" in class many times, they have some practice and experience with it when they are given a PSet, and they can rely on that to help them navigate the much more open design of a PSet.
• I haven't really thought about how to incorporate PSets into my regular Math 1 class. It feels super reasonable to at least bring in bits and pieces of the philosophy of PSets into class. By my own observance of the pressures and expectations of a standard 9th grade, Math 1 class make me less comfortable introducing such a radically different structure.
• BPs, PSets, and Mathematical Presentation
• I talked about this a little in my previous post describing how I want to do Exhibitions in my math electives. But here I talk about how PSets and BPs relate to Exhibitions in that course specifically. I think an exhibition consists of students attacking a PSet, and then presenting their work. Ideally, all of the PSets resulting in presentations are math new for everyone. 
• I imagine that I might be okay with some students deciding to do one of the BPs we already did as an exhibition. At which point they'd basically be starting the PSet at the "Probably Not Related" stage. Which is okay? But I might permit this only for the subset of BPs that I think we can meaningfully extend to PSet-level scope.
• Question Concern: By design, I'm trying to integrate all of my biggest best problems to serve as the anchoring tasks throughout these courses.
• Side note: Say a student took both of my courses. They would expect to have 20-25 BPs. Then they would experience four different exhibitions, each of which features probably 8-10 different problems. So I'm looking at 52-65 big, juicy, high-quality problems. That's a lot. I got some work to do!
• Question: Where/how am I going to find enough problems so that I can have a complete set of novel exhibition-style problems?
• Please share whatever you have! I'm looking for big, rich, deep math problems that a group of kids could conceivable spend 1-2 weeks studying it and its extensions. Comment here or tweet at me @bearstmichael
• Question: How can you consolidate a PSet, or sequence of PSets, besides student presentation?
• PROMYS really seemed like they used exams and lectures to bring about some feeling of resolution to the sequence of PSets. I've only half-done a full sequence of PCMI PSets, so can't say I really know how they do it.

I realize that this is a public reflection about some pretty non-universal ideas, so I'm not sure how helpful this post and reflection will be to other people. I hope it is! Let me know what you think on Twitter @bearstmichael.

Discussion in Math Class: Part 1 -- Why Bother?

I just finished my second year teaching. Pretty much the whole time I've been thinking hard about class discussion in math class. My teacher prep program (Boston Teacher Residency) was BIG on class discussions, so from the beginning of my career it's been a big thing I've thought about. That's not to say it's been a big part of my practice. I've got a lot of questions I'm trying to work through, a few answers, some resolution, and a bunch of unresolved issues. I'll try to present not only my current conclusions, but some of the narrative as well. I'll try to break this reflection into chunks. This is Part 1.

To be clear, when I say "classroom discussion," I mean a class full of students, all talking about a topic, or series of topics, for a sustained time (like...more than a couple minutes?). This looks mostly like one student talking at a time, and everyone else listening. The teacher may or may not be involved, and there may or may not be any other participation structures.

Experiencing Classroom Discussion as a Student

In math class, the discourse was almost entirely centered around the teacher. I never found discussions particularly educational. I'm not sure who did. For the most part, the content seemed pretty cut-and-dry. Image of a traditional classroom, the teacher's modeling and lecture provided all the information. Any small details, confusion, or minor discovery came out in the problems sets completed alone at home. I never feel the desire or need to wrestle over the math in discussion.

In my smaller math seminars later in college, where the professor explicitly invited discussion, we were all so intimidated and self-conscious that we didn't want to participate for fear of revealing how little we understood. By "we," I mean my friends and I who bonded over how lost in the sauce we felt so often. I presume there was at least a handful of people who felt, as I did in high school, that discussion was unnecessary or uninteresting.

These experience as a student led me to form the following schema about classroom discussion:

It felt like there was a pretty narrow target of students who benefitted from discussion. I would say these students are in the "Zone of Utility," where classroom discussion provides some kind of utility or benefit. They are relatively small "Goldilocks" group where the content is neither so hard it's intimidating, or so easy it's uninteresting to discuss. One assumption of this model is that understanding is one-dimensional spectrum, which is an oversimplification for sure.

What They Taught Me In Grad School

What was told to me about classroom discussion in grad school:

• It should be something that happens in almost every class, almost every day.
• It is an effective way to make sure everyone engages with the lesson objective or Big Idea.
• It is useful in an of itself, because it requires students to make sense of other people's thinking.
• It is a form of communication that can honor the discourse culture of the individuals and whole (classroom) community.

This teacher-side experience of class discussion led me to develop another model for thinking about classroom discussion, which (noticeably) isn't incompatible with the model I had developed as a student:

• Without discussion, many students can engage in a common math experience, but fail to arrive at the teacher's learning objective, these students are, as I described myself earlier, "Lost in the Sauce." Only a small subset of students managed to meet the teacher's learning objective.
• With discussion, many students can engage in a common math experience. As shown in the "Without Discussion" model, probably not all students will arrive at the teacher's learning objective independently, so the discussion after the math experience serves as a way to "funnel" most students to the learning objective.

What I learned from experience about classroom discussion in grad school:

• I didn't have a super clear idea about what a classroom discussion CAN do, and what ONLY classroom discussion can do.
• Discussion for its own sake is a palpable waste of time.
• Class discussions aren't "free learning." Just because students are talking about what they did or what someone else did doesn't mean the speaker or listener are learning from what's being said.

This discouraging experience led me to form the additional model about how my classroom discussions went (and how I thought they went down in many other classrooms):

• With ineffective discussion, many students can engage in a common math experience, but fail to arrive at the teacher's learning objective independently. These students are, as I described myself earlier, "Lost in the Sauce." Only a small subset of students managed to meet the teacher's learning objective. I then manage to lead an ineffective classroom discussion, which at best serves to validate and confirm those few students who had pretty much met the learning objective already, and at worst further alienates students who felt they had not met the learning objective.

Why bother? During my residency year and the vast majority of my first year teaching, this is what it felt like I was doing. It felt like there was no new learning happening during the discussion. Or, that is, it felt like the thing that my students were learning was that discussions are not useful for learning math. So I would bail on them. Pretty much every time. I'd rather let students keep working right up until the end of class, because I felt it was more likely that they learn something during the independent-/group-work time than during a discussion.

UPDATE (6/9/2020): A year later, I've learned a bunch. I wrote a little bit about what I've learned about discussion in math class here. I'm definitely much more willing to do it as a teacher, and have come to think of it as an advanced instructional routine, as opposed to a universal educational approach. The book 5 Practices for Orchestrating Productive Mathematical Discussions is pretty solid, and I can recommend it.

Build It Puzzles

I would like to share with you one of my favorite instructional routines that I experienced last year. A colleague recommended one version of it, I liked it, and then I saw it in a different context, and then it really engaged me in the design process. My particular interpretation of the routine is still in beta testing, and I'm using this post to share, consolidate, and reflect on all of my thinking.

Description of the Routine:

• Name: Build It Puzzles
• Objectives:
• Develop mathematical language.
• Mathematical sketching or building (what I shall henceforth call "construction," which is mostly distinct from the traditional "Euclidean" usage)
• Procedural fluency (for flexibility) in whatever content makes up the problem you use
• "Big Mathematical Ideas":
• Objects can be described by their characteristics, all of which need to be true at the same time
• The more information you have about an object, the more constraints there are, and the fewer constructions exist that meet all those constraints
• How It Works
• You provide a group of students with a set of clues. Each clue describes a different characteristic of the construction. The object can be an arrangement of colored snap cubes, a geometric diagram, a pile of sticks, pattern blocks, a Venn diagram, or some other potentially complex mathematical object.
• The students have to construct the object that is being described by all the clues. Importantly, all the clues have to be true at the same time.
• They check their final solution. If they get it right, they move onto the next set of clues.
• Lather, rinse, repeat.

Where I Found This

• I first saw this in the book "Get It Together," as it was recommended to me by Grace Evans (@grace_h_evans). This book has 100+ different problems of this design, for grades 4-12. And because they're super cool, they also give some guidance on developing your own. The problem from the book that we did was called "Build It!", which is why I have come to call all such problems "Build It Puzzles." It involved putting together 6 different colored snap cubes in a very specific way. We used it as a beginning-of-the-year task for developing group work. It
• The second time I saw this was on Andrew Stadel's blog here where he used it to work on Angle Relationships around Transversals. This was the version of the task that inspired me to to create my own problems for the routine and really work to understand what made the routine work.

What I Think the Routine Does (I'll use the CCSS Math Practices to analyze it)

• MP6: Attend to Precision
• Precise Language: Because the clues are encoded in formal mathematical language, they HAVE to learn to at least read it to understand what to do.
• Precise Diagrams: How many times have you seen a student, when asked to graph a function, to painstaking draw and label axes, when you would have accepted a much looser, sloppier graph? Really, you just need them to accurately represent the key features and the overall idea or shape. My experience with mathematical sketching is that it is "selectively accurate" and "selectively sloppy." In Build It Puzzles, students are asked to sketch (or sometimes build) something, and they are told what the important features are in the clues. As long as the sketch or construction satisfies all the clues, and uses accurate notation when necessary, students don't have to get wrapped up in "perfect" drawings.
• MP3: Construct Viable Arguments and Critique the Reasoning of Others
• Error Correction and Adaptation: If students discover that a new clue doesn't fit the construction as is, then they will need to work backwards to see where they and their group went wrong. What is really useful about this process is that "going wrong" can be something besides making a "mistake." Early in the construction process, with fewer clues in play, students will have to make some assumptions to get started. They'll probably have to tinker a bit to change it by the end, because if the early clues aren't totally deterministic, and they likely won't guess exactly right at first. This process of "guess to get started and see where it takes you" feels super authentic to the work of mathematicians. It forces students to look critically at their own work, and that of others, and look for assumptions and their consequences.
• MP1: Make Sense of Problems and Persevere in Solving Them
• Understanding the Problem: Students are quick to understand the "big picture" of the task--draw the picture being described--but can't immediately jump to an answer. You almost HAVE to dive into the problem without having an obvious end in sight. Only after wallowing in the confusion, and working through what you think you know, can you arrive at a solution.
• Sense-Making: The clues are purposefully dense and esoteric. But when students have to turn words and notation into a diagram that somehow communicates the same ideas, they will necessarily surface the degree to which it makes sense for them. Once that understanding is surfaced, it makes it easier for everyone (teacher, peers, the student) to work with it.
• Perseverance: Every time I do this routine I am impressed and delighted at the engagement and perseverance. I think this is for a number of reasons:
• Each challenge starts out easy. Most students find they can construct something that meets the first one or two clues. The fewer constraints, the easier to construct something that meets them.
• The more a student works on a problem (as long the solution still feels within reach) the more invested they become in finding the solution. As more clues come into play, it becomes a little more difficult. All the while, they are checking off all the clues they HAVE met, which keeps the end point clearly in sight for students.
• It's the kind of task that is BETTER when you do it with other people, and so it makes for a positive collaborative vibe (at least for those students who enjoy doing work with others).

Why I Think This is a GREAT Group Task

• If our goal is to teach students that collaboration is useful and worthwhile, then we need to be able to provide evidence of the fact beyond "you will be graded on your group collaboration." We need to give them tasks that get obviously better with collaborators. Any time we can do this as (math) teachers, we clearly and effectively defend the position that (mathematical) collaboration is worthwhile. I feel that this instructional routine does this, because it has essential features that make it WAY better to do with other people.
• Identifying Assumptions: This task provides automatic feedback when you have made a false assumption--not all of the clues will be satisfied. But it doesn't tell you where you made the false assumption. It can be really easy to make an assumption, thinking it's the only option, and really difficult to go back and recognize that error. Consider this example:. One student quickly drew two separate congruent triangles, labeled the vertices, and moved on to the second clue. It wasn't until much later that, during the error-finding process, another student pointed out that they had two different A's and B's, and that the triangles had to share a line segment. The other student was surprised that they had made the (now obviously erroneous) assumption that they were separate triangles.
• Separate (Physical) Perspectives: Especially with physical constructions (as with the snap cubes based challenge "Built It!" from the book), sometimes having a different physical point of view can add perspective and insight.
• Managing All the Information: You have 8-10 clues that may individually be dense, collectively confusing, and all somehow describe a separate, complex construction. At the same time you have some construction of growing complexity. Never mind that you will be looking at the diagram, constantly checking it against your new clue and all the previous ones at the same time. With so much information, and multiple representations in play, this creates space for multiple people to participate at once.
• Low Spoilage Opportunity: Because the construction is typically built slowly over time, it's difficult for one group member to just skip ahead to the solution and spoil it for everyone.

Tips for Executing the Instructional Routine

• Setting (if your construction is a sketch)
• Vertical Non-Permanent Surfaces (i.e., big whiteboards)
• Why Vertical
• Shared Space: This makes it easier for multiple people to participate, because it is a more easily shared space, making it easier for multiple to see and participate in the construction. I have done it on whiteboard tables, and it works alright, I suppose. A group can still crowd around around the same construction, but it begins to privilege the person with the marker, because the diagram is oriented towards them. But maybe it creates more space for alternative physical points of view. 
• Big Space: I have done a Transformations one on smaller paper-sized grids, because they would need a grid. This was the best I could do with what I had, since I didn't have a coordinate grid whiteboard. It is very hard to include more than one person (much less 3-4) on a single piece of paper.
• Why Non-Permanent
• Flexible (easily erasable) Space: By design, students will be led to make potentially false assumptions, which they will have to go back and change. Drawings with dry erase markers are quickly, easily, and cleanly erased, lowering student anxiety about making the drawing perfect the first time.
• For more on VNPS, see: some research, the person who I think got it rolling, a teacher who documented their own discovery project of making a vertical classroom, some helpful tips, and my former colleague who blogged about it and taught me about it in the first place.
• For constructions that aren't sketches, try to make sure that the medium is flexible and easy to edit. The Get It Together book uses various media like Snap Cubes, toothpicks, and a big Venn diagram with cutouts of the objects to be arranged. These also work well, though I will say that the thing I didn't like about the Snap Cubes was that it was really easy for one student to take control of the process by being the person that holds the blocks.
• Small Groups
• 2-3 is ideal. Definitely more than one student, because multiple perspectives encourage flexibility, breaking through fixed thinking patterns. I like 3 because I want to be able to check in on all of my groups multiple times, and 14 different pairs is way too many to confer with. I have seen groups of 4 work well, but only sometimes, if they are super cohesive and functional together already. I have found groups bigger than four to be untenable. I imagine this is because there just isn't enough space and thinking to go around. The Get It Together book's structure of distributing the clues among the group members might help sustain groups of 4+. (I talk more about that structure below).
• Formatting, Distributing, and Sequencing the Puzzles
• One Packet: I like to print all the clues out, in big font, on one page. This is then kept in a clear plastic sleeve which is taped up onto the whiteboard or wall by the table. This is so that no one person is left holding the clues, essentially preventing other group members from seeing the clues and participating. I will have 4-8 challenges, each set with their own set of clues, each on a separate page, stapled together as a packet. Instructions on the first page of the packet. Giving them all the problems in one packet allows me let groups move at their own pace, while maintaining my intended sequence, and reducing the demand on my to distribute one level at a time.
• To maximize quick starts and task entry, I tend to make the first challenge pretty simple, and then let the challenges get harder as it goes along.
• As mentioned above, the book Get It Together is designed to have you print out each set of clues, and give 1-2 clues to each group member. Each group member is then responsible for making sure that the construction satisfies their clue(s). The book is helpfully designed to this end, so that you can just copy the individual page, front and back, and cut it into card-sized clues. To be honest, this structure felt weird to me. I think this is because it felt like an inauthentic structure for collaboration. When are we naturally prevented from sharing our concrete, written down, pieces of information? Why not just pool our information and build from it, collaboratively. Additionally, if one student is misinterpreting their clue, it can stymie the whole group. That person won't receive any feedback that they have made an error until they share their clue with somebody who can help correct their misinterpretation. I love this task because it fosters authentic collaboration, and I think this particular participation structure torpedoes that. 
• What the Teacher Does
• Circulate:
• Check Solutions: 90% of the questions you get will be "Is this right?" Depending on your style and your students you can vary how helpful you are. Some potential responses include:
• "I don't know, is it?" Always fun for teachers, always frustrating for kids, really pushes them to check their own work and convince themselves that they are right.
• "No." If they're wrong, you can just let them know, and let them figure out what they did wrong. Or ask them a question to help them focus in on an error.
• "Yes." Depending on their perseverance, the time, and your expectations of how many challenges you want them to do, sometimes it's just expedient to just have them move on.
• "Check the answer key." Having an answer key or a hint bank that they can check could really alleviate the demands on you, since they'd be able to check their own work. The downside is that once a students has glanced at the solution, it can really kill the problem solving process, because even if they only use it to discover they are wrong, seeing the solution can really prime and guide their assumptions, unintentionally removing a lot of the demand of critically analyzing your own assumptions. How would you make a less spoil-y answer key?
• Help Find Errors: Usually this can be done by just encouraging them to go back through all the clues and check against the construction, one at a time. Do this a few times and they will begin to internalize it as a generally useful process. Sometimes it may be expedient to reassure them that they don't have to get rid of their whole construction and start all over.
• Correct Misunderstandings: Sometimes students may just misunderstand something, which could prevent them from finding the solution. Identifying these moments is critical. Fortunately, since this is a group activity where people will necessarily present their understandings of each of the clues, other students will also have the chance to identify and correct their peers errors. Which is almost always great!
• Provide Feedback on Participation: This is where I try to spend most of my time. Usually it sounds something like: "Hey Group 4, I noticed that two students are doing most of the drawing and talking. I wonder what we can do to make sure that everybody in this group is included." Or, "Hey Group 2, I noticed that one student is holding the clue sheet, which might make it hard for everyone to see the clues. I'm wondering how we can position the clue sheet so that everyone can see it."
• Collect Solutions
• Students sometimes feel weird just erasing their work, or taking it apart, even if you have told them that it is correct. Which makes sense! If they've worked for 25 minutes, painstakingly trying to draw this elaborate construction, and you give them a quick, "Yup! Got it! Erase it and move on," that doesn't exactly honor all the the effort they put into it. You can take a picture of it, for example. Personally, I have students take a picture of it with their phones, and text it to a Google Voice number that I made just for my students, with their group name. This has the added benefit of cataloging all the student work, in case I want to go back and look at it. Do I? Rarely. But I have the option, and (perhaps more importantly) am honoring student work by saving it (which, when efficient or easy, feels good to do).

Making Your Own Problems

• Examples:
• You can find some examples in the book Get It Together or at Andrew Stadel's blog here. I have done the routine on five separate occasions in my geometry class. I have linked all the ones that I made. Snap Cubes, Angle Relationships, Circle Sketching, Transformations, Corresponding Parts.
• Disclaimers:
• I have left mine and my colleagues comments in the document in order to show you some of our thinking about the tasks.
• Some of these problems DO have errors in the clues. I insist that you try them out on your own to try and catch them, just in case.
• You should all be able to comment on the documents yourself, so if you want to comment there with any questions, comments, concerns, or if you find any errors, go ahead and leave a comment right there on the doc! I'd love to hear your thoughts.
• Targeting Language:
• I have come to think of this as an instructional routine that is 50% vocabulary. I try to remain true to the genre of mathematical writing by giving the clues in technical language and formal notation. And true to the reading of mathematical writing, they may have to check their notes or ask each other what the different notation and vocabulary means. I try to make sure I give them enough space, time, and resources to decode and process the language of the clues.
• Apart from that, however, I try to keep the cognitive demand on the sketching and construction, and NOT on the vocabulary. So I won't typically introduce NEW vocabulary in this routine--but I could! For example, I used it my Circle Sketching puzzle to introduce the term "chord." On the first challenge, I just wrote the definition of the word right there in the clue. I then used it, in each of the following challenges. For balance, I tried to dial down the sophistication of the construction, out of consideration for the overall cognitive load.

• Designing Clues:
• The first one I made (Angle Relationships) had ~15 clues, which was too many. That one took kids 15-25 minutes to complete each challenge.The more clues, the longer it will take to complete. I try to keep it to ~6-10 minutes per challenge now.
• The more clues, the more likely it is that they will make an error on one, which could mess up the whole thing. This routine has the built in feedback mechanism of the fact that if you've made an error, it might be impossible to complete the challenge. So if you get stuck, you'll have to go back and recheck all your previous work. The more clues, and the longer you've been working on a given challenge, the more frustrating it is when you have to go back and start over because you realized you have made an error. (This "built in feedback mechanism" is something I hope to blog more about later.) I try to keep it to 6-10 clues per challenge, but vary this depending on how hard I want the challenge to be. The "Get It Together" book typically has ~6 clues per puzzle.
• You know your students best--you can change the number of clues depending on the perseverance level and attention span of your students.
• One of the objectives is developing flexibility in students' mathematical thinking. So it is good to provide clues that will force students to adapt their thinking. A non-example would be the following series of clues:
• Circle O.
• Square ABCD has all four vertices on the the circumference of O.
• Diagonal AC is a diameter of circle O.
• A student could go through these clues, add each successive clue to the diagram, and never need to go back and rethink their assumptions. Contrast this with the following series of clues:
• Right triangle ACD
• Right triangle CBA
• Triangle ACD is congruent to triangle CBA
• A, B, C, and D are on the circumference of circle O.
• The measure of arc DAB equals the measure of arc DCB.
• The measure of arc DA equals the measure of arc AB.
• Both times they are drawing the same diagram. The second set of clues, however, force the student to deduce, on their own:
• The two triangles form a quadrilateral
• The quadrilateral is inscribed in the circle
• The quadrilateral is circumscribed by circle O.
• The above three facts force B and D to be right angles.
• The arc measures force the quadrilateral to be a square.
• Additionally, they have to decide whether or not AC is a diameter, whether or not point O should be on the diagonal AC.
• Designing Constructions
• Feedback by Design: Ideally a student would look at their final construction and FEEL that they got it right, because the construction "looks" right. It should have some structure...and be satisfying. The goal here is to provide automatic feedback when they have drawn things correctly. The image should "click" together. This will also contribute to the overall sense of satisfaction at the resolution of a challenge.
• Counter-argument: having a final construction that does not present any overall big picture structure forces students to work harder to convince themselves that it satisfies all the clues. Additionally, as students get closer to the solution, the developing construction might guide or prime students in their assumptions, taking some of the demand away. You may or may not want to do this.
• Uniqueness: Make your clues lead to a unique solution. This makes it easier for you to check their work at a glance.
• Double-Check: Have another person check your clue set to confirm that it arrives at your desired construction. It is hard to "unsee" the structure of your construction once you have designed it, and so when you try to check yourself, it is easy to be guided by your own already-existing vision of the product. This may lead you to underestimate the rigor (or overestimate the uniqueness) of your construction.

For Further Development

• Extension Questions: You may or may not want to save these for the last few challenges, for the sake of your fast finishers. What I particularly like about these questions below is that they target the Big Math Idea that more information means more constraints, and results in fewer constructions that satisfy those constraints.
• Is this the only construction that satisfies all the clues?
• If yes, which clues (if any) could you take away, so that they still describe the same construction?
• If no, which clues could you add to make it unique?
• Ask them questions about their construction.
• "What is the measure of angle ABC?"
• "Are these two shapes congruent"
• How could you add a block so that..."
• Flip it the problem! Give students a construction, and ask them to write clues. This is particularly powerful if you are asking them to make a make a set of clues that describe the given construction, and only the given construction.
• Triangle Congruence: You could use this to launch triangle congruence postulates. Consider the clue set:
• Triangle ABC
• One side has length 12 units
• One side has length 6 units
• The angle in between measures 60*
• Then ask them if they all drew the same triangle. Or give them a set of clues that only gives you the measures of the angles. Did you all draw the same triangle? (No, because they are similar, but not necessarily congruent.)
• I'm not sure how to swing this, but it feels like a potentially powerful way to get the idea of "necessary or sufficient conditions," which is fundamental to triangle congruence.
• Questions I Have
• How can I make an effective answer key or hint bank that doesn't also spoil the problem by showing a solution?
• Most of the times I've done this, it has been with geometric sketching. What are the other contexts in which this works?
• How can this routine be used in other content areas?

If you read this far, wow, thank you so much. This post ended up being waayyyy bigger than I thought. I guess I have done a lot of thinking about this particular instructional routine, and want to share it with you all. Thank you for reading and supporting me! Let me know if you end up trying it out--I'd love to hear how it goes!