Wednesday, June 26, 2019

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.

My end-of-class discussions weren't going particularly well. Where any of my classroom discussions going well?

To be continued!

Sunday, June 16, 2019

Proposal: A Less-Sequential High School Math Experience

How can we design a high school math department (and graduation requirements) so that we are accurately reflecting the vastness, diversity, and non-linearity of mathematics? Consider this representation of a hypothetical high school math department (Image here):

Understanding the Map
  • Each course is a semester-long math course
  • If there are two separate arrows from A to C, and B to C, then that means either A or B are required for C, not necessarily both. If the arrows meet before C, then that means that A and B are both required for C. Consider this helpful diagram (Image here):

Graduation Requirements
To go with this more open curriculum offering, is a graduation requirement system that tries to honor both the openness of the program, and the necessity of certain math topics in high school.
  • You must pass at least eight (8) distinct courses, including:
    • Proportionality, Linearity, Financial Literacy, Probability, Correlation & Causation
  • There are four Fields of mathematics, and students must take the appropriate number of courses in each field. Some courses satisfy multiple requirements (Image here.).
    • (A) Algebra (take at least 4)
    • (G) Geometry (take at least 2)
    • (P) Applied Math (take at least 1)
    • (U) Pure Math (take at least 1)

  • You must pass any and all prerequisite courses before taking a course.
  • Students are permitted to "test out" of prerequisites provided they demonstrate sufficient proficiency, as determined by the math department.
    • If a student tests out of a course, they are not awarded the credit for that course, nor does that satisfy any Field requirements.
Logistical Comments I Have
  • The standard pushback here is that this kind of system would require teachers to teach lots of different courses in a year. More preps makes things more difficult for teachers.
  • Lots of students might be forced to take certain courses simply based on the fact that it was the only course available at a certain time in a certain semester. The bigger the school, the better the school's capacity to support all these different courses.
  • It might make more sense to default all 9th graders to starting in "Linearity," and only walk them back to "Proportionality" if necessary.
  • It would take a TON more resources to make this work if you have special populations of students who might need to be grouped (students with learning disabilities, students learning English as a second language, etc.)
  • This would work best if the other departments (ELA, History, Science, etc.) used a similar system.
Questions I Have for Y'all
  • What courses would YOU make required?
    • I thought Proportionality, Linearity, Financial Literacy, Probability, Correlation & Causation were the most important, recognizing that most high schoolers would already have proportionality from middle school, and some might already have linearity from middle school.
      • Those five courses I identified that everyone had to take are based solely upon my own value judgements of what math content I think every person should know before leaving high school. They definitely need to do more math than just those courses, but they definitely have to do at least those courses.
  • What other courses would you add?
    • These were all the courses/topics I could think of that I've either taught, seen, or felt spanned the traditional high school math curriculum.
      • Some courses I would need to figure out how to add are all the AP's and Computer Science.
      • I'd also like to add some kind of geometry-centered course where students are getting lots of geometry measurement and whatnot. Like, where in this course offering are students going to think hard about the volume of geometric solids (except calculus)?
      • I think Economics might better be split into Micro and Macro? Is one a prereq for the other?
      • If other departments get in on this, then that opens the door for requirements to be met across entire departments. For example, lots of schools consider Econ to be a social studies class. Next year we are teaching Mathematical Biology, which could count as a science credit.
    • Best case scenario, we think of the biggest reasonable course map possible, with all the appropriate relationships thought out, and then schools would pick some subset of that map.
  • Do these course descriptions make sense?
    • Some of them I don't even know how to describe (e.g., Proportionality)
    • Some of them I'm not sure are a whole semester of high school level work (e.g., Linear Algebra, Periodicity)
  • What do you think about the Fields?
    • I made the Field requirements so that they were roughly proportional to the course offerings. That is, there are way more Algebra classes in the map, so I required way more Algebra classes. But this is a reflection of my own limited ability to create course offerings. If I could think of more geometry courses, then I could make one.
    • I'm lumping statistics in with Applied Math, which would include Econ and Financial Literacy. The implication of this is that kids could take Projectile Motion or Financial Literacy and never HAVE to take any other stat classes. Except that I've already said that half of the Applied Math Field is a requirement on its own. I guess that says something about my own feelings about Applied Math in high school?
  • How does this gel with our understanding of intellectual development of kids?
    • By this model, you would have 9th graders taking Probability. You could also have 10th graders taking Linear Algebra. Is there a reason this couldn't be the case? It certainly feels a little weird, expecting 9th graders to do content that is typically assigned to 11th and 12th graders.
      • Is that because they are literally not OLD enough to do that kind of math? At which point, we should place another pre-requisite there, to slow them down? Or have an age requirement? But that feels wrong.
      • Or is that because we typically haven't taught them "enough math" until they are older? At which point, we should place another pre-requisite there, to make sure they get all the required math.
    • This would put kids of all grades/ages in the same classes. Lots of classes might be mostly grade-grouped--Linearity and Proportionality would have mostly 9th graders. Polynomials and Solving Triangles would have mostly 11th and 12th graders.
  • Where does solving equations go?
    • I think that for the most part, most of the big first "this is how you solve equations" would come out in "linearity" maybe? And then as appropriate, as you run into different contexts for solving equations, you learn new methods for solving equations?
  • How can you make sure that students are making informed decisions?
    • They would have a good idea about what each course is about, so that they can pick the classes that are the most interesting to them.
    • They would have to have a good understanding of the requirements system overall, so that they could "see" the paths and scheduling required to get to certain outcomes. Which is complicated, for sure. I mean, it's like trying to plan a major for college--only you have a major for math, science, history, ELA, and whatever else.
  • What do you think?

Thursday, May 23, 2019

Computers in Math, Part 1: Authentic vs. Inauthentic

Over break I had a brief conversation with Melynee Naegele (@MNmMath, blog: mNm Math) about the appropriateness of having computer-based standardized assessments. In particular, they raised the issue of how restrictive the medium of computers can be. It got me thinking about what a coherent philosophy about computer-use in math class can be. I started to write this blog by listing what kind of math computers should/shouldn't be asked to do. But then I realized that there is a more important guiding philosophy that I needed to think through and articulate before going into that. So I'll introduce it with an analogy:
  • "You wouldn't chop down an oak tree with a butter knife--you wouldn't butter toast with a chainsaw."
    • I mean, you could. It would suck. And if it was all you had, I guess you'd have to make do. But it would suck.
    • This is because chopping down trees is an inauthentic application of the butter knife tool. It would be an authentic application of the chainsaw tool--it's what it was made for, it's what it's good at.
  • Big Idea: Part of learning math is learning when to use what math tool.
    • What is a math tool? Here, when I say tool, I mean more or less a physical or technical devise, like spreadsheets, rulers, calculators, lasers, string, Latex, MATLAB, graphing calculators, etc. I don't mean intellectual tools like, "Try a simpler problem," or "try a different representation."
    • What do many people think the main math tools are? When a lot of people think of mathematicians, they think of someone standing at a blackboard, with chalk, just scribbling all over the place. In my math studies, this is common, in one form or another. Personally, most of my own math education has happened in pencil, on thousands of sheets of blank white printer paper.
    • How have computers-based tools been used in math?
      • The Traveling Salesman Problem, a classic graph theory problem, was ultimately solved with computers, because the nature and volume of calculations and algorithms were enormous.
      • At least two of my calculus professors were disappointed in us when we weren't proficient in MATLAB, a computing platform with powerful applications in math and science, especially with partial differential equations, which have tremendous applications in the physical sciences. MATLAB can really allow you to see and experience what happens when you change your model, conditions, and representations.
      • Almost ALL of my college math professors were incredibly bummed when we weren't able to type up our problem sets in Latex, the premier language for typing math. This is because it can be way easier to read typed out math than handwritten math.
      • Most of the common fields of applied math (finance, physical sciences, etc.) would be almost impossible without spreadsheets. Ask anyone who knows me: I'm the spreadsheets guy. The capacity to manipulate, analyze, and represent huge amounts of data is immeasurably better than what humans can do alone.
      • Modern cryptography literally exists because of computers, and our need to publicly transmit private information.
    • What if I use a tool inauthentically? You waste time, energy, and it sucks, generally.
      • What if I asked you to use a spreadsheet to find the square root of 80? You open a spreadsheet, figure out what function to use, what the syntax is, click some cells, and boom: 8.9ish. If I only asked you to do computations with spreadsheets, you'd probably think I was ridiculous for not using a calculator this whole time, and you'd have NO idea of the utility of spreadsheets. Why is that bad?
      • Because what if I THEN asked you to find the how many numbers less than 10,000 weren't divisible by a square number (square-free)? My gut tells me there's a reasonable way to do this with combinatorics, but when I first experienced this math problem, my instinct was guided by some slick spreadsheet functions, which revealed some very interesting stuff! (No spoilers!)
      • If all I knew about spreadsheets was that they could do computations, then I would be missing ALL the wonderful things that they can do, and feel forced to use less-good tools to do the job. Or worse, I might come to not feel like spreadsheets are good for anything!
  • The difference between math education and professional mathematics
    • There is definitely a difference. In math education we have to deal with things like grades, lots of standardized assessments, a criminal lack of funding and resources, and children, parents, and teachers/administrators who may not themselves be experienced mathematicians. In general, math education is a different field of mathematics, with different needs and capabilities. So the use of tools will necessarily be different.
      • So this raises a new dimension of authenticity. What are authentic math tools in the field of math education, that might not otherwise be authentic for professional mathematics. For example, one of the biggest applications of computers in math education is assessment. There aren't very many super meaningful "math assessments" that professional mathematicians engage in. I guess there's the Putnam?
    • But this difference doesn't mean that the math that children experience has to be super different from professional mathematics. The goal of math education is to teach young people how to think and do like mathematicians, because it is a useful and interesting way to think and do things. So ideally, math education would teach students how mathematicians work by asking them to do mathematical work, and have all the mathematical experiences that shape the way they think and do math, and things in general.
    • So as a teacher, when deciding what tools to teach your kids to do what math with, the degree to which you teach your kids to use the tool authentically, is the degree to which they will develop a REAL understanding of the tool, the math they did with it, and what it means decide how and when to use tools in math.
  • Conclusion: Computers don't belong in math any more or less than patty paper, calculators, pencils, chalkboards, string, and lasers. They are all tools of math, with the power to be authentically used (to great benefit), or inauthentically used (to great detriment). 

Sunday, March 24, 2019

Assessing Vocabulary and Language

I just finished a very language-heavy mini-unit on sequences in my Math 1 course, with my 9th graders. This is our first year with the CPM Integrated Math 1 curriculum, and for reasons of pacing, we pared down the unit a bit. We elected to not cover writing formal recursive rules, or writing explicit rules for geometric sequences. This left pretty much only the central question, "How can we identify and describe patterns in numbers?"

Since we were only looking at arithmetic, geometric, and quadratic sequences, we basically just needed to be able to precisely describe and extend any of those families of sequences. One of my focuses was on using vocabulary like common (nth) difference/ratio, arithmetic, geometric, quadraticterm. Having a unit with such a big language focus was a great learning experience for me, and in particular, it has led me to think hard about what it means to assess language and vocabulary. Here are some thoughts I have at this point:

What's the difference between "vocabulary" and "language"?
  • I have come to understand this difference as a manner of "scale." I think that vocabulary is word-ish-level, and language is pretty much everything bigger. An example of a language form is something like sequential sentence frames, which a student might use to describe a process: "First, I did this. Then, I did that. Finally, I did this."
What could a vocabulary and language assessment look like?
  • The big question I was trying to work through during this unit was how to know when and how I was assessing vocabulary, language, both, or neither. Consider these different assessment questions:
    • "Make an example of an arithmetic sequence."
      • I'm expecting them to write down an arithmetic sequence.
      • Vocabulary: Here, the student has to know what the vocabulary word arithmetic means, and how it creates a sequence.
    • "Find the next three terms of this sequence, assuming that it is an arithmetic sequence: 2, 4,..."
      • I'm expecting them to produce the next few terms of the sequence, no word writing necessary.
      • Vocabulary: Depending on your students, this is assessing if they know what the word "arithmetic" means.
    • "Describe this sequence: 2, 4, 8, 16, ..."
      • I'm expecting them to accurately use vocabulary like "geometric," "common ratio," and "first term," all in a mostly-grammatical sentence or two.
      • VOCABULARY & Language: They need to know those words off the top of their head, and how to use them in a sentence. They have to identify the context, and recognize that I'm not looking for "exponential," "increasing," "rate of change," or any of the more function-oriented language. Like before, you need word-level understanding of precise mathematical vocabulary (Tier 3 words). You also need language level understanding to be able to write the description. But this assessment is skewed towards vocabulary (hence the all caps), because I am thinking of language as necessarily requiring facility with the vocabulary.
    • "Use the some of these words to describe this sequence: 2, 4, 8, 16,... {arithmeticgeometric, common, difference, ratio, term}."
      • I'm expecting them to accurately use vocabulary like "geometric," "common ratio," and "first term," all in a mostly-grammatical sentence or two.
      • Vocabulary & Language: Here the students don't have to decide what vocabulary cluster of words to pick from (e.g., slope vs. common difference). Nor do they have to remember all the relevant words off the top of their head. But they DO have to know how to put them together into a sentence, which is VERY language-assess-y. But they also have to know what the words in the word bank even mean, which is a non-trivial degree of vocabulary assessment--albeit, less than if we asked them to come up with the words on their own.
    • "Describe this sequence: 2, 4, 8, 16,..."
      • I'm expecting mostly-grammatical sentences that include accurate, if not exactly precise/formal language like I've expected before. For example, a student could write something like this: "You start with 2, and then to get from one number to the next you have to multiply by 2. Then you can go on and on."
      • vocabulary & LANGUAGE: The vocabulary demand has gone down, because most of these words are pretty much what you would expect any English-speaking student to be able to use (Tier 1 and 2 words). But it also has the mathematical language rigor in that is accurately and pretty-completely describes what's going on in that particular list of numbers.
  • When is vocabulary assessment not just VOCABULARY assessment?
    • In each of the quizzes above, for my students, these vocabulary quizzes are still like...15% computation--I am necessarily assessing numbers sense and arithmetic skills. This is because students have to do some subtraction and addition to extend the sequence, or identify the common difference. Depending on your students, the computations or language themselves might be especially demanding, at which point this moves across the scale between "vocab assessment" and "computation assessment."
      • For example, if I have a bunch of students with dyscalculia, these computations could become VERY cognitively demanding, VERY quickly, making this much more of a computational assessment than a vocab/language assessment. I could move this back towards language by offering use of a calculator, numberline, or annotating the sequence for them.
      • Alternatively, if I have a bunch of ELD1 students, who are in their first year (or months) of learning English, having to work with any words in English can be very demanding. Or, if I have a student with a reading or language disability, I might have to modify my expectations so that it is developmentally appropriate.
  • Questions I still have (note my blog title)
    • What do we have to do to make a vocabulary assessment JUST a vocabulary assessment?
      • When/why could we do this, or not?
      • When/why should we do this, or not?
    • What language/vocabulary do I want which kids to actually memorize?
      • If kids had a glossary in the back of their notebooks, is that something I'm okay with them using during assessments? Or if it was on an anchor chart or something?
In a later blog post, I hope to reflect more on WHAT language forms that I have thought a lot about this year. I also hope to write about WHEN I introduce vocabulary/language. I also hope to write about what instructional routines I like to use to help students build their mathematical language/vocabulary.

Friday, August 3, 2018

The Mathematical Method

What/Why/How is Math?
I have the chance this year to teach a semester-long math elective for juniors at my school. I have pretty much full curricular freedom. The course is titled "Discrete Math," so I intend to focus on the topics of Modular Arithmetic, Number Bases, and possibly Graph Theory. I am excited at the opportunity, and am super glad I had a chance to see Joey Kelly teach two semester-long Discrete Math courses during my teaching residency.

As I sit here this summer and try to figure out what I want to teach, I have found myself slipping down the big philosophical rabbit hole of "Why?" I think that many math teachers believe that our objective is to not only teach specific math content (e.g., polynomial multiplication, estimating fractions, converting numbers to binary). We are also trying to teach students about the bigger general practices and skills that guide mathematicians. Understanding the specific content standard that the powers of i cycle every 4 powers is only long-term helpful for a kinda-narrow band of people. But teaching students to notice patterns, explore and discover truth, state and argue their position based on evidence...those are about as transferable as skills get. I believe that we are trying to teach students to think and work like mathematicians, because the way that mathematicians think and work is powerful and useful. The more math that I experience as a learner, the more I develop an understanding of what it means to think and work like a mathematician. And I have been looking for a schema that allows me to understand what that actually means.

The Common Core Standards for Mathematical Practice capture some of this, by removing mathematical work from specific content standards. I have tried to think about how to value these practices in my math teaching, alongside specific content standards. But these math practices feel disjointed, and they still haven't helped me to develop a unified "Big Picture" of "What/How" mathematicians do. But I think I have an idea of how I can begin think about this, and I want to run it by you folks.

Introducing the Mathematical Method
Scientists have a Scientific Method that reflects how they "do science." What if we had a Mathematical Method that reflects how mathematicians "do math"? The goal of developing and understanding this Mathematical Method would be to offer a vision of what mathematical work can look like, across math content. I have created a first draft of what I think this method could look like. Below, I ask some questions about this method and offer some initial responses. I would love to hear your questions and thoughts about this idea.

The Scientific Method
The Mathematical Method

Questions I Have
  • Are there any other teachers, scholars, or researchers that have done work on this kind of "Mathematical Method"?
    • These are the only two places where my research has found a discussion of a math equivalent to the Scientific Method. Both of these two articles claim that the mathematical goal of absolute proof makes the work fundamentally different from science. They don't discuss the ways in which the similarities can/should be described.
  • If math requires absolute "proof" in a way that science typically can't--does that affect the process?
    • I claim that in both math and science, you take the same big step of proof writing, where you develop an intuition as two why your conjecture/hypothesis SHOULD be true. We believe that humans are driving climate change, because it makes sense that many of the things that humans do directly affect the climate. But can we "prove" this in the same way that we can prove that the Pythagorean Theorem?
  • What are some consequences of the fact that this Mathematical Method seems similar to the Scientific Method?
    • Our science teacher friends have been teaching and advancing the Scientific Method for a long time now, and from a very early age. If we can present a parallel structure between the two methods, then we can take advantage of the fact that students already kind of understand this Mathematical Method (insofar as it resembles the Scientific Method).
    • If the methods are so similar, this can make the distinctions between the two methods more obvious.
    • If both science and mathematics seem to have a similar process, that seems to suggest that there is a broader structure of "problem solving." 
  • How is this Mathematical Method different from the Scientific Method?
    • That penultimate step of the Mathematical Method, "Develop a Conclusive Complete Proof," doesn't apply in many (any?) other fields.
    • The language of the processes is different, even if they are describing largely the same concepts (hypothesis vs. conjecture vs. thesis). More philosophically, "prove" means different things in math, science, and other fields.
  • Should we teach to this bigger "Problem Solving Process"?
    • Should we just teach a generalized "Problem-Solving Process" that can transfer between problems in math, science, and other fields? Drawing connections between big ideas within and across fields of knowledge feels useful, and allows a student to better package and organize their knowledge. This feels like part of a bigger question of what it means to "know" something?
    • What does this generalized "Problem Solving Process" have to do with "Critical Thinking?"
    • There are meaningful differences in the different problem-solving processes in math, science, and other contents. If they teach this process in science classes, in addition to teaching "general problem solving," should we do the same in math class?
  • What about content areas outside of STEM? Is there a "Historical Method?" An "Artistic Method"? A "Language Arts Method?" Do they have a similar "process"?
    • I feel like any content area in which students "solve problems" likely has some kind of general problem-solving process. I definitely don't know enough about these other content areas to begin to answer this question!
    • From my own experience in my high school English Language Arts classes, it feels like there are parts of ELA that aren't problem-centered. In particular, it seems like in parts of ELA the goal is to create or communicate something. (Personally, I always loved writing short fiction.) But there are problems posed in this process too, right? How can I make the reader "feel" what I want them to feel? How can I communicate exactly what I mean to communicate? Is there another articulated "process" that writers and artists use in this situation? This also makes me wonder: what are the things we do in math that aren't problem-centered? Which of these "processes" fit in "Critical Thinking"? Which don't?
  • How do the Math Practices fit in?
    • I think that mathematicians use the math practices throughout the Mathematical Method, some more at certain times. For example, during the Exploration phase, a mathematician will do a lot of MP8 where they "look for and express regularity in repeated reasoning." This is how they might develop their initial suspicions that something mathematical is going on that they might not yet understand. And at the end, when in the phase where they Communicate Results, mathematicians will do a lot of MP3: "Construct viable arguments and critique the reasoning of others."
    • I think science teachers have a similar set of Science Practices as a part of their NGSS standards. A lot of them feel similar to the Math Practices. For example, #2 "develop and use models" connects to MP4 "Model with mathematics." How do science teachers combine the Scientific Method with their own Science Practices?
An Example of the Mathematical Method
Maybe an example would help me to think through Mathematical Method. I will use a common Algebra II problem, based on the powers of the imaginary number i.
  1. Develop a Question: What is i^357? (Or more generally, any power of i.)
  2. Explore: use your present understanding of i to calculate some easy powers of i. What is i^2? What is i^5? What is i^6? What is i^9? 
  3. Make a Conjecture: "All odd powers of i are i and all even powers of i are -1."
  4. Test Conjecture (with strategic exploration):
    1. Strategically Explore: Test a bunch of odd powers of i. The 3rd, 5th, 7th. Maybe you remember to check the 1st power. You get -i, i, -i, i respectively, and realize that your conjecture is wrong because it doesn't account for this cycling between i and -i. You realize this same problem comes up with the even powers of i.
    2. Revise Conjecture: Odd powers cycle between i and -i, even powers cycle between -1 and 1. And you go through all four before you see a power again.
    3. Strategically Explore: Test a bunch of powers of i. This time maybe you make a table, or a diagram of some kind, and work more systematically through all the integer powers of i starting at 1.
  5. Confirm Conjecture: Your strategic exploration has shown you that you see your repeating cycle of i, -1, -i, 1. You also notice that the 1 always shows up when the power is a multiple of 4, i when the power is one after a multiple of 4.
  6. Develop Intuition As To Why Conjecture SHOULD Be True: Depending on how you have come to understand what i is...
    1. (Algebraically) i = sqrt(-1): you know that when you square i you get -1. And if you square that again, you get 1. Squaring a square is the same as taking something to the 4th power. So it's like every group of 4 i's you multiply get together, and turn into the multiplicatively useless 1. And you take away all the groups of 4, till you have 0, 1, 2, or 3 left. So every power of i is really just one of those 4 situations. (This becomes easy to articulate when you have modular arithmetic in your toolbox.)
    2. (Geometrically) i <==> quarter-rotation. Every 4 quarter-rotations you have gone back to where you started. So it's like you didn't even do anything. So you can subtract 4's from your exponent until you get to 0, 1, 2, or 3.
  7. Develop A Conclusive Complete Proof (here, I chose a more algebraic proof)
    1. For any natural number n, n=4s+r for some r in the set {0,1,2,3}. (Depending on your level of rigor, you may need to prove this as a lemma, or just cite an existing proof.)
    2. i^(n)=i^(4s+r), given
    3. i^(4s+r)=(i^4s)*(i^r) =((i^4)^s)*(i^r) by exponent laws
    4. (i^4)=i*i*i*i= -1*-1 = 1 by definition of i and some hand-wavily used axioms (definition of exponents, association, definition of additive inverses)
    5. ((i^4)^s)=1^s, from above
    6. 1^s=1 by definition of 1 as multiplicative identity
    7. Therefore i^n=((i^4)^s)*(i^r)=i^r, where r=0, 1, 2, or 3.
      1. If r=0, then i^n=i^0=1
      2. If r=1, then i^n=i^1=i
      3. If r=2, then i^n=i^2=-1
      4. If r=3, then i^n=i^3=-i
    8. Specifically, 357=4*89+1 ==> i^357=i^1=i. Q.E.D.
      1. Since I have written the rigorous proof I wanted, I may want to go back and work out some language so that it is clear that my proof results here are actually saying the same thing as my revised conjecture).
  8. Communicate Results: publish my proof in my high school journal of mathematics. Win Fields Medal, have it stolen immediately, and so on...
I think that the "experimental phase" of the mathematical method is super under-experienced in math, and we push kids to "believe" pattens that we tell them to believe (like the patterns in the powers of i, or the volumes of prisms, or that tangent is the quotient of sine and cosine). The degree to which students are forced to innovate their own explorations, develop their own conjectures, and convince themselves and others of their findings, is the degree to which they understand and believe their own findings! And this is also the degree to which students own and live by this "problem-solving process." One design problem that math teachers face is how to make sure we pose problems that allow students to go through this process. We also have to decide when we want students to go through this process, and when we don't.

What does the #MTBoS think about this schema for thinking about what mathematicians "do"? Does this feel useful to you all? Given my own developmental space (a few weeks from starting Year 2), this feels like something I'm going to keep in the back of my head throughout this year, and then maybe next summer I'll think about what it would mean to teach to this Mathematical Method.

In general, this feels like a tremendous opportunity for math teachers to collaborate with their teacher friends outside of the math world. Where is the #STBoS? I should definitely talk to science teachers and see how they teach and support the Scientific Method. Does this mean that math class can/should be filled with labs in the same way that science classes are (ideally) filled with labs where you are actually "doing science"? What do our teacher-scientists have to say? What about the #ELATBoS? How does this include understanding history? Art? Literature?

Sunday, July 29, 2018

Classroom Norms to Start the Year

Out here in Boston, here at the half-way point in the summer, I'm starting to look forward at my big classroom norms for next year. When I say "classroom norms" I mean the guiding philosophical principles that can be referred to and cited to guide how we work and function as a classroom community. Here, I would like to reflect on where I got them, how I launched them, and how they bore out during the year. I will then talk about how I plan to change them for the coming school year. Throughout this post I have some questions highlighted in green for you readers, and I'd love to hear your feedback on if you're interested in sharing! You can reply to me on Twitter @bearstmichael, or comment here on this blog. I also provide a link later on so you can comment directly on the Google doc where I am drafting my norms for next year.
My norms from last year, which I kept on laminated 8.5"x11" posters at the front of the room:

1) 2)
3)  4)

Where I Got My Norms. I got all but the second one from my mentor teacher Joey Kelly. He had a few others, but these were the biggest three that I took away from his class. The second one I made, because I know that my class is very group-work oriented, and I place a lot of responsibility on groups to stay and work together. 

Clearly these are all pretty broad norms. This contrasts with some of the specific norms that I've seen in other classrooms like Always Show Up on Time or Always Check Your Answer. In general my working philosophy on classroom norms is to have fewer of them, because when norms get really specific I feel like I need more of them to cover my all my bases.

Another norm that I initially considered was Other People are People -- They Deserve Respect. I wasn't ready to deliver and talk about this norm, so I bailed on it the night before I was supposed to introduce it, replacing it with Mistakes Are Useful -- We Learn From Them

How I Launched My Norms. In each of the first four days of my classes, I would focus on a different norm and use a math task that I felt embodied that norm. That way they experience the norm firsthand before I ask them to talk and think about it. Then, during the final closure phase of the class, I would connect the day's task to this norm, and then ask them some follow up questions in an exit ticket. Some tasks felt very connected to their associated norm, and others felt shoe-horned into the norm.

When picking tasks to pair with norms, I was also trying to balance this narrative arc with my goal to maximize engagement and fun those first four days. Here are those four tasks, in the order they rolled out:
  1. Math Is Not a Race -- No Spoilers. I paired this with a version of Nim that I learned as 21-Takeaway. This is one of my favorite games to play with kids. It is described in detail here by Jon Orr (@MrOrr_geek), and another great variant is described here by Sasha Fradkin (@aofradkin). Especially with the one-pile version described by Jon Orr, this game connected well with the norm because eventually some students start to get the strategy but naturally don't want to spoil it (because it's a competition). I also refuse to spoil the strategy for them at the end of the day, further driving home the norm No Spoilers. I ended the day asking them to reflect on the norm with the following questions:
    1. When can it be HELPFUL to race in math class?
    2. When can it be HARMFUL to race in math class?
    3. When has someone “spoiled” something for you? How did you feel?
  2. Take Care of Each Other -- Take Care of Yourself. I paired this with the 100# Task, originally described here by Sara Vanderwerf (@saravdwerf ). The task is always fun, and kinda matched the norm. The connection I communicated was that they were more effective when they got together, communicated, coordinated, and helped each other find numbers--especially if someone noticed the trick and shared it with their group (Take Care of Each Other). Furthermore, when they were all helping each other, they were more likely to win (Take Care of Yourself). This task would be an ideal fit for a norm along the lines of Take Time to Notice--It's Helpful, but I'm not sure I want exactly that as a big norm. Follow up questions:
    1. One way I can take care of MYSELF in this class is...
    2. One way we can take care of EACH OTHER in groups is...
    3. One way we can take care of EACH OTHER anytime is...
  3. Fair is Not Equal -- You Get What You Need. I paired this with Maximaze, which I got from Joey Kelly's website problem #5 at, who originally got it from @MathCurmudgeon at his blog here. This was a stretch to connect to the norm--the problem didn't align well. The closest I got was pointing out the varied calculator use. This is, incidentally, also the class in which I orient students to the open bin of calculators at the front of my room. This would have been a better connection if it were the kind of task where students actually may or may not have used a calculator.  But when I claimed that some students used calculators, and some didn't, based on what they individually needed, this was one of those white teacher-lies we tell to facilitate the discussion, because everyone used calculators. Which totally makes sense! I know I did when I did the problem for the first time. I also gave them the option to work together, or alone, pointing that out as one way that students can advocate for the working conditions they need. Follow up questions:
    1. I (liked / disliked) working (alone / in pairs) because...
    2. I (liked / disliked) working (with / without) a calculator because...
    3. It helps me in class when...
  4. Mistakes Are Useful -- We Learn From Them. I paired this with 31derful, which is described by Sarah here (full name and Twitter handle unknown?). This task had the highest engagement of the week (and maybe top 5 for the year, to be honest), and kinda matched the norm. I described the connection by pointing out that nobody got it on the first try, and kept having to backtrack and start over as they built the array of cards. This definitely matched the Mistakes Are Useful part, but I don't think it quite landed the We Learn From Them part, because it wasn't super clear exactly how students learned from their mistakes. Follow up questions:
    1. What was a time you got “stuck”?
    2. How did you get “unstuck”?
    3. If you got to go back and re-do class today, what would you do different?
I really liked this drawn-out process of grounding each of the big norms in a math task. In this way their first interaction with the norm is experiential, that way it starts as more of a feeling than words. I I also like that they are task-based because I think that we need to do math every day. Incidentally, I love leading the year with this particular  set of tasks because they all advance the Math-as-Play idea, which is something that I try to keep as close to the center of my classroom experience as possible. I also like having less than a handful simple, easy-to-say norms. The easier they are to exercise in many situations, the quicker/deeper/more often they become a part of my students' day-to-day experience.

I should note hear that this process of rolling out classroom norms is pretty different from another process that I have seen and experienced in many classrooms. Other classrooms might work as a class to brainstorm and collaboratively develop a set of norms by which to collectively abide. You might even develop a different set of norms for each class. You might also have more of than than just the 3-4 that you want to get across. I have participated in this process as a student, professional, and teacher.  I understand and appreciate the democratic and egalitarian philosophical foundation of this process. I'd love to hear other teachers talk about their own experience with this process. If so many teachers do it, and regard it as a best practice, then I should do what I can to learn about it and see if I should make it (at least in some way) a part of my practice.

How The Norms Bore Out Over the Course of the Year. Of the four norms, the one I used most was Take Care of Each Other -- Take Care of Yourself. This was my mantra, this was my routine response when students said that they were done with their work, or asked if they "really needed to do this, since it's not graded." This was how I couched my commitment to group work (and visibly-random groups specifically). This was what I said when encouraging students to volunteer, join clubs, sign up for summer programs, and take the harder class. Sometimes I would emphasize the first part, and sometimes the second. This was the highest-utility norm last year.

I also got a lot of great mileage out of Math Is Not a Race -- No Spoilers, because both parts of it come up so often. I also personally connect to this norm so much, because starting in high school I learned math (and did most of school) with a deeply competitive mindset. This competitive mindset pushed me far in my math learning, for sure, but really blew up in my face when I got to the more advanced math major classes my junior and senior year of college. Someday I'll share my math autobiography with you all!

But that was kind of it. Neither of the other two really came up that often, and I didn't take advantage of the times they did. Fair is Not Equal -- You Get What You Need is a norm that connects to a really big idea of "equity," which I would like to eventually make a bigger part of the explicit design and experience of my classroom. For sure, as I start to venture into the world of Special Education Math, it will become a bigger developmental focus for myself. Poor ol' Mistakes are Useful -- We Learn from Them. Another great message, but not one that I kept at the forefront of my classroom.

Changes for Next Year. Unsurprisingly, I intend to remove those last two norms (Fair and Mistakes). Below is my current draft of my norms. I am still developing them, and would love to hear what you folks think. Here is a link to the Google doc in which I am developing these norms. Feel free to make comments right there on the Google doc.

1)  2) 

3)  4)

Some of these norms are parts or combinations of last year's norms. I tried to build on the norms that I ended up using more, and dropped the ones I ended up using less. I realized that I never really asked Joey where he got his class norms, which is where I got most of mine, or why they looked the way they did. (I'll actually get to see him tomorrow and can ask him in person!) I just know I liked the flow of the two parts to each norm--it felt poetic and dramatic. Nevertheless, as I was writing this blog, I wanted to be able to defend my decision to use each of these different norms as they were, so I kept asking myself to identify exactly what it was that I liked about the norms that I liked. Ultimately, I think that I have clarified for myself my own function of the two-part form for each norm. (Yay, helpful reflective blogging!)

I realized I felt more compelled by the norms myself when they offered a kind of justification for why I was asking students to live by them. So upon reflection, I realized that the things I wanted the two parts to separately do was provide a Imperative (what I'm asking the student to do) and a Justification (why I believe they should do it). Clarifying this for myself helped me to reformat my norms so that they all had the same [Imperative] -- [Justification] structure, and (hopefully) fit together a little better.

1. Take the Time You Need -- Math is Not a Competition. This is a combination of two of my old norms. Taking the time you need was probably my biggest enactment of the Fair norm. I also found that I said Math is Not a Competition more often than Race, usually in reference to students bragging about grades. Thus, I think that it is useful to expand it to a broader idea of competition, as opposed to just races. UPDATE (4/18/19): Next year I'm going to change this to "Take What You Need." This is so that I can expand it to be about calculator use. I'm globally pro-calculator, and I've had a few disparaging and misguided student comments, like, "You needed a calculator for that?" And I think this norm can guide of that.

2. No Spoilers -- Math is Discovered. I separated the "Competition" idea from the "No Spoilers" ideas, since they only felt partially connected to me anyways, and I felt I could make the "Competition" point more effectively in the above norm. The language about which I am the least sure is the part highlighted in yellow below, the justification for "No Spoilers." It doesn't feel like it has the same tone as the other three norms. What would you write to help justify exactly the "No Spoilers" norm?

3. Take Care of Yourself -- You Deserve It, and...
4. Take Care of Each Other -- We Are Worth It. These are a expansion of my highest utility norms. I also think that dividing them gives me an opportunity to provide the justification. I am not sure how to word the justification for "Take Care of Each Other." I know I want that norm to embody the idea that we are a community, a collective, and we must help each other succeed. How would you phrase the justification of the norm, "Take Care of Each Other"?

I plan on doing roughly the same roll-out process next year, pairing each norm with a math task. I will make a later blog post with my complete plan for my first few days. UPDATE (8/21): I'm just going to link the document here to my First 5 days plan. In here are links to my slides and materials for all the days.
  1. Take the Time You Need -- Math is Not a Competition. 31derful. This is a dangerous idea, though, because it only really works if at least a couple groups end up solving the problem. Last year I'm not sure I had enough groups solving it. I'm not sure that any groups in either of my 9th grade classes got it (this is what I'm planning on doing with my 9th graders next year). If I'm going to need at least a couple groups in each class completing this problem, then I will need to develop my scaffolding questions.
  2. No Spoilers -- Math is Discovered. Nim.
  3. Take Care of Yourself -- You Deserve It. I'm not sure what math task really gets this norm across? What is the math task where taking care of yourself is an essential feature? What ideas for math tasks do you folks have? UPDATE (8/18): I will be using One Dollar Words, as described by Kent Haines (@KentHaines) at his website Grace Evans (@grace_h_evans) helped me think though this one. In order to make this task about Taking Care of Yourself, I will pair it with our general notebook setup day, where I go over grades and the syllabus. This task will be designed as a checklist task, where they have to do things like figure out who in their family has the most expensive name, or something like that. This fundamentally requires students to complete the task individually, though it is still reasonable to ask peers for help.
  4. Take Care of Each Other -- We Are Worth It. #100 Task.
To improve on my roll-out of the norms, I think I will make a bunch of business-card-sized slips of paper with that day's norm on it. Then, as students are doing the task, I can give individual students the slip of paper when they do something that exemplifies that day's norm. This way students get a little more face-time with the norm during the class. Additionally, the physical presentation of the slip of paper offers more concrete/specific feedback on their representation of that norm. This process would require me to somehow at least introduce students to the norm, so that they know what it means when they get a slip of paper. How could I do that, without making the students think too hard about the norm before feeling it during the math task? Also, I suppose I could then use the little papers as a kind of raffle ticket, and have a little giveaway at the end of the first four days. Or I could make them stickers somehow, because kids love stickers (side note: personally, I can't stand stickers).

Thank you for reading and reflecting with me on this blog post--it really helped me to reflect and prepare for this coming year. Double-thanks if you reply to any of the open questions I have highlighted above in green! I intend to make another blog post going more deeply into my plan for that first week, discussing all of the other things that I do to start the school year.

Saturday, July 14, 2018

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!