Reusable Structures

With 180 days of class and 3 different courses to teach, I can’t come up with a brand new activity everyday. As often as possible, our activity is “Doing Math,” but often, we need a bit more structure. Fortunately, there are tons of awesome websites, blog posts, and books about structures that I can use over and over as the backbones of my classroom activities. So how do I decide which ones to use and when? Here is some of my current thinking.

What comes first, the structure or the content?

My second year teaching, I decided to use the same structures every week as a warm up. For instance, I would have Always/Sometimes/Never on Tuesdays. When I was planning my Tuesday lesson, I would ask myself, what is a good Always/Sometimes/Never question for what I am teaching right now? Sometimes, it was a perfect fit, but it often felt forced. Nowadays, I ask myself, what is the best structure for this content? I think this is a harder question to answer, but I think it has led to better lessons.

How often should I use an structure?

For many structures (e.g. WODB, clotheslines, Notice/Wonder), there is a learning curve and I want my students to understand the structure before they try it with unfamiliar content. Therefore, I need to use the same structure several times before they become effective. On the other hand, I try to vary my structures so that they remain engaging – I don’t want to hear, “Ugh! This again!” Finding the correct balance is difficult.

What do I want pre-service teachers to learn from practicing reusable structures?

I have a resident teacher as part of the Boston Teacher Residency, and there is so much that a new teacher can learn from practicing structures. By trying various structures multiple times each, new teachers execute a wide variety of teacher moves and practice the same moves multiple times. And from a planning perspective, new teachers reflect on when to use each structure and learn how to use the resources of the online math teacher community (#mtbos).

Below are many of the reusable structures that I use. I tried to divide them up by why I use them.

For a warm up

  • WODB
  • Play With Your Math – As a structure, the distinctive feature is that we don’t share out strategies or review answers. Instead, I intentionally try to leave students hanging so that they might continue working on it on their own.
  • Notice/Wonder – This might be the structure I use most often; Also used to launch bigger problems.
  • #tbt – Questions that are explicitly from an earlier grade; e.g. fractions in Algebra 2; Note: doesn’t have to be on a Thursday
  • 3 Reads – I wrote about this here. I’ve seen it used in some different ways, but the essence remains. My favorite part of this strategy is not giving students the question and instead asking them to brainstorm possible questions. I sometimes use this structure to launch bigger problems.

For practice and developing fluency

  • Levels – Start with a small handout of accessible problems. When a student (or preferably, a group) is done, I stamp it and they grab the level two handout. Repeat until the bell rings. Move at your own pace. I probably overused this structure last year. Sometimes it feels like good differentiation and sometimes it feels like a competition/race.
  • War – Need to evaluate before you know who wins
  • Trashketball – Really fun, decent for actual learning
  • Row Game
  • Multiple Choice Questions with Rapid Feedback (e.g. Kahoot) – I like doing this a couple times a year because engagement is so high, but the focus on fluency and speed is a disincentive
  • Stations – with different mini activities at each table; best when the order doesn’t matter
  • Expression-Says-Means – Practice translating between a mathematical notation, what it says in words, and what those words mean. Example: \log_28 says “log base 2 of 8” and means “the exponent that goes on 2 to get 8.” I like to use this type of question on homework, on warm ups, and in stations.

For focusing on specific features of graphs

For structural thinking instead of calculation

  • Card Sort – I give students some graphs, diagrams, equations, tables, etc, and ask them to sort them. Sometimes I give them some criteria about how to sort and sometimes I don’t. The goal is to notice similarities and differences. Last year, I definitely overused this structure. Here’s an example from Sam Shah.

For modeling

For a shot of student engagement adrenaline

  • Snowball Fight – Example: each student writes a point on 3 small pieces of paper. Then ball up the paper and throw it around the room for 30 seconds. Then everyone grabs two “snowballs” and writes the equation of the line through them.
  • Desmos Activity Builder – Marbleslides in particular

Below are the reusable structures that I am familiar with but don’t use often or at all. I tried to separate them by how likely they are to become a part of my repertoire.

I have tried a few times and want to use more regularly

  • Question Stack – Good for fluency, especially in groups of 2-3
  • Counting Circles – I have enjoyed this when I’ve done it, and it always feels surprisingly necessary in Algebra 2 and beyond
  • Clothesline Math
  • Connecting Representations – Took me some time to get a feel for this last year, but I came around on it. I really like the way it structures low-stakes student presentations, where two students show off their solution, one speaking and one gesturing.
  • Worked Examples – I’ve tried giving students incorrect work and asking them to do an “Error Analysis” a handful of times, and it always felt awkward. I’d like to try giving students more correct work and asking them to make sense of why it was solved in that way.
  • Anchor Chart – Make one example big and pretty on chart paper; I have a hard time deciding when this would be worthwhile, but I love when students use them later on
  • Hunger Games – Students work in groups for a while. Then a “tribute” from each group comes to the board to work on the problem together for a bit. Then the “tributes” go back to their group and have to share what they learned at the board. Creates a unique need for conversation. Credit to CiCi for developing this strategy.

I haven’t tried yet but will ASAP

  • Variable Analysis Game – I think I too often get stuck in two variable contexts
  • Number Talks – I’ve only a couple number talks, but I really want to try a “how many” style number talk like Tina Cardone wrote about here
  • Talking Points – I’m intrigued by how the structure might disrupt some norms of academic math talk.
  • Visual Patterns – I use these a lot, but I’ve realized there are a lot of different ways to structure these, and the structure can determine what type of thinking students are doing. I want to try the structure that I saw Nicole Hansen implement this summer, which was kind of like a number talk crossed with a connecting representations.

I have tried and might try again someday if I can figure out when and how

  • My Favorite No – I’ve never really used this as a warm up like in the video, but it was a really fun way to review quizzes
  • Link Chart – I’ve used these for notes, but I’m not sure how to build independence from them
  • Jigsaw – I tried this like 10 times my first year teaching, and the only time I liked it was for trig identity proofs, where the questions were so hard that the group needed an expert
  • Headbandz – This was fun, but it didn’t feel worth all the prep required.
  • I have, Who has – I had a hard time keeping everyone engaged, especially when someone didn’t realize it was their turn
  • Estimation 180 – It felt ineffective to only use this sporadically, and I haven’t felt compelled to use it all the time
  • Contemplate then Calculate – Kinda like a number talk; haven’t really gotten a feel for when/where to do this is Algebra 2 or beyond
  • Always/Sometimes/Never – I think I just got sick of this
  • Square Puzzle – I did this with exponent rules, but the prep required was too much; For better or worse, this always ends up feeling a bit like a race

I haven’t tried, but I admire the creativity

What else should I try? What strategy do you want to hear more about? What strategies do you use most often?


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