I think that I do fairly well at engaging my students when I lecture. And my assessments require students to demonstrate in various ways that they truly understand the concepts. I have implemented some successful projects. But I am JUST PLAIN HORRIBLE at facilitating meaningful and rich mathematical discourse in my classroom. Inspired by wonderful blogs and websites, I have been trying to lecture less and get kids to think for themselves and engage in rich tasks and have meaningful discussion . . .with almost zero success so far. On Monday I decided to begin my unit on Sequences and Series by using some patterns from visualpatterns.org

These patterns are SO cool! All of the students were engaged some of the time and some of the students were engaged all of the time (thanks, Abe Lincoln), but the conversation was never lively and I felt like we never really accomplished anything . So often, when I try something new, I have a interesting activity and I know what I want the kids to do or to learn, but I have no idea how to get there. I don’t know how to plan a classroom discussion. How do I get the conversation started? How do I facilitate a student-driven activity and yet accomplish specific learning objectives? (Looking for some feedback here!)

Some things I know I did wrong:

- I didn’t start with a question that everyone could answer. I probably should have said, “Tell me something you notice about this pattern,” and then written down all of the student ideas. (any other ideas for good starter questions?)
- I revealed too much information at the beginning. The website gives the answer to the 43rd pattern and asks for the equation. I should have just shown them the pattern first without the additional information. I think students were intimidated by the question about the equation.
- I should have taken the time to anticipate possible student answers and plan out my questioning strategy ahead of time instead of winging it!

Thanks to two wonderful bloggers for their inspiration and help: Michael Fenton’s recent posts about his One Minute Makeovers of his old assessments helped me to realize that trying something new and failing is normal and is better than not trying something new at all. I can learn and grow better at this! And Dan Meyer’s post today on Teaching the “Boring” Bits was especially timely and full of helpful advice as I navigate the new and uncharted waters of facilitating meaningful classroom dialogue.

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First, don’t give up! From your last few sentences, it sounds like you don’t have any intention of doing so, but sometimes it’s nice to hear someone else say it. So, again… Don’t give up!

For general ideas about “orchestrating mathematical discussions” check out Five Practices (http://amzn.com/0873536770). It’s really, really fantastic. I can’t say I’ve mastered the content presented in the book, but it’s helped me make significant progress in visualizing how I can retain some of the “control” I’m used to having as a recovering lecturer as I shift toward more student-engaging activities and discussions.

As for how things played out for this particular discussion, here are a few thoughts (any of which could be misguided if I misunderstood what happened in class):

1. I think you might have spoiled a lot of the fun by showing the number of toothpicks and number of squares on that slide/image. Also, as a student in your class I would be a little uncertain of whether you wanted an equation for the number of squares or the number of toothpicks.

2. To launch the discussion with something everyone (or at least, most of your students) can handle, consider this approach:

a. Present the image ONLY.

b. Ask students to SKETCH the next step. (Wander around the room, glance at their sketches.)

c. Ask for two or three volunteers to DESCRIBE what they drew. (Consider writing down their descriptions, then making a sketch based on a composite of two or three descriptions.)

d. Ask students to SKETCH a basic outline of what the 27th step would look like. (If they have trouble here, ask for another volunteer or two to DESCRIBE what it would look like. Write down their description. Ask other students if they agree, disagree, think it’s specific/descriptive enough. Consider inviting two or three students to SHARE their sketches at this point, either via whiteboard, docucam, smart phone picture emailed to you, etc.)

e. Ask them to EXPLORE the sequence numerically (in a table) and LOOK FOR A PATTERN.

f. Invite students to NOTICE at least two patterns and SHARE with their neighbor, group, or however your students are arranged.

g. Ask for three or four (or five) volunteers to briefly SHARE the pattern(s) they or their neighbors noticed. (If you have some students who aren’t participating much, you can simply ask them to share the pattern someone else in their group noticed; this might encourage better listening in future sharing.)

h. Only at this point, after students have explored the pattern visually and numerically would I ask them to describe the pattern algebraically. Give them some time to look for an equation (but be specific: an equation describing the relationship between WHAT and the step number?)

i. Invite those who find a formula early to search for another algebraic way to describe the pattern.

j. Or, invite those who finish this bit early to evaluate their formula for some step number between 200 and 300. Then they should write the result on the whiteboard for everyone to see.

k. Invite students to figure out what step numbers would yield the “mystery results” their classmates have posted on the board.

Phew, that’s a lot of typing! Feel free to skip some of those steps based on time, interest, intention, etc.

3. Whether you share it only for the first few patterns, and then require students to organize their work and results on their own for subsequent patterns, or whether you take the reverse approach (giving them less structure in the first few patterns to allow them to explore the problems more freely), you might want to consider using a handout I created (which is a TOTAL RIPOFF of the one Fawn created and posted on the website. I usually like retyping handouts in Pages (a shiny word processor for Mac) before using them with my students (it allows me to tweak and adjust, both content and format). The handout I made is here: https://copy.com/jZWwY2bhFSj9cG23

4. You might also consider beginning with baby steps. I am a HUGE fan of Fawn’s website and the patterns presented there, but my students struggled last year (especially on patterns involving triangular numbers). This year I’ve had a lot more success by injecting a little more structure and intentionality in my sequence of presenting the patterns. It’s probably overly structured, but that’s me, for better or for worse (at least for now). Here are the slides I use to introduce the patterns, with six patterns that increase in difficulty from direct variation to linear, simple quadratic to more complex quadratic, to oblong (“rectangular”) numbers, and finally to triangular numbers. With these six under our belts, we are then free to tackle the just about anything posted on the website, and in any order. Here are the slides (in PDF, PowerPoint, and—the original format I created them in—Keynote): https://copy.com/Fku1THQ46fVyywuP

5. Your comment that you should have “taken the time to anticipate possible student answers” makes me think you might have already read the Five Practices…

I hope my rambling proves encouraging or helpful (or both!).

Cheers!

P.S. Thanks for the kind words about the One Minute Makeovers. It’s been a blast writing those posts, and I’m glad some people are benefitting from them in one way or another.

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MIchael, thanks so much for your post and links to your slides (which I freely appropriated). What class did you plan this activity for? Your scaffolding of the activity would be great for my weaker students but most of my Precalc students would not need that much scaffolding. I liked your extension for the stronger students who finish first. I was thinking maybe I should provide several patterns and label them as level 1, level 2, and level 3 problems and let students self-select problems based on their comfort level with the patterns. Did you do Jo Boaler’s online class last summer? She showed a video of a pattern activity and then kids had to work in groups and draw on large sheets of paper and then present their ideas to the whole class. They color coded parts of the pattern to illustrate how they developed their formula. Having said all of that, I really have to decide if spending the amount of time needed to really dig into these patterns is the best use of time for Precalculus. It might be better suited to Alg 1 or 2. Any thoughts on that?

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What I described was initially designed for my Algebra 1 class. I’ve also used it in Algebra 2, though they didn’t need as much of the scaffolding. I’m not sure about how all of this would play out in Precalculus. I’d like to see some form of Visual Patterns in all of my classes. I think it might take me a year or two to organize how things play with older students, since most of my tinkering has been with younger groups.

I like the idea of giving students an opportunity to select the level of difficulty most appropriate for them.

I signed up for Jo Boaler’s class, but didn’t end up taking it.

As for the best use of time in Precalculus, I think there is probably a healthy balance between trying it once and giving up since you didn’t get a desirable result on the first try, and slowing everything down just to focus on this for a few days or more. Is there a way to bring visual patterns in once a week for about 15 minutes? It might take 30-45 minutes to introduce it again… Ultimately it’s your call, but I think this kind of work is healthy for all ages. 🙂

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Once again, great thoughts! I struggle, like all math teachers, with time. How much time for the “have-to’s” versus the “want-to’s” and how much the “want-to’s” are really essential to accomplishing my goals and standards for the class. An additional problem this year is that switching to SBG slowed me down so much at the beginning of the year that I am 2 weeks behind where I was this time last year. Arghh! You are right. I would need time to re-introduce the topic properly and then I could do a little here and there. I’ll have to think about that. Meanwhile, I have ordered 5 Practices and will work on my skills. (By the way, I haven’t read it but I am familiar with the ideas. I can talk the talk, but I need to learn how to walk the walk!)

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You’re not alone in that boat. Keep at it, and we’ll both get a little bit better every year.

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Hi Jane! Two of your questions stood out because I think their answers might cover a larger umbrella for your other concerns. 1) “How do I get the conversation started?” I believe this is exactly the idea behind act 1 of Dan’s 3-act lessons. Maybe teachers shouldn’t be the conversation starters, the KIDS should. Our job is to put forth something that immediately engages them. They ask all the questions initially. 2) “How do I facilitate a student-driven activity and yet accomplish specific learning objectives?” Yes, you do have to mind the clock and facilitate the discussion by steering them toward those learning goals. This takes a lot of practice, Jane, not just in the craft of doing so, but in the rapport and trust of the class. Give yourself time. They need to learn respectful and productive discussions, some norms about sharing and critiquing each other. Are you familiar with Peg Smith’s book “Five Practices for Orchestrating Productive Mathematics Discussions”? It’s a skinny book but loaded with good guidelines. One of the first steps suggested is to “anticipate” kids’ answers. This is a huge step in planning. Sure, I still get answers and reactions from students that I didn’t expect, but I did expect the other 90%. Because knowing what they’ll do and say will help with the navigation of the lesson.

Most importantly, Jane, don’t give up. Baby steps. Lots of baby steps. And your genuine enthusiasm is contagious, how excited you are about a task will rub off on them. You have my email address, so please don’t hesitate if you think I can help with something or you just want to vent about an awful-no-good-lesson. We all have them!

It’ll be better next time. I know so because you already took time to write/reflect about it.

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Thanks so much, Michael and Fawn, for the helpful feedback. Both of you confirmed some of the things I had thought about doing and I will definitely check out the Five Practices book.

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This is one of the biggest and most important struggles I think there is for any math teacher. I am a huge fan of visual patterns and how it gives a sense of why we need algebra and generalization. I have been through the same experience. Often it has been with something like a Dan Meyer 3 acts lesson that I just didn’t structure well at all.

One thing I have found is that we are fighting against a student’s lifetime of being fed skills and expected to regurgitate them on demand. Many are just not used to thinking for themselves and taking their own learning to the next level.

With this in mind, one thing I have found useful in this type of activity is to dress it in structure, so I might give them a table of matchsticks to squares and give them a couple of examples where the least they have to do is count the matchsticks, opening the activity to all. I will then step it up to the 2nd pattern, 3rd, 5th, 10th, 100th, nth…. For the first few they can count but pretty soon they see that this becomes inefficient, pretty quickly. By this time they have not realized that it is an open ended activity but it doesn’t matter. Timing and being concise is still very much an art-form at this stage, though.

Looking forward to seeing your updates.

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Thanks for the great ideas!

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