In my role, I have the advantage of seeing many great teachers honing and refining their craft, all to provide the best possible experiences for their students. The dedication and professionalism that the teachers I work with continue to demonstrate is what keeps me going in my role!

One particularly interesting benefit I have is when I can be part of the same lesson multiple times with different teachers. When I am part of the same lesson several times I have come to notice the differences in the small decisions we make. It is here in these small decisions that have the biggest impact on the learning in our classrooms. For instance, in any given lesson:

- Who gets the most talk time? You, all students, same few students…?
- How much think time do you give? When?
- Do you show first, or allow time for students to make sense on their own first?
- Do you start with contexts or visuals or questions or explanations?
- Do you allow your students to use technology or no technology (computers, calculators…?)
- Do you build the problem together, or provide pre-determined questions?
- Do you expect manipulatives to be used by all, encourage anyone who wants to use them, or offered only for those who you think need it?
- Do you discuss the learning goal at the onset, during the close, or never?
- How/ when is feedback given? (if at all)
- When a student is struggling, do you ask them questions that get them to think, or ask questions that lead them to the answer?

There are so many little decisions we make (linked above are posts discussing several of the decisions). However, I want to discuss a topic today that isn’t often thought about: **Scaffolding**.

For the past few months, the teachers / instructional coaches taking my Primary/Junior Mathematics additional qualifications course have been leading lessons. Each of the lessons follow the 3-part lesson format, are designed to help us “spatialize” the curriculum (allow all of us to experience the content in our curriculum via visuals / representations / manipulatives), and have a specific focus on the consolidation phase of the lesson (closing). After each lesson is completed I often lead the group in a discussion either about the content that we experienced together, or the decisions that the leader choose. Below is a brief description of the discussion we had after one particular lesson.

First of all, however, let me share with you a brief overview of how the lesson progressed:

- As a warm up we were asked to figure out how many unique ways you can arrange 4 cubes.
- We did a quick gallery walk around the room to see others’ constructed figures.
- We shared and discussed the possible unique ways and debated objects that might be rotations of other figures, and those that are reflections (take a look at the 8 figures below).
- The 3 pages of problems were given to all (see below). Everyone had time to work independently, but sharing happened naturally at our tables.
- The lesson close included discussions about how we tackled the problems. Strategies, frustrations, what we noticed about the images… were shared.

Here are the worksheets we were using so you can follow along with the learning (also available online Guide to Effective Instruction: Geometry 4-6, pages 191-212):

While the teacher leader made the decision to hand out all 9 problems (3 per sheet) at the same time, I think some teachers might make a different decision. Some might decide to take a more scaffolded approach. Think about it, which would you likely do:

- Hand out all 9 problems, move around the room and observe, offer focusing questions as needed, end in a lesson close; or
- Ask students to do problem 1, help those that need it, take up problem 1, ask students to do problem 2, help those that need it, take up problem 2…

This decision, while seemingly simple, tells our students a lot about your beliefs about how learning happens, and what you value.

So as a group of teachers we discussed the benefits and drawbacks of both approaches. Here are our thoughts:

The more scaffolded approach (option 2) is likely easier for us. We can control the class easier and make sure that all students are following along. Some felt like it might be easier for us to make sure that we didn’t miss any of our struggling students. However, many worried that this approach might inhibit those ready to move on, and frustrate those that can’t solve it quickly. Some felt like having everyone work at the same pace wasn’t respectful of the differences we have in our rooms.

On the other hand, some felt that handing out all 9 puzzles might be intimidating for a few students at first. However, others believed that observing and questioning students might be easier because there would be no time pressure. They felt like we could spend more time with students watching how they tackle the problems.

Personally, I think our discussion deals with some key pieces of our beliefs:

- Do we value struggle? Are we comfortable letting students productively keep trying?
- Are we considering what is best for us to manage things, or best for our students to learn (teacher-centered vs student-centered)?
- What is most helpful for those that struggle with a task? Lots of scaffolding, telling and showing? Or lots of time to think, then offer assistance if needed?

In reality, neither of these ways will likely actually happen though. Those who start off doing one problem at a time, will likely see disengagement and more behaviour problems because so many will be waiting. When this happens, the teacher will likely let everyone go at their own pace anyway.

Similarly, if the teacher starts off letting everyone go ahead at their own pace, they might come across several of the same issues and feel like they need to stop the class to discuss something.

While both groups will likely converge, the initial decision still matters a lot. Assuming the amount and types of scaffolding seems like the wrong move because there is no way to know how much scaffolding might be needed. So many teachers default by making sure they provide as much scaffolding as possible however, when we over-scaffold, we purposely attempt to remove any sense of struggle from our students, and when we do this, we remove our students’ need to think! **When we start by allowing our students to think and explore, we are telling our students that their thoughts matter, that we believe they can think, that mathematics is about making sense of things, not following along!**

So I leave you with a few thoughts:

- Do your students expect you to scaffold everything? Do they give up easily? How can we change this?
- When given an assignment do you quickly see a number of hands raise looking for help? Why is this? How can we change this?
- At what point do you offer any help? What does this “help” look like? Does it still allow your students opportunities to think and make sense of things?

**When we scaffold everything, we might be helping them with today’s work, but we are robbing them of the opportunity of thinking. When we do this, we rob them of the enjoyment and beauty of mathematics itself!**

I love reading your posts because I always leave with so much to consider. On the question at hand, I wonder if there might be other options besides the either all or nothing. I think I would want to give them the freedom to move at their own pace, to move forward as they are ready, but also not to overload students who are LD. Some of those students do better if they see work in smaller chunks (not necessarily doing less, just visually seeing less at one time). I wonder if some of those kids would feel overwhelmed being handed four sheets, each with multiple tasks all at once. I thought about the idea of giving them one page with the direction to go collect the following page from a specified location when they finished the first (which would give them a chance to get up and move a little bit). I also thought about the possibility of putting the three tasks on some sort of binder ring. Each kid gets one and they move to the next page as they finish it. I could maybe rework the visual so that fewer items are on a page at one time if there are kids in the room who need the work chunked (use this and one construct this item on a card with three cards replacing a single worksheet). The binder ring would also give me options in terms of the work. If I wanted them to completely finish one sheet I could do that. If some students were struggling, I could make conscious decisions about what I wanted them to do/skip rather than just having them not do the ones at the end..

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You are right Cheryl. There are more than these 2 options. I was trying to show the student-centred vs teacher-centred decisions. Your decision will likely still allow students to work at their own pace and still give room for productive struggle. Thanks for sharing your thoughts!

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You post really made me think again about my lesson structure, and the questions at the end are spot on.

1. Do your students expect you to scaffold evetything? Do they give up easily? How can we change that?

I teach two classes of 25 eight year olds. My students’ reaction to challenges cannot be put on a continuum. Yesterday, I was reading my students’ reflections and stumbled upon “I thought math is easy, but now I know it’s hard. I love math.” Not everyone shares her sentiments. Some students are used to being “good at math” and getting answers quickly so that when they can’t they shut down and cry. Some students are used to “not being good at math” so they want me to validate every step they take. Some students require support to get strategies to approach the challenge and get overwhelmed easily. Even if there are five kids who need me here and now, it is too much to have a meaningful conversation with one. So my daily battlefield is how do I focus on the strategies of dealing with general problem solving situation rather than specifics.

I found that Noticing and Wondering with the whole class before working in groups or independently allows to unveil some structures that might be helpful and gives space for questions that might need to be answered.

Your post made me think what would I have done with the cubes challenge. Probably notice-wonder with the whole group after the gallery walk. Give one single challenge to everyone so that kids work together/learn from each other/ don’t spend an hour choosing what they want to do. I like Cheryl’s idea with the binder ring.

Productive struggle should revolve around mathematical concepts, not the confusion about the logistics of the task.

2. When and how do you offer help? I think I am trying to give help that, again, focuses on how to get unstuck rather than particulars of the problem. I also try to drag in different manipulatives/representations/organizational tools to try and help students to see the problem from a different angle. I’m afraid I still overscaffold with a few students…and that’s why your post resonated with me so much.

Really, how do I foster independence in some of my students who really refuse to let my hand go? If it’s either them working with me or them lying on the table with their head down, where do I begin?

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In trying to be helpful, sometimes we become short-sighted. Is our goal to help students with THIS activity, or to build mathematicians? I tend to side with the more long-term goal! Like Dan Meyer says, we need to “be less helpful”.

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Your question of “where do I begin?” is really tricky! Part of the problem is that all of the experiences these students come into your room with might have led them to believe this is how they “get through” math class. Giving our students opportunities to struggle with things that require everyone to use trial-and-error… having our students do tasks that allow them to access their spatial reasoning… something that gives all of your students motivation to keep trying and not give up. It could be as simple as giving students a set of tangrams and have them recreate the square. Or offer problems where there are lots of possible answers like the 4 4s problem, and let everyone go. Students need to feel what it is like to be “stuck” and work through that feeling to keep going. I’d love to know what you will try!

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Thank you, Mark. Took me a few days to reply because I was, well, thinking. I agree I do need to be more mindful of ensuring my “entry point” is accessible for everyone, “something that will give all my students motivation to not give up”. You are right, the lack of resilience that is often addresses with more scaffolding, can be targeted through better task design to actually help kids build resilience. I think that’s the big challenge with the whole growth mindset thingy: everyone seems to agree it’s important but the idea that you need to seriously overhaul your practice in order to foster it is much more difficult to come to terms with.

I’m doing handshake problem with kids today. We already played with triangular numbers before with @gfletchy videos, I’m hoping for some “wow”s at the end. So I will start with 5 people handshakes and get kids into the groups of 5 to be able to physically model it..then we’ll compare different organizational strategies and see how we can extend it. If I am mindful about some groupings, I anticipate no resilience problems… We’ll see how it goes.

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