Once again, Tracy Zager has pushed us to think about our teaching. In her recent talk at #TMC16 Tracy asked us to consider what it means to “close the lesson”. Here is an example of a problem and a potential close, followed by some of my thoughts about how we should close any lesson.

First of all, give a problem that will help you achieve a specific goal.

Take this problem published in Marilyn Burn’s 50 Problem Solving Lessons resource:

If rounds 1 & 2 of a tug-of-war contest are a draw, who will win the final round?

Here is the full problem. Please take a minute to read through the problem and try to solve it for yourself. Which side will win round 3? How do you know? Are you sure?

Once students understand the problem and are given time to write their solution (individually or in pairs) the learning isn’t over. In fact, while answering the problem might require thinking and writing an answer requires decisions about representations, much of the learning hasn’t happened yet! Really, students have just shown what they already understand…new learning happens in the close!

#### Closing the lesson:

**Step 1 – Sharing Different Solutions**

Since this problem is open, allowing for different strategies, there is a huge potential for interesting discussions. For example, some students will create an answer similar to the one below. Using this sample there are several things that we can bring up in conversation with the class. Notice the student(s) created equations using symbols and equal signs. Also, they did something really interesting in round three (notice the brackets and the arrows). Having a conversation about substituting would be really helpful for many students. Many might not realize that this problem would be easier to solve if only acrobats and grandmas were considered. Through substituting Ivan for 2 grandmas and 1 acrobat, the final round ends up with 5 grandmas and an acrobat against 4 acrobats. Substitution requires us to really understand the equal sign and what balance means!

Other students will create things more like the picture below. They will use numbers to represent values for each of the figures. Again, conversations with the class could be about how they chose the 1 = 1.25 at the top. Or, how this information could be used later. Again, the importance of balance and the equal sign could be brought up.

Others might do things like the sample below. Here the students have decided not to draw a picture, but instead to represent the players with letters. For many students, their first introduction with algebra is where letters are all of a sudden thrown into some equations. However, these students have realized that it is far easier to represent the items with letters than draw the pictures. The conversation here could be quite useful in bringing about the need for letters! Also, noticing that their strategy is similar to the 2nd sample above might be helpful since while the strategy is the same, the values are not. How are the two student samples’ values different? How are they the same? Is one right or are they both right?

**Step 2 – Making connections / discussing big ideas**

Once a few samples have been shared, we need to make sure our students are making connections between the samples. The learning from the problem needs to made clear for all students.

We can do this by asking students to notice similarities and differences between samples, or by taking a few minutes writing down a few simple things we can take away from the problem. Either way, our students need to be involved here and the generalities we can draw from the problem need to be clear.

For example, things that can be discussed here:

- The equal sign represents balance
- We can substitute things in equations we don’t know with things we do know
- Symbols or letters can be used in equations to represent either a variable or a constant

**Step 3 – individual practice**

Now that our students have had time to grapple with this problem, and have discussed what we can learn from it together, we need to continue the learning. Part of the close includes time for students to do something with their learning. Here are a few possible ways we can include individual time to practice what has been learned.

- Provide an Exit Card or journal prompt that asks your students to show what they have learned (linked are a variety of types of exit cards you can choose between)
- Have your students create their own problem using 3 rounds where the first 2 are a tie and the 3rd round needs to be figured out (possibly have students switch problems with a classmate and find the solution)
- Relate the problem to some practice questions that will help continue your students’ thinking. Practice with some of the early Solve Me Mobile puzzles might work nicely here.

#### Some General Advice about the Close

- Predetermine which students you want to share.
- Have a goal in mind for each person/group sharing.
- Ask them specific questions, or ask questions of the rest of the class that helps you achieve your goal.
- Avoid general questions like “tell us what you did” (we don’t want the sharing to become a show-and-tell, we want rich discussions about specific things from their work).
- Use the 5 Practices for Orchestrating Productive Conversations (found here, here, and here) to make sure your close involves real discussion of the meaningful mathematics.
- Practice is important, but students still need to think and make choices.
**Closing the lesson is about bringing the learning together as a group, then individually. The SHARED experiences here are where our students can learn WITH and FROM each other so they will be ready to work independently.****The majority of the learning takes place in the close!****Closing a lesson takes time, but skipping the close is the biggest waste of time!**

Want more information about the close? Take a look at this monograph: Communication in the Mathematics Classroom

So I leave you with some final reflective questions:

- How often do you close the lesson?
- What obstacles stand in your way to closing your lessons regularly?
- Does your close look like a show-and-tell where students are not listening to those talking, or do you have rich discussions going on between many members of your classroom? How can you help increase the level of discourse in your classroom?
- How is this different than closing the lesson by taking up homework? Does this allow more opportunities for those that might struggle, or who don’t identify with mathematics yet?
- What goals do have this year with regards to closing your lessons?

**Some say they can’t afford the time to close the lesson. I say you can’t afford not to!**

I asked this question in #msmathchat and every one of us said “I’d like to (do this cool thing), but what I usually do instead is yell ‘Oh no! The bell is about to ring. Hurry! Do you get it? Pack up!’ ”

What you’re talking about here, though, doesn’t sound like the close, it sounds like the lesson. The conversation is the lesson, to me. I think of the close as being where they do an exit ticket or answer a reflective prompt in their journal. Semantics, I guess?

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First of all, the close doesn’t have to happen in the same period. Closing off a lesson is about consolidating the learning. We need to make sure our students have had time to learn the deeper pieces. As listed above, closing off the lesson could include an exit card or independent practice material.

We need to be comfortable to go beyond our constricted scheduling and continue the lesson the next day so that the close is well done!

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