How does a home workshop work?

This past week, Esther Billings, John Golden, and I presented Making Workshop Work in Mathematics at the NCTM Regional Conference in Chicago. One of the problems with presenting at these conferences is that the session description is due almost a year before the presentation. So while the program says, "explore several mathematics lessons and assignments that use the workshop model," we decided that the session would be more meaningful to participants if we worked through a single workshop, highlighted the workshop phases and research, and discussed how the workshop structure could support exploring the Standards of Mathematical Practice.

Although we provided a few minutes for participants to reflect on how they might apply what they learned about workshop to their classes, we were not explicit about using the approach on assignments. I hope to remedy this oversight by sharing a home workshop I used recently in my Teaching and Learning Middle Grades Mathematics course. This comes near the end of a unit on rational numbers.

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Doing Math Workshop (CGI Grouping Stories)

Objective: The learner will use representations to support their thinking and make their thinking visible to others as they find solutions to grouping stories involving rational numbers.

Needs: One hour and a copy of CGI Grouping Stories

Schema Activation: Reviewing Grouping Stories (no more than five minutes)

Recall that we used Grouping Stories from Cognitively Guided Instruction to provide context to multiplying and dividing integers. Review the different stories provided below and consider what it might look like as learners use the given contexts to compute their answers.

Buschman, L. (2001). Using Student Interviews to Guide Classroom Instruction - An Action Research Project. Teaching Children Mathematics, 8(4), 222-227.

Focus: Goldilocks Problems (no more than five minutes)

As you read through the CGI Grouping Stories, you will notice that the values in each story have been left for you to choose. The goal is to select the row of values that is not too soft (so easy that it does not require any thought on your part) and not too hard (so difficult that you would not be able to make progress without a significant amount of help). In other words, find the "just right" numbers. In the space on the right, record the representations you used to support your thinking so you can make it visible to others.

Activity: CGI Grouping Stories (no more than forty minutes)

Reflection: What? So What? Now what? (at least ten minutes)

As you look back at your efforts, pick the one record that best demonstrates your ability to use representation to support and share your thinking.

• What support did the representations provide as you worked on this story?
• So what would you want others to see as you share your efforts?
• Now what does this mean for you as a teacher - as you consider designing rational number computation lessons?

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The next class period, we usually have a math congress where some of the students share their work with their peers and discuss their reflections. If there is any interest, I will try to provide some generic examples of these in a future post. Please leave your interest or your questions in the comments.

Can we just flip the homework?

The lesson has been taught and there are a few minutes remaining in the class period. Of course, this means that students can get a head start on their homework. I sit in on a lot of lessons in my role as an instructional coach for preservice and inservice teachers and the scene is nearly always the same. As I walk around the class during these final minutes, I typically see the following:

Why do they start at the beginning? Is it because "it is a very good place to start," or is it because that is how they have been trained?

The way most assignments are structured, the items get harder the farther in you get. I understand that this progression can provide support to students as they build on prior success, but I do not think they see it that way. Ask them and I believe you will find that they do it in order because the assignment implicitly suggests that they do it in order. They have been disempowered to make meaningful choices about what items to try.

What if instead we asked the students to look over the items and identify those that they might need peer or teacher support on? These are the items the students could work on during the closing minutes while they have support available. They can finish the easier problems on their own. Dare I say it? We could flip the assignment.

This could also serve as an assessment for the teacher. Knowing which items students considered challenging could offer insight into the effectiveness of the lesson or inform future instruction.

A change like this represents one of the subtle shifts we encourage teachers to make in their practice. We do not always need to make big moves to offer students a chance to make choices and, therefore, take more responsibility for their learning. A slight change can increase the likelihood of student engagement without requiring a lot of extra planning or preparation. This is another example of educational sustainability for both teacher and learner.

Which problem is "just right" for you?

Today on my Twitter-stream there was a great deal of discussion about how we use homework in math class. Being a university math educator, I have more freedom than most K12 teachers when it comes to assigning homework but that doesn't mean that I am any less concerned about this topic. Homework is a important memory for many math learners and we teachers need to consider carefully how we use it.

If our schools require us to assign homework from the text but we have some freedom in what it looks like, it is time for us to work our magic as problem solvers. John Golden and I team-taught a course for preservice teachers a few years ago and modeled some ideas of adding to pre-existing items from a text. We called them "just right" problems in reference to the NCTM article Vygotsky and the Three Bears. Here's what the original items looked like:

from Scott Foresman – Addison Wesley Math [5^th Grade]

And here is how we tried to make them more thinker friendly:

Using these existing textbook items to demonstrate the processes associated with doing math. Pick one or two of the following to explore:

• Do either the odds or the evens – your choice. Why did you pick the evens (odds) to work on?
• Look over all the items. Which five do you consider the easiest? What makes them easy? Which five do you consider the hardest? What makes them hard?
• Pick an addition item that is just right (not too hard and not too soft). Solve the item using two of these three methods: using manipulatives, drawing a picture, or developing a real world context. Compare the two methods you selected.
• Pick a subtraction item that is just right (not too hard and not too soft). Solve the item using two of these three methods: using manipulatives, drawing a picture, or developing a real world context. Compare the two methods you selected.
• Pick one item to solve and write a metacognitive memoir that describes your cognitive efforts.
• The answer to one of the items is nineteen-twenty-fourths; which item is it? (Be sure to keep a record of your thinking)
• Which answer is closest to one? How can you be sure? (Be sure to keep a record of your thinking)
• Put the items in order based on their answers from least to greatest. (Be sure to keep a record of your thinking)
• Write your own ‘just right’ problem related to this review.

Which problem is "just right" for you?

Is there another problem we could add to this list?