She started by stating that selecting a good task is an important first task. We need to select tasks that live up to their potential. Her session is based on her book, Five Practices for Orchestrating Productive Mathematics Discussions – what you need to do before and during the exercise to promote the reasoning and sense making in the exercise. I will note that I had heard a couple of really good recommendations for this book (one in another session) and it was written for teachers. The people who had read it said that it was easy to read and understand and well worth it.

**The Five Practices (+)**

0. Setting Goals and Selecting Tasks - was added after a discussion with a coach about that it was important to do this before beginning. They had already been calling the rest the Five Practices, so now she refers to them as above.

1. Anticipating

2. Monitoring

3. Selecting

4. Sequencing

5. Connecting

Below I have gone through what information she had posted on the slides (which were helpful) along with parts of an example that she had us work through to see the Five Practices in play.

**0. Setting Goals**

Involves –

• Identifying what students are to know and understand about mathematics as a result of their engagement in a particular lesson

• Being as specific as possible so as to establish a clear target for instruction that can guide the selection of instructional activities and the use of the five practices

It is supported by –

• Thinking about what student will come to know and understand rather than only on what they will do

• Consulting resources that can help in unpacking big ideas in mathematics

• Working in collaboration with other teachers

Example Goals –

Realize that examples are not enough to show that a claim is always true

Recognize that there are many different ways to prove

**0. Selecting Tasks**

Involves –

• Identifying a mathematical task that is aligned with the lesson goals

• Making sure the task is rich enough to support a discussion (i.e. a cognitively challenging mathematical task)

It is supported by –

• Setting a clear and explicit goal for learning

• Collaboration

Example task – Prove that the sum of two odd numbers is always even

**1. Anticipating**

Likely student responses to mathematical problems

Involves considering –

• The array of strategies that students might use to approach or solve a challenging mathematical task

• How to respond to what students produce

• Which strategies will be most useful in addressing the mathematics to be learned

It is supported by –

• Doing the problem in as many ways as possible (YOU doing the problem)

• Doing so with other teachers

• Drawing on relevant research

• Documenting student responses from year to year

Examples –

Thought students might draw a picture, build a model, create a logical argument, use algebra

Thought students might use examples and not know what else to do

Thought students might use the same variable in the representation of each odd number

**2. Monitoring**

Students’ actual responses during independent work

Involves –

• Circulating while students work on the problem and watching and listening

• Recording interpretations, strategies, and points of confusing

• Asking questions to get students back “on track” or to advance their understanding

It is supported by –

• Anticipating student responses beforehand

• Using a recording sheet (We had a packet with this in it. It had headings of Strategy, Who and What, and Order.)

• On the recording sheet – under strategy, fill in what you came up with in the anticipating step

The recording sheet serves as a record of who’s doing what and helps you to decide what to share with the class. It also helps keep track of strategies that students are using and you can use that data to help form groups.Also provides a historical record for you to look at the next time you do this lesson.

Examples –

Teacher asked a lot of questions to understand what they were doing, to help students to access relevant knowledge, and to suggest an avenue to pursue. Teacher left the groups with ideas to pursue and gave them the space and time to pursue them. Teacher also suggested resources that would help students make progress.

Teacher didn’t tell students what to do or how to do it.

**3. Selecting**

Student responses to feature during discussion

Involves –

• Choosing particular students to present because of the mathematics available in their responses

• Making sure that over time all students are seen as authors of mathematical ideas and have the opportunity to demonstrate competence

• Gaining some control over the content of the discussion (no more “who wants to present next”)

It is supported by –

• Anticipating and monitoring

• Planning in advance which types of responses to select

**4. Sequencing**

Student responses during the discussion

Involves –

• Purposefully ordering presentations so as to make the mathematics accessible to all students

• Building a mathematically coherent story line

(don’t start with the most abstract solution first – you need to build to it)

It is supported by –

• Anticipating, monitoring, and selecting

Example – went from concrete to abstract, left the groups with issues for the end

**5. Connecting**

Student responses during the discussion

Involves –

• Encouraging student to make mathematical connections between different student responses

• Making the key mathematical ideas that are the focus of the lesson salient

It is supported by –

• Anticipating, monitoring, selecting, and sequencing

• During planning, considering how students might be prompted to recognize mathematical relationships between responses

Examples –

All of the solutions were connected to each other and key ideas of number theory (e.g. an even number is divisible by 2, can be represented algebraically)

Teacher encouraged student to question each other. Teacher asked students to relate their work to the work of other students in the class. Teacher created homework that requires students to make sense of the arguments and reasoning of others. Teacher established a classroom climate in which students feel they can contribute.

**What’s the point –**

Good tasks are a critical starting point for developing student capacity to reason, but tasks alone are not enough.

I really liked this session. I felt like I left with a better understanding of how to promote some discussion in my classroom. I have a lot to learn about that yet, and I will definitely be purchasing the book she has co-written on the subject, especially after hearing good recommendations from others who have read it.

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