In respecting the plurality of us all, we’re all right, in our own way, in making sense of the world around us. When we come into the math situation, we’re looking at multiple solution pathways… I’m looking at what strategies learners have to get 'un-stuck.'

## Mentor Interview

In this reflection, Fran Dickinson identifies how he develops a climate of mutual respect that supports students’ multiple pathways of solving, uses formative and summative assessment strategies to understand his learners, and creates supports to scaffold students’ learning and help all learners gain points of entry to a task. He describes how his school uses the Problem of the Month, hosting gallery walks, engaging parents, and encouraging all students to go as far as they can with their mathematical understandings, seeing how ‘deep you can go’. The math practices serve as a ‘homing beacon’ for him to ensure that his students gain access to the big ideas of math.

## See how Fran's students demonstrate many mathematical practice standards in one lesson:

### 1. Make sense of problems and persevere in solving them

Fran Dickinson leads a number talk on an input/output table and graph, asking “What’s my rule?” In this clip, he generates an output number after students offer an input number. The students discuss whether or not 0 is a possible input for the table and graph. This clip is also indicative of MP 3.

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Dickinson then leads a lesson on numerical patterning, in which learners are asked to investigate a numeric pattern and to generalize what they see happening as the pattern grows. In this clip, the learners individually review a copy of two students' work on the MARS "Buttons" task and then discuss what they know or a question that they have about the sample work. This clip is also indicative of MP 2.

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Fran Dickinson’s students sketch and color code their pattern drawing and use tiles to show what is staying the same and what is changing. In this clip, one student explains to another how blue tiles represent the “old tiles,” or the tiles from the previous stage. This clip is also indicative of MP 5.

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### 2. Reason abstractly and quantitatively

Fran Dickinson leads a lesson on numeric patterning, helping students to investigate a numeric pattern and to generalize what they see happening as the pattern grows. In this clip, Dickinson describes the importance of individual think time before he asks his students to discuss the relative strength of two different approaches to a patterning task. One pair discusses the numbers within the sample strategy, and Dickinson repeats back their conversation to the whole group, telling his students, “I’ve heard two really good questions about Learner B’s strategy. One was, what are all these 3’s? and Kelcey’s question was, what about this 4? Where’s the 4 coming from?” This clip is also indicative of MP 1.

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### 3. Construct viable arguments and critique the reasoning of others

Fran Dickinson leads a number talk on an input/output table and graph, asking “What’s my rule?” In this clip, he continues a class conversation about input and output numbers. Dickinson notes that “It was interesting to hear all of the different opinions of how to state the rule. I think this illustrates where we were as a group as far as our familiarity with algebraic expression goes.” For example, the students discuss “3 groups of x versus x groups of 3.” Dickinson also models whole-group strategies for consensus and disagreement, which he explains as “silent signals.” This clip is also indicative of MP 1.

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### 5. Use appropriate tools strategically

Fran Dickinson leads a lesson on numeric patterning, helping students to investigate a numeric pattern and to generalize what they see happening as the pattern grows. In this clip, Dickinson tells his students that “the first step is to do a pictorial representation… I want you to play around with the tiles, and sketch out what you see happening in those first three patterns, but I want you to pay attention to color-coding. You’re free to use those tiles like I said, or markers if you need them, I can make those available as well.” This clip is also indicative of MP 1.

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### 6. Attend to precision

Fran Dickinson leads a number talk on an input/output table and graph, asking “What’s my rule?” In this clip, he wraps up the number talk, and the learners mention many different ways of representing the rule: x3 – 3, times 3 minus 3, 3x – 3. Dickinson notes that “So we’re doing a lot of talking about this rule. What is the rule? Can we write a rule here?” As the students respond, Dickinson notes some disagreement among the student responses and asks his students to explain their thinking to each other. This clip is also indicative of MP 3MP 7, and MP 8.

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In the closure section of his numerical patterning lesson, Dickinson chooses student pairs to present their thinking. His selection shows a progression of presenters that increases with sophistication and accuracy. He notes that “This ramping up allows learners the best chance to wrap their minds around the conversation that ensues...Note how we end with some clear disequilibrium in the room, yet we do have a bit of closure. I purposefully do not ‘give an answer,’ especially since the nature of this investigation was dissecting two different solutions.” This clip is also indicative of MP 3.

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### 7. Look for and make use of structure

In the closure section of a number talk, Fran Dickinson works with his 5th/6th grade students to identify many different ways of representing the rule: x3 – 3, times 3 minus 3, 3x – 3. The students discuss the rule and the best way to represent it, making connections to their mathematics textbook in their conversations. This clip is also indicative of MP 3MP 6, and MP 8.

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### 8. Look for and express regularity in repeated reasoning

In the closure section of a number talk, Fran Dickinson works with his 5th/6th grade students to identify many different ways of generating a rule to govern the inputs/outputs of a chart: various students offer x3 – 3, times 3 minus 3, 3x – 3. The students discuss the rule and the best way to represent it, making connections to their mathematics textbook in their conversations. They discuss the various ways of representing the rule and evaluate the reasonableness of the parameters of the problem. This clip is also indicative of MP 3MP 6, and MP 7.

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