How do we invite students to the mathematics, and explicitly signal to kids that they have ideas that matter in math class?

*By Max Ray-Riek*

In this series of blog posts, the first of which is available here, we’re exploring how, in order to be successful in a problem-based classroom, students have to shift their thinking about what being a good math student looks and sounds like. What do you notice about your own students’ beliefs about how they should participate? What are you curious about now, as you think about what it takes for students to be successful in a problem-based classroom?

In this post, I’m going to focus on the ways that different instructional components in IM curricula—units, lessons, and activities—all begin with a “launch” that explicitly invites students into the mathematics.

Here is a reminder of one model for problem-based instruction that we use at IM:

IM tasks are designed to have “low floors” or low barriers to entry. In other words, we want every kid to be able to get started, whether they have unfinished learning from prior grades or not. How do we do that?

Many of our units and lessons begin in familiar contexts, or in situations kids have reasoned about before. Unit 2 of the forthcoming Algebra 1 curriculum begins with kids considering what they would need to know to figure out the cost of ordering pizza for their whole class. Other times, we begin with a concrete experience that kids can refer back to throughout the unit, like how unit 2 of grade 6 begins with actually mixing different batches of juice mix and comparing which ones taste the same and which ones have a stronger flavor.

These contexts are an invitation to the mathematics because we are explicitly inviting kids to bring their reasoning and understanding into the situation, and then to make it mathematically formal later in the unit. We want kids thinking about the pizza party to talk about things like whether they should order from a place that has more expensive but yummier pizza, and we want kids making juice mix to talk about how they like to make the juice at home because their dad makes it too watery. We want kids making connections between the ideas they have about price and size and flavor to math.

Other units and lessons begin in purely mathematical contexts, but we still invite students into the mathematics in ways that invite them in as noticers, wonderers, and reasoners. Routines such as the Notice and Wonder routine, used in all of IM’s curricula, make this invitation explicit to students. We often invite students to record and share what they notice and wonder about an image or context during the warm-up of a lesson or launch of an activity.

An example of an inviting mathematical context comes from unit 1 of grade 8. The unit begins with some “dancing” shapes, and we ask students to describe how the shapes move in each step of the dance. The invitation comes because all you need to do to be successful is notice and say what you noticed. We add more math formality later, with words to describe certain moves, and eventually precise definitions of those moves. But we want kids to see the moves as moves, as steps in a dance, to bring what they already know about describing motion in space.

So the curriculum is designed with an invitation in mind, and it can be easier to see in those first lessons in a unit, which are often fun and hands-on. But not every lesson begins with food or dancing. How do we keep that invitation going, and what can teachers do to engage the kids who already believe that mathematics has closed the door, turned off the lights, and put away the doormat?

The principle behind these invitations remains the same: we try to invite everyone to notice, wonder, and reason—and share what they noticed, wondered, or reasoned about. We try to invite students to share their ideas and insights before we add any additional math formality.

#### Next Steps

In the next post, we’ll explore more about getting kids from noticing and saying what they notice, to having ways to tackle the problems in a problem-based curriculum.

For now, I invite you to look at the warm-ups and activity launches in the next lessons you’re going to teach. How can we show that we know that *all* kids have something to say about the problem?

- Could kids tell a partner something they noticed before you call on a few kids to share, or turn kids loose to work on the problem?
- Could every kid vote on something or make a prediction?
- Could you share just an image or story without the question and have kids share what they wonder? (Math Language Routine 3: Co-Craft Questions)

And then take a look at how kids are being set up to connect their ideas with the math that’s coming:

- Are there opportunities for kids to bring in reasoning they’ve done in another context, such as at the store?
- Are there opportunities for kids to look at something concrete like a shape or a diagram? Or to notice a pattern?
- Are the questions in the warm-up or the launch more informal (What do you think will happen? Which do you think is bigger/better?)

##### Max Ray-Riek

**Max Ray-Riek **is the Director of 6-12 Professional Development at Illustrative Mathematics and worked as a writer on the high school curriculum. Before coming to Illustrative Mathematics, Max worked for The Math Forum, focusing on fostering problem solving and making student thinking the center of math class. He is the lead author of *Powerful Problem Solving: Activities for Sense Making with the Math Practices. *Max is a former secondary mathematics teacher (and before that, preschool teacher) who finds the art and discipline of valuing each student’s ideas, and supporting all students to see themselves as people who can listen, connect, and act mathematically, to be the hardest, most rewarding, and most important work in the world. Max lives in Philadelphia with his wife, with whom he fosters puppies who will grow up to be trained to be service dogs.

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