Status & Mindset Interventions
In her book Strength in Numbers: Collaborative Learning in Secondary Mathematics, Ilana Horn writes: “Judgements about who is smart based on prior achievement or social categories violate a fundamental principle of equity and are consequential: learning is not the same as achievement” (2012, p.20). The resources below were curated to help you redefine "smarts" in math, disrupt status divisions, develop growth mindsets, and foster a collaborative math community.
Anticipatory Planning
How often does your planning for math involve searching for the "best" problem and then thinking about how you want to teach the problem? It's safe to say this is how most of us approach(ed) lesson planning. The problem with this approach, however, is that it is teacher focused and neglects to consider how students might perceive and respond to the problem. Conversely, anticipatory planning focuses planning efforts on imagining how students might respond to a problem and using that information to plan questions that will push and clarify student thinking and build understanding by sequencing and connecting approaches students are already using.
If you've ever tried to facilitate constructivist math learning in your classroom and it fell short of your expectations, it's likely because the key factor, anticipatory planning, was missing! The template below can help you prepare to facilitate constructivist math learning in your classroom. Grab a planning buddy and give it a try!
Anticipatory Planning
How often does your planning for math involve searching for the "best" problem and then thinking about how you want to teach the problem? It's safe to say this is how most of us approach(ed) lesson planning. The problem with this approach, however, is that it is teacher focused and neglects to consider how students might perceive and respond to the problem. Conversely, anticipatory planning focuses planning efforts on imagining how students might respond to a problem and using that information to plan questions that will push and clarify student thinking and build understanding by sequencing and connecting approaches students are already using.
If you've ever tried to facilitate constructivist math learning in your classroom and it fell short of your expectations, it's likely because the key factor, anticipatory planning, was missing! The template below can help you prepare to facilitate constructivist math learning in your classroom. Grab a planning buddy and give it a try!
Launch, Explore, Discuss
At the elementary, middle, and high school levels we use a "Launch, Explore, Discuss" lesson structure which allows us to build class routines for exploring a rich problem in a student centered way. The launch, explore, discuss framework forefronts student thinking and supports understanding and discussion of student problem solving strategies (as opposed to procedures, answer getting, or speed). Discussion of mathematical ideas is student generated and situates students as the mathematical authority in the classroom.
A launch, explore, discuss lesson structure is also used in Cognitively Guided Instruction (CGI). CGI is a student-centered approach to teaching math at the elementary level that starts with what students already know and builds on their natural number sense and intuitive approaches to problem solving. When we use a CGI launch, explore, discuss lesson structure we focus on understanding student thinking, asking smart questions, and supporting students in listening to and thinking about each other's ideas.
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Launch-Explore-Discuss in Action
Why Use a Launch, Explore, Discuss Lesson Structure?
Karen Underwood, one of our elementary MAIC teachers reflects on why she uses a launch explore discuss lesson structure with her 3rd grade students. Karen is in her 3rd year of teaching at the time of this video.
Former eighth grade teacher Katerina Milvidskaia and her former student, Sarah, discuss the benefits of student centered instruction and unpack the student experience. Katerina was in her 7th year of teaching at the time of this video.
The Importance of Anticipatory Planning
Karen discusses the importance of anticipatory planning and thinking about potential student problem solving strategies.
Katerina discusses the importance of anticipatory planning and thinking about potential student problem solving strategies.
Elementary Problem Launch
A typical story problem launch contains the following elements: the story, the retell, and the check for conceptual understanding. Below are two videos from fall and spring of the same 3rd grade class. The first video provides an overview of the launch and includes teacher and student commentary. The second video (from earlier in the year) includes footage and details of how to launch a story problem.
Karen and her students model and discuss a typical story problem launch. Includes footage of a launch from April of the spring semester and student commentary.
Footage from Karen's class engaging in a story problem launch during October of the fall semester. Includes details for how to launch a story problem.
Elementary Explore Phase
During the explore phase, students engage in independent problem solving while the teacher circulates and observes student thinking. Based on the anticipatory planning, the teacher is looking for student problem solving strategies that will help illuminate the mathematical goal of the lesson. The key teacher moves during the explore phase are to encourage students to make their thinking visible and to sequence the student strategies to be shared to launch the discourse.
Karen discusses how she sets up the explore phase and how she engages with student thinking. This video includes footage of the explore phase from a lesson in April of the spring semester.
Footage from Karen's class engaging in problem solving during the explore phase from a class in October of the fall semester. Includes details about how Karen encourages students to make their thinking visible, as well as questioning strategies to understand how students are thinking about the problem.
Elementary Problem Discourse
The discourse is the heart of constructivist learning. Through dialogue, students make sense of different problem solving strategies and have the opportunity to compare, contrast, and connect their classmates ideas to their own. The key teacher moves during the discussion are to ask students to compare and contrast the different strategies and ask open ended questions that get at the mathematical goal for the lesson.
Planning Tools & Resources
Books & Articles
Secondary Launch
Launches at the middle and high school level can vary. Below are two videos of sequential problem launches from an April lesson in Katerina's 8th grade class.
Katerina launches class with a claim from students from a previous class during a unit on transformations. Students record their thinking and questions related to the other students' claim and prepare to share their ideas.
Katerina launches with a claim about a pre-image and an image during a unit on transformations. The launch is followed by independent work time where students explore the claim and generate questions.
Secondary Explore
During the explore phase, students move into group work to work on the problem together. Group norms or group roles can be used to promote equitable student interaction and collaboration. The teacher circulates and reinforces group norms/roles, listens to student thinking, and engages in accountability quizzes to monitor evolving student understanding. Based on the anticipatory planning, the teacher is looking for student problem solving strategies that will help illuminate the mathematical goal of the lesson and planning which group strategies will be shared and the best sequence to launch a class discussion.
Sarah, Katerina's former student shares about group roles and the importance of the Explore phase of problem solving.
Secondary Math Discussions
Facilitating a whole class discussion is challenging for most teachers. A number of strategies can be used to support students in listening and making sense of each other's thinking, including pair shares, making student thinking visible for the class to consider, orienting students to engage with each other's ideas, and developing a culture where students feel safe sharing their ideas.
Katerina shares how class discussions often take more than one class.
During a whole class discussion, Katerina encourages students to make their thinking visible and to consider each other's ideas.
Attending to Status
Status issues often crop up in group work or in whole class discussions. If students have experienced more traditional classroom teaching practices they may have internalized that to be good at math means to get the right answer quickly, and that if they struggle it means they are "not a math person". Students also bring in ideas from our surrounding culture which often depicts mathematicians as white and male. Without a counter-narrative, students can internalize these ideas causing some students to step up during math discussions and others to fade into the background or act out in other ways to maintain status with their peers.
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Some ways to elevate academic status during group work or discussions include: having a student restate another student's mathematical idea, having a student present their thinking to their group or the class, having the class pair share about a student's idea, and sharing a student's mathematical idea with the class.
Katerina uses several pedagogical moves to elevate different students' academic status during a whole class discussion.
Karen and her students discuss what happens during the discourse phase of a story problem. This video includes footage of a discourse from a lesson in April of the spring semester.
CGI: Discourse
Footage from Karen's class engaging in a whole class discussion/discourse from a class in October of the fall semester.