In EL Education’s book Learning that Lasts, authors Ron Berger, Libby Woodfin and Anne Vilen define deeper instruction,” that is, wanting “all students to learn deeply...to feel energized and empowered by what their minds can do and to flexibly apply their knowledge to new situations.” Their three indicators of deeper instruction include work that challenges, engages, and empowers. This can be seen in further depth in Table I.1 Indicators of Deeper Instruction on page 9 of the book. When analyzing this table (see link), it is clear that in order to achieve deeper instruction - instruction that empowers, energizes and challenges students - ALL learning must include active student participation and leadership in conversation.

Mathematics instruction is no exception. All recent research (including that already linked in recommendations 1 through 6 from experts like NCTM, the writers of the Math Practice Standards in the Common Core, and EL Education’s own Core Practices) indicates that discourse is a core component of high quality math instruction. Teaching through discourse allows students to create deeper ideas and be more engaged and empowered in the math classrooms.

See some EL Education video examples of the power of mathematics discourse in action:

• Workshop 2.0 Lesson Structure with former Anser Charter School 5th Grade Teacher Giselle Isbel

• CGI Lesson Structure through Discourse with Polaris Charter Academy Teacher Mona Iehl

• Balanced Math Discourse Structure with Polaris Charter Academy Teacher Mona Iehl

Discourse should be used to unpack, critique and understand other students’ work. It can take a few forms:

• Individual Conferencing - how does one student develop in their ability to speak about math?

• Small Groups of Students

• Whole group, student-led discourse that builds to coherent and collective synthesis

All three methods should be approached and planned for differently.

Individual Conferencing

Also called “conferring” this is quick 1:1 conversations with students as they are grappling that is meant to guide their individual growth. For more on conferring see this great overview and tools for support here.

Additionally, a new book: In the Moment by Jen Munson was just released in 2018 that unpacks this concept for mathematics even further.

Small Groups

Small Group discourse is a huge leveler for strategic student development in their mathematics thinking and ability to discuss math. Small groups should be rolled out as a structure within the classroom with key expectations explained and reinforced multiple times.Some key components to consider incorporating into your structure:

• Group roles and norms

• Reflection and accountability

• Protocols that aid the flow (and connect to the task - e.g. how to analyze another group’s work)

• A process for regularly analyzing other students’ work (e.g. task cards). Consider something that needs all classroom voices, that incorporates predefined roles, or pre-designed, structured challenges)

• Size of group - this will vary based on the age of your students and may even vary based on the alchemy of your particular classroom from year to year. Experiment to see what size group results in the most active and productive involvement.

Whole Group

Whole group, student-led discourse can be particularly difficult to do effectively. It is not as simple as giving students a prompt and inviting them to talk. It is a challenging facilitative process that requires detailed planning, and a deep vision for learning goals. Successful mathematics discourses result in the following:

“Effective mathematics teaching engages students in discourse to advance the mathematical learning of the whole class. Mathematical discourse includes the purposeful exchange of ideas through classroom discussion, as well as through other forms of verbal, visual, and written communication. The discourse in the mathematics classroom gives students opportunities to share ideas and clarify understandings, construct convincing arguments regarding why and how things work, develop language for expressing mathematical ideas, and learn to see things from other perspectives (NCTM 1991, 2000).”

This result is not easy to obtain and takes two major components:

1. The development of a culture of discourse where students feel comfortable to take risks in participating, make mistakes, respectfully disagree, etc. (For more in depth unpacking of this idea see this video from School 21 in Stratford, London, UK which includes material from across content areas)

2. Purposeful planning and expert facilitation that ensures that students ideas are valued but the mathematical goals at the heart of the lesson are still uncovered, synthesized and mastered by as many students as possible. (For more in depth explanation for this see this summary of the book The 5 Practices for Orchestrating Productive Mathematics Discussions by Smith & Stein).

   • Consider using anticipatory frameworks to guide your whole-group discussions:

     • Ex: Anticipatory Frameworks - all tasks

     • Specific Content Anticipatory Frameworks:

       • Algebraic tasks

         • See Black Copy here

         • See Examples of Completed AFs (one per problem type)

       • Fraction Equal Sharing Tasks

3. Synthesis & Constructing Knowledge long term.

   • Tool for Implementation of Synthesis K - 5

   • Tool for Implementation of Synthesis 6 - 12

It is essential that all of these components are unpacked with teachers and supported in order to ensure that teachers and students are successful in implementation.

It is a challenge for most mathematics classrooms to move beyond rote memorization and up the Bloom’s Taxonomy pyramid into the realm of deeper instruction. Without guidance and supportive planning, teachers are not sure how to engage and empower students through mathematics discussions and so attempts at a “discourse” become a ping-pong match or a game of guess what’s in the teachers head.

In order to ensure that a true discourse is implemented, strategic steps must be taken:

Prior to Attempting a Mathematics Discourse with Students:

1. Teachers should experience discourse as a learner with appropriate tasks (can go hand in hand with content and pedagogical development for teachers described in Recommendations 5 and 6).

2. Teachers should map out ways to build the culture of discourse in their classroom. This includes implementing:

   • Practical moves like talk stems, silent signals, strategic seating, etc. (see other examples in this Planning tool)

   • Nuanced culture building which includes developing trust, a growth mindset about learning mathematics, and safe space for risk taking.

3. Teachers should plan (or co-plan) for the ways they will build towards a clearly defined synthesis goal, while keeping students ideas central. See an example planning tool here: Building towards synthesis support one pager for more on why a synthesis goal is essential. This planning includes:

   • Choosing a learning goal

   • Choosing a meaningful task that will achieve that goal

   • Anticipate students solving of that task from many angles including addressing possible misconceptions

When Implementing a Mathematics Discourse with Students:

1. Teachers should be following the 5 Practices recommendation for implementation in order to both keep the mathematics goal clear and deeply listen to students ideas:

   • Launch the task in an engaging way (that does NOT reveal the grapple concepts)

   • Monitor students while they solve (collecting notes that will help to facilitate the discourse)

   • Select particular students to share their thinking with the group

   • Sequence those students’ shares in a way that will build towards your math goal

   • Facilitate the discourse where teachers connect students’ ideas and pose rich moments and questions for turn and talk and debate that will help to respect student voice while also driving at the mathematics goal.

   • Synthesize the learning to reach a clear moment of purpose. Students or teacher name any of the following: new learning, new goal, reaffirmed goal, conjecture, etc. that allows for all students to be clear on where the crew is aiming to go in mathematics work from here.

   • Assess students’ mastery of the math concept or conversation with an exit ticket or notes from their speaking.

After Implementing a Mathematics Discourse with Students

1. Teachers should be supported in reflecting on:

   • Their classroom culture of discourse

   • Their ability to achieve their clear mathematics learning goal or synthesis moment

   • Students success in the lesson gathered in a form of assessment data (for example an exit ticket)

2. Math Culture guides and Teachers can also use a reflection tool like this: Rubric to Evaluate Mathematics Discourse to evaluate their mathematics discourse.