What Your Students are Thinking

How do you know that your students are thinking?  How do you know what EACH ONE of your students is thinking?  I encourage the valuable implementation of mathematical discourse communities to model problem solving, promote higher depth of knowledge and the necessity for justification of mathematical arguments, promote the use of accurate academic vocabulary coupled with growth of language for English Language Learners, allow students to build their confidence in mathematics, and provide opportunity to formatively assess students throughout.  This formative assessment is what your students are thinking.


mathematical discourse community (MDC) includes productive mathematics discussions, centered around common problem situations in which the environment is safe enough for all students to contribute in multiple means.  MDCs are accessible for all learners, including those exhibiting gaps in prerequisite knowledge or academic vocabulary.

Though the benefits are great, building a mathematical discourse community in a classroom is a challenging feat for educators.  Quality should trump quantity and conceptual understanding is a goal worth seeking.

Lynn Liao Hodge and Ashley Walther describe 4 initial practices to build a foundation of productive discourse in mathematics classrooms in their article Building a Discourse Community: Initial Practices:

  • Use a more open task
  • Support think, pair, and revoice/compare
  • Offer three ways to participate
  • Define a contribution

In my work to design and support curriculum for PK-12 mathematics, I recognize the need for a balance between scalability and fidelity.  That is, what structures I put into place in my curriculum that can be utilized horizontally (throughout the school year) and vertically (throughout the courses) by educators, and in doing so, can be done well – accurately, efficiently, and effectively.  I choose to implement a few structures in order to support teachers to do them well.  3-Act Tasks, Notice and Wonder, and Which One Doesn’t Belong? are three such structures I work to embed in my curriculum.

Screen Shot 2017-07-17 at 12.26.46 PM.png

Hodge and Walther refer to open tasks as those with embedded student choice, multiple paths to solve, the necessity to justify solutions, all within a real-world scenario.  Related to this, I strive to embed inquiry-based learning experiences, in the structure of 3-Act Tasks.  I focus on 3-Act Tasks as a means for clarity, consistency, and opportunity for professional learning related to the implementation in the classroom.

When not deep within a 3-Act Task, it is important to continue to support the discourse community in the mathematics classroom.  I argue Notice and Wonder and Which One Doesn’t Belong? are two structures that continue this work to allow conversation, comparison and justification by students about mathematics concepts.

When teachers use the Notice and Wonder structure, it causes them to pause.  It also pushes the ownership of the knowledge to the students.  The conversation begins when a prompt of some sort is shared with the students.  This prompt may be an image, an object, a graph, a situation, or any other representation that leads to interpretation.  The simple question, “What do you notice?” is open enough to encourage contributions.   Based on the work of Hodge and Walther, it is advised to encourage participation by more than a single student through think, pair, revoice/compare.  I challenge you to avoid choral responses as a go-to with Notice and Wonder.  This excludes nearly all students in the room, allows students to hide behind their classmates, and narrows the scope of the conversation.  To move beyond verbal-only contributions, I challenge you to support sketches, gestures, and other non-spoken notices and wonderings.  Then, when the conversation lends itself to transition to the wonderings, the question, “What do you wonder?” is used.  Read more about Notice and Wonder here.

Screen Shot 2017-07-10 at 2.21.51 PM.png

When teachers use the Which One Doesn’t Belong (WODB) structure, it also causes them to pause.  Wait time is a valuable commodity in the mathematics classroom!  WODB also pushes the ownership of the knowledge to the students.  Similar to Notice and Wonder, a prompt is used, but this time it includes multiple items of some sort.  This prompt is usually a set of images, but does not have to be.  Any set of images, objects, graphs, situations, or other representations that lead to interpretation can be used for WODB.  The simple question, “Which one doesn’t belong?” is open enough to encourage contributions – with justification.   Seek participation by many students (ALL) through think, pair, revoice/compare.  I challenge you to avoid choral responses as a go-to with Which One Doesn’t Belong?.   Hodge and Walther say to offer three ways to participate.  What three ways will you support in your classroom?  What tools will you encourage your students to use to communicate about mathematics?  Sketches, gestures, and other non-spoken ways to identify and justify Which One Doesn’t Belong? builds the mathematical discourse community in your classroom.  Need a place to start with Which One Doesn’t Belong?  Try this or this.

Screen Shot 2017-07-10 at 1.48.40 PM.png

Focusing on the means to contribute, I encourage reflection on our English Language Proficiency Standards.  That is, opportunities for students to listen, speak, read, and write about mathematics during the learning process as well as while they are demonstrating their understanding.  These four modalities should not be seen as a checklist, because listening is not hearing, speaking is not repeating, reading is not looking, and writing is not copying.  I bring you back to the initial questions on this post:

How do you know that your students are thinking?

How do you know what EACH ONE of your students is thinking?