Balanced Mathematics

As the leader of mathematics curriculum and instruction in my district, it is imperative that I clearly articulate the vision for teaching and learning mathematics and support it with a guaranteed, viable curriculum that includes practical, actionable items for the classroom.  As I work to complete a new curriculum rooted in the beliefs outlined in the Visioning Document and in response to the results of a recent, deficit model audit, I recognize the need for a succinct balanced mathematics plan.  This model of lesson design, instruction, assessment, and learner support will be evidenced throughout curriculum documents and professional learning opportunities for educators and campus administrators.  Successful implementation will include district-wide common understanding of the purpose and implementation of the balanced mathematics model.

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Within a balanced mathematics program, classroom structure supports intentional, responsive lesson design and facilitation of learning experiences in order to guide all students to success in mathematics.  

The components of the CISD Balanced Mathematics Plan include:

  • Conceptual Understanding: Learners understand mathematical ideas, make connections to other topics, and are able to transfer thinking to new situations in order to solve problems. Conceptual understanding builds to procedural fluency.
  • Inquiry:  Formulated by educators and learners, compelling questions are developed and subsequently guide inquiries into concepts and problems related to specific learning outcomes. (Supporting English Language Learners: Inquiry)
  • Fluency:  Procedural fluency is demonstrated by students as they carry out procedures flexibly, accurately, efficiently, and appropriately.  Fluency in the mathematics classroom is built on a foundation of conceptual understanding, strategic reasoning, and problem solving.  Mental math and estimation are aspects of fluency developed within balanced mathematics. (link to NCTM position statement)
  • Discourse:  Mathematical Discourse Communities are fostered to support students to make and test conjectures, question, and extend concepts in a welcoming classroom environment with student-created norms aimed for conceptual understanding.  (Link to article, Supporting English Language Learners: Academic Talk)
  • Intervention & Acceleration: In order to respond to the academic needs of all learners, mathematics content is intentionally designed and delivered in small group instruction with progress monitoring toward specified goals. (Supporting English Language Learners: Student Ownership of Learning)
  • Evidence of Learning: Learners demonstrate conceptual understanding or skill development through multiple modalities.  This evidence is created for learners to track their own progress and for educators to formatively monitor learner progress throughout the content.  In addition, the accumulation of evidence of learning builds toward a learner’s portfolio in a summative manner.

Components of this Balanced Mathematics Plan are evidenced in curriculum documents.

  • Conceptual Understanding: Curricular resources are organized into the Concrete/Pictorial-Representational-Abstract learning continuum.  This sequence of instruction builds a thorough understanding of mathematical concepts as learners progress developmentally, building upon prerequisite knowledge and extending to subsequent ideas.  Conceptual understanding is built through low-floor, high ceiling learning experiences.  That is, the content is accessible to all learners, including those who may demonstrate gaps in procedural skills, as the problem may be solved accurately through less efficient methods.  These less efficient methods would be used by a learner who has not yet attained procedural fluency in that given process.  High ceiling tasks allow for extensions for learners ready to make connections deeper in the content or beyond the scope of the course.  These tasks do not limit thinking with one-step, simplified processes.
  • Inquiry: Whole group or small group learning experiences following the 3-Act Task format. (link to post by Dan Meyer, link to post by Mary Kemper) The structure of 3-Act Mathematical Tasks provide built-in opportunities for learners to build background knowledge, connect to prior learning experiences, access the mathematics without the barrier of language, and extend thinking, as appropriate.
  • Fluency: Station-based or other individual learning experiences intentionally designed and accessed by learners to reinforce or extend conceptual fluency within the zone of proximal development.  Learners archive evidence of work and are held accountable for their fluency growth.  Specific practice to attain fluency should be intentionally designed, based on the current level of understanding of the learner and the next developmentally appropriate goal.
  • Discourse: Prompts such as What do you notice? and What do you wonder? as well as Which one Doesn’t Belong? are used to promote academic discourse and specific talk moves such as I agree with ___ because ___ are used to promote accountable talk related to the concepts within the unit of study.  Structures such as Number Talks provide opportunities for mathematical discourse by all learners during a whole group or small group experience.
  • Intervention & Acceleration: All students set SMART goals related to concepts as appropriately aligned within the developmental progression of mathematics.  For those learners identified as At-Risk, this small group instruction as guided mathematics fits within the CISD Mathematics RtI Program.  For those learners not identified as At-Risk, setting goals, receiving guided mathematics instruction related to the goal, and monitoring progress toward the goal, supports growth as well toward acceleration.
  • Evidence of Learning: Learners create evidence of understanding as aligned to the CISD Mathematics Transfer Goals as they display, explain, justify, and communicate mathematical ideas and arguments using multiple representations, including symbols, diagrams, graphs, and language as appropriate with precise mathematical language.  Learners use technology, as appropriate, to demonstrate understanding using verbal and visual articulation with annotation apps (such as See Saw or Explain Everything).

The components of the CISD Balanced Mathematics Plan are implemented in a synchronous manner.  Inquiry-based experiences include opportunities for mathematical discourse and result in evidence of learning, for example.  A mutually exclusive mathematics plan would not maximize the potential to balance the intentional, responsive design and facilitation of learning experiences to guide all learners to success in mathematics.


3-Act Task: Counting Coins

This 3-Act Task involves skip counting by twos, fives, and tens in order to determine the value of a collection of coins.

Act One

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What did you notice?  What do you wonder?

What questions come to your mind?

What is a reasonable estimate?

Act Two

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What information do you need to know in order to answer the question?

Act Three

Coins Act Three

What was the solution?

How does your solution compare to your estimate?

Three Acts Recording Sheet