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.



Less is More: Interpretation of Graphs

Inspired by recent conversation through blogs and on Twitter, I have been exploring analysis of visual images by removing barriers.  The barriers to interpret graphs include the details of numbers and labels.  These specific details are what makes the graphs precise and impact the interpretation, so this gradual reveal allows learners to access the graph as a whole at the appropriate time, not all at once.  In addition, the numbers and labels are intentionally revealed so attention is focused on those as the discussion promotes.  The structure of notice and wonder supports learners to interpret the graphs without being confined by the details of the graph.  That is, by allowing learners to consider the graph first, the learners are allowed to construct meaning in anticipation of transfer to new and unique situations.

The order of gradual reveal includes: the graph of the data with trend lines (that does not show specific points), the graph of the data with specific points, then the graph of the data with labels, the data in the chart.

The example below is related to Stock Market data.   The concepts behind the Stock Market are abstract to young children.  By removing the details on the graph to allow them to focus without clutter makes the math accessible, even to abstract ideas.  This promotes inquiry and fosters the mathematical discourse community.  The meaning making in this structure is the value that cannot be overlooked.  Guiding learners to make connections and interpret representations and abstract images supports them as they look to transfer this understanding.

Step 1: Graph with Trend Lines

What do you notice?  What do you wonder?

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Step 2: Graph with Specific Points

What do you notice?  What do you wonder?

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Step 3: Graph with Labels

What do you notice?  What do you wonder?

Screen Shot 2017-04-25 at 9.26.38 AMStep 4: Data in Chart

What do you notice?  What do you wonder?

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The barriers removed in this structure allow learners to access the mathematical content.  I continue to explore removing barriers in curriculum and instruction including:

  • removing the barrier of language
  • removing the barrier of disengagement
  • removing the barrier of limits
  • removing the barrier of time
  • removing the barrier of prerequisite knowledge

We must design for the margins.  For whom is the curriculum designed if not for all?






Why 3-Act Tasks are Powerful for English Learners

As I continue to study accessibility in education, specifically Universal Design for Learning, structure and strategies for supporting learners served through Special Education and Gifted and Talented Education, and our English Learner population, I have come to the (obvious) conclusion that curriculum development and design involves a complicated, intricate web of planning long before the first words are added to a blank canvas of direction for educators.  It comes down to scalability.  What format/structure/strategies should be embedded in the curriculum (that may be applied horizontally and vertically throughout) so that the resources are seen with clarity and may be supported through efficient and effective professional learning?

Before I can move forward, I need a plan.  This is a huge opportunity and I want to make the most of it.

Recently, my perspective on curriculum development and design has been impacted by a Ted Talk by Todd Rose and various articles and information related to Universal Design for Learning.

  • Todd Rose: The Myth of Average (TEDx): Mr. Rose shares the story of the U.S. Air Force design of the cockpit  for the average pilot and makes a connection to an education system’s design for an average student.  He challenges us to to wonder: Does there exist an average student?  If not, then if we design for the average student, we design for nobody.  We must recognize that students vary in each dimension and we must respond accordingly.  As Mr. Rose says – Ban the average. Design to the edges.
  • Universal Design for Learning: By planning from the beginning for curriculum to be accessible to as many learners as possible, the need to retro-fit lessons and materials are minimized.  (Hunt & Andreasen, 2011).

Enter: The 3-Act Task.

First, a little background on the format of 3-Act Tasks (more info here):

  • Act One: The learners are provided with an image, video, or other piece of information that introduces a conflict.  This conflict may be that the learners disagree with their classmates, or with themselves.  That is, what they see may be contrary to what they believe, mathematically.  Key: Use as little text as possible, because once you tell the learners something, you cannot untell them.
  • Act Two: The learners determine what tools are needed to resolve the conflict or problem.  They must either request these tools or develop these themselves.  Key: Provide as little as possible, because once you give the learners something, you cannot ungive them.
  • Act Three: The learners resolve the conflict.  This is when the solution is determined and compared to the initial idea for reasonableness.  A reflection occurs to foster metacognition and add to the learners’ toolboxes to use in future problem solving.

Now, for the connection to our English Learners:

According to Beyond Good Teaching: Advancing Mathematics Education for ELLsthere are five Guiding Principles for Teaching Mathematics to English Learners.  I argue the inquiry model of 3-Act Tasks in mathematics (done well) support these guiding principles.

1.  Challenging Mathematical Tasks

All learners, ELs included, need to experience mathematical tasks at a high level of cognitive demand.  This article (link) articulates the four levels of Depth of Knowledge according to Norman Webb.  The levels are neither developmental (this means even our youngest learners can experience strategic and extended thinking) nor sequential (learners need not experience level 1 before tackling level 2, etc.).

One of my favorite 3-Act Tasks is Volcano.  Challenging, indeed!  There are multiple entry points to solve this problem.  That is, the learners may successfully solve the problem in various means – not only by using a single algorithm.  They may draw a picture, look for a pattern, make a table of data, or use abstract formulas to solve.  Because the task contradicts the obvious method of solving (2-dimensional measurement rather than 1-dimensional), I argue this is a level 3 DOK task.  And thus, engaging!

2.  Linguistically Sensitive Social Environment

Such a learning environment fosters extensive educator-supported interactions in all forms among the learners and the educator (this includes between the leaners themselves).  This atmosphere is safe, allowing for opportunities for learners to ask questions and seek knowledge related to language.  Careful attention is paid to providing a low-stress classroom.

During the Volcano task, the town name “Tarata” and the abbreviations for hour and minute should be clarified to avoid barriers that may occur.  Though the language of this task is very minimal, even the three aforementioned terms could cause a stressful environment for ELs to manage and therefore may be unable to engage optimally in this learning experience.  The educator should kindly check in with ELs and provide a safe environment for them to ask questions, either to other learners or to the educator.

3.  Support for Learning English While Learning Mathematics

A Mathematics Discourse Community (MDC) supports all learners’ communication about mathematics.  This includes reading, writing, speaking, listening – all about the mathematics of the task.  Mathematical discourse builds the language while learning the mathematics concepts.

Technology integration while experiencing the Volcano task provides an opportunity to capture the process of problem solving.  Using a voice recorder app such as Voice Record Pro allows learners (specifically ELs) to capture the metacognition involved in the problem solving process.  Note taking apps such as Notes or Notability allow learners to capture images within notes and have an embedded microphone feature to include verbal reflection as well.  Once learners have engaged in the MDC and practiced articulating the mathematics of the task in a safe environment, they may use technology tools to capture their language.

4.  Mathematical Tools and Modeling as Resources

Within Act Two of the inquiry experience, learners recognize the necessity for specific tools to support the problem solving process.  These tools may include measurement tools, images, or diagrams as scaffolds for the task.

In the Volcano task, handouts include maps with city names and additional video footage of the situation.  Additional models may include recreations of volcanic eruptions (as concrete representations) and connections to concepts of area, distance, and other measures.  Educators and learners may create/reference anchor charts as tools as well.

5.  Cultural and Linguistic Differences as Intellectual Resources

As members of a learning community, the cultural and linguistic differences among the class are valuable resources, seen as a commodity to be used to collaboratively solve problems and feed into the MDC.

Related to the Volcano task, the MDC within the classroom should foster input from all learners, supporting the collective insight of the members.  Strategies such as a Socratic Seminar build learners’ discussion and listening skills and provide opportunities to contribute to the MDC.

3-Act Tasks are more than engaging, high quality math problems.  Their structure, with intentional implementation of MDCs and technology integration, supports the Guiding Principles for Teaching Mathematics to English Language Learners.  Done well, 3-Act Tasks support implementation of the English Language Proficiency Standards and open the doors of problem solving to all learners.  All Learners.

Then, those learners who have positively, successfully tackled inquiry experiences continue to build background knowledge and a positive perspective on mathematics learning.  And, we know the value of mindset.


Hunt, Jessica and Janet Andreasen. 2011. “Making the Most of Universal Design for Learning.” Mathematics Teaching in the Middle School. 17: 167-72.