Our goal at Castlewood is to provide mathematics instruction that will encourage students to become accurate, efficient, and flexible problem solvers. It is our belief that the rapidly changing technological advances have created a fluid and dynamic world for this generation of students. We can no longer predict and plan for the problems that these students will need to solve when they enter the work force. Therefore, we must make sure that our students have the core knowledge, and the skills to apply that core knowledge, to a variety of situations that are known and unknown to us at this time.

At the center of our mathematics instruction, are authentic problem-solving opportunities that present a significant cognitive challenge. For the sake of this document, a mathematical problem may be a hands-on exploration of a mathematical concept, a multi-step task, shorter problems with a single answer, or looking for patterns and/or repeated reasoning in discussions of arithmetic strategies. Students must have the chance to struggle with meaningful problems, discuss possible solutions with their peers, create mathematical arguments, and place these arguments before a group of their peers who can provide validation and feedback. During these problem-solving opportunities, as per NYS CCLS, students should demonstrate their ability to:

  1. Attend to problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.

In order for students to have an opportunity to demonstrate these skills, teachers must carefully plan authentic problem-solving opportunities (APSOs) that are rigorous, motivating, and meaningful to the students. APSOs should have multiple entry points to provide access for students of all ability levels, because we believe ALL students should have access to a rigorous curriculum. Failing to provide rigor in mathematics today, is failing to prepare students for the 21st century.



In order to craft appropriate APSOs, teachers spend time carefully analyzing hard data (standardized test scores), soft data (classroom observations of activities and student talk), and student work samples to identify students’ strengths and weaknesses. Reflecting on these strengths/weaknesses in relation to the NYS CCLS and Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics by Randall I. Charles, teachers must identify the next steps students should take in a particular strand of mathematics. APSOs should allow students to apply a previously studied concept in a new and unique way. APSOs should allow students to express their solution through abstract reasoning, mathematical modeling, and connections to other areas of mathematics or real life applications. Students should be able to communicate these solutions verbally or in writing.


Crafting rigorous and appropriate APSOs requires careful advanced planning on the part of the teacher. After careful analysis of various types of data, teachers plan a course of instruction that will address the appropriate next steps for each student. When planning, teachers should reflect on the content to be taught, the scaffolds that will be needed for the struggling learner, enrichment for the advanced student, instructional strategies to present the concepts, and methods for presenting clear expectations of high quality work to the students. Teacher Teams’ common planning time can support this work by creating knowledge packages (described in Small Steps, Big Changes by Conifer and Ramirez) to outline mathematical strands/concepts in a way that makes planning easier.


The ultimate goal of careful data analysis and diligent planning is to provide an exciting and cognitively engaging experience that motivates students to fully participate in the study of mathematics. Key factors to create such an environment include:

  • Teacher dominated direct teaching should not be more than 15-20 minutes.
  • Teachers must provide a safe and comfortable environment where risk taking in problem solving is respected and admired.
  • By providing appropriate think time, teachers demonstrate the importance of thoughtful and careful reasoning over speed (in all cases except fact recall.)
  • All students should have time to struggle with multistep problems.
  • The mathematics classroom should provide opportunities for students to explore mathematical ideas independently, in partnerships, small groups, and as a whole class.
  • Where appropriate, students should have control over the task/activity they complete or the method or presenting what they have studied/learned.
  • All students should have access to their own toolkit of supplies to be used, as they deem necessary, when engaged in problem solving opportunities. This toolkit may include calculators to be used when appropriate. Access to computers, iPads, or other forms of technology should be accessible and encouraged when appropriate.
  • Students should be encouraged to prove their thinking abstractly and pictorially. Students should be able to explain how these two methods are connected to express a full understanding of the idea being explored at the time.
  • All students should be expected to understand why procedures and algorithms work.
  • Students should be invited to present mathematical arguments before a group.
  • Students should be encouraged to add onto a fellow students’ mathematical argument by extending their ideas/making a connection from the original idea to a different mathematical principal or application.
  • Students should be encouraged to question their peers during discussion time when clarification is needed. Students should not be completely dependent upon the teacher for clarification of students’ thoughts.
  • Students should be taught to monitor their own thinking and self-assess their own work.
  • All students should be fluent in the language of mathematics and be expected to use correct terminology when making a mathematical argument.
  • All discussions should eventually lead to larger mathematical ideas that can be connected to prior learning or future mathematical explorations.
  • Even skill-based instruction (i.e. memorization of math facts and precision of skill with algorithms) should lead to discussions of larger ideas such as patterns observed, repetition in reasoning, properties of operations, etc…


Such a strong focus on student-centered discussion requires teachers to be adept at crafting questions that pose various levels of cognitive challenges and promote discussion. Four Depth of Knowledge (DOK) levels were developed by Norman Webb as an alignment method to examine the consistency between the cognitive demands of standards and the cognitive demands of assessments. These four levels are described as:

  • Level 1 – Focus on specific facts, definitions, details, or using routine procedures. Does not require deep content knowledge to respond to an item. (i.e. 4×3= or define rhombus)
  • Level 2 – Focus on applying skills and concepts, explaining how or why, making decisions, interpreting in order to respond. (i.e. what operation would you use)
  • Level 3 – Focus on reasoning and planning in order to respond. Complex and abstract thinking is required. Often need to provide support for reasoning or conclusions drawn. (i.e. Which is the most efficient strategy to solve this problem?)
  • Level 4 – Requires complex reasoning, planning, and thinking (generally over extended periods of time). Students may be asked to relate concepts within the content area and among other content areas. (i.e. When given two different income scales, determine which job is more profitable over a given period of time.)

All four levels have an appropriate place at various times in the mathematics classroom. For example, Level 1 and 2 questions are useful to assess student understanding in order provide specific feedback and plan next steps for the learner. However, to promote discussion and strong problem solving skills, teachers need to incorporate Level 3 or 4 questions in to the lesson plan. Most lesson plans only have one or two questions that are considered to be Level 3 or 4; however, it is difficult to spontaneously generate them during instruction. Therefore, it is important to have these questions formulated during the planning process.



In short, at Castlewood we strive to develop accurate, efficient, and flexible problems solvers by providing authentic and rigorous problem solving opportunities that lead to engaging student-to-student discussions facilitated by teachers through carefully planned questions. Thoughtful data analysis provides a clear understanding of students’ needs, which becomes the foundation of carefully planned lessons with appropriate supports and access points to authentic problem solving opportunities. During these opportunities, teachers will guide students to a deep conceptual understanding of mathematics that can be expressed concretely (by using manipulatives), pictorially (including bar models), abstractly, verbally, or in writing for to allow ALL students to participate in rigorous academic experiences that will prepare them for the challenges of the 21st century.