NTN Methodologies Coaching Resource Research

1. Problem Solving

Encourages students to apply mathematical reasoning and strategies to complex, real-world, and abstract problems.

Research Support:

  • Reflective thinking enables students to analyze their problem-solving processes, leading to improved outcomes.
  • The Common Core State Standards for Mathematics highlight that mathematically proficient students “check their answers to problems using a different method, and they continually ask themselves, ‘Does this make sense?'”
  • Problem-based learning (PBL) enhances problem-solving skills and promotes deeper understanding of mathematical concepts.
    • Hmelo-Silver, C. E. (2004). Problem-based learning: What and how do students learn? Educational Psychology Review, 16(3), 235-266.
  • Structured problem-solving activities improve long-term retention and conceptual understanding.
    • Jonassen, D. H. (2011). Designing Problem-Based Learning Environments. Educational Technology Research and Development, 59(2), 159-176.
  • Collaborative problem-solving strengthens mathematical reasoning and improves student engagement.
    • Artigue, M., & Blomhøj, M. (2013). Inquiry-based learning in mathematics education: Important themes and findings. ZDM, 45(6), 925-937.
  • Students who engage in structured problem-solving activities develop metacognitive skills essential for future learning.
    • Schoenfeld, A. H. (2013). How we think: A theory of goal-oriented decision making and its educational applications. Routledge.

2. Concrete-Pictorial-Abstract (C-P-A) Approach

Supports conceptual understanding by transitioning students through hands-on manipulatives, visual models, and abstract representations.

Research Support:

  • The C-P-A approach is a highly effective method that develops a deep and sustainable understanding of mathematics in pupils.
  • Using a Concrete-Pictorial-Abstract progression allows students to build strong mathematical foundations by linking concepts across different representations.
    • Witzel, B. S., Riccomini, P. J., & Schneider, E. (2008). Implementing CRA with secondary students with learning disabilities in mathematics. Intervention in School and Clinic, 43(5), 270-276.
  • Studies show that the C-P-A approach enhances mathematical reasoning abilities and problem-solving performance.
    • Kurniawan, H., Budiyono, B., Sajidan, S., & Siswandari, S. (2020). Concrete-Pictorial-Abstract Approach on Student’s Motivation and Problem-Solving Performance in Algebra. Universal Journal of Educational Research, 8(7), 3204-3212.
  • Scaffolded visual models in the Pictorial stage help bridge the gap between hands-on learning and abstract mathematical understanding.
    • Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of Virtual Manipulatives on Student Achievement and Mathematics Learning. International Journal of Virtual and Personal Learning Environments, 4(1), 35-50.

3. Formative Assessment

Uses ongoing assessment to provide immediate feedback and adjust instruction to meet diverse student needs through differentiation based on student responses and learning progress.

Research Support:

  • Formative assessment practices have been shown to raise the level of student attainment, increase equity of student outcomes, and improve students’ ability to learn.
    • OECD. (2005). Formative Assessment: Improving Learning in Secondary Classrooms.
  • Ongoing feedback through formative assessment enhances student engagement and ownership of learning.
    • Black, P., & Wiliam, D. (2010). Inside the Black Box: Raising Standards Through Classroom Assessment. Phi Delta Kappan, 92(1), 81-90.
  • Differentiated instruction, guided by formative assessment, supports diverse learners and improves achievement.
    • Tomlinson, C. A. (2017). How to Differentiate Instruction in Academically Diverse Classrooms. ASCD.
  • Formative assessment strategies like peer review and self-assessment improve metacognition and problem-solving skills.
    • Brookhart, S. M. (2013). How to Give Effective Feedback to Your Students. ASCD.

4. Systematic Review

Reinforces previous learning through structured review cycles while ensuring students build fluency with conceptual understanding.

Research Support:

  • Spaced repetition and systematic review improve long-term retention of mathematical concepts.
    • Rohrer, D., & Taylor, K. (2016). The effects of distributed practice on retention in mathematics learning. Journal of Educational Psychology, 108(6), 974-988.
  • Systematic review practices in programs like Mathematics Mastery improve conceptual understanding through structured repetition.
  • Cumulative review sessions contribute to deeper learning and greater conceptual retention.
    • Pashler, H., Rohrer, D., Cepeda, N. J., & Carpenter, S. K. (2017). Enhancing learning and retarding forgetting: Choices and consequences. Psychonomic Bulletin & Review, 24(6), 187-193.
  • Developing fluency with understanding supports transfer and problem-solving in new contexts.
    • National Research Council. (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press.
  • Procedural fluency develops most effectively when grounded in conceptual understanding and strategic practice.
    • Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362.
  • Frequent practice with feedback enhances both fluency and retention of math skills.
    • Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38(4), 293–304.

5. Making Thinking Visible

Encourages students to articulate and reflect on their reasoning using discourse, written explanations, and visual models.

Research Support:

  • Making student thinking visible through collaborative learning enhances critical thinking skills.
    • Hmelo-Silver, C. E., & Barrows, H. S. (2008). Facilitating collaborative knowledge building. Cognition and Instruction, 26(1), 48-94.
  • Using classroom discussion and mathematical discourse helps students refine their reasoning and justify solutions.
    • Chapin, S. H., O’Connor, C., & Anderson, N. C. (2009). Classroom Discussions: Using Math Talk to Help Students Learn, Grades K-6. Math Solutions.
  • Visible learning strategies, such as those outlined by Hattie, emphasize the importance of making student thinking visible to improve learning outcomes.
    • Hattie, J. (2008). Visible Learning: A Synthesis of Over 800 Meta-Analyses Relating to Achievement. Routledge.
  • Thinking routines normalize the practice of making thinking visible, helping students develop metacognitive strategies.
  • Observational assessments allow educators to gather student thinking in real time, improving instructional decisions.

6. Collaboration/Collaborative Pairs

Engages students in meaningful mathematical discussions and group problem-solving to strengthen understanding.

Research Support:

7. Inquiry-Based Learning

Promotes exploration, questioning, and discovery as students construct their understanding of mathematical concepts.

Research Support:

  • Inquiry-based learning fosters a deeper understanding of mathematical concepts, problem-solving skills, and student engagement.
  • Inquiry-based teaching strengthens cognitive and attitudinal dimensions, fostering habits of mathematical inquiry.
  • Inquiry-based instruction in science and mathematics classrooms promotes students’ positive attitudes toward these subjects.
  • Implementing inquiry-based learning in college mathematics has been linked to improved student outcomes, including skills, cognitive gains, and positive attitudes.
    • Laursen, S., & Hassi, M. L. (2012). Inquiry-Based Learning in College Mathematics.
  • Inquiry-based learning environments using mobile devices in math classrooms have been shown to enhance student engagement and learning outcomes.
    • Song, D., Kim, P., & Karimi, A. (2012). Inquiry-Based Learning Environment Using Mobile Devices in Math Classroom.
  • Stanford Mobile Inquiry-based Learning Environment (SMILE) integrates technology to facilitate inquiry-based learning in mathematics, promoting active student participation.

Identify the question (Step S)

Explain the meaning of what the problem is asking (Step S)

Analyze information (Step O)

Line up a plan (Step L) Use/understand multiple strategies/representations to solve (Step V)

Evaluate and reevaluate progress throughout problem solving situations

Check to see if the answer made sense (Step  E)

Check for accuracy (Step E)

Check for reasonableness (Step E)

Justify reasoning  with others

Coaching Practices

  • Introducing a cognitve pathway/SOLVE Training
  • MLR/Irs/HYS : Three Reads, Capturing Quantites, stronger and clearer, Discussion Supports,  5 practices, Distributive Summarizing
  • Operation Word Wall Creation
  • Estimation training/discussion
  • “Just In Time” Content Training (CPA)
  • Modifiying existing pathway to beef up to SOLVE
  • MTP 2:Implement tasks that promote reasoning and problem solving.
  • MTP 3: Use and connect mathematical representations.
  • MTP 4: Facilitate meaningful mathematical discourse.
  • MTP 5:Pose purposeful questions.
  • MTP 7:Support productive struggle in learning mathematics.
  • MTP 8:Elicit and use evidence of student thinking.

Create a plan to solve a problem based on the meaning of the quantities


Use context clues to solve using symbols (operation word wall)


Determine reasonableness of solution within context of problem


Justify math strategy used as most efficient method

Coaching Practices

  • MLR/Irs/HYS : Three Reads, Capturing Quantites
  • Operation Word Wall Creation
  • Estimation training/discussion
  • MTP 3: Use and connect mathematical representations
  • MTP 4: Facilitate meaningful mathematical discourse
  • MTP 5:Pose purposeful questions
  • MTP 7:Support productive struggle in learning mathematics
  • MTP 8:Elicit and use evidence of student thinking

Formulate arguments to provide evidence surrounding answers


Exchange thoughts with others to defend answers


Classify correct from flawed logic


Use discourse to clarify the approach of others

Describe relationship of facts in a problem

Represent real world situation mathematically

Apply facts to find solution to a problem

Reevaluate and redesign plan as needed when solving

Coaching Practices

  • MLR/Irs/HYS : Discussion Supports, 5 practices, Distributive Summarizing
  • Operation Word Wall Creation and Concept Based Word Wall
  • “Just In Time” Content Training (CPA)
  • Creating access points for students/Scaffolding
  • MTP 2:Implement tasks that promote reasoning and problem solving
  • MTP 3: Use and connect mathematical representations
  • MTP 4: Facilitate meaningful mathematical discourse
  • MTP 5:Pose purposeful questions
  • MTP 7:Support productive struggle in learning mathematics
  • MTP 8:Elicit and use evidence of student thinking

Demonstrate ability to choose proper tool(s)


Use tool(s) appropriately (ex. ruler)


Identify limitations of tool(s)

Use tool to guide their discovery of the concept

Coaching Practices

  • Standards unpacking for rigor components with emphasis on representations and concept development
  • “Just In Time” Content Training (CPA)
  • MTP 3: Use and connect mathematical representations
  • MTP 8: Elicit and use evidence of student thinking
  • Unpacking/planning for district provided lessons/unit
  • Identifyig things as tools that are not traditionally tools

Communicate reasoning with others

Use correct mathematical language

Apply calculations correctly

Appropriately apply correct symbol use and labeling

Coaching Practices

  • MLR/Irs/HYS : Stronger and clearer, Discussion Supports, 5 practices, Distributive Summarizing
  • Operation Word Wall Creation and concept based word wall
  • Use of Graphic Organizers
  • “Just In Time” Content Training (CPA)
  • MTP 4: Facilitate meaningful mathematical discourse
  • MTP 5:Pose purposeful questions
  • MTP 7:Support productive struggle in learning mathematics
  • MTP 6:Build procedural fluency from conceptual understanding
  • MTP 8:Elicit and use evidence of student thinking

Recognize the significance of the patterns/structures/properties of numbers and/or problem constraints


Make connections between different representations to deepen/support understanding


Deconstruct problem into easier parts to solve


Revisit problem identified (Step S) to revise plan as needed (Step L)

Coaching Practices

  • “Just In Time” Content Training (CPA)
  • Standards unpacking
  • IR Capturing Quantities, Number talks, Three Reads, 5 Practices
  • MLR – Compare and Connect
  • Problem Solving
  • Intructional Shifts – with focus on coherence – Coherence Map
  • MTP 2:Implement tasks that promote reasoning and problem solving.
  • MTP 3: Use and connect mathematical representations.
  • MTP 4: Facilitate meaningful mathematical discourse.
  • MTP 5:Pose purposeful questions.
  • MTP 7:Support productive struggle in learning mathematics.
  • MTP 8:Elicit and use evidence of student thinking.

Reference prior knowledge in learning

Develop rule/formula based on similarities and patterns

Evaluate steps of problem solving strategy throughout process (for reasonableness) 

Determine if the rule/formula works in every situation

Coaching Practices

  • MTP 2:Implement tasks that promote reasoning and problem solving.
  • MTP 3: Use and connect mathematical representations.
  • MTP 4: Facilitate meaningful mathematical discourse.
  • MTP 5:Pose purposeful questions.
  • MTP 7:Support productive struggle in learning mathematics.
  • MTP 8:Elicit and use evidence of student thinking.
  • “Just In Time” Content Training (CPA)
  • IR – 5 Practices
  • MLR: Collect and Distplay
  • Problem Solving
  • Standards Unpacking
  • Unpacking/planning for district provided lessons/unit
  • Vertical Planning
  • Coherence Map

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