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4 Rules & 3 Dimensions for Achievement in Supplemental Mathematics

Schools are under pressure to lift educational performance and graduation outcomes for all students. For students who are struggling with mathematics at the elementary and early secondary grade levels and need something more in addition to core instruction, providing enrichment, remediation, and intensive intervention in the curriculum is critical to driving higher outcomes.

Rules for Achievement

Most importantly, research posits that supplemental math intervention resources which provide a variety of multi-modal resources can best provide differentiated instruction. They need to accommodate various learning and expression styles to support the needs of a diverse student population. Next, supplemental intervention resources should easily integrate with various classroom implementation models, including:

  • Special Education and Pull-Out Intervention
  • Inclusion Classrooms
  • Blending & Personalized Learning
  • Math Workshop
  • Traditional Differentiation
  • Homework, After-School, & Summer School

These programs should align to regular instructional models, and enable teacher-led, independent learning, and engaging small-group activities. More specifically, they need to provide:

  • Activities that can be completed in whole group, small group or individually
  • An appropriate mix of teacher-guided instruction, assignable student-driven lessons and games
  • A balance between tactile, online and print-based learning moments
  • Opportunities for students to make real world connections and deepen their understanding

Finally, supplemental intervention resources should be grounded in the science of learning, based on extensive research and trials.

Three added dimensions complement core instruction

Edtech authorities agree that to be effective at a basic level, supplemental intervention resources should empower teachers to deliver at a minimum, these three capabilities:

1. Interactive, multi-modal learning experiences

2. Hands-on learning experiences

3. Game-based learning experiences

Interactive, Multimodal Learning Experiences. A multimodal learning environment uses two modes — verbal and non-verbal — to present information to students along with graphic representations that are directly relevant to the concept/skill being taught. Multimodal supports can also take the form of gesture, physical manipulatives, virtual manipulatives, or kinesthetic activities. According to university researchers Moreno and Duran, the most effective learning environments combine verbal and non-verbal representations of knowledge, and use a combination of visual and auditory inputs.

Hands-on learning experiences. Students need opportunities for hands-on, tactile learning experiences in the math classroom. Educators can use manipulatives and concrete examples to develop each student’s understanding across a continuum of concrete, representational and abstract reasoning skills. Using manipulatives and tasks that feel familiar and applicable to the students’ world outside of the classroom can also help learners connect disparate concepts, especially those that are more abstract. This holds true for all learners, but is particularly important for struggling learners and striving students who have low self-efficacy or anxiety with math.

Game-based learning experiences. Research shows that games can help all learners, regardless of ability. However, games can be especially beneficial for struggling learners. This is because some of the task-related anxiety of mathematics appears to be diminished when games are incorporated into instruction, as when supplemental resources are applied to complement regular math instruction. Games provide students with opportunities to drive their own learning, develop problem solving skills and receive immediate feedback. They also encourage productive struggle and persistence in the face of temporary setbacks, according to the NCTM. As researchers continue to explore the complexities and specific design features that optimize game-based learning, the most accepted way to implement games is in combination with other more traditional and research-based approaches.

Explore how Arrive Math Core Booster incorporates the research and best practices in supplemental mathematics discussed above.


  1. Moreno, R., & Mayer, R. (2007). Interactive multimodal learning environments. Educational Psychology Review, 19(3), 309–326.
  2. Moreno, R., & Durán, R. (2004). Do multiple representations need explanations? The role of verbal guidance and individual differences in multimedia mathematics learning. Journal of Educational Psychology, 96(3), 492.
  3. National Council of Teachers of Mathematics (NCTM). (2007). Effective Strategies for Teaching Students with Difficulties (Research Brief). Retrieved from
  4. Gundogdu, M. (2013, August). The impact of manipulatives on middle school special ED students’ learning integers (M.A. Thesis). California State University, Long Beach. Retrieved from
  5. Moore, S. D. (2014). Why teach mathematics with manipulatives? (ETA Hand2Mind Research Summary). ETA Hand2Mind. Retrieved from
  6. Ernest, P. (1986). Games. A rationale for their use in the teaching of mathematics in school. Mathematics in School, 15(1), 2–5.



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