LearnZillion’s K8 Math Curriculum offers many opportunities for students to model their mathematical thinking using manipulatives. A list of manipulatives or models to use can be found in the “About This Lesson” tab of each lesson. Many of the instructional videos also feature explanations that include representations of manipulatives.
LearnZillion Math Manipulative Kits for grades K5 supplement LearnZillion’s standardsaligned, taskbased math lessons. Providing access to manipulatives and models increases student engagement in the language and communication of mathematical ideas.
Classroom kits are packaged in durable plastic bins and contain materials for up to six groups of students. Individual manipulatives are also available for purchase.
Why are manipulatives important?
Manipulatives are physical or virtual objects that can be used to help students solve realworld and mathematical problems (e.g., tangrams, baseten blocks, number cubes, cards, rulers, counters, pattern blocks, cubes). Manipulatives offer multiple visual representations of ideas, helping students to engage in doing, understanding, and modeling with mathematics. Working with manipulatives deepens conceptual understanding, adds meaning to procedural skills and fluency practice, and increases student ability to apply understanding to new problems as they make sense of them and persevere in solving them.
The National Council of Teachers of Mathematics recommends the use of manipulatives for teaching mathematical concepts at all grade levels. In particular, studies have found that manipulatives are a valuable tool in the ConcreteRepresentationalAbstract (CRA) instructional sequence. This form of instruction moves students from concrete manipulatives to pictorial representations of those manipulatives, and finally to abstract concepts.
Providing access to manipulatives and models also increases student engagement in the language and communication of mathematical ideas. Students construct viable arguments and critique the reasoning of others when they have greater accessibility to the language needed to describe their thinking. Manipulatives offer English Language Learners (ELLs) and other special populations greater access to language as they afford a physical representation of a mathematical idea or solution. Terminology becomes easier to understand for ELLs and other special populations and, as the terms are used in the context of a model, increases student ability to attend to precision with vocabulary.
How does the curriculum integrate manipulatives and the CRA sequence?
Twothirds of the lessons in the curriculum are taskbased. These lessons, when accompanied by the recommended manipulatives, strongly support the CRA sequence by:
 Challenging students to engage in a problem in their own terms, using manipulatives and/or representations (drawings, models, equations) to make sense of the task and experiment with possible solution pathways.
 Eliciting class discussion that, when supported by the “Task Implementation Guide,” ensures that students compare and contrast solution pathways and, in the process, move from the concrete and representational to the abstract concept at the heart of the lesson.
Additionally, LearnZillion’s three lesson types (Conceptual Understanding, Fluency/Procedural Skill, and Application) provide students with opportunities to engage in an extended CRA sequence several times over the course of the year. The curriculum supports students as they first develop a concrete understanding of the Key Concept, better equipping them to understand those mathematical concepts at the abstract level. For example:
 In Conceptual Understanding lessons, students have multiple opportunities to engage in conceptualizing a Key Concept, including using physical materials, representing their thinking through models and drawings, and applying their learning to new and more abstract situations.
 In Procedural Skill/Fluency lessons, the mathematical concept is formalized and modeled in a more abstract level, using numbers and mathematical symbols as well as models and drawing representations. Students then have ample opportunity to deepen their understanding through practice.
 In Application lessons, the mathematical concept is once again modeled and students continue to build their understanding of the mathematics, representing it more often using abstract representations such as expressions and equations. Students then extend their understanding by applying it to a new realworld or mathematical situation.
Teachers play an important role in helping students understand the concepts that manipulatives represent. When teachers are able to represent math concepts using manipulatives themselves, they are more likely to use manipulatives intentionally during instruction and provide feedback to students as they make connections between representations and concepts. In many lessons, our videos play the role of modeling concepts with manipulatives or other representations after students have had the opportunity to work out the task. This helps the students link concrete materials to abstract concepts. As students represent problems using models, they also develop more structured strategies for solving problems. For example, we recommend that teachers make a variety of manipulatives available to students so that they can independently select the representation that they think works best as they're working through the task. This supports productive struggle and SMP.5 (use appropriate tools strategically).
References
Fogelberg, Ellen, Carole Skalinder, Patti Satz, Barbara Hiller, Lisa Bernstein, and Sandra Vitantonio. 2008. Integrating Language and Math: Strategies for K–6 Teachers. New York: The Guilford Press.
Hiebert, James. The struggle to link written symbols with understandings: An update. Arithmetic Teacher, vol.36, no.7
Hudson, Pamela, and Susan P. Miller. 2006. Designing and Implementing Mathematics Instruction for Students with Diverse Learning Needs. New York: Pearson.
Humbert, Katie, and Vicki M. Samelson. 2010. “How First Graders with Low Language Skill Solve Math Word Problems.” Communication Connection: Newsletter of the Wisconsin SpeechLanguageAudiology Association. Kimberly, WI: Wisconsin SpeechLanguage Pathology and Audiology Association.
Kosko, Karl, and Jesse L. M. Wilkins. 2010. “Mathematical Communication and Its Relation to the Frequency of Manipulative Use.” International Electronic Journal of Mathematics Education5 (2): 79–90. http://www.iejme.com
Morin, Joseph E., and David J. Franks. 2009. “Why Do Some Children Have Difficulty Learning Mathematics? Looking at Language for Answers.” Preventing School Failure 54 (2):111–18. Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics: An International Journal, 47(2), 175–197.
Moyer, P. S., Niezgoda, D., & Stanley, J. (2005). Young children’s use of virtual manipulatives and other forms of mathematical representations. In W. J. Masalaski & P. C. Elliott (Eds.), Technologysupported mathematics learning environments (pp. 17–34). Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.
Steen, K., Brooks, D., & Lyon, T. (2006). The impact of virtual manipulatives on first grade geometry instruction and learning. Journal of Computers in Mathematics and Science Teaching, 25(4), 373–391.
Swan, Paul, and Linda Marshall. 2010. “Revisiting Mathematics Manipulative Materials.”Australian Primary Mathematics Classroom 15 (2): 13–19.

LearnZillion Math Manipulative Kits for grades K5 supplement LearnZillion’s standardsaligned, taskbased math lessons. Providing access to manipulatives and models increases student engagement in the language and communication of mathematical ideas. ...