Suggestions for Integration of Manipulatives
Manipulatives are a helpful tactile tool for supporting students in strengthening their conceptual understanding of concepts and for supporting students in making sense of the concepts and skills (Boaler, 2016; NCTM, 2014). Use of manipulatives also relates to the Standards for Mathematical Practice (http://www.corestandards.org/Math/Practice/). For manipulatives to be an effective part of instruction, the integration of the manipulative must be intentional and explicitly linked to the goal and content of the lesson. There must be a specific and clear instructional purpose and goal for the manipulative that is communicated to students. The manipulative must be explicitly linked to the content being learned. If multiple manipulatives are being used for one concept, there must be explicit connections made among the representations to strengthen student understanding and flexibility with applying concepts (NCTM, 2014).
As an example of using multiple manipulatives to strengthen number sense, consider the introduction of counting and building the concept through connections to counting by tens and hundreds (this flow may occur across one or two grades and not necessarily occur all within one grade level). Counters and linking cubes are particularly helpful for counting and the concept of a one to one correspondence with counting, including counting all, counting on, and subitizing. The items can then be grouped into tens, introducing the idea of counting by tens. This activity can be done with a tens frame and linking cubes (as an example). Counters can then be introduced (if not previously introduced) as another tool for counting in place of the linking cubes. From making tens and counting by tens, the progression is to then make a hundred and count by hundreds. As part of this transition, the ten frame can be turned on its side and transitioned to a place value mat. The place value mat can initially be used (for lower numbers) with counters or linking cubes, with the eventual transition to place value blocks (or place value disks). Rekenreks can also be introduced as part of the transition and flow of these concepts to further strengthen the connection among concepts and tools. The point of this example is to be explicit with the manipulatives used, consider the flow and sequencing of the manipulatives used, and be explicit with connecting one concept to the next.
When using manipulatives, build in time for students to explore and play with the tool. This time allotment is particularly important if the manipulative is new to students. Students are curious and want to see what they can do with the new item. Emphasize that it is not a toy but a tool; however, it is essential to give them time to play with it. I find students are more focused during the activity if they have a few minutes at the beginning to explore the manipulative on their own. I find this to be helpful regardless of age of the students.
Guidelines should also be provided to students for proper use of manipulatives. It is often helpful to co-create these guidelines with students as students are frequently more cooperative with guidelines they helped to create. The guidelines should include how to properly use the tool, the importance of being focused on the lesson objective, a consequence for not following the guidelines, and possibly an incentive, such as, if the lesson and activity are completed with time remaining, students will have an opportunity to explore the tool independently or with their peers. If the consequence for not following the guidelines is removal of the manipulative, be sure to replace it with an alternative activity that achieves the same instructional objective. Use of the manipulative is about the lesson and the learning, not the manipulative itself, and removal of the manipulative should not result in loss of learning opportunity for the student.
Sometimes there are concerns when using manipulatives with pieces on the floor, students losing pieces, noise with some tools, etc. I've seen teachers give a piece of felt fabric to each student (like a 9 x 12 rectangle) and students are expected to keep the pieces on the felt. I've also seen teachers give each student a shoe box top that can be used as the work mat for their manipulative.
Teacher modeling is additionally important for manipulatives. There are times when it is appropriate to have the activity be open-ended, but there are times when the instruction is explicit and use of the manipulative may need to be modeled to the students. For example, when using base ten blocks to teach addition or subtraction with re-grouping, I find it helpful to do the activity with students, meaning I use a document camera or other device to demonstrate use of the manipulative to support the re-grouping as part of a collective class discussion about re-grouping. I often have students tell me how they think the manipulative should be re-grouped, I model it for everyone to see, and make adjustments based on student input. Students are doing the same individually as part of the collective learning.
Some Favorite Manipulatives
I love using manipulatives and I always look for different ways to integrate them as part of connecting conceptual understanding across math content. You will see that I use a few of the same manipulatives across different math concepts. Many of these tools are adaptable to different contexts and understanding. Following is a list of suggestions by concept (list is not all-inclusive). Many of these manipulatives can be found in both virtual and tactile forms. Teacher discretion is best when determining which version of the manipulative will best support student learning in the classroom.
Number Sense --> Place value, counting, patterns --> manipulatives can support building the conceptual foundation and flexible thinking, including decomposing and recomposing numbers and quantities, patterns, connections to hundreds charts and number bonds
Linking cubes Ten frames
Base ten blocks Counters
Dominoes Pattern blocks
Decimal operations grids Rekenreks
Dice Place value mats may be helpful with manipulatives
Computation --> Operations with whole numbers, decimals, and integers --> manipulatives can strengthen understanding of grounding computation in place value; can also be used to support connections to number bonds and strengthening number sense through computation
Base ten blocks Counters
Dominoes Dice
Decimal operations grids Linking cubes
Place value mats may be helpful with manipulatives, particularly for re-grouping concepts.
Fractions, Ratios, & Proportions --> Use of manipulatives to introduce, build, and strengthen student understanding of fraction concepts is critical as fractions present to students a different to think about numbers (in relation to whole numbers); includes computation with fractions
Linking cubes Pattern blocks
Dominoes Fractions bars
Fraction circles Dice
Measuring tapes
Geometry & Measurement --> Use of manipulatives can support logical thinking, relationship of shapes and pieces, connections to vocabulary, and understanding two-dimensional and three-dimensional shapes
Linking cubes Pattern blocks
Tangrams Geostix
Measuring tapes Rulers
Yardsticks Meter sticks
Probability --> Manipulatives can strengthen student understanding of concepts that may seem abstract
Linking cubes Pattern blocks
Dominoes Dice
Probability spinners
References & Recommended Resources
Bay-Williams, J., & Kling, G. (2019). Math fact fluency: 60+ games and assessment tools to support learning and retention. Alexandria, VA: ASCD.
Boaler, J. (2016). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. San Francisco, CA: Jossey-Bass.
Diller, D. Math work stations: Independent learning you can count on, K-2. Portland, ME: Stenhouse Publishers.
Kallick, B., & Zmuda, A. (2017). Students at the center: Personalized learning with habits of mind. Alexandria, VA: ASCD.
Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM.
Taylor-Cox, J. (2016). Math intervention: Building number power with formative assessments, differentiation, and games - grades PreK-2. New York, NY: Routledge.
Taylor-Cox, J. (2016). Math intervention: Building number power with formative assessments, differentiation, and games - grades 3-5. New York, NY: Routledge.