I’m asked frequently in my consulting work what lessons teachers can implement for using manipulatives in their classrooms. My response is always the same regardless of the grade level. What math concept do you want to explore? Students need to engage with math in a way that makes patterns and structures visible and they do that with both physical and virtual manipulatives. Manipulatives help move students from concrete to the abstract (numbers). As teachers, we can learn a lot from observing students working with manipulatives. How are they engaging with the manipulatives? Do they understand how to use them correctly to represent a problem? Can they manipulate them in multiple ways? These things allow us to better understand where students are in the development of mathematical reasoning which allows us to adjust our instruction to better accommodate each student. We also need to be mindful of what manipulatives we choose. We want to use the right manipulatives at the right time so they can be used to encourage students to progress through the mathematical reasoning process.
Pam Harris is a consultant from TX and she has made this powerful graphic that shows the K–12 journey for our students.
Children begin their math journeys by first orally counting and then learning that those words can represent the amount of objects in a collection. By observing children as they count objects, we can see whether they are counting all the objects when they are arranged in rows, scattered, and even in a circle as well as if they are saying the number names in the correct order. Clements and Sarama of www.learningtrajectories.org have found in their research that there are 20 levels of counting! We can ask the children, once they have finished counting, how many are there to see if they know that the last number they said is the quantity there (cardinality). Then, we can move those same objects into different orientations and ask them again how many to see if they realize the same amount is still there (conservation of number). This work on cardinality and conservation of number will be important as students begin to put manipulatives on organizational structures like ten-frames and move those objects around to think flexibly and enter into the next phase of reasoning which is additive thinking.
On their way to developing basic fact fluency, students will progress through several phases from counting every object in both addends, counting on from one addend, using derived facts of ones they know to determine the ones they don’t know, and finally achieving mastery. We can tell where students are on this journey by showing an image to students on a rekenrek and asking them how many beads are there. As students explain how they know, we will see that some will count all the beads, some will start with one amount and count on the other group, and others will notice there are facts like doubles or using a benchmark of 5 and 10 that they can use to more efficiently determine how many are there. This is the phase of additive thinking that we want to facilitate. Using manipulatives such as Cuisenaire® Rods, rekenreks, and linking cubes together with ten-frames will allow our students to model the strategies they used to more efficiently arrive at their answers.
The power of this work in the basic fact fluency is that those very same strategies can be used when students are working on the content standards of their grade level. Too often our students are getting caught in the counting phase of reasoning within their work with multidigit numbers. Let’s take a moment to think about the thought processes of the students in a traditional algorithm to add 98 + 17. Students typically begin in the ones place and determine 8 + 7 yet very often students have not developed fluency yet and fall back to a comfortable place by counting on the 7 from the 8. They then do the same thinking in the tens place. So essentially, students have only thought about basic facts and are even counting to determine the sums. Yet, if they had explored their basic facts by using strategies such as breaking apart one addend to make a friendlier expression they may have thought they could rename 98 + 17 into 100 + 15 which is much easier for their brains. By making the thinking visible within the work of basic facts using manipulatives, students are able to visualize the same thinking process within the context of multidigit numbers and even decimals and fractions down the road. This is so very powerful!
Using manipulatives benefits all learners and helps students build the neural pathways in their brains to conceptually understand the mathematics content all along their journeys. By continuing their use throughout the entire K–12 math journey, we will be creating students who are engaged, curious, flexible, risk-taking, and confident mathematicians!
Ann Elise is NH certified K-8 Educator as well as NH Elementary Math Specialist. She’s an educator for twenty years in the roles of classroom teacher, K-5 Math Coach, adjunct faculty member for Plymouth State University, and currently Bureau of Education and Research presenter, contributing author to Fluency Doesn’t Just Happen with Addition and Subtraction, and an independent elementary math consultant providing training virtually. Her passion is working with educators to help them implement best practices within the three basic pillars of classroom math instruction that encourage growth mindset messages: math fact fluency, word problem structures, and understanding progressions of the standards.