The math content in grades 3–5 forms a bridge between the additive thinking of K–2 and the proportional reasoning typically begun in grade 6. Very often our students are caught in a previous phase of reasoning within the content of the grade-level standards. For example, they are counting when working on the math facts within an addition or subtraction algorithm. Or they are skip counting which is additive thinking within the content of multidigit multiplication problems. Both physical and virtual manipulatives have the power to help students see the structure of problem situations as well as visualize the number relationships as they work toward developing strategic thinking all while building number sense. Many students are unintentionally locked out of learning math because they are unable to follow the procedures and rules. Using manipulatives, however, will help us make the math accessible for all learners. All students have visual parts to their brain that can be developed as we explore concepts concretely, pictorially, and then connect those representations 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.
As students enter 3rd grade, they have had exposure to thinking additively with both addition and subtraction and can compute sums and differences in flexible ways. For example, when asked 98 + 54, some students may think about breaking apart the 54 into 2 and 52 so that the 2 can join the 98 to make 100. So, the 98 + 54 can be renamed as 100 + 52 which is so much easier for our brain. For subtraction, when asked 92 – 88, rather than students removing the 88 from the 92, they may choose to use addition to solve subtraction and realize that the distance between 88 and 92 is 4. This same thinking can apply with the multidigit addition and subtraction work of grade 3-5 as well.
Grade 3 is when students are introduced to the operations of multiplication and division. They explore equal group representations as well as arrays. Once again, the key is to explore these problem types with manipulatives that allow students to see the underlying structure of the story situations. Rather than looking for keywords which tend to lead students astray, they will visualize the structure of the story problems and then have the flexibility to calculate the answers in ways that make sense for their brains. The heart of multiplication is the distributive property of multiplication where we can break our factors apart and find partial products in our calculations of products. Math tools like Cuisenaire® Rods and Rekenreks provide the visuals needed to develop a conceptual understanding of these methods which can then be used with larger numbers, decimals, fractions, and polynomials way down the road.
On their way to developing basic fact fluency with multiplication, students will progress through several phases from counting each object in each equal group, skip counting multiples, using derived facts of ones they know to determine the ones they don’t know, and finally achieving mastery with understanding. 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, others will skip count the amounts on each row, and others will notice that when there are 4 rows, they could use their doubles to determine the product. Not only will this help students with their x4 math facts, but they will be learning that they can now multiply anything by 4. This is the phase of multiplicative reasoning that we want to facilitate. Using manipulatives such as Cuisenaire® Rods and Rekenreks will allow our students to model the strategies they used to more efficiently arrive at their answers and build their own flexibility with the numbers.
In addition to the work on numbers and operations with whole numbers, in grades 3–5 students are introduced to fractions and decimals. Once again students will need manipulatives to concretely see the quantities and understand the operations with them. Fraction tiles and Cuisenaire® Rods are powerful for allowing students to see that fractions are composed of unit fractions. Having the fractions on the fraction tiles with the fractional amounts printed on them provides scaffolded support to some learners, but I would encourage you to either purchase the tiles without the numbers or to turn them upside down and have students reason their way through how much the tiles are worth. This also allows you the freedom to change the size of the whole. Students must always be thinking about equal sized pieces and how many of those pieces they have. These understandings can then be carried forward as they compute with fractions and decimals. It seems counterintuitive that multiplying a number by a fraction would end up with a product less than the other factor, yet by using manipulatives students will develop their understanding of the why.
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.