Manipulatives help students see and touch the math, which can help make connections to abstract algorithms and formulas. My colleague Jamie and I constantly discuss the idea of students using concrete manipulatives to help them get a conceptual understanding of the math. Research has shown us that students who use manipulatives in their math classroom usually outperform those who do not. But as Douglas H. Clements said, “Manipulatives do not guarantee success.” It’s the students’ thinking about the manipulative that matters most.
When I plan to teach a new math concept to students, Jamie and I often discuss what the best manipulatives will be to help students make connections and notice patterns and relationships with that particular math concept. We always want to use concrete, representational and abstract thinking in our lessons (CRA: How They are Connected and Why it Matters). When we think about “concrete,” we know that not all manipulatives are created equal, so it’s important to know our students and where they are in their mathematical reasoning.
And not all manipulatives are concrete.
Many teachers would define concrete manipulatives as objects that can be touched or moved to help students understand the math. But this definition does not consider which manipulatives help students with concrete thinking. Some manipulatives are more concrete than other manipulatives.
Let’s consider these manipulatives. All of the following manipulatives are great to help a student further their mathematical thinking, but it’s important to realize that we may need to start with a more concrete-thinking manipulative with students.
Clocks It’s great to have students use individual clocks to help them understand time, but are clocks really concrete? Students can hold and manipulate clocks, but they are pretty abstract. Before students manipulate with clocks, they should work with counters or blocks and string to understand that a clock actually consists of two number lines.
Base ten blocks Yes, students can build a number with these manipulatives. But if a student cannot unitize (combine parts to make a unit), then do they understand the full stick means one ten? Probably not. Students making a trade without understanding that the ten blocks glued together is the same quantity as 10 single units broken apart will actually be more confused by the use of base ten blocks. We should use unifix cubes first and have students build a unit of five and a unit ten.
Fraction strips When teachers first start teaching fractions, they may give students premade fraction strips or fraction bars. Or teachers may walk students through creating fraction strips, but if they simply tell students how to cut and create them, the students will use the manipulatives in a rote or procedural manner. Instead, students should start with their own concrete drawings to understand how to create equal parts of a whole.
Place Value Disks These manipulatives are great for students to use to build numbers, but only after a student fully understands the magnitude of place value. Place value disks do not show magnitude because they are all the same size.
Money Students need to understand what the coins represent, but like place value disks, this manipulative doesn’t show magnitude. One way to help students see the value of the coins is to start with using the coins with unifix cubes.
There is a continuum of manipulatives, and some manipulatives are more concrete than others. Before deciding which manipulative to use, consider the continuum of concrete thinking and decide which would be best for each of your students. Consider differentiating the use of manipulatives as one way to differentiate your instruction.
An example of the Continuum of Concrete Thinking with Manipulatives for Teaching Place Value:
Building Concrete-Representational-Abstract components into our math lessons is essential. When planning which concrete manipulatives to use in our lesson, we need to first determine where our students are in their thinking. Then we can choose manipulatives that will help our students connect their concrete thinking to the abstract math.
