Manipulative Activities

How do you know students deepen their understanding while using manipulatives?

It is not easy for some students to move from contextualized mathematical problems to the realm of abstract symbols. Manipulatives offer a bridge between real-life situations and decontextualized algorithms, allowing students to experience kinaesthetically and visually what they might not otherwise understand with symbols in isolation. Manipulatives help turn rote procedures into meaningful steps toward a solution, making the process and the outcome of equal importance. With this depth of understanding, students are more likely to persevere at attempting to solve unfamiliar problems, even if the solution is not evident to them at the outset. 

How do you know if students can transfer their understanding from manipulatives to other situations?

Once students have develop a deeper understanding of a mathematical concept, they will be able to explain not only how they come up with an answer, but why. This ability to defend their understanding and to critique the understanding of others provides teachers with the necessary evidence to support the conclusion that students are able to transfer their knowledge to other situations.  

How can you assess that understanding or growth?

Students may gradually move away from using manipulatives to solve problems once a concept is more deeply understood. Context becomes less necessary and is, instead, used more as a means of confirming that the results of an algorithm make sense. Students may also develop a more sophisticated means of using the manipulatives. For instance, attribute blocks might be sorted by new criteria, such as the number of straight edges or right angles. Students' ability to estimate and perform mental math may also show improvement. 

When students work in groups, how do you hold each youngster accountable for learning?

Individual accountability can be held by having students express their reasoning either orally or in writing. They should be asked to not only supply their groups' answer, but also the rationale behind how the answer was obtained. 

When students work in groups, how do you assess each youngster's depth of understanding?

A class of students will likely contain multiple levels of understanding for any given activity. The teacher must be prepared to challenge each student to the best of their abilities. This requires differentiated instruction and assessment. If a student is developmentally  capable of more advanced mathematical understanding, he or she must be challenged to achieve it. That said, a teacher's choice of assessment must be sensitive enough to detect incremental change in the understanding of less advanced students. 

How are you improving students' problem solving skills with the manipulatives? 

By encouraging the use of manipulatives in mathematical problems solving, we are offering students yet another means of coming to understand and solve problems. A student who has a deep understanding of how and when to use the tools at their disposal will likely be more persistent at solving problems. Manipulatives also allow students to move from the abstract, to the concrete, and back again, allowing students to double check their use of algorithms, detect structure, and recognize repeated reasoning in the process of finding a solution. Finally, manipulatives offer yet another means of mathematical communication, which helps teams of individuals work toward a common mathematical goal.