The “why” is easy! Years of research indicate students gain deeper conceptual understanding when they have teachers who engage in specific instructional practices. The recommendations form NCTM’s Principles and Standards for School Mathematics suggest that teachers employ 8 specific instructional practices. They include:
A Problem Solving approach
Children are Active Learners in the classroom
The use of Concrete materials
Cooperative work in small groups
Discussion of ideas
Justification of thinking
Writing in class lessons
The common core state standards were born out of this type of research and the 8 mathematical practices were framed around these ideas and the Strands of Mathematical Proficiency from the National Research Council.
The “How” of using math takes more time.
During our day together we explored the use of concrete, pictorial, and web based tools in the classroom. It is essential that teachers immerse themselves in the use of manipulatives to solve problems in order to understand their value. Typically, the teachers I work with did not have much experience with manipulatives as students or in their pre-service methods classes. This should be an element of preparation for any manipulative – based lesson. Teacher need to be sure students will use the manipulative to deepen understanding rather than merely mimicking a rote procedure they have learned previously. Teachers also need to be organized with their objectives for the lesson, have materials prepared, and questions ready to guide the learning experience.
We also discussed helping students move from the concrete tool, through the pictorial representation, to the abstract form of mathematics using numbers and symbols which is what the 4th CC Mathematical Practice, modeling with mathematics, requires.
Suggestions we discussed included:
Encouraging the use of a variety of strategies at all times. Children should share thinking with the teacher, with a partner, in small and whole group allows these strategies to
Set aside time to do mental math activities and have students share their strategies. Let children know that working “in their heads” is valued. Start with simple problems without access to manipulatives, or paper and pencil. Encourage “individual think time” to allow everyone to process at their own pace, before sharing answers and strategies.
Choosing prompts and questions carefully. When children struggle, instead of prompting them to use a manipulative, ask them what they know about the problem, what is important in the problem, and what the numbers in the problem represent. When students have trouble explaining how they have solved a problem ask questions like, “What number did you start with?” or “What did you do next”. When students solve with manipulatives, ask them to explain what they did mentally without recreating the actions with the manipulatives.
Create a safe environment and a spirit of risk taking in your math classroom. Help students understand that mistakes are not fatal but opportunities to learn. Examine student work where students have made mistakes (remember to remove names or even rewrite the work) to show that mistakes can teach us.
Encourage students to challenge themselves by using another method to solve the same problem. This encourages flexible thinking and gives them something else to try if a favored strategy doesn’t work in a particular situation.
Connect the concrete to the abstract. When students have had lots of practice with a manipulative, help them see where their use of the tool connects to the abstract method. Use appropriate questioning to elicit some of the algorithm from the students whenever possible.
The “Why” and “How” of using math tools and pictorial representations in our classrooms is clear…..for the sake of our children, I encourage you to learn more and try it!