In his book, "Solving for Why," education professor John Trapper explains that students gain a deeper understanding of mathematical problems if they are asked to work on problems in three different ways: concretely, through the use of physical items such as place value blocks, pattern tiles and measuring tools; representationally, by creating models with drawings, charts, diagrams and graphs; and abstractly, with traditional number equations. When students understand mathematical processes on a fundamental level, they will be more prone to seeking creative ways to approaching future mathematical problems.
From multiplication tables to trigonometric equations, memorization can be useful. However when students are presented with manipulatives, they are able to learn through discovery and gain a deeper understanding. For example, with twelve beans and an egg carton, students can be instructed to place an equal number of beans in any compartment. A typical question might be: how many combinations can you make by putting one bean in each compartment or two beans each in six compartments? With a balance, wooden blocks and a paper bag, students can learn how to write algebraic equations.
At the representational level, students show how they completed the problem. With the egg carton multiplication and division problem, students can draw pictures of their cartons or use dots and dashes to keep count. They can use numbers in lists or charts -- 1 in 12, 2 in 6, 3 in 4 and so on -- or write out the solution with words. At the abstract level, mathematical equations are derived. For the egg carton activity, students show that 1 x 12 = 12, 2 x 6 = 12, 3 x 4 = 12 and so on. Or, students can use blocks, including those hidden in a bag, to invent ways to solve for x. With the blocks, students can show that x + 3 = 5 or x + 4 = 6 and eventually solve for x.
Throughout each step, encourage students to share ideas. Keep a running tally to challenge them to keep coming up with new mathematical techniques. In communicating with peers, students clarify their thinking and get feedback. In one technique, called think-pair-share, students first work independently -- think -- with a partner -- pair -- and then with the larger group -- share. Your students may be accustomed to being told exactly how to solve problems and being rewarded for their precision in following directions. Always follow through with assessments that reward the mathematical reasoning process, not just the right answer.