THE CHALLENGE (Should You Choose to Accept It)  So, how does one create a situation in which children learn the meaning and learning of math? Even more important, how does one make it interesting and engaging to children? In “Teaching for Understanding: Guiding Principles,” I pick up three principles that I see as especially important in reaching these goals. First, not all students will get the same thing out of the same experience. Second, ideas are grasped partially at first, and a period of confusion in a natural part of the process. Third, students need to learn persistence in solving problems.   These ideas create the main frustration I have with teaching: assessment. How can I fairly test the entire class while knowing that not everyone will have picked up the ideas yet, and knowing that being chastised in the form of grades can eventually deter kids from pursuing understanding? How can I create a feeling of success for the kids while still honestly rating their progress compared to their class and their grade level?

 One way that I use models with kids is with proving answers. I think my fifth graders are sick of me saying, “Prove it!” to them, but they are learning how to create proof of what they feel is the correct answer. I think this is extremely important, for many reasons. They learn to rely on themselves and trust their answers. Also, they often discover that their original prediction of how the problem would work out is wrong, but in attempting to prove it, they found the path to the right answer. It is also an important life skill to know how to improvise in many situations, and models are more flexible to situations than rote questions and answers are.

 Next, the students move through the processes by considering the real world (“What do I see in front of me?”), to representation (“How can I model this on paper?”), to abstract reasoning (“How can I say this in words?”), and then back to making formulas. One experience I had with this was a one-on-one with a student that had fallen behind. The class had been using definitions they wrote in their math journals for the last few weeks to solve problems dealing with range, mode, mean, and median. This boy had gotten bored during the first introduction (which was the teacher reading each definition and rather quickly explaining them with no demonstration) and not written all of the definitions, and there fore missed every five-a-day question on the topic. I finally sat down with him and, by going through each of the five steps, retaught the lesson. We drew models, talked about things we had seen that were similar, wrote new definitions for each word, and practiced with a set of numbers. At the end, he said, “That’s all I was supposed to be doing? This is easy.” By taking the time to go through each step of the process, a lasting impression of the concepts is created. This is, after all, the reason for teaching, right? I plan to introduce ideas in the way that I worked with the student, hopefully to create more understanding immediately, and less misunderstanding later.