Make math problematic

As we think about how to teach students mathematics, we should reflect upon the following questions: What is the purpose of teaching mathematics? Is it to have children memorize how to do something? Or is it to teach math so students can understand the concepts well enough to be able to apply them to other situations? Hopefully, you think the latter. Teaching math means we help students to understand the math they are learning, if there is no understanding going on, are they really learning? If a child can recite their multiplication facts, what does this tell us? Nothing, only they have a good memory. We need to teach for understanding and not merely for knowing how to do something. To teach for understanding, we need to create classrooms that promote this type of learning. Borgioli (2008) state that “in classrooms that promote teaching and learning mathematics with understanding (instead of memorizing abstract algorithms and mimicking teachers’ solution paths), students collaboratively engage in making sense of mathematical tasks in their own ways and communicate their thinking with one another.” (p.186) She then goes on to say that one of the three critical pieces to create this type of classroom is in the actual tasks that teachers pose or give to students. Borgiolo (2008) suggest that “the tasks be complex and problematic for the learner and involve meaningful problem solving as opposed to simple drill and practice exercises” (p.186).
Van de Walle and Lovin (2002) describes it best as they state “the single most important principle for improving the teaching of mathematics is to allow the subject of mathematics to be problematic for students. That is, students solve problems not to apply mathematics but to learn new mathematics. When students engage in well-chosen problem-based tasks and focus on the solution methods, what results is new a understanding of the mathematics embedded in the task. When students are actively looking for relationships, analyzing patterns, finding out which methods work and which don’t, justifying results, or evaluating and challenging the thoughts of others, they are necessarily and optimally engaging in reflective thought about the ideas involved”. (p.11)
Boaler (2002) states that “teachers who teach math through a problem based approach, believe that the open-ended approach they used was valuable for all students and that it was their job to make the work equitable accessible” (p.60). Students need to be able to have access to the problem and for them to enjoy success. Boaler (2002) states that students, who were taught in a problem based approach, had very different ideas about math, than those that learned math in the traditional way. Students who were taught through problem solving thought math was active, and “did not regard math to be a rule-bound subject involving set methods and procedures that they needed to learn” (p.77) All students need to think this way, so they can begin to apply what they know, and use this to construct new mathematical knowledge.
Engaging students in rich, contextual problems gives them something to talk about. These discussions can be prompted and guided by appropriate teacher questioning. One can argue that providing students with these worthwhile problems is the most important piece of this puzzle, and they are probably right. If students do not have something to talk about, they won’t. If there is no talking, then no type of questioning would spark their interest to have mathematical discussions. No talking, no learning, simple as that. So how is it done? How are these types of tasks created? What are the elements of these types of tasks that are open ended, having multiple entry points, so all students can experience success?