I have noticed that in many classrooms, students in mathematics seem to want to know the "steps". I think I know why they jump to this automatically in that they want to get done and get the "right" answer but I'm not certain why this exists. I know math can be a creative endeavor in many ways. I wonder if people had more creative experiences with mathematical concepts and ideas if there wouldn't be quite as much math anxiety and people almost bragging that they are "bad at math". I wonder if we focused on opening up the processing of math where students could explore mathematical concepts in their own way, using their own processes, if more students would enjoy and understand math.
I reflected on a class a couple years ago where we presented an open-ended problem involving a proportional relationship to students and asked them to come to a solution using their own process. Students were at first resistant since we didn't tell them how to do this. We persisted in asking them what they did know about the problem and to think about how they might show their solution. Students worked on their own white boards and then "presented" their solutions to their peers. The students shared pictures, tables, graphs, etc. and all had a little different perspective on how the relationship worked and how they would explain it to another person. With just a little encouragement from us, valuing all ideas that were brought out in that class period, they explored different ways of thinking on many problems throughout that school year. This worked well!
What I wished would have gone a little better is to push this thinking even further..... we can use pictures and tables and graphs but how might we use concrete objects, manipulatives or ANY other concrete object, to explain mathematical concepts in ways that make sense to the individual. Also, how might we use technology more effectively to allow students access to exploration of math. I also want to explore fostering a culture of creativity in mathematics so that students are free to tinker with different ways of understanding mathematical ideas in order to connect these ideas with their own learning styles and ways of thinking.