Unproductive Beliefs

I recently attended "Mathematics in the Mountains" in Flagstaff Arizona. The conference was being held for the first time and organized by Arizona Association of Teachers of Mathematics. It was held on the campus of Northern Arizona University. It was also a great time to be away from the valley as it was the first 100+ degree day of the season. Many more to come very soon.

One of the sessions I attended was "Principles to Action: Ensuring Mathematical Success for All, NCTM's New Blueprint for School Mathematics" led by Jane Gaun of the Flagstaff Unified School District. All day I was looking forward to attending this session. I had previously watched the video of Steven Leinwand talk on the "Principles to Actions" at NCTM. When I saw that there was going to be a session based on the "Principles to Action" I was excited. Especially after seeing the passion with which Mr. Leinwand spoke of the book.

During the session, the presenter asked us to complete an activity. We were given a set of cards with a short phrase on it. We were then asked to sort the cards into two categories - those that are productive beliefs and those that are unproductive. We were to complete the task as a group and discuss any we disagreed on. It led to some rich conversation. It caused me to think about some of the common beliefs that are roadblocks to allowing all students access to high quality, math education. I hope that by writing some of these statements below a conversation can begin about these beliefs and how we can start hearing and seeing the productive beliefs more and the unproductive beliefs less. The statements are below and are in no particular order. Please comment on which ones you think are productive and which are unproductive with a rationale.

Let the conversation begin! Consider putting your response on Twitter using #PrinciplestoAction.

1. The role of the student is to be actively involved in making sense of mathematics tasks by using varied strategies and representations, justifying solutions, making connections to prior knowledge or familiar contexts an experiences, and considering the reasoning of others.
2. Students can learn to apply mathematics only after they have mastered the basic skills.
3. Mathematics learning should focus on developing understanding of concepts and procedures through problem solving, reasoning, and discourse.
4. The role of the teacher is to tell students exactly what definitions, formulas, and rules they should know and demonstrate how to use this information to solve mathematics problems.
5. All students need to have a range of strategies and approaches from which to choose in solving problems, including, but not limited to, general methods, standard algorithms, and procedures.
6. Students can learn mathematics through exploring and solving contextual and mathematical problems.
7. Mathematics learning should focus on practicing procedures and memorizing basic number combinations.
8. The role of the teacher is to engage students in tasks that promote reasoning and problem solving and facilitate discourse that moves students toward shared understanding of mathematics.
9. An effective teacher provides students with appropriate challenge, encourages perseverance in solving problems, and supports productive struggle in learning mathematics.