Teachers sometimes perceive students as trying to "trip up" the teacher with questions the teacher cannot answer. Mathematics students need to see what it looks like to do real mathematics. Teachers must be willing to give students these types of apprenticeship experiences. This work should be accompanied by the questions you ask yourself each time a next step is encountered (e.g., "Is it time to factor and why?" "Would a graph of this equation help?" "Should I add any segments to this diagram?"). Try a method and, if it fails, backtrack and start new pathways. Outline your intuitions about what the answer might be and how a solution might be reached. When students ask a question that you cannot answer immediately, try working on it in front of the class. Students will become more comfortable "talking to themselves" if they see the teacher doing likewise. I encourage them to take over that job themselves and remind them that uncertainty should not be allowed to lead to paralysis. They thought I was being helpful, yet I point out that all I did was prompt them to continue. I repeat my question, and they do one more step until the problem is solved. I ask them, "OK, what do you do next?" They do one more step and, once more, stop. After showing me the first step, they would freeze. Consider the following teacher report: On numerous occasions, students have come for extra help because they were stuck with a multi-step problem. A lack of confidence plays a role in this immobility, and sometimes the needed level of internal dialogue is strikingly simple. Students often become stuck, not because of any impediment, but because they have stopped moving forward. Encourage them to set up their own internal dialogue in which they continually ask themselves, "What do I know?", "What do I need to know?", and "What techniques do I have for bridging the gap?" As you ask them these questions, also make explicit what you are doing and why: You are asking them the questions you would ask yourself in a comparable position. Encouraging them to identify the cause of their "stuckness" (e.g., "I have too many variables," "I dont see any pattern in this sequence") is frequently all they need in order to focus on, and resolve, their difficulty. Often, students stop in the middle of a task but do not try to characterize what has occurred that has stopped them in their efforts. First, you might ask if they can clearly state what they are seeking to determine, or if they can figure out why they are stuck at that stage in the process. When students ask questions, the usual response should also be a question. The challenge, when stuck, is determining why one is stuck. Teachers and mentors need to help students understand that there are many specific ways to be stuck, and that, for each barrier, there are associated methods for becoming unstuck. They are stuck because they believe that being stuck is an amorphous and hopeless situation. The most common form of "stuck-ness" faced by students stems from their failure to identify the obstacle to their progress. Progress can also halt when we are unable to determine what knowledge would be useful to apply at a particular point. One way to get stuck is to ask questions that are beyond our background to solve or are not entirely clear. Sometimes it is appropriate to congratulate a student for being stuckit means the student has tackled a worthwhile challenge and gotten to a meaningful point. Problem-solving involves being "stuck." If a task does not puzzle us at all, then it is not a problem it is an exercise. GETTING STUCK, GETTING UNSTUCK≼OACHING AND QUESTIONING The best thing to do is to sit tight and see if things improve. To keep the same position or be unwilling to move or act.
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