How to Think Like a Mathematician

  1. I try to explain concepts in more than one way. 
  2. I investigate, make guesses and test my ideas.
  3. I try to put concepts into formats that I can relate to – visual, verbal, symbolic.
  4. I work with others to investigate problems and work out our ideas.
  5. I  verbalize, or draw, or explain, or describe my ideas.
  6. I try to communicate ideas in more than one way.
  7. I look for the same idea inside different situations. 
  8. I try to decide if a procedure is appropriate or if it is not.
  9. I invent new procedures or modify old ones if it seems necessary.
  10. I try to express my ideas in writing, speaking, or with pictures.
  11. I try to learn from errors rather than fear making them.
  12. I test out new ideas I have come up against situations I am familiar with.
  13. I listen to other’s mathematics ideas and try to connect them with my own ideas.
  14. I check the reasonableness of my results when trying to solve problems.
  15. I check to see if my strategy is unproductive for what I am trying to do.
  16. I make guesses based on information I have gathered.
  17. I use inductive reasoning to assist me in making conclusions or decisions.
  18. I look for connections between ideas in different contexts.
  19. I try to invent new ideas when none come to mind. 
  20. I try to picture problems in some way to make them simpler.
  21. I record my thoughts so that I can pick up where I left off.
  22. I use notes I take to recall my ideas later on.
  23. I try to make decisions about what information is relevant.
  24. I test out ideas using mathematical strategies like “trying a smaller number,” or “using zero.”
  25. I test out ideas using mathematical strategies like “writing down what I know about the problem.”
  26. I test out ideas using mathematical strategies like “putting the numbers in order.”
  27. I test out ideas using mathematical strategies like “looking for a pattern.”
  28. I test out ideas using mathematical strategies like “breaking the problem into smaller parts.”
  29. I test out ideas using mathematical strategies like “trying extreme cases.”
  30. I try to see new ideas as extensions of familiar ideas, rather than unconnected facts.
  31. When I am unsure of what to do in a problem, I make assumptions that allow me to proceed.
  32. I try to use my imagination to invent new ways to approach problems.
  33. I re-think ideas that I have seen before when presented with new evidence.
  34. I reflect upon my previous ideas when I am presented with new ones.
  35. I connect ideas to things I am familiar with.
  36. I choose from different tools when presented with a problem-solving task.
  37. I look for different ways (graphical, numerical, verbal, symbolic) to express the same concept. 
  38. I try to rely on reasoning rather than on someone else to tell me that I am correct.
  39. I try to organize my thoughts, whether in writing or in speaking.
  40. I continue to work on problems even when my first attempts are not successful.
  41. I try to see if a new idea I have learned in one situation works in other situations.
  42. I don’t fear to ask questions about a new situation.
  43. I talk things out with others.
  44. I try to make things fit into a pattern or a system.
  45. I repeat a process over and over if it makes sense to me.
  46. I listen to what others have to say about my ideas.
  47. I don’t give up – I look for resources.
  48. I try to put other’s explanations into terms that make sense to me.