Our Thinking About Geometry

Mathematical Habits of Mind are productive ways of thinking that support the learning and application of formal mathematics.  A major premise of these materials, drawn from our previous work*, is that the learning of mathematics is as much about developing these habits of mind as it is about understanding established results in the discipline called "mathematics."  Further, we believe that the learning of formal mathematics need not precede the development of such habits of mind.  Quite the opposite is the case, namely, that developing productive ways of thinking is an integral part of the learning of formal mathematics. 

 We believe that instruction can be shaped to foster the development of such habits of mind in students.  That is the core principle of the FGT materials. For these materials, we have put together a Geometric Habits of Mind (G-HOM) “framework” considering several habits that seem to be critical to developing power in geometric thinking.  The list isn’t meant to be comprehensive.  However, by learning to attend to these several habits, in their own, in their colleagues’, and in their students’ work, teachers can become better prepared to help students succeed in geometry.

We have developed (and continue to develop) our G-HOM framework iteratively as we receive feedback and investigate data from our field-testing of the project. The Geometric Habits of Mind we promote include the following processes:

• Reasoning with relationships: Actively looking for relationships (e.g., congruence and similarity), within and between geometric figures, in one, two, and three dimensions, and thinking about how the relationships can help your understanding or problem solving.  Internal questions include: "How are these figures alike?" "In how many ways are they alike?"  "How are these figures different?" "What would I have to do to this object to make it like that object?"
• Generalizing geometric ideas: Wanting to understand and describe the "always" and the "every" related to geometric phenomena.  Internal questions include: "Does this happen in every case?"  "Why would this happen in every case?" "Have I found all the ones that fit this description?" (Emphasis on 'all the ones')  "Can I think of examples when this is not true, and, if so, should I then revise my generalization?” "Would this apply in other dimensions?"
• Investigating invariants: An invariant is something about a situation that stays the same, even as parts of the situation vary. This habit of mind shows up, for example, in analyzing which attributes of a figure remain the same and which change when the figure is transformed in some way (e.g., through translations, reflections, rotations, dilations, dissections, combinations or distortions). Internal questions include: "How did that get from here to there?" "What changes? Why?"  "What stays the same? Why?"
• Sustaining reasoned exploration:  Trying various ways to approach a problem and regularly stepping back to take stock.  Internal questions include: "What happens if I (draw a picture, add to/take apart this picture, work backwards from the ending place, etc.….)?"  "What did that action tell me?"  “How can my earlier attempts to solve the problem inform my approach now?”