This document discusses promoting reflective inquiry in mathematics through developing "Habits of Mind" in students. It lists 12 habits of mind that mathematical thinkers should develop, such as understanding which tools are appropriate to solve problems, being flexible in thinking, and engaging in metacognition. It also lists the 8 Standards for Mathematical Practice from the Common Core State Standards. The document was then used in a math education class, where students worked through example problems to develop these habits of mind and presented problems to their peers for feedback.

1. Jamalee Stone, Rhonda Airheart, Sean Bialas, Jeremy Elsom,
Jennifer Johnson, Becca Myers, Jinhua Tan

2. Promoting Reflective Inquiry in Mathematics

3. Mathematical thinkers with well-developed CCSSM Standards of Mathematical
“Habits of Mind” * Practice
1. Understands which tools are
appropriate when solving a problem 1. Make sense of problems
2. Is flexible in his or her thinking and persevere in solving
3. Uses precise mathematical definitions
them.
4. Understands there exist multiple 2. Reason abstractly and
paths to a solution quantitatively.
5. Is able to make connections between 3. Construct viable
what one knows and the problem arguments and critique
6. Knows what information in the the reasoning of others.
problem is crucial to its being solved
7. Is able to develop strategies to solve
4. Model with mathematics.
a problem 5. Use appropriate tools
8. Is able to explain solutions to others strategically.
9. Knows the effectiveness of algorithms 6. Attend to precision.
within the context of the problem
10. Is persistent in his or her pursuit of a 7. Look for and make use of
solution structure.
11. Displays self‐efficacy while doing 8. Look for and express
problems regularity in repeated
12. Engages in meta‐cognition by reasoning.
monitoring and reflecting on the
processes of conjecturing, reasoning,
proving, and problem solving
*This definition was developed by Dr. Jim Lewis (UNL) and Mark Driscoll’s
book, Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10.

4. • On the index isosceles right triangle
provided, please write down 1-2 of your
favorite “habits of mind” problems that
you may have used in your teaching.
• Please share with one of the persons
sitting in your proximity.
• As a table, share the problems that came
to mind.

5. Thank you Google! 

6. • Will This Boat Float? -> Crossing the River
• Shapes of 4 Triangles -> Ninja Star Boomerang
• The DVD Bandit Dilemma -> Coconut Problem
• The Pool Tiling Problem ->The Border Problem
• A Peachy Problem -> The Mango Problem
• An Open and Shut Case -> The Locker Problem
• Donovan’s Donut Dilemma -> Golden Apples

7. • To work with material aligned with the age group we
will teach
• More experience teaching content to others.
• Gives us experience getting up in front of the class and
practice asking questions.
• Gives us a chance to work the problems and present
them.
• So we can learn to differentiate instruction.
• Helps us think in different ways
• To get different perspectives on how other people think
and also teach…

8. • GoogleDocs spreadsheet used for HOM sign-up.
– Partner and date
• Problems were posted in D2L content area for
students to review and select.
• Partners met with me at least four days in
advance to review the HOM problem, discuss
the solution(s) found, and provide an overview
of how they planned to facilitate the lesson.
• HOM lesson was videotaped (by me, not a
professional )

9. • Peers used an index card to indicate what went
well with the lesson, and at least one
suggestion for improvement. Comments typed
and shared.
• Students scheduled a time where they could
both watch the video in my office.
• While watching the video, partners took notes
of what they noticed while watching
themselves.
• Between peer reflections and the video
viewing, noticeable issues are addressed… 

16. Peer Review Comments
•What kind of things would you offer students with less
experience in math to help guide them along?
•More interaction with groups would be beneficial.
•I think that you could do better on not correcting right away.
Lead us to it, not show us.
•Be more sure of the equations and what you are saying so you
don’t have to start over.
•Too long to work on the experiment and not enough time for the
proof.
•When walking around and seeing how everyone is doing, make
sure to ask everyone what they are thinking.
I was never asked a question.

17. • I mentioned that I wanted online
discussion to include Jordy, but what
other reasons do you think I use online
discussion with a face-to-face class?
• Why do you think I respond to your
discussion posts via email rather than
posting to the discussion board?

20. Jami.Stone@bhsu.edu
(402) 613-0136