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I like this  idea from Chapter 1 of the book -  "Aristotle said that flourishing comes through the exercise of virtue" and that virtues are the "qualities of character" and "habits of mind that will allow [a person] to live a more fully human life and experience the best of what life has to offer".

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Chapter 2 compares space exploration with mathematical exploration. It develops the virtues of imagination and creativity - not words normally associated with mathematics. I appreciate the idea of exploration of ideas and using imagination and creativity as tools to approach math problems and make sense of the world.

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Play builds hopefulness...
Play builds confidence in struggle..
Math play cultivates patience...
Math play builds the ability to change perspectives... as well as an openness of spirit that contributes to community.

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I keep thinking back to the Saturn example, first because of how the rings may appear stationary  but upon more investigation are found to be orbiting, in flux. It's appropriate to mathematical thinking, with multiple ways to approach, view, model, with understanding changing over attempts and time. Also beautiful to me was the approachability of beginning to understand resonance with something as straightforward as a 2:1 ratio, a reminder that mathematical and scientific concepts are for all, not to be postponed until some elite math level is attained.

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Throughout the "meaning" chapter, the phrase "real world" kept echoing in my thoughts. We've likely all been asked "When will I use this in the real world?" Of course, classrooms are part of the real world, so question answered at the outset. But also, this chapter's examples serve as reminders that mathematical thinking may lead to number patterns applied in engineering, coding, baking, and the fields that readily come to mind when people think of applying math. Or mathematical thinking may emerge in untold numbers of other ways, not so easily traced back to specific concepts from the math curriculum.

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"To do mathematics means more than just learning the facts of mathematics - it means seeing oneself as a capable mathematical learner who has the confidence and the habits of mind to tackle new problems."
-How do I help my high school students develop this confidence and mindset? Do I let them ponder this quote to at least let them think about the idea?

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"To do mathematics means more than just learning the facts of mathematics - it means seeing oneself as a capable mathematical learner who has the confidence and the habits of mind to tackle new problems."
-How do I help my high school students develop this confidence and mindset? Do I let them ponder this quote to at least let them think about the idea?

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I like this idea of being captivated by the mysteries of math and sending out probes to explore and test our ideas. It was nice to notice that exploration has setbacks and that we can be undeterred by them in our mathematical explorations.

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This chapter reminded me of why I teach math. I believe that math teaches you to be a better problem solver and thinker. I tell my students that I would rather practice solving problems with numbers and variables than in real life situations. For me, learning math has taught me how to approach most problems in my life. Yes, I get emotional and forget my problem solving steps sometimes. But usually I can rationally look at a situation and try to look at the bigger picture and how to solve the problem I have been presented with. I think about things and take time to research or reflect on past experience. Math has taught me to do that and I hope that I can help me students see that math can teach them that too.

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My boys are 7 and 9 and typically enjoy math. This chapter made me wonder if I could present pattern problems to them so that they could learn to explore and discover new ideas about math.

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I like the idea that meaning comes from from understanding the relationship between things. Words have meaning because of their relationship to other words. Events have meaning in the context of other events. Math also shows the relationships between things - numbers, ideas, and patterns. I like the quote from Henri Poncare that talks about science, but the same is true for math. "Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house."

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"Play should play a prominent role in mathematical learning. Everyone can play...Everyone can have a meaningful experience in mathematical play." I wonder how much of what I do as a teacher of mathematics is play? Could I make math education more interesting, fun, and meaningful for my students if I considered this while planning my lessons/activities?

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I read an article about the "discovery" of non-Euclidean Geometry.  It was made by exploring the idea "What if what we always thought was true isn't always true in every circumstance?" 

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I read an article about the "discovery" of non-Euclidean Geometry.  It was made by exploring the idea "What if what we always thought was true isn't always true in every circumstance?" 

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Su says, “A disposition toward mathematical beauty is the engine of mathematical persistence.”  😌 so my challenge is to get a collection of tasks that are helpful in revealing mathematical beauty to those who play with them.

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The idea of there being four types of mathematical beauty was interesting to me. Seeing the mathematical beauty "through art, through music, through patterns, through cultural artifacts, through rigorous arguments, through the elegance of simple but profound ideas, [and] through the wonderous applicability of these ideas to the real world in many different fields." I normally think of mathematical beauty as sensory beauty - patterns I can see in art or nature, the patterns and harmonies in music - but there are other types of mathematical beauty.

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The idea of there being four types of mathematical beauty was interesting to me. Seeing the mathematical beauty "through art, through music, through patterns, through cultural artifacts, through rigorous arguments, through the elegance of simple but profound ideas, [and] through the wonderous applicability of these ideas to the real world in many different fields." I normally think of mathematical beauty as sensory beauty - patterns I can see in art or nature, the patterns and harmonies in music - but there are other types of mathematical beauty.

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"Trust is at the heart of a desire for permanence." Deciding who and what can can trust can be scary and can change over time, but the permanence of mathematics can be like a refuge of stability, a yardstick to measure our progress, and a foothold to rely on. It is something permanent that we can trust.

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What kept coming to my mind was the John Keats quote from Ode on a Grecian Urn,  "Beauty is truth, truth beauty - that is all Ye know on earth, and all ye need to know"   Maybe it came to my mind because we also read the chapter on beauty this month.   Perhaps the beauty in mathematics comes from it's truth. Something from this chapter than stood out to me was the need for deep knowledge compared to shallow knowledge. I think deep knowledge involves knowing why something is true and not just accepting something as true with limited understanding.  Something to consider as we are teaching our students.

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Lucy Stewart
...Beauty appears in mathematics.. - in many forms - though it is less appreciated there because many have not had a chance to experience it. or they have experienced it but not recognized the experience as mathematical. (p.69)

In the attached link are a couple of student samples from an activity meant to ease them in to studying quadratics.  This activity was supposed to help students understand that a curve can be derived from a series of straight line constructed on the coordinate plane.  I was amazed at the artistic beauty in some of the results.  some of the results.

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Lucy Stewart
...Thre are many ways to climb a rocky cliff; Euclid and Hilbert chose different footholds as starting points... (p.96)

The 2nd semester of Integrated Mathematics is the geometry half of the course. The students get to crack open their new book, volume 2. I usually put a map of the hood up on my screen and circle the Dollar Tree and ask students to write down directions on how they would get there. After sharing out the different ways to get to the dollar tree I ask students to reflect on the fact that there are many different ways to get to the same place in the end and that's how proving angle theorems are. Students use patty paper to prove all the angle theorems in the unit then I show the traditional 2 column proof with statements and reasons.

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Lucy Stewart

...For a math explorer, deep investigation of truth is a habit. To really know truth is to investigate it at many different levels -- to see the whole truth from multiple perspectives...explorers seek multiple ways of knowing the truth...to see deeply how they are in concordance with one another. (p. 106)