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Week 1: Chapters 1 - 3, Question #4 What do you find to be creative or beautiful about mathematics?

What do you find to be creative or beautiful about mathematics?

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Jason Walker's picture

When I teach exponents, I love to have student create fractal trees.  I usually have them use markers, but I've experimented with online generators.  Here is an example:  http://www.visnos.com/demos/fractal.  I think it quickly drives home how much of the biology that composes their cells and life all around them can be explained through exponential growth.  Plus, the visuals can be quite stunning.  

ecappleton's picture

Jason,

Thanks for sharing the visnos tool. Wow! That is so cool. What kinds of biological growth do you connect it to? I'm thinking mitosis and growing bacteria population, but wondering what other examples to use.

Eric

Jason Walker's picture

Another example would be population growth.  Given unlimited resources and space, human, fruit fly, or even elephant populations will increase at exponential rates.  Folks who make math problems love to explain exponential population growth using rabbit populations in Australia. . . do a Google search and be amazed!  

Robbin Rekemeyer's picture

Hi Jason! Thanks for the fractile link.

Amy Vickers's picture

Thanks, Jason!  Until now, fractals have felt a little out of reach for me as a concept.  If anyone else is in that boat, here is a one-pager:  http://fractalfoundation.org/fractivities/WhatIsaFractal-1pager.pdf

Definitely beautiful!

shepardjma's picture

I teach a combined ELL and ABE class as part of a technical I-BEST program, so we always have a wide variety of math ability/knowledge/backgrounds.  (Yes I've had refugee students with math degrees in my class before, which was quite intimidating!) Today we talked about percentages, which is fun because there are so many different ways to think about and calculate percents.  I wrote a % problem on the board and asked, "What do you think? ...What can we do with this? ...Try something and see if it makes sense!" and all of the different methods and strategies offered was certainly beautiful.  Its great to see each student attack it from their own starting point.  I love when students insist on showing their work on the board, and then start to look for connections between their method and their neighbors' method and help each other build understanding.  I think this is both what is beautiful about math and what is beautiful about being a math teacher!

Robbin Rekemeyer's picture

I agree wholeheartedly. Being able to aproach problems in different ways and explaining their work to peers is what makes math creative.

Kim Szczepanski's picture

Today I used percents within function tables. For my function rule I had one that was 100% increase, 50% increase, and 50% decrease. The students had a couple of the input and output values. Then they had some just input that they had to find the output. Then they had the output that they had to figure the input. It took them quite a while to work backwards on the 50% increase problems. I would not tell them how to get there. They had to study the tables they had and look for any patterns they could use to work backwards. They did finally find a way. For my adults, this was a very hard process and they were pretty proud at the end of class.

Sarah Sirois's picture

I love the patterns of math found in nature, a sunflower, a spider's web, pine cone just to name a few. The evidence of Fibonacci's sequence and the Golden Ration really amaze me. When I taught 5th and 6th grade, we explored the Fibonacci sequence and I remember students' fascination with the diiferent natural objects and the symmetry within them. 

 

Marcy Cordova's picture

I find that the patterns and history of math are beautiful. For example, I give my adult students a variety of tools to measure the diameter and circumference of circular objects such as plates, hula hoops, etc. Some want to "cheat" and use pi, but I tell them that pi is not one of the tools they can use. We then have them divide the circumference by the diameter, and we put all of the results in a chart. The fact that all of their answers are usually pretty close to 3.14 is amazing and beautiful. We talk about how math was explored throughout history and how pi was developed. We talk about how much easier it is to use pi than trying to measure the circumference, because most of them struggled in the activity with measuring it accurately. 

Part of what I find beautiful with mathematics is that people were thinking about the same problems thousands of years ago, all over the world, and working towards finding solutions. Looking at how the mathematical ideas developed over time is fascinating. 

MarkTrushkowsky's picture

Marcy, 

Yes!

When I first started teaching math I really struggled with the inquiry-based explorations I was able to do with history and science that I just didn't know how to do in math because it had never been presented to me in that way (at least not in math classrooms). I don't remember where I found it, but I did the discovery of Pi lesson essentially as you described it and it really changed me. 

All I offer students is string, rulers and dozens of circles (dimes, clocks, records, cans, hula hoops, oranges, etc). There is a lot of great messiness as students have to make decisions about units, how to be as accurate as possible putting string around a circle, approximate their measurements, etc. They do the measurements and then we put all the data together in one chart and look for patterns. I have never had a student talk about pi (I don't use the word circumference until after the activity), but they always say things like "Wait. It's like the distance around every circle is about three times bigger than the distance across the middle of the circle" or "The distance across the middle of the circle is about one-third the distance around the edge of the circle". Then we make predictions about both the diameters and circumferences of circular objects we haven't measured that we have in the room and can check. Then the conversation extends to what we can do with that number. It's true for the planet Earth! It's true for the orbit of the Earth around the sun (which I know is slightly elliptical, but just barely)! 

To define Pi merely as 3.14 or 22/7 is to miss the beauty that there is a relationship that is true for any and all circles in all of creation.

In her book, Becoming the Math Teacher You Wish You'd Had", Tracy Zager writes:

"When students generate real mathematical understanding, they are creating math anew. Each time a child has an aha! moment about place value, makes a connection between fractions and division, or discovers rotational symmetry in a flower, that child is reinventing mathematics. Whether or not other mathematicians have had the same idea before is completely irrelevant to that student. The inverse relationship between addition and subtraction may be settled mathematics to us, but it's an open problem for every single young mathematician we teach." (p. 32)

There is a lot of beauty I have found in patterns and connections in math I've worked on, but the beauty of the fact that we can recreate these moments of discovery in our classrooms, connecting our students to how humans have used math to make sense of the world is incredible.

 

 

donnawparrish's picture

Being somewhat of an artist, I certainly appreciate what we typically think of in terms of beauty in mathematics. There is much mathematical beauty in color theory, mixing paints to obtain specific colors, proportionality in photographs and design, Escher-type drawings/tessellations, on and on...but today my brain is focusing on multiple pathways to solution.  I found saw this graphic today (Earth Day) Image result for multiple pathways to solutionsand felt it was a great way to visualize the concept.  As an inexperienced teacher many years ago, I spent many hours helping students memorize, for example the many steps in typical long division. Now I approach teaching such concepts as Jo Boaler suggested in the opening video - share as many ways as you can to get a quotient. When learners find there are usually many ways to get to a desired result and that their creative thinking will be celebrated/rewarded, they become better thinkers and risk-takers and thus become better learners.    

ecappleton's picture

I ran across a lovely example of teaching using multiple strategies. The Japanese educator Akihiko Takahashi demonstrates a Teaching Through Problem-Solving (TTP) lesson using various ways of counting arrangements of dots. At the end of the lesson he says, "You can express a pattern different ways, because it leads to a different way of thinking. An equation, a math sentence is a way to show your thinking."

http://www.lessonresearch.net/ttp (float over the steps of the lesson to see the video links)

Patricia Helmuth's picture

Eric,

Thanks for sharing that link. Even though I kind of already knew the direction that each video was going to go, I was fascinated by the ease in which Akihiko Takahaski demonstrated the art of problem solving with a simple activity (which would absolutely work in an adult ed class). Once I started watching the first video, I had to watch all four! I really like the way that he empowered the students to own what they learned, all the way through the lesson, but especially at the end of the fourth video with those words that you quoted above.

Patricia

NancyT's picture

I really like this graphic.  I have always said in math there is a question and an answer but multiple ways to get from one to the other, it all depends on how you make sense of it.

sadkins2009's picture

I love the comments that have been shared. I find joy in the expression of math - I love to quilt which is a great representation of math. So many times our students don't appreciate the math around them in the everyday world.

Duane Dorion's picture

What I love the most about math is that creativity is never ending.  There are so many ways of teaching math from use of manipulates, to making manipulates, to using music, video's etc... There are so many ways to teach math and its never ending.  We can always find new ways of teaching concepts and we can totally learn for each and every colleague.  I love to have professional development and ask other teachers how they teach different concepts.  I have learned a lot from my colleagues.  With adult education, I feel we are better equipped to teach our students more ways to understand math in different ways than a lot teachers who are teaching it procedurally all the time.  I feel we have more opportunity for professional development than other math teachers in secondary schools.

Duane Dorion's picture

What I love about teaching the most is probably coming up with my own curriculum.  I was an engineer before I became a teacher and there was really not a lot of creativity involved.  Everything that a company did it was  in sequence and you basically started at point A and went to Z.  There were computer programs that you used and there wasn't a lot of creativity.  With teaching if your working in the right place creativity in teaching is the most exciting.  I work a lot with manipulative and with visualization with my students.   I want my students to see the abstract and also see how you can put this together in your every day life.  I have a lesson on fractions but it's with recipes.  The recipes are standard recipes and I ask questions, what would happen if I only wanted half of the recipe, double or triple the recipe.  The student now have to figure out a lot of computations of fractions.  The students really enjoy hands on activities like this for sure.  

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