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# Week 3: Arts, math & many answers (January 30 - February 5)

In discussing Week 2 live meeting, SandyG posted "Ten Lessons The Arts Teach" by Elliot Eisner. Kathy Cianciola writes about one of the lessons:

I see art and math like yin and yang.  They can fit together, yet they are very different.  For instance the idea that in art, "questions can have more than one answer"... I don't really see that in math, usually there is one answer.

The last phrase describes the usual situation, but it's been shifting toward the new balance between yin and yang. Hence your task, should you accept it:

1. Pose a math question to which every student in the class can give a different correct answer.

2. Find a math art piece (picture, sculpture, video of a dance, music, etc.) that you can use to introduce your question and inspire the students to more beautiful and meaningful answers.

• I would have students get in groups and come up with a short dance combination. I think I would use this for older grades like 4th and 5th. I want them to write about the different shapes and angles they created with their bodies relating it to math. They will perform their dances in front of the class and maybe teach the class too. I will use this video to introduce the assignment.

• Thanks for the video - I added it to the course's playlist at http://www.youtube.com/playlist?list=PL84E80E3C18F033D9

My colleague and friend Malke Rozenfeld is doing interesting things with dance and math - check it out if the topic is of interest: http://www.mathinyourfeet.com/

Are you a dancer, Keisha?

• Thanks for the 'Math in your feet' link. I really enjoyed exploring the website. This is something I'm really interested in. Yes I am a dancer. I've been dancing since I was about 4 years old. I really enjoy it and wish to use it in my teaching career.

• I really liked this idea of involving dance with math. Looking for new ideas about math and dance I found this article about a program called Mathdance. -http://www.sciencedaily.com/videos/2008/0503-do_the_math_dance.htm

"Students can create their own movement patterns. For many, the experience helps them connect with numbers they may never have understood before. "You're dancing something that is in three's, for example, a Waltz, it has a different feeling because it's an odd number of beats than three fours, which has a very even feeling," Dr. Schaffer explained."

Ideas like your's is and examples from Mathdance could make math much more interesting and help children be more excited about it. It can also help visual  or spatial learners learn more easily.

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• Your ideas for teaching the concept of infinity to children seem quite good. Certainly, I think, if you can make such an abstract concept a little less abstract and more visible, it is easier for the child to conceptual the idea.  I read an interesting article written about the concept of explaining infinity to a child: http://www.googolpower.com/content/media/articles/mom-ill-love-you-til-infinity.  In the article Susan Jarema offers some others ways to teach the concept.  The ideas include asking a child what the largest number is they can think of and then asking them about that number plus 1 and continuing this pattern until the child realizes there will always be a plus 1.  She offers approaches in terms that children could understand.  I especially like the exercise with two mirrors.

In a 2006 study published by Pehkonen, Hannula, Maijala, and Soro, it was concluded that "Boys give better answers than girls in tasks dealing with infinity" (p. 351).  The authors found that most primary children are very interested in the conceptof infinity, and they enjoy discussing the subject.  Pehkonen et al (2006) state, "questions on infinity may also come into light. Infinity awakes curiosity in children already before they enter school:  preschool and young elementary school children show intuitions of infinity. However, this early interest is not often met by school mathematics curriculum, and infinity remains mysterious for most students throughout school years" (p.345).

Studying students in a Finnish school, Pehkonen et al (2006) concluded that “students on grades 5–9 seem to have a finitist rather than a nonfinitist or an infinitist point of view in questions of infinity” and “students use intuitively the same methods for the comparison of infinite sets as they use for the comparison of finite sets. Although students have no special tendency to use ‘correct’ Cantorian method of "one-to-one correspondence," they are prone to visual cues that highlight the correspondence.  For example, students tend to match set {1, 2, 3…} more easily with the set {12, 22, 32 …} than with the set {1, 4, 9 …}” (p. 346).

Though this study shows that students may not truly grasp the concept of infinity, it seems to me that if teachers are prepared for this discussion, it is a concept that would interest students and excite them.  In addition, the concept of infinity could prepare students for abstract thinking in other curriculum areas as well.  Playing with numbers and challenging a students’ imagination is a great way to open their minds to concepts beyond the classroom confines.

"There is no branch of mathematics, however abstract, which may not

someday be applied to the phenomena of the real world."

-- Nicolai Lobachevsky

Jerema, S. (n.d.). Mom, I'll Love You 'Til Infinity. Retrieved March 2012, from Googol Learning: http://www.googolpower.com/content/articles/mom-ill-love-you-til-infinity

Pehkonen, E., Hannula, M., Maijala, H., & Soro, R. (Eds.). (2006). Infinity of numbers: How students understand it. Proceedings 30th Conference of the International Group for the Psychology of Mathmatics Education, 4, 345-352.

• A question that first came to mind that has more than one answer to me was about money. You could give your students all different coins and ask them to make a dollar. Some may have the same answer but if given enough coins, the answers are almost endless.

I struggled to find a piece of art to go with this idea. I was in search of a picture of shapes. Such as one that maybe a square but has all different shapes inside of it to make the square. I tried googling all different things but found nothing that captured what I was imagining.

• Using various combinations of coins is obviously a great way of getting more than one answer in math because you could use 4 quarters, 2 dimes and 5 pennies to make \$1.25, or 1quarter, 8 dimes and 4 nickels to make \$1.25....I agree,"The possibilities are practically endless." This activity could also be very hands-on and relevent by having students actually pay for items. Taking the whole class to the school cafeteria where they could use various combinations of coins to make actual purchases would be an ideal way for children to experience the process of counting out coins.  However, since this idea is a bit unconventional, perhaps obtaining a menu from the school cafeteria, and allowing students to "pay" for said "items" would work better. Students could first choose "items" from the menu and work together in pairs helping eachother to add up costs, and experience using different combinations of coins in a concrete way.

Another variation of this would be to have a special day  where you and students bring in small trinkets (with parental approval / cooperation), and set up "shop" where students could use the coins to pay for items of their own choosing. I think this would be fun for the whole class because most children love trite plastic novelties.  I have a box full of them at my house which I would gladly donate to a worthy, educational cause. Since kids also seem to enjoy interactive lessons that take them away from the standard classroom set-up, I believe this activity would be enjoyable and conducive to learning.

* You could also come up with a monitary amount and play a game to see how many different combinations of coins the class could discover that will equal that amount.

• Lessons with money are always so much fun! A few years back I was teaching ESL and as part of our discussion of money, we all walked to a little grocery story that was a block from the school to continue our discussion about value, less/more, and we added in some vocab building as well.  It was a lot of fun andd seemed to help the students tremendously. Within the confines of the school, the cafeteria is a great idea like Kathy suggested or some schools have a student store, too.  You can also have your class sponser a bake sale during lunch (if your school still allows such things.  Mine does.) and the students will get real experience making change and working with prices.  The money can then be donated to a good cause.

•

Students draw a simple shape or picture with defined points on a quadrant of a graph.  They then draw symmetrical images across x and y axes, rotate around origin, and translate it in x and y directions. They can then color in their objects and make it something special to them.

• I did something like this in high school and I loved it.  I am not artistic nor very good at graphing so I thought hti sporject was seting me up to fail. However I found that even I, the unartistic, no good at math girl, could ake something so interesting. I think once your students realize that anyone can do this, because each end product is an individual representative of their own skill, thye will love it too and all while learning some graphing knowledge.

Other projects that this one reminded me of are making snowflakes and tesselations. When I first looked at your picture I though patterns not necessarily graphing. Snowflakes are a great "secret" math activity, the students won't even know they are learning math through patterns. Also tesselations, for older students, illustrate patterns.

Mirror images are another activity that popped into my mind. I remember doing them with actual double mirrors when I was in elementary school, they were purple. I think a lesson with mirroring would most closely represent what you are trying to depict here.

• This is my first time seeing something like this. I really like this activity! It's a neat way to teach students about angles, symmetry, and how to graph points. Your comment and Carolyn's comment remind me of this online game I saw at one of my field work observation. http://www.innovationslearning.co.uk/subjects/maths/activities/year3/symmetry/shape_game.asp It's a game where you have to find the lines of symmetry.

• A really nice collection of power point lessons for elementary school students by Behrooz Parhami:

• sandy,

## I really like your quilt idea. If each child designed their own unique hexagon you could eventually join them together creating a huge wall hanging for the classroom.  This is using the child's inner sense of creativity in making their own original designs.  Plus it is intrinsically motivated, and modern, and you are meeting the challenge of coming a math activity which does not have just one answer.

• At our winter holiday gathering, I asked the math circle kids ( 6-9 yr olds) what burning questions they have about math and science.

Here is a sample list:

Who invented numbers?

Who invented math?
When was math first used?
Are there any unsolved math problems?

What is calculus?

What is Pythagorean theorem?
Why is something in the power of three "cubic"?
Is there a formula 2x3x4x5x....for any given number of elements?
Who invented parenthesis and why they did so?

Do figures with rotational symmetry have central symmetry?

What was the first invention?

How do people invent rules for board games?
How was the paper invented?
How did people learn to make lamps?
How to make computers?

How to make shiny metals?

How many parrots are in the world?
What was the first butterfly in the world, and how big it was?

What is the biggest tree in the world?
Are there more animals on land or in  water?
Did the comet crashing on Earth made the dinosaurs die?
What can you do to live forever?
Why our small eyes can see so big objects?

When did the world begin?
Who was the first person on earth?
Where can you find blue and red rocks?
When and how did people find out about space?
How can the sun melt the snow?
What makes light go so fast?
How does a car stay on a hill and not go back?

How is math used in astronomy?
How to measure the distance from one planet to another?
How big is the universe?

Are galaxies really part of space?

*******************************

Here you can see our math circle kids asking their questions:

And here are our high school students teaching a math circle for younger kids:

Best regards,

Julia Brodsky

•

Do we base our understanding of art and nature off of math? Or do we frame our understanding of math on the laws of nature and observable physical properties?

I thought of this question from my prior research and interest in the Fibonnacci sequence and it's applications in nature. In learning the sequence, I realized how deeply math and nature fit together, and how the world supports both. My math question is rather philosophical, yet I have a few different ways that I could adjust this for children. I could ask them to take pictures or simply brainstorm ways that math is in nature (ex: number of petals on flowers, perfect rectangles/circles, etc.). Then we could talk about where this comes from, perhaps even bringing in hands-on material for the kids (ex: pinecones, leaves, flowers, etc.) This would be a way to introduce the philosophy of the subject without overwhelming the students. I believe that it's important for children to be constantly questioning the world around them, and I want to make curiosity valued in my classroom.

• I think I misunderstood this assignment when I initially posted my response.

To use the literary arts in a math lesson, the teacher could read a short story and ask the students to count how many times they hear a specific word.  Another example might be to use a story similar to The Three Bears in which Goldilocks tries different, chairs, beds, and porridge bowls.  The students could be asked how many different things did Goldilocks try.  They could add the objects, but that could also show how multiplication works.

Music lends itself very well to math lessons in the counting of rhythms and notes.

Using the examples of my classmates for visual arts, I found this site: http://mathcrush.com/math_art_worksheets.html where there are exercises which require students to complete the math problem in order to shade certain areas on a picture to make a work of art.

Another activity using art might be to use a picture to teach 1:1 correspondence: How many blue squares?, etc.

• Sandy, I think your tasks integrate math and arts well. I am not sure the student questions you pose can have many answers. For example, there is just one correct answer to how many blue squares there are, or how many times a word comes up with a story.

I love the Goldilock story for math, too. What questions can you ask about it that have many possible answers (all correct)?

Your original question, before you edited the entry, was:

"Some educators suggest moving away from the straight subject area approach to involve the identification of a central theme and to ask what each subject area can contribute to it.  Rather than having separate subject area classes such as English, math, and science, subjects would be completely mingled. Do you think it is possible to teach math in this way, or do you think math should be an isolated subject taught independently of other content area?  Why?"

This is a teacher question in the current form - we could rephrase it for kids - but the "Why?" part is a good example of a question with (infinitely) many correct answers!

• Yes! I 100% admit that I did not understand what you were trying to get at here, and even after the posting about the rice on the chessboard (because I had never heard of that before), I still had no idea. I'll keep throwing out ideas until I understand the concept you're going for here.

With the colored tiles, I could ask the children to rearrange the colors to make a pattern.  Each child might come up with a unique pattern, but the concept of patterning would still be taught.

For the 3 Bears story, I could ask number questions that aren't directly answered in the story such as "How many bites did Goldilocks take from Papa's bowl, mama's bowl, & baby's bowl? How many all together? How did you come up with that number?" Because it isn't in the story, each child might have their own idea.  I could ask ask how tall the bears are. We could look at measurements, and the students could ask their parents who big they were at ___ age and compare that to how big they assigned baby bear.  They could measure mom & dad, too, and compare/find the difference with Mama Bear and Papa Bear.

Perhaps an activity like a quilting lesson could have an infinite number of possibilities.  Each student's quilt will be unqiue, but they will also be using geometry.  The lesson below concentrates on the hexagon, but it could be modified, I think, to include any shape as long as there aren't any "gaps" in the quilt.

## Objective

• Working individually, students will create and color a one-patch quilt design based upon the regular hexagon. A one-patch quilt is made by using only one geometric shape that is repeated throughout. The hexagon is a good choice because it can be "cut" in a variety of ways including isosceles trapezoids, rhombi, isosceles triangles, equilateral triangles, and kites.

• Students will choose a design, draw the pattern on the included worksheet, and color it to highlight the quilt design.

## Suggested Time Allowance

One to two class periods, possibly with some time needed outside of class.

## Math Skills to Highlight

• Recognizing the properties of regular polygons, particularly the regular hexagon

• Recognizing the properties of an equilateral triangle

• Measuring angles

• Finding the sum of the measures of interior angles of triangles and quadrilaterals

• Identifying lines and/or points of symmetry

• Identifying congruent figures

• "Why?" is such a good question overall. Some think it's the most important math question one can ask. I hope examples from others will help - that's why everything in this course is posted openly...

To provide another example, I tend to pose a lot of "maker" questions - inviting kids to figure out how to make something. For example, I asked my 6-7 year old group, simply, "How can you make a circle?" They came up with many, many different ways, including tracing round things, just doing it by hand approximately, first folding a paper through the middle many times, like you would for a snowflake, and then cutting it across (this leads to nice Greek math or calculus exploration), using string attached at one end, using a compass and so on. The follow-up questions were:

- How do you know it's a circle?

- How can you make your method more precise?

• what if we did not use real numbers system in our life?
• if you want to describe what mathematics is to an alien, how do you do that?

http://www.naakamaruddin.com/2011/07/what-if.html

• I think both of these questions could be the foundation for a great lesson. This would be great for the childrens' creativity. For either one, the class could brainstorm ideas. For the first, the kids could brainstorm what we would use instead of numbers. You could discuss counting on fingers and toes and ask the class to try counting without thinking of numbers.

I think the second one has even more potential. The class could brainstorm the most important aspects of math and make everything very simplified. In essence, they would be teaching someone else, which is a great way to review material. Then, perhaps they could each write a letter to an alien, explaining what math is, and talking about their experience with it. It could be a great reflective experience, and I'm sure the students would come up with some great ideas!

• Here is my favorite "infinite number of right answers" exercise:

I don't know what kind of art would inspire students on a problem like this. Perhaps something to help them think geometrically (not just in numbers)? Or perhaps just examples of students who have gone before:

(But most of those are not elementary problems. I had middle school and high school students that year.)

•

This colorful image is a digital interpretation, fine-arts print, by Wingsdomain Art And Photography, and could be used to introduce my activity to students.

In an attempt to meet the challenge of creating an activity with the possibilty of each student giving a different, but correct answer, I realized that If the answers are going to vary, then some part of the question will need to vary.  So I came up with an idea which utilizes both estimation and multiplication.

Materials:  It involves gumballs, and a gum ball machine, but a large clear jar or container could also be used.

Activity:  The children are asked how many gumballs are in the gumball machine, or jar.  This can be a guess or an estimation. Each child will have a different answer. Once the children write down their answers, they are asked to think about how many gumballs would be in 5 gumball machines, based on their first answer, and then how many might be in 10 gumball machines etc...

This will make children aware that estimation is something we can use when all the facts are not available.  This activity will also give children practice in multiplication.  I think that the idea of using gumballs is a very kid-centered idea, and will make this activity one they will not forget.

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• Kathy, you ran into the same issue, on two different sites, while trying to insert this picture. The first is Facebook being a sort of a "black hole" for media - it does NOT support open Web protocols that would allow us, for example, to link a picture from a Facebook collection. The second site, Fine Art America, also does not provide direct link to the image they host. I used the program called Jing to save the image. I believe it to be fair use, but it's really a grey area.

• Thanks, it looks like the image worked in one of my posts, only it came out quite small.  I'm still learning how this works.

• One of them could be the classic grains on a chessboard problem

Source: http://capitalforte.blogspot.com/2011/06/bite-size-science-mere-trifle.html

The other one I can think of is to use a slight variant of the proposed task. Instead of seeking answers, get the kids to frame different facets of the probelm - One of the interesting posts I could pull out is this : http://blog.mrmeyer.com/?p=5990

We may need a simplified version of this for younger kids but the idea in my view is powerful -So we start with the video and  in this case ask a simple question like "How long will it take for the jar to fill up?" leads to a number of other facets that we need to know before we can answer the question - While this is not necessarily an art piece, I do think it aligns with our objective here.