Pizza Creations: Thinking About Ingredients, Combinations, and Sequencing

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On the Mountain, it is cool enough to bake in the oven during the Summer months.  This is also a great time to make something together as a family, like pizza.   In our family, we have many different preferences when it comes to toppings.  Some of us like a plain cheese, some of us like pepperoni, and some of us like to put everything on it to jazz it up. This gives our family an opportunity to talk about the different combinations of pizza we can create, if you have a finite number of ingredients.

Let’s say you have cheese, pepperoni, olives, chicken, and tomatoes to choose from for toppings. You can easily make a cheese pizza, or a cheese and pepperoni pizza, or a cheese and pepperoni and olive pizza, or even a cheese and pepperoni and olive and tomato pizza.  How many different varieties can you make?

Thinking about the different varieties, or combinations, you can make is actually preparing the child to think and do the Mathematics.  At this point, you can just ask them to think, estimate, and calculate the number of combinations you can make.  They can draw pictures, or write each combination out.  For this age, I would suggest talking it out and creating some of those combinations with the kids.  What you are giving them is experience in working this out.  Be mindful of how many ingredients you work with though.  Start simple and then add more ingredients to choose from at another time.  Most importantly, have fun!

Something to Think About: 

When making pizza and talking about the number of pizzas you can make, just take about five minutes with your child.  You can take more time if you want, but make it enjoyable, not like a quiz.  There will be a question on whether or not double toppings count as a different pizza.  The question is, by adding double toppings does this create a different tasting pizza?  This is a great question to discuss with the kids!

Keep in mind that you may have a child who can think this all in her/his head.  You may have a child who enjoys drawing out the pictures, or listing the combinations.  You may have the child who wants you to buy lots of pizza dough so that she/he can actually create all those pizzas to find out.  For this child, keep the number of ingredients to three!  We all learn differently and we all must honor the child on her/his learning.

Think about sequencing. What do you do first when making a pizza?  What do you do next? When do you add the cheese, or other toppings? This is another great way to have the little ones work on their sequencing and critical thinking.  Make this experience fun.  The end result is to create a great pizza, memories, and learn about Math!

 

copyright 2018 Christina Grossman. All Rights Reserved

Look Up to the Clouds and Name What You See: Harnessing Imagination to Build Stronger Thinking in Mathematics

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I see a dragon on his back, a troll, a moose head/reindeer head, and some kind of duck doing the back stroke. Ha! 🙂 What do you see?

With Spring and Summer bringing beautiful clouds, this is a great time to plop on the ground and look up at the sky!  Have you ever played the game of naming clouds?  Do you know that this might be great for your creative eyes to stretch and have some fun?

Just go outside and take a few minutes to look at some clouds and take turns with your child as to what they see and you see.  Name some shapes too!  Have lots of giggles and smiles!  Your child might be able to see their favorite cartoon character, or a favorite animal.  Here are some clouds that I thought were fun look at:

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To me, these clouds look like sun glasses.  My sons sees stretched out silly putty!  What do you see?

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A little later in the morning, the same cloud from above transformed to this!  I see two space ships.  My older son sees two eyes.  What do you see?

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These are special clouds as you may be able to see a fairy or two up in the sky 🙂

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The bigger cloud may look like a seahorse or dragon looking down.  What do you see?

 

Something To Think About:

In Mathematics, the patterns, or solutions that we need to find are not always going to pop up at you and say “Here I am!”.  I have used a lot of creativity and imagination to solve some mathematical problems.  The more opportunities we take time to be creative in what we see, the easier it will be to see even the hardest patterns and solutions.  It is helping yourself and your child to see and think outside of the box.  It will also make it easier to explain it to someone else.  We also need to remember that some of the most innovative advances in science have also come from authors of science-fiction.  Creativity inspires!  🙂

There is always more than one way to add, there is always more than one way to multiply, and there is always more than one way to approach a problem.  Being an example to our children in this way can really help them adapt to challenges, seeing things differently, and knowing it is great to try another approach.

Remember, creativity and imagination go hand in hand with Mathematics!  Having this attitude and belief will help us see just how attainable Mathematics really is to all of us.  Enjoy being with the clouds!

copyright 2018 Christina Grossman. All Rights Reserved

 

 

Cupcakes of Good Measure: Another look into Counting, Division, and Fractions.

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The kids are getting older to the point of doing more things to help out in the kitchen.  This has given them a sense of empowerment and joy in cooking and baking.  For this baking experience, one needed to distribute 24 baking cups into the cupcake baking tins. This task was for my little one to do, to give her a chance to practice on her counting.  The other child was given the responsibility of measuring the ingredients and placing them in the bowl.  Then there was mixing and distributing the mixture into the cupcake holders, my other child. 🙂

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When it was time to distribute the mixture into the cupcake holders, we needed to talk about how much to fill the holder.  If it was filled to the top, the cupcake would overflow when baking.  If filled to little, then it might over baker, or burn.  So we decided to fill it up about halfway.  That worked. 🙂 We also filled some a third of the way and that worked as well, but we needed to keep an eye on them to prevent over baking.

After putting everything into the oven, waiting for them to bake, and letting them cool, it was time to decorate.  Each child was able to decorate given the icing, sprinkles, and their own different piping tips with icing bags to use too! Before they were able to begin decorating, they needed to figure out how many cupcakes each would have equally if there are 24 to distribute.  To help my little one figure it out, the two older ones distributed one cupcake at a time to each other until there were no more to distribute.  So the result was that each have 8 cupcakes to decorate any way they wanted.  We did this outside to make clean up easier.  🙂 It was a good day!

Something To Think About:

Giving each child a task to do for a single project, like baking, gives them an opportunity to contribute.  The task does not need to be daunting, or feel like they are in a lecture about Mathematics.  When you cook, bake, or do a project together, point out the math they are doing and ask them if they were having fun doing it!  For a bit of advice, do no more than pointing out three to four things they are doing in math.  For kids, they want to experience things too. 🙂

In baking, I point out the importance of following the directions because it is an exact science.  Baking, in my opinion, does not have many allowances to veer off the path because you are working in an area of chemistry.  There are substitutions, but you need to research those substitutions, or you may get goop or a something as hard as a rock for your result.

The ingredients, whether dry, or wet, need to be added a certain way in order to react properly.  Over mixing can cause too much air to be added to the batter and may not come out right in the oven.  There is a difference between baking powder and baking soda, but both are chemical agents to help the batter to rise.

There also is working with expansion when heat, from the oven, is applied to the mixture.  That is why it is so important to talk to the children about how much mixture should be put into the cupcake holder.  Describe how much batter should be added by using what fraction of the cupcake holder should be filled.

Baking might be looked at such a trivial task to do and not so complicated, but it is really a great math and science project talking about how each ingredient plays an important part.  As always, enjoy what you are doing so that the children enjoy also.  The more experiences we are given in working with mathematics, the less intimidating it will become in the future.  My goal is to create a space where mathematics is not for the chosen few to understand, but to make it accessible for all to be successful in because it is that important and beautiful!  :). Enjoy!

copyright 2018 Christina Grossman. All Rights Reserved

 

 

 

Secret Codes, Multiplication, and Making Memories: Deciphering Clues on the Mountain.

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It is time to have the kids explore the mountain.  One likes to decode, the other one needs help in multiplication, and the younger one needs to remember a clue from the book I read to her. What better way to get them moving out and about than a scavenger hunt!

The idea was to write four different clues. The first clue, both boys could figure it out.  This idea came from a childhood of mine, “The Secret Three”.  The boys had to place the secret note in front of a mirror to read it.  The second clue gave one a chance to decipher a simple code as he really enjoys doing them. Clue #2 had to do with assigning numbers to each letter of the alphabet.  It eventually led them to their favorite place to go sledding.

The next clue needed to help my other son with his multiplication. In order to do this, Clue #3 had to do with numbers assigned to letters, but those numbers were products that needed to be calculated through multiplication.  The answer led them to the next clue, which was where we had our fairy garden outside.

The younger one needed to feel she was able to help out by remembering an important part of a book I read to her. It also helped her with rhyming words. 🙂  The answer to the next clue had to do with finding it in a secret room in the house and eventually being led to the prize.  It was a lot of fun and took about 30 minutes for them to get the prize, cookies and hot chocolate! Yum! Here are pictures of each clue:

Something to Think About:

There is a new excitement to teach children how to code on the computer. What I feel is a great introduction though, is to go back to old school secret codes and scavenger hunts. The kids really need to get intrigued by wanting to decipher something and it builds on their critical thinking. By getting them started to learn how to decipher different types of codes and clues, it really gets them to understand how symbols and numbers can be associated to different commands when they get to using code on the computer.  This activity also gets my kids to go outside and explore the mountain, so this also ties into the importance of being in Nature. I plan to do more of this with them in the future!

The one thing I would do differently is to give them each a different clue to take them on their own scavenger hunt and to color code them to help them keep track. There was a time when one was working on the code while the others played nearby.  It worked, but I really wanted them to all be engaged at the same time.

The important thing is to get them excited about math and using it outside, especially if it involves cookies and hot chocolate as the end all of prizes! Enjoy!

copyright 2018 Christina Grossman. All Rights Reserved

 

 

Critical Thinking and Sledding: How to Get the Most Fun Out of Snow.

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On Mount Lemmon, the kids are understanding that there are different stages of snow, or forms of snow.  Lightly dusted snow can melt quickly once the sun comes up.  Snow that has been there for a while can be hard as a rock.  They must waste no time when the snow is just perfect because it might not be like that the next day, or even later in the day.

For this particular day, we were going to check out a favorite area to go sledding.  They had full faith that there was the perfect snow for sledding and they were right!  The rest of the afternoon was filled with pure blissful fun!  However, it was the next morning that would prove to be the challenge.

The next morning, the kids wanted to try sledding again, but there was not any “new” snow. It was the same patch that was used the previous day.  It was also windy the previous night which moved a lot of debris.  As usual, the kids went to the top of the path to sled down and it was a disappointment.  So I began asking them some questions to help in their thinking of what was potentially slowing them down.  Here is a list of things we came up with:

  1.  Pinecones, pine needles, and rocks were on the path, which slowed them down.
  2. A smoother surface would help them sled down faster
  3. The layer of snow was not as thick as the previous day because it was pressed down from yesterday’s sledding and no new snow had fallen.
  4. To sled down, someone had to push the person on the sled for a bit to get going, or having mom pull them down worked too!

The kids still had fun and they helped each other out with getting all the debris out of the way, taking turns pushing each other, and tried to lay flat on the snow board sometimes to go faster.  It was their second time sledding and I think they understand how to think critically to get the most fun out of snow.  I also did my own turn in sledding down the incline to show them what I can do to sled faster to see if that would work for them too!  So where is the math in all of this?

Something to Think About:

In Mathematics, critical thinking is really important.  When you come up with a problem that has a lot of “problems” in it, trying to look at what is keeping you from solving it out is important!  When you can critically look at a situation, without getting upset or feeling frustrated, to see what is slowing you down in solving the problem in the first place, you will become more successful.  It also takes practice.

Since sledding is not intimidating to the kids, it was easier for them to think critically and not give up so quickly.  The more times the kids can do this for things that are “no big deals” to them, the easier it will be to show them how to apply this for any sort of math problem.  It is also important to purposefully tell them that these are the same tools of thinking to apply in any math problem they encounter.

Here are a few questions to ask your child when they are having trouble with solving a math problem:

  1. What is the word problem asking you to do?
  2. Have you done this type of math problem before? If you have, how did you solve that one? (this is like thinking back to when the snow was perfect for sledding the day before)
  3. What is slowing you down? Is there too much information given to you?  What do you really need to solve this problem? (this is like the pinecones and rocks in the path)
  4. Would you like me to help you, or have your brother or sister help you? (this is like me showing them how I sled down the incline to show them what could work)
  5. What if I showed you how to do a similar problem and then you can see if what I did would help you solve the next one?

In reality, the first time we directly apply critical thinking to a situation may not mean that they will be able to transfer that when working with math problems.  It takes time, practice, and directly referring back to a situation in context.  Be patient, everyone learns differently, and most of all, have fun!  Mathematics really should not be presented as something difficult to learn. It was not made to be that way.  Have fun with it!  Enjoy!

copyright 2018 Christina Grossman. All Rights Reserved

It’s All in the Cards: Working with Numerical Operations in One Game!

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While I worked at a university to offer professional development to other teachers, my colleague and I offered a workshop on card games to help with computational skills.  We both explained that it is best to have the children understand the concepts first before doing these games, because if not, they would just be games and not supplementals for learning.  It should be fun and have a connection to meaning as well.  What is great about playing with cards is that, as parents, we do not need to have any set up time to do this.  It also gets us to do something as a family and gets us at the dinner table to interact.  It has great benefits too with siblings, having them play together.

I think many of us parents want to know how to help our child in “knowing the facts”, but sometimes the only resource we have is printing off worksheets from the internet.  What if you had a deck of cards, or number cards, at home that could do just that, help your child with numerical operations like addition, subtraction, and multiplication?  Well, you can 🙂

Have you ever played the card game “War”?  Well, it is much like that.

Instructions:

  • You need 1 deck of cards
  • Or number cards
  • Take out the Joker cards if you have a deck of playing cards and do not use the 0 card in the number cards (you can if you wish)
  • Ace cards count as a 1.  King, Queen, and Jack cards count as 10 (you may want to take these out at first as it may be confusing).
  • Each player (2 players at most) gets 7 cards each.
  • Count to three and each player puts out 1 card on the table, face up.
  • The first player who can subtract the cards (larger value-lesser value) gets the pair
  • The player with the most pairs, wins
  • To make this more with learning and helping to remember the answers, have the player say it aloud to get the pairs after the first run.
    • Example:  “10 minus 6 is 4”, or “10 times 4 is 40”.
    • Once the winner get the pair, have them both write out the equation, or written sentence of the math problem.
      • Example (1):  Ten minus six is four, or 10 minus 6 is 4 (written sentence of what they said verbally).
      • Example (2):  10-6=4 (written equation)
  • Note:  Have the players play this with just the cards for a few tries.  Then have them play one to two rounds with verbally saying the math problem and/or writing it down.  Make if fun first!  You might want to include some crackers or popcorn to use as tools to figure out the subtraction too!

Something To Think About:

At this moment, my son understands the concept of subtraction and counts with his fingers from time to time.  Counting with fingers is a strategy that helps the child with subtraction.  I tend to look up in the air to one corner to think about the subtraction because I am visualizing it in my head and that is a strategy.

There are a number of ways to think about subtraction whether you build a model, draw a picture, verbally explain your thinking, act it out, or write an equation on it to compute.  Sometimes we need blocks, fingers, or pictures to help us think about certain operations like subtraction, addition, and so forth.

Always remember that we can do this game as adults to help us strengthen our memory as well.  Doing this card game is never a reflection on our intelligence.  Regardless of where we are at in life with mathematics, from time to time, we need a bit of a mental warm up.  If stretching and warming up your muscles before exercising is recommended, then so should doing some mental exercises for math!  🙂  Enjoy and have fun!

copyright 2018 Christina Grossman. All Rights Reserved

Cookie Cutters and Snow: A Winter Exploration with Shapes

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There is snow outside, but what if there is not enough to make a snowman? Get out those cookie cutters and make some shapes!

It was still a cold day and the kids wanted to play with the snow. At this point, the snow was more like chunky ice, but still manageable.  So I gave my youngest cookie cutters to make shapes with the snow and it was a lot of fun!  This was one of those great explorations that took of little to no time for prepping and more time for exploring!

Something To Think About:

With using cookie cutters, the shapes become more 3-dimensional and is transformed from their 2-dimensional point of view.   For some eyes, going from 2-dimensional to 3-dimensional can be a challenge. The more explorations one can do with going from looking at a 2D heart to a 3D heart, the better. For those circle shapes, show how a circle in 3D looks like a snowball, a sphere! If you have some empty tissue boxes, or shoe boxes, fill those up as well with snow and see what they can build with them.

Would you like another exploration? Look up videos of people building igloos with snow/ice.  It might be amazing for the little ones to see how blocks of snow can create something that looks like a hemisphere shape. 🙂

Remember to make this a fun mathematical exploration. This is not only a time to learn about math, but a time to get together and make mathematical memories! Enjoy!

 

copyright 2018 Christina Grossman. All Rights Reserved

 

 

 

 

 

 

 

Baking with Arrays: Cookie Multiplication and Division Made with Love

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This particular batch of cookies are shortbread cookies and the filling is from using the last bit of jams that we had in the our kitchen. 🙂   

The weather is finally getting cooler and we can begin using the oven for baking!  Baking is a great time to learn about mathematics!  The best part of it is making arrays with the cookies.   An array is an arrangement of objects, pictures, or numbers in columns and rows.  An array is especially easy to make when we place the cookie dough onto the cookie sheet.  The reason for me even doing this is because I am working with my oldest son on how to think about multiplication in several different ways.  One way to think about multiplication is by making arrays of columns by rows to find the total product.  The most important part of multiplication, for me, is also learning how to apply the multiplication to everyday things in his life.

Before baking the cookies, the instructions on the box/recipe will tell you how many cookies the mixture will make.  Ask the question out loud about how many dollops of cookie dough can fit onto one oven sheet.  For this particular batch, we were able to make 30 cookies.

After the cookies are baked, the next question I ask my oldest is how many rows do we have on the cookie and how many columns.  For this particular one, we have six columns and five rows.  How many cookies do we can by multiplying 6 and 5?  To get this answer he can count out the total cookies that are on the cookie sheet, or have a chance to figure it out in a different way.

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Here comes another situation with having more than one child in the family, how to distribute/divide the cookies equally?  We have three kids in our household who have three different tastes.  One would like to have strawberry jam as a filling.  The second child does not like anything on the cookie.  The third child likes apple butter as a filling.  The conversation is now about division and distribution.  The kids then need to decide how many cookies each of them will have and if it needs to be the same number of cookies.  They need to figure out how to divide thirty cookies by three kids equally.  So they did!  The results are the pictures below. 🙂

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Something To Think About:

With your child, you may want to do fewer columns and rows. You can point out small arrays within the bigger array of cookies.  For example, you can make ask how many cookies are in a 2 by 3 array.  How about a 3 by 3 array?  Look at the shapes of these arrays too!  When you make a 2 by 3 array, is this a rectangle, or a square?  When you make a 3 by 3 array, is this a rectangle, or a square?  The answer is that a 3 by 3 array will make a square and visually will make the square number of 9!  Check out a 4 by 4 array on the cookie sheet, is this also a square?  So is 16 a square number, which is 4 multiplied by 4?  How cool is that!

In any case, make this a fun and tasty mathematical exploration.  Always remember that this is not only a time to learn about math, but a time to get together and make something of love! Enjoy!

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Copyright 2017 Christina Grossman. All Rights Reserved

Symmetry, the Bilateral Kind: Looking into Nature to Find the Balance

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As we approach Autumn, things will be slowing down for us on the mountain and in the desert.  This is a very important time to still be able to go outside into Nature and find the balance.  Take pictures, enjoy the time outside and when it starts to cool down.  Notice what the plants are showing you in terms of symmetry, especially the bilateral kind.

Yes, there are different types the symmetry, but what we are all familiar with is the type that we know when we can “cut” something in half, or see the same image reflected on the other side, or even think of it as the mirror image.  However young or young-at-heart we are, this is something that we can all enjoy learning about!

Go for a walk soon and figure out what is symmetrical.  This would be also a great time to put together a math nature journal too!  Take this mushroom above, where would you place the mirror to reflect the same image?  The place that you have the mirror is called the line of symmetry.  How many lines of symmetry could this mushroom have?  One? Two? More?  Go with those questions when you go for a walk outside.  Don’t have a mirror? Take a pencil or a straight stick with you.  Place the stick where you would a mirror and eyeball it.  Where’s the line of symmetry?  🙂

If you are not able to go outside right now and explore, print out the pictures from the website, or place them up on the screen.  Get out a mirror and see where the mirror can be placed so that the same image can be seen.  Play with it!  In some cases, it might not be exactly symmetrical, but close enough.

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Something To Think About:

Have a conversation with your child, or with someone else about what it means in Nature to be symmetric.  What does a symmetric plant tell the pollinators: the bees, the hummingbirds, and the butterflies?  Is it still alright for something to be not symmetrical?  Yes! There are many beautiful examples of what is not symmetrical, or what we call asymmetrical.  Everything in the Mathematical world has a place and importance, so asymmetrical is also wonderful and tells us a great many stories too!  Hmmmmm, I think I see a future post here!

Learning about Mathematics is not about getting to the finish line first, it is actually getting there and having an understanding and an appreciation for it.  The more we can connect with how our world is tied to Mathematics, the more we can see the patterns unfolding and even the importance of balance. Also remember that a good rule when we go out to explore is that the only thing we take are pictures, drawings, notes, and memories.  It is best to leave the plant in the natural habitat.

copyright 2017 Christina Grossman. All Rights Reserved

 

However You Slice It, There is Always Something Mathematical to Learn From Watermelon: Shapes and Application

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There is not enough to be said about the Watermelon.  When we think of summer, the Watermelon is the iconic fruit of sunshine and a carrier of water that so many of us treasure!  If you just take a moment to look at the Watermelon that you eat, it tells you a story through shapes.

Take a look at the picture above and look at the naval.  That navel is the center of a beautiful multi-pointed star that tells the story of how it began.  This watermelon started out as a flower, as most fruits do.  So as not to forget where the Watermelon came from, there is this beautiful light green and white star on it.  Also take a look at the “vein-like” features of the skin and compare this to the leaves of the plant from which it came from.

At this time, I am explaining this to my little one and telling her about the star and that this is an imprint of the flower that the Watermelon came from.  Her response was, “So this came from a flower?  So I eat flowers?”  Yes!  🙂

(Note:  This might be a good time for you to look up the blossoms of a watermelon plant right now and compare it with the picture above of the watermelon )  

 

 

Take a look at the seeds in the picture above, which I have found to be a treasure hunt in itself to find watermelons with seeds in them still.  The shapes of the seeds are oval, but they also look like water droplets and remind us of how much this wonderful fruit needs so much water to grow and retains it too!

Now take a look at the Watermelon slice below. I sliced this width-wise and saw this beautiful result!  I see spirals and curves and this slice being partitioned in at least three parts.  What do you see?  What story does this Watermelon show you on how it grew?

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Application:  

Lastly, what if you have a small-sized Watermelon and many little hands to eat it?  How would you slice this up?  I learned this great way of slicing through someone on the internet.  This is a great way for little children to be able to hold the slice in one hand and be able to finish it before getting another piece.  What shape is the piece of Watermelon now?  A rectangle? A rectangular prism?  A rectangular prism with a curved edge?  I am always an advocate for trying out different names to call a certain shape or object.  There is a place for exact names of shapes of objects, but I like to give some opportunity for the eyes to explore beyond and see what other possibilities there are.  🙂  And all with a slice of Watermelon!  🙂

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Something To Think About: 

Something as much as a fruit can be looked upon with a Mathematical eye, not a Mathematical eye of analysis, but an eye of wonder!  Everything tells a story, however you slice it.  Enjoy!