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

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:

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

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?

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

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

 

 

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

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.

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

Cookie Cutters and Snow: A Winter Exploration with Shapes

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

 

 

 

 

 

 

 

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

 

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.

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

 

Are You Taller Than a Cow Parsnip?: Measuring in Units of Nature

Cow Parsnip (Heracleum lanatum)

I am so grateful for such a wonderful paradise on Mount Lemmon, being so close to the desert, and being surrounded by a beautiful community.  What a better place to go learn mathematics because there is math everywhere on the mountain.  What a great time to be up on Mount Lemmon and go exploring!

One of my favorite plants on the mountain is the Cow Parsnip.  These are my favorite wonders of the area!  Cow Parsnips remind me of a child where they begin as small little ones and grow above and beyond 5 feet tall, or close to it.  So I guess the big question is, are you taller than a Cow Parsnip?

What is great about being outdoors is that plants do not know what inches, centimeters, or feet are.  In Nature, there is an abundance of resources use to measure and one of these can be the Cow Parsnip of course! So go for a wonderful walk and look for one of these beauties.  Stand up next to one and measure yourself.  Who is taller, you or the Cow Parsnip?

Take the kids out for a walk and see how they measure against these wonderful giants.  Start using the words of comparison like, “shorter than”, or “taller than”.  How many of you would take to be as tall as the Cow Parsnip?  What about the blossoms?  How many blossoms would it take to be the length of your hand?  How about the width of your hand?  Take a look at those leaves!  How many of your hands would it take to be the same length, or width of those leaves?

Something to Think About:

This is cute and all, but how does learning Mathematics really measure up to the “real” world?  Take a child, for instance, maybe that child is sad because he/she hasn’t grown as tall as everyone else, but take a look at the Cow Parsnip with how small the Cow Parsnip began.  The Cow Parsnip starts out tiny and then shoots up to be taller than any other flowering plant I know.

Success can be measured using blossoms and stalks of plants, or the number of tree rings a tree can have in a lifetime.  Nature has so many opportunities where we can learn and explore in Mathematics.  What I would like everyone to get out of these posts is that the world of Mathematics encompasses more than what we can imagine.  The earlier we start our kids and grandkids in interacting and learning from Nature, the more they will have more reason to protect what is treasured by the rest of us.

copyright 2017 Christina Grossman. All Rights Reserved

Diversity Exists in Nature To Teach Us a Lesson: Counting, Pattern Finding, and More!

I could not resist in getting our home a bunch of beautiful red Tulips.  If you have read the other posts about looking at flowers in Nature to look for patterns, well this one is not any different.  Tulips taught me a lesson about diversity.

Fascinated about the black and yellow six-pointed star, I counted the points on the Stigma of the flower.  Three points on the Stigma, then six Stamen, then six red petals on the Tulip.

In Nature, there are flowers demonstrating these patterns of either doubling the original number from the Stigma, or simply repeating the original number from the Stigma.  However, this second Tulip (pictured below) surprised me because this Tulip demonstrated the same doubling pattern, but a different original number.

 

Looking at this particular Tulip, the Stigma has four points, not three points.  Then that quantity of four is doubled to eight Stamen and eight petals!  Wow!  I love Nature!!!! Yes, I realize that these Tulips were not cultivated in the wild, but this one grew to be much different then the other red Tulips.  So this hit me with a great example of showing kids and ourselves as to how Nature and Mathematics shows us that Diversity still exists even when we think we are all the same.

These two Tulips have the same color pattern, have the same green stem, and are both called Tulips.  However, both are still different and unique from each other, which is beautiful!

So how is this important with Mathematics?  Well, just by understanding how to count, how to compare and contrast, and how to find patterns, that can simply take you to a lesson of learning and appreciating Diversity.  Yes, Nature has an abundance of beautiful flowers of different shapes, sizes, and colors!  Each of them have their own purpose of making the world function and be beautiful! How exciting it is to further see that even when we think a bunch of flowers look the same, Diversity still exists!

This is why Mathematics is so very important and near and dear to my heart.  Mathematics is not just solely about numbers, but extends to our very own life and life lessons that we might encounter from day to day.  What a wonderful way to introduce this particular lesson to your child about how Nature can beautifully teach us about the importance of Diversity and celebrating the Diversity!  🙂

 

Something To Think About:

The world of Mathematics is in your home and outside, so go explore what new lessons are out there!  Always remember that it does not matter who crosses the finish line first, it is actually getting there that is more important!

If you would like to see the article describing number patterns on flowers, go to

https://learningmathwithmom.wordpress.com/2016/07/01/theres-math-in-them-there-mountains-spending-time-outside-exploring-math-and-building-vocabulary/

copyright 2017 Christina Grossman. All Rights Reserved

Concentricity: Triangles, Circles, and Even More Shapes in Our Food, Life, and in Nature

Mount Lemmon, AZ

 

Concentricity is something that others would say is too early to teach the little ones about.  I strongly disagree on this note.  Basically, Concentricity happens when there is a common center shared among other circles or spheres.  What I have found are some other examples that go outside of the rule of circles.  With the kids, we have found examples with triangles and ovals.

So by trying to keep this short and sweet for this month’s post, look at the examples below.  When Concentricity happens out in Nature, it is so easy to just point that moment out to our children, regardless of age.  This happens outside our home, inside the home, and in our food.  I hope this encourages you to go find more examples where you live. 🙂

Concentric Circles: 

Tree rings are about the most common one that all of us can identify as they are concentric circles.  The sliced beets you see below are candy-striped beets and they taste delicious!

 

Concentric Triangles:

This Pine is a wonderful example of concentric triangles. So go out to the forest and see what the trees can teach you! 🙂 Let us not forget the strawberries too!

Palisades on Mount Lemmon, AZ

Strawberries 🙂

 

Concentric Ovals and More: 

Kiwi is just as delicious and a great tool to show Concentricity.  This beautiful Agate can be debated on whether this shape is an oval, or an ellipse, or a five-sided figure.  Either way, I see the Concentricity here too 🙂 Do you?

 

 

Something To Think About:

How about squares? When do concentric squares happen?  How about other shapes? Where do you see them?  Think about it and start exploring!  That is the beauty about Mathematics as it is out there and there are so many opportunities for more than one answer because that is just how it is.  Just because a definition of Concentricity only includes circles, or spheres does not mean that there is no likelihood of an exception for that definition.  When we are able to discover and see the exceptions when they happen, that is where the heart and beauty of Mathematics exists.  These interesting surprises are the most important to point out to the children!  and to you!

Learning about Geometry, especially Concentricity, at such an early stage in life gives them the opportunity to see it everywhere.  Once a child, or any person of any age, is able to see the Geometry, it becomes more tangible when it is time to learn more in the classroom.  It then becomes more relevant and more important to them to learn because Mathematics is even more interesting than rules and procedures, it is about exploring and experiencing what happens outside your door.

Have fun learning and exploring together! 🙂

copyright 2016 Christina Grossman. All Rights Reserved

Stars in the Forest, Not Just in the Sky: Geometric Stars in Plant Life.

 

Scarlet Gilia (Ipomopsis aggregata)                                                                                                                                           Copyright 2016 Christina Grossman. All Rights Reserved

On Mount Lemmon, this is the first time I have seen a Scarlet Gilia.  It is so beautiful and unlike any flower that I have seen on the mountain.  This particular Scarlet Gilia was located on Turkey Run across from the community center.  After having the gift of encountering this, it encouraged me to share this out with the rest of you and your kids that Stars do exist in our world, not just in the sky.

For the little ones, stars just may be the easiest shape to learn because they are so special looking.  So why not begin with teaching them about things that are shaped like stars and point them out in the plant life and in food!

Up on Mount Lemmon, you have the beautiful Scarlet Gilia that is a five-pointed star.  The Cow Parsnips that bloom earlier in the summer look like spheres from far away, but look closer and you see little individual blossoms that are five-pointed stars!  As these bloom until October, Richardson’s geraniums on the mountain are these beautiful dark lavender blossoms that are also five pointed stars.  So go take a walk with the kids, or even for yourself and go find these beautiful stars during the day.

For other plants, it depends on where you live.  If you live in cooler areas, you may be able to look at the Lilies of the area and notice that their blooms are six-pointed stars.  Take the Hollyhock.  Look from within the bloom and you will see this beautiful green five-pointed star in the center.  In the desert, look for the Aloe Vera plants and look from above to see the star.

In food, we have stars that form on the top tomato plants (five-pointed stars), onion blossoms (six-pointed stars), and on the pomegranate fruit (six-pointed stars).  If you cut an apple horizontally, you get to see a star there as well (five pointed star)! So get the kids in the kitchen and explore which fruits and veggies have stars in them, or cut them into stars.

Hollyhock (Alcea rosea) Ft.Ticonderoga, NY                                                                                  Copyright 2016 Christina Grossman All Rights Reserved

 

Aloe (Aloe vera) Tucson , AZ                                                                                                                 Copyright 2016 Christina Grossman All Rights Reserved

Cow parsnip (Heracleum lanatum) Mount Lemmon, AZ                                                           Copyright 2016 Christina Grossman All Rights Reserved

Tomato Plant Tucson, AZ   Corpus Christi Catholic Church Garden                                                                                                                         Copyright 2016 Christina Grossman All Rights Reserved

Richardson’s geranium (Geranium richardsonii) Mount Lemmon, AZ                                 Copyright 2016 Christina Grossman All Rights Reserved

Wood Lily (Lilium philadelphicum) Queensbury, NY                                                                   Copyright 2016 Christina Grossman All Rights Reserved

 

Pomegranate Fruit Tucson, AZ                                                                                                            Copyright 2016 Christina Grossman All Rights Reserved

Onion Blossom Queensbury, NY                                                                                                          Copyright 2016 Christina Grossman All Rights Reserved

 

Something To Think About:

To learn about shapes and to teach them to your young child, it really is not difficult.  You either need to look at the food you eat, or look at the plants outside.  Begin with teaching them about stars because they are everywhere you are,  up in the sky or growing out of the ground.  It just takes a moment to open your eyes and the eyes of your children so that you get to see and enjoy the beautiful geometries, the beautiful shapes that are already around you.  🙂

Learning about geometry at such an early stage in life gives them the opportunity to see it everywhere.  Once a child, or any person of any age, is able to see the geometry, it becomes more tangible when it is time to learn more about geometry in the classroom.  Geometry then becomes more relevant and more important to them to learn.

Have a wonderful time searching for the stars!

copyright 2016 Christina Grossman. All Rights Reserved

 

 

Taking a Closer Look at Nature’s Flower, the Carnation: Counting and Comparing

It was a very hot day of over 100 degrees outside and we needed something to do in the house.  After having carnations left over from a bouquet, I made the decision to use them to teach my children and my nephew about the parts of the flowers and counting.  For this, I gave each child a carnation, a piece of paper, tape, and glue sticks.

The question that I gave them was to find out if the number of petals for each flower were the same as the other carnations.  The other task I wanted them to answer was to count the number of stamen and compare it to the number of petals as well.

So the children began taking a part the flower petals and spreading them out.  All I have to say is how beautiful everything smelled in the room! As you look at the photos, some of them had 37 petals, while another had 36 petals.  The numbers were all within range of each other.  It was really interesting for them to work carefully in pulling apart the carnation flower from the stem and then finding out how difficult it was to pull the stamen and stigma apart from each other.

For the number of stamen and comparing this to the numbers of petals, I will have you figure this out with your own children at home.  This idea came from my time taking photos of the flowers on the mountain close to us.  Sometimes the comparison was equal, while other times it was double or half of the other.  The idea for doing this and observing is to acknowledge that there is a relationship here.

This activity took about 10-20 minutes to do.  Most of them could do this on their own, but sometimes needed help in taping, or glueing this onto the paper.  We did let the petals dry and we put the rest of the flowers into the compost.

Something To Think About:

On the photo above, you see the number 28 and then the number 36.  For this, I asked one of the them to guess the number of petals he had on his paper.  He guessed 28 and then we counted them together to get 36.  So his estimate was not too far off from what the actual count was.  Estimating is a wonderful skill to have, so even doing this is a great example of how to have your child practice estimation. When you go get blueberries, or raisins in a box, have the child estimate the number and then count them and eat them!

Remember, doing mathematics at home does not have to be fancy.  You can go outside and look at the flowers and count petals.  If you have cut flowers at home and would like to get some more use out of them, think about doing this activity.  You do not even need to have paper and glue handy.  Just have them take a part the flower and start counting.  🙂  Take a closer look around you and start counting and comparing.  🙂 Below are more examples of the work the children did that day!

copyright 2016 Christina Grossman. All Rights Reserved