Radials in Nature: Plant Life That Reflects the Starburst, Sunburst, and Nature’s Fireworks.


Wheeler Thistle (Cirsium wheeleri)  Rose Canyon Lake, AZ                                    Copyright 2106 Christina Grossman. All Rights Reserved

There are so many beautiful teachable moments up on the mountain and everywhere else, when it comes to plant life.  As children, we are taught about circles and squares and triangles, but what about radials-starbursts and sunbursts?

Radials are something that begin in the center and bursts out from there, like fireworks.  They capture our attention.  They make us stop and look and admire the beautiful way that it grows.  Some could be easily described as growing in a circular pattern, or like a globe.  This is true, but it could also be described as a radial pattern.

Look at the photo above of the Wheeler Thistle.  This flower is blooming at Rose Canyon Lake near Mount Lemmon, AZ.  From afar, it looks more like a globe, or a sphere.  When you look a little closer, it looks like fireworks!  Most of the grasses on Mount Lemmon also grow from the center and branch out as well. It is really amazing just how many radial forms are out there.

When you are walking through the desert, look at the Aloe Veras and Agaves.  Take a moment to look from above and look down to the center. They all begin from the center and grow outwards in a fireworks pattern.

Is it enough for my small child to just know what a triangle is, a square, a circle, a rectangle, a heart?  The answer is yes.  If you can show more shapes at an earlier age, then you can open their eyes to see a lot  more than just what we are normally taught.  In some cases, things do not necessarily belong just with circles, or just with triangles.  Depending on how you see the plant for that moment, it could be both, or something entirely different.  This is how mathematics is, observing the behavior and seeing for what it is for that one moment.

Take the Blue Globe Glow Thistle.  Once the petals have fallen off and I look from above the blossom, I could say that this is a radial pattern, something that looks like fireworks.  However, when the petals are still there, as shown below, I could say that this looks like a sphere, or little stars on a sphere.  It just depends on how you look at this flower.  Nature shows us how many different names and answers we can come up with by the way we look at things.  In many cases, problems in mathematics may have more than one answer, just like this Blue Globe Glow Thistle. 🙂

Take a look at the other examples of radials by scrolling down this page.


Blue Globe Glow Thistle (Echniops bannaticus) Fort Ticonderoga, NY                                                                          Copyright 2016 Christina Grossman. All Rights Reserved


Blue Globe Thistle (Echniops bannaticus) Fort Ticonderoga, NY                                                                                    Copyright 2016 Christina Grossman. All Rights Reserved

More Examples of Radials in Nature:



Dill Flowers (Anethem graveolens) Fort Ticonderoga, NY                                                                                                                                        Copyright 2016 Christina Grossman. All Rights Reserved


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



Swamp Thistle (Cirsium muticum) Queensbury, NY                                                                                                     Copyright 2106 Christina Grossman. All Rights Reserved



Queens Anne’s Lace (Daucus carrot) Queensbury, NY                                                                                                    Copyright 2016 Christina Grossman. All Rights Reserved



Grass on Mount Lemmon, AZ                                                                                                            Copyright 2016 Christina Grossman. All Rights Reserved


Grass on Mount Lemmon, AZ                                                                                                          Copyright 2016 Christina Grossman. All Rights Reserved

Something To Think About:

So there you go, radials are another shape, or growth pattern in plants that we can teach our children about and ourselves.  I will continue to say that Mathematics is everywhere.  The Mathematics out in Nature is free and teaches us that there is more out there than we are taught in the classroom and there is always more than one answer to something, it just depends on how we observe for that moment in time.  If you do not have plants growing where you live, print these out, or point them out to your children.  Have fun with your kids outside as much as you can!


For more information about Fort Ticonderoga in New York State, especially the beautiful King’s Garden, or to donate, please go to http://www.fortticonderoga.org

Copyright 2106 Christina Grossman. All Rights Reserved

Hearts, Hearts, and More Hearts: A Shape We Can Find in Nature


Common Yellow Wood Sorrel

If you have good weather today, go take your child outside and look for hearts!  Hearts actually exist in Nature and are the shapes that will brighten any person’s eyes.  Find hearts in the Cacti, the Jasmine leaves, strawberries, Wood Sorrel, or your potatoes.  The potato is one that my mother found in her bag of potatoes.  It was exciting to have the kids receive this from her. 🙂  Hearts are certainly everywhere.

My children, including myself, have a new tradition of looking for heart rocks, rocks shaped like hearts.  We either keep them to remember our memory together, or give them to each other to show how much we care.  Heart rocks really rock in our family!



Something to Think About:

When exploring Mathematics in our world, it does not need to be anything formal when we are with our kids.  The purpose is to see, touch, experience, smell, and feel the Mathematics in our world.  Mathematics is everywhere and what better classroom than in the home and outside. 🙂

Remember, it is not about getting to the finish line first, it is about actually getting there.  Whether you are an adult, or a child, it is always a good day to learn the mathematics, especially if it means spending more quality time with your children. 🙂

Copyright 2016 Christina Grossman.  All Rights Reserved




There’s Math in Them There Mountains!  Spending Time Outside, Exploring Math, and Building Vocabulary




Mount Lemmon, AZ . Copyright 2016 Christina Grossman. All Rights Reserved

(On Mount Lemmon, Arizona )

Summer brings grandkids, our own kids, nieces, and nephews up to the mountain.  This is the greatest time for kids to explore and enjoy the mountain life. Even we can be kids at heart and enjoy what the mountain has to offer! But what’s that you say? There is math on these mountains?  Of course!

As a mother and a mathematics teacher, I take every opportunity to teach my children about math, even on Mount Lemmon.   Here are a few ideas of how to look at the natural landscape of our beloved sanctuary for the little ones:

Shape Watching on a Nature Walk  

    1. Small Pine trees are growing all over and some are the right height for young children to see the top of them.  Have them look at the top, looking down, and ask them what they see?  Do they see baby pine cones growing, they are called cones for a reason because cones are shapes.  Also, look at the way the needles are growing, they form a spiral!


      Pine. Copyright 2016 Christina Grossman. All rights Reserved

    2. While you are looking at the Pines, look down below because the Ferns are growing.  Did you know that they grow from unraveling from a spiral? When the leaves stretch out, they are little triangles too!


      Fern. Copyright 2016 Christina Grossman. All Rights Reserved

    3. There are many tree stumps around, so take a closer look.  What shape do you see?  We see circles, tree rings. Together, they are concentric circles, like a bullseye.  What if you dropped a rock in the stream, would that make “tree rings”, or concentric circles in the water too?


      Tree Stump Showing Tree Rings. Copyright 2016 Christina Grossman. All Rights Reserved

    4. What about when the Common Dandelions (Taraxacum officinale)come out?  The parts that we like to call Dandelion wishes are small spheres.  If you look closer, each little part of the white “Dandelion wish” are little parabolas.


      Common Dandelion . Copyright 2016 Christina Grossman. All Rights Reserved

Counting on Nature to Teach Us Numbers

    1. Let us look at the Richard’s Geraniums (Geranium richardsonii) and Cranesbill (Geranium caespitosum) that will be growing soon.  Count the number of stamen of the flower and count the number of petals on the flower.  Is one double of the other? Is one half of the other? Are they the same?

      Cranesbill. Copyright 2016 Christina Grossman.All Rights Reserved


      Richard’s Geranium. Copyright 2016 Christina Grossman. All Rights Reserved

    2. Take a look at the Big Leaf Periwinkle (Vinca major).  Looking at the center, you see a pentagon, then out of that, another pentagon, and then the five petals.  This flower is all about the number five 🙂


      Big Leaf Periwinkle. Copyright 2016 Christina Grossman. All Rights Reserved

    3. When the Common Sunflowers (Helianthus annuus) start growing, count the number of petals they have.  Do you get an odd or even number of petals?  How many?  What about the flowers on the Common Mullein (Verbascum thapsus), count the number of petals on one little flower.  Do you get an odd or even number of petals?  How many?



Something to Think About:

These are simple things to open the mind of the child, and the child-like hearted ones we all are, to seeing the mathematics out in Nature.  Even if this takes five minutes to the whole entire day, everyone went outside, explored mathematics, and learned and used new vocabulary. This is a great way to show relevance of learning shapes and numbers because it is out there in our world.  🙂

Also, a many thanks to Frank S. Rose’s work on Mountain Wildflowers of Southern Arizona as it was a great resource to me to finally learn the names of the wonderful plant life we see on Mount Lemmon, AZ.

copyright 2016 Christina Grossman. All Rights Reserved


Spirals: Looking for Them in Nature and Creating Art with Them


Fern on the mountain that grows out from a spiral. © Christina Grossman. All Rights Reserved


The Summer season is approaching and we are at the moment between the end of Spring and the beginnings of Summer.  For most areas, it is time to go out and explore.  This would be a great time either go look for spirals, or create them with your children.

There are so many, many things that grow in spirals.  You can talk about the solar system that we live in to your child and show how the planets, starting from the center that is the Sun and show how the movement is in a spiral, while each planet does move around the Sun in an elliptical motion, the whole system moves inward, out.  Because we cannot take to the outskirts of our solar system, we can show them how living things in the world we live in also grow in a spiral fashion.  We can also make spirals for art. 🙂


Look at the Fern, in the picture above.  The way that this Fern grows is first starting in a Spiral and then opening up by unraveling itself.  If you look closely to the ends of this Fern, you can see little tiny spirals that are waiting to unravel.  I am sure there is a better scientific way to describe this, but I am here to show you how you can teach your child about different shapes, which includes spirals 🙂



Sea Shell from the ocean that I used for a Mandala for water ©Christina Grossman All Rights Reserved


There are many shells that have circular spirals and are fun to look at and find on the beach.  There are also shells that are more conic, cones, that form in a spiral fashion.  So, if you live near a beach, or a craft store, or dollar store, go see what you can find.



Fern on the mountain that grows out from a spiral. © Christina Grossman. All Rights Reserved


We get a lot of things from the nearby dollar store for arts and crafts and such.  These glass beads are so beautiful to use to make art with, especially spirals.  This gives the child and you a chance to figure out how to space things out so that everything is symmetric, or balanced on both sides.  For children who are not able to draw spirals in the beginning, this gives them an opportunity to create one with stones.  Glass beads are not required, but you get the idea.  You can even print this one out and start from the first light blue glass bead from the center and start singing the ABCs.  You have enough light blue beads to complete the whole song. 🙂



Fern on the mountain that grows out from a spiral. © Christina Grossman. All Rights Reserved


There are some leaves, that when they fall off the branch and begin to dry, they make spirals.  If you are living in the Southwest, you might be more familiar with these leaves.  They are all over our yard and they dry up in little beautiful spirals.



A vine from our Carolina Jasmine in our yard.  ©Christina Grossman. All Rights Reserved


These vines have these yellow trumpet like flowers that hummingbirds, butterflies, and bees love when they are in bloom.  The interesting way that Carolina Jasmine grows is how the vines spiral out when they grow.  This is how the Carolina Jasmine grows in different directions, whether on a lattice, around a tree, or attaching to a bush.  The Carolina Jasmine does not have any “grippers” or “creepers” to attach to a wall or a tree, it actually grows in spirals, as you can see on the picture above.  All you need to do is just pay attention.  🙂


Fern on the mountain that grows out from a spiral. © Christina Grossman. All Rights Reserved


Cabbage!  Slice up your red/purple cabbage to reveal that not all spirals are formed by circles.  If you look at the center, you see a five-pointed star.  On the outer perimeter of the star is a pentagon.  This pentagon rotates and rotates and you see a spiral as well 🙂 Give this to your child to trace the spiral with their fingers.  If you don’t have cabbage, print this picture out, or show this to them on the screen!


Spiral art made from Sycamore leaves from smallest to biggest. © Christina Grossman. All Rights Reserved


This photo of the spiral made from Sycamore leaves is not a new addition to this blog.  This was a short activity to involve my children in when working on the idea of smallest to biggest.  Do this with dried leaves, fresh leaves, flowers, cereal, or rocks.  The kids really liked this one, but you may find that keeping this as few of leaves as possible will guarantee that the kids will finish it.


Something To Think About:  

How will this prepare them for Kindergarten, or for a mathematics readiness test?  It does something even more important than that.  You showing them that the world they live in has mathematics gives them relevance and excitement.  Learning mathematics and exploring mathematics is not all about formulas, it is really about understanding how much our world lives in mathematics.  Whether it is how a plant spreads out in growth, the food we eat, or artwork that we create, the more involved we feel in something we are learning, the more we lead ourselves to a better understanding of it.

My oldest son might just say “Ya, ya, I know it’s a spiral,” but I will keep doing this over and over when I get the chance to just take a moment to point things out in his world.  It is like showing a child where vegetables come from and starting a garden, or picking from a garden.  It offers an opportunity to become more connected with the world.  So go outside and have some fun and take a few moments to point out the spirals that you see.  Enjoy the time you have with your children and make it fun. 🙂


copyright 2016 Christina Grossman. All Rights Reserved




Celebrating Autumn From Smallest to Biggest and Spiral

This kids are 1,4, and 5 at this point last. There are leaves everywhere from the Sycamore tree. So where there is nature, there is math. 🙂

This was really a great time for us to be together outside and looks for leaves.  My older ones were particular about the way the leaves looked. It took a bit of time, but it was worth it. There was a lot of comparing and contrasting to see which one was smallest to biggest, but it came naturally, as it should.

The boys were helping me put the spiral together with the leaves, but the rest of the leaves were calling out to them to play. 🙂 I finished the rest of the spiral and they did get to see that we can make art out of nature. That same day, one made a horse out of leaves, while the other one was still trying to find the smallest leaf and the perfect leaf. 🙂

What Math Concepts Were Learned?


  1. Biggest to Smallest, Smallest to Biggest
  2. Compare and Contrast
  3. Problem Solving
  4. Spatial Reasining when putting together the spiral

Something to Think About:

When Autumn comes, there is a lot to explore and play with. There are rocks, sticks, leaves, acorns, and such to compare and contrast. Then make art with it!

Remember, it is not that you get there first in learning, it is that you get there in the first place.

copyright 2015 learning math with mom

copyright 2015 learning math with mom

Biggest to Smallest: Comparing Things from Nature in any Season! Even in Color and Shape



Here in the desert area, we do not normally have Autumn weather like other parts of the country during the months of September through November, but some of the trees do change color and drop their leaves. At our house, we have a tall Sycamore tree with leaves of various sizes that really should not be in the desert, but has survived all this time through tons of watering and care. So, I took this opportunity for this Autumn Season to have the boys pick up some of the fallen leaves and pick their favorites.

At this time, my boys are 4 and 3 years old. Picking up leaves, for them, is really not too much to ask for from mommy. J As we brought them in, we talked about the colors they saw on each of the leaves, counted the number of points on each leaf, and I was wondering if they would be able to choose the biggest leaf to the smallest leaf. Looking at the picture above, there are only seven leaves, which is a good number of one thing to compare for a 4 and 3-year-old. Any more than ten might be a bit much, especially if it is the afternoon before nap time.

Placing one leaf on the floor, I asked them if this was the biggest leaf. It was not. I kept asking questions and listened to what they had to say. As this kept going back and forth, back and forth, it took about five to eight minutes to get what you see on the picture. Here are questions that I asked the boys that resulted in the above picture:

  1. Which one is the biggest leaf, this one, or that one? Why?
  2. Which one is the next biggest leaf?
  3. Which of the leaves is the smallest? How can you tell?
  4. What about this leaf? Is this bigger or smaller than the first leaf on the floor? (you can overlap the leaf, one on top of the other, to see which one is bigger or smaller).
  5. What if we did smallest to biggest, what would that look like?



For the rest of the world that has a real winter, here are some ideas that you might want to take advantage of when comparing things in nature from biggest to smallest. Although most of the fallen leaves are gone or underneath snow, it would be a great time to also talk about color and why some leaves are still green and on the tree, instead of gone or brown.

  1. Pinecones! If you have different conifers near you, take some fallen pine cones and compare them. Take a look at the way the pine cones are created. Do you see spirals? What shapes do you see?
  2. Snowballs! Before a snowball fight, make some that are small and big and compare them.       This is a great way to start exploring 3-D shapes! What solids/3-D shapes can you make with snow and how? What 3-D shapes are easier to make than others and why?
  3. Icicles! With this one, you need to be careful and very cautious because icicles can be really dangerous to be underneath. From a distance, you can compare the length of the icicles and also look at what shapes they are. This is a special treat because you can explore these from inside a warm house too!





Spring is my all time favorite season because it is a time where everything comes back to life and it is cool and warm enough to go outside and explore! Why not look at different things outside and compare them. Make it like a scavenger hunt to find different flowers, leaves, and rocks. Then bring them all together and compare. What colors do you see? What shapes do you see?

  1. Flowers
  2. Green leaves
  3. Vegetables in the garden
  4. Plants




This special season can be a way to get out of the heat and explore things in different parts of the country on a vacation, or even at the local zoo.   Compare things from biggest to smallest and you will find that your children will have an interesting way of thinking about what big is and small is and tall is. It also depends if something is standing upright or on the ground flat. If you can go to the zoo or look up pictures of animals, compare the size, colors, and shapes of the animals as well 🙂

  1. Sea Shells
  2. Flowers
  3. Zoo Animals

Where is this on the Schedule of Activities from 18 months to 3 years of age?


  1. Wednesday on Big and Small
  2. Mondays on Shapes


Where is this on the Schedule of Activities from 3 years to 5 years of age?


  1. Monday on Shapes
  2. Tuesday on Measurement (if you are using a device to compare measurements, like a ruler)
  3. Thursday on Geometry of 2D and 3D
  4. Tuesday ( that I will soon add on Tuesdays is comparisons)


  1. Thursday on Geometry of 2Dand 3D
  2. Tuesday ( that I will soon add on Tuesdays is comparing)



Something to Think About:

Mathematics is not always learned on paper or with technology, though these things are great in their own moments. The reason why most of us explore the world of mathematics is because nature provides this outside. Nature, in every season, is a playground of math waiting to be played and explored. So, take advantage of what is outside your door. Even if it is that single tree outside, rocks, or flowers. Go outside with your kids and start asking questions. Show your kids how to look at things around them and to take notice of the math that is there. Sure, I know we need to clean the house or we are busy trying to figure out how to manage everything, but just take a moment to go outside and breathe. These moments are not only a way of learning about the mathematics, but making memories with your kids. These are priceless moments with our kids.

As an idea, even if you cannot go outside, or have a local zoo in your area, just print out the pictures in this blog or read this with your kids and start comparing; start asking questions. At least this gives you time together and gives you time to slow down and breathe, making memories. 🙂

copyright 2014 learning math with mom

copyright 2014 learning math with mom