Our spring dandelion days create hours and hours of exploration, inquisition and just plain observation!  There was plenty of math happening everywhere we turned.  One day we collected dandelions just to see how many we could collect.  The five year olds collected more than one hundred, while the two year olds were happy with six.  That is developmentally appropriate math right there!  When we find ourselves in these nature-based outdoor classrooms, the learning is always developmentally appropriate and always child centered.  It is the beauty of learning in a place that gives us everything we need. “When you look at a field of dandelions, you can either see a hundred weeds, or a hundred wishes.”
We had plenty of math vocabulary going as we searched for the longest and shortest dandelions.  We looked at the circumference of the flower, and made flower bracelets out of them.

We discussed the number of pedals.  We found the pattern of petals and I introduced them to the term “Fibonacci”, a number pattern that we often find in nature.  On this day, the flowers were too tall to spend much time on patterns.  We kept finding longer and longer stems, some with flowers, some with wispy white seed heads!  We discovered that we could divide the stems in half! Wait! We could even split them into four sections! Would it be possible to blow through a stem? Would it make a whistling sound? Would it taste bitter?  The investigation and process of discovery with dandelions was fast and furious, yet lasted for hours. This was math at their level, on their timetable.  Why would we rush this?I watched as some very young children could subitize better than their older friends.  Subitizing is the ability to “see” a small number of objects and know how many are there without counting.  When we roll a dice, we don’t need to count the pips, we know the number when we see it! Some children seem to grasp this concept with ease while others need to work with it a bit more.  We subitize a lot in our program, and being the math geek that I am, it just fascinates me to watch the difference in learning styles as this concept becomes effortless!

Giggles and screams of discovery were filling the hillside as our dandelion math morning took on a life of it’s own.  They began to classify, grouping according to length or size of the flower head.  I watched as a game developed of who could find the tallest one.  When you are yards away from your friend AND on a hill, it is hard to distinguish until you pick it and compare sizes! Then the realization comes that you need to pick at the very, very bottom of the stem!  This was a concept that was way beyond the comprehension of some of our younger friends, as hard as their peers tried to teach them.

Ah, the beauty of multi-age groups.  The beauty of allowing learning to enter as the brain and physical development allow.  The beauty of friendships and childhood on a sunny spring day, when all the stars align and the learning comes so naturally.

There are few manipulatives out there that are as interesting and beautiful as a wooden set of Cuisenaire® Rods.  Developed 75 years ago by Belgian teacher Georges Cuisenaire these “rods” come in beautiful colors in varying lengths.

Using Cuisenaire® Rods to compare length is as simple as putting shorter rods next to longer rods and seeing how your children observe those differences.  Although these manipulatives were designed for a very specific purpose (units of 1, 2, 3, etc.) I think  it is far more likely that children will explore the rods by laying them out, standing them up, and comparing them.

Most young children will be able to identify which rods are shorter and which are longer, especially when they are laid out next to each other.  It is far more difficult for children to compare several rods of differing lengths simultaneously.  Putting many of them in order from shortest to longest is really challenging because it asks children to think about 2 things at the same time; which rod is shorter than these – but longer than the others?

If you look carefully at the above photo, you can see that the units of 1 are white and the units of 2 are red, 3 are green and so on.  They provide a visual representation of number units, up to 10, or for today’s purposes shortest to longest.

Compare numbers.

• CCSS.Math.Content.K.CC.C.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.¹
• CCSS.Math.Content.K.CC.C.7 Compare two numbers between 1 and 10 presented as written numerals.

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When small groups of objects are presented to young children, unless they are very different (1 toy car next to 10 toy cars), they may not be able to see the difference in quantity right off the bat.  Remember, they are confused by appearances so if you put 3 large cars next to 5 small cars, the child may believe that the 3 large cars are “more” than the 5 small cars, because they look like more i.e., they take up more room so in the child’s mind that is “more”.

There is a developmental process that needs to shift for the child to conserve quantity.  This often does not happen until the kindergarten year, and sometimes even after that.  However, that doesn’t mean that we shouldn’t provide children with as much experience as possible in comparing quantity.

When distributing items to children, ask them to compare who got more, who got the same and who got less.  Using their counting skills and one-to-one correspondence, they may be able to count the items and determine the answers.  If they are 3 and under, they may simply guess.  If they are able to count, you can use a number line to help them see which number is bigger or great and therefore, “more”.  The best way to reinforce the above concepts is to compare numbers frequently throughout the day, so you can maximize children’s exposure to the concepts, use as many visual cues as you can so they can access different ways of knowing, and give them credit for trying.

Make sure you provide lots of written numbers to be “read” around the room.  If your program is filled with written words, try and match the words that children are exposed with the same quantity of written numerals.  This will make a huge difference in gaining familiarity with written numerals.

Learning Standard C – Begin to make reasonable estimates of numbers.

There is only one Benchmark for Learning Standard C

6.C.ECa  Estimate number of objects in a set

The Example Performance Descriptors are:

Make reasonable estimates of small quantities of objects (e.g., goes “four” when asked how many peach slices are in the bowl).

Tell whether a set is more or less than 5.

Tell whether a set is more or less than 10.

This is interesting to me since I know that children like to “guess” at how many, or how old, or how long… However, we often see that children are not quite successful at most of their estimates.  Remember, if a child is confused by appearances, that objects that are “big” may appear to be “many” and objects that are “small” may appear to be “few”.

With small sets, young children will have more success at estimating.  However, looking at a group of around 10 objects and being able to “tell whether a set is more or less than 10” is a fairly lofty expectation.  I can barely do that, depending on the objects in the set.  I often have to count.  Now, if they mean that I can reasonably know that 2 objects in a set are less than 10 and 100 objects in a set are more than 10, I can do that.  It is when the set is around 10, that I would have difficulty estimating without counting.

I also think it is important for young children to verify if their estimate is correct.  You can do this with simple counting.  Frequent experience with estimation and counting will support both Learning Standard B and C.

Learning Standard D

Compare quantities using appropriate vocabulary terms.

The Benchmarks

6.D.ECa   Make comparisons of quantities

6.D.ECb  Describe the comparison with appropriate vocabulary, such as more, less, greater than, few, equal to or same as.

The Example Performance Descriptors

Match sets of things that go together 1-1 and determine whether one set has more, less, or an equal amount (e.g., compare the number of napkins to place setting at the table).

Demonstrate an understanding of equal when dividing materials (e.g., divide cars equally between self and friend).

Use appropriate vocabulary to make comparisons of quantity (e.g., acknowledge that another child has more blocks).

“More” and “less” are really interesting mathematical constructs for young children because they are deeply important to the egocentric child.  Who has “more” and who has “less” cookies or Legos matters to the egocentric child.  They care deeply about themselves and this construct feeds right into that part of their psyche.

Remember, using mathematical vocabulary, whenever you have an opportunity will reinforce the acquisition of the terms as well as the absorption of the meaning.

My husband told me that his father had a system for dividing fairly for he and his sister.  If there was one cupcake left in the house his father would tell them that one had to cut and the other had to choose.  That system required the cutter to think about dividing as equally as possible, fully knowing that if there was one side that was bigger, the chooser would select it.  We use this system in our house and, for the most part, it works.

Children also have a deep sense of justice and fairness.  This mathematical concept will also appeal to that system of “right” and “wrong”.