Saturday, October 10, 2015

Number Sense: More than just counting and writing numbers

number sense and counting

As children enter kindergarten, they bring with them many prior experiences, one of them being counting or what we know better as reciting number sequences. This is very similar to children singing the alphabet as it is easy for us to confuse this experience with numerical knowledge. Numerical knowledge has four interrelated aspects that help children develop early number concepts and number sense.

The Four Interrelated Aspects Are:

 number sense and counting

  1. Number sequence. The names and the ordered list of number words
  2. One-to-one correspondence: Counting Objects by saying numbers words in a 1:1 correspondence with the objects
  3. Cardinality. Understanding  that the last number word said when counting tells tells how many objects have been counted
  4. Subitizing. Quickly recognizing and naming how many objects are in a small group without counting. Young Children can recognize and name quantities of objects that are less than four without counting (Clements & Sarama, 2007)
These numerical aspects are crucial in the primary grades (Pre-K through 2nd grade), if students do not develop these aspects or number sense, they will struggle with addition, multiplication, and everything else that revolves around numbers. Therefore, we must keep in mind that, "Children must construct these ideas through a variety of experiences and activities. They cannot be forced. As Children work on each of these aspects of early counting, their understanding about counting is continually refined. For example, children will learn how to count  (matching counting words with objects) before they understand that the last count word indicates the amount of the set or the cardinality of the set. We can at the same time use subtilizing to emphasize the notion of cardinality and to help emphasize the notion that counting tells "how many." ( Van de Walle, Lovin, Karp, & Bay-Williams, 2014) So how do we develop number sense?

Number Sense, Developed by Building Number Relationships
Number sense is something that grows and develops over time, Howden (1989) described number sense as a "good intuition about numbers and their relationships. It develops gradually as a result  of exploring, visualizing, and relating them" to each other. Number Relationships also has four parts, that will be discussed in another blog post.  At a glance, they are:

number sense and counting

In essences math lessons must revolve around these 4 aspects and 4 relationships. If we give children these opportunities, they will develop number sense while acquiring a conceptual understanding of mathematics! What it may look like....

Number Sequence & 1:1 Correspondence

number sense and countingnumber sense and counting

After reciting numbers with videos, songs, dances, chants, poems, flash cards, give students the opportunity to explore, visualize, and manipulate what they just RECITED, SANG, etc..! 
Using number cards, students count up to that number while clapping or jumping once and only once.   Then they can build it using connecting cubes or any other manipulative. TIP: always start off with concrete lessons, then move on to abstract activities (i.e. worksheets) If you notice a student can't clap or jump for each number being said, they're most likely still struggling with the number sequence, and they will not be able to match counting words with objects, claps, or jumps (1:1 correspondence)
Here are more examples of more concrete activities:

number sense and countingnumber sense and counting

Again these 4 aspects are interrelated; they don't have to be practiced in isolation. Children can explore, visualize, and relate these aspects all at the same time. Cardinality & Subtizing may look like this: 
number sense and countingnumber sense and counting

number sense and counting

Give students activities where they "make the number." Making a number revolves around the "last number said or how many objects have been counted." 


Subitizing develops spatial relationships: "many children learn to recognize the dot arrangements on standard dice, therefore similar instant recognition can be developed for patterns in spatial relationships" ( Van de Walle, Lovin, Karp, & Bay-Williams, 2014). Eventually it will aid in counting on, or learning the combinations of numbers. When I first introduce this activity, I also let them "make the number." Simply show the dotted card for 3-5 seconds and they "make" the same quantity. Before you know it, they will just subtize.... quickly recognize the quantity without counting! Dots must be arranged in specific order and color to make patterns recognizable. (dot patterns for dot plates) 
number sense and counting

As children develop number sense, more than just counting, they will be be adding and multiplying numbers (in kindergarten) in the non-traditional algorithm way, and like me, you'll be amazed! When you see this happening, be assured that they are developing a conceptual understanding of mathematics! Van de Walle, et al, affirms, "that the principal tool children poses and will use as they construct these relationships, is counting, initially, then, you will notice a lot of counting,  and you may wonder if you're making progress. Have Patience! Counting will become less and less necessary as children construct new relationships and begin to use more powerful ideas." 


Van de Walle,  J., Lovin, L.,  Karp, K., & Bay-Williams, J., (2014) Teaching Student-Centered Mathematics. Developmentally appropriate instruction for grades Pre-K-2. 2nd Ed. USA: Pearson Education. 

Clements, D.H., & Sarama, J., (2007) Early Childhood Mathematics Learning. In F.K. Lester, Jr. (Ed.), Second handbook on mathematics teaching and learning (pp. 461-555). Charlotte, NC: Information Age Publishing.

Howden, H. (1989). Teaching number sense. Arithmetic Teacher, 36(6), 6-11.

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