In Part 1 of this series I discussed how important number sense is to a child's development along with early number sense skills and how they relate to the Common Core. In Part 2 I discussed the importance of hierarchical inclusion, magnitude, and subitizing. In Part 3 I discussed part/whole relationships (a concept which will probably fill the entire kindergarten year). The whole of the math Common Core (except for geometry and measurement) fall under the umbrella of number sense, and today we will discuss some of the concepts of number sense that are on the horizon for kindergartners.
Compensation is the ability to play with numbers. It is the understanding that if 5+5=10 then 6+5 must be 11 because 6 is one greater than 5 and so the sum must be one greater than 10. Or that if 5+5=10 then 6+4 must also equal 10 because 4 is one smaller than 5 and 6 is one larger than 5. This is a complex skill that some kindergartners will not be ready for, but some children may begin to use compensation, and teachers should feel free to conduct Number Talks introducing compensation.
The following video is an example of a 2nd grade Number Talk involving compensation. Number Talks involving compensation in kindergarten would obviously be less complex.
Compensation strategies can be used in the following Common Core standards:
CCSS.MATH.CONTENT.K.OA.A.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
CCSS.MATH.CONTENT.K.OA.A.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
CCSS.MATH.CONTENT.K.OA.A.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
CCSS.MATH.CONTENT.K.OA.A.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
CCSS.MATH.CONTENT.K.OA.A.5 Fluently add and subtract within 5.
Unitizing is a child's ability to see numbers in groups. It is an ability they use to see that there is simultaneously 1 chair with 4 legs, to count by 2's with understanding, to hold one number in their head when counting on, or to begin grouping numbers into tens.
Grouping numbers into tens is especially significant, because "A set of ten should play a major role in children's initial understanding of numbers between 10 and 20. When children see a set of six with a set of ten, they should know without counting that the total is 16. However, the numbers between 10 and 20 are not an appropriate place to discuss place-value concepts. That is, prior to a much more complete development of place-value concepts (appropriate for second grade and beyond), children should not be asked to explain the 1 in 16 as representing "one ten". The concept of a single ten is just too strange for a kindergarten or an early first grade child to grasp. Some would say that it is not appropriate for grade 1 at all. The inappropriateness of discussing "one ten and six ones" (what's a one?) does not mean that a set of ten should not figure prominently in the discussion of the teen numbers" (Walle and Lovin 2006).
(This book by Walle and Lovin is one of my very favorites for teaching mathematics. You can get it through this affiliate link)
In kindergarten, the goal is not to formalize unitizing, but to begin to help students see numbers in groupings. The following standard requires unitizing:
CCSS.MATH.CONTENT.K.NBT.A.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
Let's dissect this standard to figure out exactly what it is and what it is not asking you to evaluate. According to the standard, children should compose and decompose a number in the teens into a group of tens and some ones. Nowhere in the standard is it required for the teacher to use place value language (as Walle and Lovin discourage) but to make representations of the teen numbers using objects and drawings. Although the language of the standard includes the terms "ten ones and one, two, three, four, five, six, seven, eight, or nine one" this vocabulary is there for the teacher, the standard itself specifically requires "understanding" from the student.
Watch the following videos, if kindergartners can see the images and identify the teen number that is represented, then they have met the conditions of the Common Core Standard.
The Landscape of Learning
We should keep in mind that these concepts of number sense do not describe a linear progression of understanding.
"Historically, curriculum designers... analyzed the structure of mathematics and delineated teaching and learning objectives along a line... [f]ocusing only on the structure of mathematics leads to a more traditional way of teaching--one in which the teacher pushes the children toward procedures or mathematical concepts because these are the goals. In a framework like this, learning is understood to move along a line. Each lesson, each day, is geared to a different objective, a different "it." All children are expected to understand the same "it," in the same way, at the end of the lesson. They are assumed to move along the same path; if there are individual differences it is just that some children move along the path more slowly--hence, some need more time, or remediation. As the reform mandated by the National Council for Teachers of Mathematics has taken hold, curriculum designers and educators have tried to develop other frameworks. Most of these approaches are based on a better understanding of children's learning and of the development of tasks that will challenge them." (Catherine Twomey Fosnot and Maarten Dolk 2001)
(Cathy Fosnot is an excellent resource for mathematics teaching. Here is her book in this affiliate link.)
According to Cathy Fosnot, a child's development of number sense looks less like a line and more like the following chart, developing in a way that she describes as the "Landscape of learning".
"The paths to these landmarks and horizons are not necessarily linear. Nor is there only one. As in real landscape, the paths twist and turn; they cross each other, are often indirect. Children do not construct each of these ideas and strategies in an ordered sequence. They go off in many directions as they explore, struggle to understand, and make sense of their world mathematically... Ultimately, what is important is how children function in a mathematical environment (Cobb 1997)--how they mathematize." (Fosnot and Dolk 2001)
Because number sense development is nonlinear, the best activities we can use in our classrooms will weave together different components of number sense and engage children on multiple planes of development. Many of the links in these posts will lead you to some excellent books with activities in them, that do exactly that.
But we must be honest, many of the textbooks that have been adopted are more concerned with checking off the Common Core standards than developing the understanding behind them, much less teaching in a non-linear fashion. Therefore, we, as teachers need to be more discerning lesson planners using textbooks and workbooks only as a resource to teach the Core in the way we know best, instead of letting textbook companies dictate to us how the Core should be taught.
In our next post we will discuss the development of mathematical representation. Be sure to check it out!