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Number Sense And Counting Essay

Success in early math has lifelong implications

Researchers have found that children arrive at school on the first day of Kindergarten with wildly different levels of math knowledge. As Clements and Sarama (2011) point out, for example, “some six-year-olds have not acquired mathematical knowledge that other children acquire at three years of age” (p. 968).  These differences in children’s initial understanding of math have long-term implications for their success in school and in life, as “preschool children’s knowledge of mathematics predicts their later school success into elementary and even high school” (Clements & Sarama, 2011, p. 968).

There is broad consensus on what math knowledge matters

The good news is that researchers have identified a specific, well-defined set of concepts and skills that can make the difference between children’s success and failure in mathematics in the early years (Griffin, Case, & Siegler, 1996).  This knowledge is anchored in three key insights (Griffin, 2004):

1)   Numbers represent quantities

2)   Spoken number names (“one,” “two,” etc.) and formal written symbols (1, 2, 3, etc.) are just different ways of referring to the same underlying quantities

3)   The quantities represented by the symbols have inherent relationships to each other (7 is more than 5, for example) and it is this property of the quantities that allows us to use the symbolic number representations to solve certain kinds of problems (putting objects in order, counting to determine how many objects are in a set, etc.)

This network of concepts and skills constitute what is called Number Sense.  Happily, these research-based insights are embodied in the most recent guidelines for teaching mathematics in both the Common Core State Standards(National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010) and the Principles and Standards for School Mathematics published by the National Council of Teachers of Mathematics (2000).

We know how to support children’s Number Sense development

Case, Griffin, and Siegler (1994) found that children who have a well-developed Number Sense are able to succeed in early math (and beyond), while children who don’t are at much greater risk of falling increasingly further behind.  They also demonstrated that virtually any child could develop Number Sense when given access to a well-designed, focused intervention that provides opportunities to explore and discuss key concepts, make connections between different concepts, and develop their understanding at an appropriate pace and following an appropriate conceptual and developmental sequence (Griffin, 2004).

But Number Sense doesn’t develop by accident!

Case, Griffin, and Siegler (1994) demonstrated that every child can develop Number Sense – that’s the good news.  As I mentioned previously, however, it has been found that many children don’t.  Why not?  Because (as other research has established) Number Sense does not develop by accident or even as a side effect of engaging in informal activities such as puzzles or songs that appear on the surface to be related to math.  As Clements and Sarama (2011) caution:

[People] often believe they are “doing mathematics” [with young children] when they provide puzzles, blocks, and songs. Even when they teach mathematics, that content is usually not the main focus, but is embedded in a fine-motor or reading activity. Unfortunately, evidence suggests that such an approach is ineffective, owing to a lack of explicit attention to mathematical concepts and procedures along with a lack of intentionality to engage in mathematical practices (p. 968).

In short, although every child can learn Number Sense, not every child will unless we intentionally and systematically support that learning on an individual basis.

Native Numbers is an adaptive, research-based Number Sense curriculum

Native Numbers is an adaptive, mastery-based Number Sense curriculum grounded in research and delivered in the form of an iPad app. The adaptive features provide a highly individualized learning experience, enabling learners to move quickly through material they already understand and to spend as much time as they need to develop emerging concepts and skills.  The fact that it is mastery-based means that a learner has to demonstrate a minimum level of competency on each concept or skill before being exposed to more complex activities that depend upon that understanding. (You can see a video overview and demonstration of the app here.)

The curriculum contains twenty-five activities that are organized into five subskills.  These subskills are defined based on the Number Sense research mentioned previously as well as the relevant Common Core State Standards and NCTM standards.  Specifically, the subskills developed in Native Numbers are:

  • Number Concepts: Connect number words and numerals to the quantities they represent
  • Number Relations: Develop a sense of whole numbers and their relations, across different representations (“one”, 1, one turtle, etc.)
  • Number Ordering: Understand relative position and magnitude of whole numbers
  • Counting
    • Understand ordinal and cardinal numbers and their connections
    • Count with understanding and recognize “how many” in sets of objects

Native Numbers has been well-received by children, teachers, and parents

Initial feedback on Native Numbers from teachers, reviewers, and individual parents has been positive.  TeachersWithApps.com, for example, is a review site that tests apps in classrooms with children and bases their reviews on those data.  Here is what they have to say about Native Numbers:

TWA spent over a week with severely limited students to experience this app to its fullest. We were wildly impressed with the progression of how and when new concepts are introduced. We loved witnessing the eureka moments when you could see light bulbs going off as the kids played away. I doubt the developers knew just how addicting the app would be. It is fast paced and repetitive and loads of fun!

And Dr. Karen Mahon, a learning scientist and instructional designer who reviews educational apps, had this to say:

The learning tasks are engaging, with consistent feedback for correct and incorrect answers, and the program automatically levels up as the learner makes correct responses.  This makes it more fun and interesting for kids….and MUCH more interesting for a reviewer like me!  But fun and interesting aside, adapting to the performance of the learner allows every learner to be successful, wherever a learner falls along the continuum of skills.

We are eager to learn from our learners, and the people who support them

At Native Brain, we are committed to developing tools that help all children succeed in school and in life.  We draw on the best available research on learning and teaching to empower and support not only the learners but also all of the people who share that purpose, including parents, teachers, administrators, policymakers, and researchers.  We constantly seek evidence of impact and data that can help us refine our offerings to increase their efficacy.  To that end, we invite your feedback, suggestions, and ongoing dialogue.  Please feel free to contact us at http://www.nativebrain.com/contact.

References

Douglas H. Clements and Julie Sarama (2011). Early Childhood Mathematics Intervention. Science, 333(6045), pp. 968-970.  Digital version available online via: https://portfolio.du.edu/portfolio/getportfoliofile?uid=216781

Sharon A. Griffin, Robbie Case, Robert S. Siegler (1996). RightStart: Providing the Central Conceptual Prerequisites for First Formal Learning of Arithmetic to Students at Risk for School Failure.  In K. McGilly (Ed.) Classroom Lessons: Integrating Cognitive Theory and Classroom Practice, pp. 25-50. Cambridge, MA: MIT Press.

Sharon Griffin (2004). Teaching Number Sense. Educational Leadership, 61(5), pp. 39-42.  Digital version available online via: http://www.ascd.org/ASCD/pdf/journals/ed_lead/el200402_griffin.pdf

National Council of Teachers of Mathematics (2000). Math Standards and Expectations: Number and Operations (Pre-K – 2 Expectations).  In Principles and Standards for School Mathematics. published by the National Council of Teachers of Mathematics.  Retrieved from: http://www.nctm.org/standards/content.aspx?id=7564  See also Number and Operations Standard for Grades Pre-K – 2, retrieved from: http://www.nctm.org/standards/content.aspx?id=26848

National Governors Association Center for Best Practices, Council of Chief State School Officers (2010).  Common Core State Standards for Mathematics (see esp. Kindergarten standards related to number pp. 6-11).   Washington, D.C.: National Governors Association Center for Best Practices, Council of Chief State School Officers.  Digital version available online at: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

 

Eddie Gray & David Tall are two British researchers who worked with students, aged 7 to 13, who had been nominated by their teachers as being low, middle or high achieving students. All of the students were given number problems, such as adding or subtracting two numbers. The researchers found an important difference between the low and high achieving students – the high achieving students solved the questions by using what is known as number sense – they interacted with the numbers flexibly and conceptually. The low achieving students used no number sense and seemed to believe that their role was to recall and use a standard method even when this was difficult to do. For example, when students were given a problem such as 21-16 the high achieving students changed the numbers into 20 -15 but the low achieving students counted backwards, starting at 21 and counting down, which is extremely difficult to do. After extensive study of the different strategies the students used the researchers concluded that the difference between high and low achieving students was not that the low achieving students knew less mathematics, but that they were interacting with mathematics differently. Instead of approaching numbers with flexibility and using ‘number sense’ they seemed to cling to formal procedures they had learned, using them very precisely, not abandoning them even when it made sense to do so. The low achievers did not know less but they did not use numbers flexibly – probably because they had been set on the wrong path, from an early age, of trying to memorize methods and number facts instead of interacting with numbers flexibly (Boaler, 2009).

This page includes many different resources to help students develop number sense. Fluency without Fear summarizes the damage caused by timed tests, and shows ways to teach math facts conceptually and with student enjoyment. The Number Talks video is an extract from my online class “How to Learn Math for Teachers” showing the very important pedagogical strategy of number talks. Fractions with Sense Making is a video of a great teacher – Cathy Humphreys – teaching fractions to middle school students.