Structuring

 “Many pupils who struggle with arithmetic have a tendency to count in ones.  What is a normal stage of development for most children becomes a crutch for pupils with poor number sense.  Pupils who continue to rely on this unsophisticated and laborious strategy well beyond the stage at which counting is appropriate or efficient have fallen into the ‘counting trap’.”  (Bird, Ronit. (2009). Overcoming Difficulties with Number: Supporting Dyscalculia and Students who Struggle with Math, Thousand Oaks, California:Sage Publications, page 9.)
Using the tools mentioned last week, students should learn to identify:
  • Dice/Domino Patterns
  • Irregular dot patterns (up to 7)
  • Combinations to 5
  • Combinations to 10 and Doubles 1-5
  • Combinations to 20 and Doubles 6-10
Here are a series of videos that work on structuring
According to Minnesota State Standards, students should know –
  • Combinations to 5, by the end of Kindergarten
  • Combinations to 10, by the end of 1st grade
  • Combination to 20, by the end of 2nd grade.
Carol Johnson, Resource Teacher at Cedar Park, has been working on structuring to 10 with her students.  As an assessment, she told them that they had 1 minute to write down the combinations that equal 10. (5+5, 6+4, etc.) It gave her good information on who knows the combinations and who needs more work.

Structuring

“It is critical that students’ early arithmetical thinking progresses from being based on counting by ones to being based on structuring numbers (Bobis, 1996; Fuson, 1992; Steffe and Cobb, 1988; Wright 1994; Young-Loveridge, 2002).” (page 51. Wright, Ellemor-Collins, Tabor. (2012).  Developing Number Knowledge: Assessment, Teaching & Intervention with 7-11-Year-Olds Washington D.C.:Sage.)
In a recent study “researchers found that students who could identify and work with sets of numbers in 1st grade performed significantly better on a test of functional numeracy given years later in 7th grade, which assesses math skills needed for the workforce. By contrast, students’ ability at the start of school to perform arithmetic based on counting absolute values of numbers had little bearing on their later math skills.”  You can read more about this study here http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0054651
How can we help students develop this important skill?
  • Provide opportunities to see and work with quantities in groups
    • Dice
    • Dominoes
    • Finger Patterns
    • 5-, 10-, and 20-Frames
    • Rekenreks (Math Racks)
    • AL Abacas (AVMR Course 2 Kit)
    • 100 bead string (AVMR Course 2 Kit)
    • Arrays
  • When students are using a counting strategy (this is a stage of mathematical problem solving that all students go through), the second addend, for addition, and the subtrahend,  for subtraction, should not be larger than 5.
  • Tools that will make students reliant on counting by 1’s (TouchPoints, Number Lines) should be avoided. Empty or Open Number Lines, which promote using groups, will be discussed in a future Monday Math Message.

 

Here is a video for students to practice subitizing (recognizing a quantity without counting)  http://www.youtube.com/watch?v=_dVqV5ZEhSc