Number sense

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Number sense

  1. 1. Number Sense <ul><ul><li>“… a well organised conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms&quot; (Bobis, 1996).  </li></ul></ul><ul><ul><li>The National Council of Teachers (USA, 1989) identified five components that characterize number sense: </li></ul></ul><ul><ul><li>number meaning </li></ul></ul><ul><ul><li>number relationships </li></ul></ul><ul><ul><li>number magnitude </li></ul></ul><ul><ul><li>operations involving numbers </li></ul></ul><ul><ul><li>referents for numbers and quantities  </li></ul></ul>
  2. 2. Number Sense <ul><li>Researchers have linked good number sense with skills observed in students </li></ul><ul><li>proficient in the following mathematical activities: </li></ul><ul><li>mental calculation (Hope & Sherrill, 1987; Trafton, 1992) </li></ul><ul><li>computational estimation (for example; Bobis, 1991; Case & Sowder, 1990) </li></ul><ul><li>judging the relative magnitude of numbers (Sowder, 1988) </li></ul><ul><li>recognizing part-whole relationships and place value concepts (Fischer, 1990; Ross, 1989) </li></ul><ul><li>problem solving (Cobb et.al., 1991) </li></ul>
  3. 3. Value vs Quantity  (Smart by Shel Silverstein) My dad gave me one dollar bill 'Cause I'm his smartest son, And I swapped it for two shiny quarters 'Cause two is more than one! And then I took the quarters And traded them to Lou For three times -- I guess he don't know That three is more than two! Just then, along came old blind Bates And just 'cause he can't see He gave me four nickles for my three dimes, And four is more than three! And I took the nickels to Hiram Coombs Down at the seed-feed store, And the fool gave me five pennies for them, And five is more than four! And then I went and showed my dad, And he got red in the cheeks And closed his eyes and shook his head-- Too proud of me to speak!                                                      
  4. 4. How Numbers are Used <ul><li>Measurement </li></ul><ul><li>Nominal numbers </li></ul><ul><li>Identifying small quantities without counting </li></ul><ul><li>Relative position </li></ul><ul><li>Amounts or units determined by measuring </li></ul><ul><li>Numbers that name or identify </li></ul><ul><li>Rote counting </li></ul><ul><li>Rational counting </li></ul><ul><li>Cardinal numbers </li></ul><ul><li>Ordinal numbers </li></ul><ul><li>Sequence of numbers </li></ul><ul><li>One-to-one </li></ul><ul><li>correspondence </li></ul><ul><li>How many in a set </li></ul><ul><li>Subitizing </li></ul>
  5. 5. Components of Numeracy by Tom Richards Students must have a strong proficiency in all of these components for mathematical literacy. <ul><li>1. Counting </li></ul><ul><li>     a. Forward sequences </li></ul><ul><li>     b. Backward sequences </li></ul><ul><li>     c. Numbers before </li></ul><ul><li>     d. Numbers after </li></ul><ul><li>2.  Numeral Identification </li></ul><ul><li>     a. Single digit </li></ul><ul><li>     b. Double digit </li></ul><ul><li>     c. Triple digit </li></ul><ul><li>3.  Numeral Sequencing </li></ul><ul><li>     a. Ordering numerals </li></ul>4. Subitizing          a. Recognizing a set of items             as a quantity 5.  Cardinality      a. The relationship between            the numeral and its value               (*** - there are 3 *'s) 6.  Concept of Five and Ten           (anchor numbers)      a. Five and ten frames - patterns      b. Combinations of five and ten
  6. 6. Visualization <ul><li>&quot;Visualization is fundamental to learning in general and to mathematics in particular.  The ability to visualize is a key component of a person's intellectual competence.  It is one of the most important intellectual abilities.&quot; </li></ul><ul><li>                                             Dr. Yeap Ban Har </li></ul><ul><li>1. Subitize - remember what you see </li></ul><ul><li>2. Negative Space - visualize what is missing </li></ul><ul><li>3. Transform an image - move it in your mind </li></ul>
  7. 7. So, what is Subitizing ? <ul><li>the rapid, accurate, and confident judgments quantities of small numbers of items </li></ul>http:// www.youtube.com/watch?v =hK0E1hjW2ws
  8. 8. Relational vs. Instrumental Understanding Instrumental: Rule: Invert and multiply. Relational: What to do and  WHY ! One rectangle
  9. 9. Relational vs. Instrumental Understanding Instrumental: Rule: Invert and multiply. Relational: What to do and  WHY ! One rectangle divided into fourths.
  10. 10. Relational vs. Instrumental Understanding Instrumental: Rule: Invert and multiply. Relational: What to do and  WHY ! What is ? There are 2 groups of 1/4 in1/2. of the rectangle
  11. 11. Math Talks (Look and Talks) <ul><li>Are language based </li></ul><ul><li>Provide varied, purposeful practice with how numbers are used </li></ul><ul><li>Develop part-whole thinking </li></ul><ul><li>Follow the C-P-C approach </li></ul><ul><li>Offer an avenue for communication – learning math out loud </li></ul><ul><li>Help students mentally count all, on, and down </li></ul>
  12. 12. &quot;If you want your children to be intelligent, read them fairy tales.  If you want them to be more intelligent, read them more fairy tales.”                                                                  - Albert Einstein
  13. 13. Number Stories Goldilocks and the Three Bears
  14. 14. Number Stories The Three Little Pigs
  15. 15. Number Bonds <ul><li>Number bonds help students: </li></ul><ul><li>Understand part-whole relationships </li></ul><ul><li>Compose and decompose numbers </li></ul><ul><li>See inverse relationships </li></ul><ul><li>Develop algebraic thinking </li></ul><ul><li>Master addition and subtraction bonds (facts) through 10. </li></ul><ul><ul><li>Ten is an anchor number. </li></ul></ul><ul><ul><li>Use of number bonds continues throughout the curriculum. </li></ul></ul><ul><li>Build relational understanding, which is foundational for building mental math skills. </li></ul>
  16. 16. Teaching Number Bonds <ul><li>Teach how numbers go together </li></ul><ul><li>Math Stories </li></ul><ul><ul><li>Create addition stories </li></ul></ul><ul><ul><li>Create subtraction stories </li></ul></ul><ul><li>Number bond stories </li></ul><ul><ul><li>Without symbols </li></ul></ul><ul><ul><li>With symbols </li></ul></ul><ul><li>Sentence Frames </li></ul><ul><li>Making Ten </li></ul>10 = + 9
  17. 17. Objective: Students understand and illustrate the concept of taking a number apart and then putting it back together - NOT memorize specific facts. <ul><li>Concrete Objects   </li></ul><ul><li>always start with the concrete </li></ul><ul><li>things that can be pulled apart, pushed back together, then pulled apart in a different way </li></ul><ul><li>real world bonds </li></ul><ul><ul><li>students, matchbox cars, box of crayons, fruit, beans, sticks, cubes, etc. </li></ul></ul><ul><ul><li>must have similar attributes - size, shape, color </li></ul></ul><ul><ul><li>must use multiple different objects </li></ul></ul>How do you teach Number Bonds?
  18. 18. <ul><li>Abstract </li></ul><ul><li>learn mathematical symbols </li></ul>How do you teach Number Bonds? <ul><li>Pictorial </li></ul><ul><li>2D representations of former concrete objects </li></ul><ul><li>Game cards </li></ul><ul><li>Pictures </li></ul><ul><li>Smartboard </li></ul>
  19. 19. Sentence Frames <ul><li>There are __________ birds. </li></ul><ul><li>__________ of the birds are __________ and __________ of the birds are __________ </li></ul>
  20. 20. Sentence Frames become Number Bond Stories
  21. 21. Ten Frames (Making Tens) <ul><li>Using a variety of manipulatives to make ten </li></ul><ul><li>Using 10 frames </li></ul><ul><li>Making tens games </li></ul>http:// www.youtube.com/user/mathinfocus?blend =22&ob=5
  22. 22. Place Value <ul><li>Place Value materials: </li></ul><ul><ul><li>Place value strips </li></ul></ul><ul><ul><li>Place value disks </li></ul></ul><ul><ul><li>Place value charts </li></ul></ul><ul><li>Activities include: </li></ul><ul><ul><li>Identifying numbers </li></ul></ul><ul><ul><li>Modeling numbers </li></ul></ul><ul><ul><li>Showing “more than” and “less than” </li></ul></ul>http://www.schooltube.com/video/91d8446891b086b58e00/Addition-Place-Value
  23. 23. Mental Math <ul><li>Prerequisites: </li></ul><ul><ul><li>Counting forward and backward </li></ul></ul><ul><ul><li>Understanding more than and less than </li></ul></ul><ul><ul><li>Recognize numbers made in different ways (early number bonds) </li></ul></ul><ul><li>Early Mental Math: </li></ul><ul><ul><li>Thinking out loud </li></ul></ul><ul><ul><li>Math stories </li></ul></ul><ul><ul><li>Number bonds </li></ul></ul><ul><ul><li>Modeling (visualization) </li></ul></ul><ul><ul><li>Remember: early mental math starts with manipulatives and math talk </li></ul></ul>
  24. 24. Mental Math <ul><li>Prerequisites: </li></ul><ul><ul><li>Counting forward and backward </li></ul></ul><ul><ul><li>Understanding more than and less than </li></ul></ul><ul><ul><li>Recognize numbers made in different ways (early number bonds) </li></ul></ul><ul><li>Early Mental Math: </li></ul><ul><ul><li>Thinking out loud </li></ul></ul><ul><ul><li>Math stories </li></ul></ul><ul><ul><li>Number bonds </li></ul></ul><ul><ul><li>Decomposing numbers </li></ul></ul><ul><ul><li>Modeling (visualization) </li></ul></ul><ul><ul><li>Remember: early mental math starts with manipulatives and math talk </li></ul></ul>
  25. 25. Operation Examples (Notice emphasis on mental math) <ul><li>Addition: http://www.schooltube.com/video/776b2bf587f44952a35a/Mental-Math </li></ul>Subtraction: http:// www.youtube.com/watch?v =3OxvXyc3oIs Multiplication: http://www.schooltube.com/video/226df98736454079bbcb/Partial-Products

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