Mathematizing: to reason in a mathematical manner, to think or express ideas mathematically
Progressive Mathematization: increase range of numbers, increase task complexity, distance the setting
Progressive Mathematization: increase range of numbers, increase task complexity, distance the setting
Verbal: what we say "five"
Quantity: the amount that we see * * * * * (should teach 1st, it's the only aspect that is the same in every language - it's universal)
Symbolic: 5
To be proficient, math students need a complete understanding of the numbers around them. For many that are struggling, one or more of these dimensions are missing or flawed.
What is the difference between Numeral Identification and Numeral Recognition?
What is the difference between Numeral Identification and Numeral Recognition?
*Numeral Identification: stating/knowing the name of a given numeral in random order (ex. what number is this, what shape is this) "can verbally express the name"
*Numeral Recognition: selecting a said numeral from a randomly displayed group of numerals (ex. point to the 14, find the hexagon) "it looks familiar"
*Numeral recognition develops before numeral identification.
Counting is quantity, FNWS is saying numbers/number sequences
Conceptualize or conceptually-based: imagine/visualize (ex. child can visualize a dot pattern for 4)
Subitizing: the immediate, correct assignation of a number word to a small collection of perceptual items
- Perceptual Subitizing: recognition of a number pattern
- Conceptual Subitizing: recognizing a number pattern as a composite of parts and a whole, different arrangements lead to different decompositions of that number - when you see in groups and chunks (teach with irregular patterns and 2-color dots)
- Subitizing may include recognition of the following: spatial patterns (dot arrangements), temporal patterns (attaching number to sounds), kinesthetic (finger patterns), rhythmic (hand-clapping)
- Sequential Finger Patterns: raising an appropriate number of fingers one at a time to build the finger pattern
- Simultaneous Finger Patterns: raising an appropriate number of fingers as one action
- Numerosity: the number of items in a collection
- Composite: see numbers in groups, combine/partition, manipulate to +/-, skip count
- Numerical Composite (Stage 3): when a number word can stand for a collection of items, while maintaining the numerosity of the collection (ex. 8 consists of 1-2-3-4-5-6-7-8)
- Conservation of Number: the number of objects (quantity) remains the same even when they are arranged differently(ex. there will always be 5 items no matter the arrangement, 5 is 4 and 1 or 2 and 3)
- Cardinality: understands that the last number counted tells the whole of the collection/set (can tell "how many" when asked without immediately recounting)
- Temporal Sequences: sequences of sounds and/or movements
What is the difference between Number word sequences and Counting?
*Number word sequences: orally count, say a sequence of numbers
*Counting: quantity, 1:1 touch item or imagine (conceptualize) the item (ex. imagine dots)
- Counting all (counting from one three times): 4+5 (child count 4 fingers on one hand from one, then five fingers on the other hand from one, then counts all fingers from one)
- Discrete Counting: think in terms of counting individual objects (ex. 12-3...take away 12, take away 11, take away 10, so the answer is 9)
- Interval Counting: think in terms of jumping on the number line (ex. 12-3...12...11-10-9)
- Iterate: to mentally repeat without gaps or overlaps (think about units of measurement, line up end to end, edge to edge), using the same thing repeatedly (one paper clip to measure a long line, adding 3's each time to skip count)
- Interval vs Discrete a_______________b, Do you count a and b as the measurement or the one jump from a to b? When counting-on, 4+2, with fingers, is 4 the first finger (4, 5 so the next number would be the answer "6"- discrete) or is 5 the first finger ("5, 6" is the answer- interval)? Don't force a child to be one way or another.
- Perceptual Replacements: establishing items to count prior to solving the task (using fingers, floor tiles, etc)
- Re-presentation: a mental replay of a prior experience (reusing fingers to count)
- Verbal representation: 7+3=__ "8 is 1, 9 is 2, 10 is 3"
- Counting-On (Counting-Up-To, counting-up-from): 4+5 (start at 4, continue counting to 9, in missing addend tasks 4+_=6, counts from 4 up to 6)
- Counting on is still unitary
- Count-down-from: a strategy used by children who have attained at least a stage 3 in addition/subtraction and typically used to solve removed item tasks (ex. 11-8: 10-9-8-7-6-5-4-3, the answer is 3)
- Count-down-to: regarded as the most advanced of the counting-by-ones strategies and used by children who have attained Stage 4 in addition/subtraction (ex. 11-8: 10-9-8, the answer is 3)
- Double Counting: 8+3, after counting from one to eight (figurative counter), the child may say, "one more is 9, two more is 10, three more is 11."
Structuring is like Phonics in Literacy. It is extremely important!
Structure: making (combining) and breaking (partitioning) numbers without counting by one
*structuring=not counting
Factors: the two #'s being multiplied
Structure: making (combining) and breaking (partitioning) numbers without counting by one
*structuring=not counting
Factors: the two #'s being multiplied
Unitizing: understanding that ten can be counted as one unit
Decuple: "ten times as great" - numbers that are multiples of 10 to 100 (10, 20, 30, 40, 50, 60, 70, 80, 90)
Decade: a group of 10 numbers that go with a decuple (10, 11,12,13, 14, 15, 16, 17, 18, 19) (20, 21, 22, 23, 24, 25, 26, 27, 28, 29)
Centuple: "one hundred times as great" - numbers that are multiples of 100
Facile: skillful (no errors in solving and strategy use), ask yourself, "Do I believe I can walk away from that skill and my kids will still be okay, they've got it"
Compensation: 15-_=11 (solved b/c 15-10=5, so just add 1, knowledge of 10s and 1s structure)
Conceptual Place Value (CPV): being able to flexibly increment and decrement by ones and tens, and later by hundreds; it is critical knowledge for developing facile mental computation (BK Developing Number Knowledge, pg. 77)
Students...
-develop a sense of the relative sizes of numbers
-learn ways of relating multi-digit numbers to each other
-learn to organize numbers mentally in terms of ones, tens, hundreds, and so on
Hierarchical Inclusion, "the nesting of numbers": numbers are nested inside of each other and the number grows by one each count (9 is inside 10 or 10 is the same as 9+1) (there is a 1, 2,3, 4, 5 "hiding" within a 6)
*usually don't attain until 6 years old
*will struggle with Place Value later if they don't get this
Decuple: "ten times as great" - numbers that are multiples of 10 to 100 (10, 20, 30, 40, 50, 60, 70, 80, 90)
Decade: a group of 10 numbers that go with a decuple (10, 11,12,13, 14, 15, 16, 17, 18, 19) (20, 21, 22, 23, 24, 25, 26, 27, 28, 29)
Centuple: "one hundred times as great" - numbers that are multiples of 100
Facile: skillful (no errors in solving and strategy use), ask yourself, "Do I believe I can walk away from that skill and my kids will still be okay, they've got it"
Compensation: 15-_=11 (solved b/c 15-10=5, so just add 1, knowledge of 10s and 1s structure)
Conceptual Place Value (CPV): being able to flexibly increment and decrement by ones and tens, and later by hundreds; it is critical knowledge for developing facile mental computation (BK Developing Number Knowledge, pg. 77)
Students...
-develop a sense of the relative sizes of numbers
-learn ways of relating multi-digit numbers to each other
-learn to organize numbers mentally in terms of ones, tens, hundreds, and so on
Hierarchical Inclusion, "the nesting of numbers": numbers are nested inside of each other and the number grows by one each count (9 is inside 10 or 10 is the same as 9+1) (there is a 1, 2,3, 4, 5 "hiding" within a 6)
*usually don't attain until 6 years old
*will struggle with Place Value later if they don't get this
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