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GOOD TO GREAT
Number Sense
Mas-cate and Ruminate
1. When two even numbers are mul-plied, the product will always be:
A. Even B. Odd C. Unknown
2. When two odd numbers are mul-plied, the product will always be:
A. Even B. Odd C. Unknown
3. When an even number and an odd number are mul-plied, the product will always be:
A. Even B. Odd C. Unknown
Why Number Sense?
• Number sense in kindergarten and 1st grade predict math achievement through 3rd and 5th grade
• Kindergarten-‐entry number knowledge is a beMer predictor of overall academic achievement in 3rd and 5th grade than early reading skills or aMen-on
• An accurate mental number line in 3rd grade predicts math achievement on statewide tests
• 5th grade frac-ons and division mastery predicts math achievement in high school
Sources: Jordan et al. 2009; Geary, 2011; Duncan et al., 2007; Claessens et al., 2009; Booth and Siegler, 2006; Siegler et al., 2012
What if our students’ performance in number sense equaled geometry?
What Is Number Sense?
• Coun-ng – Cardinality – Es-ma-on
• Number Knowledge – Ordinality – Magnitude
• Number Opera-ons – Number combina-ons – Math facts
How many kiMens?
Mental Number Line
Children “o]en fail to appreciate an integral aspect of the number system—that the whole numbers are evenly spaced along the number line. Instead, it seems that in young children’s mental representa-on of magnitude, small values are far more different from each other than larger values; for example, 1 and 2 are more different than 8 and 9.”
Source: Ashcra] and Moore, 2012
0 10 5 1 2 3 4 6 7 8 9
0 10
Number Line Es-ma-on 0-‐10
0 100 50 10 20 30 40 60 70 80 90
0 100
Number Line Es-ma-on 0-‐100
0 1,000
Number Line Es-ma-on 0-‐1,000
0 1,000 500 100 200 300 400 600 700 800 900
Number Line Accuracy
• About one-‐half of kindergarten students have an accurate 0-‐10 mental number line
• About one-‐half of first grade students have an accurate 0-‐100 mental number line
• About one-‐half of fourth grade students have an accurate 0-‐1,000 mental number line
Sources: Siegler & Booth, 2004; Siegler & Opfer, 2003; Bertelef et al., 2010
0 10
8 + 2 = 10
0 10
0 10
8
2
0 100
10 x 3 = 30
0 100
0 100
10
10
0 10 20 30
Frac-ons
“Learning about frac-ons requires children to recognize that many proper-es of whole numbers are not true of numbers in general and also to recognize that the one property that unites all real numbers is that they possess magnitudes that can be ordered on number lines.”
Source: Siegler et al. 2013
0 1
0 1 ½ ⅓ ⅔ ¼ ¾ ⅕ ⅘ ⅗ ⅖ ⅙ ⅚ ⅛ ⅜ ⅝ ⅞
What Frac-on?
Understanding Number
“There are paMerns within number combina-ons and rela-onships between them. For example, paMerns in combina-ons of 6 include 3 + 3 = 6; therefore 4 (which is 1 more than 3) + 2 (which is 1 less than three) = 6, 5 + 1 = 6, so 1 + 5 = 6, and so forth. Children with an intui-ve grasp of number paMerns can readily derive answers from known combina-ons to solve unknown ones. This ability, in turn, helps them master or become fluent with number combina-ons.”
Source: Jordan et al., 2006
Number PaMerns 1
Number PaMerns 2
Number Combina-ons
How many kiMens again?
Addi-on Strategies
Single Digit • Coun-ng all (with or without fingers or manipula-ves,
verbal or nonverbal) • Coun-ng on (from the larger number if commuta-vity is
understood) • Retrieval from memory • Derived facts (memory and number paMerns)
Mul-ple Digit • Decomposi-on (making 10s, friendly equa-ons) • Standard algorithm
Source: Geary, 1994
Addi-on Accuracy
99% of 1st graders used more than one strategy and 62% used at least 3 strategies • Coun-ng all (50% error rate) • Coun-ng on (17% error rate) • Decomposi-on (8% error rate) • Retrieval from memory (4% error rate)
Source: Ashcra], 1992
Friendly Addi-on Equa-ons
0 20 10 2 4 6 8 12 14 16 18
20 + 3 = 23 17 + 6 = 23
Standard Par-al Sums
437 +86 523
1437 +86 400 110 13
523
Addi-on Regrouping Errors
Source: Geary, 1994
Finger Coun-ng
• Strong posi-ve correla-on with solving addi-on problems in kindergarten
• The correla-on decreases over K-‐2 • Small but significant nega%ve correla-on with solving addi-on problems at the end of second grade
• Children from low-‐income families use their fingers to count less than others in kindergarten but more in second grade
Source: Jordan et al., 2008
Subtrac-on Strategies
• Coun-ng up (from subtrahend to minuend) • Coun-ng down (usually reserved for problems where the minuend is much larger than the subtrahend, e.g. 27-‐3)
• Addi-on reference (fact families) • Decomposi-on (10s, friendly equa-ons) • Retrieval from memory
Source: Geary, 1994
Friendly Subtrac-on Equa-ons
0 20 10 2 4 6 8 12 14 16 18
20 – 11 = 9 17 – 8 = 9
Subtrac-on Regrouping Errors
Source: Geary, 1994
Mul-plica-on Strategies
Single Digit • Count equal groups (arrays, repeated addi-on) • Rules (zero property, iden-ty property) • Derive from known facts (especially from doubles) • Retrieve from memory
Mul-ple Digit • Decomposi-on (distribu-ve property) • Place value (par-al products) • Standard algorithm (long mul-plica-on)
Source: Geary, 1994; Lampert, 1986
Mul-plica-on Accuracy
92% of 3rd graders used more than one strategy on basic mul-plica-on facts (0-‐9) and 65% used at least 3 strategies • Wri-ng the problem (39% correct) • Sets of tally marks (55% correct) • Repeated addi-on (59% correct) • Retrieval from memory (78% correct)
Source: Siegler, 1988
Retrieval Errors
• Near misses: 10% larger or smaller • Opera-on confusions: add instead of mul-ply
• Table errors: correct answer to another mul-plica-on problem
• Table-‐related errors: another mul-ple of one of the operands
Source: Geary, 1994; Lampert, 1986
Friendly Mul-plica-on Equa-ons
(20 + 4) × 6 =
24 × 6 = 144
(20 × 6 ) + (4 × 6 ) =
120 + 24 = 144
Standard Par-al Products
47 × 32 94
1,410 1,504
147 × 32 1,200 210 80 14
1,504
2
Mul-plica-on Regrouping Errors
Source: Lampert, 1986
Division Strategies
• Trial and error (mul-ply the divisor by a succession of numbers)
• Grouping (for 12÷4, make 4 groups and add one to each group un-l there are 12)
• Repeated addi-on of the divisor • Mul-plica-on reference (fact families) • Retrieval from memory (but not much)
Source: Geary, 1994; Robinson et al., 2006
Problem Size Effect
• Math facts with larger addends or mul-pliers are more difficult to retrieve from memory
• Math facts with larger addends or mul-pliers appear less frequently in math textbooks
• Problem difficulty or fewer associa-ons with larger math facts stored in memory?
Sources: Hamann & Ashcra], 1986; Ashcra] & Christy, 1991; Dehaene & Mehler, 1992
Number Sense is Conceptual
• Early understanding of number rela-ons and opera-ons provides support for learning complex calcula-on procedures
• Children who understand concepts tend to solve problems more effec-vely across mathema-cal domains and age ranges
• Conceptual knowledge of frac-ons predicts gains in procedural knowledge of frac-ons
• Conceptual knowledge of inversion supports but does not guarantee development of strong calcula-on skills
Sources: Jordan et al., 2009; RiMle-‐Johnson & Siegler, 1998; & Siegler et al., 2013; Gilmore & Bryant, 2006