Number Flashcards
Newborns number discrim
Newborns and 4-7 months Can do 1/2/3, not 4/6
Surprised by bad math
reach for missing ball in box for 0/1/2/3 can’t do 4
explain by tracking rather than counting
Counting vs tracking
counting as magnitude is sloppy -> approximate number system ANS, also true linguistic counting
Object Tracking = no symbols, just rep the objects, size limit of prob 3 (can track 3 moving balls)
Do infants have ANS
Yes, can do ratio tests of number dots on screen
can do 8 v 16, not 8 v 12
their ratio isn’t as fine tuned yet, gets better as older
but have ANS ratio!
ANS and regular math skills
Better ans (lower weber ratio) = better math skills, training adults doens’t help sadly
Counting principles
constraints on counting
one to one correspondence (ling label per number)
stable order (said in same order 1,2,3,)
cardinality (last number said is size of set)
Lacks of constraint
order irrelevance (doesn’t matter what order count objs in)
abstraction (can count any thing)
Learning counting constraints
Learn 1-1 correspondence label by doing routines (give cookie each person)
learn stable order in many behs (alphabet/songs)
Learning cardinality… hard
3yo don’t know that last number is size, when aske dhow many, start counting again! even if say “hand me 5” don’t really know to count until 5, just grab a bunch
CAN GRAB 1/2, but guess at >3. learn cardinality around 4yo
Help with numbers from lang?
succeed on 6 in 4 out ball reaching task if label numbers
still fail on 4 in 3 out, so still approximate oddly 19 months
what is lang has numbers in grammar (arabic dual marker)
- enter and stay in 2 knowing stage longer (hi whorf)
Adding strategies and experiment
guess
sum (start from 1 count on both hands. add 2 hands)
shortcut-sum (start from 1 and count once)
min (start from larger number and count up)
retrieval (know it already 5+5=10)
children asked to do add problems
eventually used min strategy. took longer on trials right before used strat
discovered min after shortcut-sum
DEVO ISN’T ABOUT REPLACING TOOLS BUT ADDING TOOLS TO INVENTORY TO KNOW BEST WHEN TO USE EACH
Conceiving of larger number
logarithimic number line, can do small distinct, larger less distinc
more linear in 2nd grade
BUT 2nd grade log again when set to 1000 number, 4th grade linear to 1000
BUT don’t respond proportionally to clustered dots
Training large # estimation
better number line better at addition
play coutning board games inc. performance on number ID and counting to 10
Fraction competence
Fractions HARD 1/4 smaller than 1/3 but bigger number!
teachers don’t know how to teach it
Notion of infinite divisibility
children under 10 may say matter can be divided to nothing
notion of divisibily may be root of trouble. half say can get to 0 if keep divifing (wrong)
Fractions require conceptual change thats why hard
