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Academic Skills I - Mathematics

I. Some basic considerations

A. holding back

1. age may be relevant for conservation-type skills, but was not a concern with regard to academic skills such as early reading and math

B. standardized testing

1. can predict success in school or current knowledge level

C. process approach

1. how do children allocate attention?
2. how do children choose strategies?
3. what causes individual differences?

II. Counting

A. early competency
1. cardinality
a. absolute numerical size
b. development

1. in the first 6 mos - 1 vs 2, 2 vs 3
2. age 3-4: 4 from 5 or 6
c. processes
1. subitizing: perceptual process
2. subtraction & addition of 1

2. counting
a. principles
1. one-one
2. stable order
3. cardinal
4. order irrelevance
5. abstraction
b. by age 5 complete understanding

1. perhaps facilitated by Sesame Street

3. ordinality - relational
a. more or less develops between 12 - 18 months
1. reinforcement studies
b. by 4 -5 use with larger number sets
4. most 5 year olds
a. can count to 20
b. know the relative sizes of numbers 1 to 10
c. work with basic number principles

B. 3 types of competence
1. procedural, conceptual, utilizational

III. Basic Arithmetic

A. use of varied strategies for solutions

1. direct retrieval, counting from one, counting on, making inferences, guessing
a. even college students use strategies 30% of the time for simple addition!

b. involves the prefrontal cortex, motor cortex, parietal lobe, and other areas in both hemispheres

B. strategy development

1. some successful problem solving by Kindergarten a. influenced by educational TV? 2. extends to early multiplication

3. changes in speed and accuracy

a. greater reliance on direct retrieval after a few years of adding/subtracting and one year of multiplication/division

b. also more sophisticated and faster strategies

4. some cross-cultural similarities

a. faster development in East Asia and Europe

C. strategy choice
1. backup vs. retrieval strategies
a. choices seen even in young children
b. preference for accuracy
2. retrieval will be used for more familiar problems
a. a build up of association increases the likelihood of direct retrieval
b. flat vs. peaked distributions

D. applications to teaching

1. it's okay to use one's fingers!

a. children who use counting strategies in 1st grade are likely to use direct retrieval in 2nd grade
2. individual differences
a. two potential dimensions: peakedness of distributions & criterion for decision

b. good students, not so good students, perfectionists

1. present as early as 1st grade

2. predicts differences in performance for perfectionists on standardized vs. untimed testing

E. mathematical disabilities

1. characteristics

a. 6% of students are classified as math LD

b. difficulty in retrieving correct answers

1. use counting from one

2. use backup strategies poorly and slowly

3. rarely use retrieval

2. reasons

a. limited exposure to numbers
b. limited working-memory capacity
c. limited conceptual understanding

F. conceptual competence: principles
1. inversion principle: a + b - b = ?
a. develops between 6 and 9

1. size of b is related to time to solve

b. but not till 11 can they ignore b
2. mathematical equality: 3 + 4 + 5 = + 5 a. = is seen as a command rather than as equality even to 4th grade
b. success correlates with correct speech and pointing
c. discrepancy between the two is easier to remedy than when both are at a low level

G. context effects: word problems

1. memory demands are a challenge
a. small numbers do not necessarily help
1. interpretation precedes calculation

2. unfamiliar contexts a. e.g. Brazilian street children
1. could solve all problems using their own goods, many problems using goods from other stands, but less than half without a sales context!

IV. Complex arithmetic

A. buggy subtraction algorithms

1. error analysis identifies incongruence between problem solving methods and conceptual understanding
a. correct subtraction procedure but with bug

1. zero subtraction & no borrowing

2. less common in Korea

B. fractions

1. 1/2 + 1/3 = 2/5; common error even at community colleges
2. need to think of magnitude
a. errors present even in adolescents & adults
3. extends to decimal fractions
a. 2.86 < 2.357 because of digits after the decimal
b. but vice versa, too

C. algebra
1. symbol manipulation ? application or understanding
a. e.g., a · (b+c) ? a + (b · c)
2. some strategies to check appropriateness of procedures
a. inserting actual numbers
b. citing a rule
1. but, may not be an accurate one
2. first year engineering students interpreted "there are six times as many students as there are professors" as 6S = P!

V. Computer Programming

A. hypothesized to increase programming and general skills
1. Papert’s LOGO
2. most successful with mediated instruction
a. how to use + feedback
b. explicitly making analogies, pointing out general programming concepts

B. Klahr & Carver (1988)
1. instruction in debugging could transfer
2. 8-11 yr olds were faster with instruction
C. transfer also seen in younger children 1. problem-solving with LOGO led to increased scores on analogical reasoning tests in 2nd graders

Academic Skills II — Reading & Writing

I. Reading

A. Chronological approach (Chall, 1979)

1. Stage 0 (birth to 5): prerequisite skills incl. identifying letters, writing one’s name, and reading a few words (e.g., ALL, TIDE)

2. Stage 1 (1st & 2nd grade): phonological recoding skill development. learn letter names and sounds

3. Stage 2 (2nd & 3rd grade): increased fluency, but still not reading for learning

4. Stage 3 (4th - 8th grade): reading to learn

5. Stage 4 (high school & beyond): comprehension of written material from multiple viewpoints

B. Pre-reading Skills

1. some are effortlessly acquired
a. left-right (in English), line-to-line organization, spaces signal words

2. letter perception

a. in most contexts, orientation does not affect meaning

b. whereas identification of letters in K predicts reading scores to grade 7, teaching letter names is not sufficient

3. phonemic awareness

a. words consist of separable sounds

b. phonemic awareness in preschool predicts reading achievement in early grades

c. can be taught; leads to better reading/spelling up to 4 yrs later

d. may be related to exposure to nursery rhymes

1. predicts later performance at age 3
2. also instructionally related

4. precocious readers

a. some children spontaneously pick up reading around 2 or 3
1. related somewhat to IQ
2. other differences

a. greater verbal knowledge and larger WM spans
b. mastery of prereading skills
c. interest in reading
3. does not negatively affect later performance
4. predicts later skill through grade 6
a. teaching does not produce the same correlation

C. Learning to read 1. identifying individual words
a. 40% of those with poor WI skills reported that cleaning their room was more fun than reading!
b. two methods

1. phonological recoding
a. emphasis of the phonics approach
b. important for word attack skills
1. for the 3,000 words in a beginning reader, 70% occurred fewer than 5X, 40% occurred only once
c. supports later visually based retrieval

2. visually based retrieval

a. emphasis of the whole-word approach

b. there is not always a progression from PR to VBR
1. Budweiser & STOP as first read words!
a. can occur even outside of context c. relies on multiple cues: context, particular letters, familiarity
1. present even in 1st grade

2. strategy choice

a. decision process similar to math facts b. back up strategies when direct retrieval fails or not attempted
1. sounding out, asking an adult or other child
c. phonics approaches are better than whole-word approaches
d. plodders versus explorers

3. dyslexia
a. phonological dyslexia - trouble with phonological recoding
1. evident with pronounceable non-words & exceptions
2. poor reading can extend to adulthood, teaching helps
a. strategies for circumventing
1. draw analogies with known words, try alternative pronunciations, identify parts that are known
b. improving phonological recoding

D. Comprehension 1. 4 components
a. lexical access, proposition assembly, proposition integration, text modeling
b. listening comprehension often exceeds reading comprehension up to about 8th grade

2. what develops in comprehension?
a. basic processes
1. automatization of word identification
2. increased working memory capacity

b. strategies

1. adjusting speed and carefulness to difficulty of text
a. skimming rarely before age 14

c. metacognitive knowledge

1. better readers are better monitors than poor readers
2. some of their strategies

a. rereading, slowing down, visualizing, concretizing
d. content knowledge
1. relevant prior knowledge
2. focus on causal relations

a. 8 yr olds emphasize causes within an episode, 14 yr olds link across episodes
3. story schemas

3. instructional implications

a. make sure that background knowledge is supplied
b. reciprocal instruction (Palinscar & Brown, 1984)

1. 7th graders’ reading skills were at grade level, but comprehension was delayed
2. emphasized comprehension monitoring
a. summarizing, clarifying, questioning, anticipating future questions 1. after 20 sessions, 60% of statements accurately captured the main idea
2. comprehension went from 20% to 80%
3. still present 6 months later, transferred to other subjects

3. corroborating evidence from other research

II. Writing

A. subprocesses
1. initial drafting & revision
2. necessary skills even in adulthood (19% on memos, letters, etc.)
B. Initial drafting - demands of unfamiliar topics
1. must activate relevant info and pull it together
2. often a list of facts rather than an organized presentation
C. Initial drafting - demands of multiple goals
1. variety of goals with little feedback from others
2. knowledge telling approach
a. answer directly & write down all of relevant info
b. often very brief in length (about 1/2 page)
3. improving writing involves considering multiple goals a. Berieter & Scardamalia (1987)
1. used decks of cards with common sentence openers
2. led to more interesting essays
4. knowledge transforming approach
a. combines deciding what to say and how to say it 1. part 1: analysis of subject, adopting a point of view
2. part 2: moving back and forth btn content area and writing strategies; comparisons of intent with reality
b. evident in time to begin writing
1. college students take more time than 5th graders

2. take more time with longer essays

D. mechanical requirements 1. includes forming letters, spelling words, punctuation, capitalization
2. Berieter & Scardamalia (1982) tested 4th & 6th graders

a. typical-writing, slow-dictation, standard-dictation
b. typical-writing < slow-dictation < standard-dictation
3. similar results with word processing
a. some transfer back to writing by hand, too

E. revision
1. problems
a. not always done, may not improve
2. identifying weaknesses a. correcting another’s writing: sentences were missing, contradictory, or uninterpretable
1. identified by 25% of 4th graders, 60% of 6th graders

b. egocentrism in correcting one’s own writing

1. less able to notice referential ambiguities than grammatical errors

2. but waiting doesn’t really help

a. quality was the same for 4th - 12th graders after a week
3. correcting weaknesses

a. spontaneous recognition = good corrections
b. 6th graders could cope with other identified errors