equity and mathematics education - university of … and mathematics education ... mathematics...

James C. Kennedy Institute for Educational Success Position Paper #2

Douglas H. Clements and Julie Sarama January 16, 2015

Equity and Mathematics Education

A kindergartner stepped up to research assistant grinning and showing her "Happy Birthday" crown. The assistant asked how old she was. The girl stared without responding. "Can you show me on your <ingers?" The girl slowly shook her head "no."

As we saw in our first Kennedy Institute position paper

(“Education and Equity”), students in some groups, such as those

from low-income communities, demonstrate significantly lower

levels of achievement. This is especially true of achievement in the domain of

mathematics (1-5). In this position paper, we address issues regarding equity in

mathematics achievement and education, including students who live in poverty and who

are members of linguistic and ethnic minority groups (the next position paper will

address students who have mathematical disabilities or difficulties).

Poverty and Minority Status

The U.S. mathematics education system needs substantial improvement. The

mathematics achievement of American students compares unfavorably with the

achievement of students from many other nations, even as early as first grade and

kindergarten (6). Some cross-national differences in informal mathematics knowledge

appear as early as three to five years of age (7, 8). Children in East Asia and Europe learn

more advanced math than most children in the U. S. are taught (9).

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Further, children from some groups come to school with fewer experiences in

mathematics than others. For too many, these differences do not disappear. In the U.S.,

they increase (10). That is, the "achievement gap" does not close; it widens (9). This gap

is most pronounced in the performance of U.S. children living in economically deprived

urban communities (11-15). There is no age so young that equity is not a concern. The

achievement gaps have origins in the earliest years, with low-income children possessing

less extensive math knowledge than middle-income children of pre-K and kindergarten

age (3, 14, 16-20). As one example, the ECLS-B found that the percentage of children

demonstrating proficiency in numbers and shapes was 87% in higher SES families but

only 40% among lower socioeconomic status (SES) families (21, but this involved simply

reading numerals and so the report does not provide useful details). These differences

start early and widen (22).

Equity demands that we establish guidelines for quality mathematics education

for all children.

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Children from some groups come to school with fewer experiences in mathematics than others. Yet they have full potential to learn.


What Accounts for These Differences?

It is is not parents’ IQ or children’s ability to learn (see our Position Paper #1.)

The key factors in one study were the educational level attained by the child's mother and

the level of poverty in the child's neighborhood (23). These are distinct factors, with

income having a direct effect on the child and also an effect that is mediated by the

parents’ interaction with children (e.g., higher, compared to lower, income parents

providing more support for problem solving, 24, 25). Similarly, an analysis of the ECLS

data shows that SES indicators and the number of books in the home both strongly

predict math (as well as reading) test scores (26, also controlling for these substantially

reduces ethnic differences). Low SES children score .55 standard deviations below

middle-SES children and 1.24 standard deviations below high-SES children.

Differences in parents’ beliefs may also play a role. Compared to middle-income parents,

lower-income parents believe that math education is the responsibility of the preschool

and that children cannot learn aspects of math that research indicates they can learn (7).

Also, low-income families more strongly endorsed a skills perspective than middle-

income families, and the skills and entertainment perspectives were not predictive of later

school achievement. What was predictive was the “math in daily living” perspective

adopted by more middle-income parents (27). Such deleterious effects of living in low-

resource communities are more prevalent and stronger in the U.S. than other countries

and stronger in early childhood that for other age ranges.

Equity Concerns Start Early

Consider these two children. Peter was at the highest levels of competence in

number. He could count beyond 120, state the number word before or after any given

number word, including those in the hundreds. He could also read those number words.

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Finally, he could use counting strategies to solve a wide range of addition and subtraction

tasks. Tom could not count. The best he could do is say “two” for a pair of objects Asked

for the number after "six," said "horse." After one, he said, come "bike." He could not

read any numerals.

Both Peter and Tom were beginning their kindergarten year (28).

A large-scale study of this gap, a survey of U.S. kindergartners found that 94% of

first-time kindergartners passed their Level I test (counting to 10 and recognizing

numerals and shapes) and 58% passed their Level 2 test (reading numerals, counting

beyond 10, sequencing patterns, and using nonstandard units of length to compare

objects). However, 79% of children whose mothers had a bachelor's degree passed the

Level 2 test, compared to 32% of those whose mothers had less than a high school

degree. Large differences were also found between ethnic groups on the more difficult

Level 2 test (30). Differences appear even in the preschool years.

Other analyses from the same large ECLS showed that children who begin with

the lowest achievement levels show the lowest growth in mathematics from Kindergarten

to the 3rd grade (31). The authors conclude that more time on mathematics for these

children in preschool is essential.

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Some children have acquired number knowledge before the age of 4 that other children will not acquire before the age of 7 (29).


These studies have a clear message. If high-quality mathematics education does

not start in preschool and continue through the early years, children are trapped in a

trajectory of failure (32).Another study combined the two types of comparisons. Results

showed that mathematical knowledge is greater in 3- and 4-year-old Chinese children

than in American middle-class children and greater in American middle-class children

than in 3- and 4-year-olds from low-income families (7).

Young children from low-income families show specific difficulties in

mathematics. Too often, they do not understand the relative magnitudes of numbers and

how they relate to the counting sequence (12). They have more difficulty solving addition

and subtraction problems. Working-class children in the U.K. are a year behind in simple

addition and subtraction as early as 3 years of age (33). Similarly, U.S. low-income

children begin kindergarten behind middle-income children and, although they

progressed at the same rate on most tasks, they ended behind and made no progress in

some tasks. For example, although they performed adequate on nonverbal arithmetic

tasks, they made no progress over the entire kindergarten year on arithmetic story

problems (34). Further, lower-class children were more likely to show a “flat” growth

curve for the year.

A recent survey of preschoolers’ competencies (35) revealed that children from

preschools serving middle-SES populations outperformed those serving low-SES

populations on the total number score and most individual subtests. The particular

subtests that showed significant differences were, with few exceptions, those that measure

more sophisticated mathematical concepts and skills. In number, there we no significant

differences for simple verbal counting or recognition of small numbers (36, 37). There

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were significant differences on object counting and especially more sophisticated

counting strategies, comparing numbers and sequencing, number composition, arithmetic,

and matching numerals to dot cards. In geometry, there were no significant differences on

the simple tasks involving shape and comparison of shapes. There were significant

differences on representing shapes, composing shapes, and patterning.

Other research from across the world confirms the finding that there is greater

variation in number knowledge among young children of lower SES background (29) and

that there is a definite trend for students from lower SES backgrounds to perform at a

lower level, and this was more apparent for the difficult items (38, see also 39). This was

especially so for beginning kindergartners in the verbal counting and numeral

recognition. On the other hand, the most advanced children were the least well served.

They were learning nothing throughout the entire kindergarten year. They did not

advance in reading multi-digit numerals throughout their first grade year.

Similarly, lower-income preschoolers lag behind higher-income peers in the

earliest form of subitizing, spontaneous recognition of numerosity (40). They often lack

foundational abilities to classify and seriate (41). Older children entering first grade

showed a smaller effect of familial factors on computation than on mathematics concepts

and reasoning. Majority-minority contrasts were small, but parents’ economic and

psychological (e.g., high school graduation) resources were strong influences (42).

Early research indicated that such problems have existed for decades, with serious

negative effects (22). The first year of school has a substantial influence on the

trajectories of young children’s knowledge of number. Black children gained less than

white children in this study, and the gap widened over a two-year period. Transitions to

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school, and recovering from initial gaps in learning, may be more problematic for black

children than white children.

Into kindergarten and the primary grades, lower-income children use less-adaptive

and maladaptive strategies more than middle-class children, probably revealing a deficit

in intuitive knowledge of numbers and different strategies (12, 15). Most 5- and 6-year-

old low-income children are unable to answer the simplest arithmetic problems, where

most middle-income kindergartners could do so (12). In one study, 75% of children in an

upper middle class kindergarten were capable of judging the relative magnitude of two

different numbers and performing simple mental additions, compared to only 7% of

lower-income children from the same community (12, 43). As another example, about

72% of high, 69% of middle and 14% of low-SES groups can answer an orally-presented

problem, “If you had 4 chocolate candies and someone gave you 3 more, how many

chocolates would you have altogether?” Low-income children often guess or use other

maladaptive strategies such as simple counting (e.g., 3 + 4 = 5). They often do this

because they lack knowledge of strategies and understandings of why they work and

what goal they achieve (15) However, given more experience, lower-income children use

multiple strategies, with the same accuracy, speed, and adaptive reasoning as middle-

income children.

In summary, the SES gap is broad and encompasses several aspects of

mathematical knowledge: numerical, arithmetic, spatial/geometric, patterning, and

measurement knowledge (35, 44). The reason for this gap appears to be that children

from low-income families receive less support for math development in their home and

school environments (7, 14, 45, 46). Public pre-K programs serving low-income,

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compared to those serving higher-income, families provide fewer learning opportunities

and supports for mathematical development, including a narrower range of mathematical

concepts (47, 48). Lack of resources is the main problem, but research indicates it is not

the only explanation. There are also differences in attitudes, motivations, and beliefs that

need to be addressed (10). For example, "stereotype threat" —the imposition of societal

biases such as the lower ability mathematics of Blacks or woman to learn mathematics—

can have a negative influence on the performance of the threatened groups (10). We need

research on whether this affects young children and how this and other problems can be

avoided.

Further, quality is lower in classrooms with more than 60% of the children from

homes below the poverty line, when teachers lacked formal training (or a degree) in early

childhood education, and held less child-centered beliefs (49). An analysis of the large

ECLS data set found that black children had made real gains in mathematics knowledge

upon entering kindergarten—but that over the first two years of school, they lost

substantial ground relative to other races (26). These differences are on arithmetic—

addition, subtraction, and even multiplication and division—rather than lower-order

skills. There are insufficient resources in these settings to address the needs of the

children.

Conclusions and Implications

Thus, there is an early developmental basis for later achievement differences in

mathematics: Children from different sociocultural backgrounds are provided different

foundational experiences (7). Programs need to recognize sociocultural and individual

differences in what children know and in what they bring to the educational situation.

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Knowledge of what children bring should inform planning for programs and instruction.

Extra support should be provided those from low-resource communities. We must meet

the special needs of all children, especially groups disproportionately under-represented

in mathematics, such as children of color and children whose home language is different

that that of school. All these children also bring diverse experiences on which to build

meaningful mathematical learning (50). The younger the child, the more their learning is

enhanced by contexts that they find relevant and meaningful. There is no evidence that

such children cannot learn the mathematics that other children learn. Too often, children

are not provided with equivalent resources and support (13). They have different and

inequitable access to foundational experiences, mathematically-structured materials such

as unit blocks, technology, and so forth. The settings in which children from different

sociocultural backgrounds are served too often have fewer resources and lower levels of

high-quality interaction. They also have less to support their physical and mental health

(51). The needs of children with physical difficulties (e.g., hearing impaired) and learning

difficulties (e.g., the mentally retarded) must also be considered. There is a critical need

for everyone involved with education to address this problem, so that children at risk

receive equitable resources and additional time and support for learning mathematics.

This does not mean we should treat children as if they were the same; it means equivalent

resources should be available to meet the needs of children who differ in myriad ways,

including socio-culturally and individually (e.g., developmentally delayed and gifted

children). This is important, as knowledge of mathematics in preschool predicts later

school success (52-54). Specific quantitative and numerical knowledge is more predictive

of later achievement than are tests of intelligence or memory abilities (55). Those with

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low mathematics in the earliest years

fall farther behind each year (16, 56,

57).

Students in Linguistic and Ethnic Minority Groups

Children who are members of linguistic minority groups also deserve special

attention (58). Although teaching specific vocabulary terms ahead of time, emphasizing

cognates, is a useful approach, vocabulary alone is insufficient. Teachers need to help

students see multiple meanings of terms in both languages (and conflicts between the two

languages), and address the language of mathematics, not just the "terms" of

mathematics. Building on the resources that bilingual children bring to mathematics is

also essential. For example, all cultures have "funds of knowledge" that can be used to

develop mathematical contexts and understandings (50). Further, bilingual children can

often see general mathematical idea more clearly than monolingual children, because,

after expressing it in two languages, they understand that the abstract mathematical idea

is not "tied" to given terms (see 5). In general, then, "talking math" is far more than just

using math vocabulary.

Final Words

Children who live in poverty and who are members of linguistic and ethnic

minority groups need more math and better math programs (32). They need programs that

emphasize the higher-order concepts and skills at each level, as well as base knowledge

and skills (26, 35). What programs address these problems? We will describe several

research-based programs in a future position paper (for the early years, see 59).

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More children are in deep poverty in the U.S. than other countries. The effects are devastating (24).


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