GUIDING CHILDREN'S LEARNING OF MATHEMATICS

28 Part 1 Guiding Elementary Mathematics with Standards

• students who live in poverty and those who do not;
• students who have not been successful and those who have been

successful in school and in mathematics.

The Board of Directors commits the organization and every group
effort within the organization to this goal.

In a recent position statement (2005) NCTM restates and extends
this commitment and challenge (http://www.nctm.org):

Every student should have equitable and optimal opportunities to learn
mathematics free from bias—intentional or unintentional—based on
race, gender, socioeconomic status, or language. In order to close the
achievement gap, all students need the opportunity to learn challenging
mathematics from a well-qualified teacher who will make connections
to the background, needs, and cultures of all learners.

In this chapter you will read about:
1 Ways to include diverse cultural aspects in mathematics teaching
2 The role of ethnomathematics in teaching mathematics
3 Strategies for teaching mathematics to English-language learners
4 Teaching strategies for students with learning disabilities
5 Characteristics of students who are gifted and talented in

mathematics
6 Multiple intelligences and learning styles

Equity in Mathematics Learning pared to enter many occupations. The social and
economic implications for individuals and society
Whether teachers work in a self-contained setting as a whole are great when students are not nurtured
or as specialists in mathematics, they have a profes- in mathematics.
sional responsibility to provide a mathematics-rich
environment for all—boys and girls, students with In this chapter we con-
limited English proficiency, members of all cultural sider equity in mathe-
and ethnic groups, students who are physically and matics learning from sev-
learning challenged, students who are gifted and eral viewpoints: gender,
talented, and learners of every style and orienta- ethnicity, limited English
tion. Meeting the needs of every student is not easy, proficiency, technologi-
but every teacher must work toward that goal. When cal equity, special needs,
students fail to develop their full potential in math- giftedness, and learning
ematics in elementary school, they have difficulty styles. Each of these groups presents its own chal-
with mathematics in later grades. Those in high lenges for the classroom teacher, challenges that
school who lack the prerequisite understanding must be met to ensure that every child in each of
and skills for college and university study are unpre- these groups is able to learn interesting and valu-
able mathematics.

Chapter 3 Mathematics for Every Child 29

Gender pected to be, do, think, and feel. This learn-
ing process is called gender socialization.
Only a few decades ago one of the truisms in educa- Differences in treatment and expectations
tion was “Math is for boys, English is for girls.” Not . . . influence the kinds of skills each sex de-
that girls didn’t achieve in mathematics—some did. velops, how confident children feel as learn-
But the expectation was that boys should do well ers, and even what intellectual risks they are
in mathematics. Test scores in high school and col- willing to take. . . . Girls learn to approach
lege enrollment figures supported the truism. In the math and science with greater uncertainty
minds of many, mathematics was the domain of and ambivalence than boys, with inadequate
men, not women. practice and uncertainty in particular skill
areas (like spatial skills) and, more generally,
In recent years evidence put this “truism” to rest with conflicts about competence and inde-
forever. Girls can do mathematics as well as boys, pendence. (Davenport, 1994, pp. 1–2)
and recent test scores support that stand. The most
recent test results of the National Assessment of Ed- Even when teachers treat students equally, de-
ucational Progress (2005) show only a slight differ- rogatory remarks from male students about female
ence in mathematics scores between boys and girls students’ capabilities and interests in mathematics
(girls’ score, 278; boys’ score, 280). The difference have negative effects on females. Female students
between the genders has remained about the same may downplay their interest and skills when they
over the past decade, and scores for both have con- reach middle school because of beliefs that females
tinued to rise. Thus, as a percent of the mean test are not or should not be as good in mathematics
score for boys and girls, the difference between the as males. Gallagher and Kaufman and Campbell
genders continues to shrink, to the point that any dif- provide a fuller discussion of causes and effects of
ferences are now statistically insignificant. A meta- gender differences in mathematics (see Gallagher &
analysis of recent mathematics scores on the SAT Kaufman, 2004; Campbell, 1995).
shows no difference in girls’ and boys’ scores (Bar-
nette & Rivers, 2004). Girl’s enrollment in mathemat- Even though these societal attitudes fly in the
ics courses in high school and college continues to face of facts and educational policies, they do have
rise but still trails boys’ enrollment figures. an effect on young women. Fewer girls enroll in up-
per-level mathematics courses in high school or
Although the facts show no gender differences become mathematics majors in college. One study
in mathematics achievement, there are still reports found that 9% of girls in grade 4 expressed a dis-
of girls being treated differently from boys in math- like for mathematics. By grade 12 the percentage
ematics classes in subtle ways. In the past teachers had risen to 50% (Chacon & Soto-Johnson, 2003).
at all levels devoted more classroom time to helping A long-term study followed students from middle
boys. Boys were encouraged more than girls to think school through high school for a period of 17 years.
through a problem, as opposed to being given direct The findings showed that although girls had higher
cues or directions for solving it. Teachers tended to grades, they had a lower interest in math than boys
call on boys more than girls during class discus- at every grade level and their interest in math de-
sions, and boys were praised more frequently than clined every year from grade 7 through high school
girls (Huber & Scanlon, 1995; Sadker & Sadker, 1994; (Steeh, 2002). Clearly much more must be done to
Tapasak, 1990). Although today’s classrooms show ensure that our young women remain confident
less of this gender distinction, there is still work to in their mathematics ability and that they develop
be done. Even parents’ attitudes reflect this distinc- and maintain a desire to pursue more advanced
tion in mathematics. Parents tend to attribute their mathematics.
girls’ mathematics achievement to hard work, but
the boys’ achievement is attributed to natural talent There are always going to be differences between
(Gershaw, 2006; McCookLast, 2005). the genders in classrooms. Training little boys to be
more like girls in their decorum or little girls to imi-
Children learn that being a boy or a girl tate boys in their energy level is counterproductive.
makes a difference in what they are ex- Both genders benefit the classroom. It is important

30 Part 1 Guiding Elementary Mathematics with Standards

to recognize those bene- have a poorly developed spatial sense are given the
fits and to help all students opportunity to improve and extend that ability, they
gain from them. quickly erase any deficit within a few grades. Thus
the single-gender difference in mathematics may be
One stereotypical dif- overcome with a focused effort in the primary and
ference between boys and elementary classrooms. See Chapter 17 for activities
girls that can benefit boys that develop spatial sense.
in mathematics is boys’
risk-taking tendency. A In closing, gender is no barrier to achievement
stereotypical elementary in mathematics. Girls and boys bring various talents
school classroom can find and abilities to their mathematics, which serve to
boys out of their seats, nois- enrich a mathematics classroom. At one time math-
ily interacting with others, ematics may have been only for boys, but not any-
and taking and issuing dares to each other. Most of more. More than ever, mathematics belongs to ev-
the girls are composed, on task, and following the eryone, and the role of the classroom teacher is to
teacher’s directions. When these behaviors are ex- ensure that both girls and boys believe and achieve
tended to mathematics, they can work to the boys’ in mathematics.
benefit. Consider the approach to nonroutine prob-
lems that each type of student will take. Students Ethnic and Cultural Differences
who have practiced only the routine algorithms (a
set of instructions used to solve a problem, such as Students from ethnic or cultural groups different
the procedure for long division) and solution strat- from the dominant culture may encounter inequita-
egies presented by the teacher in class may find ble treatment. According to Secada (1991), a math-
that such routine approaches are not helpful with ematics curriculum that fails to reflect the lives of
nonroutine problems. Risk-takers who have tried children from culturally different groups may stereo-
alternative approaches, used guessing, and created type mathematics as belonging to a few privileged
their own solution strategies are in a position to try groups. In addition, in the United States expectations
something different and so are not intimidated by a for achievement in mathematics are sometimes not
problem that is unfamiliar and cannot be solved by as high for these students as for Asians and white
the routine algorithms learned in class. A teacher students. Gloria Ladson-Billings (1995, p. 38) illus-
who is alert to such stereotypical tendencies will trates this difference:
help girls confront and solve nonroutine problems.
At the same time the teacher will assist the boys to White, middle-class students are treated as
master classroom procedures and strategies when if they already have knowledge, and experi-
they are explored. ence instruction as apprenticeship. However,
Research suggests that gender affects spatial African American students often are treated
sense. The ability to view objects, mentally ma- as if they have no knowledge and experi-
nipulate them, and move between two and three ence instruction as teaching.
dimensions is related to gender. Among children,
only 17% of girls achieved the average score that When students are apprenticed, they are
boys did on tests of spatial sense (Linn & Petersen, afforded the opportunity to perform tasks
1986). Further research has supported this gender that they have not fully learned. . . . In the
factor in spatial sense (Greenes & Tsankova, 2004; classroom, this apprenticing is played out by
Levine et al., 1999). If no intervention is done, then teachers treating white, middle-class stu-
the gap in spatial sense becomes larger as children dents as if they are competent in areas they
move from grade to grade, until the difference be- are not. They are treated as if they come
gins to affect other areas of mathematics learning. with knowledge (which they do).
The solution is simply to teach spatial sense in the el-
ementary classroom along with other mathematics However, African American young-
topics. When students (regardless of gender) who sters often are treated as if they have no
knowledge.

Thus, as “empty vessels” they must be
filled.

Chapter 3 Mathematics for Every Child 31

Ladson-Billings noted that African American stu- Grounding mathematics in meaningful contexts
dents received more teacher-directed lessons in for children is crucial; otherwise mathematics be-
specific knowledge and skills and fewer opportuni- comes a dance of symbols and abstract numerical
ties to engage in problem-solving situations that re- relationships.
quired independent thought and action.
Putting mathematics
No evidence supports in meaningful settings in-
any claim that African volves more than chang-
American, Latino Ameri- ing the names of people
can, Native American, in word problems to re-
Asian American, or any flect different ethnic back-
other group of students grounds. It involves prob-
lacks the ability to learn lems in real-life settings
mathematics. What is that children from differ-
lacking in many instances ent ethnic backgrounds
is a common background for learning mathemat- will find compelling. Chil-
ics. Many students live in socioeconomic environ- dren can be a source of multicultural mathematics
ments that offer different background-building ex- material. A teacher who knows the children in the
periences. Ladson-Billings (1995) cites a specific classroom and their backgrounds can place math-
example of how socioeconomic circumstances in- ematics in meaningful contexts for them. When
fluence students’ thinking: A problem asks, “Which mathematics is couched in significant situations for
is a more economical way to commute to work, a children, they begin to understand that mathemat-
bus pass that costs $65 a month or a one-way fare ics is a valuable subject, one that has real worth
for $1.50?” The answer depends on experience. One in their personal lives, as reflected in their ethnic
student might consider one-way fares better be- background. Generic settings that can appeal to
cause a parent’s transportation cost would be $60 children from various ethnic backgrounds include
when commuting to a single job for 20 workdays native cooking and recipes, crafts involving mea-
each month. Other students could view the pass as surement and geometry, designing and building a
less costly than individual fares, if a monthly pass house, shopping for native foods and goods. Read-
would allow for unlimited trips between home, em- ing multicultural stories with mathematics themes is
ployment, shopping, and other important sites. So- an effective way to fold multiculturalism into math-
cioeconomic conditions and a narrow range of ex- ematics class. Table 3.1 is a list of some multicultural
periences may impede early learning, but they are storybooks with mathematics themes to them.
not reasons for believing that students are less able Storytelling can also feature multicultural
to learn mathematics and should aim for less lofty themes. An effective storyteller can be more engag-
goals than other students. In keeping with the key ing than a story in a book. These stories or folktales
point for teaching diverse students, it is important can come from a variety of sources, including the
to provide all children with a fair and equitable op- students themselves (see Goral & Gnadinger, 2006).
portunity to learn mathematics. An advantage of storytelling is that the teacher can
adjust the story to emphasize a particular point or
Multiculturalism mathematics concept. In addition to stories, songs
that feature mathematical concepts or numbers
When students are from different cultural or eth- may also be part of mathematics class. The list in
nic backgrounds, introducing multicultural aspects Table 3.2 is adapted from Galda and Cullinan (2006).
into the mathematics classroom can be construc- It contains songs and folktales that feature a specific
tive. Multiculturalism in mathematics suggests using number.
materials from many cultures to explain mathemat-
ics concepts, to apply mathematical ideas, and to Ethnomathematics
provide a context for mathematics problems. Thus
there is more to multiculturalism than hanging post- In 1985, Uribitan D’Ambrosio introduced the term
ers on the wall or focusing on a few special holidays. ethnomathematics. He used it to refer to mathemat-

32 Part 1 Guiding Elementary Mathematics with Standards

TABLE 3.1 • Multicultural Books with a Mathematics Theme

Anansi the Spider: A Tale from the Ashanti, Gerald McDermott, 1987 African
The Black Snowman, Phil Mendez, 1989 African American
Count on Your Fingers African Style, Claudia Zaslavsky, 2000 African
Count Your Way Through Africa, Jim Haskins, 1989 African
Fun with Numbers, Jan Masslin, 1994 Mayan and African
The Girl Who Loved Wild Horses, Paul Gobel, 1978 Native American
Grandfather Tang’s Story, Ann Tompert, 1990 Chinese
The Hundred Penny Box, Sharon Bell Mathis, 1975 African American
Legend of the Indian Paintbrush, T. de Paola, 1988 Native American
A Million Fish . . . More or Less, Patricia McKissack, 1992 Native American
The Patchwork Quilt, Valerie Flournoy, 1985 African American
Popcorn Book, T. dePaola, 1978 Native American
The Rajah’s Rice, Dave Barry, 1994 Indian
Sadako and the Thousand Paper Cranes, Eleanor Coerr, 1993 Japanese
The Story of Money, Carolyn Kain, 1994 Multicultural
Story Quilts of Harriet Powers, Mary Lyons, 1997 African American
Thirteen Moons on the Turtle’s Back, Joseph Bruchac, 1992 Native American
Two Ways to Count to Ten: A Liberian Folktale, Ruby Dee, 1988 African
The Village of Round and Square Houses, Ann Grifalconi, 1986 African

ics as seen through a cultural filter. Ethnomath- velopment of accounting and the algorithms we use
ematics is the “study of the interaction between today. Consequently, ethnomathematics reflects
mathematics and human culture” (Johnson, 2006). situations that led to the development of mathemat-
All mathematics developed from a cultural need, ics in various cultures. Ethnomathematics informs
regardless of the historical setting. Geometry con- teachers and students about how mathematics was
cepts grew out of a need to measure land, build developed in response to societal needs, and it is
dwellings, and find one’s way between villages and shaped by cultural settings and issues today. Thus
cities. The rise of commerce set in motion the de- ethnomathematics delves more deeply into the in-
terplay between mathematics and culture than does
TABLE 3.2 • Folk Songs and Folktales multiculturalism.
That Feature Numbers
Every culture developed mathematics to fit a
Number Folk Songs and Folktales particular need. Children illuminate these develop-
One ments by asking their parents how they used math-
Two Puss in Boots ematics in their native land, under what situations
Three they learned to use mathematics, and how they use
Jorinda and Jorongel mathematics in their everyday lives. Rather than
Four Perez and Martina sanitize mathematics from all cultural influences,
Six mathematics becomes the shaper and tool of all
Seven Three Wishes civilizations. Minority students are encouraged
Three Little Pigs by and involved in mathematics that is intimately
Twelve Three Billy Goats Gruff connected to their cultural background. Table 3.3
Three Little Kittens presents 10 ways of including culturally relevant ac-
Goldilocks and the Three Bears tivities in the classroom. These topics and related
projects can motivate students from diverse ethnic
Bremen Town Musicians backgrounds to learn mathematics.
Four Gallant Sisters
Multicultural themes should be integrated into
Six Foolish Fishermen the mathematics curriculum. They could be the
settings that introduce mathematics concepts, il-
Seven Blind Mice luminate those concepts, or summarize the study
Her Seven Brothers of a particular concept. Multiculturalism is not an
Seven with One Blow
Snow White and the Seven Dwarfs

Twelve Dancing Princesses
Twelve Days of Christmas

Chapter 3 Mathematics for Every Child 33

TABLE 3.3 • Activities That Draw on the Cultural and Ethnic Background of Children

1. History of Mathematics

Biographies and anecdotes about Chinese, Hindu, and Persian mathematicians can be sources of interesting, humanizing
stories for children. Teachers can relate these stories to students at appropriate times during mathematics class. Children
could also act out the stories, playing the roles of mathematician and others in these minidramas (see Johnson, 1991, 1999).

2. Number Systems

Studying the numerical systems of the Egyptians, Native Americans, or the Maya is broadening for all children. Children can
make posters describing the other systems. They can also change the prices on price tags or in circulars to show prices in
other numerical systems.

3. Counting Language

Learning to count in the language of another student can be a means of introducing multiculturalism into the classroom.
Many languages have a counting system that is much more logical than ours. In many languages, 11 is expressed as “ten-
one,” 12 is “ten-two,” up to 19, “ten-nine”; 21 is “two-ten-one,” 22 is “two-ten-two,” 31 is “three-ten-one,” and so forth.
Children can teach the class to count in their native language. They could also show their counting words in a visual exhibit
such as a poster or table display.

4. Algorithms

The algorithms we use for computation may be different in another culture. Ask children to share their algorithms. Russian
peasant multiplication or lattice multiplication are interesting and effective alternatives to algorithms commonly used in the
United States. Students can explain the algorithms that they or their parents use that are alternative algorithms to the ones
studied in class. A homework assignment might be to compute using an alternative algorithm.

5. Problem Contexts

Teachers might provide contexts for story problems that are drawn from settings and activities familiar to other cultures.
Students can write out meaningful contexts or problems to use in class.

6. Multicultural Literature

Folktales or foreign literature may be used to teach mathematics in context. Children can ask parents and relatives to relate
these tales to them. The children can then bring the tales to class.

7. Art

Symmetry patterns or artworks from different cultures may be used to promote geometric understandings. Navajo basket
designs, African mandalas, or Asian needlepoint designs are full of symmetry and geometric figures. Some students may
have native artwork at home that they can bring to class. Children can produce artwork that resembles the native art. Chil-
dren can also cut out examples of native art in magazines and assemble collages to represent the native artwork.

8. Recreational Mathematics

Games such as Pente, GO!, Sudoku, or Mancala involve strategies that promote logical thinking. One station of an activity
center can be stocked with these different games.

9. Calendars

Calendars from different cultures include different names for months, days of the week, and frequently different ways of
recording the date. Foreign names might be included in the daily calendar reading. Several foreign language calendars could
be displayed around the classroom, and different students assigned to read the date each day.

10. Notation

It can be beneficial to have children show and explain their different notations for familiar representations. For example,
∠ ᭝
some Latin American children will correctly write ABC for ∠ ABC or QRS for ᭝QRS. Children can share the notation they

learned in their native land at appropriate times during mathematics class. Groups of students could design posters to show

the notations that other cultures use.

34 Part 1 Guiding Elementary Mathematics with Standards

add-on to supplement the study of mathematics. It is Students Who Have Limited
the vehicle for the very study of mathematics. English Proficiency

Every mathematics The United States has a well-deserved reputation as
classroom has its own cul- a melting pot of races, cultures, and nationalities.
ture, either one that recog- In the nation’s classrooms nearly 6 million students
nizes and prizes the vari- have limited English proficiency (LEP). By some es-
ous ethnic backgrounds timates, within 20 years most students in American
of its students or one that schools will not speak English as a first language. As
ignores all cultural back- the number of such students has grown, educational
grounds and sets mathematics in a culturally neutral policies for these students have changed. More and
context. To best encourage ethnically diverse stu- more states are eliminating bilingual or foreign lan-
dents, the teacher must acknowledge their cultural guage classrooms in favor of schooling LEP students
and ethnic backgrounds and use those backgrounds in regular English-speaking classrooms. The result is
to enliven and enhance their study of mathematics. that more and more teachers will have LEP students
Mathematics may be the first subject area that in class.
culturally diverse students begin to learn, but teach-
ers cannot assume that students with computational The first language of
proficiency understand number and operational LEP students is intimately
concepts. Certainly, many children from all cultural tied to their cultural or
backgrounds have learned pencil-and-paper com- ethnic background. It is
putations. However, focusing exclusively on compu- the language spoken at
tations deprives the student of language and con- home and in their social
cept development needed for problem solving and community, and it is the
reasoning. Problem solving, best done in the con- means by which LEP stu-
text of real-world story problems, should be based dents communicate with
on children’s experiences. Zanger (1998) reports a family, peers, and neigh-
classroom project of collecting and publishing math bors about their neighbor-
problems written by students and parents. At first hood and community, all
the parents were wary, but when the book was pub- of which are immersed in
lished, it became an immediate hit with students, their native culture. Many
who now had problems related to their experiences. LEP students might speak
Students unable to express themselves fully in Eng- English only at school.
lish might write out problems in their first language LEP students learn English over several years.
accompanied by graphic depictions. These prob- Table 3.4, adapted from the Virginia State Board
lems can then be translated by another student or of Education, gives some sense of how long it can
faculty member. When the teacher writes real-world take for LEP students to communicate effectively in
problems, the problems should be concise, clear, English.
and free of slang or unfamiliar idioms. The timeline in Table 3.4 assumes continuous liv-
Many school districts provide professional devel- ing in the United States. Many LEP children return to
opment for teachers who work with cultural, ethnic, their native lands every year, sometimes for weeks
and language diversity. A new teacher should take at a time. During their visit they will rarely speak
advantage of these professional opportunities. Un- or hear English, and so such visits may delay the
derstanding the principles of good mathematics is a development suggested in the table. With Table 3.4
foundation, but additional preparation provides the as a guide, teachers can begin to understand why
background to become more sensitive to the spe- a child who can carry on a perfectly good English
cific cultural and ethnic groups in the local district conversation in the playground can rightly claim he
and to become familiar with programs for language cannot understand the English discussions in the
learners offered in and beyond the classroom. classroom. By some estimates, instructional class-

Chapter 3 Mathematics for Every Child 35

TABLE 3.4 • Time Needed for Language Acquisition

State of Language Acquisition General Behavior of Students Who Have Limited English Proficiency

Silent/Receptive Stage • Point to objects, act, nod, or use gestures
• 6 months to 1 year • Speak hesitantly
• 500 receptive words
• Produce one- or two-word phrases
Early Productive Stage • Use short repetitive language
• 6 months to 1 year • Focus on key words and context clues
• 1,000 receptive/active words
• Engage in basic dialogue
Speech Emergence Stage • Respond using simple sentences
• 1–2 years
• 3,000 active words • Use complex statements
• State opinions and original thoughts
Intermediate Fluency Stage • Ask questions
• 2–3 years
• 6,000 active words • Converse fluently
• Understand grade-level classroom activities
Advanced Fluency Stage • Read grade-level textbooks
• 5–7 years • Write organized and fluent essays
• Content area vocabulary

SOURCE: Virginia Department of Education (2004).

room English lags behind conversational English by copy material from the board or overhead screen.
up to 5 years. An added factor is that mathematics Instead, prepare a handout for them (actually for the
has its own language, which is quite different from whole class) so they can focus on discussions and
conversational English. Words and terms that chil- explanations and not on the task of accurate copy-
dren hear only in a mathematics class include de- ing. In a similar light, when writing on the board,
nominator, numerator, quotient, isosceles, greatest print rather than use cursive.
common factor, and composite, to name a few. As
a consequence, mathematics English is even more Extralinguistic clues are particularly helpful to
challenging than general classroom English. LEP students. Appropriate gestures, facial expres-
sions, examples and nonexamples, and models or
A number of teaching strategies can help LEP manipulatives can illuminate a discussion that might
students in the classroom. For example, speaking otherwise be unintelligible to LEP students. When
slowly and clearly and carefully pronouncing each introducing new terms or vocabulary, teachers can
word is helpful to LEP students. In spoken and written write these on the board beforehand and point to
speech teachers should avoid slang, colloquialisms, the specific word as it is being discussed, along with
and other unusual linguistic expressions, including some model or action to clarify its meaning, such as
contractions. It can be beneficial to LEP students pointing to a picture of a rectangle when that term
to use familiar phrases and vocabulary, especially is used.
when introducing new concepts. That way they are
able to focus on the mathematics involved and not A well-meaning teacher may use childish vocab-
have to wrestle with the English content. Sometimes ulary and an extremely slow pace in conversations
a simple alteration in speech patterns can be advan- directed to LEP students. LEP students can perceive
tageous. Instead of using pronouns, teachers can these childish speech patterns and may react nega-
use full referents. Not “What is its area?” but “What tively to them. Avoid speaking louder than normal
is the circle’s area?” Do not expect LEP students to to LEP students, as some people do when assist-
ing a foreign tourist with directions. LEP students

36 Part 1 Guiding Elementary Mathematics with Standards

can sense this and may react against it. If a teacher pect. For example, the
makes any of these errors, recognizing the error and number 3.14 is written 3,14
correcting it is the key. by some cultures, whereas
the number 3,452 is writ-
Grouping is another way to support learning En- ten 3.453. We write $1 but
glish. English learners can work in small groups to read this as “one dollar,”
solve story problems or open-ended problems. An with the leading symbol
all-English-learning group allows children to com- ($) read last. However,
municate freely with one another about the problem we read 5¢, 3Ј, 4 lbs, and
and its potential solutions. At other times, English 6 yds from left to right, as
learners might be grouped with English speakers the symbols appear. Some LEP students read right to
who can model procedures and behaviors appropri- left, so the number 86 means “sixty-eight” to them.
ate for the mathematics classroom. English learners A simple expression such as 8 Ϭ 2 is read several
can also be grouped with bilingual children. Such different ways in English:
a grouping validates their culture and supports the
maintenance of their first language while learn- 8 divided by 2
ing mathematics. As language learners gain con-
fidence, they may make an oral presentation to a 2 goes into 8
small group, where they can make use of gestures,
intonations, and visual aids. 2 divided into 8

Another aspect of teaching mathematics to LEP 8 divided in half
students involves the mathematics itself, the words
and symbols of mathematics. There are many terms Consider how confusing this simple expression and
in mathematics that have entirely different mean- others can be to LEP students. Becoming aware of
ings in conversational English. A list of these words the subtle aspects of language is an important first
includes: step for teachers to help LEP and other students be-
come successful in mathematics.
Right Volume Left
Square Face Variable E XERCISE
Mode Median Round
Give four English expressions that are equivalent to
Native English speakers can tell from the context
which meaning of the term is suggested, but not LEP the mathematics expression 7 ؊ 2. •••
students. They need time to develop the ability to
discern what meaning such words convey. Assessing LEP students in mathematics can be
challenging. Is the difficulty with a mathematics
Consider mathematics homophones, such as concept or a homework problem due to a language
one/won, whole/hole, cent/scent, and arc/ark. Even issue or the mathematics involved? At times LEP
a native English-speaking child needs assistance children might frame an explanation in their native
to discern the correct word and meaning. LEP stu- language. Another child or a teacher might then
dents require more time and experience before they translate the answer. This enables the child to com-
can do so. Then there are soundalike words that are municate without the burden of trying to write in a
difficult for LEP students to distinguish: line/lion, new language, and the teacher has a more accurate
three/tree, leave/leaf, graph/graft, four/fourth, and understanding of the child’s progress.
angle/angel. In all these cases the teacher who pays
careful attention to the mathematics vocabulary Many LEP students learn different algorithms
seeks to clarify any confusion before it becomes for the four basic operations in their native lands.
problematic. It may be more advantageous to allow LEP students
to continue to use the algorithms they have already
Another aspect of mathematics to consider is learned rather than insist that they adopt the famil-
the symbols of its language. Many symbols are not iar algorithms we use in the United States. Allow-
as universally used and accepted as we might ex- ing LEP students to use their native algorithms also

Chapter 3 Mathematics for Every Child 37

validates the students’ cultural background, as sug- may work from home and cannot give up valuable
gested in the section on multiculturalism. For exam- computer time. Teachers may interview students to
ple, students from some Latino cultures will perform assess their home computer situation. See Chapter
two-digit subtraction as shown here. 6 for information about how useful the computer,
calculator, and the Internet are for helping children
83 8 13 learn mathematics.
Ϫ47 Ϫ5 7
6 What about children who do not have any Inter-
3 net access at home? What can be done to ensure
83 Ϫ 47 ϭ 36 that all children have appropriate opportunities to
use the Internet? A teacher can arrange for students
This algorithm avoids regrouping. Instead, 10 units to use the classroom or school library computers
are added to the units of the larger number, and one during the school day or before or after school or a
10 is added to the tens digit of the smaller number. computer in the public library. Children once went
Because both numbers were increased by 10, the to the library to use the encyclopedias and other
difference between the resulting values is the same reference texts. Library opportunities for children,
as it was for the original values. however, must be carefully considered. To require
a child to visit a public library several times a week
It is important to remember that LEP students seems unfair when others in the class can use a
may be further advanced in their mathematics than computer at home. Even one public library visit in
the rest of the students in the class. They simply a two-week period may be burdensome because of
need to develop a command of English that will en- location or transportation. Teachers can get all the
able them to contribute to the class and progress in facts involving their students before deciding what
their mathematical knowledge. With your help they is reasonable for out-of-school computer use. The
can succeed. central tenet is that all students should have equi-
table and appropriate opportunities to succeed in
Technological Equity mathematics.

Technological advances have changed the land- Students Who Have Difficulty
scape of mathematics education. Calculators, com- Learning Mathematics
puter software, and the Internet have had profound
effects on the teaching and learning of mathemat- Difficulty learning mathematics denotes a wide
ics. These new and emerging technologies will have range of impediments that students must deal with.
a yet undetermined effect on mathematics teaching One child may have a disability in reading, whereas
and learning. We advocate taking appropriate ad- another may have difficulty in mathematics; one
vantage of existing technologies to improve math- child with disabilities may be hyperactive, whereas
ematics learning. another may be quiet and withdrawn. Difficulty
learning mathematics can stem from a variety of
The positive effects of technology must be sources, including emotional, learning, and cogni-
viewed through the lens of equity. Not all students tive disabilities. In other words, children with learn-
have the same access to the technologies we discuss ing disabilities are a heterogeneous group (Mercer,
throughout the text. Many schools provide calcula- 1992, p. 25).
tors and software programs. The question of equity
arises with access to the Internet. Many students The Individuals with Disabilities Education Act
can use a computer at home to explore the Internet (IDEA), a federal law, requires that children with
and to try some of the interactive lessons in later disabilities have free public education in the “least
chapters. However, Internet access is not univer- restricted environment.” Rather than being placed
sal. According to a recent Education Week survey, in isolated special education settings, more and
about 1 in 5 students does not have a computer at more students with disabilities are being included in
home (Education Week, 2006). Not all children with the regular classroom, sometimes with instructional
a computer at home have Internet access. For some
children with Internet access, one or both parents

38 Part 1 Guiding Elementary Mathematics with Standards

support from aides or special education teachers. ing, practicing, and drilling the same math fact,
Inclusion means that teachers work with students procedure, or algorithm many times to achieve the
with a variety of cognitive abilities, learning styles, ability to use it without prompting. For example,
social problems, and physical challenges. The chal- all children practice the algorithm for multiplica-
lenge is to find strategies that maximize learning for tion with whole numbers, so they learn it and can
all learners. develop the habit of using it automatically, without
having to think through each step of the algorithm
The learning expecta- every time they use it. Alternative algorithms, dis-
tions and nature of the in- cussed in Chapter 12, may be less confusing for
structional support for stu- many students who struggle with traditional algo-
dents with special needs rithms. Students with special needs will require
is outlined in an individu- more time with such an algorithm, careful and re-
alized educational plan, peated modeling of the algorithm by their teacher,
commonly referred to as and many repeated practice sessions to achieve a
IEP. Based on individual- measure of mastery. However, it is not necessary to
ized goals, the teacher insist on mastery of math facts or a particular algo-
may modify instructional rithm before children with leaning disabilities can
techniques and assign- move forward with their mathematics learning.
ments as described in the
plan. Growing evidence Students with disabili-
shows that active engagement of students is as ties can also benefit from
beneficial for students with learning disabilities as math cards. Students write
for those without diagnosed disabilities. Table 3.5 their own math facts on
outlines types of problems associated with students an index card, or they
who have learning disabilities. write information to help
For many children with special needs, especially them with a mathematical
those with learning and cognitive disabilities, their concept or process. For
difficulty comes from the fact that their processing example, the math card shown on page 39 was used
time is slower than that of regular children. They by a child with special needs to help him recall the
may require more time to understand a mathematics steps in the long division algorithm. He had previ-
concept, apply it, and make it theirs. One approach ously engaged in developmental activities to build
to help children access new mathematics might be his understanding of division but needed help to
termed overlearning. Overlearning involves learn- complete the steps in the division algorithm.

TABLE 3.5 • General Learning Disabilities: Area of Disability

Academic Social-Emotional Cognitive

Poor reading skills Lack of motivation Short attention span

Inadequate reading comprehension Easily distractible Perceptual difficulties

Problems with math calculation Inadequate social skills Lack of motor coordination

Math reasoning difficulties Learned helplessness Memory deficits

Deficient written expression Poor self-concept Problem-solving hurdles

Listening comprehension problems Hyperactivity Difficulty evaluating one’s own learning processes

SOURCE: Adapted from Mercer (1992, p. 53).

Chapter 3 Mathematics for Every Child 39

Divide Read
Multiply Draw a picture of what is
Subtract happening.
Compare What is the question?
Bring down DoI need the exact answer?
What operation do I need?
Do I use a calculator, pencil,
or mental math?

Figure out

Check my answer

Another math card (adapted from Bley & Thorn- cases the cards continue to serve as a security blan-
ton, 2001) helped a student relate mathematical pro- ket, a quick reminder to students of what they have
cesses, symbols, and English expressions. already mastered. Still other children will require
the math cards for all their academic lives and may
Add Plus ϭ ᮍ use similar cards for life outside the classroom.
Subtract Minus ϭ ᮎ
Multiplication Groups of ϭ ᮏ Many students with special needs have difficulty
Division Divide By ϭ ᭺Ϭ with their visual perception. The page of a standard
mathematics textbook might be an overwhelming
Note that the colors help the child relate each swirl of colors, pictures, and symbols. Students
operation and symbol to the appropriate English might cover up the nonessential part of a page when
representation. solving a problem from their book. They might use
a piece of cardboard with a rectangular opening
Another math card helped one student to solve cut in it. The opening could be shifted around the
word problems. She listed each important step for page to reveal a problem, diagram, or example.
solving every word problem she confronted. Children with visual difficulties or trouble with fine
motor skills will find some of the algorithms difficult
These math cards and similar learning aids because of the requirement of precise placement
should be available to students whenever they need of digits as they use the algorithm. The algorithm
them. Sometimes children outgrow them and do for multiplication with whole numbers requires
not require the prompts the cards provide. In other

40 Part 1 Guiding Elementary Mathematics with Standards

78 children to line up columns of dition, many gifted and talented students are at risk.
ϫ5 2 numbers, which can prove chal- Surprisingly, the dropout rate for gifted and talented
students (about 20%) nearly matched the dropout
lenging. One possible solution rate of the general student population (about 25%)
in several studies (Ruf, 2006; Renzulli & Park, 2002;
to children’s difficulty is to pro- Schneider, 1998). Thus gifted and talented students
deserve a place in a chapter on equity.
vide a grid for them to use, as
At one time, intelligence test scores were used as
shown here. the sole determiner of gifted and talented students;
today, identification is based on multiple criteria.
There are many Internet resources for special- Laurence Ridge and Joseph Renzulli (1981) identi-
fied three clusters of traits: above-average general
needs children. These sites contain lesson plans or ability, task commitment, and creativity. Table 3.6
lists the characteristics of each cluster. Test scores,
links to other sites with lesson plans that might be teacher observations, and work samples are often
used to evaluate these characteristics. In the past,
used with students with special needs: qualified gifted students were eligible for various
services, ranging from special programs and activi-
http://mathforum.org/t2t/faq/faq.disability.html ties during the school day to enrichment classes and
http://www.teachingld.org/teaching_how-tos/ extracurricular programs.
math/default.htm
http://www.ricksmath.com The federal No Child Left Behind Act has had
http://www.col-ed.org some unexpected consequences for gifted and tal-
http://www.learnnc.org/lessons/ ented education. The act focuses attention on under-
http://www.mathforum.com achieving groups, seeking to raise their mathemat-
http://www.kidsdomain.com ics performance, but it ignores gifted and talented
http://www.mathsolutions.com students. As of 2004, 22 states did not contribute any
http://www.teachersfirst.com funds to gifted and talented programs, and 5 oth-
http://www.arches.uga.edu ers earmarked only $250,000 to support gifted and
http://www.crayola.com/educators/lessons talented education. At present, only 2 cents of every
http://www.teachers.net/lessons/ $100 in federal funds for education is dedicated to
http://www.yale.edu/ynhti/curriculum/units gifted and talented learning (Bilger, 2004). An ex-
http://www.remc11.k12.mi.us/bcisd/curriculum.html ample of how gifted and talented funding is drying
http://www.successlink.org up can be found in Illinois. In 2002, Illinois provided
$19 million for gifted and talented programs. In
Physical Disabilities 2004, all gifted and talented funding was eliminated
(Shermo, 2004).
http://www.sedl.org/about/termsofuse
.html#access Gifted students are far more likely to be in a reg-
http://www.rit.edu/~comets/bibliopage1.htm ular classroom for the entire school day. Although
gifted students will be able to achieve basic mathe-
Gifted and Talented Students matics without any special attention, these students
who are gifted in mathematics deserve as much at-
Why are students who are gifted and talented in- tention to their needs as any other group of students
cluded in a chapter on equity in mathematics? After to meet their full potential in mathematics.
all, if these students are gifted, do they really need
any assistance to achieve? The point here is not to In addition to the characteristics listed in Table
help gifted and talented students meet a specific 3.6, classroom teachers should also recognize other
achievement level that is established for all students. pertinent characteristics of gifted students in math-
Rather, the point is to meet their specific needs. All ematics. Gifted students may immerse themselves in
children, including gifted and talented children, de- one or two mathematics topics intensely for a period
serve a challenging mathematics curriculum, one of time. They often have original ways of thinking
that will advance their knowledge of mathematics
at a pace that will continue to motivate them. In ad-

Chapter 3 Mathematics for Every Child 41

TABLE 3.6 • Characteristics or Traits of Gifted and Talented Students

Above-Average General Ability Task Commitment Creativity

Accelerated pace of learning; Highly motivated Curious
learns earlier and faster

Sees relationships, Self-directed and perceptive Imaginative
readily grasps big ideas

Higher levels of thinking; Accepts challenges, may be Questions standard procedures
applications and analysis highly competitive
easily accomplished

Verbal fluency; large vocabulary, Extended attention to one area Uses original approaches and solutions
expresses self well orally and of interest
in writing

Extraordinary amount Reads avidly in area of interest Flexible
of information

Intuition; easily leaps Relates every topic to own area Risk taker, independent
from problem to solution of interest

Tolerates ambiguity Industrious and persevering

Achievement and potential High energy and enthusiasm Masters advanced concepts in field of
have close fit interest

SOURCE: Ridge & Renzulli (1981).

and a willingness to try different approaches and in- characteristics. Additional characteristics, listed in
ventive problem solutions. Despite the stereotype of Table 3.7, were developed by Russian educator Igor
gifted children as ideal students, they are individu- Krutetskii, who found that gifted and talented stu-
als with the full range of behaviors, including be- dents in mathematics exhibited a number of these
ing stubborn, messy, forgetful, or rebellious. Traits characteristics to a high degree.
of gifted students—such as questioning authority,
extreme imagination, and absolute focus on one After identifying potential gifted and talented stu-
topic—often dismay teachers but may be signs of dents in mathematics, what helps them reach their
boredom and a need for additional responsibility full potential? In a word, challenge. If gifted children
and challenge. are not challenged by the curriculum early in their
school years, they will equate smart with easy. As a
Gifted students do not learn concepts and skills result, challenges and hard work will feel threatening
automatically; they need to participate in a planned, to their self-esteem. Many gifted children sit through
systematic mathematics program in which they lessons that they fully understand and could easily
learn basic mathematical concepts, develop rea- teach to the rest of the class.
sonable proficiency with basic facts and algorithms
and other skills, and apply their knowledge to solv- TABLE 3.7 • Krutetskii’s Characteristics of
ing problems. However, gifted students may learn Mathematics Giftedness
at a rapid pace and need additional opportunities
beyond the basic curriculum. Resourcefulness Flexibility
Economy of thought Use of visual thinking
To challenge and provide quality mathematics Ability to reason Ability to generalize
to gifted and talented students, teachers need to Mathematical memory Ability to abstract
identify gifted and talented students in the class. Enjoyment of mathematics Mathematical persistence
Test scores and grades might assist in identifying
these students, but they are not the only identifying SOURCE: Krutetskii (1976).

42 Part 1 Guiding Elementary Mathematics with Standards

Challenges for gifted students can take many and talented students be used sparingly as tutors or
forms. A well-meaning teacher might provide extra teachers.
problems to keep a gifted student on task, but this
serves only to punish students who complete an as- All the preceding discussion applies to large-
signment quickly. In this case gifted and talented group instruction or individualized assignments.
children quickly learn that their hard work results When forming small groups with gifted and talented
only in more work to do. Instead of assigning more students, teachers should place one gifted student in
of the same types of problem, challenge problems a group with regular students or form a group com-
such as brainteasers might be used to spark atten- posed of all gifted and talented students. When a
tion and motivate gifted students to move beyond gifted student is in a small group with regular stu-
the typical problems at hand. In many cases gifted dents, both benefit. Regular students benefit from
students could prepare a presentation to the class of observing the effort and time on task that many
their findings or solution strategies. At a minimum gifted students exhibit. They also benefit from the
they could explain their results to the teacher in insights gifted children have and the conclusions
written and/or oral form. that they make. The gifted children benefit by de-
veloping interpersonal skills and by framing their
When assigning homework to a class, some mathematics concepts into formal expressions that
teachers allow students to try the most difficult they can communicate to others. By doing so, they
problems first. If students can solve the most dif- solidify the foundations for building even more ad-
ficult problems, then they do not need to solve all vanced mathematical understandings.
the problems on the assignment. In this instance all
students should have the opportunity to try solving When only gifted students work together in a
only the difficult problems. It may be surprising to small group, they have the opportunity to work with
see who tries to do so. At times students who suc- peers who are able to assimilate and organize in-
cessfully solve the most difficult problems are not formation quickly and completely. They can work
the ones the teacher might have identified as gifted with group members who are able to conceptual-
and talented. ize relationships and make conclusions as quickly
as they can. As a group, much should be expected
Another approach used by some teachers pro- of them. They should produce excellent results, far
vides the opportunity for gifted students to test out beyond those expected of other groups. In addition,
of a unit of study. For example, a gifted student might they should make a class presentation of their find-
try the unit test at the start of the unit. If the child ings so that the entire class can benefit from their
scores high enough, then she might spend math- concerted efforts.
ematics time on a project or exploration in lieu of
the unit topic. This will prevent gifted students from In closing, having gifted students in the class-
spending time in what they do not need, review of room requires a teacher to ensure that their educa-
a topic that the rest of the class must practice. As tional needs are met, just as the needs of all other
suggested earlier, the results of their explorations children are met. At times, gifted students can be
might be reported to the class and certainly to the frustrating, charming, and inspiring, all in the same
teacher. day. That is simply who they are, and it is the chal-
lenge for a classroom teacher to help these children
A common strategy that many teachers use is achieve and meet their potential in mathematics.
to have gifted and talented children tutor or assist
students who are having difficulty with a particular Individual Learning Styles
topic. It is true that both can benefit from such an ar-
rangement; the struggling students receive individu- Mathematics teachers work not only with students
alized help, and the gifted students explicitly explain of many ethnic and cultural backgrounds, language
the mathematics at hand, perhaps formalizing their differences, and learning capabilities but also with
innate understanding of a topic. However, gifted students with individual learning styles. Recent re-
and talented students are in school to advance their search and theories about learning point out differ-
own mathematics, not to teach others, which is the ences in children’s learning styles and strengths.
teacher’s responsibility. We recommend that gifted

Chapter 3 Mathematics for Every Child 43

Multiple but not at all comfortable writing poetry or speak-
Intelligences ing before an audience. A person with the natural-
ist intelligence may be comfortable around plants
The theory of multiple but have little interest in animals. Linguistic (verbal)
intelligences was first intelligence has been the mainstay of traditional
proposed by Howard schooling. Even logical/mathematical intelligence,
Gardner in 1983. In Frames strongly related to problem solving, reasoning, and
of the Mind: The Theory mathematical thinking, receives little attention in
of Multiple Intelligences, classrooms where teacher-directed lessons are the
Gardner asserts that peo- primary means of instruction.
ple are capable, or even
gifted, in different ways. Gardner, Kagan and Ka- In real-world activities the intelligences are inte-
gan, and other advocates of multiple intelligences grated, and several may be needed to complete com-
believe that no single description of intelligence plex tasks. Carpeting a house requires a number of
exists. Intelligence tests focus on only one type of intelligences: drawing a picture of each room (spa-
learning, typically verbal knowledge, but individuals tial), measuring each room (bodily/kinesthetic),
are “smart in different ways” (Kagan & Kagan, 1998, calculating square yards and cost (logical/mathe-
p. xix). Gardner initially identified seven intelligences: matical), and discussing choices and making selec-
linguistic, logical/mathematical, spatial, body/kin- tions (linguistic and interpersonal). By offering all
esthetic, musical, interpersonal, and intrapersonal. children a rich curriculum with a variety of hands-
He has since added an eighth intelligence: natural- on problem-solving activities, teachers provide the
ist. Figure 3.1 gives a short description of each of the variety needed to appeal to and develop all the
eight intelligences. intelligences.
In each intelligence area, individual differences
also occur. Individuals may feel more comfortable E XERCISE
with some of the intelligences and less comfortable
with others. Differences may occur within an intelli- Take a Multiple Intelligences Test at http://www
gence; a person might be comfortable writing prose .bgfl.org/bgfl/7.cfm?s‫؍‬7&m‫؍‬136&p‫؍‬111,resource_

list_11. Were the results what you expected? •••

Multiple Intelligences Learning Styles

INTRAPSEeRlfSONAL LINGUISTIC LOGICAL-MATHEMATICAL Learning styles is another way of describing how
Word people receive, process, and respond to their world.
Smart Learning modalities is one way of describing
Smart learning. Do learners prefer to process informa-
tion visually, auditorily, kinesthetically, or in some
INTERPERSONAL C combination? Do learners prefer to learn individu-
AB ally or in groups, with quiet or background noise, in
People Logic structured or open-ended assignments (Carbo et al.,
Smart Smart 1986)? Other style descriptions focus on specific as-
pects of learning and personality on a continuum.
12 3 Table 3.8 lists eight different learning styles with
the related adjectives at the extreme points of the
MUSICAL Music Nature NATURALIST continuum.
Smart Smart
Based on present research and understanding of
Body Picture individual differences, teachers cannot design les-
Smart Smart sons specific to the styles and intelligences of each
BODILY-KINESTHETIC SPATIAL child. Instead, effective teaching must allow for
learning needs of a wide variety of learners. Variety
Figure 3.1 Gardner’s eight multiple intelligences in teaching approaches, use of materials, focus on

44 Part 1 Guiding Elementary Mathematics with Standards

TABLE 3.8 • Learning Styles and Descriptors verse classroom. A balanced approach encourages
student thinking and creativity, autonomy of thought
Orientation Dependent Independent and action, and meaningful learning in many con-
Control Internal External locus texts. Design of worthwhile and challenging math-
Stimulation High need Low need ematics activities is the core of teaching.
Processing Random Sequential
Thinking Concrete Abstract E XERCISES
Personality Introvert Extrovert
Response Emotional Rational What kind of learner are you? Discuss your
Time Impulsive Reflective strengths and weaknesses as a learner with a

meaning, and relating to individual needs and inter- classmate. •••
ests are key ideas in teaching the many students in
any classroom. The characteristics of good teach- Do you believe students should be able to concen-
ing are congruent with constructivist principles, re- trate on their strong areas and avoid their weak
search on effective teaching, and strategies for a di- areas, or should students develop their skills in all

the intelligences? •••

Summary without physical disabilities. What effect do you
believe the children portrayed in textbooks have on
Students have special learning needs in mathematics. Stu- the children who use those books?
dents with disabilities may need classroom adaptations 2. At what level of schooling—early elementary,
to support their learning. Gifted and talented students intermediate grades, junior high, senior high, or col-
need additional challenges. Students who have limited lege—do you believe the best mathematics teaching
English proficiency will require accommodations to sup- occurs? Why?
port their mathematics learning. Multiple intelligences 3. Do a ministudy of yourself. Which of the eight intel-
and learning styles underscore the need for variety in ligences discussed in this chapter do you think best
teaching and learning activities in the classroom because characterize you? What personal characteristics
all students have individual learning characteristics. We prompted you to name these intelligences?
no longer try to teach all students the same way, reason- 4. How would you characterize your learning style?
ing that equal treatment for all students means equitable 5. Print out the abstracts for three to five research
treatment. Supreme Court Justice Felix Frankfurter said, articles dealing with gifted students. What did you
“There is no greater inequality than the equal treatment learn from reading these abstracts that is useful to
of un-equals.” Treating all children the same does not you as a teacher? Which articles would you like to
ensure that they are all treated fairly. Treating children read in full?
fairly requires providing all of them with opportunities 6. What is the difference between ethnomathematics
to reach their full potential in mathematics regardless and multiculturalism?
of what individual accommodations are needed. Only
when children’s needs are met can it be said that they Teacher’s Resources
are treated equally.
Alcoze, Thom, et al. (1993). Multiculturalism in mathe-
Study Questions and Activities matics, science, and technology: Readings and activities.
Reading, MA: Addison-Wesley.
1. Look through a recent elementary school mathemat-
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male/female, majority/minority, and children with/ ed.). Upper Saddle River, NJ: Pearson.

45

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ematics education of African-American children. Reston, Research in Mathematics Education (JRME) has shifted
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dren do, to a constructivist approach, focusing on how
National Council of Teachers of Mathematics. (1999). they think. Battista and Larson give practical sugges-
Developing mathematically promising students. Reston, tions for using research to improve instruction.
VA: National Council of Teachers of Mathematics. Carroll, William M., & Porter, Denis. (1997). Invented
algorithms can develop meaningful mathematical pro-
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equity. Reston, VA: National Council of Teachers of invented algorithms, which allows students to make
Mathematics. sense of the mathematics they are doing.
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Secada, W. G., & Edwards, C. (Eds.). (1999). Changing are no algorithms for teaching algorithms. Teaching
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Changing the faces of mathematics: Perspectives on in- to formalize an algorithm, and that the most effective
digenous peoples of North America. Reston: VA: National resource for making instructional decisions is students
Council of Teachers of Mathematics. themselves.
Kerssaint, Gladis, & Chappell, Michaele. (2001). Captur-
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Americans. Reston, VA: National Council of Teachers of Mathematics problems centered on what is relevant
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46 Neel describes how multiculturalism is enhanced
when students bring in cultural artifacts that become
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procedures of recent immigrant students. Mathematics
These games and activities come from around the Teaching in the Middle Grades 7(6), 346–352.
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ematics curriculum in grades 1–8. The book, which Perkins discusses awareness of the differences in
introduces children to the ethnic heritage of others standard notation and algorithms of students in the
and encourages an appreciation of cultural diversity, class when planning lessons and activities.
contains convenient, reproducible activity pages for
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Using multiple intelligences to master multiplication.
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Children Mathematics 7(9), 500–502. One theory of learners claims eight different intel-
ligence strengths for children. Knowledge about these
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diverse students. dren’s learning of multiplication.

Moldavan, Carla. (2001). Culture in the curriculum: En- Zaslavsky, Claudia. (2001). Developing number sense:
riching numeration and number operations. Teaching What can other cultures tell us? Teaching Children Math-
Children Mathematics 8(4), 238–243. ematics 7(6), 312–319.

Numeration systems and alternative algorithms Mathematics from other cultures can be used to
from different cultures can enrich the study of math for great advantage in the classroom to illuminate various
elementary school students. mathematical concepts.

Neel, K. (2005). Addressing diversity in the mathemat-
ics classroom with cultural artifacts. Teaching Math-
ematics in the Middle School 11(2), 54–58.

CHAPTER 4

Learning
Mathematics

hen states and local school districts develop curricula using
the NCTM principles and standards, they answer the ques-
tion, What mathematics should students know? Accord-
ing to the standards in Principles and Standards for School

Mathematics (National Council of Teachers of Mathematics,
2000), students need conceptual knowledge and skills in numerical opera-
tions, geometry, measurement, data analysis, probability, and algebra
with an emphasis on problem solving and application in meaningful
contexts.

The next question for teachers is, How should I teach mathematics?
The NCTM principle of teaching and the principle of learning address this
question. Knowledge about learning and effective teaching are based on
learning theory and research and are verified in classroom practice. Teach-
ers who understand how children grasp concepts in mathematics provide
instructional experiences that support the needs of the learner and the
demands of the content. The learning needs of a diverse student population
reinforce the need for teachers to understand how students learn mathemat-
ics. In this chapter we review theories of learning and research on learning
mathematics.

In this chapter you will read about:
1 Learning theories and their implications for elementary mathematics

instruction
2 Research on teaching mathematics and recommended instructional

practices

47

48 Part 1 Guiding Elementary Mathematics with Standards

Learning is a complex cognitive process. For more E XERCISE
than 100 years psychologists have observed people
as they mastered skills; from their observations Can you think of something you have learned
they formed new concepts and developed learning through repetition? What happened when you
theories to explain this complex process. No single continued to practice the skill or knowledge? What
theory explains all the nuances and complexities
involved in learning. However, understanding vari- happened when you stopped practicing? •••
ous learning theories provides a foundation for the
choices that teachers make in their teaching. A brief Cognitive Theories
synopsis of learning theories as they relate to teach-
ing mathematics reinforces information from child Cognitive theories share a common belief that men-
development or educational psychology in the con- tal processes occur between the stimulus and re-
text of mathematics instruction. sponse. Mental processes, or cognitions, although
not directly observable, result in highly individual-
Theories of Learning ized responses or learning; therefore human beings
learn by creating their own unique understandings
Behaviorism from their experiences. The major differences be-
tween behaviorist and cognitive theories are sum-
Behaviorism is a theory of learning that focuses marized in Table 4.1.
on observable behaviors and on ways to increase
behaviors deemed positive and decrease behaviors Different cognitive theories explain learning
deemed undesirable. In the late 19th century the by emphasizing different aspects. Some cognitive
“mental discipline” theory of learning influenced theories focus on how complex learning proceeds
the way mathematics was taught. According to this from one level or stage to the next. Cognitive-
theory, the mind is like a muscle and benefits from developmental theories, such as those of Jean
exercise, just as muscles do. Early in the 20th century Piaget, Richard Skemp, and Jerome Bruner, propose
stimulus-response theory explained that learning levels of successively more complex intellectual
occurs when a bond, or connection, is established understanding or conceptualization. Other theories
between a stimulus and a response. Thorndike, Pav- explain learning through the functions or mecha-
lov, Skinner, and others demonstrated the effects of nisms that are involved. Information-processing
different conditioning plans on a variety of animals. models compare learning to computer functions,
Positive reinforcement, such as rewards of food and brain-based theories explain learning in terms
or water, led animals to perform a task again and of how the brain receives, stores, and retrieves infor-
again. Animals could be trained to respond to a cer- mation. Constructivism is another term associated
tain stimulus by prior rewards. with cognitive theories. Regardless of the particular
model used, cognitive theories center around the
Behaviorism has a long history in teaching, as idea of constructing meaning from experience.
many teachers subscribe to the stimulus-response
theory by “exercising” the brain. Drill and practice Key Concepts in Learning
of facts and mathematical procedures is based on a Mathematics
belief that repetition establishes strong bonds. Since
the 1930s researchers and theoreticians have chal- Focus on Meaning
lenged the stimulus-response theory as too simplis-
tic to explain all learning. If learning occurs only as In the 1930s the meaning theory of William
a response to a stimulus, how can people create new Brownell challenged the mental exercise and train-
words, new art, new music, new inventions, or even ing focus of the stimulus-response theory. Brownell’s
new theories? Cognitive and information theories meaning theory suggested that children must un-
explore how learning is influenced by language and derstand what they are learning if learning is to be
culture, individual and social experiences, intention permanent. When children generate their own solu-
and motivation, and neurophysiological processes. tions to problems while investigating the meanings
of mathematical concepts with manipulative materi-
als and other learning aids, they are demonstrating

Chapter 4 Learning Mathematics 49

TABLE 4.1 • Comparison of Behaviorism and Cognitivism

Behaviorism Cognitivism

Principal concepts Stimuli, responses, reinforcement Higher mental processes (thinking, imagining,
problem solving)

Main metaphors Machinelike qualities of human Information-processing and computer-based
functioning metaphors

Most common research Animals; some human research subjects Humans; some nonhuman research
subjects subjects

Main goals To discover predictable relationships To make useful inferences about mental
between stimuli and responses processes that intervene to influence and
determine behavior

Scope of theories Often intended to explain all significant Generally more limited in scope; intended to
aspects of behavior explain more specific behaviors and processes

Representative theorists Watson, Pavlov, Guthrie, Skinner, Hull Gestalt psychologists, Bruner, Piaget,
connectionist theorists

SOURCE: Lefrancois (2000, p. 194).

the meaning theory (Brownell, 1986). Marilyn Burns, During the preoperational stage (age 2–3 to
a leading mathematics educator, also emphasizes age 6–7), children gradually change from being
the importance of meaning in learning mathemat- egocentric and dominated by their idiosyncratic
ics. She suggests that teachers must do “what makes perceptions of the world to beginning to become
sense” rather than teaching by rote (Burns, 1993). aware of feelings and points of view of others in
their world. Children develop symbol systems, in-
E XERCISE cluding objects, pictures, actions, and language,
to represent their experience. Blocks can be build-
Can you think of something you “learned” even if ings or trucks, cups or plates, people or numbers.
you did not understand it? How did that experience Representing ideas and actions with objects is an
make you feel? How do you feel when you learn important step toward understanding pictures and,
later, symbols. Children’s concepts of number and
something you really understand? ••• space start with concrete objects and interactions
with peers and adults.
Developmental Stages
During the concrete operational stage (ages
Piaget described learning in four stages: sensorimo- 7–12), children master the underlying structure of
tor, preoperational, concrete operational, and for- number, geometry, and measurement. Work with
mal operational. The sensorimotor stage occurs concrete objects is the foundation for developing
between birth and age 2–3 years. Foundations for mathematical concepts represented with pictures,
later mental growth and mathematical understand- symbols, and mental images. Children learn about
ing are developed at this stage. For instance, chil- classification systems based on attributes of objects,
dren learn to recognize people and things and to events, and people and how they are alike and dif-
hold mental images when the people or things can ferent. They gradually consider multiple attributes si-
no longer be seen. This ability, called object per- multaneously: The cube is red, rough, thick, and big;
manence, is essential for recalling past experiences the triangle is yellow, thin, smooth, and small (Fig-
to connect with new experiences. Rather than “out ure 4.1). Children recognize actions that are revers-
of sight, out of mind,” children need to be able to re- ible or inverses, such as opening and closing doors
member events, objects, and ideas even when they or joining and separating sets. Addition and subtrac-
are no longer present.

50 Part 1 Guiding Elementary Mathematics with Standards

Figure 4.1 Attributes of geometry blocks Social Interactions

tion are reversible because one reverses, or undoes, Lev Vygotsky believed that interactions between
the other. Children learn to think about parts and the learner and the physical world are strongly in-
wholes needed in fractions and division. Manipula- fluenced by social interactions; his theory is called
tions of objects and pictures develop into mental social constructivism. According to Vygotsky
images and operations as children internalize those (1962), learning is enhanced as adults and peers
actions. provide language and feedback while learners pro-
cess experiences. The zone of proximal develop-
Starting at ages 11–13, more sophisticated ways ment is just beyond the learner’s current capabilities
of thinking about mathematics, including propor- but can be reached with assistance from adults or
tional reasoning, propositional reasoning, and cor- peers. Scaffolding occurs when adults or peers sup-
relational reasoning, begin and continue to develop port learners while they construct meaning from
during the teenage years and into adulthood. For- their experience.
mal operational thinking enables children and
adults to form hypotheses, analyze situations to Concrete Experiences
consider all factors that bear on them, draw conclu-
sions, and test them against reality. Richard Skemp describes learning at two levels: ex-
perience and abstraction. Interaction with physical
Through the stages of learning and interaction objects during the early stages of concept learning
with objects, events, and people in their world, provides a foundation for later internalization of
children construct meaning for new experiences ideas. Later, physical experiences are processed
in relation to old ideas and experiences. Complex again at the abstract level. An underlying structure,
mental structures, or schema, represent the unique or schema, allows prior learning to serve as a basis
understanding of how the world works. As new ex- for future learning. Time for reflection and opportu-
periences are assimilated, or taken into the mental nity to use knowledge are essential for organizing
framework, they are compared to existing schema. thoughts.
If they do not correspond, they create a state of dis-
equilibrium. Disequilibrium ends when learners rec- Similar ideas about using physical models for
oncile new experiences through accommodation conceptual learning were proposed by Dienes
or by modifying their understanding. Piaget saw (1969). His work with manipulatives, such as base-10
learning as a continual process of assimilation and blocks (Dienes blocks), convinced him that learn-
accommodation. Confusion and making mistakes ing improved when concepts were shown in “mul-
in the process of assimilation and accommodation tiple embodiments” rather than a single representa-
are natural and necessary parts of constructing new tion; place value can be demonstrated with base-10
schema. blocks in addition to bundles of stirring sticks, bean
sticks, and Unifix cubes.

Levels of Representation

Jerome Bruner (1960) was interested in how chil-
dren recognize and represent concepts. Like Dienes,
Bruner advocated discovery learning and learning
through hands-on activities. Bruner’s first stage of
representation is enactive, suggesting the role of
physical objects in learning. The second level is
ikonic, referring to pictorial and graphic represen-
tations. Finally, the symbolic level involves using
words, numerals, and other symbols to represent
ideas, objects, and actions. Bruner’s levels of rep-
resentation are related to both Piaget’s stages and
Skemp and Dienes’s ideas about the need for physi-
cal experiences in the development of meaning.

Chapter 4 Learning Mathematics 51

Procedural Learning neural network, or connectionist, model has appeal
because it mirrors the structure of the human brain
Learning a procedure, how to do something, is and accounts for the dynamic nature of learning.
another important aspect in mathematics. James Neural networks are similar to the constructivist
Hiebert and Patricia Lefevre (1986) define proce- idea of schema. Which memories are stored and
dural knowledge as recognizing symbols and learn- how they are retrieved in the brain are apparently
ing rules. Recognizing symbols is illustrated when controlled by the learner’s emotions, motivations,
a child identifies “ϩ” as a plus or addition sign but and intentions. Deciding what is important to re-
does not know what addition means. Learning member and making meaning from experience in-
the rules and steps of an algorithm is procedural crease retention and retrieval. Strong emotions also
learning. If students learn procedures without their appear to aid memory. The brain mechanisms be-
meanings, they often use the procedures at inap- hind human emotions, motivations, and intentions
propriate times. “Invert and multiply” is a procedure are not fully understood, although many cognitive
that is often misapplied because students do not un- scientists continue to investigate brain anatomy and
derstand when or why the procedure makes sense. functioning.

Conceptual knowledge provides meaning for pro- Pattern Making
cedures. Conceptual knowledge is a schema built
on many rich relationships. A student who knows In the past 30 years new technologies designed for
when joining sets is needed and follows the symbols medical diagnosis have enabled researchers to “see”
and procedures in solving a problem demonstrates which parts of the brain are active during learning
both conceptual understanding and procedural activities. According to Leslie Hart, learning occurs
knowledge of addition. Estimation skills draw on because the brain is built to find patterns and to
understanding the concepts behind procedures. If make connections between experiences. “Learning
asked to find the square root of 950, a student might [is] the extraction from confusion of meaningful
say, “I don’t remember the steps, but the answer is patterns” (Hart 1983, p. 67). Hart suggests six prem-
a little more than 30 because 30 ϫ 30 is 900.” A stu- ises about how the brain is built to learn:
dent who has only memorized the procedure for cal-
culating square roots will say, “I don’t remember the 1. The brain is by nature a magnificent pattern-
rule, so I don’t know the square root of 950.” detecting apparatus, even in the early years.

Short-Term to Long-Term Memory 2. Pattern detection and identification involves
both features and relationships and is greatly
Although cognitive scientists agree that cognitive speeded up by the use of clues and categoriz-
processes occur between stimulus and response, ing . . . procedure[s].
exactly how learning occurs is still a subject of
lively debate. One cognitive theory—information 3. Negative clues play an essential role.
processing—uses a computer as a metaphor for
learning. Instinctual behavior is similar to read-only 4. The brain uses clues in a probabilistic fashion,
memory (ROM) in a computer; it is preprogrammed. not by . . . adding up.
Random access memory (RAM) is short-term mem-
ory that the computer receives and stores temporar- 5. Pattern recognition depends heavily on what
ily but does not retain when turned off. Learning be- experience one brings to a situation.
comes permanent when new ideas and experiences
are transferred from short-term memory to long-term 6. Children and youngsters must often revise
memory, where it is stored for later retrieval and use. the patterns they have extracted, to fit new
Information is stored in the computer’s permanent experiences.
memory based on the needs and choices of the
computer user. Instead of being a passive receiver of stimuli, the
brain is an active processor of information. How the
Lefrancois (2000) describes the similarities and brain turns experience into knowledge is still being
differences between learning by humans and learn- explored, but the formation of synaptic and dendritic
ing by computers with models and metaphors. The connections between brain cells appears to be the
biological mechanism of learning. Regardless, the
conclusion is that the brain makes sense of experi-

52 Part 1 Guiding Elementary Mathematics with Standards

ences by finding relations and connections between Research in Learning and
old and new information (Jensen, 1998, pp. 90–98). Teaching Mathematics
Children process experiences into knowledge and
skills because they have human brains. Theories about learning are developed and tested
through research studies on how children learn and
Thinking to Learn how teachers teach mathematics. Through educa-
tional research, educators and psychologists ask
In reviewing the implications of brain research for and answer questions about how children learn
teaching, Eric Jensen (1998) found two elements and how teachers can improve their effectiveness
important: “First, the learning is challenging, with in teaching. Research describes, explains, and pro-
new information and experiences. . . . Second, there vides information about learning and teaching.
must be some way to learn from the experiences
through interactive feedback” (pp. 32–33). If learn- By reading research and research summaries,
ers are engaged in novel, complex, and varied expe- teachers learn more about how children learn
riences, they become critical thinkers and problem and what techniques can be used to improve their
solvers. Interactive feedback includes both internal learning. With electronic databases and search en-
and external information: what children tell them- gines, teachers can find research studies on almost
selves and information received from adults, peers, any topic in teaching or learning mathematics. The
and interactions with the physical world. Physical ERIC Clearinghouse for Science, Mathematics, and
and social interactions provide new information for Environmental Education is the major source of
the construction of linguistic, social, scientific, and research information about mathematics with the
mathematical knowledge. AskERIC database (available at http://ericir.syr.edu).
For example, by entering the key words fractions,
The metaphors for learning from computer and elementary mathematics, and research, a complete
brain models connect with constructivist theories of bibliography of research studies and other articles
learning: provide research about teaching and learning
fractions. Figure 4.2 shows the abstract of a study
• People create meaning by finding relationships from the ERIC database that describes misconcep-
and patterns in their experiences. tions about fractions. A search on “research” into
“problem solving” strategies yields an abstract
• What people learn and how they learn and how (Figure 4.3) that explains kindergartners’ problem-
long they remember depend on unique motiva- solving processes. After reading the abstract, teach-
tions, intentions, and emotions of individual ers can decide whether they want to read the entire
learners. study.

Human beings undoubtedly learn some things In educational journals and full-text online ser-
in behaviorist ways. The hot stove is a stimulus vices, teachers can find complete studies that in-
that evokes a strong, immediate, and long-lasting clude questions they are trying to answer, the sub-
response in the unwary toddler or adult. Repeat- jects who participated in the study, how data were
ing a telephone number keeps it active in short- collected and analyzed, and conclusions or an-
term memory. But humans use more sophisticated swers drawn from the data. InfoTrac College Edition
learning strategies than instinct and repetition; they and EBSCO are other database services that allow
perform unexpected and complex cognitive tasks searches and offer many full-text articles.
when they learn concepts, act creatively, and solve
problems. Cognitive theories conjecture that people E XERCISE
transcend their immediate physical sensations and
think. Cognitive theories are generalizations about Read the research abstracts in Figures 4.2 and 4.3.
learning because individuals learn in idiosyncratic What questions were the researchers trying to an-
ways based on their experiences, intentions, social swer? What did they find out, and how could that be
interaction, and maturity. The theories are complex
because human learning is complex. useful to you as a teacher? •••

Chapter 4 Learning Mathematics 53

Title: Hispanic and Anglo Students’ Misconceptions in Mathematics

ERIC Digest

Author(s): Mestre, Jose

Publication Date: March 1, 1989

Descriptors: *Concept Teaching; *Error Patterns; *Hispanic Americans; *Mathematical Concepts; *Mathematics Instruc-
tion; *Misconceptions; Anglo Americans; Concept Formation; Elementary Secondary Education; Student Attitudes

Identifiers: ERIC Digests

Abstract: Students come to the classroom with theories that they have actively constructed from their everyday
experiences. However, some of these theories are incomplete half-truths. Although such misconceptions interfere
with new learning, students are often emotionally and intellectually attached to them. Some common mathematical
misconceptions involve: (1) confusion between variables and labels, with failure to understand that variables stand
for numerical expressions; (2) mistakes about the way that an original price and a sale price reflect one another;
(3) misconceptions about the independent nature of chance events; and (4) reluctance to multiply fractions. Hispanic
students display some unique mathematical error patterns resulting from differences in language or culture. In addition,
linguistic difficulties increase the frequency with which Hispanic students commit the same errors as Anglo students. Since
students will not easily give up their misconceptions, lecturing them on a particular topic has little effect. Instead, teachers
must help students to dismantle their own misconceptions.

One effective technique induces conflict by drawing out the contradictions in students’ misconceptions. In the three steps
of this technique, the teacher probes for qualitative, quantitative, and conceptual understanding, asking questions rather
than telling students the right answer.

In the process of resolving the conflicts that arise, students actively reconstruct the concept in question and truly overcome
their misconceptions.

This digest contains 10 references. (SV)

Research abstract on misconceptions about fractions

Title: Models of Problem Solving:
A Study of Kindergarten Children’s Problem-Solving Processes
Author(s): Carpenter, Thomas P., and Others
Source: Journal for Research in Mathematics Education v24 n5 p428-41 Nov 1993
Publication Date: November 1, 1993
ISSN: 00218251
Descriptors: *Cognitive Processes; *Cognitive Style; *Heuristics; *Problem Solving; *Word Problems (Mathematics);
Addition; Division; Interviews; Kindergarten; Learning Strategies; Mathematical Models; Mathematics Education;
Mathematics Instruction; Models; Multiplication; Primary Education; Schemata (Cognition); Subtraction
Identifiers: Representations (Mathematics); Mathematics Education Research
Abstract: After a year of instruction, 70 kindergarten children were individually interviewed as they solved basic,
multistep, and nonroutine word problems.
Thirty-two used a valid strategy for all 9 problems, and 44 correctly answered 7 or more problems. Modeling provided a
unifying framework for thinking about problem solving. (Author/MDH)

Figure 4.3 Research abstract on kindergarteners’ problem-solving processes

Reviews of Research for professional educators in their journals: Teach-
ing Children Mathematics, Journal of Educational
Single research studies provide clues about learn- Research, Journal of Research in Mathematics Educa-
ing and teaching, but they do not allow researchers tion, and Elementary School Journal. The Handbook
to draw strong conclusions by comparing and con- of Research on Mathematics Teaching and Learning
trasting similar studies. Professional organizations (Grouws, 1992) and the Handbook of International
make research summaries and syntheses available

54 Part 1 Guiding Elementary Mathematics with Standards

TABLE 4.2 • Recommendations Based on Research for Improving
Student Achievement in Mathematics

1. The extent of the students’ opportunity to learn mathematics content bears directly and decisively on student
mathematics achievement.

2. Focusing instruction on the meaningful development of important mathematical ideas increases the level of
student learning.

• Emphasize the mathematical meanings of ideas, including how the idea, concept, or skill is connected in multiple ways
to other mathematical ideas in a logically consistent and sensible manner.

• Create a classroom learning context in which students can construct meaning.
• Make explicit the connections between mathematics and other subjects.
• Attend to student meanings and student understandings.

3. Students can learn both concepts and skills by solving problems.

Research suggests that it is not necessary for teachers to focus first on skill development and then move on to problem
solving. Both can be done together. Skills can be developed on an as-needed basis, or their development can be
supplemented through the use of technology. In fact, there is evidence that if students are initially drilled too much on
isolated skills, they have a harder time making sense of them later.

4. Giving students both an opportunity to discover and invent new knowledge and an opportunity to practice
what they have learned improves student achievement.

Balance is needed between the time students spend practicing routine procedures and the time they devote to inventing
and discovering new ideas. To increase opportunities for invention, teachers should frequently use nonroutine problems,
periodically introduce a lesson involving a new skill by posing it as a problem to be solved, and regularly allow students to
build new knowledge based on their intuitive knowledge and informal procedures.

5. Teaching that incorporates students’ intuitive solution methods can increase student learning, especially
when combined with opportunities for student interaction and discussion.

Research results suggest that teachers should concentrate on providing opportunities for students to interact in problem-
rich situations. Besides providing appropriate problem-rich situations, teachers must encourage students to find their
own solution methods and give them opportunities to share and compare their solution methods and answers. One
way to organize such instruction is to have students work in small groups initially and then share ideas and solutions in a
whole-class discussion.

6. Using small groups of students to work on activities, problems, and assignments can increase student
mathematics achievement.

When using small groups for mathematics instruction, teachers should:

• Choose tasks that deal with important mathematical concepts and ideas
• Select tasks that are appropriate for group work

Research in Mathematics Education (English, 2003) dren learn and connecting important content to the
collect current reviews of research on important top- needs and interests of the learner. Galileo wrote,
ics for teaching and learning. ERIC digests provide “You cannot teach a man anything; you can only
short summaries of research and research-based help him find it within himself.” Constructivist
best practices. teaching has roots in the theories of Piaget, Bruner,
Skemp, and Vygotsky and is consistent with current
Based on a review of research, Grouws and Ce- research on teaching and learning. Research is the
bulla (2002) made 10 recommendations (available foundation for the recommendations about learning
at http://www.ericse.org/digests/dse00-10.html) with direct and teaching found in the Principles and Standards
application for teachers of mathematics. Current re- for School Mathematics (National Council of Teach-
search is also available through the website of the ers of Mathematics, 2000). NCTM continually up-
National Center for Improving Student Learning and dates research through its publications such as Les-
Achievement in Mathematics and Science (available sons Learned from Research (Sowder & Schappelle,
at http://www.wcer.wisc.edu /ncisla/). 2002). Gerald A. Goldin (1990, p. 31) cites six themes
on which teaching mathematics in a constructivist
Recommendations given in Table 4.2 support manner is based:
cognitive theories about learning. Constructivist
teaching depends on teachers knowing how chil-

Chapter 4 Learning Mathematics 55

TABLE 4.2 • Continued

• Consider having students initially work individually on a task and then follow with group work where students share
and build on their individual ideas and work

• Give clear instructions to the groups and set clear expectations
• Emphasize both group goals and individual accountability
• Choose tasks that students find interesting
• Ensure that there is closure to the group work, where key ideas and methods are brought to the surface either by the

teacher or the students, or both

7. Whole-class discussion following individual and group work improves student achievement.

It is important that whole-class discussion follows student work on problem-solving activities. The discussion should be
a summary of individual work in which key ideas are brought to the surface. This can be accomplished through students
presenting and discussing their individual solution methods, or through other methods of achieving closure that are led
by the teacher, the students, or both.

8. Teaching mathematics with a focus on number sense encourages students to become problem solvers in a
wide variety of situations and to view mathematics as a discipline in which thinking is important.

Competence in the many aspects of number sense is an important mathematical outcome for students. Over 90% of
the computation done outside the classroom is done without pencil and paper, using mental computation, estimation,
or a calculator. However, in many classrooms, efforts to instill number sense are given insufficient attention.

9. Long-term use of concrete materials is positively related to increases in student mathematics achievement
and improved attitudes toward mathematics.

Research suggests that teachers use manipulative materials regularly in order to give students hands-on experience that
helps them construct useful meanings for the mathematical ideas they are learning. Use of the same materials to teach
multiple ideas over the course of schooling shortens the amount of time it takes to introduce the material and helps
students see connections between ideas.

10. Using calculators in the learning of mathematics can result in increased achievement and improved student
attitudes.

One valuable use for calculators is as a tool for exploration and discovery in problem-solving situations and when
introducing new mathematical content. By reducing computation time and providing immediate feedback, calculators
help students focus on understanding their work and justifying their methods and results. The graphing calculator is
particularly useful in helping to illustrate and develop graphical concepts and in making connections between algebraic
and geometric ideas.

SOURCE: Grouws & Cebulla (2002).

1. Mathematics is viewed as invented or constructed 5. Effective teaching occurs through the creation of
by human beings; it is not an independent body classroom learning environments that encourage
of “truths” or an abstract and necessary set of the development of diverse and creative
rules. problem-solving processes in students.

2. Mathematical meaning is constructed by the 6. Teachers must consider the origins of math-
learner rather than imparted by the teacher. ematical knowledge to understand that such
knowledge is constructed and that mathematical
3. Mathematical learning occurs most effectively learning is a constructive process.
through guided discovery, meaningful applica-
tion, and problem solving rather than imitation A constructivist teacher develops opportunities
and reliance on the rote use of algorithms for for children to activate prior knowledge and to ac-
manipulating symbols. quire, understand, apply, and reflect on knowledge
(Zahorik, 1995). Teachers who understand both
4. Study and assessment of learners must occur the structure of mathematics and the processes of
through individual interviews and small-group learning create classrooms that lead to success for
observations that go beyond paper-and-pencil the learners.
tests.

56 Part 1 Guiding Elementary Mathematics with Standards

Teachers and Action Research Children Mathematics are based on practices that
teachers deem successful. For example, Ambrose
Many teachers do not realize that by asking and and Falkner (2001) report on students’ thinking, un-
answering questions about student learning and derstanding of concepts, and use of geometry vo-
effective teaching, they conduct research in their cabulary as a result of building polyhedra. In some
classrooms. Teachers describe, explain, and try issues of Teaching Children Mathematics a problem
methods to improve their students’ learning. They for students is presented, and several months later
assess student learning by collecting data, draw- a follow-up article shows different student solutions
ing conclusions, and changing their techniques to with a discussion of various strategies. This feature
increase learning. This process, called classroom gives teachers insight into student thinking and
action research, is central to improving teaching models classroom research.
and learning. Many articles in the journal Teaching

Summary 2. Compare the recommendations for teachers from re-
search by Grouws and Cebulla (2002) with the “key
Learning theories and research provide understanding of ideas” from cognitive theories. Which points match
learning and teaching processes for teachers. Although between them and which do not?
behaviorist theories and practices of drill and practice
have often dominated mathematics instruction, new re- Teacher’s Resources
search supports a more cognitive approach to teaching.
The cognitive approach recognizes the importance of Benson, B. (2003). How to meet standards, motivate
individual differences in motivation, emotion, experi- students, and still enjoy teaching. Thousand Oaks, CA:
ence, language, and culture and how these differences Corwin Press.
can influence how and what students learn. Key ideas
for teaching based on cognitive theories include focus Clements, D. (2003). Learning and teaching measure-
on meaning, developmental stages in learning, social ment. Reston, VA: National Council of Teachers of
interaction, concrete experiences with multiple repre- Mathematics.
sentations, learning meaningful procedures, building
long-term strategic memory, and finding patterns. As indi- Kirkpatrick, J., Martin, W. G., & Schifter, D. (2003). A re-
cated by cognitive research and theories, teachers need search comparison to principles and standards for school
to provide instruction that engages students actively in mathematics. Reston, VA: National Council of Teachers
constructing mathematical concepts and building skills of Mathematics.
in a meaningful context.
Posamentier, A., Hartman, H., & Kaiser, C. (1998). Tips
Through journals and technology teachers have ac- for the mathematics teacher. Thousand Oaks, CA: Cor-
cess to the latest research about learning and teaching. win Press.
Specific studies or reviews of research that summarize
and synthesize current research are found in journals and Ronis, D. (1997). Brain-compatible mathematics. Thou-
online. Teachers also ask and answer questions in their sand Oaks, CA: Corwin Press.
own classrooms as they assess student learning and de-
termine which techniques are most effective for learners. Sowder, J., & Schappelle, B. (2002). Lessons learned
from research. Reston, VA: National Council of Teachers
Study Questions and Activities of Mathematics.

1. Bransford and colleagues (1999, p. 108) said, “The Tate, M. (2005). Worksheets don’t grow dendrites. Thou-
alliance of factual knowledge, procedural profi- sand Oaks, CA: Corwin Press.
ciency, and conceptual understanding makes all
three components usable in powerful ways. Stu- Tomlinson, C., & McTighe, J. (2006). Integrating differen-
dents who memorize facts or procedures without tiated instruction and understanding by design: Connect-
understanding often are not sure when or how to ing content and kids. Alexandria, VA: Association for
use what they know, and such learning is often quite Supervision and Curriculum Development.
fragile.” Is this approach more behaviorist or cogni-
tive in nature? Explain your answer. Wolfe, P. (2001). Brain matters: Translating research into
the classroom practice. Alexandria, VA: Association for
Supervision and Curriculum Development.

CHAPTER 5

Organizing
Effective
Instruction

ffective teaching strategies in mathematics emerge when
learning theories, research on teaching and learning, and
successful teaching practices are taken into consideration
and drawn upon. Constructivism is the theory that best

explains the complexities of learning mathematical
concepts and procedures in a developmental process. By exploring
materials and discussing interesting mathematics-rich problems, students
construct mathematical concepts and develop skills. In a challenging and
supportive classroom, teachers organize mathematical tasks and encour-
age student engagement and communication by the way they manage
space, time, and materials. In this chapter we describe how teachers make
plans and implement instruction and how the decisions they make foster
conceptual and procedural knowledge.

In this chapter you will read about:
1 Characteristics of successful teachers
2 Long-range and short-range planning based on curriculum objectives
3 Teaching approaches and strategies that support learning
4 Cooperative learning to encourage student achievement and

communication
5 Organization of time, space, and materials that support learning and

student interaction

57

58 Part 1 Guiding Elementary Mathematics with Standards

What Is Effective • Teachers create a positive learning environment
Mathematics Teaching? that supports critical and creative thinking.

New and experienced teachers ask two questions Using Objectives to Guide
about their mathematics teaching: Mathematics Instruction

1. What concepts, skills, and applications should Three classroom vignettes illustrate how teachers at
children learn? different grade levels adapt lessons to match the con-
tent with the level of students. The general objective
2. What are the most effective ways to develop for each lesson is for students to collect data and
these concepts, skills, and applications? represent information on a bar graph with a title and
key and to draw conclusions based on the data.
The first question, about the expected content and
processes of mathematics, can be answered by re- First-Grade Graphing
ferring to the state and district learning objectives, Mr. Gable plans a lesson on classification and rep-
which are often based on the NCTM standards pre- resentation for first-graders using the objective “Stu-
sented in Chapters 1 and 2. The answer to the sec- dents will sort 20 wooden blocks into groups by
ond question, about teaching content and process color and display the information on a bar graph.”
skills, is further developed in Part 2 of this text. Students gather in a circle on the floor. Mr. Gable
places a red flower on red construction paper, a
Effective teaching depends on the knowledge blue glove on blue paper, and a yellow fire engine
and skills that teachers bring to instruction decisions on yellow paper. Students notice the objects match
and the choices teachers make in implementing the color of the paper. He asks students where new
this pedagogical knowledge. Teachers must make objects belong, and children place them on the cor-
many decisions about the way they teach. Teachers responding color. Students talk about which color
plan worthwhile mathematics experiences, interact has the most or least objects, how many more red
with children while they are learning, and monitor has than yellow, and so forth (see Figure 5.1a).
student’s progress. Successful teachers understand
how children learn and vary their teaching based (a) (b)
on group and individual needs. Because no single
instructional approach works for every child and Figure 5.1 (a) Object graph and (b) bar graph
every concept, effective teachers build a repertoire
of instructional skills and techniques with charac- Then Mr. Gable gives each pair of students 20
teristics suggested by theory, research, and practice, blocks of three colors and matching construction
such as those cited in Chapter 4. paper. He asks them to sort the blocks by color and
to count them. After placing the blocks in groups,
• Teachers engage children actively in learning. students move the blocks on 1-inch graph paper and
color in the squares on the paper (see Figure 5.1b).
• Teachers encourage students to reflect on their
experiences and construct meaning. After putting graphs on a bulletin board, Mr. Ga-
ble asks questions: “How many graphs show seven
• Teachers invite children to think at higher cogni- or more red cubes? Are there any graphs with fewer
tive levels. than three of a color? Which graph has the most red
cubes?” Activities and lessons over several weeks
• Teachers help children connect mathematics to focus on classifying and graphing.
their lives.

• Teachers encourage children to communicate
their ideas in many forms and settings.

• Teachers constantly monitor and assess students’
understanding and skill.

• Teachers adjust their instruction to fit the needs,
levels, and interests of children.

Chapter 5 Organizing Effective Instruction 59

Third-Grade Data Collection dents collect data, organize them, and display them
Mrs. Alfaro plans a lesson with this objective: “Given on graphs with titles and a key. The activity is con-
centimeter-squared paper, students will collect data cluded when groups show their graphs and report
on a topic of their choice and construct a bar graph, conclusions drawn from them.
including a title and key, showing the results.” She
asks students to write their favorite sandwich on Students poll students in other classes to com-
2-inch sticky notes with their initials. Students put pare their class results with those for other groups.
their sticky notes on a white board with others of the
same type: peanut butter, peanut butter with jelly, Sixth-Grade Rainfall Graph
grilled cheese, bologna, bacon and tomato, and Ms. Clary’s students have had experiences with bar
“other.” After grouping, they line the sticky notes up, graphs and know how to determine the mean aver-
starting at the base. age for a set of numbers, so she wants them to work
independently on graphing for an interdisciplinary
An overhead transparency of graph paper shows social studies and science unit on weather. The per-
labeled columns with the six types of sandwiches. formance task is, “Given a chart of the monthly rain-
Initials of each child are transferred to a correspond- fall for the last 10 years, students will use a computer
ing square on the transparency. Finally, Mrs. Alfaro spreadsheet to compute the mean average rainfall for
asks students to supply a title; the group decides on each month, create a bar graph showing the mean av-
“Yum-Yum: Our Favorite Sandwiches.” She writes, “1 erage monthly rainfall, and write a short paragraph in
square ϭ 1 sandwich” beneath the title and labels it which they describe weather trends over the 10 years
the “key” for the graph (see Figure 5.2). using mean averages and extremes.” She allows the
students to pick any city in the United States using
The next day, students work in groups of three data from almanacs and the Internet. After entering
to poll each other about other favorites: pizza, ice the data on the computer, students use the average
cream, sports teams, pets, singing groups, colors, function on the spreadsheet and create bar and line
and dream cars. Over two or three days, the stu- graphs. Students take turns working as computer
consultants if problems occur.

After a week, each student has a table and bar
graph of mean rainfall (see Figure 5.3) and an essay

Rainfall of Seattle

Monthly
Average

38.7

Figure 5.2 Yum-Yum: Our Favorite Sandwiches J FMAMJ J A S ON D
graph Months

Figure 5.3 Rainfall graph

60 Part 1 Guiding Elementary Mathematics with Standards

about the weather pattern in the city. As they look TABLE 5.1 • INTASC Beginning Teacher
for similarities and differences in rainfall patterns Standards
across the United States, they identify weather pat-
terns such as droughts, events such as El Niño and 1. Content Pedagogy
La Niña, and hurricanes. The teacher understands the central concepts, tools of
inquiry, and structures of the discipline he or she teaches
Using standards and objectives as beginning and can create learning experiences that make these
points, these three teachers demonstrate character- aspects of subject matter meaningful for students.
istics for successful teachers through their flexibility
and adaptation. The content objectives and skill lev- 2. Student Development
els are met with different tasks and materials, but all The teacher understands how children learn and develop
challenge students and encourage thinking. and can provide learning opportunities that support a
child’s intellectual, social, and personal development.
Becoming an Effective Teacher
3. Diverse Learners
The three teachers from the vignettes exhibit excel- The teacher understands how students differ in their
lent teaching techniques. However, acquiring all approaches to learning and creates instructional
the qualities of effective teaching does not occur opportunities that are adapted to diverse learners.
overnight; teaching is a continuous improvement
process. Ten principles for preparing new teachers 4. Multiple Instructional Strategies
were adopted by the Interstate New Teacher As- The teacher understands and uses a variety of instruc-
sessment and Support Consortium (INTASC). The tional strategies to encourage student development of
INTASC standards, listed in Table 5.1, outline what critical thinking, problem solving, and performance skills.
beginning teachers should know and be able to
do. These statements provide a framework for new 5. Motivation and Management
teachers entering the profession and are adapted The teacher uses an understanding of individual and
by many states for new teachers. In practice, the 10 group motivation and behavior to create a learning
standards are not 10 separate practices. Combined environment that encourages positive social interaction,
they represent what good teachers know and do on active engagement in learning, and self-motivation.
a daily basis.
6. Communication and Technology
E XERCISE The teacher uses knowledge of effective verbal, non-
verbal, and media communication techniques to foster
Using the INTASC standards or the teacher educa- active inquiry, collaboration, and supportive interaction in
tion standards in your state, describe how the the classroom.
teachers in each vignette exhibited the characteris-
7. Planning
tics of effective teaching. ••• The teacher plans instruction based on knowledge of
subject matter, students, the community, and curriculum
Using Objectives in Planning goals.
Mathematics Instruction
8. Assessment
Teachers coordinate and manage many instruc- The teacher understands and uses formal and informal
tional decisions as they create a classroom that en- assessment strategies to evaluate and ensure the con-
courages children’s mathematical thinking. tinuous intellectual, social, and physical development of
the learner.
• How do teachers plan lessons and units based on
curriculum objectives? 9. Reflective Practice: Professional Growth
The teacher is a reflective practitioner who continually
• What teaching approaches and strategies engage evaluates the effects of his or her choices and actions
students in worthwhile learning activities? on others (students, parents, and other professionals
in the learning community) and who actively seeks out
Many states and school districts adopt or adapt the opportunities to grow professionally.
NCTM standards when writing their curriculum goals
10. School and Community Involvement
The teacher fosters relationships with colleagues, par-
ents, and agencies in the larger community to support
students’ learning and well-being.

and performance objectives. Wording of objec-
tives varies, but the scope and sequence are often
similar. Performance objectives, also called instruc-
tional objectives and learning targets or bench-
marks, describe what students should know and be

Chapter 5 Organizing Effective Instruction 61

able to do as a result of instruction. By knowing the TABLE 5.2 • Long-Range Plan (Grade 4):
expectations for the grade level they teach, as well Mathematics
as objectives for earlier and later grades, teachers
see how content and skills develop over time. By September
aligning the curriculum, teams of teachers on the Review geometry concepts and terminology—symmetry,
same grade level and across grade levels in a school
coordinate their objectives, instructional practices, 2-D and 3-D figures.
and assessments so that all children have a positive Introduce area and volume measurement.
learning experience.
October
Long-Range Planning Review concepts and basic facts for four operations.
Develop algorithms for two- and three-digit numbers for
In a long-range plan teachers outline how instruc-
tional topics and objectives for the year are devel- four operations.
oped and sequenced. Planning across the year al- Introduce estimation and calculator strategies for larger
lows time for development, review, and extension
of topics and balances instructional time so that numbers.
all standards receive attention and build on one
another. With increasing emphasis on accountabil- November
ity, teachers also consider when state assessments Continue estimation and algorithms.
are scheduled and make sure that content is devel- Review and extend common fractions—concepts,
oped and reviewed before the test. A fourth-grade
teacher’s preliminary plan for the year is shown in equivalence, comparisons.
Table 5.2. A long-range plan is subject to change Introduce addition and subtraction with fractional
to take advantage of learning opportunities and
learner needs. numbers concretely.

Unit Planning December
Review and extend linear, area, and volume measure.
Teachers create two- or three-week instructional Review addition and subtraction models of fractions.
units based on the long-range plan that included Review and extend data gathering and graph making.
varied lessons and activities. A three-week instruc-
tional unit on addition and subtraction of two- and January
three-digit numbers in third grade might include Introduce probability using common and decimal
eight directed teaching/thinking lessons, six learn-
ing center and computer-based activities, three fractions.
games, and two investigations or student-centered Continue estimation and algorithms for four operations,
projects. Multiple learning experiences provide for
varied learning styles, interests, and abilities within including fractions.
the topic. Unit planning also considers various stu- Introduce transformational geometry and coordinate
dent groupings from whole class to small group to
individual learning. geometry.

Learning activities are sequenced so that new February
concepts and skills are introduced and developed Introduce statistics and interpretation of graphs.
in a logical way. A real-life or imagined problem cap- Review and extend estimation and algorithms with whole
tures children’s interest. Good instructional prob-
lems motivate learning and have mathematical con- and fractional numbers.
cepts and skills embedded in them. A teacher starts Review and extend measure of volume and weight
a lesson or unit with problems based on a children’s
book, newspaper article, children’s lives, or manipu- (with science).
latives. How Many Snails: A Counting Book (by Paul
March
Continue decimal fraction concepts (money), statistics,

and probability.
Review place value and extend to decimal fractions.
Introduce measurement of angles.

April
Review and extend four operations with decimal

fractions.
Review and extend probability concepts.
State assessments.

May
Review and extend data collection and analysis.
Review and extend transformational geometry.
Review and extend estimation and algorithms for four

operations.

Giganti) introduces counting, classification, subtrac-

tion, and fractions by asking students to count sets

of objects with different attributes. A newspaper ar-

ticle states that the state is running out of telephone

numbers and brings up questions about how many

62 Part 1 Guiding Elementary Mathematics with Standards
Figure 5.4 Area of a triangle

Height Height

Base Base

Area ϭ base ϫ height Area ϭ 1 of base ϫ height
2

telephone numbers are possible in any area code. of a rectangle or a triangle by counting square units
This discussion includes large numbers, multiplica- is a solution strategy for that shape. However, noting
tion, and combinations. that the area of rectangles is determined by multi-
plying the length by the height is a generalized rule.
Teachers also consider the sequence of activi- Finding that the area of a triangle is half the area
ties in a unit. Teachers introduce new concepts with of its related rectangle extends the first rule (Fig-
simple examples and move to more sophisticated ure 5.4). As children take specific solutions and gen-
concepts as children become successful. Patterning erate a formula for the area of all triangles, they are
might be explored with A-B repetition first, then A-A- thinking algebraically. Figure 5.5 summarizes some
B, then A-B-C and A-B-C-B. Measurement of length is important sequences that teachers consider in plan-
modeled with paper clips and a chain of paper clips ning their units to maximize student learning.
before the ruler is introduced. Students compare
and classify the size of angles on triangles. Little, PROBLEM SOLVING
big, and corner angles are later renamed “acute,
obtuse, and right” angles and are compared to an- Problem Concept Skill Solution New problem
gles found in different figures. As students gain skill
and confidence, situations and vocabulary become COMPLEXITY
more complex.
Simple Complex
Effective teachers incorporate different levels of
representation in lessons moving from concrete ob- SITUATION
jects to pictures to symbols and eventually to mental Specific General
images and operations. Mathematical concepts are
developed using stories, models, pictures, tables REPRESENTATION
and graphs, and number sentences or formulas. This
sequence of introducing material to students builds Concrete Pictorial Symbolic Mental
understanding of essential attributes of mathemati-
cal concepts. Realistic contexts give way to sym- Figure 5.5 Developmental sequence for activities
bolic representations. Mental computation, number
sense, reasonableness of answer, estimation, and Daily Lesson Planning
spatial sense are indicators of students’ ability to
reason abstractly. The daily lesson plan focuses on the activities pre-
sented in a single day in the context of the unit plan
Students also develop from specific to general as well as the long-range plan. Instructional decisions
solutions. Learning the blue-blue-red pattern with on a daily lesson plan are specific, because teach-
blocks is specific, but patterning is a fundamental
thinking skill that extends beyond learning one
pattern. Identifying and labeling many patterns in
many situations and many forms (A-B-C, A-B-C-B, A-
A-B-C) creates a generalized skill. Finding the area

Chapter 5 Organizing Effective Instruction 63

ing optimizes student learning. Teachers ask, “What
types of activities should I plan for today?” to guide
children toward mastery of the objectives for the
unit. Each day contributes to attainment of the unit,
and teachers have several approaches available.

Varying Teaching Approaches Figure 5.7 Pattern blocks show fractional parts.

Some teachers believe that standards or curricu- ativity and critical thinking through open-ended
lum objectives limit their creativity and flexibility in tasks. Students create new uses and new rules with
teaching mathematics. Although standards guide in- games and objects. Informal activities allow stu-
structional goals, effective teachers choose the tech- dents to individualize their learning. The teacher
niques and activities that address student learning serves as a guide and facilitator while students work
styles, preferences, intelligences, and needs. Meet- informally.
ing all students’ needs requires varied instructional
approaches, including informal or exploratory activ- • Individual student’s needs and interests can be
ities, directed teaching/thinking lessons, and prob- addressed by offering choice as part of informal
lem-based projects or investigations. activities. Different children learn at different lev-
els of understanding with the same materials. In-
Informal or Exploratory Activities formal activities encourage independent thought
and autonomous action, and they provide variety
When planning daily lessons, some lessons should in the instructional approaches in the classroom.

provide time for students to informally explore con- • By selecting the materials and the tasks care-
fully, teachers extend engaged learning time for
cepts. Work with manipulatives, playing games, and students. While teachers work with individuals or
small groups, students in the classroom explore
setting up learning centers are informal approaches materials and activities related to the objectives
using directions from printed materials or task
that have many benefits for student learning. Infor- cards. Task cards coordinated with manipulatives
challenge students to complete an activity. For
mal activities with manipulatives prepare children example, a task card on patterning with colored
shapes may leave several blank spaces for stu-
for directed teaching episodes and investigations dents to complete the pattern with pattern blocks.

with materials. Children are fascinated by new ma- Informal activities provide valuable learning for all
students when teachers select appropriate materials
terials, so experienced teachers provide students and tasks, monitor and assess student work, discuss
the concepts being encountered, and introduce
exploratory time with materials before using them vocabulary related to the activities. If students es-
tablish a purpose for exploratory activities, they are
in directed activities. When students have worked more likely to see them as important. Children in all
elementary grades, as well as middle and secondary
with materials before a directed lesson, they con- levels, need exploratory experiences as new con-
cepts and materials are introduced. Informal activi-
centrate on the content of the ties, learning centers, and games extend skills and
topics from a previous grade or earlier in the year.
lesson rather than on the nov-
Directed Teaching/Thinking Lessons
elty of the materials. Manipu-
A different approach to daily planning is a directed
latives have mathematics con- teaching/thinking lesson. Some people see direct
teaching as teacher-controlled or even scripted les-
cepts and skills embedded in

them. When children spontane-

ously build a stair-step pattern

Figure 5.6 Cuise- with Cuisenaire rods (Figure
naire rod stairs 5.6), they are modeling the plus-
demonstrate ϩ1. 1 rule for learning basic facts:
1 ϩ 1 ϭ 2, 2 ϩ 1 ϭ 3, 3 ϩ 1

ϭ 4, and so on. Arranging pattern blocks is an explo-

ration of proportionality and fractional parts (Fig-

ure 5.7). Informal experiences give students a head

start on understanding because they have already

experienced a concept through their hands-on ex-

ploration. Finally, informal explorations invite cre-

64 Part 1 Guiding Elementary Mathematics with Standards

sons, but the directed teaching/thinking lesson veloped. A mystery box might contain something re-
in mathematics is interactive and incorporates ma- lated to the lesson, and students guess the contents;
nipulative materials, visual aids, discussion and for example, an orange in the mystery box might
argument, and interesting performance tasks to introduce spheres. A recent newspaper headline or
encourage thinking. A directed teaching/thinking personal experience might pique student interest in
lesson can be taught to small groups or the whole a new topic. A number puzzle gets students think-
class. Small groups are often more effective because ing about mathematics. These activities motivate
the teacher closely observes children’s work with students to shift their attention from a previous ac-
materials, watches their eyes, and listens to their tivity, such as recess or reading, to mathematics and
comments. activates prior knowledge for the concept or skill be-
ing addressed.
During a unit, the teacher uses the directed
teaching/thinking format to introduce, reinforce, In the second phase of a directed teaching/think-
and extend key concepts and skills several times ing activity, students develop understanding of new
while working toward student mastery. A lesson plan content. According to Gagne’s plan, the teacher ex-
serves as a road map for a well-thought-out lesson plains and models the new content or skill and the
with a destination (the instructional objective) and student models what the teacher has done. How-
a route (procedures needed to master the objec- ever, a more constructivist approach, described
tive). Robert Gagne (1985) describes a well-planned in Chapter 4, places the teacher as a facilitator of
lesson in nine events that can be grouped into three student learning. In this role the teacher prepares a
phases: preparation, presentation, and practice. problem or situation that engages students’ thinking
about an important mathematical concept or skill.
Preparation (Motivation and Preparation: Rather than tell the students what to do and think,
Setting the Stage) the teacher asks questions and elicits student ideas.
The performance task requires the student to think
• Gain the learner’s attention. mathematically. When children rearrange blocks to
show regrouping for two-digit addition or roll two
• Inform the learner of the learning objective number cubes and record the results on a graph,
or task. the activity itself carries the content. Students learn
by doing mathematics. In this way the critical attri-
• Recall prior experiences, information, and butes of the concept are developed by the learner.
prerequisite skills. As the activity progresses, the teacher labels the ac-
tions with more formal vocabulary and introduces
Presentation (Instructional Input and Modeling) symbols as they are needed.

• Present new content or skills by demonstrating As teachers check for understanding, they dis-
and describing critical attributes. cover that some students need more explanation
and examples, whereas others are ready for guided
• Provide guided practice with additional practice. During guided practice, students work in
examples. pairs or independently on problems that extend
their thinking and skill. The teacher monitors student
• Elicit performance on a learning task. work and gives corrective feedback or guiding ques-
tions as needed. As students progress, the teacher
• Check for understanding. asks students to generate their own problems. Stu-
dents who show understanding of the concept or
Practice (Independent Practice) skill are ready for the third phase; others need more
time and examples.
• Provide feedback and assess performance.
In the third phase of the directed teaching/think-
• Apply content or skill in new situation. ing lesson the teacher still monitors student work but
the student works more independently. Independent
In the first phase the teacher creates the mind set for practice is appropriate when students understand
learning. A stimulating introduction helps students
focus their attention on the concept being explored.
Teachers have many ways of creating an expectancy
or anticipation for learning: a children’s literature
book on a concept, a manipulative for modeling, or
a problem situation related to the concept being de-

Chapter 5 Organizing Effective Instruction 65

Seven-step lesson cycle

Assess readiness

Setting the stage Instruction
• Motivate • Demonstrate skill
• State learning objective • Label concepts
• Relate to prior knowledge • Define terms and symbols
• Model operations
Check for understanding
• Ask questions Guided practice
• Observe operations • Students demonstrate skill
• Adjust problems • Students extend concepts
• Students work samples
• Students repeat operations

Independent practice
• Students practice skill or concept

— learning center
— games
— computer
— seatwork

Assess mastery
• Ask questions
• Observe operations
• Give tests

the skill or content (Figure 5.8). Students may also the essential question, teachers and students con-
practice and extend their new concepts and skills nect each day with the unit goals. Without focus on
through problems, projects, and investigations. the central concept, individual lessons become dis-
jointed. The NCTM standards and expectations help
E XERCISE teachers find the central ideas for planning.

The lesson cycle in Figure 5.8 is often called the Lesson plans can be written in many formats, but
Hunter lesson model, named for educator Madeline they typically include four or five common elements:
Hunter, who popularized it. Discuss the lesson cycle instructional objectives; step-by-step procedures for
with a classmate, and compare it to Gagne’s events introducing, guiding, and generalizing the concept
or skill; materials needed; evaluation techniques;
of instruction. ••• and modifications for special learners. The blank
lesson plan format in Figure 5.9 is based on the
Writing Plans for Directed Teaching/Thinking lesson cycle; a sample lesson plan is shown in Fig-
Lessons. When writing lesson plans, teachers re- ure 5.10. In Chapters 8 through 16 lesson plans and
alize that all activities support the core concept of activities for each content standard are presented.
the unit. Whether called the focus, the big idea, or

66 Part 1 Guiding Elementary Mathematics with Standards

Figure 5.9 Teacher Lorenz Date 10/24/2007
Blank lesson plan form Instructional Objective:
Materials:

Motivation and preparation: setting the stage

Instructional input and modeling

Check for understanding and guided practice

Independent practice

Extend learning

Modifications for individual learners

E XERCISE with the teacher providing technical assistance and
support. Sometimes, students may require specific
Many activities and lesson plans can be found on instruction on a mathematics concept related to
the Internet or in teacher manuals that accompany their problem.
elementary mathematics textbooks. Find three les-
son plans on the Internet and determine how well Investigation or projects reflect classroom or real-
they match the lesson plan format in Figure 5.9. world situations. In Ms. Hale’s fifth-grade class pass-
ing trains were visible from the classroom window.
What is similar and what is different? ••• One student kept a log of the trains and their sched-
ules on the north and south routes. With encourage-
Problem-Based Projects ment the student researched the trains and discov-
and Investigations ered coal was the cargo, where the coal was coming
from and where it was going, the speed of the train,
Another instructional approach is problem-based and how scheduling prevented train wrecks. A re-
projects or investigations. While investigating a tired train engineer served as a resource person for
problem, students explore and apply concepts and the project. The culmination of the investigation was
skills, expand ideas, and draw conclusions. Investi- a paper, posters, and a presentation to classmates. A
gations can be initiated by the teacher or by students

Figure 5.10 Teacher Lorenz Date 10/24/2007
Sample lesson
plan for Instructional Objective: Students will solve story problems using subtraction of 3-digit numerals by
subtraction modeling decomposition with manipulatives and recording the regrouping.

Materials: Have students prepare 3 ϫ 5 cards in learning center in packs of 100s, 10s, and 1s with rubber
bands. Distribute 1000 3 ϫ 5 cards for each group of 4, so that each group has some of each amount.

Procedure: Motivation and preparation: Setting the stage

• Review place value with dice game, placing numerals to make the largest 4-digit number.

• Model several subtraction stories with two digit numbers.

• Tell stories to match the numbers in the problems. Students may use the 100s chart to find the
answers

37 83 95
Ϫ22 Ϫ48 Ϫ73

• Introduce the regrouping and renaming subtraction model with packs of cards and have students
follow. “You had 37 cents and spent 22 cents on a pencil.”

• Put 3 packs of 10 and 7 single cards on the board tray. Remove 22 of them. Ask how many are still
on the tray.

• Model several other problems.

• Compare the answers to answers using the 100s chart.

• When students demonstrate understanding of regrouping, tell a story with larger numbers: “In
January, Mrs. Cardwell ordered 1000 Valentine cards for her card shop. On Monday, February 1, she
still had 862 cards for sale. By Friday, February 5, she only had 374 cards on the shelf. How many had
she sold during the week?”

Instructional input and modeling
• Ask students to show how many cards she had at the start with the card sets.
• Ask students to remove enough cards to show what she had on Monday using regrouping.
• Then ask them to show how many cards she sold during the week.

1000 862
Ϫ862 Ϫ374

138 488

• After a few minutes, ask groups to share their solutions and process.

• If students used subtraction processes other than decomposition, such as equal addition or
compensation, accept these solutions and have students discuss different approaches.

Check for understanding and guided practice
• Provide several more problems.
• Ask students to tell a story for each subtraction example and model it with cards. They also may use
one of the other processes to see if they get the same answer.

125 284 284 731
Ϫ73 Ϫ143 Ϫ198 Ϫ374

• Monitor work with cards and written work.
• Ask students who are modeling accurately to make up several more problems together.
• For students who are showing difficulty, plan a small-group reteach. Students who are catching on

can pick three problems from page 93 to work. Go back and review simple subtraction.

Extend learning with a journal problem

How many cards do you think Mrs. Cardwell should have on hand for the rest of the Valentine season?
Justify your answer.

Modifications for individual learners
Decide if students are ready to work three problems from page 93 or to wait another day until process
is stronger.

For Mable: Pair with Sy Ning.

For Sean and Tran: Introduce virtual blocks on computer and have them teach others.

For Jeffri: Be alert to frustration. Simplify problems as needed.

68 Part 1 Guiding Elementary Mathematics with Standards

teacher who is aware of student interests can make vestigation (including resources they will use) and
time for such projects. Other students might be in- finally report on what was learned. Investigations
terested in horses, dinosaurs, or astronauts, which provide curriculum differentiation for gifted and tal-
could generate investigations that lead to mathemati- ented students; however, all students benefit from
cal questions and concepts as well as interdisciplin- an investigation approach. Some students complete
ary ideas. several investigations in a year; others may work
on only one or two. Investigations on different top-
• How fast can different horses run? ics can go on simultaneously. Although curriculum
goals are established by standards, problem-based
• How big were dinosaurs? How much did they eat? investigations provide a way that students can push
the curriculum. Providing time and support for
• How fast does the shuttle fly when taking off and project-based learning stimulates students to be in-
while in orbit? dependent learners.

Classroom projects provide investigation opportuni- Integrating Multiple Approaches
ties. Ms. Shivey asked first-graders to design board
games for practicing addition and subtraction facts. Because children learn mathematics in many ways,
They played commercial board games to become as discussed in Chapters 3 and 4, teachers must
familiar with game structure. Using poster board, teach using various approaches. When teachers vary
dice, index cards, and colored pens, they designed their instruction with informal activities, directed
and constructed games, played their game with teaching/thinking lessons, and investigations, they
peers, and revised the game. A two-week creative provide for students who have different learning
project resulted in many weeks of fact practice as strengths and needs. Within a unit, teachers want
well as problem solving, communicating, and rea- to balance exploratory, teacher-guided, and project-
soning. Although penalties were favorite design fea- type activities.
tures, the first-graders learned quickly that nobody
wanted to play games that were impossible to win. Each instructional approach invites children to
construct mathematical knowledge and to develop
Newspapers are sources of problem-based proj- skills. Informal experiences build background and
ects. When the newspaper reported that a new provide practice and maintenance of skills. Directed
area code was needed, students wanted to know teaching/thinking lessons provide engagement and
why. They discovered that telephone companies re- interaction as teachers and students develop con-
served thousands of telephone numbers for future cepts, skills, and vocabulary. Investigations allow
use by new subscribers and for new technologies, students to pursue mathematical and interdisciplin-
such as cell phones and fax machines. This inves- ary projects beyond the predetermined curriculum.
tigation led to population statistics and work with Individual students may need specific modifications
large numbers. Another article showed how far dif- based on their individual plans, but many times ac-
ferent animals could jump compared to their body commodations that work for one student can also
length: fleas, rabbits, kangaroos, and the best hu- help other students.
man jumpers. Interest in jumping animals led to in-
terest in world records and then led to activities in Delivering Mathematics
linear measurement. Instruction

Because investigations are open-ended, student The success of each instructional approach is influ-
success is based on whether they complete their task enced by other decisions that teachers make about
and demonstrate the target knowledge and skill. As- mathematics instruction and its delivery. New and
sessment reflects the learning goal, the choice of experienced teachers ask questions about the
suitable sources of information, the completeness amount and types of practice and homework they
and accuracy of information, and the manner in should assign. They also make decisions about how
which it is summarized and reported. Many teach- to organize students for instruction. In addition, they
ers use a K-W-H-L chart for organizing investiga- encourage mathematical conversations by the type
tions. Students list what they know, what they want
to know, and how they are going to conduct the in-

Chapter 5 Organizing Effective Instruction 69

of questions they ask. Decisions on these topics in- • Use a variety of practice activities, including
fluence how they teach and the learning environ- games, flash cards, and computer programs.
ment they create.
• Avoid comparison of students. The time re-
Practice quired to master the facts differs from student to
student.
Practice is often equated with intense, isolated drills
of number facts. In a constructive classroom, prac- • Allow children to monitor their own progress and
tice has a broader meaning. Practice in a construc- record their improvement.
tivist sense means having many rich and stimulating
experiences with a concept or skill rather than rep- Homework
etition of the same experience. The concept of area
is developed when students measure various sur- Homework seems like a rather simple issue, but
faces using tiles, cards, or sticky notes of different homework practices are often complicated by is-
sizes. Understanding of area measurement is not ex- sues such as the age of the child, the demands of the
pected to be a one-time event but an accumulation content, and the support and expectations of par-
of related experiences. Informal games and learn- ents. Schools may adopt formal policies about the
ing center activities provide multiple experiences in nature and amount of homework at different grade
learning. The trading game described in Chapters 9 levels to provide some consistency in practice, but
and 10 helps students at different levels come to new many homework policies are unwritten. New teach-
understandings of how numbers work. ers should ask about the homework policies in a
school.
Guided and independent practice built into di-
rected teaching/thinking lessons allows the teacher Homework activities are usually of two types: ex-
to observe and question students during learning. By ploratory or conceptual activities and independent
assessing students, teachers decide whether some practice of skills. At-home exploratory or concep-
need additional developmental work and which tual activities connect school mathematics to home
ones are ready for independent practice or projects. and back. Kindergartners might bring an object or
Only after students have demonstrated understand- picture that is “round” or “square” from home. Third-
ing of concepts and proficiency is independent prac- graders can find how far they travel from home to
tice appropriate. Unless students are 70–80% profi- school on the bus, in the car, or by walking. Sixth-
cient, practice may reinforce incorrect thinking. If graders could look at the calories on food pack-
sufficient time is spent on conceptual development ages and create menus using a balanced food plan.
with a variety of activities and teaching approaches, Teachers also develop take-home backpacks with
students build strong understanding, which reduces mathematics games, puzzles, or books to stimulate
time and energy spent with practice. exploratory or practice activities.

When students understand number operations Independent practice or drill activities often
and develop strategies for learning facts, most are come from worksheets or textbook pages. Students
successful with fact practice. Without understand- should understand what they are to do so that they
ing or strategies, however, most find fact practice can complete the sheet with little assistance. Par-
frustrating and defeating. Timed drills are cited as a ents complain when students do not know what to
major source of mathematics anxiety. By following do and when the homework takes too long to com-
some simple guidelines, teachers can avoid harmful plete. More is not better. If students understand and
aspects of drill: are successful, 10 to 20 examples are sufficient. If
students cannot complete 10 or 20 examples, assign-
• Develop concepts and skills before independent ing 50 or 100 creates frustration for students and par-
practice is started. ents. Some teachers set a time limit on homework,
such as “work as many examples as you can in 10
• Emphasize understanding and accuracy as the minutes.”
most important outcome. Speed is a secondary
goal and varies from student to student. Responding to homework is another issue with
many different opinions. Some teachers insist on
• Keep practice sessions short, perhaps 5 minutes. grading homework, while others check for accuracy

70 Part 1 Guiding Elementary Mathematics with Standards

and understanding. By removing the emphasis on experienced teachers and the least opportunity to
grading, teachers and students look at homework experience a full and balanced curriculum.
differently. Students who check their own home-
work can see what they got right and wrong. Self- The equity principle advocates a balanced, stim-
assessment allows them to find out what they ulating curriculum for all students. When children
understand. With a quick check of homework, have the opportunity to work together to learn and
teachers identify strengths and weaknesses of solve problems, all are advantaged. Using multiple
the class or individuals. A mistake on homework instructional approaches increases interest in learn-
or seat work provides an opportunity to correct a ing with lessons that draw on students’ background
misunderstanding. experiences. Informal activities and projects in-
crease enrichment of the curriculum, greater per-
Grouping sonalization of instruction, and student interaction
and discussion.
Flexible grouping means that the teacher uses dif-
ferent instructional groups appropriate to the task or Cooperative Group Learning
situation. Students can work independently, in small
groups, or in the whole-class setting. Groups are not Cooperative learning is a grouping strategy that is
fixed but are reorganized periodically so children designed to increase student participation by capi-
can work with a variety of peers on different tasks. talizing on the social aspects of learning. In math-
Each grouping format has advantages and disadvan- ematics, students cooperate while working together
tages. Students working alone gain independence on a geometry puzzle, measuring the playground,
but miss the opportunity to discuss, explain, and or reviewing for a test. Spencer Kagan (1994) identi-
justify their work with peers. Whole-class instruc- fied basic principles for implementing cooperative
tion seems efficient because all the students appear learning successfully: positive interdependence, in-
to be learning the same thing, but individuals may dividual accountability, equal participation, and si-
not be engaged with the group task at all. When multaneous interaction. Positive interdependence
whole-group instruction is the only instructional means that team success is achieved through the
format, some students are unchallenged and bored successes and contributions of each member. In-
while other students are lost. Small groups of three dividual accountability requires that team mem-
to six students have many instructional advantages. bers be held accountable for contributions and re-
Students of similar abilities or interests can work to- sults of the team effort. Every student on the team is
gether for a unit or project. responsible for learning the content and skill, and
the team is accountable for all the team members’
Flexible grouping avoids being successful. Equal participation means that
the dangers of tracking, team members have equal opportunity to partici-
or grouping students by pate; an activity is not dominated by one member,
ability and keeping stu- nor can members choose not to participate. In tra-
dents together for long pe- ditional instruction, interchanges are often between
riods. Tracking frequently the teacher and one student at a time. Coopera-
works to the detriment of tive learning involves several students at the same
lower-ability students, with time—simultaneous interaction.
instruction that deempha-
sizes concepts and over- In cooperative learning students are organized in
emphasizes isolated skills. groups, or teams, that are heterogeneous so that stu-
Tracking also isolates dents with different skill levels, ethnic and cultural
the lower-ability students backgrounds, socioeconomic status, and other char-
from the modeling and acteristics work together. English-language learners
support of more proficient work with bilingual students or English speakers.
students. The lower-track Teachers balance teams with higher-, middle-, and
students often have the lower-skill students. Simpler, shorter mathematics
least qualified and least tasks, such as covering the desk with tiles or play-
ing a game on addition facts, may be done in pairs.

Chapter 5 Organizing Effective Instruction 71

Cooperative-learning activities may be done with 1
two to six students, but many teachers prefer teams 11
of four because they are small enough for active par-
ticipation by everyone and large enough for diver- 1
sity of members and roles. Larger teams result in stu-
dents’ withdrawing from tasks and observing rather 4 1 2
than participating. Many teachers assign specific 44 42 22
roles within the teams, such as recorder, reporter,
materials manager, teacher contact, and convener. 4 3 2
Students rotate roles so they all gain social and or-
ganizational skills. 3
33
Team building and cooperative skills are also es-
sential to cooperative learning. Team members bond 3
by sharing who they are, where they have been, and
what they aspire to become. Choosing a team name, Figure 5.11 In Jigsaw members of the home team join
designing an insignia, or creating a motto encour- an expert team to learn something new, then return to
ages team pride and identity. Students learn about teach their teammates what they learned.
each other and appreciate the skills and experience
each one brings to the task. Interpersonal skills are angles, students become experts on acute angles,
developed through cooperation with teammates. right angles, obtuse angles, and measuring angles.
Students learn to participate actively and equally, When all parts of the content come together, stu-
to share and rotate responsibilities, and to report dents gain a complete understanding of angles.
results of group activities. An effective teacher pro- Jigsaw places responsibility on each team member
motes cooperative skills by modeling them, select- to contribute to the overall completion of a divided
ing students to role-play for practice, and holding task, story, or chapter.
discussions and evaluations of cooperative efforts.
Students learning English benefit from coopera-
Kagan (at http://www.cooperativelearning.com) de- tive learning because they use and hear English in
scribes 20 cooperative learning structures for dif- a less threatening context than whole-class discus-
ferent purposes from complex to simple tasks. Stu- sions. Cooperative groups accommodate students
dents unfamiliar with cooperative learning begin with learning disabilities as well as gifted and tal-
with simple activities and structures for a specific ented students. Students with learning disabilities
task for a short time. In Pairs/Check two students benefit from interaction with other students. Stu-
solve problems, check each other, and correct each dents proficient in mathematics extend their un-
other if needed. For Think/Pair/Share individuals derstanding when they teach others and exchange
work individually on a problem or task, then discuss ideas with team partners. For students with limited
their results with a partner, and finally share with mathematics proficiency, group cooperation sup-
all members of the team to see if all team members ports their learning.
understand the content or process.
Cooperative learning also benefits students from
A cooperative structure of intermediate com- different cultural and ethnic backgrounds. Lee Little
plexity is Numbered Heads Together. When a learn- Soldier (1989) says that cooperative learning sup-
ing task and objective is presented, each team is ports the Native American value of learning based
responsible for all members’ mastering the content on cooperation rather than competition. She notes
or skill. The teacher calls on team members by num- that “traditional Indian families encourage children
ber. If the team member answers a question or dem- to develop independence, to make wise decisions,
onstrates a skill, the team earns points. In Jigsaw and to abide by them. Thus the locus of control of
students leave their home team to become part of Indian children is internal, rather than external, and
an expert group. Each team member becomes an they are not accustomed to viewing adults as authori-
“expert” on one aspect of the content or skill and ties who impose their will on others” (pp. 162–163).
teaches the home team (Figure 5.11). If the topic is

72 Part 1 Guiding Elementary Mathematics with Standards

Carol Malloy (1997), in a discussion of African Asking good questions is an important teaching skill
American students and mathematics, observes, “In- to develop. Without preparation, teachers tend to
structional models that permit students to partici- ask questions that have short, factual answers. Good
pate in cooperative rather than competitive learn- questions challenge students to think at a deeper
ing activities will allow students to take advantage of level. The teacher can write several open-ended
their community focus and interdependence as well questions in a lesson to elicit student understand-
as their preference for learning through social and ing and misconceptions. After asking higher level
affective emphases” (p. 29). The structure of coop- questions, teachers should also allow enough time,
erative groups means that all students have opportu- called wait time, so that one or more students can
nities to participate, make contributions, and share respond and engage in discussion.
the results of their work. Instead of every person for
him- or herself, cooperative learning is based on the Managing the Instructional
Three Musketeers philosophy of one for all, and all Environment
for one. The group succeeds only when all the part-
ners succeed. Managing the instructional environment also influ-
ences instruction. Teacher decisions about the time,
Cooperative learning has been the subject of space, materials, textbooks, and other resources for
research for more than 30 years. Research and re- teaching can encourage or discourage student in-
views of the literature about cooperative learning by teraction and increase or decrease independence
Slavin (1987, 1989) attest to its benefits for student and responsibility of children managing their own
achievement. Of 68 studies on cooperative learning learning.
that were judged to be adequately controlled, 72%
showed higher achievement for cooperative-learn- Time
ing groups. Cooperative approaches are effective
over a range of achievement measures and different Teachers seldom say that they have too much time
students; the more complex the outcomes (higher- for teaching. Careful planning helps teachers maxi-
order processing of information, problem solving), mize engaged learning time for active learning.
the greater the effects. Marzano and colleagues Long-range planning helps teachers balance topics
(2003), in an extensive review of research on in- throughout the year. Standards have a scope and
structional strategies, found that cooperative learn- sequence of content and process, but teachers allo-
ing was highly effective in raising student achieve- cate time for introduction, development, and main-
ment an average of 30 percentile (Marzano, 2003). tenance of concepts and skills throughout the year.
Teachers also balance time for mathematics with
Encouraging Mathematical other subjects. If mathematics is scheduled for 1 hour
Conversations daily, the teacher has about 180 hours to develop
knowledge and skills. Every minute is precious.
Teachers encourage or discourage talk and thinking
with the questions they ask children. While watching Teachers maximize time by integrating math-
informal activities, checking for understanding in a ematics with different subjects. Measuring tempera-
lesson, or guiding an investigation, teachers need ture or mass addresses science and mathematics
to ask more open-ended questions. Open-ended standards. A graphing lesson that compares state
questions ask students to explain their reasoning. or country populations includes mathematics skills
Closed questions call for short, specific answers. and social studies content. Interdisciplinary plan-
ning means instructional time does double duty
Closed Questions Open-Ended Questions
If mathematics lasts 50 minutes or an hour, an
What is the area How do you determine effective teacher maximizes student engaged time
of the rectangle? the area of the rectangle? with a variety of activities by using a parallel sched-
Does Cereal x cost more Would you buy Cereal x ule (Figure 5.12). Some students participate in di-
or less than Cereal y? or Cereal y? Why? rected lessons while others work in learning centers,
What is the formula for Given this triangle, how on projects, on the computer, or on seat work. The
the area of this triangle? could you determine its teacher teaches a whole-group lesson, reteaches
area? Is there more than with a small group, assesses individuals, and moni-
one way to show it?

Chapter 5 Organizing Effective Instruction 73

Figure 5.12 15–20 minutes 30–45 minutes
Parallel-activities schedule

What the Whole-Class Monitor Small-Group Assessment Closure
Reteaching
Teacher Does Lesson

What the Whole-Class Exploration and Investigation Closure
Students Do Lesson *Learning Centers
*Games *Seatwork
*Small Group *Computer
*Individual

Assessment

tors student work during the period. The teacher pro- if student desks are in rows, teachers may allow stu-
vides closure to the math period by reviewing major dents to move their desks together for activities or to
points and discussing issues and questions raised work together on the floor or at a table. Changing the
by students. Offering a variety of different learning room arrangement is natural as children progress
activities is important whether teachers plan a fixed and new topics are introduced. Displaying student
mathematics period, an integrated day, or a parallel- work on bulletin boards also reinforces the topics
activity schedule. being studied. Students can post samples of work
or alternative answers to problems. Then the bulle-
Space tin board contributes to discussion of mathematics
concepts and ideas.
How teachers organize space in the classroom also
influences the classroom climate. A classroom with Learning Centers
desks lined up in rows with the teacher’s desk in
front implies a teacher-centered approach. If desks Learning centers are areas in the classroom set
are arranged in groups of four or tables with centers up with materials that students can use in their
and materials displayed around the room, the room learning. With learning centers students engage in
invites more student interaction (Figure 5.13). Even informal or exploratory work, review and maintain

Figure 5.13 Class- Shelves
room arrangements: Table
(a) traditional class-
room; (b) classroom Rug Shelves
with table and rug
for group work,
games, or manipula-
tives; (c) classroom
with tables for proj-
ects and learning
centers; (d) math-
ematics laboratory

(a) (b)
Shelves

Table Table

Shelves Table Computers

Table Table Table

Table Table

Shelves
(c) (d)

74 Part 1 Guiding Elementary Mathematics with Standards

prior skills, and conduct problem-based investiga- notebooks and keep records of the profit and loss
tions. Students can work independently and/or have in the store.
specific tasks that are required. Some learning cen-
ters contain a variety of manipulatives that can be Task cards or other assignment options in the learn-
used every day in mathematics instruction: calcu- ing center encourage independent work. For prob-
lators; games, puzzles, and books; paper, crayons, ability the assignment shown in Figure 5.14 might be
and marking pens. Other centers support specific posted. The task card for measurement illustrated in
instructional objectives. For studying geometry, the Figure 5.15 guides students’ measurement activities
center might include geometry puzzles, geoboards, with a centimeter ruler. Advance planning and se-
tangrams, and pattern blocks. lection of materials is essential for implementation
of learning centers, but because they do not require
• A data center may contain spinners, number constant supervision by the teacher, the teacher
cubes, a clipboard for data collection, and a com- can be engaged in other instructional or assess-
puter for entering data, analyzing, and displaying ment tasks. However, rotating and updating materi-
graphs. als in the center is important because new options
increase students’ interest.
• A sand or water tub with plastic containers of
various sizes and units provides exploration of Manipulatives
volume and capacity.
Manipulatives, sometimes called objects to think
• A game center includes games and puzzles for with, include a variety of objects—for example,
developing strategy, practicing number facts, or blocks, scales, coins, rulers, puzzles, and contain-
learning geometry terms. ers. Stones or sticks were probably the first manipu-
latives used; today the choice of materials seems
• Computers are set up with software selected to endless, with educational catalogs full of materials
support the instructional goal or maintain skills that support mathematical thinking (see Appendix
from earlier units. B). Manipulatives can be as simple as folding paper
to demonstrate fractional parts or as elaborate as a
• A classroom store may be adapted to different classroom kit of place-value blocks. The value of a
instructional purposes. Play money and empty manipulative is in its use. Skilled teachers help stu-
product boxes are all that primary-grade children dents discover mathematics embedded in the mate-
need to play store while they learn about the rials. Matching the manipulatives to the concept or
exchange of money for goods. Third- and fourth- operation being taught is the key for using them to
graders pay and make change, practicing subtrac- scaffold students’ understanding.
tion skills. Fifth- or sixth-graders can set up a real
store in which students buy school pencils or

Figure 5.14 Exploring Probability
Probability assignment for
learning center Each group is to select one probablility device: spinner, die, or coin. Follow the directions and
record the number of times for each result.

■ If you choose a spinner, spin it 50 times and count how many times the pointer stops at
each color.

■ If you select a die, roll it 50 times and count how many times each number lands on top.

■ If you have a coin, toss it 50 times and record the number of heads and tails.

When you have completed the task, discuss the results, make a table or graph showing the
results, and write a statement that tells what you have observed. Was this the result you
expected, or did it surprise you?

This activity should take about 15 minutes. You may repeat this assignment three times
with different probability devices. Place the results of your experiment, with conclusions, in
your folder.

Chapter 5 Organizing Effective Instruction 75

Figure 5.15 Using Centimeters
Task card for measurement
Use the centimeter ruler to measure the items and answer the questions. Use a com-
plete sentence to answer each question. Compare your answers with those of another
student. If there are any discrepancies, measure the objects again to resolve them.

1. How many centimeters long is the plastic pen?

2. How many centimeters long is the yellow spoon?

3. How many centimeters long is the film box? How wide is it? Is the end of the film
box shaped like a square?

4. How many centimeters long is the strip of blue paper?

5. What are three things in the room that are less than 20-centimeters long? How long
is each thing?

Manipulatives can be packaged for easy dis- by determining how well any materials connect to
tribution and compact storage. Plastic shoeboxes standards and learning expectations of the state
and zip bags are often used for these reasons. Be- and district. Effective teachers start with the stan-
cause schools cannot buy every manipulative for dards, not the textbook chapters, to balance topic
every teacher; sharing resources is essential. With coverage of them. Relying on a textbook for a long-
a central mathematics storage area for the school range plan may put too much emphasis on a topic of
or grade level, teachers can check out materials as limited value and reduce treatment of other topics.
they teach a topic. Checking the developmental sequence of skills is
important. One teacher found that measurement to
Some people believe that manipulatives are just the quarter inch was presented before students had
for younger children or for slow learners. However, studied fractional parts. She introduced fractions
manipulatives support learning of concepts by creat- before starting the measurement lesson.
ing physical models that become mental models for
concepts and processes. Many manipulatives can Textbooks and other materials are effective only
be used for several mathematics concepts. Teachers if they serve the learning needs of the students. Even
need to know what concepts are being developed when a textbook includes informal and exploratory
by the manipulative so they can encourage students activities to open a lesson, some teachers start with
to develop the concept or skill. Table 5.3 shows the sample problems. Whether teachers use the devel-
connection between mathematical concepts with opmental lesson from the textbook or create their
appropriate manipulatives. own, concept development is essential. Eliminating
exploratory developmental activities undermines
Textbooks and Other Printed Materials children’s understanding of the core concept.

Although recent textbook series claim to address NCTM and many commercial publishers provide
the NCTM and state standards, research shows that supplementary materials that support the standards.
they are only partially successful. Written to ap- The number of lessons, games, and activities avail-
peal to the widest audience, textbooks include a able on the Internet is almost limitless; however,
little of everything. Used in a conceptual teaching many materials posted on the Internet have not un-
plan, textbooks provide units and lessons with ac- dergone editing or a selection process. Reviewing
tivities, games, extension activities, and evaluation activities is required to match the content and learn-
strategies. Teachers exercise professional judgment ing needs of the children. Vendors can also provide

TABLE 5.3 • Mathematical Concepts and Their Appropriate Manipulatives

Angles Cuisenaire rods, fraction models, decimal squares, color
Protractors, compasses, geoboards, Miras, rulers, tiles, 10-frames, money
tangrams, pattern blocks
Odd, Even, Prime, and Composite Numbers
Area Color tiles, cubes, Cuisenaire rods, numeral cards, two-
Geoboards, color tiles, base-10 blocks, decimal squares, color counters
cubes, tangrams, pattern blocks, rulers, fraction models
Patterns
Classification and Sorting Pattern blocks, attribute blocks, tangrams, calculators,
Attribute blocks, cubes, pattern blocks, tangrams, two- cubes, color tiles, Cuisenaire rods, dominoes, numeral
color counters, Cuisenaire rods, dominoes, geometric cards, 10-frames
solids, money, numeral cards, base-10 materials, poly-
hedra models, geoboards, decimal squares, fraction Percent
models Base-10 materials, decimal squares, color tiles, cubes,
geoboards, fraction models
Common Fractions
Fraction models, pattern blocks, base-10 materials, Perimeter and Circumference
geoboards, clocks, color tiles, cubes, Cuisenaire rods, Geoboards, color tiles, tangrams, pattern blocks, rules,
money, tangrams, calculators, number cubes, spinners, base-10 materials, cubes, fraction circles, decimal squares
two-color counters, decimal squares, numeral cards
Place Value
Constructions Base-10 materials, decimal squares, 10-frames, Cuisenaire
Compasses, protractors, rulers, Miras rods, math balance, cubes, two-color counters

Coordinate Geometry Polynomials
Geoboards Algebra tiles, base-10 materials

Counting Probability
Cubes, two-color counters, color tiles, Cuisenaire rods, Spinners, number cubes, fraction models, money, color
dominoes, numeral cards, spinners, 10-frames, number tiles, cubes, two-color counters
cubes, money, calculators
Pythagorean Theorem
Decimals Geoboards
Decimal squares, base-10 blocks, money, calculators,
number cubes, numeral cards, spinners Ratio and Proportion
Color tiles, cubes, Cuisenaire rods, tangrams, pattern
Equations/Inequalities; blocks, two-color counters
Equality/Inequality; Equivalence
Algebra tiles, math balance, calculators, 10-frames, Similarity and Congruence
balance scale, color tiles, dominoes, money, numeral Geoboards, attribute blocks, pattern blocks, tangrams,
cards, two-color counters, cubes, Cuisenaire rods, Miras
decimal squares, fraction models
Size, Shape, and Color
Estimates Attribute blocks, cubes, color tiles, geoboards, geometric
Color tiles, geoboards, balance scale, capacity containers, solids, pattern blocks, tangrams, polyhedra models
rulers, Cuisenaire rods, calculators
Spatial Visualization
Factoring Tangrams, pattern blocks, geoboards, geometric solids,
Algebra tiles polyhedra models, cubes, color tiles, Geofix Shapes

Fact Strategies Square and Cubic Numbers
10-frames, two-color counters, dominoes, cubes, Color tiles, cubes, base-10 materials, geoboards
numeral cards, spinners, number cubes, money, math
balance, calculators Surface Area
Color tiles, cubes
Integers
Two-color counters, algebra tiles, thermometers, color Symmetry
tiles Geoboards, pattern blocks, tangrams, Miras, cubes,
attribute blocks
Logical Reasoning
Attribute blocks, Cuisenaire rods, dominoes, pattern Tessellations
blocks, tangrams, number cubes, spinners, geoboards Pattern blocks, attribute blocks

Measurement Transformational Geometry: Translations,
Balance scale, math balance, rulers, capacity containers, Rotations, and Reflections
thermometers, clocks, geometric solids, base-10 Geoboards, cubes, Miras, pattern blocks, tangrams
materials, color tiles
Volume
Mental Math Capacity containers, cubes, geometric solids, rulers
10-frames, dominoes, number cubes, spinners
Whole Numbers
Money Base-10 materials, balance scale, number cubes, spinners,
Money color tiles, cubes, math balance, money, numeral cards,
dominoes, rules, calculators, 10-frames, Cuisenaire rods,
Number Concepts clocks, two-color counters
Cubes, two-color counters, spinners, number cubes,
calculators, dominoes, numeral cards, base-10 materials, Concepts matched with manipulatives developed by National Supervisors of
Mathematics (1994).

Chapter 5 Organizing Effective Instruction 77

many written and computer-based instructional tings that stimulate children’s interest and thinking.
resources that can be used in planning and imple- In Chapters 8–20 children’s books on each content
menting exciting instruction. topic are listed. Bibliographies of children’s books
organized by concept are available on several Inter-
Children’s Literature net sites, with activities to extend the concept.

Only 20 years ago few children’s books that focused E XERCISE
on mathematics were available beyond simple
number books. Now concept books on every math- Find a bibliography of mathematics concept books
ematical topic are a great addition to the materials with an Internet search. Compare the topics to the
teachers can use for teaching and learning. Books
introduce concepts in realistic or imaginary set- NCTM content and process standards. •••

Summary master content. Cooperative learning benefits students
with learning disabilities, gifted and talented students,
Effective teachers actively engage students in meaning- ethnic and cross-cultural groups, and English-language
ful mathematics tasks using a variety of instructional ap- learners because all students are supported in a positive
proaches, including informal and exploratory activities, social climate focused on learning.
directed teaching/thinking lessons, and projects and
investigations. Exploratory activities involve students in Decisions regarding grouping and teaching strategies
independent work with manipulatives and games and al- connect to classroom decisions about time allocation,
low them to develop an intuitive understanding of con- space arrangements, and choice of materials. Because
cepts through their experiences. In directed teaching/ teachers have limited time, careful scheduling is required
thinking lessons teachers guide students through a series to provide time for different instructional approaches,
of interactive experiences. Lessons are not lectures and activities, and groupings. Flexible seating arrangements
are not passive in nature; they require students to engage encourage group cooperation and sharing. Learning
in a problem or situation that requires development of centers contain learning materials and assignments for
mathematics skills and concepts. Projects and investiga- investigations, projects, and independent tasks. Manip-
tions are good extensions of learning that allow students ulatives are an essential part of teaching mathematics
to refine their understanding and apply skills. Using a va- because they help students develop physical models
riety of instructional approaches allows students with dif- to represent mathematics operations. All students need
ferent skills, interests, and needs to be challenged. Unit the benefits of manipulatives in learning mathematics.
planning, weekly planning, and daily planning provide a Textbooks can be valuable resources when teachers use
structure to build concepts in a logical sequence of les- them to support developmental learning. They contain
sons and activities that move from simple to complex, lessons using manipulatives and provide many sugges-
concrete to symbolic, and specific to generalized. tions for games and enrichment. Teachers must review
textbooks and other resource materials carefully to make
Teachers confront many issues and make many or- sure they support the scope and sequence of their curric-
ganizational and management decisions that influence ulum. Children’s literature that addresses mathematical
the learning environment. Decisions are based on un- concepts is available for all topics. Children’s literature
derstanding state and local curriculum standards and helps students relate mathematics to real or imagined
providing a safe and stimulating environment for learn- situations. Lists of books and lesson plans are available
ing. The role and amount of practice and homework are on the Internet.
practical issues. Homework can consist of conceptual
or practice activities. Practice should be assigned only Study Questions and Activities
if students have developed a skill, or else they will be
frustrated and will practice the skill incorrectly. 1. Select one performance objective from your state or
district objectives. Find a lesson plan on the Internet
Teachers decide whether to teach large groups, small that might be used to teach the objective. Does the
groups, or individuals, depending on the needs of stu- lesson plan include all the parts of the model lesson
dents on a specific lesson. Cooperative-learning groups plan?
require students to work together to complete tasks and