Within our K-12 classrooms, we find a symphony of different voices with respect to race, ethnicity, culture, language, exceptionality, social class, gender, age, religion, gender preference, learning styles, genetic make-up, and individual experiences. Such student demographics continue to shift, affecting both schools and teaching (Hodgkinson, 1994). In contrast, our teaching force remains predominantly white, middle-class females (Davidman,1997; Smith, 1987). With an increasingly culturally diverse and technological society, it is essential that all children be provided equitable opportunities to master the mathematical skills essential for social and economic success. Yet, in research and educational reports, curricular and instructional elements of bias are recognized for minorities, females, and children of poverty as an ongoing concern in classrooms with respect to their poor performance in mathematics (Campbell, 1995; National Research Council, 1990, 1989; National Science Foundation, 1994; Oakes, 1990;Rechin, 1994; Secada, 1992).
Mathematics has been traditionally viewed as a discipline where success is limited to a minority as opposed to a majority of children. During the past several years there has been a conscious effort by mathematics educators to change this view of mathematics to an orientation that focuses on making mathematics accessible and enjoyable for all children. Reform efforts address topics such as the need for relevance by virtue of providing real-life applications, collection and organization of data, and problem solving as opposed to rote memorization of procedures. In addition, the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), prepared by the National Council of Teachers of Mathematics, while not directly addressing cultural diversity issues, advocates instructional practices that include the use of manipulative materials, cooperative work, communication of mathematical ideas in everyday language, and writing about mathematics. It seemed like a natural ‘fit’ that cultural diversity and mathematics join together to make mathematics truly a discipline for ALL. In light of these facts, two teacher educators united to help teachers in urban elementary classrooms meet the needs of our ever-changing student population. At the forefront of their shared beliefs were 1) all children can learn and should be afforded the opportunity to do so, 2) a repertoire of best practice must be at the forefront to facilitate essential knowledge, attitudes, and skills for all children, 3) mathematics should be facilitated through emphasis on more problem-solving, hands-on activities, interactive learning experiences, and alternative assessment, and 4) education is the responsibility of all people, not simply the classroom teacher.
Two primary goals, success in math and acceptance of self and others’ differences and similarities, were at the heart of the teacher educators’ efforts. It was important for each child to feel successful and accepted, to recognize his/her potential and ability to learn, and to recognize the relevance of mathematics both within and outside the classroom. With this in mind, all lessons and activities highlighted relevant math curriculum and experiences and provided equitable 1) student acceptance and representation through dialogue to understand differences and similarities of people with respect to attitudes, knowledge, and behaviors, 2) verbal and interpersonal interaction patterns and expectations [praise, feedback, questioning, encouragement, proximity], and 3) opportunities for critical thinking, problem-solving, and conceptual understanding through cooperative, hands-on activities to construct meaning. The instructional vehicles for teaching multicultural education and basic math concepts and skills were game- formatted activities, using hand-made wooden manipulatives.
Creating and maintaining a positive relationship between the teacher educators and children was a primary consideration. To establish and maintain rapport, make connections between prior and new levels of understanding and perceptions, and reinforce content, it was essential to have high expectations and to use smiles and humor, positive animated verbal and nonverbal behaviors, close proximity, and personalized experiences. The teacher educators were friendly to each child, while simultaneously establishing required guidelines for participation in hands-on, cooperative learning activities. Prior to each lesson, expectations for both instructors and students were shared, and each childs hand was shaken by the instructor as a sign of mutual agreement and trust to perform at his/her best. The teachers agreed to excite children about learning math through real-life applications, and the students agreed to give 100% attention and effort to master the math concepts and skills. The few students who at first felt a little awkward shaking hands or making direct eye contact with the teachers were not coerced, but rather were continually encouraged to participate and use math in a fun and meaningful way.
Creating a Milieu of Understanding and Acceptance of Self and Others
To reinforce the acceptance and worth of each child and dispel the myth that math is a Western invention, contributions by females, minorities, and other cultures in the field of math were infused throughout the lessons. High expectations were held for each child as he/she was encouraged to work and learn together in cooperative learning groups. Through informal discussion and assessment of the children’s present attitudes and knowledge about where they see themselves and others in society, efforts were made to reinforce the value of all persons and to help the students construct links to new information. The idea of infusing issues on diversity should be an everyday objective, not a once a month or annual topic [i.e. Black History Month; study of different cultural foods and dress], whether directly or indirectly related to the content.
During two math activities, meaningful discussions evolved about race and gender equity. Using basketball goal manipulatives to teach fractions, the children were quick to recognize the imbalance and/or invisibility of White males in the National Basketball Association (NBA) in comparison to Black males, and the less-regarded status of the Women’s National Basketball Association (WNBA) female teams in comparison to male teams. To reinforce the idea of equity and reaffirm that the game has no boundaries with respect to color or gender, a person’s skill level, knowledge of the game, and commitment to play in an organized sport were emphasized. Using cross-cultural research, it was explained that the stages of physical growth do not vary by race, although the tempo [quickness, speed, agility, strength] may vary due to genetic make-up of an individual or the circumstances in which a child is reared, Black or White, male or female (Berk,1993). Athletes such as Michael Jordan, Larry Byrd, Lisa Lesley, and Rebecca Lobo, along with members of the children’s own classroom, served as examples to affirm this research. When the children were asked to select from among the male and female students in their class who they thought would perform the best at shooting a basketball, the results indicated possible change in stereotypical thinking that basketball is a male- oriented game only. And, using rubber band propelled racecar manipulatives to teach the metric system, the teacher educators infused information on the number of women presently designing and/or racing racecars.
Lesson Implementation to Reinforce Acceptance of Self and Others, Teamwork, Problem-solving, and Critical Thinking
Piaget defined play as a natural and inherent characteristic of individuals across cultures, asserting that children respond spontaneously to game-like activities (de Menlendez & V. Osterbag, 1997). Thus, games were viewed as vehicles for knowledge and skill development and attitude formation. In each game, opportunities were created within cooperative learning groups to allow the children to construct their own understanding of a specific mathematical concept and/or skill. Use of hands-on manipulatives [i.e. hand-made wooden basketball goals and racecars] resulted in learning at the concrete level, tapping visual, auditory, and tactile modalities.
Organizing the children into cooperative learning groups promoted ideas of acceptance, getting along, sharing ideas, and working as a team to solve problems. Through group interaction, the children were involved in the socialization process of dialoging with one another about, and through, math. Working in groups, the children learned about math, themselves, and others as they problem-solved together and recorded their findings. Students having limited experience working in cooperative groups, limited problem-solving opportunities, and limited success in math required more structure and supervision in the cooperative learning setting. The heterogeneous groupings minimized documented harmful effects of ability grouping and poor-quality instruction usually found in lower-ability homogeneous groupings (Jones, 1993; Oakes, 1985). Within each cooperative group, female, minority, and low SES students were actively involved to counter the fact that such children are too often given fewer opportunities and less encouragement to learn mathematics (Dossey, 1993). According to Slavin (1986), Allport (1992), Catsambis (1994), and Rech (1994), cooperative learning 1) promotes self-esteem, motivation, and achievement for female and minority students, 2) improves student attitudes toward their classmates, particularly those from backgrounds differing from their own, and 3) lessens boundaries created by gender and race, reducing prejudice when cross-cultural contact situations are cooperative, a feeling of equal status prevails, similar goals are shared, and the contact has the approval of parents, teachers, and other authority figures.
Within each lesson, individual tasks were created to promote competence and acquisition of a specific math skill or concept, and to allow each child to be an active learner and integral member of a problem-solving math team. In playing games with varied hands-on manipulatives, each child had a specific task to complete. The tasks were adapted to meet the needs of a particular game and area of math content. During the games, the children alternated tasks until each one had a turn performing every task. Using worksheets, the children assigned the tasks before the games to insure a smoother transition when time for them to perform a new task.
In one game, hand-size wooden basketball goals were used to teach a lesson on fractions. The object of the game was to score points by releasing a miniature basketball tied to a plastic spoon, wedged in the base in front of the goal. The students pull the spoon backwards and release it with force, thus propelling the ball toward the goal. Each child in the cooperative groups had a task to complete: serve as the timekeeper [timekeeper], shoot the basketball [shooter], count the number of shots taken [shot counter], count the number of shots made [goal counter], and record the information [statistician]. The end objective was for the children to write fractions by identifying the number of shots made in relation to the number of shots taken. The students recorded data on worksheets provided for use during the game. While playing a desk-top game of basketball, the students simultaneously learned to write fractions, to conceptually understand what the parts of a fraction represent, and to see the use of fractions beyond textbooks and math worksheets. Using the fractions recorded on the worksheets, the children were asked which student had the best score. This question set the class up for their next lessons, comparing fractions to determine the larger or smaller in value, percentages, and decimals. The game can be adapted to address other skills according to the questions posed to the children [i.e. how many goals did you make, how many more goals did you make than other group members, what was each players shooting average/percentage]. Process skills such as comparing and contrasting are important in these lessons, as well as the ability to collect, analyze, and communicate data.
In another game using racecars practice estimation using metric units of measurement, the tasks were modified: one student served as the timekeeper [timekeeper], one raced the car [racer], one calculated the distance [estimator], one measured the distance [measurer], and one recorded the data [statistician]. The object of this game was to accurately estimate the distance a hand-size wooden racecar traveled after being released on the floor. The back wheels of the racecars were attached to a rubber band that, when wound tightly, would propel the car forward upon the release of the wheels. While playing a game with racecars, the students simultaneously learned metric measurement [i.e. centimeters, decimeters, meters]. The children would estimate the distance a racecar traveled and then would use a ruler (meter stick) to find the actual distance. After several runs, the children’s estimations became more accurate as a conceptual understanding of each unit of measurement was internalized. This lesson is easily adapted for many units of length in both the English and metric systems.
Reinforcing the Rationale for Math Competency
To experience the usefulness of mathematics outside the confines of the classroom, the lessons demonstrated, through play, the use of mathematics in the real world. The game-like format and use of basketball goals and racecars offered an accessible approach for each child to learn and use mathematics, promoting competence within the ability levels and experiences of the children involved. Depending on the individual classroom and math content being taught, the children actively played games using manipulatives and learned to apply basic math skills and concepts. Whether playing a game with wooden manipulatives or role playing in simulation activities [i.e. buying merchandise in a store, restaurant, entertainment facility; tracking a savings account; planning a vacation; performing the tasks of a bank teller or cashier], the goal is to provide a way in which students can master basic math skills and concepts and simultaneously see the value of such knowledge beyond their classroom doors. Each child was able to see himself or herself as a mathematician, capable of problem solving and success in math, working independently in different roles and then collectively to apply math. The students were able to see the value of math in everyday living, beginning with the world of sports. The children constructed a rationale for why he/she needs certain math. For example, in the basketball lesson, the children realized the need for further math skills to determine who had the highest shooting percentage. Each child wanted to know who won the competition, thus was eager to learn the associated math.
At the close of each lesson, the cooperative groups were asked to identify in writing and to share orally the reasons they saw their team as being a success and to also report the group data obtained from the activity. It was inspiring to read the thoughts of children reared in an inner-city setting, children who many believe can not learn and who are persistently blamed when their cognitive performance is below that expected by teachers and administrators. Overall, the student comments were indicative of the students giving, as agreed to, 100% attention and effort as individuals and a group. The children saw their lack of shots as being due to a lack of concentration, and possibly hurrying through their shots, not a lack of skill. None of the groups blamed a team loss on a particular team member’s lack of skill, and each group consistently complimented each member’s genuine attempt to play well and to complete their assigned tasks correctly. The children truly enjoyed the math activities, performing their individual tasks correctly and efficiently, and working cooperatively as a team. Comments such as, “We did our best…We didn’t pick on each other for making a mistake…We didn’t argue about who was to do each task…We worked together…Everyone followed the rules…I like working in a team…I like learning math this way…I had fun doing math…[and] I understood” affirmed the success of the learning experience. Equally important were the comments, “I made a new friend…I didn’t know she was so funny…I want to work in his group again…She plays basketball like a pro…[and] I didn’t think girls played with racecars” affirming acceptance of others.
The oral presentation of the group data provided the children with an opportunity to communicate mathematical ideas and procedures. Obtaining and analyzing data is but one step in demonstrating mathematical literacy. Clearly expressing mathematical concepts gets everyone in the ‘conversation’. Thus, mathematics becomes a language used and understood by all.
Commitment to the Vision
As our society becomes increasingly more diverse and technologically based, greater emphasis will continue to be focused on acceptance of others and math education. Mathematics, a universal language spoken in all cultures, is a discipline that allows children to view and appreciate the similarities and differences that exist among all people. It is a vehicle that promotes problem solving, communication, logical reasoning, and relationships. The classroom milieu created by teachers can encourage or discourage a child to ask questions or share answers, thus building or destroying a child’s confidence in their mathematical abilities. The authors emphatically believe that all children can learn, and thus further believe that all educators must promote understanding and acceptance for all voices of diversity and implement research-based curriculum and pedagogy to increase access to and achievement in mathematical literacy. [Note: Worksheets and wooden racecar and basketball manipulatives are available through Educational Horizons, P. O. Box 56557, Jacksonville, Florida, 32241 or email Jan Bosnick at firstname.lastname@example.org for information].
by Patty Adeeb of Nova Southeastern University and Janet Bosnick of University of North Florida