Reform in Mathematics Education:
Rethinking the Curriculum
Melissa A. Bono
The mathematics classroom has traditionally been a place where students listened quietly as the teacher lectured about the proper way to complete math problems. Through continuous independent practice in memorizing basic facts and word problems that lend themselves to a key word approach, the pedagogical goal was that students would develop automaticity and proficiency in the skills being taught. Students who encountered difficulties would receive additional help and practice to increase the accuracy and speed of their computations.
As educators try to meet the ever-increasing demand that students demonstrate competence in math as well in the ability to solve problems, recent reform in mathematics education calls for changes that alter dramatically the way math is being taught in schools (National Council of Teachers of Mathematics, 1989). Students are still expected to listen quietly as the teacher presents, works, and discusses mathematical concepts. Struggling students are still required to receive extra assistance to increase their understandings. However, the teacher is no longer the only source of information. Students are being encouraged to engage actively in the pedagogical process by solving real-life mathematical problems and explaining their mathematical reasoning to the teacher and their classmates.
Students spend much of their time in a reform-based mathematics classroom solving challenging problems that entail open-ended answers or involve the use of different strategies. To reinforce math concepts, teachers introduce students to an array of math tools and manipulatives, for example, calculators, geometry templates, and unifix
cubes. To succeed in a reform-based mathematics classroom, students must carefully listen and explain their mathematical reasoning through conversations among their teacher and their peers to construct personally meaningful understandings of mathematical concepts (Baxter, Olson, & Woodward, 2001).
Educational researchers, teachers, parents, and school board members have raised many questions about the effects of reform-based mathematics upon student achievement, particularly upon low achievers. As Baxter, Olson, and Woodward have noted: “An underlying assumption of the reform is that the new mathematics pedagogy and curricula are effective for all students, including low achievers” (2001, p. 530). Research has suggested, however, that the standards adopted by National Council of Teachers of Mathematics (NCTM), released in 1989 and revised in 2002, provide little guidance and modifications for students who are at risk (Baxter, Olson, & Woodward, 2001, p. 530).
A study by
Carroll (1998) investigated the level of geometric understanding of
students
when exposed to Everyday Mathematics,
a reform-based mathematics program developed by The University of
Chicago’s
Mathematics Project. Carroll’s
experimental study used a paper-and-pencil method to determine the
geometric
knowledge of fifth and sixth graders using Everyday
Mathematics. The researcher administered a pretest and a posttest at
the
start and end of the year to 20 fifth and sixth grade classes using the
reform-based program, 10 of which were used as comparison classes.
Carroll’s
findings indicated an increase in the level of fifth and sixth grade
geometric
knowledge when practicing Everyday
Mathematics. Students using the reform-based program outperformed
their
comparison groups in both a pretest and a posttest (1998, p. 8).
Two-tailed t
tests indicated that these differences between the groups were
significant
(Table 1).
Table 1
Mean Correct (and Standard Deviation) on Pretest and
Posttest
Grade
Pretest
Posttest
5
9.0 (3.9) 11.8
(4.5)
10.2 (4.0) 14.7
(5.3)
6
10.5
(3.6) 13.5 (4.4)
13.8 (3.4) 16.9 (4.1)
Note: indicates a significant difference (p <
0.01), on t tests between groups at the
same grade; e.g., fifth grade
UCSMP and comparison on the same test. On the
pretest, T-statistics on a pretest
were T (165) = 4.25 for fifth graders and T (244)=
5.77 for sixth graders. On
a posttest, T-statistics were T (165) = 6.31 for fifth
graders, and T (244) = 5.62 for
sixth graders.
On the pretest,
Carroll noted two
differences at the fifth grade level and three significant differences
at the
sixth grade level on the pretest. Students exposed to Everyday
Math scored higher than comparison sixth graders on the
pretest, with a mean percent correct of 47% and 42% respectively.
Posttest
results depicted a similar pattern, with Everyday
Math students significantly outperforming the comparison students at
both
grade levels. Data indicated two significant class differences at the
fifth grade
level and two significant class differences at the sixth grade level
(1998, p.
5).
In a second study, Noyce
and Riordan (2001) conducted a post-treatment study using matched
comparison
groups in a quasi-experimental design. The researchers compared
standardized
test scores of fourth and eighth grade students using Everyday
Math and Connected
Mathematics to the scores of students using a magnitude of curricula
that
represent the norm instruction in Massachusetts. The researchers found
that
students attending schools using a reform-based mathematics program
outperformed those using a more traditional curriculum. All differences
between
the target curriculum and the comparison curriculum were statistically
significant (p < 0.001).
The results also indicated that longer exposure to either of the programs in the schools was associated with greater student test scores. Noyce and Riordan noted: “Connected Mathematics students in the first two or three years of school implementation performed 4.0 points better than their counterparts, whereas Connected Mathematics students in the fourth year of implementation performed 5.5 points better. Similarly, Everyday Mathematics students in the first two or three years of school implementation scored 2.5 points better than their counterparts, whereas Everyday Mathematics students in their fourth year of school implementation scored 5.7 points better” (2001, p. 383). In light of these findings, Noyce and Riordan argue that Everyday Mathematics and Connected Mathematics are two effective programs for all students at the bottom, middle, or top of the achievement spectrum (p. 386).
Believing
that the NCTM Standards (1989) provide little, if any, guidance about
how
teachers might modify the standards for students who are at risk of
academic
failure or have a learning disability in math, Baxter, Olson, &
Woodward
(2001) studied the reactions and responses of students who struggle in
math.
To test for
understanding and involvement among low-ability students, the
researchers
focused on 16 qualified participants in five third-grade classrooms
using reform-based
instruction. Through 34 classroom observations of the students, peers,
and
teachers and several interviews with the teachers, the researchers
analyzed the
involvement of low achievers in whole-class discussions and pair work,
essential parts of this type of instruction.
This one-year qualitative study inquiring into the responses of low achieving students in reform-based mathematics classrooms indicated that in all five math classrooms there were only three occasions out of 34 observations when low achievers volunteered answers during class discussions, verbalizing only one-word answers or remaining silent while their partners spoke. “The target students were observed playing quietly with a small object in their laps, staring out the window, writing on a piece of paper, or avoiding eye contact with the teacher” (Baxter, Woodward & Olson, 2001, p. 537). The target students were more actively engaged in pair work than in whole-group discussions. However, they chose a more supportive, nonmathematical role of organizing materials and listening to their peers’ ideas while their higher ability classmates responded with detailed and accurate mathematical explanations. The data analysis of the observations and interviews from the five classrooms identified the social, verbal, and cognitive challenges low achievers faced in the reform-based math classrooms.
This gloomy finding regarding the minimal involvement of low achievers need not be viewed as a complete failure for reform-based mathematics programs. In fact,
Baxter, Olson, and Woodward utilized the data to paint an optimistic image about how teachers and parents can better engage the target students in a reform-based setting (2001, pp. 543-545).
Teacher awareness of the social, verbal, and cognitive demands that reform-based mathematics curricula put upon low achieving students is a critical first step in molding students into more active participants of reform-based mathematics instruction. Educational researchers have discussed findings reporting that both the grouping of students and the use of manipulatives in the classroom have positively influenced the involvement of low-ability students during the math lessons. How students respond to a teacher’s structural and instructional strategies depends largely upon how the teacher steers the students. Ma (1999) notes, for example, that the teacher must make clear connections between manipulatives and mathematical ideas in order to promote understanding as one teacher did by forming “ad hoc” groups consisting of 8-11 low achieving students who were chosen based on the goal of the teacher’s lesson that day. After reviewing definitions such as parallel lines and rays during an “ad hoc” lesson, students demonstrated the various terms with their arms, pieces of yarn, and geoboards. “We can make these form right angles, one low achiever stated, as he spontaneously started counting the degrees of the other angles in the intersecting yarn (Baxter, Olson, &
Woodward, 2001, p.
541). In working with geometric terms, and
building
conceptual
understandings
of important math ideas, the easily distracted target students were able
to
respond to the teacher’s request and were eager to answer her questions while constructing conceptual understandings that were meaningful to them as learners of math.
Additional help and extra support both inside and outside of the classroom from teachers, parents, and classmates appeared to help in monitoring the students’ progress and behavior. To enhance the development of a child’s math skills and concepts, they should be encouraged to practice and revisit new and previously learned computational skills and word problems, found in the child’s workbook or “math journal.” Since Everyday Mathematics provides many opportunities for students to master a topic, it is not necessary to dwell on one skill area if a child does not master it immediately. (The University of Chicago School Mathematics Project, 2002, p. 29). The University of Chicago School Mathematics Project notes: “Staying too long with a topic may help some students attain temporary mastery, but for maximum long-term retention, it is best to follow the basic structure of the curriculum as it is written” (2002, p. 29).
Teachers and parents need to assist students in speaking about their mathematical knowledge and ideas as well as encourage them to explain how they arrived at their answers. Through the students’ individual conjectures and reasoning, they are to make sense of the math concepts being studied. The more practice the students get in explaining why they are doing math, rather than just following the rules of math, the easier time they will have in meeting the high verbal, social, and cognitive expectations and challenges that reform-based instruction presents. The University of Chicago School
Mathematics Project notes: “Because verbalizing often clarifies concepts, talking about mathematics is an important part of thinking about mathematics” (2002, p. 4).
While some elementary and middle school educators will continue to favor the traditional, old fashioned way of teaching math, others strongly view mathematics as an opportunity to mold children into problem-solvers who can make relationships through explorations, reasoning, and exchanging ideas with others, which are essential skills in real life. A number of studies have concluded that reform-based mathematics curricula do improve student mathematical understanding. Elementary and middle school students are capable of developing stronger mathematical understandings through daily exposure to
reform-based math programs such as Everyday Mathematics and Connected Mathematics.
While the social, verbal, and cognitive gap regarding low-achievers is an important issue, these results suggest that achievement for all students can be reached by changes in the classroom structure and instruction, and the extra assistance in school as
well as at home from kindergarten onward (Baxter, Olson, & Woodward, 2001, p. 545).
Therefore, it would be misguided policy to turn away from reform-based mathematics
curricula. Instead, these curricula should be investigated further to discover how
mathematics teachers might engage struggling students as more active participants of this
type of instruction.
The questions asked in the rising tide of debate about the reform in mathematics education have remained unresolved. While different groups have strong, varying opinions about what constitutes high-quality mathematics education, research suggests that exposure to reform-based mathematics curricula such as Everyday Mathematics and Connected Mathematics are associated with a greater score advantage and a deeper understanding of geometric knowledge and mathematical reasoning for elementary and middle school students. Results have suggested that low-achieving students display a lack of social and verbal interactions in reform-based mathematics classrooms, which need to be addressed if these students are to benefit from this type of instruction. With this exception, most reform-based mathematics programs have demonstrated their effectiveness with a fair degree of consistency.
References
instruction on low achievers in five third grade classrooms. The Elementary School Journal, 101, 529-546.
Carroll, W. M. (1998). Geometric
knowledge of middle school students in a
reform-based
mathematics curriculum. School
Science and Mathematics, 98, 188.
Retrieved October 3, 2002, from http://web4.epnet.com/citation.asp.
Lappan G.,
Fey J., Fitzgerald W., Friel S., &
Phillips E. (2002). Getting to Know
Connected Mathematics: An Implementation Guide. Illinois:
Prentice Hall.
Ma, L.
(1999). Knowing and Teaching
Elementary Mathematics. New Jersey: Lawrence
Erlbaum
Associates.
National
Council of Teachers of Mathematics. (1989). Curriculum and
evaluation
standards for school mathematics. Reston, Virginia.
Noyce, P.
E., & Riordan J. E. (2001). The impact
of two standards-based mathematics
curricula on student achievement in Massachusetts. Journal for Research in Mathematics Education, 32, 368-393.
The
University
of Chicago School Mathematics Project. (2002). Everyday
Mathematics
Teacher’s reference manual. Columbus, Ohio: McGraw-Hill.