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February 1, 2012
Vol. 69
No. 5

Math Groups That Make Sense

Instructional Strategies
Picture a mathematics lesson in a typical 4th grade class, where students' math achievement levels range from 2nd to 9th grade. The teacher, who believes in an egalitarian classroom and differentiated instruction, begins the class with a minilesson on long division. Students then work in three groups, attempting to solve the problem 365 divided by 24. The teacher races from small group to small group providing support.
In five minutes, the most-skilled group has solved the problem, and the students head to the computers for more challenging work; the middle group is still working on the problem; and the students in the least-skilled group begin to clown around, hoping that joking will provide a cover for confusion. The conscientious teacher sits down with the struggling group, simplifies the problem, and leads them to a solution. In the meantime, one of the computers freezes. Some of the students from the middle group begin to work on math centers; a few others argue about their solution, without the benefit of teacher input. Eventually the teacher fixes the computer and begins to check in with the arguing students. Ten minutes remain. The teacher reviews the problem on the board and then assigns homework targeted to the needs of each group.
That night, the teacher will correct the homework from the previous day, directing some students to check in with her in the morning or during a study period so she can review their mistakes and correct misconceptions. She will note areas in which each student struggled so she can adapt the work for the next day. In her spare time, she will plan her language arts and social studies lessons.

Challenges of a Heterogeneous Classroom

If you've spent substantial time in a classroom, you'll recognize this scenario as all too common. Something is clearly wrong with this picture.
We know that students learn best when they experience just the right amount of challenge, not too much or too little. In the classroom we've described, it's almost impossible to provide that kind of challenge for each student, even some of the time. In a heterogeneous class, students who lack facility with mathematics are often inclined to hide and to participate minimally. Ashamed of their struggle with math, they may disengage, preferring not to try rather than be seen (or see themselves) as unsuccessful.
Differentiating with technology is not the whole answer, because we also know that students learn best when they work with someone, preferably a teacher, who can ask the right questions at the right time to help them move to the next level of understanding. A good computer program may lead a student to the next skill level but not as effectively as a person with a social connection who can urge the student to grapple with understanding. The least-skilled group needs a different approach than the more advanced groups. Students in the most challenged group often have subtle working-memory, processing, or attention challenges best recognized by a skilled and experienced teacher.
Probably the most important problem with the heterogeneous classroom is that the students know who's the most talented and who's the most challenged; in fact, they could probably list the students in the class in order of ability. Mixing the gifted with the most mathematically challenged does not create an egalitarian classroom, any more than mixing expert swimmers with non-swimmers (while hoping that no one drowns and that everyone learns something) would create an egalitarian swim lesson. Does such mixing create role models that enhance learning, or does it make self-conscious preadolescents do everything possible to hide their inadequacy?

Tracking, Detracking—or Neither?

A study of tracking and detracking in Massachusetts middle schools conducted by Tom Lovelessexamines the history, characteristics, and consequences of these practices. Tracking has lost favor in recent years because of concerns about inequity. In tracked classroom settings, the youngest and least experienced teachers were frequently assigned to the lowest achievers, thereby compounding the problem of students falling further and further behind. The stigma of placement in underachieving groups and the feeling of failure stifled motivation for these students.
Detracking, however, has produced mixed results. Loveless found that compared with middle schools that maintained two or three tracks, detracked middle schools had fewer students scoring in the top 10 percent on the mathematics section of the Massachusetts Comprehensive Assessment System (MCAS) and had more students scoring at the failing and needs improvement levels. These results suggest that detracking may harm high-achieving students while producing no significant benefits for lower-achieving students.
Despite such evidence, tracking in mathematics at the elementary level continues to be viewed unfavorably. And it's undeniable that assigning students to a lower track can foster continued cycles of low achievement.

A Different Approach

To avoid some of the pitfalls associated with tracking and detracking, we suggest a different strategy, which we call guided choice. We employed guided choice for 12 years in the 4th and 5th grades at the Philadelphia School, an independent school in Philadelphia, Pennsylvania, serving approximately 400 students in grades preK–8. We are now introducing guided choice in the upper grades at Friends School Haverford, another independent school in Pennsylvania, which serves 111 students in grades K–6 (soon to be expanding to grade 8).
Guided choice enables our teachers to provide the right level of challenge for each student without demoralizing lower-achieving students. This strategy is also pedagogically efficient and less stressful for the teacher.

Helping Students Choose Their Level

Let's assume that a school has two 4th grades and two 5th grades, for a total of four teachers in these grades. All 4th and 5th grade students, after a week of math review in the beginning of the school year, are given a comprehensive test of skills and comprehension. Teachers not only score the tests but also review them to be sure that a low score was caused by a lack of understanding rather than careless arithmetic.
On the basis of each student's performance on the test, the teacher notes which group she believes to be most appropriate for that student. Copies of the corrected test (but not the teacher's preliminary judgment) are given to the students. Scores within the test are broken down by four areas: whole numbers, fractions, decimals and percents, and pre-algebra.
Students receive a list of available math groups, identified by the topic on which each group will focus (the same topics as those in the test). They review their performance on the test and select the math group where they believe they belong. Students and teachers generally reach the same conclusion. If not, the teacher and the student meet and together decide which group offers the right amount of challenge.
Any student who wants to move to a more advanced group is given the text and assignments of the next group to use at home (ideally, with parents or other mentors) and is offered periodic instruction from the teacher during study periods. From time to time (for example, quarterly) students may move to a more advanced group if they perform well enough on those sections of the test that reflect the extra, more advanced work. With realistic feedback, support, clear goals, and some control over placement, students are more often motivated to achieve, more thoroughly insulated from the sense of being judged and found wanting, and more optimistic about their prospects.

Providing Appropriate Instruction

Each of the four 4th and 5th grade teachers teaches one of the math groups. The lowest-achieving group begins with a focus on whole numbers, the next group with fractions, the next with decimals and percents, and the highest group with pre-algebra. The least-advanced group is often smaller so that students can receive extra time and attention to overcome obstacles to success.
Although the groups are not completely homogeneous, the range is narrowed; and the teacher's preparations, lessons, activities, and discussions can be more focused and thorough. Although the mathematics curriculum spirals (for example, fractions may be touched on in the whole-numbers group and then revisited in greater depth and breadth in the fractions group), the continuity of the curriculum, moving from whole numbers to parts of wholes to algebraic expressions—is preserved.
It is essential that within all the groups, mathematics instruction is constructivist and based on problem solving, with an appropriate measure of factual information and algorithm instruction. For example, a teacher might typically engage students in a lesson on fractions by having them work with manipulatives to complete a puzzle, to answer a provocative questions, or to deploy a successful game strategy. Poor teaching will undermine even the savviest placement strategies, excellent curriculum, and terrific resources.

Results at the Philadelphia School

In our experience at the Philadelphia School, the most advanced group often began with about 17 percent of the class and grew to about 24 percent by the end of each school year. As students moved along through the topics of the curriculum, about 10 percent routinely asked to take extra work home and to meet with teachers during workshop or study periods.
Over 12 years, we found that a significant number of 5th graders went on to middle school prepared for an advanced pre-algebra curriculum, a small number of them were placed in an advanced arithmetic group, and the middle group was comfortable with a standard 6th grade pre-algebra curriculum. Five-year averages from 2000 to 2005 on the Educational Records Bureau exams showed that just .8 percent of our students scored below average, 32 percent scored in the average range, and 67.2 percent scored above average (with 32 percent scoring at the 90th percentile or above). These data confirm that increased time and attention devoted to the group in the needs improvement category did not limit the achievement of the most talented mathematicians.

One Student's Story

The story of one particular student, Jack, illustrates the process and potential benefits of guided choice for a student who finds mathematics challenging.
Although he had experienced a strong, hands-on, constructivist 3rd grade math program, Jack scored quite low on the beginning-of-the-year 4th grade math assessment. He was sure that he was ready for the fractions group: After all, he had studied whole numbers for three years. But as he looked at his test results and reviewed the errors with his teacher, Jack recognized that he had not mastered whole number operations. His teacher discussed his options with him, and together they looked at the material that each math group would study. On the basis of these discussions, Jack placed himself in the whole-number group.
Jack spent his 4th and 5th grade years grouped with other students who found mathematics difficult. In this context, Jack was gradually emboldened to feel himself a leader. He became a frequent participant in class discussions and activities and a helper to his mathematics group peers. He continued his progress in middle school, and he pursued mathematics courses through trigonometry and precalculus in high school.
Guided choice promoted not only Jack's success as a mathematics student, but also his disposition toward mathematics. He knew where he stood. He participated in choosing his group. As he progressed, he preferred to be a stronger member of a math group that felt more manageable. Because his mathematics instruction was the right fit for him, he remained engaged in mathematics. We believe that Jack's experience was the norm rather than the exception.

Opportunity for All Students

At a time when funding for gifted education is being reduced or eliminated but a competitive global economy continues to make higher mathematics performance essential, we must find effective ways to meet the needs of talented students without shortchanging students who require more help to succeed. Differentiated instruction and technology are not replacements for consistent teacher time and attention.
Guided choice can produce the benefits of homogeneous grouping while protecting students who find math challenging from becoming demoralized and disinterested. The beauty of guided choice is the trust placed in students to take greater control of their education. We believe that guided choice gives each student an opportunity to excel.
End Notes

1 Loveless, T. (2009). Tracking and detracking: High achievers in Massachusetts middle schools. Washington, DC: Thomas B. Fordham Institute.

Author bio coming soon

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