The Heterogeneous Classroom

What Is Meant by Heterogeneous?

It seems as if you cannot read an article or attend an inservice on mathematics education reform without hearing the terms heterogeneous classroom and untracking. According to Webster's New Collegiate Dictionary, heterogeneous means "consisting of dissimilar ingredients or constituents: mixed." The movement in education is away from grouping of students by perceived ability level--tracking--and toward a heterogeneous learning environment, where students with different mathematical maturity and development levels are in the same classroom.

Instead of helping students, sorting and tracking them according to ability institutionalize failure in mathematics. However, placing students in heterogeneous classes and groups and teaching the same old curriculum will not solve the problem. . . . The curriculum must be untracked just as the school structure must be untracked. A multidimensional curriculum will be accessible to more students and more interesting and more valuable to the most mathematically sophisticated. ¹

A heterogeneous classroom coupled with a curriculum written to engage all students creates the ideal.

Why IMP Believes in Heterogeneous Classes

Our educational system needs to broaden the range of students who learn mathematics. The heterogeneous classroom promotes access to genuine mathematics for a larger pool of students than does a system based on ability-level tracking.

The IMP curriculum is designed to be used with heterogeneous classes, and thus to make the learning of a core mathematics curriculum more accessible, especially to those groups, such as women and minorities, who traditionally have been underrepresented in college mathematics classes and math-related fields.

A curriculum built around complex, open-ended problems can be explored at many levels of sophistication. The central problems in IMP units have a richness that will challenge the brightest student, yet their concreteness allows all students to do meaningful mathematical work.

How to Work with a Heterogeneous Class

Your Own Expectations

First on the agenda for working with a heterogeneously grouped class is to confront the expectations created in all of us by conventional conceptions of intelligence--conceptions that have led to ability-level grouping. We need to believe that all our students are capable of learning mathematics and, as a group, are rich in their differences.

Students' Expectations

You will probably have some students, previously identified as "gifted," who don't want to be in a class with "normal" students. You will probably also have students who have never enjoyed or succeeded in math and now feel intimidated in a class that includes all the "smart kids." In order to work with both groups, you need to convey the idea that a variety of backgrounds and learning styles will prove to be a benefit, not a detriment, to the learning process. To take full advantage of the various learning styles and backgrounds in your IMP classroom, foster as much communication among students as possible. Provide a learning environment where students are encouraged to present their methods and ideas as well as to listen thoughtfully to the presentations of others. Provide a model, showing how to ask thoughtful questions when trying to understand another's point of view. The heterogeneous classroom needs to provide an environment where cooperation for the common good is highly valued. Help students build an appreciation of each other's differences and encourage them to learn from other approaches and points of view.

Supplemental Problems

When you work with students who have a wide variety of math backgrounds, there may be times when discrepancies in learning arise. The supplemental problems in each unit can help you deal with these situations; they were created in response to requests from IMP teachers. These teachers asked for problems, written with the IMP style and philosophy, that could be used when students showed a need for more experience or more challenge when they approached a topic in the unit.

Using the supplemental problems often requires planning ahead. As you look over the next week of a unit, ask yourself, "Which lessons are likely to involve wide discrepancies in student response?" and "How can I meet the needs of different students?" The teacher guide will often give you guidance, since it indicates where in the unit each problem fits best.

There are two types of supplemental problems.

Reinforcements: The reinforcement problems exist for those times when your students struggle with a concept in the unit. Since they come to you from various backgrounds, some of your students may need to investigate a topic from approaches besides those provided by the basic unit. You may even find that, at some point in time, your whole class needs more work on a concept. The reinforcement problems provide such additional experience.

Extensions: There may be times when students understand a concept and want more challenge. The extension problems are provided for those students who are ready to take concepts from the IMP curriculum farther than the basic unit does. Extension problems give students greater depth of understanding of topics in the current unit, rather than having them "accelerate" to material that appears later in the curriculum. In this way, they will gain appropriate challenge and enrichment and yet each new unit will be fresh for them.

Whenever you use supplemental problems, be cautious of tracking within your IMP classroom. Students should be in on the decision as to which type of supplement, if any, they work on. You should avoid giving them a sense that you are labeling them one way or the other. Let it be known that those who need reinforcement this time are not necessarily the same students that will need it next time and that all students can tackle the extension problems, not just those who the teacher feels are "capable."

Revision of Work

Students in a heterogeneous class will not vary only in their mathematics backgrounds; they will also vary in their writing ability. One way to work with these differences is to encourage revision of written work. This will benefit students who find it difficult to express ideas. Also, if a student has not solved a particular problem or completed an assignment, this will allow the student to show what he or she learned from the class discussion of the activity.

It is possible for all students to meet high standards; some simply have to work harder to get there. Opportunities to revise their work provide such students with a chance to learn from others and to improve upon their initial attempts.

Getting Students Started

For a variety of reasons, including weak English-language skills, students may sometimes have trouble getting started on an assignment. One key to getting all the students involved in a problem or activity is ensuring that each student has access to the task at hand. To give students access, you may want to have a student read the directions aloud and then have each group discuss or rewrite the task in their own words. You may throw out an open question about how to get started. You may even let the students get to work on an activity, then stop them after five minutes for group reports on where they are headed. Your goal should be to ensure that all students at least start everything you assign.

The Honors Option

A heterogeneous mathematics classroom may include students who were previously labeled "Gifted" or "Honors" and placed in separate classes. As a result, there may be parental or administrative pressure to provide an opportunity for students to have an "Honors" designation on their transcripts.

You can provide this option within your heterogeneously grouped class, offering it to every student in your IMP class, not just a select few. For example, you can have students elect to attempt some combination of the extension problems. You will need to set clear criteria for the quantity and quality of work needed for a student to receive the "Honors" designation at the end of the grading period.

You can enhance your IMP classroom by having those who do extra work make presentations to the whole class on their findings. Or you may prefer to keep this activity separate, providing regular time outside of class for students who are working on the extension problems to meet and share ideas.

¹Mathematics Framework for California Public Schools(Sacramento, CA: California Department of Education, 1992), p. 62.