Heterogeneous Classrooms

Diane Kruse, Division Three MST

August, 2001 

Any class, even in the most tracked, regimented, sequential mathematics program, will have students with a range of abilities and ages in it.  In a classroom where every single student presumably has the same algebra background, there will be students who remember almost every procedure that they have used, students who remember fragments but cannot recall them when needed, and students who got through algebra with a D+ and never really internalized anything.  The classic math teacher lamentation is that the teachers in earlier grades must not have taught fractions or factoring or linear inequalities, because their students certainly cannot use them!   

Teaching in a heterogeneous classroom required me to make peace with this issue once and for all, and to recognize as a pipe dream the notion that all students will remember every math technique they have ever seen.  Once I acknowledged this, I was able to accept that part of my job in any math classroom will always be review.  Mathematics is somewhat sequential, though not nearly as much so as many of us believe.  It makes sense to acquire solid skills in algebraic manipulation and the properties of geometric shapes before studying calculus, but whether one studies exponential functions before or after quadratics or descriptive statistics seems less crucial.  As we design our mathematics curriculum at Parker, we use a “spiral” approach that takes our classes through topics in algebra, geometry, statistics, and probability each year, with built in review and increasing complexity along the way. 

I teach in the Division III mathematics program, and by this stage we expect students to have acquired a solid foundation in algebraic manipulation, basics of geometry, and descriptive statistics.  The courses I teach are organized more clearly around traditional mathematical themes, such as Advanced Algebra (the semester-long study of functions and their graphs), Trigonometry, and Statistics.  Students who take my Advanced Algebra – Trig course sequence may be using it as a pre-calculus sequence, or they may be taking their final course before graduation and focusing on strengthening their algebra skills.  Several of my students this year were on IEPs.  I need to accommodate the needs of this entire range of students within the context of these courses.    

To teach this wide range of students effectively, I have begun planning my units with a clear structure, deciding ahead of time what information my “beginners” should strive to master and what concepts will be explored by more advanced students.  I include all students when introducing the more complex work, because I think that students benefit by sometimes listening to conversations that are over their heads (and I am never 100% certain who will “get” the material—sometimes someone really surprises me).  The unit structure is as follows:  

·        Introductory Problem and Essential Question.

Typically, on the first day of a new unit I will pose a problem that gets students thinking through issues and asking questions.  For instance, when I teach the unit circle, my opening problem asks students to determine the height of a rider on a ferris wheel.  After exploring for a while, some students generally begin to use their right triangle trig, and we can then transition through discussion to the essential question for the unit: How can we use trigonometry to model oscillating motion?

·        Core topics of unit

These activities and explorations will build on the work we did with the opening problem.  Sometimes I will review previous mathematics that will be useful during the unit, sometimes I will pose a series of increasingly complex problems that allow students to develop and deepen their understanding of the phenomenon, and sometimes I will review and teach relevant vocabulary.

·        Reinforcement/additional practice OR extensions

At this point, students begin using their new knowledge with a variety of new problems.  I usually choose quite a range, so that students who need to practice their beginning understanding get the repetition that they need, while those who “get it” can do more challenging work.

·        Final Project/Unit Problem

As a performance based school, Parker assesses students based on their ability to do real work with the knowledge they develop.  Students in my math classes usually have a choice of final project for each unit, with varying levels of difficulty.  I will sometimes “nudge” students to make a choice that I think is appropriate, but usually my students are already choosing work at an appropriate level. 

·        Quizzes/Tests along the way

For more traditional feedback on procedural skills (such as finding the lengths of triangle sides using trig), I will give quizzes.  Ideally, these quizzes cover material that I hope every student will master. 

Many of the assignments that I give to all students can be done with varying degrees of depth and content mastery.  For instance, when I teach new functions, students complete lab investigations exploring the effects that the different constant values have on the shape of the graph.  Some students can only document the patterns that they see, while others can explain in depth the reasons why those patterns occur.  As students mature mathematically, the level of explanation I expect from them increases.   When I give a unit project or unit problem, I often give a general assignment that my students make specific through their own interests.  When we study projective geometry techniques, students design their own mural that must accurately create the illusion of a three-dimensional scene; the complexity of the mathematics involved depends on the content of the painting.   When we study logarithms, students complete a research project where they must ask and answer a question about logarithms by conducting their own research and reading their sources for information.  Questions range from “How does the Richter Scale measure earthquake intensity?” to “Why do logarithmic spirals occur in nature?”  Students then present their findings to the class in 10-minute mini-lessons.   

To teach effectively in this context, I rely on several specific strategies.  Some days I do self-paced skills work, where different students have different practice worksheets in front of them, and I circulate to offer help or to encourage students to help each other.  Often I present open-ended problems with challenging extensions, so that when a group of problem-solvers finishes ahead of the others, I have a set of planned questions to push their thinking.  When I offer student choice on individual projects, I have a good set of recommendations in mind so that I can help students generate ideas, and we often engage in group brainstorming (this also helps to generate excitement about the work).  And when I check in with students about their work, I use different coaching for different kids.  Some students get more structured help, and more specific instructions; some get only more questions to push their thinking.  The longer I have students, the more I try NOT to answer their questions directly. 

Assessing major work for mathematical problem-solving and mathematical communication rather than for the “right answer” reminds me to look at the students’ overall growth in these areas, not just at their understanding of the unit circle or logarithms.  When focusing on the “big goals” of encouraging students to organize known information, choose appropriate mathematical tools, carry out their work accurately, and verify that their results are correct, I am really teaching broad critical thinking skills that will apply to all aspects of their lives.  When I focus on teaching students to use appropriate mathematical vocabulary and multiple representations of quantitative information, and to communicate not only their answers but the implications of their results, I am teaching them the conventions that govern the use of mathematics in adult life.   

Although planning courses with the deliberate intent of accommodating heterogeneity has been challenging, I have experienced some inspiring successes that make it worthwhile. 

·        Students seem to be learning more math overall than they would if I insisted on mastery of each concept before moving on.

·        The cyclical approach to revisiting concepts allows students to review, relearn, or finally learn a topic they have seen in prior years.  They eventually actively seek connections between current material and their past coursework.

·        The “higher” the level of math that students study, the greater their overall problem-solving sophistication.

·        Advanced material (and maturity) provides incentive for revisiting and really learning more basic topics.  For instance, students decide that they really want to get faster at factoring, or solving for x efficiently.

·        Students don’t get the sense that they have “done enough math” once finished with geometry.  Almost everyone feels obligated to take four years of math, even if they “hate” it. 

Despite these successes, I still face significant challenges, and sometimes catch myself longing for a class of honors Algebra II.  My weakness as a teacher has been that I focus more on struggling students than on those who are really talented—sometimes my advanced students are bored and should be using their time better.  The emphasis on individualizing sometimes creates unrealistic expectations for me and for my students, and some days I have the sense that rather than having one class I am conducting 17 personal lessons.  Special needs students, who don’t have a resource hour, sometimes need more help than I remember to give.  We all need to be more vigilant about making accommodations, and we continue to work on this.