Tiered Lessons: One Way to Differentiate Mathematics Instruction

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Adams, C. & Pierce, R.

Gifted Child Today

Prufrock Press

Vol. 27, Issue 2, pp. 50-65

2004

This article is about differentiation. Due to the broad range of academic needs among students, teachers find themselves in a dilemma. The Burris Laboratory School outlines how teachers can reach all the students in their classrooms when they are academically diverse, have special needs, are ESL learners or have some combination of any or all of these factors.

The movement toward inclusion has impacted classrooms by requiring teachers to respond to a broader range of academic needs. How can we possibly reach all the students in our classrooms when they are academically diverse, have special needs, are ESL learners, or have some combination of any or all of these factors? An answer to this question lies in differentiating instruction. Working in the Burris Laboratory School, an inclusion school using a resource consultation model to serve the needs of all its students, we have found that using tiered lessons is a viable method for differentiating instruction.

What is Differentiation?

Although differentiated instruction is not a new idea, the differentiation movement has recently taken center stage as a means of meeting the needs of all students in the classroom. It is an organized, yet flexible way of proactively adjusting teaching and learning to meet students where they are and help all students achieve maximum growth as learners (Tomlinson, 1999). Instruction may be differentiated in content/input, process/sense-making, or product/output according to the students’ readiness, interest, or learning style. By content, we mean the material that is being presented. Processactivities help students practice or make sense out of the content, while product refers to the outcome of the lesson or unit, such as a test, project, or paper. Readiness refers to prior knowledge and a student’s current skill and proficiency with the material presented in the lesson. A student’s interest may be assessed with an interest inventory for the particular topic being studied or by an individual conversation with the student. Many teachers use the theory of multiple intelligences to characterize learning styles (Armstrong, 1994; Gardner, 1993; Martin, 1996).

Essential elements for successful differentiation include specific classroom management techniques addressing the special needs of a differentiated classroom, planned use of anchoring activities, and flexible use of time, space, and student groups. In a differentiated classroom, the management plan must include rules for working in a variety of configurations. You can only work with one group or individual at a time. Therefore, we have developed two critical rules that thwart chaos and preserve sanity. The first is “Use six-inch voices,” meaning that students should modulate their speaking level so that their voices can only be heard six inches away. The second rule is “Ask three before me.” If students need assistance completing a task or come to a stumbling block in a lesson and you are not available, they should find three other students to ask before they may interrupt you. If their three peers cannot answer the question, the student has permission to interrupt you. Adding the caveat that the student should also bring along the three students who were asked will nearly eliminate the chance that you will be interrupted except in extreme cases. Anchoring or “sponge” activities are provided for students to use when they are waiting for you to assist them before they can go any further or at the beginning of the class period to get them ready to work. A wide variety of materials and resources can serve as anchoring activities (see our Web site athttp://www.bsu.edu/teachers/services/ctr/javits/Instruction/anchoring.htm for a listing of books that have great activities for anchoring). Flexible grouping arrangements such as pairs, triads, or quads, as well as whole-group and small-group instruction, create opportunities to meet individual needs. A flexible use of time allows lessons to proceed to their natural conclusion, rather than being carried out in set blocks of time. The desks or tables should be arranged in such a way as to facilitate group work, as well as wholeclass groupings that encourage sharing of ideas.

A variety of instructional strategies, including compacting, learning contracts, cubing, and tiered lessons, can be used to differentiate instruction (for a discussion of these and other strategies, see Gregory & Chapman, 2002; Heacox, 2002; Smutney, Walker, & Meckstroth, 1997; Tomlinson, 1999; Winebrenner, 1992). It makes sense to alert your administration and the parents that you will be trying some new strategies in the classroom in case there are questions.

The tenets of differentiated instruction support both the Equity Principle and the Teaching Principle of the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000). These principles direct us to select and adapt content and curricula to meet the interests, abilities, and learning styles of our students; to recognize our students’ diversity; and to encourage them to reach their full potential in mathematics.

What is a Tiered Lesson?

Tomlinson (1999) described tiered lessons as “the meat and potatoes of differentiated instruction.” A tiered lesson is a differentiation strategy that addresses a particular standard, key concept, and generalization, but allows several pathways for students to arrive at an understanding of these components based on their interests, readiness, or learning profiles. A lesson tiered by readiness level implies that the teacher has a good understanding of the students’ ability levels with respect to the lesson and has designed the tiers to meet those needs. Think of a wedding cake with tiers of varying sizes. Many examples of lessons tiered in readiness have three tiers: below grade level, at grade level, and above grade level. There is no rule that states there may only be three tiers, however. The number of tiers we use will depend on the range of ability levels in your own classroom since you are forming tiers based on your assessment of your students’ abilities to handle the material particular to this lesson. Students are regrouped the next time you use tiering as a strategy. Hence, the idea of flexible, rather than static, groups is essential.

No matter how you choose to differentiate the lesson—readiness, interest, or learning profile—the number of groups per tier will vary, as will the number of students per tier. You are not looking to form groups of equal size. When you form groups based on the readiness needs of individual students, Tier I may have two groups of three students, Tier II five groups of four students, and Tier III may have one group of two students. When the lesson is tiered by interest or learning profile, the same guidelines apply for forming groups: Different tiers may have varying numbers of students. Even when students are already homogeneously grouped in classes by ability, there is still variance in their ability levels that must be addressed.

To take a closer look at the anatomy of a tiered lesson, we have included a mathematics lesson (see Figure 1) that was developed as part of the Javits Grant, Project GATE, a federally funded partnership between the Indianapolis Public Schools and Ball State University, both in Indiana. When developing a tiered lesson, we have found the eight steps described below useful.

First, identify the grade level and subject for which you will write the lesson. In this case, the grade level is first and the subject is mathematics.

Second, identify the standard (national, state, district, etc.) you are targeting. A common mistake for those just beginning to tier is to develop three great activities and then try to force-fit them into a tiered lesson. Start with the standard first. If you don’t know where you are going, how will you know if you get there? The author of this lesson has selected the Content Standard “Number and Operations” of the National Council of Teachers of Mathematics’ (2000)Principles and Standards for School Mathematics (pp. 78–88).

Third, identify the key concept and generalization. The key concept follows from the standard. Ask yourself, “What big idea am I targeting?” In this example, it is to understand and represent commonly used fractions. While there are many concepts that could be covered under the standard chosen, this lesson addresses only one. The generalization follows from the concept chosen. Ask, “What do I want the students to know at the end of the lesson, regardless of their placement in the tiers?” In this lesson, all students will develop their understanding of fractions as representing parts of a whole.

Fourth, be sure students have the background necessary to be successful in the lesson. What scaffolding is necessary? What must you have already covered or what must the student have already learned? Are there other skills that must be taught first? Before engaging in this lesson, students have been exposed to halves and thirds. Fractions (halves/ thirds) have been introduced to the students, and they have illustrated them with pictures. There are several literature books that illustrate fractional parts using food that could be used to introduce the lesson.

Fifth, determine in which part of the lesson (content, process, product) you will tier. You may choose to tier the content (what you want the students to learn), the process (the way students make sense out of the content), or the product (the outcome at the end of a lesson, lesson set, or unit—often a project). When beginning to tier, we suggest that you only tier one of these three. Once you are comfortable with tiering, you might try to tier more than one part in the same lesson. This lesson is tiered in content.

Sixth, determine the type of tiering you will do: readiness, interest, or learning profile. Readiness is based on the ability levels of the students. Giving a pretest is a good way to assess readiness. Students’ interest in a topic is generally gauged through an interest survey, while the learning profile may be determined through various learning style inventories. In this lesson, the author chose readiness.

Seventh, based on your choices above, determine how many tiers you will need and develop the lesson. When tiering according to readiness, you may have three tiers: below grade level, at grade level, and above grade level. If you choose to tier in interest or learning profile, you may control the number of tiers by limiting choices or using only a few different learning styles. For example, tiering on all eight of Gardner’s multiple intelligences in one lesson may not be a good place to start, so choose only a few, such as logical-mathematical intelligence, spatial intelligence, and linguistic intelligence. (For further information on multiple intelligences in an easy-tounderstand format, see Wahl, 1997). For this lesson, students are placed in one of three tiers based on their ability to work with halves and thirds as assessed by the teacher through observation.

Differentiation means doing something different—qualitatively different. Make sure you keep this in mind when tiering the lessons. Second, be sure that students are doing challenging, respectful, and developmentally appropriate work within each tier. In other words, no group should be given “busywork.” We don’t want one group doing blackline practice sheets and another doing a fabulous experiment.

Notice in this lesson that all three tiers are working on fractions. Students in each tier use paper shapes to divide. However, the activities for each tier in the sample lesson, beginning in Tier I and moving through Tier III, differ from concrete to abstract and from simple to complex, to use Tomlinson’s Equalizer word pairs (Tomlinson, 1999).

Finally, develop the assessment component to the lesson. The assessment can be formative, summative, or a combination of both. You may use some means of recording observations of the various groups, such as flip cards or sticky notes. You could develop a rubric for each tier based on the particular product that is created. You may give a formal paperand- pencil test. Whatever it is, choose your assessment based on your needs and your lesson design.

In this lesson, the teacher observes the students as they share their answers and jots down notes for a formative assessment of each student. For example, which child is struggling with the concept? Which child is moving rapidly and accurately through the material? Whose answers show more thought and insight? Answers to these and other questions will assist you in determining who needs reteaching and who is ready to go beyond the material presented. A formal assessment is not used here since the standards emphasize that students should have “informal experiences [with fractions] at this age to help develop a foundation for deeper learning in the higher grades” (NCTM, 2000, p. 83).

When this lesson was taught, the students were engaged during the entire lesson. The lesson was introduced by reading the book Eating Fractions (McMillan, 1991). Students were placed in groups based on their level of readiness to interact with the content. Four students did not have a clear understanding of halves and fourths. These students needed a more concrete activity and were placed in Tier I. Another 12 students could recognize halves and thirds and were ready to complete the Tier II activity. They were placed in four triads. Two students had in-depth knowledge of halves and thirds and were placed in Tier III. This pair worked at a more abstract level, and the questions they were asked required them to use different critical thinking skills than the other two groups. Tier I and Tier II students were provided with activities from the book Fractions(Watt, 2001) to use as anchoring activities if they finished early or were waiting for the teacher’s assistance. The anchor for Tier III students was Apple Fractions (Pallotta, 2002), which introduced fifths through tenths.

The second sample lesson (see Figure 2) is tiered in process according to learning style. In this case, students are grouped heterogeneously based on one of two learning preferences: kinesthetic or visual. The same eight steps for tiering a lesson apply in this case. In the second lesson, notice that the activities are at relatively the same level of complexity. This would be the “layer cake” model as opposed to the “wedding cake” model used when tiering according to readiness.

Final Thoughts

Time, energy, and patience are required to learn to differentiate instruction effectively in an academically diverse classroom. In addition, you need administrative and peer support, as well as professional development over extended periods of time; therefore, don’t expect to have a differentiated classroom by Monday morning. Start small: Choose a favorite lesson in your next unit and differentiate it according to the needs of your students. Seek the expertise of specialists such as special and gifted education coordinators, media specialists, and others with whom you can collaborate to improve instruction in the academically diverse classroom.

For more information on tiering, contact the Center for Gifted Studies and Talent Development, Ball State University (BSU)http://cms.bsu.edu/academics/centersandinstitutes/giftedstudies.

Author Note

Research for this article was supported under the Javits Act Program (Grant R206A980067) as administered by the Office of Educational Research and Improvement, U.S. Department of Education. Grantees undertaking such projects are encouraged to express freely their professional judgment. This article, therefore, does not necessarily represent positions or policies of the government, and no official endorsement should be inferred.

References

Armstrong, T. (1994). Multiple intelligences in the classroom. Alexandria, VA: Association for Supervision and Curriculum Development.

Gardner, H. (1993). Multiple intelligences: The theory and practice. New York: BasicBooks.

Gregory, G. H., & Chapman, C. (2002). Differentiated instructional strategies: One size doesn’t fit all. Thousand Oakes, CA: Corwin Press.

Heacox, D. (2002). Differentiating instruction in the regular classroom. Minneapolis, MN: Free Spirit.

Martin, H. (1996). Multiple intelligences in the mathematics classroom. Palatine, IL: IRI/SkyLight.

McMillan, B. (1991). Eating fractions. New York: Scholastic. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author.

Pallotta, J. (2002). Apple fractions. New York: Scholastic.

Smutney, J., Walker, S., & Meckstroth, E. (1997). Teaching young gifted children in the regular classroom. Minneapolis, MN: Free Spirit.

Tomlinson, C. A. (1999). The differentiated classroom: Responding to the needs of all learners. Alexandria, VA: Association for Supervision and Curriculum Development. Wahl, M. (1997). Math for humans.

Langley, WA: LivnLern Press.

Watt, F. ( 2001). Fractions. New York: Scholastic.

Winebrenner, S. (1992). Teaching gifted kids in the regular classroom. Minneapolis, MN: Free Spirit.

Permission Statement

This article is reprinted with permission of Prufrock Press, Inc.http://www.prufrock.com/.

This article is provided as a service of the Davidson Institute for Talent Development, a 501(c)3 nonprofit dedicated to supporting profoundly gifted young people 18 and under. To learn more about the Davidson Institute’s programs, please visitwww.DavidsonGifted.org.