# V.5 #4 Mathematics - The Process of Differentiating Mathematics Instruction: Addressing Individual L

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In this era of standards and increased accountability, one challenge plaguing teachers is how to meet the needs of the diverse learners in their classrooms. Mathematics reform has recommended changes in what and how mathematics is taught (NCTM 2000; 2006). The recent Common Core State Standards (CCSSO, 2010), which aim to set high expectations for all students, indicate that it is important to find ways to help all learners to achieve mathematics proficiency in areas such as conceptual understanding, procedural fluency, strategic and adaptive mathematical thinking, and also in acquiring a productive disposition (Baroody, 2011). Accordingly, mathematics instruction should be meaningful, inquiry-based, and purposeful.
**Differentiated Instruction**

Toward the beginning of every semester, I ask my preservice teacher candidates to define differentiated instruction and talk about its salient characteristics. Many of these candidates initially believe that it is a way to address students’ needs by breaking down the material to target three types of students: those who struggle, those typical learners on grade level, and those who may be more advanced and need additional challenges. In short, they tend to think in terms of three compartmentalized groups of students: low, middle, and high. However, this is a very narrow conceptualization of differentiated instruction. I try to broaden their perspectives away from three compartmentalized groups to think about students as individuals who have a range of learning styles.

According to Tomlinson and Imbeau (2010), differentiated instruction is an individual focused approach to teaching. Some key elements include content (knowledge, understanding, and skills students should learn), process (how students come to understand and make sense of the content), and product (how students demonstrate what they come to know, understand, and are able to do after an extended period of time). It is very common to focus on differentiating the content, but I encourage my teacher candidates to also focus on differentiating the students’ process of learning and the teachers’ process in how the classroom is structured and curriculum is enacted. It is important for teachers to incorporate multiple representations of mathematical ideas; students need to hear mathematics, see mathematics, say mathematics, touch mathematics, manipulate mathematics, write mathematics, and draw mathematics (Shih, Speer, & Babbitt, 2011). Students with learning difficulties rarely learn from seeing or hearing the material only (Shih, Speer, & Babbitt, 2011) and other modalities must be incorporated. I utilize the Theory of Multiple Intelligences (Gardner, 1983; 2011) to have my teacher candidates think about ways to plan and structure mathematics lessons and activities to target individual learning styles across a range of learners, and away from the three compartmentalized groups. This article focuses on four of the eight intelligences: bodily-kinesthetic, naturalist, spatial, and intrapersonal. Some activities to address those learning styles in the classroom are provided.
**Bodily-Kinesthetic Learners. **To target the needs of bodily-kinesthetic learners, teachers must find ways to use the whole body or involve hands-on experiences for students. Some mathematics experiences that support this learning style include using the students as manipulatives and providing tactile learning experiences. With this approach, students actively become involved in their learning experiences and they can understand problems more conceptually rather than procedurally.

Instead of drawing 2-D pictures to demonstrate early addition and subtraction, utilize the students as manipulatives. Consider the following problem: Jessica has 5 fewer pennies than Fernando. Jessica has 8 pennies. How many pennies does Fernando have? Have students act out the problem with real pennies. Although they see the word “fewer” and likely associate it with subtraction, this word problem really utilizes addition. With a dramatization of the problem, students will likely gain greater conceptual understanding because they can visualize its meaning.

Engage in human place value activities. When working on numbers in the ten thousands, for example, have five students come to the front of the class and give them each a number. The class has to determine what number the students have formed by how they are placed. Let’s say the number formed is 54,192. Next, students can shift places and the class needs to determine the new number (e.g., Ask the students with the 5 and 9 cards to switch places; what number is now formed?) This activity can be used in a range of grade levels to target whatever place value students are working on.

If you are working on polygons, have students read The Greedy Triangle (Burns, 1994). Then students can create polygons out of straws, pipe cleaners, popsicle sticks, playdough, and even themselves. Through tactile learning, students can gain an awareness of attributes of polygons.

**Naturalist Learners.** To address the needs of naturalist learners, teachers must find ways to bring nature and the outdoors into the learning environment. Consider using rocks, shells, or leaves as manipulatives to illustrate important mathematics concepts. Sometimes these natural manipulatives can replace fancy store bought ones as a cost cutting measure in these economic hard times.

In the fall, take a leaf walk and have students collect leaves. They can enhance geometric thinking by sorting them by attributes such as color, size, and texture.

Use these natural manipulatives to aid with simple addition and subtraction or more complex problems involving fractions, decimals, and percents (e.g., Out of the leaves collected, what fraction is red? Express this fraction as a decimal and percent).

Think about having a class pet or plant which students can observe and record growth over time. This promotes learning measurement skills in a real life context.

**Spatial Learners. **To address the needs of spatial learners, teachers should incorporate visual aids and color. Think about ways to combine mathematics and art.

Have students creatively design flags to demonstrate fraction concepts. Consider this prompt: You are asked to design a flag for a country you will run. The only stipulation is that it needs to contain 1/3 of one color, 2/4 of another color, and 1/6 of a third color.

Create tessellations using pattern blocks. Teachers can photograph the tessellations to display around the classroom. Students can also write about their tessellations.

**Intrapersonal Learners.** To address the needs of intrapersonal learners, teachers need to consider how to evoke personal feelings or memories. These students may not feel comfortable in front of a large group of people, but can showcase their mathematical knowledge in other formats.

Consider using journals, which are not used often enough in mathematics, to ask students to write about their feelings toward mathematics (e.g., Do you consider yourself good in mathematics; why or why not?)

Use mathematics journals regularly where students can write about how they solved certain problems. This promotes mathematical discourse on a more private level.

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Benefits of Utilizing Multiple Modes of Inquiry to Address Diverse Student Needs**

The aforementioned are just a sampling of mathematics activities that teachers can employ to reach a range of learners to make mathematics meaningful for them. It is important to go beyond traditional paper and pencil experiences to encourage inquiry-based learning; this allows students deeper engagement with mathematics. Because how students feel about a subject impacts their learning, it is important for educators to expand their process of teaching (modes of inquiry) with unique learning styles in mind. With greater engagement, students are more likely to retain the material which could enhance their achievement in mathematics.
**References**

Baroody, A. J. (2011). Learning: A framework. In F. Fennell (Ed.), Achieving fluency: Special education and mathematics (pp. 15-57). Reston, VA: National Council of Teachers of Mathematics.

Burns, M. (1994). The greedy triangle. New York, NY: Scholastic, Inc.

Council of Chief State School Officers [CCSSO]. (2010). Common Core State Standards for Mathematics. Washington, DC: CCSSO.

Gardner, H. (1983; 2011). Frames of mind: The theory of multiple intelligences. New York, NY: Basic Books.

National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics [NCTM]. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.

Shih, J., Speer, W. R., & Babbitt, B. C. (2011). Instruction: Yesterday, I learned to add; today I forgot. In F. Fennell (Ed.), Achieving fluency: Special education and mathematics (pp. 59-83). Reston, VA: National Council of Teachers of Mathematics.

Tomlinson, C. A., & Imbeau, M. B. (2010). Leading and managing a differentiated classroom. Alexandria, VA: ASCD. Joan Gujarati, Ed.D., is an Assistant Professor in the Department of Curriculum and Instruction at Manhattanville College in Purchase, New York where she teaches the childhood mathematics methods courses. She is a former elementary school teacher and Math Teacher Leader. Dr. Gujaratiâ€™s research interests include early childhood and elementary mathematics education, teacher beliefs and identity, teacher quality, effectiveness, and retention, and curriculum development. Dr. Gujarati can be reached at Joan.Gujarati@mville.edu