Graphic Organizers for Teaching Advanced-Level Mathematics: Understanding Negative Integer Exponents & Solving Linear Equations
Category: Mathematics
Grade Level: Secondary Grades
1. What is the purpose of using Graphic Organizers in advanced-level mathematics?
Graphic organizers are visual-spatial displays used to organize knowledge and represent relationships among pieces of information. Due to their visual-spatial nature, they require less reliance on language skills. While other techniques used in mathematics instruction, such as the use of manipulatives and graphs, also rely little on language skills, their applicability to advanced-level mathematics (ie. algebra) has been questioned (Ives & Hoy, 2003). The effectiveness of graphic organizers in improving reading comprehension has therefore prompted the use of graphic organizers in teaching advanced concepts in mathematics.
2. With whom can they be used?
In advanced mathematics, graphic organizers can be used by teachers to complement regular classroom instruction. Graphic organizers might be particularly useful for students with learning disabilities who have weaker language skills and stronger spatial and nonverbal reasoning skills.
3. How does one adapt Graphic Organizers to mathematics instruction?
Many different types of graphic organizers have been developed for reading comprehension and other language instruction. Consequently, when developing graphic organizers to use in mathematics, one can look to literature on reading comprehension and other literature on strategies for creating graphic organizers (eg. Winn, 1991). However, there are three important points to remember when applying graphic organizers to advanced mathematics:
I) In mathematics, the content of graphic organizers should no longer be verbal elements, such as words, phrases, and sentences. Rather, content should be mathematical in nature, consisting of numbers, other types of symbols, expressions and equations.
II) Advanced mathematics skills emphasize understanding concepts, patterns, and processes, as opposed to memorization of numbers, expressions, and equations. Consequently, the goal of using graphic organizers in higher-level mathematics should be to help students recognize and understand relationships between elements. Therefore, within graphic organizers, the way in which elements are spatially arranged should reinforce these relationships.
III) Graphic organizers should be used to complement, not substitute for regular classroom instruction. Teachers should explicitly teach concepts while relating them to graphic organizers.
4. What are some examples of Graphic Organizers that can be used in higher-level mathematics?
Suggestions for graphic organizer use in the areas of negative integer exponents and solving linear equations are presented below.
Understanding Negative Integer Exponents
The goal of this graphic organizer is to present negative integer exponents as a meaningful concept which naturally builds on the students’ previous knowledge of positive exponents. The graphic organizer is used to draw attention to the relationship between positive integer exponents and negative integer exponents. The teacher starts out by engaging students in an interactive discussion of information related to positive integer exponents. Such information is visually laid out in a column of increasing exponents and multiplying values, as shown in Figure 1. The teacher encourages the students to point out patterns in this column. For example, as one goes down the column, exponents decrease by one and values are divided by 2. Negative integer exponents are then introduced in a second parallel column, as shown in Figure 2, and students are encouraged to apply the rules used with positive integer exponents (ie. as exponents decrease by 1, divide the previous value by 2). The teacher then encourages students to look for more patterns in the organizer. It becomes apparent that exponents in the right-hand column are the opposite of exponents in the left-hand column and that values in each column are reciprocals of each other. With this technique, concepts are communicated by the relative positioning of elements within the organizer. [For more details and specific scripts regarding this method, see Ives & Hoy (2003).]
Solving Systems of Three Linear Equations with Three Variables
The frame suggested for this graphic organizer is shown in Figure 3. The frame is divided into cells. Again, the teacher engages the students in an interactive discussion as they are guided through the various steps to solving the equations while using the graphic organizer. The equations are first placed in the cell in the top left corner of the graphic organizer (see Figure 4). They are then worked through from cell to cell in a clockwise direction. Equations are combined in the top row and solved in the bottom row. The Roman numerals on top of each column correspond to the number of variables within the equations being worked. Thus the Roman numerals allude to one of the goals of solving linear equations – combining equations so that they contain fewer variables. Figure 5 represents the completed graphic organizer. The relative positions between elements, as well as the relative positions between elements and the frame, help to emphasize relationships and concepts important in understanding how to solve linear equations. [For more details and specific scripts regarding this method, see Ives & Hoy (2003)].
5. In what types of settings should Graphic Organizers be used?
Teachers can use graphic organizers to facilitate classroom learning. Once understood, they can be applied independently by students.
6. To what extent has research shown Graphic Organizers to be useful in higher-level mathematics?
The effectiveness of graphic organizers in teaching higher-level mathematics has been verified informally in classrooms. Systematic research on the graphic organizer for solving linear equations (described above) is currently underway to determine whether its use can lead to improvement on mathematics achievement measures for students with learning disabilities (Ives & Hoy, 2003). Initial evidence from this study suggests that students find the graphic organizers useful.
References
1. Ives, B. & Hoy, C. (2003). Graphic Organizers Applied to Higher-Level Secondary Mathematics. Learning Disabilities Research & Practice, 18(1), 36-51.
2. Kim, A., Vaughn, S. Wanzek, J. & Wei, S. (2004). Graphic organizers and their effects on reading comprehension of students with LD. Journal of Learning Disabilities, 37, 105-118.
3. McEwan, S. & Myers, J. (2002). Graphic organizers: Visual tools for learning. Orbit, 32(4), 30-34.
4. Winn, W. (1991). Learning from maps and diagrams. Educational Psychology Review, 3, 211-247.
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Information on Types of Graphic Organizers
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Reviewed by: Carly Guberman
26 = 64
25 = 32
24 = 16
23 = 8
22 = 4
21 = 2
Figure 1. Left column of a graphic organizer for teaching negative integer exponents (Ives & Hoy, 2003).
26 = 642-6 = 1/64
25 = 322-5 = 1/32
24 = 162-4 = 1/16
23 = 82-3 = 1/8
22 = 42-2 = 1/4
21 = 22-1 = ½
20 = 1
Figure 2. Completed graphic organizer for teaching negative integer exponents (Ives & Hoy, 2003).
III II I
Figure 3. Blank graphic organizer for solving systems of linear equations in three variables (Ives & Hoy, 2003).
IIIII I
2x +4y+2z=16-2x-3y+z=-5
2x+2y-3z=-3
Figure 4. Graphic organizer for solving systems of linear equations in three variables as it may appear after the original equations have been entered (Ives & Hoy, 2003).
IIIII I
2x +4y+2z=16-2x-3y+z=-5
2x+2y-3z=-3 / y+3z=11
-y-2z=-8 / z=3
2x+4(2)+2(3)=16
2x+14=16
2x=2
x=1 / y+3(3)=11
y+9=11
y=2 /
z=3
Figure 5. A completed graphic organizer for solving systems of linear equations in three variables (Ives & Hoy, 2003).