Graphic Organizers for Teaching Advanced-Level Mathematics: Understanding Negative Integer

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 I) In mathematics, the content of graphic Organizers in advanced-level mathematics? organizers should no longer be verbal elements, Graphic organizers are visual-spatial displays such as words, phrases, and sentences. Rather, used to organize knowledge and represent content should be mathematical in nature, relationships among pieces of information. Due consisting of numbers, other types of symbols, to their visual-spatial nature, they require less expressions and equations. reliance on language skills. While other techniques used in mathematics instruction, II) Advanced mathematics skills emphasize such as the use of manipulatives and graphs, understanding concepts, patterns, and processes, also rely little on language skills, their as opposed to memorization of numbers, applicability to advanced-level mathematics (ie. expressions, and equations. Consequently, the algebra) has been questioned (Ives & Hoy, goal of using graphic organizers in higher-level 2003). The effectiveness of graphic organizers mathematics should be to help students in improving reading comprehension has recognize and understand relationships between therefore prompted the use of graphic organizers elements. Therefore, within graphic organizers, in teaching advanced concepts in mathematics. the way in which elements are spatially arranged should reinforce these relationships. 2. With whom can they be used? In advanced mathematics, graphic organizers III) Graphic organizers should be used to can be used by teachers to complement regular complement, not substitute for regular classroom instruction. Graphic organizers might classroom instruction. Teachers should be particularly useful for students with learning explicitly teach concepts while relating them to disabilities who have weaker language skills and graphic organizers. stronger spatial and nonverbal reasoning skills. 4. What are some examples of Graphic 3. How does one adapt Graphic Organizers Organizers that can be used in higher-level to mathematics instruction? mathematics? Many different types of graphic organizers have Suggestions for graphic organizer use in the been developed for reading comprehension and areas of negative integer exponents and solving other language instruction. Consequently, when linear equations are presented below. developing graphic organizers to use in mathematics, one can look to literature on Understanding Negative Integer Exponents reading comprehension and other literature on The goal of this graphic organizer is to present strategies for creating graphic organizers (eg. negative integer exponents as a meaningful Winn, 1991). However, there are three concept which naturally builds on the students’ important points to remember when applying previous knowledge of positive exponents. The graphic organizers to advanced mathematics: 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 between elements and the frame, help to interactive discussion of information related to emphasize relationships and concepts important positive integer exponents. Such information is in understanding how to solve linear equations. visually laid out in a column of increasing [For more details and specific scripts regarding exponents and multiplying values, as shown in this method, see Ives & Hoy (2003)]. Figure 1. The teacher encourages the students to point out patterns in this column. For example, 5. In what types of settings should Graphic as one goes down the column, exponents Organizers be used? decrease by one and values are divided by 2. Teachers can use graphic organizers to facilitate Negative integer exponents are then introduced classroom learning. Once understood, they can in a second parallel column, as shown in Figure be applied independently by students. 2, and students are encouraged to apply the rules used with positive integer exponents (ie. as 6. To what extent has research shown exponents decrease by 1, divide the previous Graphic Organizers to be useful in higher- value by 2). The teacher then encourages level mathematics? students to look for more patterns in the The effectiveness of graphic organizers in organizer. It becomes apparent that exponents in teaching higher-level mathematics has been the right-hand column are the opposite of verified informally in classrooms. Systematic exponents in the left-hand column and that research on the graphic organizer for solving values in each column are reciprocals of each linear equations (described above) is currently other. With this technique, concepts are underway to determine whether its use can lead communicated by the relative positioning of to improvement on mathematics achievement elements within the organizer. [For more details measures for students with learning disabilities and specific scripts regarding this method, see (Ives & Hoy, 2003). Initial evidence from this Ives & Hoy (2003).] study suggests that students find the graphic organizers useful. Solving Systems of Three Linear Equations with Three Variables References The frame suggested for this graphic organizer 1. Ives, B. & Hoy, C. (2003). Graphic is shown in Figure 3. The frame is divided into Organizers Applied to Higher-Level Secondary cells. Again, the teacher engages the students in Mathematics. Learning Disabilities Research an interactive discussion as they are guided & Practice, 18(1), 36-51. through the various steps to solving the 2. Kim, A., Vaughn, S. Wanzek, J. & Wei, S. equations while using the graphic organizer. (2004). Graphic organizers and their effects on The equations are first placed in the cell in the reading comprehension of students with LD. top left corner of the graphic organizer (see Journal of Learning Disabilities, 37, 105-118. Figure 4). They are then worked through from 3. McEwan, S. & Myers, J. (2002). Graphic cell to cell in a clockwise direction. Equations organizers: Visual tools for learning. Orbit, are combined in the top row and solved in the 32(4), 30-34. bottom row. The Roman numerals on top of 4. Winn, W. (1991). Learning from maps and each column correspond to the number of diagrams. Educational Psychology Review, 3, variables within the equations being worked. 211-247. Thus the Roman numerals allude to one of the goals of solving linear equations – combining Websites: equations so that they contain fewer variables. Information on Types of Graphic Organizers Figure 5 represents the completed graphic http://www.sdcoe.k12.ca.us/score/actbank/torganiz.htm http://www.writedesignonline.com/organizers/ organizer. The relative positions between More Information on Graphic Organizers Can Be Found elements, as well as the relative positions In Recommendations For “Writing/Spelling” On This Website: http://www.oise.utoronto.ca/depts/hdap/report_writer/Spe Reviewed by: Carly Guberman ll.htm

26 = 64 III II I 25 = 32

24 = 16

23 = 8

22 = 4

21 = 2 Figure 3. Blank graphic organizer for solving Figure 1. Left column of a graphic organizer for systems of linear equations in three variables teaching negative integer exponents (Ives & (Ives & Hoy, 2003). Hoy, 2003). III II I 2x +4y+2z=16

-2x-3y+z=-5 26 = 64 2-6 = 1/64

25 = 32 2-5 = 1/32 2x+2y-3z=-3

24 = 16 2-4 = 1/16

23 = 8 2-3 = 1/8

22 = 4 2-2 = 1/4 Figure 4. Graphic organizer for solving systems 21 = 2 2-1 = ½ of linear equations in three variables as it may appear after the original equations have been 0 2 = 1 entered (Ives & Hoy, 2003). Figure 2. Completed graphic organizer for teaching negative integer exponents (Ives & III II I Hoy, 2003). 2x +4y+2z=16 y+3z=11 -2x-3y+z=-5 z=3 -y-2z=-8 2x+2y-3z=-3 2x+4(2)+2(3)=16 y+3(3)=11 2x+14=16 y+9=11 2x=2 x=1 y=2 z=3 Figure 5. A completed graphic organizer for solving systems of linear equations in three variables (Ives & Hoy, 2003).