A review of popular approaches to teaching mathematics that emphasize nonverbal skills, such as using manipulatives or problem–solving schema, shows that they are often not immediately applicable to some important areas of secondary algebra. However, graphic organizers in various forms have been widely suggested and researched as an intervention approach to improve reading comprehension. In this article, suggestions for modifying graphic organizers to make them more applicable to teaching higher–level mathematics concepts and procedures are proposed. Using an appropriately modified graphic organizer to teach higher–level mathematics skills may help students with relatively weak verbal skills and strong nonverbal reasoning skills to be more successful in mathematics. This subgroup of students with learning disabilities has been consistently identified in several schemes for subtyping mathematical disorders developed from empirical evidence. Excerpts of two sample scripted lessons based on this approach, one for the solving of systems of linear equations in three variables and one for the concept of negative integer exponents, are included.
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References
1.
AlvermannD. E., & BoothbyP. R. (1986). Children's transfer of graphic organizer instruction. Reading Psychology: An International Quarterly, 7, 78–100.
2.
AlvermannD. E., BoothbyP. R., & WolfeJ. (1984). The effect of graphic organizer instruction on fourth graders’ comprehension of social studies text. Journal of Social Studies Research, 8, 13–21.
3.
AlvermannD. E., & SwaffordJ. (1989). Do content area strategies have a research base?Journal of Reading, 32, 388–394.
4.
AusubelD. P. (1960). The use of advanced organizers in the learning and retention of meaningful verbal behavior. Journal of Educational Psychology, 51, 267–272.
5.
BakkerD. J., BoumaA., & GardienC. J. (1990). Hemisphere–specific treatment of dyslexia subtypes: A field experiment. Journal of Learning Disabilities, 23, 433–438.
6.
BaroodyA. J. (1996). An investigative approach to the mathematics instruction of children classified as learning disabled. In ReidD. K., HreskoW. P., & SwansonH. L. (eds.), Cognitive approaches to learning disabilities (3rd ed., pp. 545–615). Austin, TX : Pro–Ed.
7.
BarrenR. F., & SchwartzR. M. (1984). Graphic postorganizers: A spatial learning strategy. In HolleyC. D. & DansereauD. F. (eds.), Spatial learning strategies (pp. 275–289). New York : Academic Press.
8.
BeanT. W., SingerH., SorterJ., & FrazeeC. (1986). The effect of metacognitive instruction in outlining and graphic organizer construction on students’ comprehension in a tenth–grade world history class. Journal of Reading Behavior, 18, 153–169.
9.
BenderW. N. (1996). Teaching students with mild disabilities. Boston, MA : Allyn & Bacon.
10.
BleyN. S., & ThorntonC. A. (1995). Teaching mathematics to students with learning disabilities (3rd ed.). Austin, TX : Pro–Ed.
11.
BoothbyP. R., & AlvermannD. E. (1984). A classroom training study: The effects of graphic organizer instruction on fourth graders’ comprehension. Reading World, 23, 325–339.
12.
BrightG. W. (1986). Using manipulatives. Arithmetic Teacher, 33, 4.
13.
Cain–CastonM. (1996). Manipulative queen. Journal of Instructional Psychology, 23, 270–274.
DeLucaJ. W., RourkeB. P., & Del OttoJ. E. (1991). Subtypes of arithmetic–disabled children: Cognitive and personality dimensions. In RourkeB. P. (Ed.), Neuropsychological validation of learning disability subtypes. New York : Guilford.
16.
DeshlerD. D., EllisE. S., & LenzB. K. (1996). Teaching adolescents with learning disabilities. Denver, CO : Love Publishing.
17.
DeshlerD. D., & LenzB. K. (1989). The strategies instructional approach. International Journal of Disability, Development and Education, 36, 203–224.
18.
DeshlerD. D., & SchumakerJ. B. (1993). Strategy mastery by at–risk students: Not a simple matter. Elementary School Journal, 94, 153–167.
19.
DunstonP. J. (1992). A critique of graphic organizer research. Reading Research and Instruction, 31, 57–65.
20.
DunstonP. J., & RidgewayV. G. (1990). The effect of graphic organizers on learning and remembering information from connected discourse. Forum for Reading, 22, 15–23.
21.
EarleR. A. (1969). Use of the structured overview in mathematics classes. In HerberH. L. & SandersP. L. (eds.), Research in reading in the content areas: First year report (pp. 49–58). Syracuse, NY : Syracuse University, Reading and Language Arts Center.
22.
EstesT. H., MillsD. C., & BarronR. F. (1969). Three methods of introducing students to a reading–learning task in two content subjects. In HerberH. L. & SandersP. L. (eds.), Research in reading in the content areas: First year report (pp. 40–47). Syracuse, NY : Syracuse University, Reading and Language Arts Center.
23.
FiedorowiczC. A. M., & TritesR. L. (1991). From theory to practice with subtypes of reading disabilities. In RourkeB. P. (Ed.), Neuropsychological validation of learning disability subtypes. New York : Guilford.
24.
FueyoV., & BushellD.Jr. (1998). Using number line procedures and peer tutoring to improve the mathematics computation of low–performing first graders. Journal of Applied Behavior Analysis, 31, 417–430.
25.
FusonK. C., & WillisG. B. (1989). Second graders’ use of schematic drawings in solving addition and subtraction word problems. Journal of Educational Psychology, 81, 514–520.
26.
GearyD. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362.
27.
GearyD. C. (2000). Mathematical disorders: An overview for educators. Perspectives, 23, 6–9.
28.
GriffinC. C., & TulbertB. L. (1995). The effect of graphic organizers on students’ comprehension and recall of expository text: A review of the research and implications for practice. Reading & Writing Quarterly: Overcoming Learning Difficulties, 11, 73–89.
29.
GrossenB., & CarnineD. (1990). Diagramming a logic strategy: Effects on difficult problem types and transfer. Learning Disability Quarterly, 13, 168–182.
30.
HarrisC. A., MillerS. P., & MercerC. D. (1995). Teaching initial multiplication skills to students with disabilities in general education classrooms. Learning Disabilities Research & Practice, 10, 180–195.
31.
HegartyM., & KozhevnikovM. (1999). Types of visual–spatial representations and mathematical problem–solving. Journal of Educational Psychology, 91, 684–689.
32.
HutchinsonN. L. (1993). Effects of cognitive strategy instruction on algebra problem–solving of adolescents with learning disabilities. Learning Disability Quarterly, 16, 34–63.
33.
HynesM. C. (1986). Selection criteria. Arithmetic Teacher, 33, 11–13.
34.
JitendraA. K., GriffithC. C., McGoeyK., GardillM. C., BhatP., & RileyT. (1998). Effects of mathematical word problem–solving by students at risk or with mild disabilities. Journal of Educational Research, 91, 345–355.
35.
JitendraA. K., & HoffK. (1996). The effects of schema–based instruction on the mathematical word–problem–solving performance of students with learning disabilities. Journal of Learning Disabilities, 29, 422–431.
36.
JitendraA. K., HoffK., & BeckM. M. (1999). Teaching middle school students with learning disabilities to solve word problems using a schema–based approach. Remedial & Special Education, 20, 50–64.
37.
JonesE. D., WilsonR., & BhojwaniS. (1998). Mathematics instruction for secondary students with learning disabilities. In RiveraD. P. (Ed.), Mathematics education for students with learning disabilities (pp. 155–176). Austin, TX : Pro–Ed.
38.
KennedyK. M., & BackmanJ. (1993). Effectiveness of the Lindamood Auditory Discrimination in Depth program with students with learning disabilities. Learning Disabilities Research & Practice, 8, 253–259.
39.
KiewraK. A., DuboisN. F., ChristianD., & McShaneA. (1988). Providing study notes: Comparison of three types of notes for review. Journal of Educational Psychology, 80, 595–597.
40.
KoscL. (1974). Developmental dyscalculia. Journal of Learning Disabilities, 7, 164–177.
41.
KoscinskiS. T., & GastD. T. (1993). Use of constant time delay in teaching multiplication facts to students with learning disabilities. Journal of Learning Disabilities, 26, 533–567.
42.
LarsonR. E., BoswellL., KanoldT. D., & StiffL. (2001). Algebra I. Evanston, IL : McDougal Littel.
43.
MacciniP., & HughesC. A. (2000). Effects of a problem–solving strategy on the introductory algebra performance of secondary students with learning disabilities. Learning Disabilities Research & Practice, 15, 10–21.
44.
MarshL. G., & CookeN. L. (1996). The effects of using manipulatives in teaching mathematics problem–solving to students with learning disabilities. Learning Disabilities Research & Practice, 11, 58–65.
45.
MastropieriM. A., & ScruggsT. E. (1994). Effective instruction for special education (2nd ed.). Austin, TX : Pro–Ed.
46.
MattinglyJ. C., & BottD. A. (1990). Teaching multiplication facts to students with learning problems. Exceptional Children, 56, 438–449.
47.
McIntyreS. B., TestD. W., CookeN. L., & BeattieJ. (1991). Using count–bys to increase multiplication facts fluency. Learning Disability Quarterly, 14, 82–88.
48.
MeeseR. L. (1994). Teaching learners with mild disabilities: Integrating research and practice. Pacific Grove, CA : Brooks/Cole.
49.
MercerC. D., & MillerS. P. (1992). Teaching students with learning problems in mathematics to acquire, understand, and apply basic mathematics facts. Remedial and Special Education, 13, 19–35, 61.
50.
MillerS. P., & MercerC. D. (1993). Using data to learn about concrete–semiconcrete–abstract instruction for students with mathematics disabilities. Learning Disabilities Research & Practice, 8, 89–96.
51.
MillerS. P., & MercerC. D. (1998). Educational aspects of mathematics disabilities. In RiveraD. P. (Ed.), Mathematics education for students with learning disabilities (pp. 81–95). Austin, TX : Pro–Ed.
52.
MontagueM. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem–solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230–248.
53.
MontagueM., ApplegateB., & MarquardK. (1993). Cognitive strategy instruction and mathematical problem–solving performance of students with learning disabilities. Learning Disabilities Research & Practice, 8, 223–232.
54.
MooreD. W., & ReadanceJ. E. (1984). A quantitative and qualitative review of graphic organizer research. Journal of Educational Research, 78, 11–17.
55.
MosesR. P., & CobbC. E.Jr. (2001). Radical equations: Math literacy and civil rights. Boston, MA : Beacon Press.
56.
NaglieriJ. A., & GottlingS. H. (1997). Mathematics instruction and PASS cognitive processes: An intervention study. Journal of Learning Disabilities, 30, 513–520.
57.
PadgetS. Y. (1998). Lessons from research in dyslexia: Implications for a classification system for learning disabilities. Learning Disability Quarterly, 21, 167–178.
58.
RiceG. E. (1994). Need for explanations in graphic organizer research. Reading Psychology: An International Journal, 15, 39–67.
59.
RiveraD. P. (1998). Mathematics education and students with learning disabilities: Introduction. In RiveraD. P. (Ed.), Mathematics education for students with learning disabilities (pp. 1–31). Austin, TX : Pro–Ed.
60.
RobertsonJ. (2000). Neuropsychological intervention in dyslexia: Two studies on British pupils. Journal of Learning Disabilities, 33, 137–148.
61.
RobinsonD. H. (1998). Graphic organizers as aids to text learning. Reading Research and Instruction, 37, 85–105.
62.
RourkeB. P. (1989). Nonverbal learning disabilities: The syndrome and the model. New York : Guilford Press.
63.
RourkeB. P. (1993). Arithmetic disabilities, specific and otherwise: A neuropsychological perspective. Journal of Learning Disabilities, 26, 214–226.
64.
RourkeB. P., & ConwayJ. A. (1998). Disabilities of arithmetic and mathematical reasoning: Perspectives from neurology and neuropsychology. In RiveraD. P. (Ed.), Mathematics education for students with learning disabilities (pp. 59–79). Austin, TX : Pro–Ed.
65.
SaxonJ. H.Jr. (1997). Algebra 2: An incremental development (2nd ed.). Norman, OK : Saxon.
66.
SharmaM. C. (1986). Progress of Dr. Ladislav Kosc's work on dyscalculia. Focus on Learning Problems in Mathematics, 8, 47–119.
67.
StellingwerfB. P., & Van LieshoutE. C. D. M. (1999). Manipulatives and number sentences in computer aided arithmetic word problem–solving. Instructional Science, 27, 459–476.
68.
SwaffordJ., & AlvermannD. E. (1989). Postsecondary research base for content strategies. Journal of Reading, 33, 164–169.
69.
SwansonH. L., HoskynM., & LeeC. (1999). Interventions for students with learning disabilities: A meta–analysis of treatment outcomes. New York : Guilford Press.
70.
ThompsonP. (1994). Concrete materials and teaching for mathematical understanding. Arithmetic Teacher, 41, 556–560.
71.
UttalD. H., & DeLoacheJ. S. (1997). Manipulates as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18, 37–54.
72.
WeaverJ. F. (1950). Some areas of misunderstanding about meaning in arithmetic. Elementary School Journal, 51, 35–41.
73.
WillisG. B., & FusonK. C. (1988). Teaching children to use schematic drawings to solve addition and subtraction word problems. Journal of Educational Psychology, 80, 192–201.
74.
WinnW. (1991). Learning from maps and diagrams. Educational Psychology Review, 3, 211–247.
75.
WoodD. A., RosenbergM. S., & CarranD. T. (1993). The effects of tape–recorded self–instruction cues on the mathematics performance of students with learning disabilities. Journal of Learning Disabilities, 26, 250–258, 269.
76.
ZawaizaR. W., & GerberM. M. (1993). Effects of explicit instruction on mathematics word–problem–solving by community college students with learning disabilities. Learning Disability Quarterly, 16, 64–79.
