The use of visualization for learning and teaching mathematics

Transcription

1 Te use of visualization for learning and teacing matematics Medat H. Raim Radcliffe Siddo Lakeead University Lakeead University Tunder Bay, Ontario Tunder Bay, Ontario CANADA CANADA Astract In tis article, ased on Dissection-Motion-Operations, DMO (decomposing a figure into several pieces and composing te resulting pieces into a new figure of equal area), a set of visual representations (models) of matematical concepts will e introduced. Te visual models are producile troug manipulation and computer GSP/Cari software. Tey are ased on te van Hiele s Levels (van Hiele, 1989) of Tougt Development; in particular, Level 2 (Informal Deductive Reasoning) and level 3 (Deductive Reasoning). Te asic teme for tese models as een visual learning and understanding troug manipulatives and computer representations of matematical concepts vs. rote learning and memorization. Te tree geometric transformations or motions: Translation, Rotation, Reflection and teir possile cominations were used; tey are illustrated in several texts. As well, a set of tree commonly used dissections or decompositions (Eves, 1972) of ojects was utilized. Introduction and Background Wy Visualization? In te literature, visualization as een descried as te creation of a mental image of a given concept (Kosslyn, 1996). As suc, and from te teacing point of view, visualization seems to e a powerful metod to utilize for enancing students understanding of a variety of concepts in many disciplines suc as computer science, cemistry, pysics, iology, engineering, applied statistics and matematics. Specifically, tere are many reasons tat sustantiate te use of visualization for learning and teacing of matematics at all levels of scooling, from elementary to university passing troug te middle and ig scool levels. Te literature also indicates tat te activity of seeing differently is not a self-evident, innate process, ut someting created and learned (Witeley, 2000; Hoffmann, 1998). As cognitive science suggests, we learn to see; we create wat we see; visual reasoning or seeing to tink is learned, it can also e taugt and it is important to teac it (Witeley, 2004, p. 3; Hoffmann, 1998). Tus, teacers wo ave learned and ecame skillful in te use of visualization and seeing to tink would e ale to reinforce matematical concepts and improve te learning process in te classroom. Te literature furter suggests tat rain imaging, neuroscience, and anecdotal evidence confirm tat visual and diagrammatic reasoning is cognitively distinct from veral reasoning (Butterwort, 1999). Also, studies of cognition suggest tat visuals are widely used, in a variety of ways, y mat users and matematicians (Brown, 1998). Moreover, Witeley (2004) stated tat I work wit future and in-service teacers of matematics: elementary, secondary and post-secondary. Tey are surprised to learn tat modern astract and applied matematics can e intensively visual, comining a very ig level of reasoning wit a solid grounding in te senses (p. 1). Visualization as Justification and Explanation Visual justification in matematics refers to te understanding and application of matematical concepts using visually ased representations and processes presented in diagrams, computer grapics programs and pysical models. Tere are several distinct caracteristics of visual justification (reasoning) in many disciplines: 496

2 Visual justification in solving prolems is central to numerous fields eside matematics suc as statistics, engineering, computer science, iology and cemistry. Visual reasoning is not restricted to geometry or spatially represented matematics. All fields of mat contain processes and properties tat provide visual patterns and visually structured reasoning. Cominatorics is very ric in visual patterns. Algera and symolic logic rely on visual form and appearance to evoke appropriate steps and comparisons. Visually ased pedagogy opens matematics to students wo are oterwise excluded. Studies suggest tat students (and adults) wit autism and dyslexia may rely more on visual reasoning tan veral reasoning (Grandin, 1996; West, 1998; Witeley, 2004; Gooding, 2009). Types of Visual Representations Tis article covers te following visual representations, (1) Diagrams, (2) Computer grapics programs and (3) Pysical models. (1) Diagrams: Visually ased Representations and Processes Visually ased representations and processes are utilized in a variety of mat sujects. Tis article will focus on te following sujects: a) Geometry ) Functions and Trigonometry c) Numer Patterns, and d) Algera. a) Geometry: Due to space limits, te focus will e on visual representations for te derivation of area formulas of all commonly used polygons in scool matematics. Tus, a numer of examples will e presented. Trougout tese derivations, Dissection-Motion- Operations, DMO, were utilized. Te DMO process consists of two components: (1) Decomposition of a sape into parts y Dissection operations (vertical, orizontal, olique), (2) Composition of te parts into new sapes of equal area troug Motion operations (translation, rotation, reflection). DMO were primarily introduced for 2-D sape transforms among polygons (Raim, 1986; Raim, Bopp, & Bopp, 2005; Raim, Sawada, & Strasser, 1996; Raim & Sawada, 1986, 1990); tey were extended for 3-D prisms (Raim, 2009). In 2-D, te area of te rectangle is taken as given, A = ase eigt; it can e verified y te graps in Figure 1. square unit Figure 1: Based on te square unit, te Area of te rectangle = Examples Below are a numer of examples for visual representations in 2-D. Example 1: Figure 2 elow sows were te area formula of any triangle, Area of = ½, did come from troug DMO applied on eac type of te triangle. 497

3 1/2 1/2 Figure 2: Troug DMO, Area of = area of rectangle = ½ Example 2: Figure 3 elow sows were te area formula of a trapezoid did come from. Te visual representations elow sow tat te Area of te trapezoid = area of te resulting. Tat is, Area of te trapezoid wit eigt and ases & 2 = area of wit eigt and ase ( + 2 ). Tus, area of te trapezoid = area of = ½ = ½ ( + 2 ). 1/ Figure 3: Area of te trapezoid = area of = ½ ( + 2 ) Example 3: Figure 4 elow sows te romus visual representations of its area derivation troug DMO were X = orizontal diagonal and Y = vertical diagonal of te romus. 2 Y 1/2X 498

4 Y 1/2Y Figure 4: Area of te romus = area of rectangle = ase eigt = ½ X Y ) Functions and Trigonometry: Example 4. In tis example, f(x), g(x) and (x) were given elow using GSP. By animation troug GSP, as point C travels along te circumference of te circle B, te measures for te distance AC and angle ABC vary and te graps of te tree functions will correspondingly get in motion simultaneously. Vital oservale information aout te caracteristics of eac of te functions f, g and wit te relationsips among tem will e availale. For example, students would oserve wat would appen wen C coincides wit A? X Figure 5: As C moves, te graps of f, g and get in motion revealing crucial properties c) Numer Patterns: Among many visual numers patterns, te Symmetric Multiplication is particularly attractive (Posamentier, Smit, & Stepelman, 2009). Consider te tree symmetric multiplication patters sown in (A), (B) and (C) elow. Students can multiply (A), (B) and (C) y conventional means (calculators for cecking). After tey attempted to use conventional metod, tey may welcome a more elegant solution y considering te romic metod in (A) and (B) and explore te pyramid metod in (C). Te intention of tese patterns is to introduce interesting properties of numers multiplication. d) Algera: Example 5. Visual representations: Signs multiplications. 499

5 2. Computer Grapics Programs: E.g., GSP and Cari. For Figure 5 content, GSP was used. 3. Pysical Models: A pysical model to justify (x + y) 3 = x 3 + 3x 2 y +3xy 2 + y 3 will e displayed wenever presenting tis article. Finally, over te centuries, matematicians, pilosopers and some artists ave recognized and igligted te artistic aspects of matematics. G. H. Hardy ( ), a well-known matematician at Camridge University once stated: a matematician, like a painter or a poet, is a maker of patterns. If is patterns are more permanent tan teirs, it is ecause tey are made wit ideas (O Daffer & Clemens, 1992, p. 12). References Brown, J., (1988). Pilosopy of matematics: Introduction to a world of proofs and pictures. Routledge. Butterwort, B., (1999). Te Matematical Brain. Macmillan. Eves, H., (1972). A survey of geometry. Revised Edition, Boston: Allan and Bacon Inc. Gooding, D., Dimensions of creativity: Visualization, inferences and explanation in te sciences, Retrieved July 1, Retrieved July 1, Grandin, T., (1996). Tinking in Pictures. Vintage Books, New York. Hoffmann, D., (1998). Visual Intelligence: How we create wat we see? Norton. Kosslyn, S., (1996). Image and Brain. MIT Press. O Daffer, P., & Clemens, S., (1992). Geometry: An Investigative Approac. Menlo Park, CA: Addison-Wesley. Posamentier, A., Smit, B., & Stepelman, J., (2009). Teacing Secondary Scool Matematics (8 t Ed). Toronto: Allyn & Bacon. Raim, M. H., (2009). Dynamic geometry software-ased Dissection-Motion-Operations: A visual medium for proof and proving. Learning and Teacing Matematics Journal. Sumitted. Raim, M. H., Bopp, D., & Bopp, J., (2005). Concrete modeling of sape transforms troug Dissection- Motion-Operations (DMO): A journey among sapes in 2D & 3D plane geometry. International Journal of Learning. Volume 11, Raim, M. H., (1986). Laoratory investigation in geometry: A piece-wise congruence approac: Preliminary propositions, applications, and generalizations, International Journal of Matematical Education in Science and Tecnology. 17(4), Raim, M. H., Sawada, D., & Strasser, J., (1996, Marc). Exploring sape transforms troug cut and cover: Te oy wit te ruler. Matematics Teacing Journal, 154, Raim, M. H., & Sawada, D., (1990). Te duality of qualitative and quantitative knowing in scool geometry. International Journal of Matematical Education in Science and Tecnology, 21(2), Raim, M. H., & Sawada, D., (1986). Revitalizing scool geometry troug dissection motion operations. Scool Science and Matematics Journal, 3, West, T., (1997). In te Minds Eye. Prometeus Books, Amerst, New York. Witeley, W., (2004). Visualization in Matematics. Witeley, W., (2000). Dynamic geometry programs and te practice of geometry. van Hiele, P. M., (1989). Structure and insigt: a teory of matematics education. Orlando: Academic Press. 500