VISUAL PREFERENCES IN MATHEMATICS TEACHING AND LEARNING

1. VISUAL PREFERENCES IN MATHEMATICS TEACHING AND LEARNING Norma Presmeg Illinois State University

2. PROTOCOL FOR A RESEARCH STUDY Chapter 1: Introduction. Describe a real problem. Preliminary statement of research questions. Chapter 2: Situate the problem in theoretical contexts (like searchlights that illuminate the problem). End of Chapter: State research questions in detail. Chapter 3: Methodology (descriptive and argumentative discourse). Description of pilot study (which tests instruments - not for collection of data). Description of the main study, including a very explicit description of the procedures.

3. Chapter 4: Description of the results, from the data. Analysis of the results (argument), relating this analysis to Chapter 2 and the literature. How did the study contribute to the literature? Chapter 5: Implications for practice, relating these implications to Chapter 1 and the issues which were of concern. Self-critique: in retrospect, how could the study have been improved? Suggestions for future research.

4. Three early visualizers

5. Senior Aptitude Tests battery of the Institute for Psychometric Research Four tests loaded on SR-O (spatial relations and orientation) and Vz (visualization), namely, Pattern Completion, Figure Series, Spatial 2D and Spatial 3D. Stanine scores: 9 - 4%, 8 - 7% of population. Anton 8, 8, 9, 8 Robin 8, 9, 9, 9 Mark 8, 9, 9, 9

6. Central research issue: To understand more about the circumstances which affect the visual pupil’s operating in his or her preferred mode, and how the teacher facilitates this or otherwise (Presmeg, 1985).

7. Definition of a visual image A visual image is a mental construct depicting visual or spatial information.

8. Definition of a visualizer A visualizer is a person who prefers to use visual methods (including visual imagery) to solve problems that are capable of solution by visual and nonvisual methods.

9. Theory: Krutetskii’s types 1.Analytic: very strong verbal-logical component predominating over a weak visual-pictorial component; spatial concepts weak; cannot use visual supports in problem solving, but feels no need. 2. Geometric: very strong visual-pictorial component, predominating over an above average verbal-logical component; spatial concepts very good; can use visual supports in problem solving, and feels a need. 3. Harmonic: strong verbal-logical and strong visual-pictorial components in equilibrium; spatial concepts good. Subtype a (abstract harmonic): can use visual supports in problem solving, but prefers not to. Subtype b (pictorial harmonic): can use visual supports in problem solving, and prefers to do so.

10. “The fact is that the graphic schemes used by these pupils are a unique synthesis of concrete and abstract… A graphic scheme is quite an abstract form of visuality - the graphic expression of mathematical relationships is abstracted and generalized. Although it does constitute a definite type of visuality, its visuality and concreteness are still specific and relative… In a drawing - a graphic scheme - a schematization and a certain generalization of a visual image occur.”

11. “Supporting thought by, and even ‘binding’ it to, such a generalized visual image cannot prevent generalized thinking. In such a case, this image is in a certain sense the ‘bearer’ of the sense and content of an abstract concept. The ‘geometer’ pupils feel the need to interpret a problem on a general plane, but for them this general plane is still supported by such images. In this they differ from pupils of little ability - for whom visual images really bind thinking, push it onto a concrete plane, and hinder the interpretation of a problem in general form” (Krutetskii, 1976, p. 325-326)

12. Mathematical Processing Instrument: A A-1. One day John and Peter visit a library together. After that, John visits the library regularly every two days, at noon. Peter visits the library every three days, also at noon. If the library opens every day, how many days after the first visit will it be before they are, once again, in the library together? A-2. Altogether there are eight tables in a house. Some of them have four legs and the others have three legs. Altogether they have 27 legs. How many tables are there with four legs? A-3. A straight path is divided into unequal sections. The length of the second section is half the length of the first section. What fraction of the whole path is the first section?

13. MPI: Section A (continued) A-4. Only four teams took part in a football competition. Each team played against each of the other teams once. How many football matches were there in the competition? A-5. At each of the two ends of a straight path a man planted a tree, and then every five metres along the path (on one side only) he also planted another tree. The length of the path is 25 metres. How many trees were planted on the path altogether? A-6. One day a third of the potatoes in a storeroom were taken out of it. If 80 kilograms of potatoes were left in the storeroom, how many kilograms of potatoes were in the storeroom at first?

14. MPI Section A: Solutions for A-1 Solution 1: I drew a diagram representing days after the first visit. John Peter From the diagram it can be seen that once again they will be in the library together six days after the first visit. Solution 2: I used the same method as solution 1, but in my mind. Solution 3: I used examples, e.g., start on Monday, …together again on Sunday, i.e., six days later. Solution 4: John visits the library again on the 3rd, 5th, 7th day, etc. Peter visits on the 4th, 7th day, etc. So they will be together on the 7th day, i.e., 6 days after the first visit.

15. Solutions for A-2. Solution 1: I solved the problem by trial and error, e.g., by drawing a grid with three columns. 3(x4) plus 5(x3) = 27. Solution 2: I used symbols and equations, e.g., let the number of tables with four legs be x. Then 4x+3(8-x) = 27. Then I solved, and found that x = 3. Solution 3: I drew a diagram representing the tables´ legs, and then grouped them into groups of 4 and groups of 3. Solution 4: I drew 8 tables, first as if they all had 3 legs, and then I added legs until the total number of legs was 27. Solution 5: Reasoning that 9 tables would be needed if all had 3 legs, I redistributed the legs of the extra table. Thus 3 tables have four legs, and 5 have three legs. Solution 6: I wrote down numbers 1-8. With 4 under each one the total was 32. I adjusted with 3s to get 27.

16. Solutions for A-3. Solution 1: I drew a diagram representing the path. From the diagram, the first section is 2/3 of whole path. Solution 2: Same method as solution 1, but in my mind. Solution 3: Reasoning that there are 3 equal parts, the first section contains 2 parts and the second 1. Thus the first section is 2/3 of the whole path (using no imagery or diagram at all). Solution 4: I solved this problem using examples. Suppose the length of the first section is 50 meters. Then the second is 25 meters, and the whole is 75 meters. This means that the first section (50 meters) is 2/3 of the whole path.

17. Solutions for A-4. Solution 1: I solved the problem by reasoning. Each team played against 3 other teams once. 4 teams x 3 matches, i.e., that gives 12 matches. But in that way each match has been counted twice, so divide 12 by 2, giving 6 matches. Solution 2: I listed all the match pairs and then counted them. If the teams are A, B, C, and D, then the match pairs are AB, AC, AD, BC, BD, and CD. There were 6 matches. Solution 3: I drew a diagram representing the matches, and then counted them. There were 6 matches.

18. Solutions for A-4 (continued). Solution 4: I solved this problem as in solution 3, but I saw the diagram in my mind (and did not draw it on paper). Solution 5: I reasoned as follows: because there were four teams, there could be only two matches in each round. Altogether there were three rounds, in order for each team to play each of the other teams once. That gives 3x2, which is 6 matches. Solution 6: I drew a matrix or grid representing the matches.

19. Solutions for A-5. Solution 1: I solved the problem in this way. Every 5 meters along the path a tree was planted. This means that the path was divided into 5 equal parts (25÷5). Every part corresponds to one tree, but there is an extra tree at the end, i.e., six trees. Solution 2: I imagined the path and the trees, and then counted the trees in my mind, and found 6 trees. Solution 3: I drew a diagram representing the path and the trees, and counted 6 trees.

20. Solutions for A-6. Solution 1: One third of the potatoes were taken out, so two thirds were left in the storeroom. So the amount left was twice the amount taken out. 80 kg was left, so 40 kg was taken out. Thus 80+40=120 kg were in the storeroom. Solution 2: I used symbols and equations, e.g., let the amount of potatoes at first be x kg. One third of x were taken out, thus two thirds is 80. So x=120, i.e., 120 kg. Solution 3: I drew a diagram representing the potatoes, and could see that there were 80+40, i.e., 120 kg of potatoes. 80 kg one third Solution 4: I used the same method as solution 3, in my mind (and not on paper).

21. Frequency distribution of visuality scores

22. Mathematical Processing Instrument: Section C C-1. The distance that a tourist traveled by train is 150 km longer than the course he traveled by steamer, and 750 km longer than his journey on foot. Determine the length of his entire trip if it is known that the distance he covered on foot was 1/3 the distance he covered by steamer.

23. MPI: Section C (continued) C-2. What time is it now if the time that has passed since noon constitutes a third of the time that remains until midnight?

24. MPI: Section C (continued) C-3. A boy walks from home to school in 30 minutes, and his brother takes 40 minutes. His brother left 5 minutes before he did. In how many minutes will he overtake his brother?

25. MPI: Section C (continued) C-4. An older brother said to a younger, “Give me eight walnuts, then I will have twice as many as you do.” But the younger brother said to the older one, “You give me eight walnuts, then we will have an equal number.” How many walnuts did each have?

26. MPI: Section C (continued) C-5. Two candles have different lengths and thicknesses. The long one can burn 3½ hours; the short one, 5 hours. After burning for two hours, the candles are equal in length. What was the ratio of the short candle’s height to the long candle’s height originally?

27. MPI: Section C (continued) C-6. A train passes a telegraph pole in ¼ minute, and in ¾ minute it passes completely through a tunnel 540 metres long. What is the train’s speed in metres per minute and its length in metres?

28. Classroom aspects facilitative of use of visual imagery • Controlled, relaxed, unhurried; • Subject presented pictorially: inessential diagrams; • Use of teacher’s imagery: number forms; gesture; • Use of students’ imagery: imagery instructions; thinking in moving pictures;

29. Classroom aspects facilitative of use of visual imagery • Use of motor component: arm/finger/body movement; • use of manipulatives; • Use of color: by teacher; by students; • Teaching which is not rule-bound: use of intuition; pattern-seeking methods; delayed use of symbolism; deliberately creates cognitive conflict; shows and accepts alternative methods.

30. Teaching visuality 13 teachers were chosen based on their mathematical visuality (MV) scores. Teaching visuality (TV) scores for these teachers were based on 12 classroom aspects facilitative of visual thinking. MV and TV scores were only weakly correlated (Spearman’s rho = 0.404)

32. Kinds of visual imagery • Concrete, pictorial imagery (52) • Pattern imagery (18) • Memory images of formulas (32) • Kinaesthetic imagery (16) • Dynamic (moving) imagery (2) Parentheses: number of visualizers (out of 54) using this type of imagery.

33. Effectiveness of visual imagery • Vivid imagery of all types has mnemonic advantages. • Concrete imagery is effective in alternation with nonvisual modes such as analysis, logic. • Dynamic imagery is potentially effective. • Imagery that serves an abstract function is potentially effective: “concretizing the referent” (Allison); pattern imagery (Crispin).

34. Difficulties in imagistic thinking • The one-case concreteness may tie thought to irrelevant details, or introduce false information. • A prototypical image may induce inflexible thinking. • An uncontrollable image may persist, thus preventing more fruitful avenues of thought. • Imagery needs to link with rigorous analytical thought processes to be effective.

35. Coming home to the beauty of mathematics as it connects. “Only by the form, the pattern, can words or music reach the stillness, as a Chinese jar still moves perpetually in its stillness.” T.S. Eliot: Burnt Norton