most difficult shape for the children to cover was an equilateral triangle with a point facing down. Familiarity with the shape to be covered seemed to increase.
Mathematics Education Research Journal
1998/ Vol. 10/ No. 3/28-41
Covering Shapes- with Tiles: Primary Students' Visualisation and Drawing Kay Owens
Lynne Outhred
University a/Western Sydney
Macquarie University
Students' early area· concepts were llwestigated by an analysis of responses to a worksheet of items that involved visualising the tiling of given figures. Students in Years 2 and 4 in four schools attempted the items on three occasions and some of the students completed ten classroom spatial activities. Half the students had difficulty visualising the tiling of shapes, but students who participated in spatial activities were generally more successful iJ;l determining the number of tiles that would cover a shape. Students' drawings indicated a varying awareness of structural features such as aligimlent and tile size. Students who drew the tilings were more likely to be successful on the items involving trapezia. The tiling items were part of a test of spatial thinking/ Thinking About 20 Shapes, and scores on the overall test were very highly correlated with results for the tiling items.
Young children often hear the word area referring to place, and may think of area as somewhere to go-for example, the assembly area or the reading areawithout considering it as a region. They do not seem to realise that such regions· are two-dimensional (2D) spaces enclosed by boundaries and that they can be covered with units (e.g., sheets of newspaper). Students may have covered small regions such as desks, books, and chairs with informal units and perhaps compared the two by counting the number of units needed to cover them. Such activities, intended to be introductory to the concept of area measurement, may in fact confuse studen~s. The use of irregular shapes and informal units (e.g., potato prints) may result in the activity being dominated by counting, while ideas crucial to the concept of area measurement (e.g., overlaps, gaps, and congruent units) are ignored (Outhred, 1993). A greater appreciation of the concept of covering would seem to be necessary if older students are to calculate areas meaningfully. (See Mitchelmore, 1983, and Clements, 1994, for examples of typical student difficulties with area calculations). In observations of pre-school children covering squares, rectangles, and. triangles with smaller cut-out rectangles, squares, and triangles, Mansfield and Scott (1990) have shown that students vary in their ability to choose appropriate unit shapes, in their persistence, and in their turning and flipping tactics. The most difficult shape for the children to cover was an equilateral triangle with a point facing down. Familiarity with the shape to be covered seemed to increase success on the task. In a study by Wheatley and Cobb (1990), students were asked to cover a large square by selecting shapes from a collection comprising a square, several triangles, and a parallelogram. Some students chose only the parallelogram and tried to cover the square with it, an approach which suggested to Wheatley and Cobb that the students were matching lengths. Alternatively, students may have
Covering Shapes with Tiles
29
chosen the shape that appeared to be largest. Other errors included leaving gaps, . especially on the sides, and overlapping pieces or the sides of the square. The types of materials used to cover areas have been found to affect students' performance on rectangular area tasks. Doig, Cheeseman, and Lindsey (1995) found that students were more successful when they used wooden tiles than when they used paper squares. The use of paper squares revealed inadequate understandings of area because students were more likely to overlap or leave gaps between the paper squares. Consequently, practice· in tiling with rigid materials may not help students' understandings of area (Outhred, 1993). If mathematical concepts are to emerge, it seems important that concrete experiences of covering areas should engage students' visual imagery and . analysis (Owens, 1994b) and should also involve student-student and student'teacher interaction about the ideas needing development (Hart & Sinkinson, 1988; . Owens, 1994a). Drawing may be one way of linking experiences with concrete materials to students' mental models of tessellations. Several researchers (e.g~, Mitchelmore, 1983; Outhred, 1993; Outhred & Mitchelmore, 1992) have suggested that drawing is an important tool in developing students' knowledge of rectangular arrays and in making links to multiplication. Outhred (1993). found that many students had difficulties visualising or drawing tHings of square units to cover rectangles when the squares were only shown on adjacent sides of the rectangle or indicated by side marks, particularly for rectangles with large dimensions. Some students' drawings suggested that they did not understand what features of arrays were important in constructing tessellations of squares. Owens (1992, 1993b) found that students in Years 2 and 4 had difficulties imagining tilingsof squares, rectangles, and triangles to cover larger· shapes. For example, in the activities illustrated in Figure I, they had difficulty in predicting the number of smaller triangles that would be needed to cover the larger ones. Very few students commented on the amount of space covered when asked what was the same about different arrangements of five squares (pentominoes); nearly all focused on the number of tiles (Owens, 1993b). (a) Tangram triangles
(b) Pattern-block triangles
Figure 1. Shapes made during spatial activities (Owens, 1993b). The studies mentioned above, especially those by Outhred (1993) and Owens (1993b), emphasise the importance of spatial thinking and visualising when students cover and compare shapes. To learn about tiling, students need to identify suitable units, to transform shapes to other orientations, to recognise and
30
Owens & Outhred
partition shapes, and to identify key features of shapes (e.g., matching parts such as right angles or equal lengths). Similarities between drawings produced by students in the studies undertaken by Owens (1993b) and Outhred (1993) suggested that a further investigation of Owens' data would be warranted. The source was a larger study of students' spatial problem solving that consisted of three components: .(a) the development of a test of spatial thinking, Thinking About 20 Shapes; (b) a quantitative study of the performance of students on the test and the effects of a series of spatial activities on their test scores; and (c) a qualitative study of how students learn through spatial problem solving. Some data from part (b) will·be used in the current study. The aim of the current study is to investigate students' performance on the tiling items mcluded in the test Thinking About 20 Shapes. In particular, we explore how students visualise tilings with a variety of different units and seek to determine if their responses are correlated with general spatial thinking scores, participation in spatial activities, and spontaneous drawing. The research questions were as follows. 1.
What is the order of difficulty of the tiling items and do these, together with other items of the spatial thinking test, fit a single latent trait model? ·2. What are the effects of prior attempts on the test Thinking About 20 Shapes, with or without participation in a series of concrete spatial problem-:-solving activities, on students' responses to the tiling items? 3. How do students' spontaneous drawings of imagined tilings change over .time? 4. Is there any relation between students' tendency to draw a tiling pattern, or their success on an item, and their spatial thinking scores?
Method
Spatial Thinking Test The items of the Thinking About 20 Shapes test were developed after reviewing spatial abilities tests, classroom activity worksheets, and research studies. For full details, see Owens (1992). The test consisted of a number of worksheets printed in colour on A4 paper. The data for the present investigation comprise the responses to the worksheet of tiling items. These items required students to imagine tessellations to cover various figures with different unit shapes. Items in other sections of the test required students to recognise shapes and designs in rotated or reflected orientations, to disembed a shape within a more complex design, to complete a shape, to fold pentomino .and other shapes mentally to make three-dimensional shapes, and to recognise properties of shapes such as equality of sides and angles. Preliminary trials were undertaken and item responses analysed using item difficulty and discrimination indices. Satisfactory items were included in the final version of the test. Two different forms of the test (Form S and Form H) were
Covering Shapes with Tiles
B: "
31
Suppose you had some tiles like the shape that is under the tace. wnhout cutting or overlapping, could you tit them together to make the shape? Circle Ves or No. It yes, write the number of tiles'you need .
II 1.
3.
5.
Ves_of No
6.
Yes
or No
7.
Ves_or No
8.
Ves
or No
Figure 2. Worksheet of tiling items (Form S). used so that children sitting next to each other had different forms; in later administrations, the forms were reversed. Form S is shown in Figure 2; Form H consisted of the same eight items presented in a similar format but in a different order (6, 5; 4, 3; 2, 1; 8, 7).1 The items were introduced to the students in a class discussion using large cardboard cut-outs to illustrate the idea of covering without cutting or
1
Throughout this paper, we shall refer to items according to their item number on Form S.
32
Owens & Outhred
overlapping. Students. completed a practice example before beginning the worksheet. A score of one was given for answering correctly whether the shape could be made with the given tile. Except for Item 8, the only item for which the answer to the first part of the question was "no," a score of two was given if the student answered "yes" and gave the correct number of tiles.
Data Collection Each form of the test, Thinking About 20 Shapes, was attempted by 200 students in each of Years 2 and 4, selected from five multicultural schools in Sydney (Owens, 1992). Eighteen of these students (nine in each year selected from across the ability range) were interviewed immediately after they completed the test, and the information from this immediate recall of how they were thinking about each question was used to illuminate the data from students' test responses. From these students,.over 200 were involved in a follow-up study of the effect of spatial activities on students' spatial thinking (Owens, 1993b). Students were first matched in groups on school, year, class, and scores on the Thinking About 20 Shapes test. Some students from each group were then randomly selected to participate (working either individually or in small groups) in a series of spatial problem-solving activities based on tangrams, pattern blocks, pentominoes, and matchstick designs (Owens, 1995); the remainder acted as controls. During the ten spatial activity sessions, participants explored features of different shapes and joined pieces together to make new shapes. All the students made shapes which were enlargements of each pattern block by tessellating the pieces. Many made triangles, squares, and rectangles using either the tangram pieces or the pattern blocks (see Figure 1). Only a few made trapezia other than an isosceles trapezium. Students also made tessellations of squares with matchsticks. Students were asked how many small triangles from the tangram set were needed for each of the larger shapes-but not how many of the pattern blocks were needed for the shape enlargements. When they were asked what waS the same about all the pentomino shapes, the teacher used the terms "area" and "amount of 20 space." The spatial activities were not specifically designed to teach students to answer the tiling items. The Thinking About 20 Shapes test was given twice more after the spatial activity sessions, once as a posttest and once as a delayed posttest. No feedback was given on the correctness of their answers. Thus, all students attempted the worksheet of tiling items on three occasions over a three-month period. Results for both first and last testings were available from 137 participants and 45 nonparticipants in the spatial activities. In responding to the tiling items, 62 students made a drawing on at least one testing (21 on more than one occasion). These drawings were analysed qualitatively by looking for similarities and differences among students and over ~ time.
Covering Shapes with Tiles
33
Results
Performance on Tiling Items The difficulty levels of the eight tiling items varied slightly for the two forms, but the order of difficulty was the same (in increasing order of difficulty: Items 1 and 6; Item 2; Item 7; Items 3 and 4; Item 5; and Item 8). Two differences between the two forms may have influenced students' responses and led to differences'in item difficulty: (a) the items already attempted; and (b) the proximity of the shape to the tile (Owens, 1992). Data from the. two forms were pooled for most of the subsequent analysis. Students found Item 1 and Item 6 rather easy. On the final testing, 83% and 80% of students respectively gave the correct number of tiles, almost double the percentage for any other item. The most difficult tiling item was Item 8, with 33% .correct. This was the' only item for which a whole number of unit tiles did not cover the shape without overlap. It was clear from interviews and drawings that some students said the figure could be made with the tiles even though they were aware that the tiles would overlap or would need cutting to fit the shorter rectangular section. Table 1
Percentages of Students giving Various Responses to Selected Tiling Items at First and Ulst Testings, by Participation in Spatial Activities Item
2
3
4
5
7
Groupa
"No" response
"Yes" response, wrong number
"Yes" response, correct number
First
Last
First
Last
First
Last
P NP
30 20
21 17
42 44
39 38
29 37
40 45
P NP
45 42
35 36
27 35
32 36
28 23
32 29
P NP
52 48
36 49
19 17
16 17
28 36
48 34
P NP
57 56
53 50
15 17
17 19
27 27
30 31
P
33
26
28
25
39
48
NP
35
21
21
37
44
42
Note: The number of students completing each item varied slightly as some students did not respond to all parts of the question. ap: participants in the spatial activities, NP: non-participants.
[
~
I';
f
L r f
!
ti ~
f
34
Owens & Outhred
In the remainder of this paper, we shall concentrate on the remaining items (Items 2, 3, 4, 5, and 7), all of which required tessellations to cover the given shapes. Table 1 summarises students' responses to these items on the first and last testing occasion~. The percentages of incorrect responses indicate that covering with triangular tiles (Items 3, 4, and 5) was more difficult than with rectangular tiles (Items 2 and 7). The difficulty was particularly marked when the shape to be covered was the unfamiliar trapezium shape (Items 4 and 5). On the first testing, more than half the students thought the trapezia could not be made by tessellating the given triangles, and less than a third of them could give the correct l).umber of triangles. Although many students seemed to realise that the square (Item 7), the rectangle (Item 2), and the equilateral triangle (Item 3) could be made by tessellating the given tile, they were unable to visualise the tessellations to work out how many tiles would fit. For both the equilateral triangle (Item 3) and the square (Item 7), many students wrote 3 or 5 tiles as their answer. For the rectangle (item 2), common answers were 8 and 9, but larger answers were also givenwhich suggests that some students· disregarded the size of the tile. Students' drawings frequently indicated difficulties with size estimates. Similar results were also foundby Outhred (1993). The Rasch an~lysis (Andrich, 1988) indicated that all but one ofthe 2D items on the Thinking About 2D Shapes test, including all the tiling items except Item 8, fitted a latent trait model. (The items requiring mental folding to make a 3D shape did not fit with the 2D items.) The Cronbach alphas for the 2Ditems were 0.89 for Form Hand 0.90 for Form S, indicatirig strong internal consistency. To some ext-ent, these results are artifacts of the test construction procedure whereby test items were chosen from trial items on the basis of satisfactory item discrimination. The underlying trait could be described as 2D spatial thinking (Owens, 1992, 1993b). Multivariate analyses provided further information about the relationship of performance on the tiling items to total scores on the Thinking About 2D Shapes test. The correlation was very significant (p < 0.01) for Items 2 and 3 (rectangle and equilateral triangle) on both forms of the test and for Items 4 and 5 (trapezia) on Form S. The factors we suggested were responsible for differences in item difficulty (order of items and distance of the shape from the tile) might also have affected the relationship with the overall test score. Performance on Item 7 (square) was not significantly related to scores on the test; students' drawings suggested that it was difficult for them to visualise or draw the correct-sized rectangles to tile the square. The reason the Rasch analysis showed that this item fits an underlying trait model is that allowance was made for partially correct responses (i.e., for saying that the shape could be tiled but giving the incorrect number of tiles).
Effect ofParticipating in Spatial Activities Comparison of the last two columns in Table 1 indicates that· there was a general tendency for student performance on the tiling items to improve between
Covering Shapes with Tiles
35
the first and last testings. Students who participated in the spatial activities improved markedly more than the non-participants on Items 2 and 4. The improvement in visualising square units for a rectangle (Item 2) might be the result of visualising grids of squares during the pentomino'and matchstick-design activities as well as building large rectangles with square pattern blocks. Similarly, participants may have improved in giving the number of equilateral triangles for the trapezium (Item 4) as a result of their experiences with triangular pattern blocks, including making enlargements and making an isosceles trapezium. The' tangram sets included right-angled triangles. Most students made squares from them, but there were not enough triangles available to make a trapezium. This may be a reason why the' participants did not make any , improvement in Item 5. It should be noted that the spatial activities emphasised the use of concrete materials whereas the worksheet required visualisation. Also, few students drew copies of the designs they made during th~ spatial activities; so there was no reason to expect them to improve in their ability to use drawirigs to solve the tiling items. The improvement in participants' responses is supported by a result reported in the larger study (Owens, 1993a, 1993b). Scores on the Thinking about 2D Shapes test on the third testing occasion were analysed using initial scores on the test as a covariate. It was found that the mean score of participants in the spatial activities was significantly higher (p < 0.05) than that of the non-participants.
Children's Drawings of Tilings The students' spontaneous drawings provide further insights into their understandings of tessellations. The drawings showed the following .approaches: tiling around the sides, filling up from a corner, drawing individual tiles in rows, often sloping or getting smaller; representing rows by lines, but marking off individual tiles; maintaining good size; and, for the rectangle in Item 2, using a grid. In general, it was noted that there was reasonable consistency in the way students drew on each occasion. Additional details on the various approaches can be found in Owens and Outhred (1996). Outhred's (1993) study identified similar response categories for rectangular items, with progression from responses in which students were unable to cover the figure, to coverings in which individual squares were drawn, and finally to the use of lines to indicate rows and columns. Table 2 shows the drawings of three typical students. (The names are fictional.) The approach of drawing individual tiles beginning on an edge or at a corner is especially evident from drawings of the triangular tessellations. Rea, who had not yet established an image of rows of triangles, had particular difficulty. Nima initially drew individual tiles in rows, then joined them together using a better sized triangle. Her last attempts involve patterns and visualising without drawing. Other notable approaches were the halving of the rectangle and square by Rea and the radiating triangles drawn by Ona. Ona tried individual triangles and then looked for patterns. Nima was able to imagine the grids for the rectangle, initially pointing and later checking by drawing. Nima's third attempt
Owens & Outhred
36
Table 2
Three children's drawings of tiling items, by testing occasion Test occasion
Item 2
Item 3
Dna (Year 2) First
Second
Third
N ima (Year 2)
First
Second
Third
Rea (Year 4) First
Second
Third
Item 4
Item 5
Item 7
Covering Shapes with Tiles
37
shows only partial grid lines. Nima clearly made use of mental imagery or pointing to complete her drawing. The size of the depicted tiles improved in some instances. For example, at first Nima had difficulty drawing congruent equilateral triangles to make the trapezium. but on the third attempt she was successful. Although the size of the triangles improved on Item 3, Nima had difficulties on the last attempt and was misled by her drawing. In fact, inadequate drawings often led students to decide that the shapes could not be covered, or at least to unc3ertainty. For example, Rea apparently recognised that gaps arid variation in tile size were not appropriate and then claimed that the rectangular area could not be tiled. \
Effects ofSpontaneous Drawing on Test Performance Table 3 gives a breakdown of responses. to the tiling items on the last testing occasion according to whether the student made or did not make a drawing. The results indicate that students who drew the tessellation were more likely to decide that the figure could be tiled. However, except for Items 4 and 5 (the trapezia), they were· not more successful in working out the correct number of tiles. For Items 4 and 5, drawings may have helped students by providing either a physical diagram from which to calculate the number of tiles or a check on their visuali.,. sation of the tessellation. In Items 2 and 3 (the rectangle and equilateral triangle), the larger shapes required more tiles to cover them, and in these cases students were likely to depiCt too many tiles. In Item 7, it was apparently difficult to judge the narrow rectangular tiles However, these results have to be treated with caution as many more students solved the problem mentally than drew solutions. Table 3
Percentages of Students giving Various Responses to Selected Tiling Items at Last Testing, by Drawing Status Group
N
"No" response
Drawing No drawing
32 148
19 20
41 38
41
3
Drawing No drawing
27 154
22 38
48 31
30 31
4
Drawing No drawing
18 154
11 43
22 14
67 44
Drawing No drawing
20 159
25 56
15 18
60 26
Drawing No drawing
23 157
4 27
61 27
35 46
Item 2
5 7
"Yes" response, wrong number
"Yes" response, correct number
Note: Students who did not respond to an item have been omitted from the analysis.
42
I I r
I'
f,
f f ~
Owens & Outhred
38
For each of the five items under discussion, the students· who drew and did not draw at the third testing were compared in terms of their total score on the Thinking About 2D Shapes test. There was no significant diff.erence (p < 0.05) between the two groups. In other words, there is no evidence that better students visualised solutions and did not need to make a drawing.
Discussion The development of the concept of area is complex and this study has provided insights into components of the area concept that make it so. difficult for young students. This study, in particular the data illustrated in the diagrams in Figure 3 and in Owens and Outhred's previous paper (1996), suggests that students' responses were influenced by their cognisance of the following: (a) . maintaining tile size; (b) covering without gaps or overlaps; (c) ~ligning tiles; and (d) matching features of tiles such as angles, sides, and the triangular parts of the trapezia. Moreover, students needed to imagine or draw the relevant tessellation and to be aware of its structure. The triangle and trapezia tessellation were more difficult than the rectangular case as students had to consider the orientation of the tiling unit. Finally, students needed to be aware of the limitations of their own drawings. It seems from this study that students first consider covering a region with tiles by filling in from the sides and corners. Gradually they become more systematic, aligning tiles accurately. and attending to features such as size and shape. In the interviews, several students who had been involved in the activities with concrete materials spontaneously remarked that they had made the isosceles trapezium from equilateral triangles in class. Students improved on this item between the first and last testing, so episodic imagery (Gagne & White, 1978) may have assisted some students. Covering regions may seem a straightforward task but, when students are given specific shapes to cover, they often find this difficult. They need to recognise whether a shape will tessellate, and to partition and compare regions visually. Aspects that may affect their performance are the choice of units, shape and size of the units, the pattern of the covering, and the materials of which the units are made. Wooden tiles might be more effective for introducing the concept of covering while the use of paper tiles might elucidate students' conceptual difficulties (Doig et al., 1995). The results reported in this article indicate that the use of units other than squares would highlight the importance of covering without gaps or overlap, as well as drawing students' attention to the geometrical properties of the tiling unit. Making and then drawing tessellations of tiles of various shapes could assist students to recognise key attributes of different patterns, such as the alignment and orientation of the tiles, and how these tessellations are represented by patterns of lines and grids.
Covering Shapes with Tiles
39
Conclusion Difficulties associated with 2D spatial thinking and the complexity of tessellations for young students have been highlighted by this study. There was a slight improvement in scores on the tiling items for students who participated in 2D spatial problem-solving activities with concrete materials, such as tangrams, pentominoes, and pattern blocks, as well as some general improvement apparently resulting from practice on the tiling worksheets. The open-ended spatial activities were not specifically designed to increase performance on the worksheet items,· but the authors believe that such activities, combined with drawing and discussion, could assist students in their development of visual imagery and covering concepts. The use of drawing to develop covering concepts would seem to provide a link between activities with concrete materials and visualising tessellations of units and, therefore, may assist students to develop abstract representations of spatial arrangements. However, drawing difficulties need to be discussed so that students can appreciate that poor drawings may misrepresent the tessellation. Otherwise students may decide that shapes do not tessellate or may make inaccurate estimates because of poor drawing skills. Students need investigative learning experiences that will engage them in noting features of shapes, in analysing tiling patterns, and in assessing their drawings adequately. Such experiences would promote understanding of key attributes of tessellations-that is, that the units are all the same size and are aligned in a regular pattern. The complexity of the acquisition of counting and number concepts for young children has been shown in earlier studies (Gelman & Gallistel, 1978). This study has shown the complexity of a basic measurement concept, that of area for children. Young students progress in their conceptualisation of area measurement by developing their visualisation and knowledge of the structure of tessellations for various tiling units. They also have to develop understandings of (a) units, (b) composite units such as rows, and (c) fractional units. Limitations of their representations-modelled, drawn, or visualised-need to be discussed. Students may construct the concept of area as they develop their understanding of tessellations based on experiences with a range of tiling activities. Such activities will also lead to an exploration of ways to calculate areas and the development of area formulae for specific shapes. Further research could investigate the effects of a series of activities specifically designed to promote various aspects of the area concept and help to establish how practical activities involving tiling areas can best be used to develop the abstraction of the area concept. In addition, further study is warranted on the effects of improving students' understandings of the structures of different tessellations on their knowledge of area, on their distinguishing of area and perimeter, on their ability to estimate and measure areas, and on the development and application of formulae.
Owens & Outhred
40
References Andrich, D. (1988). Rasch models for measurement. Newbury Park, CA: Sage. Clements, M. A. (1995). The rhetoric/reality gap in school mathematics. Reflections, 20(1), 2-9. Doig, B., Cheeseman, J., & Lindsey, J. (1995). The medium is the message: Measuring area with. different media. In B. Atweh & S. Flavel (Eds.), Galtha (Proceedings of 18th annual conference of Mathematics Education Research Group of Australasia, pp. 229-234). Darwin: MERGA. Gagne, R. M., & White, R. T. (1978). Memory structures.and learning outcomes. Review of Educational Research, 48, 187-222. Gelman, R., & Gallistel, C. R. (1978). The child's ·understanding of number. Cambridge, MA: Harvard University Press.. Hart, K., & Sinkinson, A. (1988). Forging the link between practical and formal mathematics. In A. Borbas (Ed.), Proceedings of 12th annual conference of the International Gmup for the Psychology of Mathematics Education (Vol. 2, pp. 380-384). Vesprem, Hungary: Program Committee. Mansfield, H., & Scott, J. (1990). Young children solving spatial problems. In G. Booker, P. Cobb, & T. N. de Mendicuti (Eds.), Proceedings of 14th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 275-282). Oaxtepec, Mexico: Program Committee. _ Mitchelmore, M. C. (1983). Geometry and spatial learning: Some lessons from a Jamaican experience. For the Learning of Mathematics, 3(3), 2-7. Outhred, 1. (1993). The development in young children of concepts of rectangular area measurement. Unpublished PhD thesis, Macquarie University. Outhred, 1., & Mitchelmore, M. C. (1992). Representation of area: A pictorial perspective. In W. Geeslin & K. Graham (Eds.), Proceedings of 16th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 194-201). Durham, NH: Program Committee. Owens, K. D. (1992). Spatial mathematics: A group test for primary school students. In M. Stephens & J. Izard (Eds.), Reshaping assessment practices: Assessment in the mathematical sciences under challenge (pp. 333-354). Melbourne: ACER. Owens, K. D. (1993a). Factors pertinent to children's responsiveness in spatial problemsolving activities. In B. Southwell, B. Perry, & K. Owens (Eds.), Space: The first and final frontier (Proceedings of 15th annual conference of Mathematics Education Research Group of Australasia, pp. 421-431). Sydney: MERGA. Owens, K. D.(1993b). Spatial thinking employed by primary school students engaged in mathematical problem solving. Unpublished PhD thesis, Deakin University. Owens, K. D. (1994a). Concrete materials: Why they do or don't work. In D. Rasmussen & K. Beesey (Eds.), Mathematics without limits (Proceedings 6f the 31st annual conference of the Mathematical Association of Victoria, pp. 342-347). Melbourne: MAV. Owens, K. D. (1994b). Visualisation as an aspect of spatial problem solving. In R. Killen (Ed.), Educational research: Innovation and practice. (Proceedings of the annual conference of the Australian Association for Research in Education [On-line]. Available: www.swin.edu.au/AARE/conf94.html File: OWENK94.161 Owens, K. D. (1995). Imagery and concepts in mathematics: Spatial activities for the primary school [Professional development package with video]. Sydney: University of Western Sydney Macarthur, Faculty of Education.
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Owens, K. D., & Outhred, L. (1996). Young children's understandings of tiling areas. Reflections, 21(1), 35-40. Wheatley, G., & Cobb, P. (1990). Analysis of young children's spatial constructions. In L. P. Steffe & T. Wood (Eds.), Transforming children's mathematics education (pp. 161-173). Hillsdale, NJ: Lawrence Erlbaum.
Authors Kay Owens, Faculty of Education, University of Western Sydney Macarthur, P. O. Box 555, . Campbelltown NSW 2560. E-mail: . Lynne Outhred, School of Education, Macquarie University NSW 2109. E-mail:· .
