Architecture students' spatial reasoning with 3-D ...

J. Design Research, Vol. 9, No. 4, 2011

Architecture students’ spatial reasoning with 3-D shapes Ömer Akin School of Architecture, Carnegie Mellon University, 201 College of Fine Arts, Pittsburgh, PA 15213, USA E-mail: [email protected]

Ömer Erem* Faculty of Architecture, Istanbul Technical University, Taskisla, 34437, Sisli, Istanbul, Turkey E-mail: [email protected] *Corresponding author Abstract: We observe that exercises involving abstract representation of 2-D and 3-D shapes are considered by architectural educators to be an important part of early design education. Although the results have been mixed, at best, this conviction persists. The Architectural Scholastic Aptitude Test (ASAT) administered by Educational Testing Services, in the 1960s is one such well known effort that since has been abandoned. Yet the practice of using abstract design problems focusing on the kind of spatial reasoning included in the ASAT is present in the core repertoire of many introductory design studio problems. As we reported in a paper published in 1999, in architecture programmes all over the USA, freshmen still compose with basic geometric shapes in order to learn general design principles. Upperclassmen explore the virtues of the 3 × 3 × 3 grid space. Design researchers toil over the process of recognising emergent patterns based on primary shapes. Our findings in this paper indicate that the decline in skill to manipulate abstract shapes by upperclassmen is due to lack of practice or rehearsal; and this skill does not appear to be central to the learning of students of architecture. Keywords: spatial reasoning; aptitude test; representation; strategic reasoning; spatial manipulation; errors; cube arrangements; skill retardation; domain independent skills; memory rehearsal. Reference to this paper should be made as follows: Akin, Ö. and Erem, Ö. (2011) ‘Architecture students’ spatial reasoning with 3-D shapes’, J. Design Research, Vol. 9, No. 4, pp.339–359. Biographical notes: Ömer Akin is a Professor of Architecture at Carnegie Mellon University, Pittsburgh, PA, USA, since 1978. He is a well published researcher with several hundred reviewed publications; and texts that include Representation and Architecture (1982), Psychology of Architectural Design (1986, 1989) Generative CAD Systems (2005), A Cartesian Approach to Design Rationality (2006) and Embedded Commissioning (in prep). His research interests include design cognition, computer aided design generation, case-based instruction, ethical decision making, design of virtual worlds,

Copyright © 2011 Inderscience Enterprises Ltd.

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Ö. Akin and Ö. Erem building commissioning and automated requirement management. He has served as the Head of the School of Architecture and Director of the graduate programmes. Ömer Erem is a faculty member at Istanbul Technical University since 1992. He is conducting researches on spatial cognition, legibility and identity, computer aided design, space syntax, vernacular architecture and tourism building design. He is attending in architectural design studios and has participated in international conferences on urban life quality, space syntax and Mediterranean tourism. He is working on architectural education theories and has articles in international and national books on this topic. In 2009, he has studied at Carnegie Mellon University as a Visiting Scholar for six months and worked with Prof. Dr. Ömer Akin on this research about spatial reasoning.

1

Introduction

This paper is a continuation of the analysis of spatial reasoning in the field of architectural education first undertaken by Akin (1999). His review refers to the lore of starchitect power (Kamin, 2008) that surrounds this approach to design teaching: “It is hardly possible to read about F.L. Wright and not be told that his early training as a child has greatly influenced his life-long success as an architect. Froebel blocks with which little Frank was encouraged to play is usually credited to be the source of this influence. In a letter to Grant Mason, an author for Architectural Review, even Mr. Wright himself acknowledged this influence: “somehow you did get to the source of my mother’s contact with Froebel ... and you are perfectly right regarding the formative power and direction the ‘kindergarten’ gave my instincts and could beyond all else give children if properly applied.” [Pfeiffer, (1984), p.115; Akin, (1999), p.1]

Akin’s review goes on to cite several design course documents that are aimed at building the cognitive tools of situated and realistic design tasks based on the spatial manipulation exercises of simple 2- and 3-dimensional shapes [Bizios, (1991), pp.54–56 and pp.191–193]; formalisation efforts through topological patterns [Clark et al., (1982), p.58], topological formalisms (Mitchell et al., 1976), generative shape grammars (Stiny, 1975), and use of simple plan parti diagrams found in speak-aloud, design protocol studies (Akin, 1989)1. Akin concludes by stating: “presenting complex designs in the form of simple geometries or as topological entities affords us the ability to understand, and closely study properties of designs which we would otherwise be unable to do so.” [Akin, (1999), p.3] This is not an isolated phenomenon that is particular to our time. In the history of architectural education there are notable watershed moments when simple, spatial representations of architectural designs have taken centre stage for long periods of time. The Ecole des Beaux Arts curriculum [Chafe, (1977), p.82], the Bauhaus (Bayer et al., 1938) and the ASAT designed and administered for decades by the ETS of Princeton University (ACSA, 1963) come to mind. In the annals of architectural pedagogy, there are a variety of vantage points that are similar to those we reviewed above, which deal with spatial reasoning skills. Before we bring that review up to date, it may be useful to describe how we frame the concept of spatial reasoning in this study. Spatial reasoning is the ability to orient oneself in physical

Architecture students’ spatial reasoning with 3-D shapes

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settings. It consists of two skill sets, perceptual and cognitive. The former deals with managing sensory information and the latter to reason with this information in a goal directed manner. The spatial reasoning skills we consider in this study have to do with the perception of primary shapes (cube sets) and their cognition in order to manipulate them towards specific goals (fitting cube sets within a given space). We consider these skills to be consistent with skills that are covered in current architectural pedagogy as reviewed below. First we consider a compendium of essays on pedagogy edited by Salama and Wilkinson: Design Studio Pedagogy: Horizons for the Future (2007). One essay in this volume by Stephen Kendall argues that students can learn through ‘small decisions’ to deal with larger ones. This is the underpinning of the pedagogy that supports acquisition of architectural space manipulation skills through simple spatial exercises. This strategy of progressive and gradual advancement of skill is manifest in Jeffrey Haase’s essay that argues for using advancement from 2-D to 3-D spatial representations, which finds support in installation artists’ work. Mirjana Radojevic makes the argument for simple to complex in the context of virtual (computer assisted) learning environments, in which the medium is conducive to, even demanding of, breaking down complex entities into their ‘essential’ or simplest abstractions. It is a small leap from computation and the virtual to typology and the abstract in building types. Frankand and Schneekloth (1996), in their article entitle ‘Ordering space: types in architecture and design’, underscore the important role of abstraction in design pedagogy. Finally, we observe that the interest in the simplified spatial basis for architectural cognition and pedagogy has its roots in historico-philosophical works. Pont (2009) refers to the mathematical underpinnings of architectural theory and practice, alluding to the relationship between the abstract form and the complex (real) form of architecture, and the importance of this in the training of the architect.

2

Research on cognition of spatial configurations

In this section, we consider findings in the cognitive realm that relate to our interest in spatial reasoning, including subitising, rehearsal, mental rotation and an analysis of the ASAT tasks.

2.1 Spatial reasoning during subitising The literature on counting with three dimensional blocks is one of the areas in which spatial skill enhances and inhibits other cognitive functions, namely subitising, due to special configurations of 3-D cube arrangements in 2-D drawings. “Subitising is the process that people use to directly quantify a small number of objects without counting. It was found that most people consistently subitise up to four blocks”, [Akin and Chase, (1978), p.397]. Among several spatial variables, compactness had the largest effect and suggests that spatial arrangements of cubes affect perceptual grouping of sub-assemblies of cubes (Akin and Chase, 1978). This process of grouping with a small number of items applies directly to the task of interest in this study since the number of cubes per (sub-)group is less than four when all arrangements are maximally compact (Akin, 1999).

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2.2 Memory and rehearsal Subjects perceive, manipulate and solve 3D-cube-set problems primarily through the ‘working memory.’ The working memory, which has three basic components, coordinates complex cognitive activities by controlling, processing, and making the stimuli available to ‘active’ memory. The main processor ‘central executive’ is the core of all cognitive skills and is enriched by two slave systems: phonological loop that holds our verbal memory and visio-spatial ‘sketchpad’ that stores visual and spatial information (Baddeley, 2000). Cognitive skills are improved by numerous transfers of verbal and spatial information using either one or both of these capabilities. In spatial memory tasks, one usually takes advantage of the phonological loop and assigns semantic meaning to unfamiliar stimuli (Baddeley, 2000). Information on stimuli is rehearsed in the short term memory (STM). With subsequent rehearsal, this information is transferred and stored in long term memory (LTM) (Atkinson and Shiffrin, 1971). If there is appreciable complexity in arrangement of spatial items, rehearsal performance suffers. While, a matrix-like spatial distribution, where stimuli are not competing or distracting from each other, LTM use enhances spatial learning (Kemps, 1999). Organisation of spatial items during rehearsal is similar to memory organisation of words and numbers constituted as chunks (Lillo, 2004). Experts in spatial reasoning with extended rehearsal experience, like chess players, tend to discover many more relationships and develop new strategies for problem solving in addition to rules that allow them to perceive, store and reconstruct spatial relations (Craig, 1979). Extended periods of rehearsal using schema-based organisation of chunks enhances strategies for spatial reasoning problems either in 2-D like chess or 3-D like mental rotation tasks. Thus the organisation of verbal and spatial items affects memory processes and influences rehearsal performance either in a positive or negative way.

2.3 Mental rotation Mental rotation has a special place in our discussion of spatial reasoning because it seems to incorporate in the rotation act some of the basic spatial tasks that overlap with aspects of our experimental task and there is a rich lineage of research in this area. Mental rotation studies involve rotated figure comparisons. Subjects, when asked to compare the similarity or difference between two three-dimensional block arrangements, were observed to perform a mental process that is analogous to rotating one of the forms to match the other. There are three phases in the mental rotation process, where the observer: 1

searches segments on each shape that can be transformed onto each other

2

repeatedly transforms and compares the representation of corresponding segments until the internal representation is matched, and

3

confirms the congruency of remnant segments of the two shapes in order to be sure about their decision (Just and Carpenter, 1976).

In most mental rotation findings the reaction time to judge the identity of pairs increases linearly with the increase of the angle between the original and its rotated versions. If a subject rehearses on a shape, the internal representation of the shape is rotated to match


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that of the rehearsed version. Then, the mentally rotated image is compared to the memory representation of the trained shape (Cooper, 1975). This process corresponds to the entire reaction time for mental rotation. The effect of comprehensive rehearsal accounts for only 20% of the decrease in reaction time, which means that 80% of the reaction time is accounted for by rotation (Shepard and Metzler, 1971). Reaction time is also affected by the complexity of the stimulus. Time to initiate a transformation, direction of the orientation difference, the total degree of rotation, and number of mental rotations to transform can be factors that impact overall reaction time (Parsons, 1987). If the observer rehearses on shapes, reaction time increases with angular difference between the rehearsed viewpoint and the observed shape position. With rehearsal, the value of reaction time converges for all familiar viewpoints, but does not change for unfamiliar ones. The observer tends to rotate a shape to the nearest familiar viewpoint to make judgments on the mental rotation task (Tarr, 1995). Mean error rate on mental rotation decreases and rotation velocity increases with rehearsal, while reaction time remains constant for the same transformations. The skills to imagine and rehearse can interact significantly with each other. Observers with superior spatial skills are more inclined to change strategies and have a higher rotation velocity than ones without. Rehearsal helps to develop different strategies in mental rotation like double or self-body rotating. Yet, the strategy of mental rotation never disappears. Less complex shapes facilitate rotation skills more significantly than more complex ones (Leone et al., 1993). The representation strength and variations increase with rehearsal. At the beginning of rehearsal, a small number of representations are available and mental rotation reaction time is mostly spent on the rotation algorithm. After some rehearsal, a shape is learned so well that the observer is able to retrieve its form directly from memory (Kail and Park, 1990). Parsons (1987) found that a ranking of reaction time based on X, Y and Z axes – where the X-axis points towards the right of the viewer, Y away from the viewer, and Z towards up as in Figure 5 – can be substantiated. The reaction time ordered from shortest (easy) to longest (difficult) is Y (–90), X (–180), Z (–180) and Y (–180). Values for diagonal axes are higher than all types of axial rotation. Parsons emphasises three results: 1

horizontal axis reaction time is longer than vertical axis

2

shortest reaction time is in the Y-axis direction, and

3

the alteration of rotation is independent of left or right orientation.

There are dimensional strategies developed by observers performing mental rotation tasks. The 2-D representation of a 3-D object stimulates the development of two different strategies: spatial and non-spatial. The spatial strategy creates 3-D configurations of shape and mentally rotates these shapes. After creating the mental model of a shape, the observer treats the object as a 2-D pattern without any necessity to rebuild the 3-D model once again (Gittler and Glück, 1998). Rehearsing with 2-D patterns of a 3-D object decreases rates of recall error. Subjects who rehearse with a complex 2-D representation of a 3-D geometrical shape pass the similarity test with four times less trial than novel subjects without any practice. This supports a strong influence of visual experience on subsequent perception of 3-D forms (Sinha and Poggio, 1996). Ho et al. (2005) found that 2-D and 3-D spatial tasks are correlated. Observers who were good at 2-D mental rotation also showed good performance in 3-D rotation; while bad performers showed bad performance in both rotation tasks. Observers internally use a

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2-D shape identification technique called ‘feature mapping’, which is determining the 3-D form of a shape from multiple 2-D views. Feature mapping comprises three strategies: 1

2-D feature mapping and rotation

2

3-D feature mapping, and

3

rotation of mental model in 3-D.

According to Cooper and Mumaw (1985) there is no difference between subjects with high and low aptitudes for 3-D to 2-D feature mapping strategies, while a greater difference is seen in higher level 3-D mental rotation tasks strategies. Figure 1

ASAT test task of spatial manipulation

2.4 ASAT: an aptitude test Some aptitude test tasks, such as the ASAT, involve the finding of as many arrangements of pairs or triples of cubes in a three dimensional space of cubes as possible, within a set time period (Figure 1). This is the task which is studied in this paper. The task in its simplest form can be represented as a two low level cognitive operations (Figure 2), linearly executed with a simple feedback loop connecting them:


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1

Operation-1: Consider representation of triple cube-sets (CS), in a two-dimensional medium.

2

Operation-2: Place a triple CS in the stimulus CS space.

3

Feedback loop: Continue to step 1, until all cubes in the stimulus space are accounted for.

Figure 2

The representation operation

Source: Akin (1999, p.5)

An initial cube-set (CS), either a pair of cubes or a triple, is in the three-dimensional space of 2 × 3 × 3 cubes. The representation operation is repeated until the entire space is filled, which neither guarantees the creation of different arrangements nor the avoidance of redundant ones. Figure 3

The manipulation operation


An articulated version of the same task (Figure 3) includes a step called manipulation (Akin, 1999). In this case, following the initial placement, the position of the next CS is found after it is manipulated (rotated, translated, moved) with respect to the previous placement. This step follows step 2 in the above list of steps before the feedback loop takes over. This guarantees the generation of arrangements that appear different, but does not guarantee non-redundant arrangements over rotation or translation. Figure 4

The strategic reasoning operation


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In the most advanced version of the cognitive model, an additional step to avoid generating redundant arrangements of CS is needed: strategic location. The manipulation of the CS must be compared to previous solutions in order to cover new permutations rather than generating arrangements similar to previously generated ones. In this version, the flow of operations is: 1

Operation-1: Consider representation of CS, in the two-dimensional medium.

2

Operation-2: Place a CS in the stimulus CS space.

3

Operation-3: Strategically (re-)locate the CS to avoid duplication – by rotation and translation.

4

Operation-4: Manipulate placement with respect to that in the previous solution(s) attempted.

5

Feedback loop: Continue to step 1, until all cubes in the stimulus space are accounted for.

2.5 Summary of background research Subitising research indicates that CSs that are most compact take the least time to estimate (Akin and Chase, 1978). Memory and rehearsal literature indicates that extended periods of rehearsal using schema-based organisation of chunks enhances strategies for spatial problems either in 2-D, like chess, or 3-D, like mental rotation. Mental rotation literature offers the most in terms of the effect of rehearsal on spatial cognitive tasks. In some studies, rehearsal accounts for 20% improvement in reaction time. Complexity of the stimulus and the unfamiliarity of the view point are factors that degrade performance. Individuals with superior spatial skills can perform faster and rehearsal can create mental models that ease the mapping of CS arrangements. The axis of rotation applied influences reaction times, with the Y-axis being the most efficient orientation. Subjects who rehearse with a 2-D representation of a complex 3-D geometrical shape take less time to learn be trained. Finally, in order to improve performance, some observers use a 2-D shape identification technique called ‘feature mapping’, which is determining the 3-D form of a shape from multiple 2-D views. ASAT problems we use as our experimental tasks can be modelled using four basic cognitive operations: represent, manipulate, strategise, and test for completion (Figure 4). The subitising results apply to the representation operation. Findings in the areas of memory and rehearsal apply to the strategising operation. Finally, the mental rotation literature offers useful parameters for the manipulation operation.

3

Experimental design and data codification

3.1 Sampling The entire undergraduate population of the School of Architecture at Carnegie Mellon University (CMU) was used as the population for the study. This is not considered to be a representative sample of all schools of architecture since establishing reliable controls upon different curricular and pedagogic intentions is an insurmountable problem. The


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strategy of testing the entire population, rather than a smaller representative sample, was easier, more reliable and feasible to carry out.

3.2 Task We used an ASAT spatial test in which test takers were asked to arrange pairs or L-shaped sets of cubes in a larger (2 × 3 × 3), regular, gridded space (Figure 1). This task was obtained from the 1965 version of the ASAT issued by the ETS (ACSA, 1963). This test has seven sections: Section 1: architectural terminology (20 minutes), Section 2: visual arts interpretation (30 minutes), Section 3: physics (30 minutes), Section 4: solid geometry (20 minutes), Section 5: spatial reasoning – I (six minutes), Section 6: spatial reasoning – II (12 minutes), Section 7: graphic pattern completion (12 minutes). The specific stimulus used was copied and reproduced directly from the ASAT test package, Sections 5 and 6. No changes were made to these tests.

3.3 Data collection Data collection took place in the regular meeting place of each year’s lecture session of the design studio. First year students were assembled in Hamerschlag Hall – B103, second year in Breed Hall in Margaret Morrison Carnegie College building (MMCC), third year in Room-14 in MMCC, fourth and fifth years each in Room-211 of the College of Fine Arts building. All of these rooms are 80 to 100 seat auditoria, with the exception of CFA-211 which is a 35 seat classroom. Faculty conducting each class session were contacted ahead of time and given the standard explanation for the test. They were told that this is part of a study examining the spatial reasoning skills of architecture students. Students were given a similar explanation at the time of the test. Students were also told that their participation was voluntary and anyone who wished could leave without any explanation or prejudice. None did. Students were assured that the test sheets would be kept confidential. Sheets contained no requests for information that could identify any of the students.2 Several students who arrived late were told that they could not take part in the test. Packets were distributed one by one to the students, by the experimenter. Students were not permitted to turn pages over until such time that all were clear on the instructions and were instructed to turn the pages. Instructions stated that the students were to find unique arrangements and views of arrangements without creating identical views and to draw these arrangements in the given space (Figure 1). In the interest of time, they were also asked not to spend time trying to cross out redundant arrangements once they were generated. Students were given exactly six minutes to complete the first task, which has been analysed in an earlier study (Akin, 1999) and 12 minutes to complete the second task, which is the subject of this paper. Students were asked to stop exactly at the same time, prompted by a count-down from five to zero.

4

Analysis of data

In this study, data analysis consists of examining the relationship between the spatial reasoning task performed and the skill sets of the subject population grouped in academic

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years of the undergraduate programme at CMU. Since the entire population of students participated in the experiment there was no issue regarding sampling errors. Gender differences were found to be statistically insignificant in the analysis of the first task (Akin, 1999) and are not reanalysed for this study. Other factors like cultural differences were not part of the controlled variable set. Students were engaged in the experiment at the same time of day, just prior to their studio sessions, and during the same time of the academic semester.

4.1 Data tabulation – based on the new analysis In this study, the second task using L-shaped, three cube-sets (CS) from ASAT Section 6 is analysed. Before data tabulation, all possible single and unique CS positions were generated using a spread sheet application developed for this analysis. There are 80 possible, unique positions of the L-shaped CS in the 2 × 3 × 3 grid space (Figure 5). First we coded each position using an identifier consisting of a letter and a number where the letter indicates the orientation of the CS against coordinate axes of X, Y and Z, and the number indicates its position in the 2 × 3 × 3 space. Every combination of six different CS instances, where there was no spatial conflict between the located CS, signifies a valid cube set combination (CSC), or a solution to ASAT Section 6 task. Thus, each valid CSC is formulated by the combination of six unique two-character alphanumeric instances that results in a 12 character code combination. For example, J1, J2, J3, D4, D5, D6 is a valid non-overlapping CSC for a solution arrangement shown in the 2nd column of Figure 6. Figure 5

All possible single and unique CS positions

Notes: The pairs of figures A4 and A5, I4 and I5 and K4 and K5 appear to be identical. This is an illusion created by the vantage point of the 3-D diagrams. In reality one figure in the pair depicts a CS in the front of the 2 × 3 × 3 space while the other depicts one in the back.

Architecture students’ spatial reasoning with 3-D shapes Figure 6

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Valid CSC examples and analysis

Analysis was done in two phases. In the first phase repetitive solutions and errors were detected in order to score the total number of correct non-repetitive solutions. Some subjects generated CSCs that were repeats of previously generated ones. These repetitions were eliminated from the whole solution set (WS) where each CSC was counted as one solution. After omitting the repeats (R) we reached the number of all valid solutions (VS). There were CSC in which the total number of cubes in a CS was not equal to three, the drawings were not complete, or CSs were spatially overlapping. These were scored as errors (E). After excluding the E from the solution set, we obtained the total number of solutions without error (SW). Some subjects had one SW while others had 27. The following calculations were applied to above variables to find the number of errorless solutions: Whole solution set:

WS

Repeats:

R

Errors:

E

Total valid solutions:

VS = WS – R

Errorless solutions:

SW = VS – E

Using the spreadsheet application all valid errorless solutions tested by comparing a solution with a rotational variant of another solution. If a match was found, that solution was marked to have the condition of rotational identity (CR) for that axis and rotation angle. Any CSC with a rotational identity was assigned the value: ‘true’. The sum of total number of CSC with the ‘true’ value yields the total number of CSC with some rotational identity (RI). Five RIs, one for each axis, are defined for each possible axis rotation type with three dependent variables, based on: SW

Total number of solution with rotation identity (RI): If CR=TRUE then RI=

∑ CR n−1

Net number of solutions without identities (NS): NS=SW − RI where CR is the condition of having a RI. The total number of solution types that have no rotational identity (NSI) was estimated. The total number of CSCs that have NSI found by all five years is 275. The slope of the function, solutions found against years (number of solution types, NSI: year-1 = 186, year-2 = 122, year-3 = 134, year-4 = 61, year-5 = 70), is negative. NSI is different than NS, because NS is the total number of no-identity solutions found by subjects. Naturally, there are overlaps between solutions by different subjects. To reach

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the NSI number for each year, all NS codes were listed in a spreadsheet and unique ones were picked from the list by a series of Microsoft Excel operations. Through this process we estimated the number of solutions generated by each subject: μNSI, the mean number of solutions that a subject has contributed to the total NSI of his or her year. We developed similar method of calculation for each rotation-axis type. For brevity these are not included in this paper.

4.2 Descriptive statistics Table 1 shows the primary independent variable of studio years (Y), against five dependent variables of the mean number of total solutions VS, rotational solutions (SW, RI), net solutions (NS), and errors (E). The mean values corresponding to Y are significantly different and generally monotonically ordered with some notable exceptions, which will be covered in the discussion section. We expected that, for each year’s subjects (indicated by the subscripts), the direction of monotonicity for mean number of arrangements (μVS, in Table 1) and repeat solutions (μRI, in Table 1) should be ascending (positive slope), i.e., P(μ1) < P(μ2) < P(μ3) < P(μ4) < P(μ5), where P is expected probability. Table 1

Descriptive statistics of performance by each year

Studio years

1st year

2nd year

3rd year

4th year

5th year

70

45

33

21

21

1429

886

668

320

339

μVS: mean number of total solutions

20.41 (sd. = 6.403)

19.69 (sd. = 6.160)

20.24 (sd. = 7.706)

15.24 (sd. = 6.131)

16.14 (sd. = 6.183)

μRI: mean

8.24 (sd. = 6.383)

10.76 (sd. = 7.840)

11.24 (sd. = 8.624)

9.05 (sd. = 7.500)

9.52 (sd. = 6.290)

μNSI: mean

2.68

2.71

4.06

2.91

3.33

μNS: mean net number of solutions

6.41 (sd. = 4.031)

4.42 (sd. = 3.388)

6.06 (sd. = 3.102)

3.81 (sd. = 2.182)

4.43 (sd. = 2.580)

μE: mean

5.76 (sd. = 5.029)

4.51 (sd. = 4.110)

2.94 (sd. = 2.597)

2.38 (sd. = 2.819)

2.19 (sd. = 3.010)

0.285

0.252

0.162

0.168

0.128

S: total number of subjects VS: total number of solutions

number of rotated solutions

solution types

number of errors

EYEAR (E/VS)

Instead the actual values of mean number of arrangements and repeat solutions are generally in descending order. This is the most puzzling finding of the work published in 1999 and one that is replicated in the current study. The total number of subjects (S) includes all students who volunteered to take the test we administered. The number of S in each year is a function of enrolment for that year, which is descending in value due to academic attrition. Third row of Table 1 shows the mean number of solutions, μNS, for that year obtained by dividing the number of (VS,


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row two) generated by the total number of subjects (S, row one) for each year. These values present a negative slope in ascending order of years. As verified through analysis of variance (ANOVA), which we will present later, each year has done statistically significantly worse than the previous year (at confidence level of 0.01), almost in a strictly monotonic function, with a minor deviation in year two. The values in row three include solutions that are multiple views of the same arrangement with rotation. An arrangement can be rotated around a single axis creating a different 2-D pattern on the page, while the spatial arrangement is the same. For spatial reasoning in its strict sense, we consider these arrangements ‘duplicates.’ A student of architecture, or an instructor for that matter, designing a 3-D design would, more than likely, regard rotational views as different appearances of the same ‘object.’ Accordingly, we found five different versions of rotation, three around the 180-degree axis along X, Y, and Z, and two around the 90-degree axis along Y in clockwise (–Y) and counter clockwise (+Y) directions (Table 2), which we described in Section 4.1. In the last column of Table 2, we calculated the mean total number of rotations for each of the five years. Table 2

Analysis of the RIs of solutions generated in each year Mean rotation x180 RI(L) X180

Mean rotation y180 RI(L) Y180

1st

3.014

5.129

3.314

4.129

3.971

8.243

2nd

3.800

7.022

5.000

6.289

5.156

10.756

3rd

3.879

7.273

5.182

6.576

5.636

11.242

4th

3.762

6.714

4.524

5.714

4.524

9.048

5th

4.143

6.381

3.524

4.333

5.048

9.524

Year

Mean Mean Mean rotation rotation rotation z180 y + 90 y – 90 RI(L) Z180 RI(L) Y + 90 RI(L) Y – 90

Mean total rotation RI(L)

When discounted from the mean number of total solutions (μVS, in Table 1), we obtain the mean number of solutions (μNS, Table 1). Once again we see an overall negative slope with notable deviations in years three and five. Nevertheless, the fit with a linear function is strong (Table 3) with intercept of 6.402 and slope of –0.458 (Figure 7). Figure 7

Change in mean number of NS

Mean Number of Net Solutions for L-Shape Test Net number of solutions

7,0 6,0 5,0 4,0

Mean NS(L)

3,0

Linear Slope (-0.458)

2,0 1,0 0,0 1

2

3

Years in college

4

5

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The mean numbers of error and solution trials, which do not yield arrangements while conforming to the requirements of the test, present a very different picture. While the E and solutions trials, to be consistent with inferior performance per ascending years, should be increasing with each ascending year; in our findings, they do just the opposite. It shows strictly monotonic decline with the increase of years. This suggests that the number of E is not related to elimination of false moves because of greater spatial reasoning skill that may be realised with the advancement of years of study. Rather it seems to be a function of the number of trials, regardless of the skill differential. It may be the case that the skill differential is not significant enough to impact the error rate per solution trial (EYEAR).

4.3 Analysis of trends observed through descriptive statistics Next we looked at the significance of the trends that we observed in the descriptive statistics through ANOVA (Table 3). We considered five key dependent variables of performance – VS, SW, RI, E, and NS – against the independent variable number of years of study ranging from one to five. Two of the dependent variables: SW, and solutions w/ RI, were not significant in terms of the variance they appear to have. Table 3

One-way ANOVA with all dependent and independent variables

ANOVA table for independent variable year (Y) and frequency of observed dependent variables of performance Valid solutions (VS)

Solutions without error (SW)

Solutions with rotation identity (RI)

Errors (E)

Net solutions (NS)

F = 3.947

F = 1.564

F = 1.341

F = 5.786

F = 4.326

P = 0.004

P = 0.186

P = 0.256

P = 0.000

P = 0.002

(p < .01)

(p > .01)

(p > .01)

(p < .01)

(p < .01)

The remaining three dependent variables, VS, NS, and E are strongly significant, exhibiting monotonic and negative slopes. The former two support the hypothesis that with advancing years the performance of students in solving the spatial reasoning task deteriorates as is the case with the ASAT task with I-stimuli (Section 5). This suggests that the simple, abstract spatial reasoning tasks encouraged and included in architectural curricula do not become a fruitful component in the students’ skill set, in advanced years. We will discuss the potential implications of this for architectural curricula later. Here, we are interested in considering an alternative explanation for this finding. Could it be the case that the spatial reasoning with the simple, abstract spatial design skills became dormant in later years due to the demands placed on the cognitive system of the student by real, complex design problems that constitute the ultimate architectural design task? Are the cognitive mechanisms critical in one different from those critical in the other? In order to address these questions we hypothesise that there should be a significant improvement of performance between the first task performed (ASAT – Section 5 with I-stimuli analysed in Akin, 1999) and the second one (ASAT – Section 6 with L-stimuli analysed here). Due to the rehearsal provides by the first task (I-stimuli) the subjects should be performing better in the second task (L-stimuli).

Architecture students’ spatial reasoning with 3-D shapes Mean number comparisons for VS, E, SW, RI and NS between I & L shape tests Mean Number of VS for I & L Shape 25,0

20,41

19,69

20,24

20,0 Mean TS

15,24

16,14

15,0 15,09

15,13

14,79

10,0

14,81 12,19

5,0 0,0 1

2

3

4

5

Years in college Mean VS(L)

Mean VS(I)

Mean Number of SW for I & L Shape 25,0 20,0

17,30

Mean SRI

14,66

15,18 12,86

15,0 10,0

12,99

13,53

1

2

14,27

13,95 14,24

11,48

5,0 0,0 3

4

5

Years in college Mean SW(L)

Mean SW(I)

Mean number of RI 12,0 10,0

10,76

11,24 9,05

9,52

8,24

8,0 Mean RId

Figure 8

353

6,0

7,07

7,67

7,00

6,09 5,19

4,0 2,0 0,0 1

2

3

4

Years in college Mean RI(L)

Mean RI(I)

5

Mean number comparisons for VS, E, SW, RI and NS between I & L shape tests (continued) Mean Number of E for I & L Shape 25,0

Mean E

20,0 15,0 10,0

5,76

4,51

5,0 0,0

2,10

1,60 2

1

2,94

2,38

2,19

0,52 3

0,71 4

0,57 5

Years in college Mean E(L)

Mean E(I)

Mean Number of RI per axis for I & L Shape 10,0 9,0 8,0 7,0 Mean RI

Figure 8


6,0 5,0 4,0

5,1 4,5 4,0

3,0 2,0

3,0 2,7

7,0

7,3

5,6 5,2

5,6

6,7 4,5 3,8

4,4 3,9

3,8 3,2

6,4 5,9 5,7 5,0 4,1 3,7

2,7 1,8

1,0 0,0 1

2

3

4

5

Years in college Mean RI(I)X180

Mean RI(I)Y180

Mean RI(I)Z180

Mean RI(L)X180

Mean RI(L)Y180

Mean RI(L)Z180

Mean Number of NS 12,0 10,0 8,0 Mean NS

354

6,0

6,90

7,27 6,47

6,41

4,0

6,29

6,57

6,06 4,42

3,81

2,0

4,43

0,0 1

2

3

4

Years in College Mean NS(L)

Mean NS(I)

5


355

In order to test this hypothesis we compared the results obtained from the L-shapes stimuli study to those of the I-shapes stimuli study (Akin, 1999). In the latter case the following were found: 1

Hypothesised direction of monotonicity between the years in terms of the spatial manipulation variable does not hold. With years of education students’ ability to manipulate simple 3-D cube arrangements (mental rotation and translation) deteriorates.

2

The number of arrangements obtained without redundancy and number of different arrangement classes found indicate differing results. In the former case, no interaction with the number of years is indicated; in the latter case a significant difference between studio years and a counter intuitive direction of monotonicity in their values are observed. Students got worse over the years, rather than better.

3

The number of E committed by year declined monotonically. As students advance in years they are better able to represent spatial objects and make fewer E.

In conclusion, “There is statistically significant correlation between the ability to reason with (spatial arrangements) and studio years … in an architectural programme. The relationship is by and large an inverse one.” (p.16).

He went on to state: “At this time, there is no clear evidence that explains this counter intuitive result. One possible explanation is a diminishing emphasis on reasoning with simple spatial arrangements as years of architectural study increase. This can lead to a lack of practice with such stimuli. This hypothesis will be tested when the data collected during the second half of the test (Section 6) is analysed to identify a potential practice effect, particularly with upper classmen.” [Akin, (1999), p.13)

Here, we have undertaken this plan. If the reason for the decline in the spatial reasoning skill over the years is one of atrophy due to lack of use or overshadowing due to the increased attention to situated and real design learning, then we should see improvement in the second task. The rehearsal effect of having performed the first task should improve the number of solutions generated. This should be uniformly true for all five years. Analysis of the L-shaped stimulus data shows that this is indeed the case. While the error rates got better the number of solutions found, in total, with rotations and NS, all fit a linear function of negative slope. This confirms that the second spatial reasoning task solved by the same group of student’s exhibits similar characteristics (Table 4). The significance score 0.000 is less than 0.05. There is a significant difference in variances of groups. On the other hand, the number of solutions developed in the second task (L shape) is consistently higher for all three cases, VS, SW, and RI. The similarity of the plots for the two tests, one with the linear stimulus and the other with the L-shaped one is remarkable (Figure 8). This confirms our hypothesis that the performance of all subjects improved from the first task (I-stimuli) to the second task (L-stimuli). We attribute this improvement to the rehearsal effect between the two tasks. In other words, the spatial reasoning skills of students of architecture with abstract stimuli decline over the years. This is likely a result of benign inattention, in the upper years, to abstract spatial tasks and the prerequisite skills that underlie them. Alternatively,

356


it may be due to the atrophy caused by real architectural design tasks that do not rely on abstract spatial skills; or both. In any case, the performance of these students should improve, that is if the deficit is due to a lack of practice in the first place, through practice or rehearsal. The second test represents this practice or rehearsal effect, since the two tasks were given back to back in the same session. Table 4

Levene’s t-test results between the means of the I-task and the L-task Independent samples test Levene’s test for equality of variances

E

Equal variances assumed

F

Significance

t

df

48.831

.000

–7.870

390

Equal variances not assumed

5

T-test for equality of means

–7.870 321.790

Sig. Mean (two-tailed) difference

Std. error difference

95% confidence interval Lower

Upper

.000

–2.781

.353

–3.475

–2.086

.000

–2.781

.353

–3.476

–2.085

Discussion

5.1 Importance of basic 3-D manipulation skills Literature on architectural pedagogy and other sources related to spatial skills learning – such as mind expanding/exercising games and puzzles for popular consumption; computer science, artificial intelligence and information technology models of cognition; and historic-philosophical speculations of pedagogy – support strategies that starts with abstract, simple problems that arguably may provide a foundation upon which more complex, real world problem solving skills would be constructed. Like learning how to add and subtract would be indispensable skills for performing more advanced calculations like estimating mean, mode, standard deviation and variance of a given set of numeric values. In other words upper classmen would excel in higher level skills like solving complex, real world architectural problems as well as in foundational skills of abstract spatial problem solving, as a student of mathematics would retain basic arithmetical skills alongside of those that apply to statistical problem sets.

5.2 Degradation of skill in curricular programme Contrary to this assumption we found that abstract spatial skills gained in the earlier years of architectural curricula do not remain at the same levels of proficiency, let alone improve, over the years as students of architecture gain higher level skills with complex problems. This is a starling result that was observed in an earlier study with I-stimuli (Akin, 1999) and was replicated with L-stimuli in this study. A possible explanation for


357

this is that as advanced skills are gained, the simpler skills gained in the earlier years atrophy because they are no longer used, or rehearsed. This would be possible only if, unlike the basic arithmetic operations that get used as the basis of higher level statistical calculation skills, basic spatial skills do not get used or rehearsed in undertaking real world architectural problems.

5.3 Improvement of skill through rehearsal The analysis we did with the L-stimuli task, following the rehearsal provided by the I-stimuli task, shows the presence of the rehearsal effect. Subjects in the latter task (L-stimuli) did better than their performance in the earlier task (I-stimuli). If rehearsal causes better performance, then lack of performance by students in the upper years has to be due to the lack of rehearsal, or use, of abstract spatial skills.

5.4 Relationship to real design skills The fundamental implication of this result is that if we are going to measure the success of architectural curricula by the strength of cognitive-spatial skills acquired by students by the time they graduate, the role of learning about abstract spatial skill stressed in the earlier years is, at best, suspect. This is a direct challenge to the current emphasis of simple 2-D and 3-D, puzzle like design problems used in the first and second year design studios in most if not all architectural curricula.

Acknowledgements This work could not have been completed without the support of TÜBITAK (The Scientific and Technological Research Council of Turkey) and we are grateful to them. We are also grateful to the entire student body and the year coordinators of the School of Architecture, Carnegie Mellon University, for taking the time to schedule and complete the test administered in this study.

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Notes 1

2

Today the interest in spatial reasoning has reached new frontiers. Aside from mind expanding/exercising games and puzzles for popular consumption that are in the tradition of F.L. Wright’s Froebel blocks – such as Brain Training Games (‘Improve The Age Of Your Brain With Scientifically Designed Games Now!’ www.lumosity.com), Contract Window (‘Covering Innovative design combined with high degree of performance’ www.HunterDouglasContract.com), and GeoTimeGeo (‘Temporal Data Visualisation and Analysis Software’ http://www.geotime.com/?gclid=CKi_kdKN8p8CFeh_5QodqV8Ucg) – traditional disciplines of research continue to explore the role of spatial cognition in science and technology. In particular, computer science, artificial intelligence and information technology circles have joined the ones that have persisted throughout the recent decades, such as cognitive scientists, and architectural educators, two areas of close interest for the scope of this paper. The only exception to this is a student who is much older than any of the other students. He pointed this out that he could be identified from the answer to the age question. He was asked if he wanted to leave. He indicated that he had no problem with completing the test. His test results are included with the rest of the data.