Professor George Mandler died at the home in London that he shared with his wife, Professor Jean Mandler, and not far from their two sons, Peter Mandler (Professor of Modern Cultural History at the University of Cambridge and Fellow of Gonville & Caius College, Cambridge, United Kingdom) and Michael Mandler (Professor of Economics at Royal Holloway College, University of London, United Kingdom).
I will not use this space either to list or to praise George’s numerous and varied academic achievements. These achievements are well-known to many; in addition, George’s work and its significant influences on the field of experimental psychology are elegantly summarised in the memorial piece about George on the University of California, San Diego (UCSD), Department of Psychology website and also in some of the other remembrances in this collection. Furthermore, those wishing to know more about George’s reflections on his own life and times should read his book, Interesting Times: An Encounter With the 20th Century 1924-. My contribution here is intended to evoke George as I experienced him.
George was my PhD supervisor for 3 years, from 1968 to 1971, in the UCSD department of psychology that he established in 1965. When I try to characterise how he seemed to me then (and, indeed, perhaps always), two things come to mind. The first is formidable. The Chambers English Dictionary defines this word as “inspiring awe; causing fear,” and I think both of these meanings are appropriate here. George must have had a degree of self-doubt at times, but to me he always seemed supremely self-confident. During the years that I was his rather timid postgraduate student, I cannot say that I was ever totally at ease in his presence, but he always treated me and his other fledglings with generosity and respect, and gave us his full attention when we came to report on and discuss our doctoral research projects. Moreover, George had an outstandingly sharp eye for tractable research questions and a good feel for how to address them experimentally. Under his guidance, I and his other students learned how to think and behave like experimental psychologists.
The second thing that comes to mind is witty. Goodness me, he was quick and funny. For all of the many years that I knew him, through thick and thin, George could make me laugh. Once when he was staying at our house in Cambridge, he criticised me for putting the toilet paper on the holder the wrong way (he never hesitated to speak his mind on any topic). I had it placed so that the paper rolled over the top; he thought it should roll from underneath. Even after years of knowing him as a friend, I did not have the courage to argue with him, but I felt slightly miffed by his complaint and — thinking to suggest that my “error” might not be so strange — I said, “I wonder what proportion of people do it each way?” His instant reply was, “Oh, I think we’re the dominant race.” Any sense of irritation on my part dissolved in giggles. A few months before he died, he’d had a couple of missing teeth replaced, and I emailed him to say that I was looking forward to seeing his new teeth on my next visit to London. His reply was that the teeth were available for inspection by appointment. Finally, I cannot resist repeating what I consider one of his best lines, which he himself reported in his aforementioned book. Not long after B. F. (Fred) Skinner’s 1957 book Verbal Behavior was published, George was standing near Skinner at a party when another young academic asked Skinner what he was planning to do next. Before he could reply, George quipped “Collect the data on which his book is based.”
When my husband Roy and I finished our PhDs at UCSD in 1971 — by which time I had perhaps begun to relax just a little in George’s presence — our first jobs were back in Toronto, from where we had come to UCSD. Then in 1975, when Alan Baddeley succeeded Donald Broadbent as the Director of the UK Medical Research Council’s Applied Psychology Unit (now known as the Cognition and Brain Sciences Unit), Alan offered both Roy and me positions and we moved to Cambridge. In part because of George’s youthful time at boarding school in England and in part because of his strong appreciation of Europe and European culture, George had an abiding attachment to the United Kingdom. He and Jean were regular visitors to London, Oxford, and Cambridge, eventually buying a house in Hampstead. Both of their sons obtained degrees from Oxbridge universities, and both eventually settled in England, providing further impetus for George and Jean’s continuing foothold in England. As a result of this, Roy and I saw George and Jean regularly in every one of the last 41 years. We all shared a love of good food and wine and often met for dinner in London; we occasionally travelled together to Wales, Paris, and Sicily. And for all of that time, George was his formidable and witty self. I am very lucky to have known him; I respected and loved him. And although he always referred to my chosen neuropsychological research area as “hole-in-the-head” research, I hope that he respected and perhaps even loved me.
University of Cambridge, United Kingdom
Rita E. Anderson
Memorial University of Newfoundland, Canada
During the heady early days of psychology at the University of California, San Diego (UCSD), in the late 1960s and early 1970s, I and several graduate students, visiting professors, and research assistants populated George’s lab. UCSD was a relatively new campus and George had assembled a faculty complement of high-powered experimental psychologists; they had created their ideal forward-looking graduate program, one not bound by eons of tradition. We were in the vanguard, thanks to George, and loving it.
Because George was the chair of that new department and the editor of a major journal (Psychological Review), as well as involved in innumerable other activities of which I was mostly unaware, his available time for students was probably very limited. None of us noticed any lack of attention, however, because George made time for us. In addition to our weekly lab meetings, George dropped into the lab for a coffee and informal chat at least once a day. There, we got to know George as someone who liked to laugh and share stories, who was genuinely interested in our daily lives and activities, who bummed cigarettes and liked good coffee, who reacted to rejection letters from editors and then got on with it — in other words, he was a real person.
George combined his impressive intellect and interest in wide-ranging topics with an approachable and caring persona to create a potent training ground for his graduate students. Over the years, I’ve come to realize that our weekly lab meetings involved much more than project updates and opportunities for us to hone the basics of research design and analysis, to develop critical listening and thinking skills, to practice thinking outside of the box empirically and theoretically, and to learn how to make constructive suggestions. Equally important was the way that George used our lab meetings to promote respect for and bonding among lab members. Because he treated all of us as important individuals, we looked forward to our lab meetings as intellectually stimulating, fun events.
George created a family sensibility that worked remarkably well. When he and Jean sailed from Los Angeles for a sabbatical at Oxford, they invited us to a shipboard sailing party — it was great fun! To keep in touch during that year, George had the idea of using Dictaphone belts. Instead of relying totally on sterile typewritten letters (no email then), we mailed belts back and forth weekly; as soon as a new belt arrived, we would all gather around the machine to listen to George’s comments, news, and gossip. In later years, an important activity at conferences was joining George for a gabfest about the who, what, when, and how of everyone associated with the lab.
George had high standards and promoted excellence in science, communication, and respect for one another in his students. He respected our individuality and rose to the occasion of our foibles and successes, sometimes in the most humorous ways. For instance, as I was preparing to leave for a postdoc in Toronto, I quipped that if the job market bottomed out, I could always go back to waitressing; without batting an eye, George replied, “Fine, but only in one of two Viennese restaurants in Toronto.”
After all these years, I continue to learn from him; for example, when facing a difficulty with a graduate student, my first thought is: “How would George have handled this situation?” Not being George, I probably don’t get it quite right, but maybe I get close. I was privileged to know George as a mentor and as a friend; he was central to who I became, both as a person and as a professional. I thank him and Jean for so many fond memories, lessons, and laughter.
Pat Worden Benson
California State University, San Marcos
George Mandler’s recent passing brought me back to the years when I was one of his graduate students at University of California, San Diego (UCSD), in the early 1970s. George and his colleagues had founded the psychology department a few years earlier. He was shaping it to be a leading institution in the revolutionary new area of cognitive psychology. Everything at UCSD was new and up-to-date. We tested subjects on a cutting-edge PDP-8 computer that was the size of a jumbo refrigerator. We used portable calculators as large as paperback books. We carried stacks of IBM cards to the university mainframe for big analyses. We were in awe of our professors and our peers; we felt like pioneers.
George watched over us with benevolent bemusement. Our lab meetings were intense, thoughtful, productive, and filled with George’s colorful stories, gossip, and occasionally earthy language. George mentored by steering our research in the general direction of the organization of memory, making sure everyone knew what the others were doing, and encouraging us to go at it. Our ideas often were subjected to what Jean once called “Mandler troubleshooting”: When presented with an experimental design of dubious possibility, he’d tell the story of the optimist looking into a room filled to the ceiling with dung and exclaiming “there might be a pony in there somewhere!” He often advised us to “do what you think best” and was genuinely excited when something good turned up. Behind his sophisticated and worldly “Viennese uncle” persona was a man driven by intellectual curiosity and the joy of the research enterprise.
Because I was a student at the time, I had only a superficial appreciation for George’s essential role in the creation of the world-class UCSD psychology department. A dozen years later, I was fortunate to become a founding faculty member and administrator at a new campus in the California State University system. I can see echoes of George’s laissez-faire style in how I approached my role as Vice President for Student Affairs. I did my best to secure support for the division, set a general direction, made sure everyone knew what the others were doing, and got out of the way. It was then that I understood the thrill of building a new institution from scratch, and perhaps why George seemed to take such delight in his world at UCSD. Looking back at the totality of his life achievements, I count myself lucky to have spent my graduate years under George Mandler’s unforgettable influence.
London, United Kingdom
We first spent proper time with George when we went to visit him and his wife Jean at their house in San Diego in about 1995. I was 11, Anna must have been 8, and we had a lot of fun together. We explored their house, with its extensive collection of B. Traven books, abstract art, and swathes of classy beige and brown leather, evoking 1970s academia and lives of active contemplation and cultural acquisition.
One sunny afternoon George turned to us and, in his lovingly rasping voice, said, “You don’t have any Jewish grandparents left, do you? How about you adopt a couple? We could be that for you.”
Now, one of Anna’s and my favorite games was playing “offices” — we would sit at twin computers, speaking in professional jargon, and printing out important documents and pretending they were faxes (all children of academics in the 1990s were the same, I’m sure). We loved the idea of adopting George and Jean as our Jewish grandparents — the inverse age game, the control we could have over the demands of the role and, most importantly, the “important document” we could make for it.
We ran to the computer, opened a document and titled it, in big letters, “The Butterworth Sisters’ Adopt-A-Jewish-Grandparent Contract.” Below, for about three pages, we outlined the actions they were to take in this new role and what we would do for them in return, and then placed dotted lines at the bottom for us all to sign. The demands were things such as: teach us about Jewish history and culture; remember our birthdays and advise us with tips for those ages; and have things like Werther’s Originals in their pockets. We, in return, promised to tell them how our exams were going, ask them for help with difficult homework, and send them birthday cards. They were delighted, and we all signed with joy.
That contract is somewhere in our attic, and most of the agreements were never met. However, from that day forth, George and Jean invited us to call them Opa and Granny, and we would always refer to them as our adopted Jewish grandparents. Despite the agreement not being honored in an itemized way, it was still a contract we all held dear, knowing that as the sun was setting over those Californian hills, we had agreed to be in each other’s lives.
The last time we saw George, we went to lunch around the corner from their house in Hampstead. We were talking about their travels (I had always marveled at their Alaskan and Amazonian cruises and their other exciting ways of getting around the world) and I asked George what his No. 1 place was — one we should make sure we visited as well. He said it was too difficult to pick just one, and the conversation moved on. Some two courses later, George lifted his head and said very clearly “Costa Rica. The views, the geography, the potential adventure — that’s the place to go.” We didn’t sign anything with him that day, but we will definitely honor that advice.
University College London, United Kingdom
George was never interested in mathematical cognition; nevertheless, he published with Billie Jo Shebo an important and influential paper in 1982 on subitizing. What he was really interested in was the limits of conscious experience, and subitizing provided a classical route to study this. For readers who, like George, are not interested in mathematical cognition, I quote their characterization of subitizing, which, unlike many modern papers, notes its relationship with the limits of consciousness.
“The term subitizing was coined by Kaufman, Lord, Reese, and Volkmann in 1949 to describe the rapid, confident, and accurate report of the numerosity of arrays of elements presented for short durations. They noted that this process, different from counting and estimating, was restricted to arrays with 6 or fewer elements. Ever since the general awareness of some such process in the nineteenth century, the phenomenon has been a benchmark for the limited capacity of human consciousness.” (Mandler & Shebo, 1982, p. 1).
This paper is important for several reasons. First, as the authors point out, the basic phenomenon of subitizing was observed more than 100 years ago, well before Kaufman et al. coined the term (1949). In George’s characteristically scholarly way, it cites observations by Bourdon in 1908, Cattell in 1886, Jevons in 1871, and William Hamilton in his 1859 Edinburgh lectures. (George was a keen student of the history of psychology, and liked to remind people that important theoretical problems in psychology were not raised first in recent issues of Psychological Review). The question arises as to why we should be rapid, confident, and accurate in reporting small numerosities, but not in reporting larger numerosities. Does this imply separate processing mechanisms?
Kaufman et al. distinguished counting (“the usual pairing of objects with the incrementing number series”) from subitizing for numerosities less than five and “estimating” for numerosities greater than five (presented too briefly to allow serial enumeration), which they believed depended essentially on the same processing mechanism. They also concluded, as Mandler & Shebo (1982) point out, that there is no such phenomenon as “the immediate cognition of number,”; that is, they are not sensations of the same order as blue, green, red, round, or square, a position that they endorse on the basis of increasing reaction times even in the subitizing range. 
Second, it is currently widely, and fashionably, assumed that there are indeed two processes, but exactly what they are has given rise to several competing theories; however, as the paper points out,
“Kaufman et al. invoked the term subitizing for a process of arriving at judgments of visual numerosity that applies to arrays of 6 or fewer objects. They did not specify what that process might be. Some of the subsequent discussions in the literature might well have paid heed to the coiners of the term. Subitizing is defined by systematic changes in the slope for judgments of numerosity; it is not a concept disputable on theoretical grounds. To assert that people do or do not subitize requires only a demonstration that certain discontinuities can or cannot be demonstrated in the data obtained from numerosity judgments. To determine what processes are involved during subitizing is indeed a theoretical enterprise, which Kaufman et al. touched on only tangentially.” [p. 2]
Before I turn to the theories, it is worth noting three important methodological advances demonstrated in the paper. First, the standard procedure for demonstrating subitizing is to ask participants to name the number of objects on a screen (as in Figure 1). The naming time (RT) slope is relatively flat (about 50 milliseconds per additional item for adults) for numerosities less than or equal to four, and then it changes to a steeper slope (about 270 to 400 milliseconds) for numerosities greater than four. For numerosities greater than or equal to seven with exposure brief enough to prevent counting, the RT tends to an asymptote. Mandler and Shebo, as well as Kaufman et al., characterize the three slopes as subitizing, counting, and estimating.
Accurate measurement of the cognitive process will depend, in part, on how quickly each number word can be retrieved by the participant, and this will depend on the frequency of the word in the language — with the names of small numbers being more frequent than large numbers — and on how quickly the word activates a voice key, which will depend on the acoustic properties of the word. The study involved a separate experiment in which participants named a digit presented on the screen and RTs to the number of dots were adjusted in the light of this because some number names just take longer to retrieve and to activate the voice key.
The other methodological innovation was to use two types of dot stimuli — random and ‘canonical’. The reasoning was as follows:
“If it is the case that children count arrays as small as 2 or 3 [as demonstrated by Gelman & Gallistel, 1978] but that adults apparently do not, then it would be reasonable to assume that adults have developed a novel strategy for dealing with these arrays. Such a strategy would be different from that used for arrays of 4 or more.
“Our basic argument is that adults have developed canonical pattern perceptions for arrays of 2 and 3 — doublets (perceptually straight lines) and triplets (triangles). These canonical patterns develop slowly during childhood, such that 5-year-olds apparently have adopted the twoness pattern but not yet the triplet triangular pattern. Random presentation of displays of 2 and 3 will always produce the canonical pattern for 2 and frequently the canonical triangle for 3. If there is a canonical pattern for four-ness[,] it presumably is a square array, but random generation will not frequently produce such a pattern.
“We believe that if canonical patterns are not available, adults count the arrays. If the array can be ‘held in consciousness,’ that is, if it has 6 or fewer elements, it can be counted accurately after short presentations of a few hundred milliseconds. For arrays with more than 6 elements, postexposure counting fails and estimating procedures are substituted.” [p. 5]
This hypothesis was tested and supported by showing the effect of practice with novel arrays, such as the larger arrays in Figure 2. This pattern-recognition hypothesis has informed studies with typical children, typical adults, and with atypical participants. For example, an fMRI study showed a clear difference in the brain activity between numerosities less than or equal to four and those greater than four, and one possible explanation advanced by the authors of the study is what they called “the perceptual groupability of the items (Mandler & Shebo, 1982)” (Piazza, Giacomini, Le Bihan, & Dehaene, 2003, p. 1244).
Not everyone shows a subitizing effect or an advantage for canonical arrays, and this in itself is interesting and revealing about individual differences in cognition: In particular, individuals may have a severe disability in learning arithmetic because they suffer a core deficit in extracting number from sets of objects, a precondition for understanding cardinal concepts. These individuals are usually referred to as “dyscalculic” (Butterworth, 2005). For example, we found no subitizing effect for one adult dyscalculic named “Charles”; he always counted, even for two objects, and he also showed a reduced advantage for canonical stimuli (Butterworth, 1999).
In a more systematic study, the advantage for canonical stimuli is smaller or even absent with developmental dyscalculics (DDs). Ashkenazi, Mark-Zigdon, and Henik (2012) write, “It is possible that the deficit in the DD participants in processing canonical arrangements may be due to difficulties in pattern recognition.” A lack of advantage has been found in children with cerebral palsy [CP] with a low subitizing limit, and Arp and Fagard (2005) write that:
The children with cerebral palsy “Were impaired to the same extent on the pattern recognition task as on the subitizing task, and had a short visuospatial short-term memory span. These results suggest that the low subitizing limit of [children with cerebral palsy] stems from a (nonnumber dependent) lesser capacity to perceive a dot configuration as a gestalt. A low subitizing limit was almost always associated with a right-hemisphere lesion.” p89
The third methodological advance is very straightforward but rarely applied. It is that the changes of slope depend on exposure time. If you allow observers enough time to count, they will, and the change of slope will begin at about four items. With brief exposure (especially if the display is followed by a mask), counting is suppressed, or at least, discouraged, and then the change may happen later. Also with brief exposure, increases in set size beyond six scarcely affect the slope at all, as can be seen in Figure 1.
Recently, subitizing has been identified as a core resource in the development in the concept of cardinal number. Fuson, summarizing an extensive body of research, writes:
Young children’s first use of cardinal uses of number words refer to small numbers of entities and seem to rest on subitizing, the immediate apprehension of small numerosities … young children do learn to subitize at least two entities and that many children learn to label particular patterns or situations with a cardinal label … Children continue to use this pattern-based approach of seeing certain situations as patterned sums of small numbers of entities … These pattern-based small numerosities continue to play important roles in some equivalence, addition and subtraction situations. (Fuson, 1992) (1992, p. 131)
Fuson’s position incorporates the Mandlerian idea of pattern-based support for enumerating entities, though without explicit citation of the paper.
More recently, it has been proposed that humans start with two separate core systems, one for small numbers and one for large numbers. The large number system provides approximate and analogue representations of numerosity, while the small number system provides exact representations of numerosities (Feigenson, Dehaene, & Spelke, 2004).
This is contra Mandler & Shebo, and indeed contra Gelman & Gallistel, who argue for a single system of estimation that can be supported by acquiring patterns. The hypothesis of two core systems raises the question of how children are able to link what they know from the small number system to what they know from the large approximate system. The link according to Carey and colleagues, as well as many others, is language, and, in particular, the “integer list” (counting words). The idea is that children “bootstrap” (i.e., inductively generalize) from their knowledge of counting words and exact numerosities in the subitizing range to concepts of exact larger numbers (Carey, 2009; Le Corre & Carey, 2007). It is not yet clear how this bootstrapping works (see Gelman & Butterworth, 2005).
Nevertheless, there does seem to be a relationship between subitizing and the development of number competence and arithmetic. Individual differences in children’s ability to subitize are predictive of their arithmetical ability (Berch, 2005). So for example, we have shown in a longitudinal study that 7% of kindergarten children have a subitizing range of just 2 items, while 93% can subitize 3 or even 4. Our findings indicate that the 7% will continue to be slower at counting and estimating at least until the age of 11, and also worse at age-appropriate arithmetic for the whole of that period (Reeve, Reynolds, Humberstone, & Butterworth, 2012). Thus, there is a practical educational implication: This very simple test of subitizing is a reliable indicator of which children are likely to struggle to learn arithmetic.
This still leaves open the question of exactly what the subitizing mechanism is. One popular theory is that it is a “preattentive” mechanism for maintaining up to four “object files” in visual working memory for exact enumeration. This idea was originally proposed with different terminology by Pylyshyn and Trick (Pylyshyn, 1989; Trick & Pylyshyn, 1993; Trick & Pylyshyn, 1994). For small numerosities, objects are preattentively identified in parallel, but larger numerosities are enumerated sequentially in a way that demands attention.
Thus, Piazza et al. (2003) conclude:
Behavioural manipulations … support the parallel/serial dichotomy of subitizing and counting. On the one hand, performance in the counting range, but not the subitizing range, is affected by manipulations that alter the orientation of gaze and/or attention … When the parallel separation of targets from distractors is made impossible, for instance when quantifying Os among distracting Qs, subitizing disappears although counting can be largely unaffected (Trick & Pylyshyn, 1993, 1994, p. 1244).
Now giving observers the opportunity to count may suppress estimating, as Mandler & Shebo suggest. This is why in their series of studies exposure time for the displays is systematically manipulated to promote estimating and discourage counting, or, in other exposure conditions, to enable counting.
More recently, this preattentive account has been called into question. We showed that modulating attentional load affects estimation in the subitizing range, and hence provides an argument against the preattentive account (Vetter, Butterworth, & Bahrami, 2008, 2010). Using a similar methodology, Burr, Turi, & Anobile (2010) have shown again that it is subitizing and not estimation that requires attentional resources (Anobile, Cicchini, & Burr, 2015).
Mandler and Shebo anticipated this issue: “The theoretical problems to be approached concern the nature of parallel processing on the one hand and a better understanding of the limitation of the attentional/conscious capacity system on the other.” [p. 20]
Their overall conclusion is still worth taking seriously.
We have demonstrated two quite different mechanisms to explain the subitizing phenomenon by dividing the positively accelerated slope of reaction times into a combination of acquired canonical patterns and counting of the array held in consciousness. We ascribe the discontinuity between these two “subitizing” processes and the estimating used for larger arrays to the more general limitation of human attention or consciousness. Only up to 6 or 7 discrete events can be held in consciousness and counted. Larger arrays cannot be so maintained and the accuracy of counting suffers; estimation is substituted for it.
The shallow, or often flat, slope of reaction times for arrays of 1 to 3 items seems to be explained by acquired canonical patterns. As we noted earlier, a similar suggestion was made by Woodworth and Schlosberg in 1954. The fact that canonical patterns for larger arrays could be acquired by our subjects and were responded to as quickly and as accurately as the smaller arrays supports this argument. Our results also support in large part Warren’s (1897) conclusions, discussed earlier. However, we have suggested that instead of perceptive counting and inferential counting being available in addition to progressive (incremental) counting, the apparent perceptual apprehension of numerosity is probably also a case of inference. Patterns of small numbers are familiar and responded to “inferentially”; people “know” that a triangular pattern represents 3 and respond accordingly.
The notion of “mental counting,” the application of a simple counting mechanism to a representation of the array that is held in consciousness, deserves more detailed examination. If the counting mechanism operates on the conscious array in the same way that it operates on the physically presented array, then the slope for reaction times in arriving at numerosity judgments should be similar under the two conditions. (pp. 18–19)
So, why was George’s contribution to mathematical cognition inadvertent? First, as I said, he wasn’t interested in mathematical cognition as such, despite making this decisive contribution to it. I recall asking his opinion of some experiments we were carrying out to test his theory, and he just didn’t want to know, and he especially wasn’t interested in our finding of a separate neural locus for subitizing (subsequently published as (Vetter et al., 2010).
He was using subitizing just as a way of exploring the limits of consciousness, and as his paper points out, “The limitation of human consciousness in the apperception of discrete events has been known at least since the early 18th century (see Hamilton, 1859)” (p. 18). So subitizing was, in a sense, a classical approach to this issue. The final paragraph in the paper, however, was about mathematical cognition and not about the limits of consciousness:
Thus, one of the next steps in the use of the subitizing paradigm will be to study the effect of visual, auditory, and more general cognitive interference tasks on the cognition of numerosity. The distinction between the canonical pattern process and mental counting will make it possible to study interference and, indirectly, the use of general-purpose resources as they affect and are affected by these two processes separately. p21
It is disappointing to all of us working on mathematical cognition that he didn’t follow up on these intriguing ideas, because it has taken more than thirty years for others to come up with similar ideas (Arrighi, Togoli, & Burr, 2014). They should have read George and Billie Jo’s paper more carefully.
Anobile, G., Cicchini, G. M., & Burr, D. C. (2015). Number as a primary perceptual attribute: A review. Perception. doi:10.1177/0301006615602599
Arp, S., & Fagard, J. (2005). What impairs subitizing in cerebral palsied children? Developmental Psychobiology, 47, 89–102. doi:10.1002/dev.20069
Arrighi, R., Togoli, I., & Burr, D. C. (2014). A generalized sense of number. Proceedings of the Royal Society of London B: Biological Sciences, 281(1797). doi:10.1098/rspb.2014.1791
Ashkenazi, S., Mark-Zigdon, N., & Henik, A. (2012). Do subitizing deficits in developmental dyscalculia involve pattern recognition weakness? Developmental Science, 16, 35–46. doi:10.1111/j.1467-7687.2012.01190.x
Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38, 333–339.
Burr, D. C., Turi, M., & Anobile, G. (2010). Subitizing but not estimation of numerosity requires attentional resources. Journal of Vision, 10. doi:10.1167/10.6.20
Butterworth, B. (1999). The mathematical brain. London: Macmillan.
Butterworth, B. (2005). Developmental dyscalculia. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 455-467). Hove: Psychology Press.
Carey, S. (2009). Where our number concepts come from. Journal of Philosophy, 106, 220–254.
Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307–314.
Fuson, K. C. (1992). Relationships between counting and cardinality from age 2 to 8. In J. Bideaud, C. Meljac, & J. P. Fisher (Eds.), Pathways to number, children’s developing numerical abilities. Hillsdale, NJ: LEA.
Gelman, R., & Butterworth, B. (2005). Number and language: how are they related? Trends in Cognitive Sciences, 9, 6–10.
Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number (1986 ed.). Cambridge, MA: Harvard University Press.
Kaufman, E. L., Lord, M. W., Reese, V. J., & Volkmann, J. (1949). The discrimination of visual number. American Journal of Psychology, 62, 498–525.
Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105, 395–438. doi: 10.1016/j.cognition.2006.10.005
Mandler, G., & Shebo, B. J. (1982). Subitizing: An analysis of its component processes. Journal of Experimental Psychology: General, 11, 1–22.
Piazza, M., Giacomini, E., Le Bihan, D., & Dehaene, S. (2003). Single-trial classification of parallel pre-attentive and serial attentive processes using functional magnetic resonance imaging. Proceedings of the Royal Society B, 270, 1237–1245.
Pylyshyn, Z. W. (1989). The role of location indexes in spatial perception: A sketch of the FINST spatial-index model. Cognition, 32, 65–97.
Reeve, R., Reynolds, F., Humberstone, J., & Butterworth, B. (2012). Stability and change in markers of core numerical competencies. Journal of Experimental Psychology: General, 141, 649–666. doi:10.1037/a0027520
Trick, L., & Pylyshyn, Z. W. (1993). What enumeration studies can show us about spatial attention: Evidence for limited capacity preattentive processing. Journal of Experimental Psychology: Human Perception and Performance, 19, 331–351.
Trick, L. M., & Pylyshyn, Z. W. (1994). Why are small and large numbers enumerated differently: A limited-capacity preattentive stage in vision. Psychological Review, 101, 80–102.
Vetter, P., Butterworth, B., & Bahrami, B. (2008). Modulating attentional load affects numerosity estimation: Evidence against a preattentive subitizing mechanism. PLoS One, 9(e 3269), 1–6. doi:doi:10.1371/journal.pone.0003269
Vetter, P., Butterworth, B., & Bahrami, B. (2010). A candidate for the attentional bottleneck: Set-size specific modulation of the right TPJ during attentive enumeration. Journal of Cognitive Neuroscience, 23, 728–736. doi:10.1162/jocn.2010.21472
 Burr and colleagues have shown that visual numerosities are subject to adaptation just like color, and conclude that indeed numerosity is a “primary visual property” that is immediately and obligatorily grasped (Anobile, Cicchini, & Burr, 2015).
Fergus I. M. Craik
University of Toronto, Canada, and Rotman Research Institute
Given the fleeting nature of fame in our busy discipline, it is worth reminding ourselves and others what an outstanding contributor George Mandler was to our science. He was a pioneer of the cognitive revolution, publishing influential papers and books on memory and emotion. In his memory work, he was a major proponent of the dual-process theory — the idea that recognition memory involves two underlying processes, familiarity and identification. These processes are dissociable; as a common case of familiarity without identification George gave the nice example of failing to recognize “the butcher on the bus.” His work on this topic went on to stress the perceptual nature of familiarity as opposed to the conceptual nature of identification and recollection, and also addressed the notions of intraitem integration (largely perceptual) and interitem elaboration (largely conceptual). George also showed how the strengths and weaknesses of amnesic memory fit dual-process theory; he championed the reemergence of consciousness as a respectable topic, analyzed the process of subitizing, and reduced George Miller’s magical number from seven to the more plausible four. Additionally, he founded and developed the University of California, San Diego, department of psychology into one of the world’s best groups, edited Psychological Review for 6 years, and chaired the governing board of the Psychonomic Society — a major figure indeed!
Browsing through some of George’s work, I was struck by the point that, although he could be delightfully edgy in conversation and debate, he was an integrator rather than a revolutionary who cast out all old ideas. As an example, in his influential 1962 article “From Association to Structure,” he argues for a model embracing both associations and organization. He wrote: “The major notion to be presented in this paper … is the proposition that an associationist and a cognitive view of behavior may work side by side — that they are compatible.” This approach, stressing integration and building on current knowledge, is a hallmark of the Mandler style.
Another salient aspect of George’s character was his interest in the work of young researchers and his genuine efforts to encourage and promote their work. As a young faculty member working in London, I once attended a meeting in Oxford where George was presenting a paper. I had some questions afterward, but George had to hurry off. He took my name, however, and promised to call me in London — which he did (to my mild amazement), and we had a long and fruitful talk.
This initial meeting led to further exchanges and ultimately a warm friendship involving his wife, Jean, and my wife, Anne. There are many more tales — visits to La Jolla, conferences in midsummer Sweden, walks in Scotland, splendid meals in Hampstead and by the river Thames — but they can wait to be retold with pleasure in congenial company over a glass or two of good malt whisky. I miss George a lot, but am very glad to have known him.
Arthur C. Graesser
The University of Memphis
I was a graduate student of George Mandler at the University of California, San Diego (UCSD), during the heyday of the cognitive revolution in the mid-1970s. George’s major interests in cognitive psychology at that time were organization of memory, the impact of organization on recognition memory, and the role of familiarity on recognition judgments. At the same time, he was writing a book on emotions (Mind and Emotion) and editing the esteemed journal Psychological Review. There were weekly research meetings in George’s lab during which we presented data and had group commentary. Those were convivial, jovial, illuminating meetings with absolutely no pressure — the perfect combination of science, challenge, and play.
George gave me absolute freedom over whatever research I wanted to conduct. That included research adventures in other labs in cognitive psychology (with Lynn Cooper, Jean Mandler, Don Norman, and Dave Rumelhart), anthropology (with Roy D’Andrade), and linguistics (with Tim Smith), along with visiting professors (Harlan Lane and Gordon Stanley). This interdisciplinary environment planted the seeds that created the field of cognitive science. As I explored research in these other labs, I also maintained a strong research program with George that investigated memory and his emerging explorations into consciousness. Quite clearly, my research and training experiences at UCSD at that time sprouted in many directions, including discourse comprehension and conversation.
My favorite experience with George was serving as a teaching assistant for his course on emotions. I read a draft of his book Mind and Emotion and gave diligent feedback. I prepared lectures while he was out of town and passionately put everything I had into making the materials come to life. I was touched when students applauded and when one student called Jean Piaget to solicit his opinion on a developmental theory. I was intrigued by George’s theory that connected emotions to schemas, goal interruptions, and other cognitive constructs. George was in the midst of transforming the field of psychology to open everyone’s eyes to the science of consciousness and emotion. That eventually happened, and George was part of the process that brought it to fruition. At the time, however, I did not appreciate fully the significance of George’s vision and impact. I was merely a curious student who deeply respected his mentor.
Twenty-five years later I again became immersed in George’s views on emotion. I investigated the moment-to-moment emotions that occur when students read difficult material and interact with intelligent tutoring systems (the important affective states are confusion, frustration, anxiety, boredom, and flow or engagement, with a sprinkle of surprise and delight). Goal interruptions, contradictions, and discrepancies with schemas were major predictors of deep learning. George’s theoretical framework played a central role in explaining the moment-to-moment affective states and their impact on learning, as we discovered in my research collaborations with doctoral students (Scotty D. Craig, Sidney D’Mello, Nia Dowell, Carol Forsyth, Blair Lehman). So George’s research in the mid-1970s had a major impact on my research published 30 years later, with a 20-year hiatus in between.
I recall having an intense debate with George in 1974 on his theory of organization of memory. George claimed that memory was organized hierarchically and that each node in the hierarchy directly dominated only 5 1 nodes. I argued that data suggested 3 nodes rather than 5, based on empirical evidence from some of my experiments and a computational argument on minimizing search time in a hierarchy. George also had his data. . The debates were passionate and always illuminating. His penultimate argument was simply expressed: “Does it really matter whether it is 3 or 5? The fundamental issue is that there are node constraints on the organization of long-term memory.” That was the grand finale of that debate.
A couple of decades later I read Interesting Times: An Encounter With the 20th Century 1924-, George’s autobiographical reflections on science, his career, his immigration to the United States from Austria, and his life in other European countries before, during, and after World War II. I was so absorbed in the book that I read it in a single sitting. It is a must-read for anyone interested in the history of psychology and in George Mandler’s unusual life.
I still remember many of George’s edgy quips that made us think (e.g., “Data trump good ideas that come a dime a dozen.”), question (e.g., “Radical behaviorism was a minor hiccup in the history of psychology.”), and laugh (e.g., “Show me a very healthy body and I’ll show you a sick mind.”). The research vision of George Mandler spanned centuries, and he provided a mature intellectual foundation for researchers who were not entirely consumed by the research fashions of the year.
Three years ago my wife Nancy and I had dinner with George and Jean in London. His mind was as sharp as ever. I felt like I was transported back 40 years to when I was a graduate student at UCSD — sparks of creative ideas, novel slants on controversial issues, and spontaneous visceral laughter permeated the conversation.
University of Hertfordshire, United Kingdom
George Mandler was undoubtedly one of the great cognitive psychologists of the 20th Century, one who was instrumental in shaping our thinking about memory processes (activation, organisation), consciousness, cognition, and emotion for several decades. What I have started to realize and appreciate more recently, though, is how far ahead of his time he was in his thinking about memory. In several of his book chapters published in 1986, 1989, and 1994, he repeatedly was emphasizing the spontaneous nature of memory retrieval. Here is one of my favorite quotes from these chapters:
Much of human memory in everyday situations does not involve the classical retrieval studied in the psychological laboratory. We interrupt our stream of thought with ideas that suddenly “come to mind,” we are frequently “reminded” of one or another occurrence in the past, and often we are aware of memories whose apparent irrelevance to the requirements of the moment surprises us. (1986, p. 291)
Another favorite quote from George is, “Deliberate retrieval of information seems to be the exception rather than the rule. One can argue that much of the nondeliberate access to previously stored/encoded information occurs automatically or as a result of higher order spreading activation” (1989, p. 103).
At the time, these were very bold statements, which contrasted sharply with the accepted wisdom that memory was effortful, deliberate, and constructive. Even if such transient and spontaneous memory phenomena did exist, they were considered not only unsuitable for empirical research, but unimportant and uninteresting (rather like noise in a system). George A. Miller, for example, suggested that psychologists should leave involuntary memories to Proust and his fellow artists for consideration, and that “a scientist should do better to study the workaday memory that guides our plans … answers promptly when it is called[,] and stands still long enough to be measured” (1962, p.181).
How things have changed! There is an abundance of research on spontaneous cognitive phenomena, using a variety of different methods such as surveys, diaries, experience sampling, and even laboratory experiments, that enable researchers to capture and measure these seemingly transient and nonmeasurable phenomena (e.g., mind-wandering, involuntary autobiographical memories, spontaneous future thinking, intrusive memories, musical earworms). Results from these studies have begun to show that in everyday life, involuntary recall may be the norm rather than the exception — exactly in line with what George was saying some 20 to 30 years ago.
I consider myself extremely fortunate to have met George in the late 1990s. Obviously, my acquaintance with his work started much earlier, before we actually met for lunch in a nice London café near his home in Hampstead on September 17, 1997. I had not thought of the actual date for many years and had no recollection of it when it popped into my mind, except for the vague feeling of familiarity that this was the day we met. Having checked in my pocket calendar at the bottom of the drawer in my office desk, I saw that it turned out to be correct! If George were still alive, I am sure we would have discussed this incident as we usually did during our 20 or so years of lunch meetings in London. Below, I describe how George was instrumental in the conception of a new line of research on involuntary semantic memories, or “mind-pops” (his preferred term).
I completed my undergraduate and postgraduate degrees in Tbilisi, Georgia, which was part of the Soviet Union up until 1991. In March 1993, I moved to the United Kingdom (UK) as a Royal Society Research Fellow to work at Cardiff University in Wales. In 1995, I accepted a research position at the University of Hertfordshire and moved to Hatfield (north of London). At the time, my main research area was prospective memory, or remembering intended actions in the future. Everyday examples of prospective memory involve remembering to pass on a message when seeing a colleague or taking a medication at particular times. Given that in everyday life, the retrieval of a prospective memory task often seems to be spontaneous (e.g., just before 2:00 p.m. you may suddenly remember that you had promised to call a friend at 2:00 p.m.), I probably was sensitized to other spontaneous thoughts and memories that were coming to my mind.
Sometime after my arrival in the UK, I started noticing that single words, phrases, names of people, or visual images (of places, objects, etc.) would pop into my mind and would surprise me with their irrelevance to the current situation. For example, I might be preparing a cup of coffee in the kitchen and thinking about an upcoming research seminar when suddenly a word like “dilapidated,” the name of an actor like “Tom Cruise,” or a street like “Bessemer Road” would pop into my head and make me wonder: Why on earth did I think of that? Even more fascinating were the few cases in which an English word would come to my mind, but I would not know what it meant. However, when checking its meaning in the dictionary, I would realise that one of the meanings of the word matched the situation I was in when the mind-pop occurred, almost implying that my unconscious knew the meaning of the word that I had forgotten consciously.
I became more curious about the phenomenon, and once when I experienced a particularly interesting mind-pop, I started keeping a diary of mind-pops and recording their accompanying circumstances (e.g., activities I was involved in, possible triggers, thoughts at the time, etc.). I kept a diary for 4 months in 1995 and recorded 126 mind-pops. I called them “involuntary semantic memories,” as they were disconnected fragments of semantic knowledge rather than memories with contextual details (i.e., what, when, how). Having analysed the data, I confirmed my suspicions that these transient occurrences seemingly were coming from nowhere and lacked any immediate external or internal triggers. However, what became obvious was that they were not completely random, as in almost 50% of cases I was able to ascertain that the content of a particular mind-pop had been encountered directly or indirectly in the environment or in my thoughts in the recent past. For example, the name ‘Tom Cruise’ could have been mentioned on TV few days before it popped into mind. This indicated to me that the activations of representations in response to objects and words encountered in daily life persisted much longer than demonstrated in the laboratory studies of repetition and semantic priming (as suggested by George in Mandler, 1994).
By September 1997, I plucked up my courage and presented initial findings from my diary data as a poster at the British Psychological Society (BPS) Cognitive Section Conference in Bristol. I did not feel confident presenting findings from a single participant diary study on a phenomenon that nobody seemed to know about — except George, who had described it several times in his book chapters, which were my only references for the poster. Shortly before this conference, I had heard from Brian Butterworth that George lived in London for 6 months every year. Therefore, after the conference I emailed George, introduced myself, and sent him my poster. George responded and suggested we meet in London to discuss mind-pops.
Meeting someone who seemed to know everything about mind-pops was an incredibly uplifting experience, and we had a very interesting discussion about the phenomenon and potential underlying mechanisms. George immediately suggested what I needed to do next: First, repeat the diary study on myself, and second, devise a brief Mind-Popping Questionnaire and administer it to a large sample of students to validate the existence of this phenomenon in a younger population. I kept a diary and administered a questionnaire that George and I co-designed to 205 undergraduates. We presented the findings at the BPS Cognitive Section Conference in September 1998.
When the paper eventually was written in 2001, we first submitted it to the Journal of Experimental Psychology: General. It was returned to us without being sent out for review. The action editor’s view was that it did not satisfy the requirement of being an experimental work. I remember that George was quite upset by this evaluation, and he even wrote back to the editor to indicate his disagreement. We then submitted the paper to Cognitive Psychology, and after we conducted a fourth study, in which 50 participants kept a diary of mind-pops for 1 week and a diary of involuntary autobiographical memories for another week (in counterbalanced order), the paper was accepted and published in 2004, with the title “Out of One’s Mind: A Study of Involuntary Semantic Memories.”
I am absolutely certain that without George’s support and guidance, this work would have stayed in my drawer and never been published. Even in the early 2000s, work of this type was not looked upon favorably. In 2006, Simone Schlagman and I submitted a paper to the Journal of Experimental Psychology: Learning, Memory, and Cognition on how to elicit and measure retrieval times of involuntary autobiographical memories in the laboratory; it was rejected with comments that, because participants themselves pressed a button to notify that they had experienced an involuntary memory, the data was based on self-report and hence was not valid. This type of comment seemed to invalidate also all the diary studies on involuntary autobiographical memories, as they, too, are based on participants’ self-reports of their inner mental states.
I will remain forever grateful to George for his friendship, genuine interest in everything I was doing, and continuous moral support and advice throughout these years. He will be missed!
Mandler, G. (1986). Reminding, recalling, recognizing: Different memories? In F. Klix & H. Hagendorf (Eds.), Human memory and cognitive capabilities (pp. 289–297). Amsterdam, the Netherlands: Elsevier Science Publishers.
Mandler, G. (1989). Memory: Conscious and unconscious. In P. R. Solomon, G. R. Goethals, C. M. Kelly, & B. R. Stephens (Eds.), Memory: Interdisciplinary approaches (pp. 84–106). New York, NY: Springer.
Mandler, G. (1994). Hypermnesia, incubation, and mind popping: On remembering without really trying. In C. Umilta & M. Moscovitch (Eds.), Attention and performance XV (pp. 3–33). Cambridge, MA: The MIT Press.
Miller, G. A. (1962). Psychology: The science of mental life. London, UK: Penguin Books.
Donald A. Norman
University of California, San Diego
I met George Mandler in 1965 when he was the founding chair of the department of Psychology at the newly formed University of California, San Diego (UCSD), and — along with the other two members of the few-months-old department, Bill McGill and Norman Anderson — he hired me.
George was a major force in the development of UCSD, the department of psychology, and the Center for Human Information Processing (CHIP). This was in an era dominated by behaviorists, when the very concept of human information processing was unsettling and disturbing to many. CHIP thrived, as did information-processing psychology, which then morphed into cognitive psychology.
George was a mentor and a friend, scolding me when I needed it (which was frequent), acting as the editor of my book, Memory and Attention, and guiding my career. Many of my favorite publications came from our conversations. For example, his comment, “I poured myself a glass of scotch, put the glass into the liquor cabinet, and walked away with the bottle,” led to my work on human error. But even after the work was published, he chided me for explaining how an error occurs, but not why that particular error had transpired, pointing me to Sigmund Freud’s insights.
His knowledge was encyclopedic, his judgement brilliant, and his assessment of colleagues acerbic. One common pronouncement was, “In any group of people, half are stupid. Any group: Nobel prize winners, senior professors — any group.” (I have come to see the truth in that statement.)
The era that moved us from rampant behaviorism to today’s richer, more enlightened and eclectic view owes much to George. He championed modern psychology. He was an early, lonely worker in the field of emotion, which is mainstream today. He convinced me of the value of emotion, although I had no idea what to do with this for many decades until I had the insight for my book, Emotional Design: Why We Love (or Hate) Everyday Things. I should have dedicated that book to George.
His enduring legacy is to have created a great department of psychology, as well as the numerous successful students, postdoctoral fellows, and faculty whom he hired, nurtured and mentored.
University College London, United Kingdom
When George was forced to leave Austria so abruptly by himself, alone as a teenager in the late 1930s, he came to England, and although he then moved on to the United States a few years later, he retained his affection for England. I got to know him a little during the 1970s, when he and Jean spent a sabbatical in London. When a year or two later I was invited to a meeting at San Diego, he and Jean, with typical kindness, invited me to stay with them — even though I did not know them very well. The week made a very big impression on me.
The University of California, San Diego (UCSD), department of psychology, which George had set up some years before and continued to chair dynamically, was intellectually the most exciting place I had ever visited. Its Centre for Human Information Processing, set up by George with Don Norman, contained not only Don but also Dave Rumelhart and Jay McClelland, together with a remarkable set of postdocs, including, for instance, Geoff Hinton. Moreover, unlike all the heads of university departments whom I had come across in England at that time, George was no mere head hunter and committee-man who had retreated from active research. Indeed, that was just the time when he was producing his definitive two-factor model of recognition memory, still alive and well today. Many years later, when I was asked to set up a new institute in London, it was my memory of George’s double role and his example of what it might be possible to achieve that led me to accept.
At the time of my visit, George thought I might get on well with Don Norman,with whom I had briefly crossed swords 10 years or so before. He invited Don and his wife Julie one evening, and it was in chatting with Don before dinner, looking at the splendid view from the Mandler living-room window, that one of my most intellectually productive collaborations began.
Discussions that I had with George after he and Jean began to live in London for half of each year were often challenging. George was very much a cognitive psychologist. However, his great achievements in the field of memory were based theoretically on semi-phenomenological concepts like familiarity, recollection, and organization. His theoretical framework within cognitive psychology led him to be rather unsympathetic to that of an unreconstructed “boxologist” such as me, so our discussions frequently moved beyond psychological topics to ones in which we were more in tune — through cognitive science in general, particularly anthropology, to politics. Discussions on politics were particularly stimulating, as not only did he have very wide-ranging knowledge, but he also remained true to his roots in the Europe of the 1930s and 1940s by being a socialist — one much influenced by Marxism — and not a typical academic liberal. He was a true intellectual.
My memory of George, particularly in the 1970s and 1980s, was of an immensely dynamic person completely at home and self-confident in the academic world and in his major research contributions. On reflection, it is amazing what he achieved in academic life after he had been forcibly sundered from family, culture, and country at so young an age. Of course he received remarkable support later from Jean, but earlier he must have been really strong to overcome such a drastic rupture and being left to fend for himself in an alien and unsupportive culture. And through it all he retained his humanity, as illustrated by the kindness and support he showed for many younger academics. I was just one of many fortunate recipients.