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Although solving arithmetic problems approximately is an important skill in everyday life, little is known about the development of this skill. Past research has shown that when children are asked to solve multi-digit multiplication problems approximately, they provide estimates that are often very far from the exact answer. This is unfortunate as computation estimation is needed in many circumstances in daily life. The present study examined 4th graders, 6th graders and adults' ability to estimate the results of arithmetic problems relative to a reference number. A developmental pattern was observed in accuracy, speed and strategy use. With age there was a general increase in speed, and an increase in accuracy mainly for trials in which the reference number was close to the exact answer. The children tended to use the sense of magnitude strategy, which does not involve any calculation but relies mainly on an intuitive coarse sense of magnitude, while the adults used the approximated calculation strategy which involves rounding and multiplication procedures, and relies to a greater extent on calculation skills and working memory resources. Importantly, the children were less accurate than the adults, but were well above chance level. In all age groups performance was enhanced when the reference number was smaller (vs. larger) than the exact answer and when it was far (vs. close) from it, suggesting the involvement of an approximate number system. The results suggest the existence of an intuitive sense of magnitude for the results of arithmetic problems that might help children and even adults with difficulties in math. The present findings are discussed in the context of past research reporting poor estimation skills among children, and the conditions that might allow using children estimation skills in an effective manner.
Citation: Ganor-Stern D (2016) Solving Math Problems Approximately: A Developmental Perspective. PLoS ONE 11(5): e0155515. https://doi.org/10.1371/journal.pone.0155515
Editor: Bert De Smedt, University of Leuven, BELGIUM
Received: October 22, 2015; Accepted: April 30, 2016; Published: May 12, 2016
Copyright: © 2016 Dana Ganor-Stern. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: Experimental data are provided in a Supporting Information file.
Funding: This work was supported by Grant number 906/12 from the Israel Science Foundation to DGS.
Competing interests: The author has declared that no competing interests exist.
Past research on numerical cognition has focused on how people solve arithmetic problems exactly and how this ability develops with age (e.g., [1–3]). Relatively little work was devoted to computational estimation, which is the process of producing an approximate answer to an arithmetic problem, and to its development. The dearth of research on this topic is unfortunate because of the following reasons. First, computation estimation is an important skill in everyday life. It is needed when a person is faced with an arithmetic problem but exact calculation is not possible due to time or attentional limitations. In such a case, computational estimation might be used instead. It is also useful as a "sanity check", to quickly evaluate whether the result obtained through the use of a calculator or mental calculation is reasonable. Second, past research has shown that children are poor estimators, and even many adults are not good at it (see [4] for a review). Thus, exploring this topic will extend our understanding of the numerical processes taking place in everyday life, and perhaps propose ways to improve such important skills.
Most past research on computational estimation has used the estimation production task, in which participants are presented with an arithmetic problem and are asked to estimate how big the answer to the problem is without calculating it exactly (e.g., [5, 6, 7]). The estimation production task requires both conceptual and procedural knowledge. The conceptual knowledge includes the simplification principle—the understanding that simplifying a problem to an approximate solution is legitimate in some circumstances, and the proximity principle—the understanding that an estimate should be as close as possible to the exact answer [5]. The procedural knowledge relevant exclusively for estimation includes the ways to simplify a problem in order to solve it approximately and the compensation procedures to correct for the distortion produced by the simplification procedures. Children acquire the simplification principle and the relevant rounding procedures around 4th or 5th grade, and the proximity principle along its procedures even later, around the age of 15 [5, 8].
The evaluation of estimation skills is usually based on accuracy measures. Estimation is expected to produce faster responses than exact calculation but at the expense of accuracy. The question of how accurate an estimate should be to be considered accurate enough is a practical question, and the answer probably depends on the context. It seems to be agreed by all however, that estimates that are off by an order of magnitude (i.e., place value errors) are unacceptable. Such errors are especially common in estimates of results of multiplication problems. For example, providing an estimate of 320 for the problem 85 x 43 is considered a place value error. When using a set of 2 digit (D) x 2D multiplication problems, Ganor-Stern and Siegler [9], reported 52%, 28%, and 9% place value errors for 6th graders, 8th graders and adults, respectively. Such errors might reflect weak procedural or conceptual knowledge, or a working memory limitation. Importantly, in the majority of the cases, the estimates involving place value errors were underestimates, that is they were ten times smaller than the exact answer.
In addition to looking at accuracy of estimates, past research has also examined the strategies used when producing estimates to arithmetic problems. The common strategies involve rounding procedures. For example, in the context of 2-digit numbers multiplication, the common rounding procedures were rounding either one or both multiplicands down to the nearest decade, or rounding one multiplicand up and the other one down to the nearest decade. In some cases participants used post-compensation to correct for the error introduced by the rounding procedure. That is, if they rounded both multiplicands down, they would add some to the product to correct for the error introduced by the rounding down. Importantly, such post-compensation procedures were used often by adults but were rarely used by children [5–7, 10].
Thus, the emerging picture is that the estimation production task demonstrates the children's procedural knowledge on the rounding procedures, but the prevalence of their place value errors might suggest weak sense of magnitude or limited working memory resources. The children's performance in this task could reflect poor general estimation skills. If so, it could raise doubt whether estimation can be used effectively as a substitute for exact calculation, or as a sanity check for the solutions obtained through other means. Alternatively, it might reflect the inability to produce estimates rather than poor estimation skills in general.
To address this issue performance on a different computational estimation task should be explored. In the estimation comparison task a multi-digit multiplication problem is presented together with a reference number and participants are asked to indicate whether they estimate the answer to that problem to be larger or smaller than the reference number [11, 12].
Performance in terms of speed and accuracy was found to decrease for larger multiplicands (i.e., the problem size effect). It was enhanced for reference numbers that were far vs. close from the exact answer (i.e., the ratio or distance effect). It was also enhanced for smaller vs. larger reference numbers (i.e., the reference number size effect). For example, it was easier for the participants to decide that the exact answer for a problem such as 37 x 54 was larger than 400 than it was smaller than 4000 and to decide that it was smaller than 8000 than it was smaller than 4000 [12, 13]. The ratio/distance and size effects in quantity and magnitude comparisons suggest the existence of a condensed approximate representation. These effects are considered the markers of the ANS (Approximate Number system) (e.g., [14]). The ANS is a mental system of approximate magnitude representation that is universal, and appears early in development. This system is assumed to be active whenever a numerical process is taking place involving either symbolic or nonsymbolic stimuli (e.g., [14]). The presence of the ratio/distance and size effects in this task might be interpreted as reflecting the involvement of the ANS in this task [12, 13].
A study using a version of this task, in which young children were presented with non-symbolic stimuli found a similar pattern. McCrink and Spelke [15] examined the ability of 5- to 7-year-old children to estimate the result of multiplication of numerosities relative to a reference quantity. Their performance was above chance level, and it was affected by the ratio between the correct and proposed answer. The fact that the children were prior to any formal schooling suggests that their performance relies on a core, intuitive multiplication ability which is based on their ANS.
As to the underlying processes, Ganor-Stern [12, 13] identified two main strategies used by college students to solve this task. The first is an approximate calculation strategy—rounding either one or two multiplicands, multiplying the rounded numbers, and comparing the product to the reference number. An example of a self-report of this strategy for the problem 54 x 36 and the reference number of 400, might be: "It is like 50 x 40, so it is larger than the reference number". This method produces relatively high accuracy but at the cost of slower responses. It is assumed to require rounding and multiplication skills as well as working memory resources [16]. The second strategy is the sense of magnitude strategy—basing the response on an intuitive sense of magnitude without any calculation. Thus, for the same problem, if the reference number is 100, participants might indicate that "it feels much larger than this number". This strategy reflects an intuitive coarse sense of magnitude, which is presumably built on the individual life experience with multiplication. Practice in solving arithmetic problems leads to sensitivity to various aspects of these problems, (e.g.,[17, 18, 19]), one of which is the range of magnitudes possible for a specific type of problems. This intuitive sense of magnitude is further fine-tuned by the experience with the specific set of problems presented during the experiment [20]. This intuitive coarse sense of magnitude might be linked to the ANS. This strategy, in contrast to the former, is associated with less accurate responses, especially when the reference number is close, but it is faster and requires little working memory resources. A recent study using eye movement measures validated the existence of these two distinct strategies, by showing different eve movements patterns for the two strategies [20]. When the participants used the approximate calculation strategy they focused longer on the multiplicands than on the reference number, while when they used the sense of magnitude strategy their looking time was distributed equally between the multiplicands and the reference number.
The approximate calculation strategy was used more often by adults than the sense of magnitude strategy. However, most participants used both strategies, consistent with past studies (e.g., [21]). Importantly, the participants exhibited adaptivity in strategy choice [21–23], as they used the time-consuming approximate calculation strategy more often when the reference number was close vs. far from the exact answer, and thus the faster sense of magnitude strategy was less likely to produce a correct response [12, 13].
The results were interpreted as reflecting a two-stage problem solving process, in the first stage participants attempt to reach an answer using their intuitive sense of magnitude for the results of such multiplication problems. The second is a more time-consuming approximate calculation process, which took place mainly when the first stage could not provide a definitive response, such as when the reference number is close to the exact answer [12, 13].
Present Study
The goal of the present study was to examine whether the poor estimation skills exhibited by children in the estimation production task reflect poor general estimation skills, perhaps due to imprecise numerical representations, or a more task-specific weakness. For this purpose the estimation comparison task was used. To investigate development across age, 4th graders, 6th graders, and college students completed this task with a set of 2D x 2D multiplication problems. In each trial a multiplication problem was presented horizontally on the computer screen with a reference number below it (Fig 1). Participants had to estimate whether the answer to the problem was larger or smaller than the reference number, and to press the corresponding key. As in Ganor-Stern [12], the reference numbers were either larger or smaller than the exact answer, and they were either far (i.e. fifth or 5 times the exact answer) or close (half or twice the exact answer) to the exact answer. To explore strategy use, participants also provided self-reports and described how they reached their answer.
The following predictions regarding accuracy, speed and strategy use were tested.
Accuracy and speed
Performance in terms of speed and accuracy should improve with age [4, 5]. Moreover, as mentioned earlier Ganor-Stern and Siegler [9] reported that in the estimation production task, about half of the estimates produced by 6th graders were ten times smaller than the exact answer. If this poor performance reflects poor estimation skills in general, one might expect that the children's accuracy in the estimation comparison task would also be very low, even approaching chance level.
Strategy use
With age there should be more variability in strategy use and more adaptivity in strategy choice [21, 23]. Furthermore, with respect to the frequency of use of the two main strategies found in this task–the approximate calculation and the sense of magnitude, the following two possibilities were raised. Due to the training in multiplication and in rounding procedures given in school and the emphasis given to solving mathematical problems using calculation, children might be more inclined to use the approximate calculation strategy than adults, who in turn might rely more on the sense of magnitude strategy at least for far reference numbers. Alternatively, due to the children's relatively weak multiplication skills and limited working memory resources, they might rely more often on the sense of magnitude strategy than adults, who due to their life long experience with arithmetic and enhanced working memory resources, might use mainly the approximate calculation strategy.
Ninety-two participants took part in this study, thirty-two 4th graders, thirty 6th graders, and thirty college students. Four 4th graders, two 6th graders and two college students were excluded from the analysis due to chance level performance or an exceptionally long response time, leaving 28 participants in each age group. The mean age was 24.0 years for the adults group (17 females), 11.1 years for the 6th grade group (18 females), and 9.35 years for the 4th grade group (14 females). The children were from two public elementary schools in the south of Israel. The college students were from Achva Academic College and Ben-Gurion University of the Negev and they participated in the experiment for payment ($5 each). Two research assistants administered the experiment in a quiet room in school or in the laboratory.
Ethics statement
The procedure was approved by the ethics committees of the Israeli Ministry of Education and of Achva Academic College. The college students provided written informed consent to participate in this study. Adhering to the policy of the Ministry of Education IRB, the parents of the school children denied consent by returning an enclosed form.
The experiment was conducted on a Lenovo Thinkpad laptop computer with a 17-inch screen. The software was programmed in E-Prime [24].
The stimuli set was composed of forty 2D x 2D multiplication problems. Following Ganor-Stern [12], the problem set was constructed with the following restrictions. There were no tie problems. No operand had 0 as units digit. No reversed orders of operands were used (53 x 76 was not used with 76 x 53). The operands ranged between 13 and 95, and the exact answers were in the range of 768–8178. In half of the problems the larger operand was on the right, while in the other half the larger operand was on the left. Each problem was associated with 4 reference numbers: one which was about one fifth of the exact answer, one which was about one half of the exact answer, one which was about twice the exact answer, and one which was about 5 times the exact answer. Reference numbers were rounded to the nearest hundred. The problems were arranged in four lists that were counterbalanced across participants. Thus, each participant responded to only one list. Within each list, each problem appeared once; across lists, each problem appeared with each of the four reference numbers. Within each list, in half of the trials the exact answer was larger than the reference number, and in the other half it was smaller than the reference number.
The experiment was conducted individually. The participant sat about 50 cm from the computer screen. In each trial a multiplication problem appeared on the screen with a reference number below it (see Fig 1). Participants were asked to estimate whether the answer for each problem was larger or smaller than the reference number. They had to press the "L" key if they estimated it to be larger than the reference number, and the "A" key if they estimated it to be smaller. The participants were given two examples of the task, together with the corresponding correct responses, to make sure that they understood the task requirements. Participants were explicitly told that they should not solve the problems exactly, but should only estimate whether the answer was larger or smaller than the given number. The numbers remained on the screen until the participants' response. The order of trials was random. Participants were not allowed to use calculators or paper and pencils for calculation. Participants did not receive any feedback for their responses. In the first 8 trials, participants responded by pressing the computer keys only, while in the remaining 32 trials, after they pressed the response key for each trial, they were asked to describe how they reached their decision. The experimenter documented their descriptions.
The results section includes analyses of accuracy, speed, and strategy use.
Accuracy and speed analysis
Analyses of variance (ANOVA's) with age (4th graders, 6th graders, and adults) as a between-participants variable and distance between the exact answer and the reference number (far, close), and size of the exact answer relative to the reference number (exact answer > reference number, exact answer < reference number) as within-participants variables were conducted on percent error (PE) and response time (RT) of correct responses. Partial eta squared values (η2p) were calculated as SSeffect / (SSeffect + SSerror). Responses that took longer than 2.5 standard deviations above each participant mean were excluded from the analyses (less than 3% of the trials).
As can be seen in Fig 2 (top panel), PE was lower when the exact answer was far from the reference number (14.34%) compared to when it was close to it (22.14%), F(1, 81) = 32.44, MSE = 157.4, p = .0001, η2p = .29. PE was also lower when the exact answer was larger than the reference number (15.24%) compared to when it was smaller than it (21.25%), F(1, 81) = 5.78, MSE = 525.7, p = .02, η2p = .07. PE decreased with age, F(2, 81) = 2.38, MSE = 384.1, p = .07, η2p = .06 although the effect was only marginally significant. PE was 21.52%, 16.96%, and 16.25% for the 4th graders, 6th graders, and adults, respectively. The age and distance variables interacted, F(2, 81) = 3.54, MSE = 157.4, p = .03, η2p = .08, as PE in the close condition decreased with age, F(2, 81) = 4.31, MSE = 332.8, p = .02, η2p = .10, from 27.86% for 4th graders, 20.36% for 6th graders, and 18.21% for the adults, while PE in the far condition did not differ across age, F(2, 81) = 0.17, MSE = 208.71, p = .84. It was 15.17%, 13.57%, and 14.28% for 4th graders, 6th graders, and adults, respectively. Importantly, one-sample t-tests that compared accuracy to chance level (50%) found that accuracy was above chance level in all three age groups in all conditions (all ps < .05).
Similar patterns appeared in the speed analysis (Fig 2, bottom panel). RT decreased with age, F(2, 81) = 10.51, MSE = 228.74, p = .0001, η2p = .21. It was 12.28 sec for 4th graders, 11.11 sec for 6th graders, and 3.73 sec for the adults. Scheffe post hoc tests showed that the adults were faster than the children groups, which did not differ (p < .05). RT was shorter when the exact answer was far (8.56 sec) from the reference number compared to close (9.53 sec), F(1, 81) = 9.78, MSE = 8.09, p = .002, η2p = .87. RT was shorter when the exact answer was larger than the reference number (8.45 sec) compared to when it was smaller than it (9.63 sec), F(1, 81) = 14.11, MSE = 8.32, p = .001, η2p = .15. The two variables interacted, F(1, 81) = 4.69, MSE = 5.26, p = .03, η2p = .57, indicating that the difference in RT between close and far trials was larger when the exact answer was larger than the reference number. Similar to the PE analysis, the distance effect in RT was somewhat smaller for the adults compared to the children groups, although the effect did not reach significance, F(2, 81) = 1.22, MSE = 8.09, p = .30.
Strategy analysis
Classification of strategy use.
The classification of participants' self-reports into strategies was done independently by two researchers based on the experimenter verbatim, and the percentage of agreement was 97%. At the group level, the approximate calculation strategy was used in 35% of the trials among 4th graders, in 47% of the trials among 6th graders, and in 70% of the trials among the adults group. The sense of magnitude strategy was used in 62% of the trials among 4th graders, in 50% of the trials among 6th graders, and in 25% of the trials among the adults group. The rest of the trials (less than 5%) could not be classified into any of those strategies and thus were not analyzed.
An individual level analysis classified participants as consistent users of a certain strategy if they reported using it in at least 80% of the trials [25]. The distribution of participants varied with age, χ2 (2) = 12.98, p = .01, with the main difference between the adults and the two children groups, which did not differ. As can be seen in Table 1, 21 out of the 4th graders (75%) used one strategy consistently, 14 of them used the sense of magnitude strategy, and 7 used the approximate calculation strategy. Exactly the same number of 6th graders employed one strategy consistently, however the distribution of strategy use differed with 10 participants implementing the approximate calculation strategy and 11 using the sense of magnitude strategy. Sixteen adults (57%) employed one strategy consistently, with 14 of them implementing the approximate calculation strategy and only two participants using the sense of magnitude strategy. Thus, both the group level information and the individual level analysis showed a clear increase in the use of the approximate calculation strategy with age.
Strategy choice.
Since most participants in all age groups used the two strategies to solve the task, the next step was to investigate how participants decided which strategy to use for which item. Thus, the frequency of use of each strategy was entered into an analysis of variance with age, strategy, size of the reference number relative to the exact answer, and distance as independent variables. This analysis was limited to the participants that used both strategies (19 4th graders, 18 6th graders, and 22 adults). The two effects involving the strategy variable are theoretically important (see Fig 3). There was a significant interaction between strategy and age, F(2, 56) = 5.51, MSE = 54.42, p = .01, η2p = .16, indicating that the frequency of use of the approximate calculation strategy increased with age, and that the frequency of use of the sense of magnitude strategy decreased with age. Moreover, the significant interaction between strategy and distance, F(1, 56) = 24.24, MSE = 3.04, p = .001, η2p = .30 indicated that while the approximated calculation strategy was used more often when the reference number was close (vs. far), the sense of magnitude strategy was used more often when the reference number was far (vs. close).
Relationship between strategy use, accuracy, and speed
As mentioned in the Introduction, the two strategies are expected to produce different patterns of speed and accuracy, with the approximate calculation strategy expected to produce accurate but slow responses, and the sense of magnitude strategy expected to produce fast and less accurate responses [12, 13]. Moreover, the advantage of the approximate calculation in accuracy is supposed to be especially pronounced for close reference numbers.
To examine the effects of age and strategy use on accuracy and speed two analyses were conducted. The first looked at the accuracy and speed of participants who used a single strategy throughout the whole experiment. Among the 4th graders, 3 used the approximate calculation strategy exclusively and 6 the sense of magnitude strategy solely, among the 6th graders, 6 used only the approximate calculation strategy and 4 solely the sense of magnitude strategy, and among the adults, 5 used only the approximate calculation strategy and one participant used only the sense of magnitude strategy. Since there was only one adult that used the sense of magnitude exclusively the following analysis was limited to the children groups. ANOVA's with age and strategy as between-participant variables and distance and size of the reference numbers as within-participant variables were conducted on PE and RT for correct responses.
The PE analysis (Fig 4 top panel) has shown an interaction between strategy and distance (F(1, 15) = 6.60, MSE = 170.14, p = .03, η2p = .31, indicating that as predicted PE was higher for the sense of magnitude strategy compared to the approximate calculation strategy only when the reference number was close (F(1, 15) = 5.81, MSE = 347.50, p = .03, η2p = .28, (14.58% vs. 29.79%, for the approximate calculation and the sense of magnitude strategy, respectively), and not when it was far (14.58% vs. 13.75%), F(1, 15) = 0.03, p = .85. The RT analysis (Fig 4 bottom panel) has shown that as expected, RT was shorter for the sense of magnitude strategy (6.93 sec) vs. the approximate calculation strategy (19.49 sec), F(1, 15) = 26.56, MSE = 25.93, p = .001, η2p = .63, it was also shorter for the 6th (8.19 sec) vs. 4th graders (18.24 sec), F(1, 15) = 16.99, MSE = 25.93, p = .001, η2p = .53. Importantly, the improvement in speed with age was larger for the approximate calculation strategy (28.95 sec for 4th grade vs. 10.03 sec for 6th grade) than for the sense of magnitude strategy (7.52 sec for 4th grade vs. 6.35 sec for 6th grade), F(1, 15) = 13.25, MSE = 25.93, p = .001, η2p = .47. The interaction between strategy and distance. F(1, 15) = 2.93, MSE = 11.07, p = .11, η2p = .16, showing that RT was faster for the far compared to the close condition only when the sense of magnitude strategy was used, although did not reach significance, is still worth mentioning as it might be informative with respect to the origin of the distance effect.
The second analysis was conducted on participants that used both strategies. It included age as a between-participant variable and strategy as a within-participant variable. Since there were not enough trials per strategy the data were collapsed across distance and size of the reference number. The speed analysis showed an age effect, F(2, 46) = 6.09, MSE = 337.21, p < .005, η2p = .21, with adults responding faster than 6th graders (p < .01), and a strategy effect, F(1, 46) = 5.53, MSE = 103.57, p < .05, η2p = .11, with the responses based on the sense of magnitude strategy (9.80 sec) significantly faster than those based on the approximate calculation strategy (14.78 sec). There were no significant effects in the analysis of PE.
In many real life circumstances, a decision depends on a person's ability to estimate the result of an arithmetic problem relative to a reference number. Despite the frequent use of the computational estimation skill in everyday life, relatively little is known about how people solve such tasks, and how this skill develops with age. The present research is the first to examine developmental patterns in the computational estimation comparison task by looking at the accuracy, speed, and strategy use of 4th graders, 6th graders, and college students.
The developmental pattern between children and adults was reflected in three measures. First, there was an increase in accuracy with age. Importantly, this increase was limited to trials when the reference number was close to the exact answer. Second, speed improved with age across conditions. Third, the dominant strategy changed with age. In 4th grade the sense of magnitude was the most common strategy, in 6th grade the two strategies were used equally often, while in the adults group the approximate calculation strategy was the most common. The transition from frequent use of the sense of magnitude strategy into using mainly the approximate calculation strategy with age can explain why the advantage in accuracy for the adults group was limited to the close reference numbers. The faster and less attention demanding sense of magnitude strategy is likely to produce a correct response when the reference number is far, but not when it is close.
The preference of fourth-graders to use the sense of magnitude strategy is probably due to the fact that they do not fully master the calculation skills at this stage, and thus cannot use the approximate calculation strategy effectively. Furthermore, the sense of magnitude strategy puts little burden on working memory compared to the approximate calculation strategy [16]. With age, there is an improvement in calculation skills and an increase in working memory resources [26, 27], and as a consequence more frequent use of the approximate calculation strategy. Support for this idea is provided by the sharp increase in speed found even between 4th and 6th graders that was especially pronounced for children who used the approximate calculation exclusively. Further rise in speed is expected between childhood and adulthood.
This increase in speed with age can explain the seemingly paradoxical pattern of adults being faster than the children although they used more often the time-consuming approximated calculation strategy. This is due to the general advantage in speed for the adults over the children, which exists across domains but is especially large when using the approximate calculation strategy.
Although children were less accurate than adults, they performed well above chance level. This finding is somewhat surprising given their performance in the estimation production task in past studies [4, 28]. As mentioned earlier, when 6th graders were asked to produce estimates for a set of 2D x 2D multiplication problems, more than half of their estimates involved place value errors in contrast to less than 10% of the adults [9]. Thus, the present results provide a more optimistic picture of children's estimation skills, and on their ability to use them as a substitute for exact calculation or as a sanity check. What could account for this difference? The presence of the reference number might be an important factor. The reference number reduces the working memory load posed by the task [16]. In addition, the reference number might serve as an anchor, which was found to improve estimates of quantities by past research [29].
Children as well as adults were affected by the characteristics of the reference number. In all age groups performance was enhanced in terms of accuracy and speed when the reference number was far vs. close to the exact answer, and when the reference number was smaller rather than larger than the exact answer [12]. These distance and size effects are considered to be the signatures of an approximate condensed representation of magnitude. Their presence in all age groups suggests the existence of a holistic representation of numbers in the thousands range even among 4th graders.
The enhanced performance for trials in which the reference numbers were smaller than the exact answer might reflect the greater familiarity with the smaller numbers, which made it easier for the children as well as the adults to recognize that the exact answer should be larger than the reference number. Interestingly, the effect of familiarity on estimation is task-dependent. As mentioned earlier, when asked to produce estimates children often produced estimates that were an order of magnitude away from the exact answer. In the majority of the cases their estimates were ten times smaller rather than larger than the exact answer [9]. For example, children often estimated the answer to the problem 54 x 87 to be 400. Thus, the greater familiarity with the smaller numbers might have encouraged, especially the children, to estimate the result to be in the more familiar hundreds range. In the estimation comparison task, the greater familiarity with the smaller numbers improved participants' discrimination when the reference numbers were smaller than the exact answer.
Children in both age groups exhibited more rigidity and less variability in strategy use compared to adults, as using a single strategy consistently was much more common among children than among adults. Such a pattern of increased variability in strategy use over development has been reported in the literature (e.g., [2, 30]). Importantly, the children as well as the adults who used multiple strategies showed adaptivity in strategy choice, as they used the approximate calculation strategy more often when the reference number was close to the exact answer and the sense of magnitude strategy more often when the reference number was far from the exact answer. This pattern was found in the past for adults who performed this task [12, 13]; the fact that it was found here even for children as young as 4th graders is of importance because it suggests that children are not only sensitive to the distance between the reference number and the exact answer, they are also sensitive to the consequences of using each strategy and the need to match strategy use with problem characteristic (i.e. the distance of the reference number from the exact answer) for the sake of high accuracy. Past research has reported a similar pattern of adaptivity in strategy choice for children in computational estimation involving multi-digit addition [6, 7].
The effects of strategy use and age on accuracy and speed were examined in two ways. First when looking at children who used only a single strategy, the analysis has shown that accuracy was lower for the sense of magnitude strategy compared to the approximated calculation strategy only when the reference number was close, and that speed increased with age mainly for the approximate calculation strategy. The second analysis that focused on participants who used both strategies has shown that speed was affected by strategy and age, in an additive manner, while there were no effects on accuracy.
The difference between the results of the two analyses is possibly due to strategy choice contaminating strategy effects in the latter analysis. Participants who use multiple strategies were shown to choose which strategy to use for which item in an adaptive manner; they used the approximate calculation strategy more often for the close reference numbers, and used the sense of magnitude alone in the far condition [12, 13, 20]. This is the reason why accuracy was affected by strategy only when the strategy choice process was absent (e.g. when participants used only a single strategy for the whole experiment). Another way to dissociate effects of strategy use and strategy choice is by using the choice-no choice paradigm [21].
The present results showing that 4th graders prefer the sense of magnitude strategy resemble the results of adults diagnosed with dyscalculia, who also tended to use the sense of magnitude more often than controls [13]. Together, they suggest that people with calculation problems, such as dyscalculics or those who have not yet mastered the calculation procedures well (such as 4th graders), might rely on their sense of magnitude to compensate for their difficulties in calculation. This idea might be implemented in the educational setting, as suggested next.
The present study shows a developmental pattern of greater reliance on the approximate calculation strategy which reflects the application of procedures learned in school with age, and less reliance on the intuitive sense of magnitude. This pattern parallels a similar development shift that was documented in another estimation task, the number line task. In this task participants are shown a line flanked by a number at each end (e.g., 0 and 1000), and asked to place a third number (e.g., 150) on that line [31]. Numerous studies have documented a developmental shift in this task from using a logarithmic representation, which is the default basic approximate condensed magnitude representation, into using the school-taught linear representation (e.g., [17, 32, 33]). In the present study there is also evidence for a transition from using an intuitive sense of magnitude strategy into using the school-taught calculation procedures.
The present results are interpreted in the context of past works suggesting that nonsymbolic and symbolic stimuli are processed in a similar manner, possibly by the same mechanism, the ANS (e.g., [14, 34, 35]. Specifically, here the ANS was proposed to be responsible for the distance and size effects and to underlie the sense of magnitude strategy. However, alternative approaches argue that symbolic and nonsymbolic stimuli are processed separately and in a different manner. While nonsymbolic stimuli are processed approximately by the ANS, symbolic stimuli are represented exactly. This approach was recently supported by findings showing no relationship between the distance effect found for symbolic and for nonsymbolic stimuli [36].
Note that such proposals regarding the exact representation of symbolic stimuli were made with respect to numbers in the first decade [37]. The numbers used in the present context are much larger (e.g. the answers themselves are in the thousands range, while the reference numbers might be even larger) and it is very unlikely that such numbers are represented exactly. Support for the idea that the ANS underlies the distance effect and the sense of magnitude strategy is provided by the pattern showing that the distance effect in speed was present for participants that exclusively used the sense of magnitude strategy, but not for those that used the approximate calculation strategy exclusively. Although the effect did not reach significance in the present study, it was found also in past research using this task [12].
The present results together with past research using the same paradigm [12, 13, 20] suggest the existence of approximate representations of symbolic magnitudes, and an intuitive coarse sense of magnitude for the results of multiplication problems. Although the patterns seem compatible with the ANS, the symbolic approximate representation might not be based on the nonsymbolic approximate representation. Future research should explore this issue directly perhaps using brain imaging techniques.
Implications for education
Math instruction tends to put a strong emphasis on the application of algorithmic procedures [38]. The present results together with past research on dyscalculics [13] suggest that such an estimation task that recruits the intuitive sense of magnitude might be used in addition to the algorithmic methods in order to help children with difficulties in math or with poor working memory resources. In the educational setting the teachers can give the children, prior to solving complex arithmetic problems, tasks where they judge the expected answer in comparison to a reference number. Such a prior estimation procedure might help to decrease the errors in solving complex arithmetic problems via exact calculation.
Supporting Information
S1 Data. Percent error and reaction time for correct responses by age, distance, and size of the reference number.
I thank the students who participated in this study and their teachers. I also thank Orr Ben Yitshak, Loraine Sharoni, and Ofir Steinhorn for their assistance with data collection and analysis. Thanks are also due to the editor, Patrick Lemaire and an anonymous reviewer for their helpful comments on an earlier version of this manuscript.
Author Contributions
Conceived and designed the experiments: DGS. Performed the experiments: DGS. Analyzed the data: DGS. Contributed reagents/materials/analysis tools: DGS. Wrote the paper: DGS.
- 1. Ashcraft MH, Stazyk EH. Mental addition: A test of three verification models. Memory & Cognition. 1981;9:185–96.
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Siegler RS. Emerging minds. New York, NY: Oxford University Press; 1996.
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Geary DC. Children's mathematical development. Washington, DC: American Psychological Association; 1994.
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Siegler RS, Booth J. Development of numerical estimation: A review. In: Campbell J, editor. Handbook of mathematical cognition. New York, NY: Psychology Press; 2005. p. 197–212.
- 5. LeFevre J-A, Greenham SL, Waheed N. The development of procedural and conceptual knowledge in computational estimation. Cognition and Instruction. 1993;11:95–132.
- 6. Lemaire P, Lecacheur M. Children's strategies in computational estimation. Journal of Experimental Child Psychology. 2002;82:281–304. pmid:12225757
- 7. Lemaire P, Lecacheur M, Farioli F. Children's strategy use in computational estimation. Canadian Journal of Experimental Psychology. 2000;54(2):141–8. pmid:10881397
- 8. Sowder J, Wheeler M. The development of concepts and strategies used in computational estimation. Journal of Research in Mathematical Education. 1989;20(2):130–46.
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Ganor-Stern D, Siegler RS. The development of computation estimation skills. Poster presentation at the 45th Annual Meeting of the Psychomic Society; November 2004 Minneapolis, MN2004.
- 10. Reys R, Bestgen B, Ryblot J, Wyatt J. Processes used by good computational estimators. Journal for Research in Mathematical Education. 1982;13(3):183–201.
- 11. Rubinstein R. Computational estimation and related mathematical skills. Journal for Research in Mathematical Education. 1985;16(2):106–19.
- 12. Ganor-Stern D. When you don't have to be exact–Investigating computation estimation skills with a comparison task. Acta Psychologica. 2015;154:1–9. pmid:25461437
- 13. Ganor-Stern D. Can dyscalculics estimate the results of arithmetic problems? Journal of Learning Disabilities. 2015.
- 14. Feigenson L, Dehaene S, Spelke E. Core systems of number. Trends in Cognitive Sciences. 2004;8(7):307–14. pmid:15242690
- 15. McCrink K, Spelke E. Core multiplication in childhood. Cognition. 2010;116:204–16. pmid:20537618
- 16. Kalaman D, Lefevre J-A. Working memory demands of exact and approximate addition. European Journal of Cognitive Psychology. 2007;19(2):187–212.
- 17. Booth J, Siegler RS. Developmental and individual differences in pure numerical estimation. Developmental Psychology. 2006;41:189–201.
- 18. Rotem A, Henik A. Development of Product Relatedness and Distance Effects in Typical Achievers and in Children With Mathematics Learning Disabilities. Journal of learning disabilities. 2014:0022219413520182.
- 19. Rotem A, Henik A. Sensitivity to general and specific numerical features in typical achievers and children with mathematics learning disability. The Quarterly Journal of Experimental Psychology. 2015;68(11):2291–303. pmid:25633228
- 20. Ganor-Stern D, Weiss N. Tracking practice effects in computation estimation. Psychological Research. 2015:1–15.
- 21. Siegler RS, Lemaire P. Older and younger adults' strategy choices in multiplication: Testing predictions of ASCM via the choice/no choice method. Journal of Experimental Psychology: General. 1997;126:71–92.
- 22. Geary DC, Brown S, Samaranayake VA. Cognitive addition: A short longitudinal study of strategy choice and speed-of-processing differences in normal and mathematically disabled children. Developmental Psychology. 1991;27(5):787–97.
- 23. Lemaire P, Siegler R. Four aspects of strategic change: Contributions to children's learning of multiplication. Journal of Experimental Psychology: General. 1995;124(1):83–97.
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Schneider W, Eschman A, Zuccolotto A. E-Prime user's guide. Pittsburgh, PA: Psychology Software Tools, Inc.; 2002.
- 25. Siegler RS. Developmental sequences within and between concepts. Monographs of the Society for Research in Child Development. 1981;46(2):1–84.
- 26. Siegel LS, Ryan EB. The development of working memory in normally achieving and subtypes of learning disabled children. Child development. 1989:973–80. pmid:2758890
- 27. Levine SC, Jordan NC, Huttenlocher J. Development of calculation abilities in young children. Journal of experimental child psychology. 1992;53(1):72–103. pmid:1545190
- 28. Dowker A. Young children's addition estimates. Mathematical Cognition. 1997;3:141–54.
- 29. Izard V, Dehaene S. Calibrating the mental number line. Cognition. 2008;106:1121–47.
- 30.
Siegler RS. Microgenetic analyses of learning. In: K D., Siegler RS, editors. Handbook of child psychology: Cognition, perception, and language. 2. Hoboken, NJ:: Wiley; 2006. p. 464–510.
- 31. Siegler RS, Opfer JE. The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science. 2003;14:237–43. pmid:12741747
- 32. Siegler RS, Booth J. Development of numerical estimation in young children. Child Development. 2004;75:428–44. pmid:15056197
- 33. Siegler RS, Thompson CA, Opfer JE. The logarithmic-to-linear shift: One learning sequence, many tasks, many time scales. Mind, Brain, and Education. 2009;3:143–50.
- 34. Ansari D. Effects of development and enculturation on number representation in the brain. Nature Reviews Neuroscience 9(4):278–91. 2008;9(4):278–91. pmid:18334999
- 35. Lussier CA, Cantlon FJ. Developmental bias for number words in the intraparietal sulcus. Developmental Science. 2016:1–18.
- 36. Lyons IM, Nuerk HC, Ansari D. Rethinking the Implications of Numerical Ratio Effects for Understanding the Development of Representational Precision and Numerical Processing Across Formats. 2015.
- 37. Van Opstal F, Gevers W, De Moor W, Verguts T. Dissecting the symbolic distance effect: Comparison and priming effects in numerical and nonnumerical orders. Psychonomic Bulletin & Review. 2008;15(2):419–25.
- 38. Ambrose R, Baek J-M, Carpenter TP. Children's invention of multidigit multiplication and division algorithms. The development of arithmetic concepts and skills: Constructive adaptive expertise. 2003:305–36.


text:
This Starter Guide has shown us how to configure a basic Incident integration, sending outbound messages via the REST service. The elements we configured were:
- Stage table
- Process
- Integration
- Connection
- Messages
In testing our integration, we created and updated an Incident - following the data as it flowed from our ServiceNow Incident record, through the Unifi transaction message bus, to the external system's Incident record.
By Using the Unifi integration platform, we have discovered that building and testing our integration is simpler and more efficient, saving both time and money. Our hope is that this Starter Guide has been a valuable aid to you in that process.
For further information please see our technical documentation.
If you have any feedback or questions, please contact us via our website contact form.

subset: commoncrawl


text:
Milepost 705
Before our tour of the Brewster shop, we met at the Wandle House. Later, we returned there for lunch. I was particularly interested in the Wandle House for several reasons, but the biggest two were that I'm almost positive that several of my relatives would have stayed there when they worked for the Wheeling & Lake Erie and because of my having worked for the YMCA for twelve years.
The structure was originally built as primarily a dormitory YMCA. The Y and the railroads worked together in the early half of last century to provide train crews adequate overnight facilities in areas where other hotels may not consider building. After a long day on the road, train crews knew that they could get cleande up, get a good night's sleep and a good meal at the local railroad YMCA.
The building was eventually taken over by the railroad and operated until it closed when times changed and crews didn't need those types of overnight accommodations. Today, the Wandle House is the local headquarters of Brewster's historical society. Most of the building is closed off to the public but there are a few display areas in the main lobby which feature plenty of examples of the community's railroad heritage. The Station Restaurant is located off the main lobby and serves good food for a good price; I speak from experience on this one. :)
Also on the grounds of the Wandle House is this preserved Wheeling & Lake Erie caboose.
The main street in Brewster is Ohio Route 93. Since it cuts the Wheeling & Lake Erie yard in two, an underpass was built to separate highway and railroad traffic. It is funny that there seems to be so much complaining about the state spending money to "assist" private enterprise; this particular project happened quite some time ago.
Just as it was back in the early part of the last century, the headquarters of today's Wheeling & Lake Erie is located in Brewster across the street from the Wandle House. The building was built back in 1914 to the Wheeling's rather Spartan standards. It was a "no frills" building and reminded many of either a local high school building or a college dormitory. Although window air-conditioners now dot the front & sides of the building, not much else has changed over the past 100 years.
Finally, I'd be totally remiss if I didn't tell you about the Shearer Potato Chip factory located on the north side of Brewster. No, there aren't any tours but they do have a company store located there and you can pick up any of the many snack foods that Shearer's makes, including some 3-pound bags of "seconds" or surplus chips for as little as a dollar a bag! I speak from experience on this one too. :)


text:
Be Responsible Citizens, says Letter of the Week Winner
Patiliai Leqanidruaivalu, 43, strongly believes that the society must end the blame game and work together to save lives.
"We must all take that responsibility," he said.
In his winning Letters to the Editor, Change the Outcome, Mr Leqanidruaivalu voiced his opinion on the growing rate of death caused by accidents, suicides, poverty and many other illegal activities that happens in the hands of youths these days.
He highlighted that one of the contributing factors of such tragedies was negligence.
The full-time student of the University of the South Pacific said he felt the need to express his opinion because he noticed a lot of sad and disturbing news almost every day.
"I have been reading a lot of news in the papers and watching as well. They are news like infant deaths, suicide, and all the other tragic incidents so close to our hearts," Mr Leqanidruaivalu said.
"I was thinking about what would be the main cause of it.
"And what came into my mind was the problem of negligence. Generally-speaking most of these incidents are out of negligence," he said.
"My message is we cannot be ignorant. We need to look into the issues and maybe we can solve them," he added.
As a parent, Mr Leqanidruaivalu advised other parents to prioritise their children first. And as a responsible citizen, we must take it upon ourselves to talk about these issues and look out for one another, he added.
Mr Leqanidruaivalu is pursuing studies in Diploma in Leadership Governance and Human Rights.
Edited by Ranoba Baoa
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text:
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text:
Kiwi kayaking teen makes World Cup semi
Auckland schoolgirl Hannah Thomas booked a spot in her first ICF World Cup canoe slalom semifinal after finishing ninth in the repechage in Slovakia overnight.
The 17-year-old overcame some tricky gate placements on the harsh Bratislava course, posting a time of 124.40secs to easily qualify in the top-10 that made it through the second run.
She joined teammates Luuka Jones (women's K1) and Finn Butcher (men's K1) in the semifinals after a brutal day which saw some big-name paddlers miss out.
"It is a tricky course and it was hard to adapt to my new C1 boat, as well as the water levels changing each day of training," says Hannah. "But I was feeling really good on the water in both my kayak and canoe and felt solid, despite a couple of touches. I'm really happy with getting into the semifinals and I'm really excited about getting out there and racing on Sunday."
Earlier, she'd missed the seventh gate in her first C1 run to collect a 50sec penalty, as well as picking up two touches, but kept calm in her second run. She finished 36th in her first K1 run and was just 6.19secs off qualifying in her K1 repechage as well.
That K1 repechage was won by Luuka with a classy 99.73sec run, a time that would've put her sixth in the first run.
Germany's Ricarda Funk qualified quickest in the women's K1, posting a 96.51 run to finish just ahead of Slovenia's Eva Tercelj in 96.72, with world champion Jessica Fox third in 98.26.
Finn, meanwhile, qualified for his second consecutive World Cup semifinal after a clean 93.87sec time left him just 5.61secs behind France's Quentin Burgi in 17th spot. Quentin's 88.26sec effort was almost two seconds quicker than Australia's Lucien Delfour (90.22) with Poland's Dariusz Popiela completing a surprise top three in 90.47.
Finn's teammate Callum Gilbert struggled, after the highs of making the final in the opening round in London last week and finishing fifth - he picked up a touch in his first run to finish 40th in 103.52, then was just 1.07secs off qualifying in the repechage, finishing 15th. Jack Dangen was 53rd after missing a gate in an otherwise fast first run, thn 24th in his second.
Other Kiwis in action were Patrick Washer and Ben Gibb in the C1, with Ben missing gates in both his runs and Patrick picking up two touches in his first run to finish 43rd.
Several big names needed second runs to get through to the semi-finals of the men's C1, including five time Olympian Michal Martikan in front of his home crowd, and French duo Cedric Joly and Martin Thomas.
New Zealand's Kensa Randle was also 47th and 30th in her two K1 runs, while Cook Islands duo Jane and Ella Nicholas were 39th and 41st respectively in their first runs, with Jane posting a clean second run to be 18th, just behind Hannah.
Luuka will be the sole Kiwi in action tonight in her K1 semifinal, while Finn and Hannah will race tomorrow.


text: IEC doesn't care about Zuma - ANCWL Karabo Ngoepe, News24 SHARE THIS 2016-08-06 22:00 Silent protesters hold up placards while President Jacob Zuma gives his speech. (Karabo Ngoepe, News24) Pretoria – The IEC's handling of the anti-rape protest during President Jacob Zuma's speech showed the commission did not care about him, the ANC Women's League said on Saturday."They were supposed to ask the president to sit down and deal with the issue, apologise and then ask him to continue. That never happened because they don't care about the president of the country," league president Bathabile Dlamini said.She demanded that IEC chairperson Glen Mashinini apologise and explain what happened.The women did not say anything, but stood in front of the podium, facing the audience, and held up pieces of cardboard with the words "Khanga", "Remember Khwezi", "10 yrs later" and "I am 1 in 3". This was in reference to Zuma's rape trial 10 years ago. His accuser was known as Khwezi in the media. He was acquitted.Dlamini said the EFF was using women to advance patriarchy. Some of the women who were part of the protest sat at the EFF's table at the IEC results centre in Pretoria during the elections."We observed over the three days how women of the EFF were used to run around to get water and coffee and none of them spoke or represented their party during the press conference or any structure of elections," she said.As soon as Zuma's speech was over, IEC deputy chair Terry Tselane took the microphone to apologise to the president."This took us by surprise, and we really want to apologise to all of you," he said. He asked IEC spokesperson Kate Bapela to ensure that journalists did not "exacerbate the situation".'The ANC won'Dlamini dismissed the protest as an attention-seeking stunt by the EFF. The ANC was victorious and nothing would change that."The ANC has won these elections. We are not going to mourn or allow reactionaries and tyrannies who are supported by clandestine forces that pay them for any action they take to embarrass our democracy, as well as the ruling party." EFF spokesperson Mbuyiseni Ndlozi said Dlamini was trying to deflect attention from the election outcome which saw support for the ANC fall below 60% countrywide."Bathabile Dlamini must go play cards. She has no capacity to understand the historic moment that the South African electorate has produced. One of the most important things is a vote of no confidence in her and her food parcels," said Ndlozi.He said he did not speak for the women who took part in the protest.Attempts to get comment from the IEC were unsuccessful. - Find everything you need to know about the 2016 Local Government Elections at our News24 Elections site, including the latest news and detailed, interactive maps for how South Africa has voted over the past 3 elections, or download the app for iOS and Android. - News24 SHARE THIS Read more on: iec | ancwl | pretoria | local elections 2016


text:
Who Are the Top Earning Musicians?
Forbes has unveiled its annual list of the world's 100 top earning stars and several of our favorite music makers made the cut with a few cracking the Top 10.
U2, Coldplay and Ed Sheeran landed in spots 6, 7 and 9 respectively, all pulling in more than $100 million in 2017.
It should be noted Ed was the only solo artist in the Top 10, so in essence he is the top earning musician.
Calvin Harris was the world's top earning DJ at No. 50, raking in $48 million, followed by The Chainsmokers at #59 with $45 million.
The top earner overall was boxer Floyd Mayweather with $285 million followed by George Clooney at No. 2, who pulled in just under a quarter billion thanks to the sale of his Casamigos Tequila brand.
Photo: Getty Images


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text:
6 pieces of evidence that suggest Harry Potter cat Crookshanks is actually a Slytherin in disguise
Fans believe the big furry feline is actually hiding a magical secret...
R.A.B. Those initials plagued the minds of Harry Potter fans for what felt like millennia after JK Rowling first published them in Half Blood Prince in 2005.
We weren't long finding out that they stood for Regulus Arcturus Black, aka the younger brother of one Sirius Black, aka Harry Potter's godfather, though.
And through Kreacher, the Black family house elf, we learned that Regulus had died in the cave Dumbledore visited with Harry on the hunt to find and destroy Horcruxes – the objects Lord Voldemort concealed slivers of his soul in after murdering wizards and muggles.
But what if Kreacher got it wrong? What if his master didn't die after being pulled into the lake by the Inferi? What if Regulus went into hiding? And what if – just like his brother – Regulus was an Animagus?
Some fans think it's quite possible he could have been, and point the finger at a fateful ginger cat. Could Crookshanks be Regulus Black in disguise?
Let's examine the evidence…
1. We first meet Crookshanks in Prisoner of Azkaban
Hermione's cat comes into play just as Sirius Black arrives on the scene. Coincidence? Maybe.
But this is JK Rowling's world we're talking about, and as far as the fans are concerned there's no such thing as "it just so happens" when it comes to the Potterverse.
2. And we learn he's been around for a lot longer than that
When Hermione picked him out at the Magic Menagerie she was told nobody wanted him and he'd been there a long time.
Could a long time mean days, weeks, months, or perhaps even decades? Regulus is said to have died in 1972, and Hermione took Crookshanks home with her some time in 1993.
3. His name is stellar, just like Sirius Black's
The Black family was known for naming its children after constellations. Sirius was named for the Dog Star, which features in Canis Major, and he took the form of a dog when he became an Animagus.
So isn't it funny, then, that Regulus is a star featured in Leo? Y'know, that constellation that's represented by a lion, which is basically just a great big hulking cat?
4. Sirius can communicate with him
Wizards – even Animagi – can't exactly do a Doctor Dolittle and talk to all the animals so isn't it a little odd that Sirius was able to figure out how to talk to Hermione's cat?
Fans are convinced there's no way he could have worked it out if Crookshanks wasn't another secret Animagus. Hey, if Peter Pettigrew could conceal it and go undetected for decades, surely Regulus Black could too…
5. He had Scabbers sussed from day one
Crookshanks was no fool when it came to Ron's rat, barely paws-ing before pouncing on him at the earliest opportunity.
The cat seemed to have a particular distaste for the Weasley family pet and some fans say that's because the Animagus could smell a fellow transfigured former Death Eater a mile off.
6. And he's SIRIUSLY protective
When Harry tries to turn his wand on Sirius before he discovers the truth about Petter Pettigrew, Crookshanks is the first to stand in his way.
He attacks the Boy Who Lived as he reaches for his wizarding weapon and sits on Sirius' chest to prevent Harry from casting a killing spell.
JK Rowling says Crookshanks isn't an Animagus – just a VERY clever Kneazle/cat crossbreed – but when you consider the evidence, do you agree or disagree?


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text:
How to be a UK Festivals & Events Specialist
This post originally featured on KQ Magazine, which I co-ran
At KQ we're on a mission. A mission to bring you the inspiration, industry tips and low-down you need to motivate your career. Need a change? We're here for you. Feel like there's something missing in your job? We're here for you. We've been speaking to those in desirable, interesting and creative industries so you don't have to. This week, I'd like to introduce you to Dr Roxy Robinson, a UK Festivals and Events Specialist and Lecturer at Leeds Beckett University.
Dr Robinson is a self confessed events obsessive. She is also a half Persian salsa dancer battling a nightly addiction to Ted Talks. Roxy works as a lecturer, writer and events consultant, specialising in UK music festivals. This journey began in a field in Hampshire at age fifteen-and it has since brought her to a place where almost every aspect of her work today concerns music festivals; how to run them, programme them, make them safe, make them interesting. She tackles these issues weekly, in lecture theatres across Leeds Beckett University and Leeds College of Music. Her book Festival Culture and the Politics of Participation has recently been approved by Ashgate publishers, and it is set for release at the end of the year, as well as Pop Festival: Music, History, Media Culture, edited by Professor George McKay, which features a chapter (11) written by herself. On her written achievements, Roxy says, "I like to think all of this paper-pushing makes a difference in the real world of festivals — my consulting work involves working with private festival owners in the areas of creative design and marketing. I love my job!"
1) What did you want to be when you were growing up?
Important, like my Dad. I had a powerful sense of the pride he took in his work, in his own success as the founder of a construction company which he called Roxan. I wanted to experience that life-affirming sense of pride myself. I always knew I wanted to throw myself into work, like him. He encouraged me, too — though now he is beginning to do the whole 'where are my grandkids' routine.
2) What does your job entail?
My job entails a mixture of writing, lecturing and industry work — a portfolio career you might say. The constant has been the UK Centre for Events Management at Leeds Beckett University, where I have worked part time for four years as a lecturer. During that time I have also held a 3 year contract with Beacons Festival, written a book, written a chapter for someone else's book, and embarked on a new development project for Kendal Calling festival.
3) If you went, what did you study at university?
My first degree was Philosophy BA at Cardiff University. My post-graduate degree was in Events Management at Leeds Beckett (then Metropolitan) and my doctorate was in festivals research, at the School of Music, University of Leeds.
4) What other qualifications do you have?
Without wanting to sound up my own arse, three degrees are enough.
5) When you work with other people, do you like it?
Yes and no — it depends who I am working with. But mostly yes. In fact, I crave working with others. Oddly enough lecturing can be isolating — you go in, teach, come home, sometimes without speaking to someone over 19 the whole day. I need interaction with peers that challenge and motivate me. As a portfolio careerist that works autonomously, a lot of working with others is done remotely though there are obviously benefits to that too.
6) When are you happiest at work?
When I am doing what I love, and smashing it. The most recent examples of that include a Festivals Panel session I put together and chaired for the UK Centre for Events Management. It was just fantastic; a huge amount of preparation but the thrill of the session which was delivered to an audience of 200 was amazing. Another example would be consulting with Kendal Calling on developing a new area within the site for the 2015 event. All of this kind of stuff is in my blood. I don't know how or why but it is. It's not all happiness of course — it's stress too, and I feel it probably more than most, but you happen to have caught me at a high point where I'm reaping some rewards!
7) What challenges you most?
The biggest challenges yield the most satisfaction when overcome. For me, many things I wished I could do seemed completely impossible for years and years. Public speaking is an example. But throwing yourself out there, taking the risk — over and over again — does magical things over time. I have the same philosophy with most challenges that seem threatening, but are aligned to my goals and interests. This means that I literally spend half, if not two thirds of my life worrying — worrying that I am not going to make it through the next hurdle. But nothing compares to the feeling that you have proved yourself wrong. That you are not an idiot. That you did it. I'm addicted to that feeling (my friends compare me to Lesley Knope from Parks and Recreation — enough said)
8) What is the most exciting thing that's happened to you in your career?
Probably what happened two days ago. My publisher told me they were 'very pleased' with my book manuscript — the culmination of a decade of research, that was all done without knowing if any of it would ever be seen by the world. Now it is, how awesome is that?
9) Of all of your jobs, which was the worst?
This is a tough one as they all had value in some sense, even the ones that really sucked. Working at Claires Accessories as a shop girl was pretty grim. You had to clean the whole shop at the end of an 8 hour shift on your feet, unpaid.
10) What advice would you give to someone who wants to do your job?
Stop at nothing, take every opportunity you can to present yourself and your work, prepare to make many sacrifices, find a true passion and follow it, stand up for yourself, know that you are not a fraud — even if you feel like one. Be brave.
Click here to pre-order Pop Festival: History, Music, Media, Culture With Dr Robinson's chapter 'No Spectators! The Art of Participation: from Burning Man to boutique festivals in Britain'
Image used with permission from Dr Roxy Robsinson. Feature image source: Making A Modern Man


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tag:blogger.com,1999:blog-6704776389626254933.post645242083573017320..comments2020-01-19T00:28:01.842-08:00Comments on Chloe Elizabeth: How to Look After Your Mental Health Whilst at UniversityChloe Elizabethhttp://firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-28448247203276805792018-08-16T05:53:17.000-07:002018-08-16T05:53:17.000-07:00This is such an important post! I thought Universi...This is such an important post! I thought University would be all fun and games, but it took such a toll on me and was the hardest three years of my life. One thing that really helped me was giving myself a break when I needed it. Even if I had three assignments due, I would give myself the afternoon off if I was feeling too overwhelmed. I always managed to get everything done in the end, and got much better grades than I did when I was frantically trying to push my anxiety away and force myself to work.notesatduskhttp://firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-59159245692953102342018-08-16T06:49:54.000-07:002018-08-16T06:49:54.000-07:00Thank you so much for your comment! It's so im...Thank you so much for your comment! It's so important to give yourself a break. So so important! If you don't, you'll never stop! <firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-59585750406433392332018-08-16T12:39:02.000-07:002018-08-16T12:39:02.000-07:00I've just finished my first year at univeristy and...I've just finished my first year at univeristy and I can relate to how overwhelming it can be, loved this post! XMolhttp://email@example.com:blogger.com,1999:blog-6704776389626254933.post-76113478322057660832018-08-16T14:35:49.000-07:002018-08-16T14:35:49.000-07:00I needed this, I'm going to uni this year and ...I needed this, I'm going to uni this year and hoping to keep on top of my anxiety. Thanks for sharing xx
Alys
https://alysjournals.comAlyshttps://firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-53805628470635568062018-08-16T15:54:28.000-07:002018-08-16T15:54:28.000-07:00That is so exciting! Good luck Alys! xxThat is so exciting! Good luck Alys! firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-81228556165456457512018-08-16T15:54:51.000-07:002018-08-16T15:54:51.000-07:00Thanks so much Molly! Glad you liked it! <3Thanks so much Molly! Glad you liked it! <firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-70346388726909150812018-08-17T04:24:46.000-07:002018-08-17T04:24:46.000-07:00This post really resonates with me as I had to see...This post really resonates with me as I had to seek for support at university this year for my mental health. I'm so happy that universities have wonderful student wellbeing centres and have helped me stay grounded during a difficult year. I'd definitely agree and advise that you should be seeking support as soon as you don't feel 100% yourself or feel yourself slipping into a bad mental health phase. The people who help out at my uni are so lovely and I wouldn't be able to carry on without them. Lovely post Chloe xxxCharlotte Pricehttp://firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-33830856415261212362018-08-17T10:14:53.000-07:002018-08-17T10:14:53.000-07:00This is such a helpful post. I really struggled wi...This is such a helpful post. I really struggled with my mental health this year at university, and I really want to have a better year in that respect come September. My one piece of advice would be to just ask. I was terrified to talk to uni about what I'd been struggling with, but I really didn't need to be! They're so used to all of this, and now I have the right things in place to help me out academically and socially if things get bad again.

-pixieskiesblog.wordpress.com xoMegan @ Pixieskieshttp://email@example.com:blogger.com,1999:blog-6704776389626254933.post-15326258984934531702018-08-17T12:55:21.000-07:002018-08-17T12:55:21.000-07:00Thanks so much for your comment Charlotte! I defin...Thanks so much for your comment Charlotte! I definitely agree, it's better to seek help sooner rather than later. I wish I had! firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-26610061279058759572018-08-17T12:56:39.000-07:002018-08-17T12:56:39.000-07:00Thanks so much for your comment Megan. Yes definit...Thanks so much for your comment Megan. Yes definitely, just ask!! I really hope you have a better year lovely <email@example.com:blogger.com,1999:blog-6704776389626254933.post-89425240753008253172018-08-17T22:58:13.000-07:002018-08-17T22:58:13.000-07:00It is so important to look after your mental well ...It is so important to look after your mental well being, especially when you're in college. If something happens, it can drastically change your college path going forward (eg. grades going down/etc). YES on communicating with others - especially course staffs. It's OK to say no! You're not obligated to go anywhere you don't want to. Thanks for sharing these positive tips!

Nancy ♥ exquisitely.meNancyhttps://firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-43857895685313913642018-08-18T13:34:49.000-07:002018-08-18T13:34:49.000-07:00Thanks so much Nancy! I completely agree! xxThanks so much Nancy! I completely agree! email@example.com:blogger.com,1999:blog-6704776389626254933.post-16413522451097177772018-08-18T14:16:47.000-07:002018-08-18T14:16:47.000-07:00As someone who struggles with mental health and st...As someone who struggles with mental health and stresses easily and is about to start university, this was a really important post for me to read and I can't thank you enough for writing it.

Rachael x email@example.com:blogger.com,1999:blog-6704776389626254933.post-56717955762253027192018-08-19T02:08:55.000-07:002018-08-19T02:08:55.000-07:00This is a really great post for anyone about to st...This is a really great post for anyone about to start uni or maybe who's been straggling, some really good tips! :)

Lainy x https://email@example.com:blogger.com,1999:blog-6704776389626254933.post-21385282287667165052018-08-19T02:11:00.000-07:002018-08-19T02:11:00.000-07:00This is such an important post and I'm so glad you...This is such an important post and I'm so glad you've written about it! There's an unbelievable amount of pressure at university and it's so important that people who suffer from mental health issues prepare themselves

Tish | thesundaydiary.co.ukLeticia email@example.com:blogger.com,1999:blog-6704776389626254933.post-91190522389761583452018-08-19T02:13:31.000-07:002018-08-19T02:13:31.000-07:00I feel like anyone going to Uni will find this pos...I feel like anyone going to Uni will find this post helpful. In the lead up to going to university no one speaks about the mental health aspect of it but literally everyone I know who I went to uni has suffered with mental health problems.Sophiehttps://firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-82405525494331696662018-08-19T03:38:17.000-07:002018-08-19T03:38:17.000-07:00This is such a great post on a really important is...This is such a great post on a really important issue. My mental health was fine whilst at uni but found it dipped after I left! I'm doing okay now. It's all the change...it really throws you off!Rosiehttps://firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-42143110210364866452018-08-19T07:49:53.000-07:002018-08-19T07:49:53.000-07:00Thank you so much Rachael! I'm so glad you enj...Thank you so much Rachael! I'm so glad you enjoyed it email@example.com:blogger.com,1999:blog-6704776389626254933.post-54572276474507718692018-08-19T07:51:34.000-07:002018-08-19T07:51:34.000-07:00Thank you, lovely! ❤Thank you, lovely! ❤email@example.com:blogger.com,1999:blog-6704776389626254933.post-83163699038927042062018-08-19T07:52:06.000-07:002018-08-19T07:52:06.000-07:00Thanks so much Tish! I agree so much, I wasn't...Thanks so much Tish! I agree so much, I wasn't prepared for university at email@example.com:blogger.com,1999:blog-6704776389626254933.post-584451070926634792018-08-19T07:52:41.000-07:002018-08-19T07:52:41.000-07:00Thanks so much Sophie! It's really hard someti...Thanks so much Sophie! It's really hard sometimes and definitely something that should be spoken about more! email@example.com:blogger.com,1999:blog-6704776389626254933.post-67486133195143340572018-08-19T07:53:25.000-07:002018-08-19T07:53:25.000-07:00Oh bless you! I had a tipple when I left too but I...Oh bless you! I had a tipple when I left too but I wasn't too bad. It's definitely the change too I think! email@example.com:blogger.com,1999:blog-6704776389626254933.post-19488160731210925422018-08-19T08:40:33.000-07:002018-08-19T08:40:33.000-07:00This is such a beautifully written post, mental he...This is such a beautifully written post, mental health is so important and unfortunately my University seem to not have a mental health society...

FATIMAH • ROSEYGOLDBOX
email@example.com:blogger.com,1999:blog-6704776389626254933.post-14136842825253766862018-08-19T09:33:47.000-07:002018-08-19T09:33:47.000-07:00Thanks Fatimah! Mine didn't have a mental heal...Thanks Fatimah! Mine didn't have a mental health society either, but you could always set one firstname.lastname@example.com:blogger.com,1999:blog-6704776389626254933.post-9183579626709798942018-08-19T10:08:49.000-07:002018-08-19T10:08:49.000-07:00I am sorry that you had to deal with that Chloe, I...I am sorry that you had to deal with that Chloe, I too had to deal with my mental health issues in college and was one of the biggest struggles of my life. I remember the first year of school I could hardly get out of bed to get to my classes. Often times I'd show up late, and have to give the instructor some BS excuse because I was to embarrassed to tell him the truth. Once the semester was over, I finally came clean, and he was so great about it. He helped me get help too. Thanks for sharing!Kayla Pettigrewhttp://firstname.lastname@example.com


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August 6
2019 August 6
The Herald Moth is supposed to have a strong proboscis, which can pierce the skins of ripening fruits. This one, photographed by Gordon Hart yesterday in his Highlands yard, is demonstrating its skills on a yellow plum.
Herald Moth Scoliopteryx libatrix (Lep.: Erebidae – Scoliopteryginae) Gordon Hart
Mike Yip writes from Nanoose: I encountered some extra-curricular activity in my yard this morning with the Common Woodnymphs. They have been flying locally for about three weeks and are easy to find in the Garry Oak meadows around Nanoose Bay. Mylitta Crescents are also in flight again. The Dun Skipper was seen on the logging road near Rhodo Lake two weeks ago, as was the Branded Skipper (Common or Western?) at Whiskey Creek. The moth was photographed at Long Beach and was determined by Libby Avis as Euxoa sp. (Possibly E. olivia?)
Common Woodnymphs Cercyonis pegala (Lep.: Nymphalidae – Satyrinae) Mike Yip
Mylitta Crescent Phyciodes mylitta (Lep.: Nymphalidae) Mike Yip
Note the slight concavity of the forewing outer margin, characteristic of the species.
Dun Skipper Euphyes vestris (Lep.: Hesperiidae) Mike Yip
The white spots indicate that this is a female.
Branded Skipper Hesperia comma (Lep.: Hesperiidae) Mike Yip
Some "splitters" will divide this skipper into Common Branded Skipper H. comma, and Western Branded Skipper H. colorado. On this site I treat them as a single species, H. comma. Jeremy Tatum
Euxoa sp. (possibly olivia ?) (Lep.: Noctuidae) Mike Yip
Jochen Möhr's moths in Metchosin this morning:
1 Amorbia cuneanum
1 Callizzia amorata
1 Eulithis xylina
1 Evergestis funalis
2 Lacinipolia strigicollis
2 Lophocampa argentata
1 Abagrotis apposita
1 Panthea virginarius
1 Pyrausta perrubralis
Evergestis funalis (Lep.: Crambidae) Jochen Möhr
Abagrotis apposita (Lep.: Noctuidae) Jochen Möhr
Pyrausta perrubralis (Lep.: Crambidae) Jochen Möhr
Pyrausta perrubralis (Lep.: Crambidae) and Misumena vatia (Ara.: Thomisidae) Jochen Möhr
Kalene Lillico photographed this nice kitten moth outside the door of the Swan Lake Nature House this morning, August 6:
Furcula scolopendrina (Lep.: Notodontidae) Kalene Lillico


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Lutz Leonhardt was born in Voerde in 1958 and studied Music in Duisburg and Bern. He graduated in 1981 and worked as a percussionist in concert halls, theater and radio. He directed his first feature-length film, The Robber (Der Raeuber), in 1985. Between 1990-1993, he completed his first long-term adventure project, Stroll to Syracuse (Spaziergang nach Syrakus), and in 1994 started preparations for his second film Zakir and His Friends (1997).


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