In this brief, we discuss our recent work exploring how adults and peers have supported youth who participate in Hive NYC-affiliated activities. Drawing from interviews conducted with our case study youth (read more about them in our first Youth Trajectories brief), we articulate a set of 16 supportive roles—encompassing Material, Knowledge Building, Emotional, Brokering and Institutional forms of support—that our youth identified as being important to them.

We also describe how we developed visualizations of these supportive roles and the adults and peers behind them, forming maps of something we call a young person’s *social learning ecology (SLE). *These SLE maps may be used to better understand and characterize aspects of a young person’s SLE such as *redundancy of support* and *diversity of sources* that we hypothesize may be consequential to sustained engagement in certain activities. Also, comparing a youth’s SLE map at different time periods demonstrates the dynamic nature of social learning ecologies generally and how certain providers (and thus forms of support) may be more transient in a young person’s life than others.

Youth Trajectories Interim Brief #2: Mapping Social Learning Ecologies of Youth.

]]>Youth Trajectories Interim Brief #1 – Introduction to Case Portraits of Hive Youth.

]]>Advocates of

“One way of thinking about this pair of developments (learning the properties shared and not shared by different types of numbers) is as a gradual change from initially conceptualizing numbers in terms of characteristic features (salient properties of whole numbers that are not necessarily properties of other types of numbers) to later distinguishing between deﬁning features (properties of all real numbers, in particular their magnitudes) and features that apply to some but not all classes of numbers. This change is analogous to the shift from characteristic to deﬁning features in semantic development described by Keil and Batterman (1984).”

“One implication of the present theory of numerical development is that acquisition of knowledge about fractions emerges as a crucial process in numerical development, rather than being of secondary importance. Learning about fractions provides the ﬁrst major opportunity for children to learn that a variety of salient and invariant properties of whole numbers are not deﬁnitional for numbers in general.”

**If fractions are crucial for overall mathematical understanding, and if understanding magnitudes is crucial for understanding fractions, then (1) Understanding of fraction magnitudes should be strongly related to proﬁciency at fractions arithmetic; and (2) Understanding of fraction magnitudes should be strongly related to overall mathematical knowledge.**

“Children could memorize fraction arithmetic algorithms without understanding the magnitudes of the fractions being manipulated. Indeed, many mathematics educators have lamented that this is exactly what most students do (e.g., Cramer, Post, & del Mas, 2002; Hiebert & Wearne, 1986; Mack, 1995; Sowder et al., 1998). However, rote memorization without understanding tends to be inaccurate over short time periods and becomes even more inaccurate over longer periods (Reyna & Brainerd, 1991). Thus, fraction arithmetic procedures seemed likely to be more accurately remembered by children who understand the magnitudes of the fractions used in the computation than by children who do not understand the fraction magnitudes. One reason is that accurate fraction magnitude representations make it possible to estimate the results of fraction arithmetic operations and to reject implausible solutions. This, in turn, might lead children to reject ﬂawed arithmetic procedures that produced implausible solutions and to continue trying to learn a procedure that generates reasonable answers. Consistent with this perspective, Hecht (Hecht, 1998; Hecht, Close, & Santisi, 2003; Hecht & Vagi, 2010) has found repeatedly that mea- sures of conceptual understanding of fractions correlate positively with fraction addition skill. Another reason for the prediction is that knowledge of whole number arithmetic correlates positively with knowledge of whole number magnitudes (Booth & Siegler, 2008; Siegler & Ramani, 2009). If knowledge of magnitudes play the same role with fractions as it does with whole numbers, as envisioned by the present theory, similar relations between knowledge of numerical magnitudes and arithmetic would be expected with both types of numbers.”

Understanding of numerical magnitudes has been shown to be related to many other aspects of mathematical development, including counting (Ramani & Siegler, 2008; Whyte & Bull, 2008) arithmetic (Booth & Siegler, 2008; Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; Geary, Hoard, Nugent, & Byrd-Craven, 2008; Siegler & Ramani, 2009), memory for and categorization of numbers (Laski & Siegler, 2007; Opfer & Thompson, 2008; Thompson & Siegler, 2010), and mathematics achievement test scores (Booth & Siegler, 2006; Geary et al., 2007, 2008; Halberda, Mazzocco, & Feigenson, 2008; Siegler & Booth, 2004). Manipulations that improve numerical magnitude knowledge have been shown to be causally related to increased proﬁciency in arithmetic (Booth & Siegler, 2008; Siegler & Ramani, 2009) and counting (Ramani & Siegler, 2008; Whyte & Bull, 2008).

The number line task has several important advantages for measuring representations of numerical magnitudes:

- It can be used with any real number (large/small, positive/negative, integer/fraction, rational/irrational).
- It transparently reﬂects the ratio characteristics of the number system.
- The task is practiced infrequently compared to skills such as counting and arithmetic, so estimates reﬂect people’s sense of the magnitudes of the numbers rather than memorization of procedures.

Two other estimation tasks: numerosity estimation (‘‘There is 1 dot in this beaker and 1000 in this one; hold down the mouse until there are N dots in this empty beaker.’’) and measurement estimation (‘‘This short line is 1 zip long; this long line is 1000 zips long; draw a line N zips long.’’) (Booth & Siegler, 2006).

Individual differences in numerical magnitude representations have been found to be positively related to other individual differences in mathematical knowledge — arithmetic competence (Booth & Siegler, 2006; Gilmore, McCarthy, & Spelke, 2007; Halberda et al., 2008; Holloway & Ansari, 2008; Mundy & Gilmore, 2009; Schneider et al., 2008) and scores on standardized math achievement tests (Booth & Siegler, 2006; Halberda et al., 2008; Laski & Siegler, 2007; Siegler & Booth, 2004).

Link btw numerical magnitude representations and arithmetic: If learning answers to arithmetic problems is a meaningful process, accurate magnitude representations might indicate the implausibility of many answers and the plausibility of a few, producing more peaked distributions of activation around the correct answer, thus facilitating correct retrieval. Thus, **experiences that improve numerical magnitude representations not only increase subsequent learning of correct answers to arithmetic problems but also lead to errors being closer to the correct answer on trials where children err (Booth & Siegler, 2008; Siegler & Ramani, 2009).**

“Another important way in which development of knowledge of fractions magnitudes seems likely to differ from development of whole number magnitudes is in the **role of strategies**. Although strategic inﬂuences have been recognized in a wide range of problem solving and reasoning contexts (Siegler, 1996), reviews of the literature on whole number magnitude representations (e.g., Ansari, 2008; Dehaene, Dehaene-Lambertz & Cohen, 1998; Fias & Fischer, 2005; Hubbard et al., 2005) typically do not even mention strategies or strategy choices (for exceptions, see Geary et al., 2007, 2008). **The implicit assumption is that people invariably use a particular representation of numerical magnitudes and that the research task is to determine the characteristics of that representation. Sometimes, the assumption is explicit, as when Dehaene (1997, p. 78) described logarithmic representations of whole number magnitudes as occurring ‘‘like a reﬂex’’ that cannot be inhibited. Consistent with the assumption that magnitude representation is an automatic, non-strategic process, number line estimation with whole numbers is no less accurate under time pressure than without time pressure (Siegler & Opfer, 2003).”**

Study design:

- 6th and 8th graders (11- and 12-year-olds and 13- and 14-year- olds)
- Assessments of fraction magnitude knowledge: 0–1/ 0–5 number line estimation, 0–1 magnitude comparison, fraction arithmetic problems.
- Verbal reports of strategy use were obtained immediately after each number line estimation and arithmetic problem, to allow examination of relations between strategy use and speed and accuracy on each task.
- Obtained students’ mathematics achievement test scores, to examine their relation to the three measures of fraction magnitude knowledge and to fraction arithmetic proﬁciency.

We made six predictions:

- Even after years of fractions instruction, fraction magnitude representations, whether measured by magnitude comparison or number line estimation, will be quite inaccurate in both 6th and 8th grade.
- Despite information about fractions magnitudes being taught explicitly in 3rd and 4th grade (NCTM, 2007), this understanding should still be increasing between 6th and 8th grade, due to students learning about fraction magnitudes from solving problems involving proportions and percentages in those grades.
- No logarithmic to linear transition should be present with fractions, because frequency of encountering fractions (and therefore knowledge of speciﬁc fractions) is correlated minimally if at all with fraction magnitudes, at least in the 0–1 range.
- Students should use a variety of strategies to solve fraction number line estimation and arithmetic problems, and the quality of these strategies should be related to students’ accuracy and speed in solving problems, as with whole number arithmetic.
- Individual differences in knowledge of fractions magnitudes should correlate highly with success at solving fraction arithmetic problems.
- Individual differences in knowledge of fractions magnitudes should correlate highly with individual differences in overall mathematics achievement test scores.

Number line estimation task:

- participants were sequentially presented 10 number lines on a computer screen.
- 0-1 number line items: 1/19, 1/7, 1/4, 3/8, 1/2, 4/7, 2/3, 7/9, 5/6, and 12/13.
- 0-5 number line items: 1/19, 4/7, 7/5, 13/9, 8/3, 11/4, 10/3, 7/2, 17/4, and 9/2.
- One fraction was drawn from each tenth of the number line, and presentation order was random.
- Participants responded on each trial by moving the cursor to the desired position on the number line and clicking the mouse.
- Accuracy of number line estimation was indexed by
**percent absolute error (PAE).**PAE = (|Child’s Answer – Correct Answer|) / Numerical Range. For example, if a child was asked to locate 5/2 on a 0–5 number line, and marked the location corresponding to 3/2, the PAE would be 20% ((|1.5 – 2.5|) / 5).

Magnitude comparison task:

- Participants were asked to compare to 3/5 a fraction shown on the computer screen: 3/8, 5/8, 2/9, 4/5, 4/7, 5/9, 8/9, or 2/3.
- If the fraction was smaller than 3/5, the participant was to press the ‘‘a’’ key; if the fraction was larger than 3/5, the participant was to press the ‘‘l’’ key.”

Fraction arithmetic assessment:

- Participants were presented 8 problems, 2 for each of the 4 arithmetic operations: 3/5 + 1/2, 3/5 + 2/5, 3/5 – 1/2, 3/5 – 2/5, 3/5 * 1/2, 3/5 * 2/5, 3/5 / 1/2, and 3/5 / 2/5.
- One of the two problems for each arithmetic operation had operands with equal denominators. Problems appeared one at a time on the computer screen, and participants typed their answers.

Classiﬁcation of strategies:

- Strategies were classiﬁed on the basis of overt behavior and immediately retrospective self-reports.
- When overt behavior clearly indicated the child’s approach, that behavior was the basis of the strategy assessment; otherwise, the child’s self-report was used.
- The two main types of number line estimation strategies were
**numerical transformation**strategies, in which participants transformed the presented fraction to a more convenient number, and**number line segmentation**strategies, in which participants generated subjective landmarks on the number line. - Both types of strategies could be, and often were, used on a single trial; children could transform the fraction to a more convenient numerical form and segment the number line in a way that helped them locate the fraction.
- In coding numerical transformation strategies, we distinguished only between using a numerical transformation and not using one.
- The most common numerical transformations were rounding the fraction (‘‘5/9 is a bit more than ½’’), simplifying it (‘‘9/5 = 1 and 4/5, which is a little less than 2’’), or translating it into a different form (‘‘12/13 is about 90%). The reason for not distinguishing among these numerical transformations was that they overlapped and could not be reliably distinguished.
- The main number line segmentation strategies were division into halves; division into ﬁfths or whole number units (e.g., placing marks on a 0–5 number line at the estimated positions of 1, 2, 3, and 4); division into units corresponding to the denominator (e. g., dividing a 0–1 number line into sevenths to locate 4/7); ﬂawed approaches (e.g., on a 0–1 number line, reporting, ‘‘I put 3/7 near 0 because 3 rounds down’’); and none unknown (e.g., saying, ‘‘I don’t know’’).
- “The lengthy solution times on none unknown trials, roughly 10 s, suggested that children might have used some strategy, but neither self- reports nor overt behavior indicated what it was.”
- The mean of the median solution times across number lines and age groups was 9.5 s. On 0–1 number lines, the means of the medians of individual children’s solution times were 8.4 s for 6th graders and 7.3 s for 8th graders; on 0–5 number lines, they were 11.9 s for 6th graders and 10.3 s for 8th graders.
- The response times suggest that fractions number line estimation is far from automatic; rather, it appears to be a controlled, strategic process.”

“The present ﬁndings indicate that understanding of fraction magnitudes and fractions arithmetic are closely related. If learning fraction arithmetic algorithms reﬂected rote memorization, as has often been claimed (e.g., Cramer & Bezuk, 1991; Hiebert, 1986; Kerslake, 1986), there would be no reason to expect such a relation. However, the strong correlations between fractions arithmetic and all three measures of magnitude knowledge in both 6th and 8th grades indicate that conceptual and procedural knowledge of fractions are intertwined. (See Hecht (1998), Hecht, Close, and Santisi (2003), Hecht and Vagi (2010), and Schneider and Stern (2010) for similar ﬁndings.) One plausible interpretation of these results is that magnitude knowledge makes it easier to learn and remember fraction arithmetic algorithms. This might occur through children with good magnitude knowledge of fractions rejecting procedures that produce unreasonable answers, such as operating independently on numerator and denominator often does, and searching longer for procedures that produce reasonable answers. For example, children might reject the procedure that produces arithmetic errors of the form 3/5 – 1/2 = 2/3 if they recognized that subtracting a positive number can- not lead to an answer larger than the number being subtracted from. This could lead them to try other procedures and test whether they yielded plausible answers.”

“Another implication for understanding fraction arithmetic is methodological: The same strategy assessment techniques that have proved useful with whole number arithmetic also are useful for investigating fractions arithmetic. As with whole number arithmetic, individual children used a variety of fractions arithmetic strategies. Even on a single fractions arithmetic operation, strategy use varied with problem characteristics, notably with the equality or inequality of denominators. The quality of fraction arithmetic strategy use was related to both knowledge of numerical magnitudes and to overall mathematics achievement test scores. In whole number arithmetic, these strategy assessment techniques have provided a base for computer simulation models of arithmetic learning that accounted for numerous ﬁndings regarding variations in accuracy, solution times, and strategy use across problems; discovery of useful new strategies; individual differences in arithmetic proﬁciency; and changes in speed, accuracy, and strategy use with problem-solving experience (Shrager & Siegler, 1998; Siegler & Shipley, 1995). The prominence of strategy use in fractions arithmetic suggests that similar models might be applicable to that area, and that it might be possible to formulate a common model of development of whole number and fractions arithmetic.”

“A third implication for understanding fractions arithmetic is that fraction arithmetic errors often reﬂect confusion about the right strategy, together with a lack of constraints on the magnitudes of answers, rather than a consistent whole number bias or other systematic misunderstanding.”

“[whole number bias] has been said to lead to children treating numerators and denominators as independent whole numbers and operating on them independently, for example by subtracting numerator from numerator and denominator from denominator (Carpenter et al., 1981; Gelman, 1991; Kilpatrick, Swafford, & Findell, 2001). However, the present ﬁndings revealed greater variability in fraction procedures than implied by this attribution of errors to a systematic misconception.”

**“Roughly half of arithmetic errors stemmed from applying whole number algorithms independently to numerators and denominators, but a similar percentage reﬂected using parts of algorithms that would have been correct for a different fraction arithmetic operation or trying other erroneous procedures. The inconsistency of strategies even within a single arithmetic operation was striking; 40% of children correctly solved one of the pair of problems for a single arithmetic operation and erred on the other. This variability suggests that the whole number bias is only part of the problem in understanding fractions arithmetic. Rather than reﬂecting a systematic misconception, fractions arithmetic knowledge seems piecemeal; understanding of whole numbers is one source of ideas about how to solve fractions arithmetic problems, but other types of numerical knowledge are also incorporated. It also is unclear whether children who use erroneous fractions arithmetic procedures believe that those procedures are correct. They might well be skeptical about their correctness but have learned that saying ‘‘I don’t know’’ is not an acceptable alternative in school mathematics. This issue could be addressed through studies that examine children’s conﬁdence in their fraction arithmetic answers.”**

“In the United States, instruction in fractions emphasizes part-whole interpretations far more than other interpretations of fractions (Ni & Zhou, 2005; Sophian, 2007; Thompson & Saldanha, 2003). For example, students are taught to interpret 1/5 as one of ﬁve slices of pizza, but less often to think of 1/5 as one ﬁfth of the distance from zero to one on a number line (Moseley, Okamoto, & Ishida, 2007). This is quite different than the approach to teaching fractions in Japan, China, and other countries where students understand fractions better. Indeed, many teachers in the US can only explain fractions in terms of the part-whole interpretation, unlike teachers in China and Japan who also emphasize num- ber line and other interpretations (Ma, 1999; Moseley et al., 2007). Part-whole interpretations have the advantages of concreteness and accessibility. When numerators and denominators are small and positive and the numerator is less than the denominator, it is easy to think about N parts of a whole that includes M parts. For example, children have little difﬁculty understanding that if a pizza is cut into four pieces, then each piece is 1/4 of the pizza (Mix, Levine, & Huttenlocher, 1999). **However, the part-whole interpretation of fractions also has some serious limitations. Negative fractions cannot be represented in this way, it is very difﬁcult to imagine fractions with large numerators and denominators (e.g., 734/878), and improper fractions can be confusing within the part-whole interpretation, as illustrated by one learner’s reaction to being presented 4/3, ‘‘You cannot have four parts of an object that is divided into three parts’’ (Mack, 1993). Moreover, there is nothing in the operation of dividing an object into N parts that says that the size of the parts must be equal; many students fail to understand that the parts must be equal for fractions to have any consistent meaning (Sophian, 2007).”**

“emphasizing that fractions are measurements of quantity might improve learning about fractions. Indeed, a common feature of instructional studies that have yielded especially promising results in teaching rational numbers, such as work by Robbie Case and his associates, is that they emphasize that fractions are measures of quantity (e.g., Cramer et al., 2002; Fujimura, 2001; Keijzer & Terwel, 2003; Moss & Case, 1999; Rittle-Johnson & Koedinger, 2002, 2009). The present integrated theory of numerical development helps to explain the prevalence of this common feature of successful instruction: **If magnitudes are central to understanding fractions as well as whole numbers, then instruction that emphasizes magnitude understanding is more likely to succeed than instruction that does not emphasize magnitude understanding.”**

The authors tested the validity of eye-movement data as a means for investigating children’s use of the number line when solving number line estimation tasks, and as a measure of children’s developing number sense. In a cross-sectional design with children from Grades 1 to 3, they assessed (a) the accuracy of manual solutions of number line estimation tasks, (b) the accuracy of the positions ﬁxated by gaze while solving a second set of number line estimation tasks, and (c) the accuracy of responses to mental addition tasks.

Research questions:

- Is grade-related increase in children’s estimation competence only reﬂected by manual answers or also by eye-tracking data?
- Are individual differences in the accuracies of the estimated positions and in the accuracies of their eye movements during the estimation process correlated?
- Are the accuracy of the estimated positions and the accuracy of the positions ﬁxated by gaze during solution production correlated with children’s addition competence?
- To what extent does the criterion validity of the eye-movement measure increase with age?
- Does eye-movement data indicate that older children increasingly use the midpoint of the number line as an orientation point?”

Number sense: “ability to quickly understand, approximate, and manipulate numerical quantities” (Dehaene, 2001, p. 16).

“The “mental number line” is regarded as the core neurocognitive system underlying number sense (Fias, Lammertyn, Reynvoet, Dupont, & Orban, 2003; Hubbard, Piazza, Pinel, & Dehaene, 2005; Pinel, Dehaene, Riviere, & LeBihan, 2001), which in turn underlies a variety of behavioral competencies, like estimating, computing, and efﬁciently using notational systems to solve mathematical problems (Berch, 2005; Jordan, Kaplan, Locuniak, & Ramineni, 2007).” Represents the magnitudes of numbers in an analogous form.

“Arabic numerals, like 5 and 7, do not allow for any direct inference as to which of them is the one with the higher value. The same is true for number words, like 5 and 7. In contrast, when numerical magnitudes are represented by positions on a number line, one immediately grasps which of them is the higher value number. Therefore, Case and Okamoto (1996) suggested that children use the mental number line “to build models of the conceptual systems that their culture has evolved for measuring such dimensions as time, space. . . They use it to make sense of any direct instruction that they may receive regarding the particular systems that their culture has evolved for arranging numbers into groups and for conducting numerical computations” (pp. 8–9).”

*Measuring number sense:*

- See Berch, 2005; Jordan, Kaplan, Oláh, & Locuniak, 2006.
- Siegler and Opfer (2003) “asked children to estimate the positions of given numbers on an external number line where only the starting and the end points were labeled. They interpreted the patterns of estimates as indicative of children’s representation of magnitudes on their internal number line. These answer patterns relate not only to children’s competence in performing other estimation tasks, such as numerosity estimation and computational estimation, but even to children’s addition competence and general math achievement (Booth & Siegler, 2006, in press; Siegler & Booth, 2004).”
- The number line estimation task is a “practical and powerful tool applicable across a wide range of age groups. It has been hypothesized to reﬂect children’s mental representation of numbers more directly than alternative assessments of number sense do.”

“Despite its potential beneﬁts, the number line estimation task has a major drawback. Although it is easy to measure such products of children’s estimation processes as accuracy, solution times and estimate patterns, it is hard to investigate the processes themselves that children employ to construct their solutions.”

*Solution strategies:*

- [Grade 1] Counting up strategy, counting down strategy, in whole units or decades from the beginning of the number line (Petitto, 1990; Newman & Berger, 1984).
- [Older children] Start counting up from the midpoint.

“Eye-movement data can be collected with high temporal and spatial resolution (e.g., several-hundred measures per second with a spatial precision of 0.01◦ ) notwithstanding the fact that the reliability of the resulting data can suffer from technically caused measurement error and task-irrelevant ﬁxations. Compared to accuracy and speed measures, eye-tracking data potentially provide more direct evidence of the process of problem solving. Moreover, these data are more objective than self-reports or behavioral observations of strategy use.”

“Children’s increasing ability to focus their attention on the task-relevant features of a problem situation are likely to reduce the number of task-irrelevant eye movements and, thus, to increase the validity of our eye-movement measure over the three grade levels.”

“By means of eye-tracking data, Rehder and Hoffman (2005) demonstrated that in adults, increasing competence in object categorization goes along with an increasing tendency to focus attention on task-relevant characteristics of a problem situation. To guide attention to task-relevant characteristics of a problem situation and to ignore task-irrelevant features is an important part of mathematical competence (The Cognition and Technology Group at Vanderbilt, 1992). Therefore, eye movements might reﬂect individual differences in mathematical competence.” [could also guide the design of scaffolding in a math game…]

“Two studies (Green, Lemaire, & Dufau, 2007; Verschaffel, De Corte, Gielen, & Struyf, 1994) have demonstrated that eye movements validly reﬂect various strategies chosen by elementary school children and adults to solve mental addition problems.”

“the external number line, as a diagram, is an analogous and more holistic diagrammatic knowledge representation (Larkin & Simon, 1987). The validity of eye movements as an indicator of children’s competence concerning this type of external knowledge representation has not been investigated.”

Methodology:

- Children had to solve 30 trials of the number line estimation task.
- Following Rittle-Johnson, Siegler, and Alibali’s (2001) example, we coded an answer as correct if it was within an error margin of ±10% of the number line around the actual position of the stimulus on the line, and as incorrect in all other cases.
- The stimuli were selected by a pseudo-random algorithm from the natural numbers between 0 and 100.
- For the eye-tracking version of the number line estimation task, children were instructed to actively search for and focus their gaze on the correct position for each number. After 4000 ms, a marker appeared. The children were asked to decide as fast as possible whether the marker position was correct or not and to give their answer by clicking a respective button. Button clicks and reaction times were recorded automatically.
- No feedback was given to the children during the experimental trials. The order of the two number line estimation tasks was counter-balanced at each grade level.
- In addition to individual ﬁxation accuracies, we derived the frequency distribution of all ﬁxations over the number line for each grade level by computing the position of each ﬁxation on the number line and by rounding to the nearest whole value. We counted per individual and task how often each of the 101 positions of whole numbers on the line had been ﬁxated. The resulting value for each position was averaged per grade level. To enhance readability, these values were multiplied by 10,000.
- Fixation accuracy measure is based on the assumption that not only the last ﬁxation but all ﬁxations during a trial indicate children’s knowledge.

*Some results*

- Addition accuracy increases most strongly from 15 to 91% and has the highest proportion of explained variance, as indicated by 2 values.
- The effect sizes also show that grade-related increases in knowledge are more clearly reﬂected by estimation accuracy than by ﬁxation accuracy. Of the three variables, ﬁxation accuracy shows the least change and the smallest, albeit still high, proportion of explained variance.
- Signiﬁcant change in estimation accuracy and ﬁxation accuracy occurs only between Grades 1 and 2, but not between Grades 2 and 3. Differences in addition accuracy are highly signiﬁcant between each of the grade levels.
- Exploratory comparisons of (a) the last ﬁxation in each trial with (b) all previous ﬁxations in that trial conﬁrm our expectations. The average percentage of correct ﬁxations in the sample is 42.1% for only the last ﬁxations and 34.0% for all ﬁxations. Both accuracies lie well above chance level (i.e. 20%) and, thus, indicate knowledge.
- The accuracies for both types of ﬁxations correlate across individuals with r = .53, p = .017, for ﬁrst graders, r = .73, p < .001, for second graders, and r = .70, p < .001, for third graders, suggesting that the two groups of ﬁxations measure similar aspects of children’s knowledge.
- Guided by our second research question, we computed the correlations between estimation accu- racy and ﬁxation accuracy. These are r = .28, p = .212, for ﬁrst graders, r = .66, p = .002, for second graders, and r = .63, p = .001, for third graders.
- Children in all three grades ﬁxate positions near the starting point of the line, near the end of the line, and near the number 50 in the middle of the line more frequently than any other position. respectively.
- Children’s ﬁxations (a) validly reﬂect grade-related competence increases, (b) are closely related, in Grades 2 and 3, to manual solutions of estimation tasks, (c) are related, in Grade 2, to addition com- petence, (d) are very systematically distributed over the number line, and (e) replicate Petitto’s (1990) ﬁndings with respect to the use of the midpoint strategy and the counting-up strategy by students in Grades 1–3.

“The fact that grade level explains more than one third of the interindividual variance of children’s ﬁxation accuracies establishes that eye movements reﬂect children’s increasing knowledge about natural numbers, their interrelations, and ways of their spatial representation. This understanding of natural numbers lies at the very core of children’s number sense and is a prerequisite for their future acquisition of more advanced mathematical concepts (Dehaene, 1997).”

“As explained by Verschaffel et al. (1994) and Green et al. (2007), identifying trial-by-trial strategy use by means of eye-tracking data can considerably improve research on the development of mathematical strategies. For example, parallel use of a verbal and a nonverbal measure of strategy use allows researchers to investigate the interaction of explicit and implicit knowledge during strategy development (Siegler & Stern, 1998). However, distinguishing strategy use on a trial-by-trial basis by means of eye-tracking may be as difﬁcult as it is desirable. In most eye-tracking studies the data are aggregated over trials – and often over persons – because ‘eye tracking data is never perfect. The system may lose track of the pupil or the corneal reﬂection, or the observation may be simply incorrect (e.g., beyond the screen limits even though the subject is clearly looking at the screen)’ (Aaltonen, Hyrskykari, & Räihä, 1998, p. 135). “Eye-movement data are inherently noisy” (Hornof & Halverson, 2002, p. 593), due to task-irrelevant ﬁxations (e.g., when the individual is distracted by sounds from the surroundings). Individuals may make data-distorting head-movements (children more so than adults) which might even require a re-calibration of the scanning system. Finally, individuals may use peripheral vision to detect information from regions on the screen that they do not ﬁxate directly. All these inﬂuences increase measurement error, thus, making it harder to ﬁnd effects on the trial level than on the level of aggregated data (Rayner, 1998; Verschaffel et al., 1994).”

“Eye movements allow for a direct investigation of how children orient themselves in problem situations and how they direct their attention to speciﬁc features. Although previous theories tended to conceptualize mathematical problem solving as an abstract symbol- manipulation process, more recent approaches emphasize the interaction with problem situations as a highly important part of mathematical competencies (Collins, Greeno, & Resnick, 2001). For example, advocates of the situated-cognition view have argued that the ability to pick up action-relevant information from the environment is the most important foundation of competent problem solving and knowledge transfer (Greeno, 1994; Greeno, Moore, & Smith, 1993). Eye movements offer a means of investigating the dynamic and selective search for information in problem situations. Such search invokes higher-level cognitive processes, including what one knows about, and expects of, a situation (Rehder & Hoffman, 2005). More advanced analysis techniques for eye-tracking data, such as ﬁxation density maps (Ouerhani, von Wartburg, Hügli, & Müri, 2004), may become useful for deepening our understanding of these phenomena.”

References:

Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333–339.

Booth, J. L., & Siegler, R. S. (2006). Developmental and individual differences in pure numerical estimation. Developmental Psychology, 41, 189–201.

Booth, J. L., & Siegler, R. S. (2008). Numerical magnitude estimations inﬂuence arithmetic learning. Child Development, 79(4), 1016–1031.

Butterworth, B. (2005). Developmental dyscalculia. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 455–467). New York: Psychology Press. Campbell, J. I. D. (Ed.). (2005). Handbook of mathematical cognition. New York: Psychology Press.

Case, R., & Okamoto, Y. (1996). The role of central conceptual structures in the development of children’s thought. Monographs of the Society for Research in Child Development, 61(1 2), 1–295.

Collins, A., Greeno, J. G., & Resnick, L. B. (2001). Educational learning theory. In N. J. Smelser & P. B. Baltes (Eds.), International encyclopedia of the social & behavioral sciences (pp. 4276–4279). Oxford: Elsevier Science.

Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press.

Dehaene, S. (2001). Précis of “The number sense”. Mind and Language, 16, 16–32.

Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122(3), 371–396.

Fias, W., & Fischer, M. H. (2005). Spatial representations of numbers. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 43–54). New York: Psychology Press.

Fias, W., Lammertyn, J., Reynvoet, B., Dupont, P., & Orban, G. A. (2003). Parietal representation of symbolic and non-symbolic magnitude. Journal of Cognitive Neuroscience, 15(1), 47–56.

Geary, D. C., & Hoard, M. K. (2005). Learning disabilities in arithmetic and mathematics: Theoretical and empirical perspectives. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 253–267). New York: Psychology Press.

**Green, H. J., Lemaire, P., & Dufau, S. (2007). Eye movement correlates of younger and older adults’ strategies for complex addition. Acta Psychologica, 125, 257–278.**

Greeno, J. G. (1994). Gibson’s affordances. Psychological Review, 101(2), 336–342.

Greeno, J. G., Moore, J. L., & Smith, D. R. (1993). Transfer of situated learning. In D. K. Detterman & R. J. Sternberg (Eds.), Transfer on trial: Intelligence, cognition, and instruction. Norwood, NJ: Ablex.

Hornof, A. J., & Halverson, T. (2002). Cleaning up systematic error in eye tracking data by using required ﬁxation locations. Behavioral Research Methods, Instruments, and Computers, 34(4), 592–604.

Hubbard, E. M., Piazza, M., Pinel, P., & Dehaene, S. (2005). Interactions between number and space in parietal cortex. Nature Reviews Neuroscience, 6(6), 435–448.

Jordan, N. C., Kaplan, D., Locuniak, M. N., & Ramineni, C. (2007). Predicting ﬁrst-grade math achievement from developmental number sense trajectories. Learning Disabilities Research & Practice, 22(1), 36–46.

Jordan, N. C., Kaplan, D., Oláh, L. N., & Locuniak, M. N. (2006). Number sense growth in kindergarten: A longitudinal investigation of children at risk for mathematical difﬁculties. Child Development, 77, 153–175.

Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11, 65–99.

Newman, R. S., & Berger, C. F. (1984). Children’s numerical estimation: Flexibility in the use of counting. Journal of Educational Psychology, 76(1), 55–64.

Ouerhani, N., von Wartburg, R., Hügli, H., & Müri, R. M. (2004). Empirical validation of the salience-based model of visual attention. Electronic Letters on Computer Vision and Image Analysis, 3, 13–24.

Petitto, A. L. (1990). Development of numberline and measurement concepts. Cognition and Instruction, 7(1), 55–78.

Pierce, C. A., Block, R. A., & Aguinis, H. (2004). Cautionary note on reporting eta-squared values from multifactor anova designs. Educational and Psychological Measurement, 64(6), 916–924.

Pinel, P., Dehaene, S., Riviere, D., & LeBihan, D. (2001). Modulation if parietal activation by semantic distance in a number comparison task. NeuroImage, 14, 1013–1026.

Rayner, K. (1998). Eye movements in reading and information processing: 20 years of research. Psychological Bulletin, 124(3), 372–422.

Rehder, B., & Hoffman, A. B. (2005). Eyetracking and selective attention in category learning. Cognitive Psychology, 51, 1–41.

Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362.

Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75(2), 428–444.

Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations. Psychological Science, 14, 237–243.

Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy discoveries: A microgenetic analysis. Journal of Experimental Psychology: General, 127(4), 377–397.

The Cognition and Technology Group at Vanderbilt. (1992). The Jasper series as an example of anchored instruction: Theory, program description and assessment data. Educational Psychologist, 27, 291–315.

**Verschaffel, L., De Corte, E., Gielen, I., & Struyf, E. (1994). Clever rearrangement strategies in children’s mental arithmetic: A confrontation of eye-movement data and verbal protocols. In J. E. H. V. Luit (Ed.), Research on learning and instruction of mathematics in kindergarten and primary school (pp. 153–180). Doetinchem, The Netherlands: Graviant.**

**Verschaffel, L., De Corte, E., & Pauwels, A. (1992). Solving compare problems: An eye movement test of Lewis and Mayer’s consistency hypothesis. Journal of Educational Psychology, 84(1), 85–94.**

Whiteside, J. A. (1974). Eye movements of children, adults, and elderly persons during inspection of dot patterns. Journal of Experimental Child Psychology, 18(2), 313–332.”

]]>The authors used eye tracking data to investigate the development of numerical magnitude understanding in primary school children (grades 1-3). Children were presented with two number line estimation tasks; one was restricted to behavioral measures, whereas the other included the recording of eye movement data. Gaze data indicates a quantitative increase as well as a qualitative change in children’s implicit knowledge about numerical magnitudes that precedes the overt (behavioural) demonstration of explicit numerical knowledge. Thus, eye movements may reveal more about the presence of implicit precursors of later explicit knowledge in the numerical domain than more traditional approaches, suggesting further exploration of eye tracking as a potential early assessment tool of individual achievement levels in numerical processing.

“The idea that cognitive growth can be characterized as a transition from implicit forms of knowledge to more explicit knowledge provides developmental researchers with a fascinating yet challenging starting point for translating the Vygotskyan ideas into the domain of knowledge acquisition.”

“Vygotsky’s concept of what he called a child’s zone of proximal development emphasizes the need to also consider children’s learning potential instead of merely focussing on what a child has explicitly mastered at any given point in his or her development. An important question is therefore how to assess children’s learning potential, that is, how to determine their level of knowledge that is not yet explicitly expressed (Brown & Ferrara, 1985).”

Dienes and Perner (1999): fully explicit knowledge – when the person is conscious of that knowledge and is in a position to entertain second-order thoughts about it. Eye movements can be used as sensitive indicators of implicit knowledge that precede more explicit forms of knowledge.

Susan Goldin-Meadow et al. conducted a number of inﬂuential developmental studies that aim at identifying children’s implicit or ‘budding knowledge’ (Vygotsky, 1978). “Focussing on the gestures that accompany children’s problem solving processes, Goldin-Meadow et al. were able to demonstrate subtle mismatches between children’s verbal and nonverbal behaviours (Alibali & Goldin- Meadow, 1993; Goldin-Meadow, Alibali, & Church, 1993). According to the authors, gesture–speech mismatches can be indicative of a child’s speciﬁc readiness-to-learn by revealing forms of knowledge that are there at an implicit level, but not yet explicitly available to the child (Goldin-Meadow, 2000; Goldin- Meadow & Sandhofer, 1999).”

Siegler et al.: primary schoolers’ representational patterns change from a logarithmic to a more adequate linear model of numerical magnitude between the ﬁrst and the second grade. This transition is considered a crucial step that enables children to improve their achievement in a wide range of mathematical task domains.

“The question is whether the behaviourally established quantitative and qualitative changes can also be demonstrated using eye movement parameters, and, more importantly, whether eye movement measures that we assume to be able to tap into implicit knowledge add to what we already know about the development of numerical magnitude representation in children.”

Some results:

- “While in ﬁrst graders, the eyes seem to roam a wide range of numerical positions on the number line, in third graders, approximately two-thirds of all ﬁxations fall within the immediate vicinity of the respective correct positions. The distributional pattern of second graders’ ﬁxations lies somewhere between the two extremes.”
- “In-depth analyses of the ﬁts of number line ﬁxations to the logarithmic and linear models determined in the behavioural setting revealed a somewhat different picture for the eye movement data compared with the behavioural data. Although ﬁrst graders’ estimates were ﬁtted substantially better by the logarithmic model than by the linear model in the behavioural task, this does not hold true for their eye movements. Comparable to the second and third graders, for the eye-tracking data the linear model provides a better ﬁt in ﬁrst graders as well.”
- “However, tests of the residuals between the models and the gaze data revealed that in ﬁrst graders the differences in the ﬁt of the linear compared to the logarithmic functions are not signiﬁcant. In second and third graders the linear functions provide the signiﬁcantly better models.”
- “Eye movement data demonstrate that even when, on ﬁrst sight, children still appear to be stuck on a more immature level their implicit knowledge might already have undergone some subtle yet detectable changes. The eye movement data demonstrate that even in cases where no evidence of appropriate insight into a certain numerical magnitude can be found in children’s overt behavioural responses, their eye movements might still show manifestations of knowledge at work on a more implicit level. Thus, for instance, our analyses of children’s eye movements in error trials indicate that third graders seem to understand more about certain numerical magnitudes than ﬁrst or second graders even when they do not actually appear to be more knowledgeable than their younger peers on the level of overt behaviour. The eye movement data reveal that compared with children from lower grade levels, older children shift their gaze signiﬁcantly more often and also for a signiﬁcantly longer time to the respective correct positions on the number line in trials where their explicit responses are consistently false.”

“While ﬁrst graders’ overt behaviour suggests that their representation of numerical magnitude still follows an immature logarithmic model, a change towards more mature representational patterns might already be on the way, as revealed by measures that are capable to tap this emerging knowledge. There might, thus, be a transitional phase where both representational patterns exist in parallel. Such an idea is well in line with Siegler’s (1996) Overlapping the Wave model of change that proposes that during development ‘multiple ways of thinking coexist for prolonged periods’ (p. 89).”

**References to read:**

Clements, W., & Perner, J. (1994). Implicit understanding of belief. Cognitive Development, 9, 377–397.

Clements, W. A., Rustin, Ch. L., & McCallum, S. (2000). Promoting the transition from implicit to explicit understanding: A training study of false belief. Developmental Science, 3, 81–92.

Alibali, M., Flevares, L., & Goldin-Meadow, S. (1997). Assessing knowledge conveyed in gesture: Do teachers have the upper hand? Journal of Educational Psychology, 89, 183–193.

Alibali, M. W., Bassok, M., Solomon, K. O., Syc, S. E., & Goldin-Meadow, S. (1999). Illuminating mental representations through speech and gesture. Psychological Science, 10, 327–333. Alibali, M. W., & Goldin-Meadow, S. (1993). Gesture–speech mismatch and mechanisms of learning: What the hands reveal about a child’s state of mind. Cognitive Psychology, 25, 468–523.

Dienes, Z., & Perner, J. (1999). A theory of implicit and explicit knowledge. Behavioral and Brain Sciences, 22, 735–755. Garber, P., Alibali, M. W., & Goldin-Meadow, S. (1998). Knowledge conveyed in gesture is not tied to the hands. Child Development, 69, 75–84.

Goldin-Meadow, S. (2000). Beyond words: The importance of gesture to researchers and learners. Child Development, 71, 231–239. Goldin-Meadow, S., Alibali, M. W., & Church, R. B. (1993). Transitions in concept acquisition: Using the hand to read the mind. Psychological Review, 100, 279–297.

Goldin-Meadow, S., & Sandhofer, C. M. (1999). Gesture conveys substantive information about a child’s thoughts to ordinary listeners. Developmental Science, 2, 67–74.

Perner, J., & Dienes, Z. (1999). Deconstructing RTK: How to explicate a theory of implicit knowledge. Behavioural and Brain Sciences, 22, 790–808.

Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. Cambridge, MA: MIT Press.

[Karmiloff-Smith’s (1992) notion of representational re-description as a process by which ‘implicit information in the mind subsequently becomes explicit knowledge to the mind’ (p. 18).]

Garnham, W. A., & Perner, J. (2001). Actions really do speak louder than words—but only implicitly: Young children’s understanding of false belief in action. British Journal of Developmental Psychology, 19, 413–432.

Garnham, W. A., & Ruffman, T. (2001). Doesn’t see, doesn’t know: Is anticipatory looking really related to understanding of belief? Developmental Science, 4, 94–100.

Laski, E. V., & Siegler, R. S. (2007). Is 27 a big number? Correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison. Child Development, 76, 1723–1743.

Perry, M., Church, R. B., & Goldin-Meadow, S. (1988). Transitional knowledge in the acquisition of concepts. Cognitive Development, 3, 359–400.

Ruffman, T., Garnham, W., Import, A., & Connolly, D. (2001). Does eye gaze indicate knowledge of false belief: Charting transitions in knowledge. Journal of Experimental Child Psychology, 80, 201–224.

]]>The goal of successful game design is the creation of meaningful play.

Play is situated within the game but play doesn’t come from the game itself but from the way the players interact w the game in order to play it. Meaningful play emerges from the interaction btw players and the system of the game, as well as from the context in which the game is played.

Playing means making choices and taking actions. The designer should try and make the choices as meaningful as possible.

Two notions of meaningful play:

1) descriptive – meaningful play resides in relationship btw action and outcome;

2) evaluative – when those relationships are both discernible and integrated into the larger context of the game.

Players must be able to discern whether or not a choice lead them closer or farther from winning. Must know the consequences of one’s actions — this requires the player knowing the meaning of each action. Actions have consequences that are woven throughout the game (a great example of this is Go).

**Chapter 7**

Lusory attitude – the peculiar state of mind of game players

“A game is a system in which players engage in an artificial conflict, defined by rules, that results in a quantifiable outcome” (p. 80).

**Chapter 8**

What can digital technology do? Traits of most digital games:

Immediate but narrow interactivity

Information manipulation. Karen Sideman points out that with digital games, part of the play is discovering the rules (contrast w board games).

Automated complex systems. Innerworkings hidden (what Dunnigan calls the ‘black box syndrome’ of computer games)

Networked communication

**Chapter 9**

The Magic Circle – defined as a special place in time and space created by a game. Separate from, but still references the real world.

Lusory attitude – allows players to “adopt rules which require one to employ worse rather than better means for reaching an end” (cited from Bernard Suits).

“From somewhere in the gap between action and outcome, in the friction between frustrated desire and the seductive goal of a game, bubbles up the unique enjoyment of game play” (p. 98).

One should design structures that can create and support the magic circle, as well as qualities that affect the lusory attitude and the possibility of meaningful play.

At the heart of games are rules, the space of games framed as formal systems.

**Chapter 11**

“Rules are the logical underbelly beneath the experiential surface of any games” (p. 120). The formal system of a game, the game considered as a set of rules, is not the experience of the game. When looking at games from the point of view of rules, we are less concerned w player experience than w the rules constituting the experience. Rules are the ‘formal identity’ of a game” (p. 121).

Characteristics of game rules

limit player action

explicit and unambiguous

shared by all players

fixed (do not change)

binding

repeatable & portable

**Chapter 12**

Three levels of rules:

Operational – rules of play, essentially

Constituative – underlying formal structures; logical & mathematical in nature

Implicit – unwritten rules. Concern etiquette, good sportsmanship, etc.

The identity of a game emerges from the interaction btw operational and constituative sets of rules.

All three levels work in concert to generate the formal meaning of a game.

Elegant rules allow the player to focus on the experience of play rather than the logic of the rules.

**Chapter 13**

In digital games, the game rules regulate game logic, player action and outcome, scoring system, structural arrangement of the game space, etc. — the formal structure of the game.

Operational rules for digital games include the use of input devices, certain external or representational event that impact player interactivity and formal game events.

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