Mathematical expertise: the role of domain-specific knowledge for memory and creativity

In contrast to traditional expertise domains like chess and music, very little is known about the cognitive mechanisms in broader, more education-oriented domains like mathematics. This is particularly true for the role of mathematical experts’ knowledge for domain-specific information processing in memory as well as for domain-specific and domain-general creativity. In the present work, we compared 115 experts in mathematics with 109 gender, age, and educational level matched novices in their performance in (a) a newly developed mathematical memory task requiring encoding and recall of structured and unstructured information and (b) tasks drawing either on mathematical or on domain-general creativity. Consistent with other expertise domains, experts in mathematics (compared to novices) showed superior short-term memory capacity for complex domain-specific material when presented in a structured, meaningful way. Further, experts exhibited higher mathematical creativity than novices, but did not differ from them in their domain-general creativity. Both lines of findings demonstrate the importance of experts’ knowledge base in processing domain-specific material and provide new insights into the characteristics of mathematical expertise.


Mathematical structure in the Mathematical Memory items
The first three items of this task involved numerical material. (1) Pascal's triangle. The Pascal's triangle starts with the number 1 in the top row and the row below is constructed by summing the two adjacent elements in the preceding row (as such the triangular array can be expanded infinitely). In the structured condition we used the sixth, seventh and eighth row of Pascal's triangle resulting in 24 numbers. In the unstructured condition, the 24 numbers were arranged randomly. (2) Numerical series. In the structured condition numbers were generated according to the rule an = n^2+2. The number series started with n=1 and therefore with the number three and the following 14 numbers in this sequence were calculated resulting in 15 numbers to remember. In the unstructured condition the numbers were arranged randomly.  In the unstructured condition three graphs of quadratic functions, which were not reflecting each other, were presented. (5) Triangles. In the structured condition we showed three isosceles triangles on a coordinate system. Again, one triangle was reflected across the x-axis and across the y-axis. In the unstructured condition we presented three isosceles triangles varying in height which were slightly 2 shifted on the coordinate system. For analyses we manually counted how many of the previously defined points (intersections and vertices for Graphs and vertices for Triangles) were correctly drawn on the coordinate system. For the Graphs item a maximum of 15 points could be reached, for the Triangles item, it was a maximum of 9 points.
The last item used verbal material in the form of a theorem with the corresponding proof. (6) Theorem and Proof. For the structured condition we used the following sentences "Theorem: Suppose n is an integer. If n is even, n2 is also even. Proof: If n is even, we can write n as n = 2k. We then see that n2 = (2k)2 = 4k2 = 2 x 2k2. Therefore, n2 is even.". In the unstructured condition, we randomized the order of the words but kept the position of theorem and proof and the length of both approximately the same. Formulas were not separated when randomizing the order of the words. For analyses we manually counted how many words were correctly recalled, with a maximum of 36 points to be reached.

Explicit scoring scheme of the Mathematical Memory task
While the three items, which used numerical material, were scored automatically using a Python script, the two figural and the one verbal task were scored manually.
For the two figural items we manually counted how many of the previously defined points (intersections and vertices for Graphs and vertices for Triangles) were correctly drawn on the coordinate system. We decided give one point per perfectly recalled unit, but also give 0.5 points if for example in the triangle item, participants drew a line over the vertex (e.g., (1, 1)), but their vertex was at a neighboring location (e.g., (1, 0)). Further, participants had to draw the shapes they remembered using the mouse, so often they might remember the intersections or vertices correctly, but drew them slightly offset, due to limits of usability of this task. Thus, it was up to the scoring person to decide which amount of deviation still counted as correct. Due to this room for individual scoring, we decided to use three independent raters to score the two figural items. For the verbal item we manually counted how many words were correctly recalled. We decided to give one point per perfectly recalled word, but also give 0.5 points if participants dropped one word (e.g., "Theorem:") but all the other following words were correct again. In the latter case, strictly scored, all the following words would be incorrect, because they were not in the correct absolute position anymore. Due to this room for individual scoring, we decided to use three independent raters also for the verbal item. Again, objectivity was excellent for all items (Theorem and Proof -Structured: ICC = .99; Theorem and Proof -Unstructured: ICC = .99). For all further analyses, a mean score of the three raters was used, and a maximum of 36 points could be reached.

Mathematical creativity items
The problem-solving items were the following: in order to create an equality.
Overcoming fixations items were the following: (5) Cuts 5 . Participants were asked to divide a rectangle into a given number of equal parts. In the example item, the rectangle can be cut into two equal parts using one vertical line. For the next three items, participants had to give the answer for three, five, and seven parts, and had a maximum of 45 seconds for each item. The correct answer is always the number of parts minus one. In the fourth item in this task, participants were asked to divide the rectangle in nine parts. Even though eight vertical lines is a correct answer, the more creative answer, where the algorithmic fixation is broken, would be two horizontal and two vertical lines. (6) Sum and difference 5 . Participants had to find the two numbers with a given sum and difference. The example with a sum of ten and a difference of four had the answer seven and three and lead the participants to think in the content-universe of positive integers. The next three items (sum 12 & difference 4; sum 7 & difference 3; sum 8 & difference 4) enforced this fixation, and participants again had a maximum of 45 seconds for each item. In the fourth item in this task, the participants were asked to find the two numbers with a sum of nine and a difference of two.
Here the correct answer was 3.5 and 5.5, therefore participants had to overcome the self-restriction on whole numbers as answer. In both overcoming fixation tasks, participants were given one point if they overcame this fixation and no point if they gave no answer (or the algorithmic solution in Cuts task).
The Problem-posing item was the following: In this task, originally from Bicer et al. 8 , participants were presented with a figure showing in a pictographic way how many books were sold per weekday. Using this figure, they had three minutes to make up as many problems as possible. Next to the mathematical problem, they also had to write down the mathematical formula of the solution.

Explicit scoring scheme of mathematical creativity and domain-general creativity
All mathematical as well as domain-general creativity items (except for the two overcoming fixation items) were scored for fluency, flexibility and originality.
Fluency was operationalized as the number of correct answers. Flexibility was operationalized as the number of categorically different responses. Originality was judged by 5 independent raters. We decided to not rate the originality of the domain-general creativity items based on the manual of the the fluency score to reduce the confounding effect of fluency 10 . Second, we z-standardized each score to allow the building of unweighted composite cores. Third, we computed mean scores for each creativity item. Fourth, we averaged the items on a higher level leading to one score for each creativity category (problem solving, overcoming fixation, problem posing, verbal creativity, figural creativity), and averaged those to a mathematical creativity score and a domain-general creativity score.