Education Research International

Volume 2017, Article ID 9132791, 13 pages

https://doi.org/10.1155/2017/9132791

## Hands-On Math and Art Exhibition Promoting Science Attitudes and Educational Plans

^{1}University of Helsinki, P.O. Box 9, 00014 Helsinki, Finland^{2}University of Jyväskylä, Jyväskylä, Finland

Correspondence should be addressed to Helena Thuneberg; if.iknisleh@grebenuht.aneleh

Received 24 March 2017; Revised 24 July 2017; Accepted 10 September 2017; Published 18 October 2017

Academic Editor: Seokhee Cho

Copyright © 2017 Helena Thuneberg et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The current science, technology, engineering, art, math education (STEAM) approach emphasizes integration of abstract science and mathematical ideas for concrete solutions by art. The main aim was to find out how experience of learning mathematics differed between the contexts of school and an informal Math and Art Exhibition. The study participants () were 12-13 years old from Finland. Several valid questionnaires and tests were applied (e.g., SRQ-A, RAVEN) in pre- and postdesign showing a good reliability. The results based on General Linear Modeling and Structural Equation Path Modeling underline the motivational effects. The experience of the effectiveness of hands-on learning at school and at the exhibition was not consistent across the subgroups. The lowest achieving group appreciated the exhibition alternative for math learning compared to learning math at school. The boys considered the exhibition to be more useful than the girls as it fostered their science and technology attitudes. However, for the girls, the attractiveness of the exhibition, the experienced situation motivation, was much more strongly connected to the attitudes on science and technology and the worthiness of mathematics. Interestingly, the pupils experienced that even this short informal learning intervention affected their science and technology attitudes and educational plans.

#### 1. Introduction

Children start to learn mathematics long before they are exposed to formal teaching at school [1]. Nearly all children have some sense of numbers from early on, are capable of counting the basic numbers (“one, two, three, etc.”), and are proud to tell their own age. They get to know the basic geometrical shapes and objects like circles, balls, and squares in natural everyday situations. Further, they can tell the time, use money by playing shop, compare and evaluate the magnitude of figures, and strategize, for example, by playing cards. Preschool aged kids get involved with applied mathematics also through ICT and digitalization while playing computer games or using tablet and smartphone applications. This learning of mathematics most often happens unconsciously. This is typically informal learning [2], which can also be utilized in a science exhibition context [3].

However, the older the children, the more complicated the mathematical problems they encounter in everyday situations, especially when they start school. Then, it becomes crucial to exploit their natural curiosity, imagination, and willingness to play [4] in the learning of mathematics and to support them to discover the meaningfulness and worth of mathematics. According to the TIMSS 2015 study (TIMMS: Trends in International Mathematics and Science Study), half of the international fourth-grade mathematics curricula include attitudes and mention, for example, beliefs, confidence, and perseverance as well as the beauty of mathematics, developing a productive disposition toward mathematics, appreciating the practical applications of mathematics in life, and displaying a constructively critical attitude toward mathematics [5]. Some countries mention appreciation of scientific inquiry and science as a discipline or curiosity and interest.

The current science, technology, engineering, art, mathematics education (STEAM) approach underlines integration of abstract mathematical ideas to find concrete solutions and evidence by art [6]. Children have to be able to use their senses and hands-on experimentation in order to test their thinking, especially at the concrete operational stage [7]. The importance of own exploration and experience is supported by the principle of learning by doing by Dewey [8] and the key of science center pedagogy, hands-on activity by Oppenheimer [9]. In case of math learning, manipulation of materials in multiple ways allows abstract mathematical concepts to become understandable, creative problem-solving to become possible, and mathematics to become meaningful [10]. Hands-on activities and exploration involve factors that enhance creativity: the encouragement of questions and novel initiatives and the offering of opportunities to discuss and debate problems with others [11]. Usually, these are perceived as welcoming challenges by high-achieving students. The empirical results of a study by Mann [12] that explored elements of mathematical creativity in middle school students showed that the strongest predictor was math achievement; it explained one-quarter of the variance. And one half of that predicted gender, attitudes toward mathematics, and belief in one’s own creative abilities. However, mathematical giftedness does not always guarantee mathematical creativity [13]. Further, high achievement or giftedness can sometimes be connected with perfectionism, in which case a fear of failure might turn to avoidance orientation and lead to underachievement as Mofield et al. [14] stated.

Pupils have shown that, through using a hands-on method, they like learning more, learn and remember better [15], and attribute their learning outcomes more to hands-on than to traditional teaching methods, or only to seeing or to hearing things. Liking and motivation have been shown to be connected to developing mathematical metacognition, which along with reduced anxiety supports problem-solving [16]. As with the children, teachers have reported that the hands-on method has been the most effective method for their pupils [17]. These benefits of the hands-on method have been shown to apply to a diverse number of learners, from pupils with mild disabilities [18] to pupils with serious emotional disturbances [19].

In this article, mathematical problem-solving was combined with art. The learning context was a Math and Art Exhibition, and the mix of math learning and art was represented in the building of mathematical geometric models with concrete materials. These activities require visual imagery and mental rotation. According to Hope [20], the capacity and skill to create visual representations of the mental images form an essential part of the learning process. Although the immediate goal was to enhance math learning, these activities support the development of spatial skills [21] and spatial intelligence, which have been identified as important factors of school achievement in general [22].

According to Fenyvesi et al. [23], problem-solving can also be a basis for the integration of learning mathematics in transdisciplinary educational frameworks, such as STEAM integration, although the integration of liberal arts into STEM is mutually reshaping scientific education and humanities education [24]. It seems evident that, just like mathematics learning, it is recommended that science, technology, engineering, and arts education also follow problem-oriented approaches.

As the creative element and esthetic component are the inherent core of art, combining art with math learning offers an additional dimension for concretizing math concepts. Art and math have been considered to share many principles, for example, space and shape [25], but also that of esthetics, as Mack [26] discusses in his article “A Deweyan perspective on esthetic in mathematics education.” The synthesis of math and art might show the beauty of both domains and possibly in a novel light. As such, by applying art, ways of looking and observing become critical [27].

Making art,* “Kunst*,*”* requires practical skills and handicraft. However, it is also an emotional process involving play, risk-taking, and imagination. The imagination has often been undermined in teaching academic school subjects, although it is crucial for “inventive scholarship” [28]. A combination of math and art invites pupils to approach math problems from a new perspective [29] because imagination, which is closely related to art, gives the possibility of “seeing things other than the way they are”, as Eisner [28] states. When this artistic math learning process with its esthetic beauty is shared with others, it creates an emotional experience, which then is likely to support also cognitive learning and the retention of learned contents and skills [30] and the “convergence of both cognitive and emotional parts of the mind” [26]. Because mathematical problem-solving has been shown to involve affective factors [16], such as anxiety, these kinds of shared activities might ease negative experiences and feelings.

Although art in science has recently become more prominent in the move from STEM to STEAM [23], according to Hickman and Huckstep [29], the role of art, at least in math education, lacks research evidence. However, there are more recent results related to this topic [31–33].

Learning in informal contexts has often been regarded as the opposite of formal education and critical toward traditions as is depicted in Ivan Illich’s [34] classical presentation* Deschooling Society*. One of the main difficulties is that pure informal learning refuses to be categorized, and the definitions are not needed until informal learning becomes institutionalized. In this sense, it has often been described as a creative way of learning as is the case in Gardner’s [35] book,* The Unschooled Mind*, which he points out also with the element of reframing [36].

The main results related to informal creative exhibition learning underline the motivational effects. In particular, the role of situation motivation seems to be essential [37]. Also, novelty has turned out to be one of the key factors in creating interest and situation motivation in the open learning environment [33], which can be interest-based settings that motivate otherwise non-mathematically oriented pupils. The dilemma of the informal creative pedagogy is how to enhance this strong situation motivation to support its transformation into intrinsic motivation and deep-learning strategy. As such, it is also a challenge of this study and is embedded in the main research questions.

*Research Questions*. The research questions were as follows:(1)How does the pupils’ experience of learning mathematics differ in the school context and in the exhibition context?(2)How does the experienced worth of mathematics and belief of hands-on effectiveness on learning change after the Math and Art Exhibition?(3)What is the role of situation motivation and other variables on change of attitudes toward science and technology and on the future educational aspirations of pupils?These questions are analyzed in regard to gender and math achievement groups. The role of cognitive, visual reasoning, autonomous motivation, and pretest variables was controlled in the constructed SEM model.

#### 2. Materials and Methods

##### 2.1. Participants

The participants in Math and Art Exhibition were 12 to 13 years old from a city in Middle Finland (), 52% girls () and 48% boys (). The five randomly selected schools were chosen from the schools which had preregistered for the mobile exhibition. The study was conducted following the research’s ethical principles.

##### 2.2. Context of the Study: Learning Mathematics by Hands-On Activities

The context of this study was a mobile interactive mathematics exhibition Art of Math. The exhibition consisted of eleven interactive “hands-on” science exhibition objects, which the students were allowed to use, test, explore, and learn freely during a 45-minute time period. Following that, they attended a workshop (also 45 minutes) in which they were allowed to build their own structures and creatures by using and applying the small “Lego” type of plastic pipes and circles.

The 4Dframe construction system and building set was developed by Park Hogul, who is a Korean engineer and model maker originally inspired by classical Korean architecture [23] with inspirations of other mathematical and artistic approaches [33]. His concept is based upon the structural analysis and geometric formalization of building techniques utilized in the construction of Korea’s traditional, wooden buildings. The set itself consists of 2–30 cm long “tubes” and various types of “connectors”: just a small number of elegantly structured, simple module pieces made out of polypropylene, which are flexible enough for the construction of “unbreakable” modules and spatial formations as well [23]. The wealth of structural variability offered by this versatile device renders it an excellent tool for conceptualizing, modeling, or analyzing structures and topics relevant to geometry, engineering, architecture, design, or art. Due to its numerous advantages and flexibility, the 4Dframe is adaptable to a wide variety of complex educational uses [23].

The central aim underlying 4Dframe educational methodology [38] is to activate students’ familiarity with geometric structures within the context of problem-solving. This method is based upon the creative exploration of these structures and uses a step-by-step approach to scientifically analyze each stage in the construction process. The 4Dframe also provides opportunities to experiment with creative methods related to mathematical art.

##### 2.3. Measures

###### 2.3.1. Deci-Ryan Autonomous Motivation

The Deci-Ryan scale measuring autonomous motivation was based on self-determination theory (SDT). It was administrated as a pretest, and the variable was used as a covariate in the structural equation model in order to reveal the purified influence of the short-time situation motivation in the exhibition context.

The Deci-Ryan Motivation (SRQ-A: Self-Regulation Quality-Academic) scale has 32 standardized items with four Likert options: 1 = not at all true, 2 = not nearly true, 3 = somewhat true, and 4 = totally true (for the translation into Finnish, see Thuneberg, 2007). The questions correspond with the self-regulation styles on the self-determination continuum. For example, the students are asked about the reasons why they do their homework or try to answer hard questions during lessons. The summative variables forming the self-determination continuum from the external toward the intrinsic direction are as follows: external, introjected, identified, and intrinsic. Based on the formula by Ryan and Connell presented in the validation article of the SRQ-A [39], the RAI (Relative Autonomy Index) of the summative variables (i.e., external, introjected, identified, intrinsic) was calculated. The RAI describes the overall relative autonomy level of the pupil. The positive plus sign in RAI indicates that the experience is rather autonomous, and the negative minus sign indicates that one relies more on others than trusting in one-self.

The reliability of the SRQ-A was checked. It was good, Cronbach’s , 32 items.

###### 2.3.2. Situation Motivation Test

The situation motivation questionnaire consisted of 12 Likert scale items (scale: 1–5, from “totally agree” to “totally disagree”). The questionnaire was constructed by the authors, piloted in a small group of 12-year-olds, and used before this present study in other studies. The questionnaire was administered as a posttest. The items were constructed and instrumentalized in relation to extrinsic elements like “edutainment” and recommendable outer aspects. The questions explored how attractive the pupils viewed the exhibition, for example, as follows: I was able to experiment and do many things by myself; I wish I would have had a chance to stay longer at the exhibition; I would recommend the math exhibition to others. The reliability of the measure was good, Cronbach’s , 12 items.

###### 2.3.3. RAVEN Test

The cognitive measure was the visual reasoning test RAVEN Standard Progressive Matrices [40]. The test measures visual nonverbal cognitive skills [41]. It has been shown to be a reliable standardized tool for comparing an individual’s learning abilities with the age group, irrespective of sex.

In each test item, the subject is asked to identify the missing element that completes a pattern. The test contains 60 items divided into five sets (A, B, C, D, and E), each including 12 different tasks. The reliability was good, Cronbach’s , 60 items.

###### 2.3.4. Math Achievement

Math grade was used as a math achievement indicator. The scale of the school subject grades in Finland goes from 4 to 10. In addition to using the math grade as a continuous variable, pupils were categorized in math achievement quartiles, and that grouping was applied in the analyses.

*Liking Math at School Context and Liking Math at Exhibition Context.* The single variables, which were summed, were formulated by the authors and were the same in the pre- and posttest. The only exception was that in the first one the items were related to school and in the second to exhibition. Following are examples of the 14 semantic differential items:* I think that math learning at the school/at the exhibition was important/useless (scale 1–5, with 5 indicating “important” and 1 indicating “useless”), modern/old-fashioned (scale 1–5, with 5 indicating “modern” and 1 indicating “old-fashioned”), and clear/confusing (scale 1–5, with 5 indicating “clear” and 1 indicating “confusing”)*: pretest Cronbach’s , 14 items; posttest Cronbach’s , 14 items (note: in pretest, time point 1 has been abbreviated to T1; in posttest, time point 2 has been abbreviated to T2).

###### 2.3.5. Experienced Worth of Maths

The summed variable was formed out of six single variables formulated by the authors. The question was,* What do you think of the statement*? The scale ranged from 1 (not agree at all) to 5 (totally agree)*. *The statements were as follows:* Math makes my everyday life easier. Understanding mathematics supports me in many practical situations. It is important to understand mathematical phenomena. Mathematics is interesting. Mathematics is useful in many occupations*: pretest Cronbach’s , 5 items; posttest Cronbach’s , 5 items.

The next three measures were single items. It is rather unconventional to apply single-item variables, but single items have been shown to yield reliable and valid data and predict outcomes effectively in certain conditions [42–45]. They even might be more economical and suitable than multiple item measures [46–48]. The three items of our study met the prerequisites of usage of single items based on the literature [49, 50]: (1) they are concrete and simple, not multifaceted; (2) they relate soundly to the other instruments; (3) they are integral parts of (the second and third) research questions; and (4) they fit in our sample consisting of young children who are most likely impatient and not willing to answer many extra questions that only slightly differ from each other.

###### 2.3.6. Experienced Effectiveness of Hands-On on Learning

This variable in pre- and posttest was based on a single question. The question was,* What do you think of the statement*? The statement was,* By hands-on experimentation and testing I can learn effectively*. The scale ranged from 1 (not agree at all) to 5 (totally agree).

###### 2.3.7. Exhibition Enhances My Science and Technology Attitudes

This variable was based on a single question. The question was,* What do you think of the statement*? The statement was,* Due to the math exhibition my attitudes toward science and technology changed in a more positive direction*. The scale ranged from 1 (not agree at all) to 5 (totally agree).

###### 2.3.8. Exhibition Affected My Future Educational Plans

This variable was based on a single question asking about educational plans in general. The question was,* What do you think of the statement?* The statement was,* I believe that the math exhibition influences my future educational plans*. The scale ranged from 1 (not agree at all) to 5 (totally agree).

##### 2.4. Data Analysis Methods

The mean differences and the change between the pre- and posttest were analyzed by General Linear Modeling (GLM; univariate, multivariate, and repeated measures) method. The effect-size measure was partial coefficient ( > .01 small, >.06 middle, and >.14 large), which is as acceptable as the recommended generalized coefficient when only one grouping factor is used [41]. Graphical plots were used, as recommended, to illustrate the pre- and posttest levels and the change between the pre- and posttest [51].

To answer the third research question and to see how our data would fit in the theoretical model, we applied SEM, the structural equation modeling (AMOS 22). The* RAI*,* gender*,* RAVEN*,* math grade, belief of hands-on effectiveness on learning T1*,* and math worth T1 *were used as covariates to control their effects on measured posttest variables. The goodness of fit of the models was based on a test () and indices of NFI, TLI, and CFI (good fit >.90, or better > .95), RMSEA reasonable fit < .08, good fit < .05 [52]. The predictors were indicated by standardized* β*-coefficients, and -multiple correlation indicated the total variance explained.

For testing the invariance of the models across genders and across math achievement quartiles, the unconstrained and fully constrained overall models were compared and the invariance was evaluated based on the test.

##### 2.5. Missing Values

There were on average 6% missing values. The maximum likelihood method and estimation of means and intercepts were used in the SEM path analysis due to the missing values.

#### 3. Results

In Tables 1 and 2, we present the statistical descriptors of the variables and the significant differences between the boys and girls and between the math achievement percentiles. In addition, the overall and between-group change are explained in regard to the pretest situation and posttest situation.