Hours of Play:. Tell Us Where You Are:. Preview Your Review. Thank you. Your review has been submitted and will appear here shortly. Extra Content. Table of Contents 1. Mathematics and visual arts education 2. Understanding the language of mathematics and visual arts 3. The list of sources on the Math Awareness web site is a great place to start.
In this brief essay, I'll highlight a few of the possible ways to fill in the blanks above. I hope that this will stimulate you to explore many others. Mathematics produces art At the most practical level, mathematical tools have always been used in an essential way in the creation of art. Since ancient times, the lowly compass and straightedge, augmented by other simple draftsmen's and craftsmen's tools, have been used to create beautiful designs realized in the architecture and decoration of palaces, cathedrals, and mosques.
The intricate Moorish tessellations in tile, brick, and stucco that adorn their buildings and the equally intricate tracery of Gothic windows and interiors are a testament to the imaginative use of ancient geometric knowledge. The symbiosis of art and mathematics during these times as linear perspective and projective geometry were developing is one of the most striking examples of art and mathematics evolving almost simultaneously in new directions.
In the hands of an artist, computers can produce art, powered by unseen complex internal mathematical processes that provide their magical abilities. Mathematical transformations provide the means by which an image or form in one surface or space is represented in another. Art is illusion, and transformations are important in creating illusion. Isometries, similarities, and affine transformations can transform images exactly or with purposeful distortion, projections can represent three and higher -dimensional forms on two-dimensional picture surfaces, even curved ones.
Special transformations can distort or unscramble a distorted image, producing anamorphic art. All these transformations can be mathematically described, and the use of guiding grids to assist in performing these transformations has been replaced today largely by computer software. Compasses, rulers, grids, mechanical devices, keyboard and mouse are physical tools for the creation of art, but without the power of mathematical relationships and processes these tools would have little creative power.
3 Ways Using Math Can Improve Students' Drawings - The Art of Education University
Mathematics generates art Pattern is a fundamental concept in both mathematics and art. Mathematical patterns can generate artistic patterns. Often a coloring algorithm can produce "automatic art" that may be as surprising or aesthetically pleasing as that produced by a human hand.
The intricacy of these images, their symmetries, and the endless in theory continuance of the designs on ever-smaller scales, makes them spellbinding. For example, begin with an array of numbers such as a large data set, a sequence, a modular operation table, or Pascal's triangle and color the numbers in the array according to some rule. Often surprising patterns -- even art -- emerges. Recursive algorithms applied to geometric figures can generate attractive self-similar patterns.
Begin with a curve, a closed figure, or a simple spatial form, apply an algorithm to alter that figure by adding to or subtracting from specified parts of that figure, then repeat the algorithm recursively. Many nonperiodic tilings such as the Penrose tilings can also be generated automatically, beginning with a small patch of tiles and then applying a recursive "inflation" algorithm. Transformations and symmetry are also fundamental concepts in both mathematics and art.
Mathematicians actually define symmetry of objects functions, matrices, designs or forms on surfaces or in space by their invariance under a group of transformations. Conversely, the application of a group of transformations to simple designs or spatial objects automatically generates beautifully symmetric patterns and forms. In , Brewster's newly-invented kaleidoscope demonstrated the power of the laws of reflection in automatically generating eye-catching rosettes from jumbles of colored shards between two mirrors.
Periodic tessellations, whether geometric or Escher-like, can be automatically generated by computer programs [R12] or by hand, following recipes that employ isometries. Art illuminates mathematics When mathematical patterns or processes automatically generate art, a surprising reverse effect can occur: the art often illuminates the mathematics. Who could have guessed the mathematical nuggets that might otherwise be hidden in a torrent of symbolic or numerical information?
The process of coloring allows the information to take on a visual shape that provides identity and recognition. Who could guess the limiting shape or the symmetry of an algorithmically produced fractal? Additionally, teachers are advised to keep in mind that the tool is a guide and should be used in conjunction with their judgment as professional educators. Such a tool may be best suited to teachers desiring a simple, global evaluation of a book, rather than a tool for more structured planning and decision making.
While Hunsader lauded Hellwig et al. The six mathematics-related criteria included accuracy of math content, visibility of content and effectiveness of presentation, developmental appropriateness of math content, level to which the text facilitates the involvement of the reader, if the math content compliments the narrative, and how many resources would a teacher need to collect in order to use the text successfully. Hunsader stated her criteria clearly and the rubric would be straightforward to implement. She used it to assess 77 books, each coming from one of two primary grades mathematics curricula and found many of the books rated too low to recommend for use in mathematics learning.
However, her book evaluation process drew criticism. However, they noted the lack of clarity from Hunsader regarding how many reviewers scored books for her article, perhaps only Hunsader herself. They thus argued that the likelihood of multiple interpretations of books poses a challenge for the evaluation process. This may stem, at least in part, from fundamental differences in how mathematics concepts may be viewed within a book.
Exploring The Math And Art Connection: Teaching and Learning Between the Lines
As a result, they analyzed book evaluations from 30 reviewers representing mathematics professors, mathematics educators, English professors, literacy professors, and third-grade teachers. The group with the lowest agreement, while still acceptable, was the third-grade teachers. Unlike other respondents, they sometimes evaluated books given caveats for their implementation with a classroom.
Nesmith and Cooper could have developed or addressed this concern more directly; it remains a significant question. Also, notably, these researchers gave important consideration to the ideal relation between the text and illustrations in a book. Explicit presentations of mathematics in picture books may risk being perceived as pseudotextbooks Nesmith and Cooper, , and such books are well-represented among current tradebooks.
As argued in the earlier sections on communication, representation, and connections, mathematical meanings can be recognized through the words and pictures and the experiences with them, even without explicit presentations of the mathematical concepts. In the most recent evaluation scheme, van den Heuvel-Panhuizen and Elia drew upon the approaches mentioned previously to identify the learning-supportive characteristics of picture books for their use in supporting mathematics learning. It is clear from their results that the framework greatly assisted experts in recognizing the learning—supportive aspects of literature as compared to their recognition of such content without the use of the tool, but then the question remains of how it could be adopted by others, including teachers, more broadly.
This dilemma brings us now to the processes of teacher education and professional development. None of these studies compared the rated books in their implementation with children individually or in classroom settings. Given the differences in emphases of the evaluation criteria and the different approaches to promoting learning with them, such as through teacher- or adult- scaffolded discourse, as opposed to using the books as springboards to a subsequent activity, comparisons of books and their implementations may be a greatly informative line of work.
Addressing the ongoing concern that students are not adequately prepared to engage in mathematically rich problem solving and discourse has been a concern of pre-service teacher preparation programs and in-service teacher professional development for some time now. By looking specifically into geometric knowledge attainment of students and geometric content knowledge levels of teachers, which we again note has had significant gaps, we examine one component of mathematics as a whole and highlight the ways in which teachers can improve instruction and student outcomes.
- SEMS Inspection & Enforcement; Field PINCs;
- Five Candles.
- Teaching and Learning Between the Lines.
The primary concerns for professional development are the inadequate training and development available to teachers along with their use of limiting curricular materials, as this combination often results in the perpetuation of geometric misconceptions in their students Clements and Sarama, Chard et al. Incorporating appropriate interventions during times of critical development of mathematical knowledge is a preventative strategy for supporting increased, accurate content knowledge in students.
Awareness of these ideological beliefs also sparked discussion and reflection between teachers, which informed their practice. Oberdorf and Taylor-Cox elucidated the ways in which some of the more common misconceptions are passed on to students through their identification of books containing such errors.
They identified The Silly Story of Goldie Locks and the Three Squares MacCarone, as an example of a book containing misconceptions due to the mismatch between the illustration and the text, as the shapes pictured are three solids yet they are named as two-dimensional shapes: circle, square, and triangle. Presenting such mismatches may seem trivial and easily corrected; however, the reality is that students retain these misconceptions as truths and experience confusion when trying to connect this knowledge to future learning.
Without adequate experiences not only with the mathematical content, but also planning for learning with picture books, either in preparation programs or professional development teachers may not recognize such errors. During these sessions teachers are focused on specific geometric content knowledge aimed at connecting their knowledge to the understandings of their students Ball et al.
Teachers should be active participants in their development as they plan and create manipulative tools, engage in defining and discussing terminology, and examine the pedagogical implications for their students. As teachers strive to connect professional development to classroom practice, they often start by searching for supplemental materials that reinforce their newly attained knowledge. In the field of early childhood education, picture books are a familiar gateway for introducing purposeful, content-specific concepts to young children Shatzer, Teacher professional development that incorporates practicing strategies for critically evaluating geometry based literature, materials, and student content knowledge will lead to richer classroom discussion and higher quality opportunities for student learning.
Specifically, for recognizing shapes and their attributes, teachers can focus on gathering literature that presents geometric shapes in a variety of colors, sizes, and orientations while also presenting non-examples to deepen student discussion Clements and Sarama, When teachers have experiences that promote their trust in and access to high-quality, accurate literature for use in their instruction, they can pay closer attention to preventing future potential learning misconceptions in mathematics Chard et al. More specifically, professional development opportunities should be created and teachers should seek out professional development that meets their individual learning needs while also translating the targeted instructional practices into their own instructional techniques.
At times this may require teachers to seek out instruction or information on an individual level rather than in school-wide professional development. The authors provided a way to connect math and storybooks through a three-step framework of choosing, exploring, and extending the text all in one short, peer-reviewed, accessible article. The author recommended that teachers focus on first leading students in enjoying the illustrations and text and then making connections to math content areas when reviewing the text. As teachers use articles such as the two mentioned above, they are developing themselves professionally while also making an immediate improvement to their teaching.
Collaborations among professionals can not only be effective, but may indeed be essential for developing effective planning and implementation of learning experiences with picture books, based on the strong evidence provided by Nesmith and Cooper and van den Heuvel-Panhuizen and Elia Picture books can then be incorporated as another source of geometric representations that need to be critically evaluated for their geometric content knowledge. As teachers deepen their pedagogical content knowledge for geometry, they will be better prepared to guide their students during discussions of the mathematics connected through texts.
Recently, Brendefur et al.
This professional development was structured to focus on the content of four mathematical domains, number, interpreting relationships, measurement, and spatial reasoning. Teachers who participated in this professional development took with them scripted lessons to enact as centers in their classrooms.
Teachers could reference scripts as a way of seeking ongoing support after the conclusion of the professional development.
Teaching and Learning Between the Lines
The coaching model of professional development supports teachers through sustained one-on-one interaction during instructional times in the classroom and during group-centered initiatives led by coaches. Poglinco and Bach examined coaching over the course of 1 year finding that complexities within the relationship between teacher and coach can be unexpected.
These relational aspects can have an impact on the effectiveness of the coaching model as a tool for delivering professional development. The authors also point out the strengths coaching brings when well-trained individuals are implementing the practice. An example of the effectiveness of coaching is outlined by Rudd et al.
Whole-group professional development was provided for teachers with the caveat that additional coaching in the classroom would be provided after the initial professional development intervention. However, teachers did not experience the increase until the onset of a coach in their classrooms, suggesting that perhaps follow-up coaching may be key to translating professional development into practice. Poglinco and Bach warned that although coaching has been identified as an effective professional development model, teachers and schools must be informed as to the complexities before fully adopting the model.
Considering the following aspects prior to implementing a coaching model are key according to Keller : how it will be funded, how to define, select, and evaluate coaches, how principals and districts will support the coaches, and lastly the details of how often coaches will work with teachers in the schools. Regarding the use of picture books in particular, coaching can offer an ongoing relationship with the coach and potentially other teachers as well, to evaluate books and then plan, implement, observe, assess, revise, and retry learning experiences with those books.
Teachers can discuss with their coach and their peers which books contain accurate representations of shapes and descriptive text of those shapes, for instance, and which experiences with hands-on materials and the spatial environment of the classroom would be especially effective. As well prepared as any individual teacher may be, he or she could benefit greatly from well-structured, well-focused interactions with colleagues near or far.
The use of these technologies can be a means of bringing educators together even across continents to share ideas for lessons and activities, to create collaboratives of book reviews, and to be a louder collective voice demanding publishers offer high-quality picture books, not boring, mathematically impoverished, or pseudotextbooks, for mathematics learning.
The greatest need is for well-designed empirical studies to test the effectiveness of different book types narrative and nonnarrative; those with text and those wordless books, etc. For example, imagine an individual teacher coming across that statement regarding anxiety, but in isolation, and taking that statement at simple face value. Using read-aloud literature would not necessarily relieve anxiety for a child with hearing difficulties or children whose home language differed from the classroom.
And, equally important, we need books to expand the world of mathematical ideas for children beyond those they will learn in school. Researchers and educators have both the opportunity and responsibility to do so. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. National Center for Biotechnology Information , U.
3 Ways Using Math Can Improve Students’ Drawings
Journal List Front Psychol v. Front Psychol. Published online May Lucia M. Author information Article notes Copyright and License information Disclaimer. Reviewed by: Vanessa R. This article was submitted to Developmental Psychology, a section of the journal Frontiers in Psychology. Received Jan 13; Accepted Apr The use, distribution or reproduction in other forums is permitted, provided the original author s or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice.
No use, distribution or reproduction is permitted which does not comply with these terms. This article has been corrected. See Front Psychol. This article has been cited by other articles in PMC.