Combining Math and Music in Internet Education
July 8, 2009
Since May 2008, when I began a more intensive study of the math involved, I have been looking a bit at the research on the use of interactives and 'concrete vs. abstract'. These first references may need a separate page soon.
In what follows, I might try to characterize what I am doing as 'interactive illustration with integrated text' designed specifically for a fast evolving WWW to distinguish it a bit from 'manipulatives' designed for or supplemental to classroom or home schooling use.
Here's some research generally advocating manipualtives - the second is early computer, pretty much pre-internet.
http://www.nmsa.org/Research/ResearchSummaries/Mathematics/tabid/1832/Default.aspx
NMSA Research Summary
Manipulatives in Middle Grades Mathematics
(February 2009)
...Manipulatives are not restricted to concrete, hands-on materials; virtual manipulatives are hands-on models that students interact with in a virtual environment. The hands-on materials are presented as interactive tools. Students click and drag to move the materials into desired locations. Moyer, Bolyard, and Spikell (2002) described them in this way:
A virtual manipulative is best defined as an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge. Currently, virtual manipulatives are modeled on the concrete manipulatives commonly used in schools. … However, their ability to be used interactively—that is, to allow the user to engage and control the physical actions of these objects—combined with the opportunities that they offer to discover and construct mathematical principles and relationships, distinguishes them as virtual manipulatives. (p. 373)
Computer manipulatives have two advantages: (1) recording, replaying, changing, and viewing actions that encourage real math exploration and (2) the direct, immediate link between the object and the symbolic representation (Clements, 1999).
In a study of eighth graders, Meira (1998) noted that a physical object did not make mathematics more accessible, though a virtual manipulative did. Resnick and others at MIT have conducted research on digital manipulatives and have developed physical objects with embedded computation, FiMs (Froebel-inspired Manipulatives) and MiMs (Montessori-inspired Manipulatives). Zuckerman, Arida, and Resnick (2005) found that an iterative process of hands-on modeling and simulation on the computer provides students an opportunity to confront their misconceptions about dynamic behavior. Web-based manipulatives can "enhance the knowledge and understanding of learners, while creating a conceptual understanding of mathematical theories beyond the mere formulaic models of traditional mathematical coursework" (Crawford & Brown, 2003, p.176).
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http://cehd.umn.edu/rationalnumberproject/79_2.html
ABSTRACTION, GENERALIZATION, AND THE DESIGN OF
MATHEMATICAL EXPERIENCES FOR CHILDREN - (1979)
Thomas R. Post, University of Minnesota
Robert E. Reys, University of Missouri
These issues are complex. It seems unlikely that research will provide simple answers. It seems more likely that answers to such questions will need to be qualified in terms of teachers, mathematical content, and embodiments used, as well as by the pupil's ability, back- ground, and achievement level. Nevertheless, these issues are significant.
...The theoretical foundations of these ideas have been provided by the writings of Z. P. Dienes (...generally espousing the views of Jean Piaget) and E. W. Golding (1971a, 1967, 1971b).
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As adults, with some knowledge, our approach to many things is abstract. We readily ignore some characteristics in order to be satisfied that the remaining properties will do something for us ...
Children do not ignore the different characteristics of materials so readily...
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One of the basic assumptions underlying the use of physical materials is that pupils learn best through active involvement with concrete experiences.
However, this depends on the concepts involved, the age and previous experience of students, and the particular materials supplied.
One cannot say, at least with any degree of certainty, that using physical materials with active learning experiences will be effective in helping all children master all types of mathematical objectives.
Too much depends on the complex interaction among and between pupils, teachers, and materials.
May 30 , 2008
The following study generally advocates an abstract approach to math teaching.
http://researchnews.osu.edu/archive/mathed.htm
"...The authors said that students seem to learn concepts quickly when they are presented with familiar real objects such as marbles or containers of liquid, and so it is easy to see why many advocate this approach. “But it turns out there is no true insight there. They can’t move beyond these real objects to apply that knowledge,” said Sloutsky...
“We really need to strip these concepts down to very symbolic representations such as variables and numbers,” she said. “Then students are better prepared to apply those concepts in a variety of situations..."
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http://www.nytimes.com/2008/04/25/science/25math.html
".... The findings run counter to what Dr. Kaminski said was a “pervasive assumption” among math educators that concrete examples help more children better understand math..."
(ror - I am in favor of getting rid of the train problem since it brings back rather strong, unpleasant memories to the effect of "What does this have to do with me?" Burnt pizza does not sound too interesting either.
Music, on the other hand still interests me and this may be an issue of when and how rather than whether to apply such concrete examples.)
(July 7, 2009 here is something of an inconclusive follow-up on the preceeding article -
http://mathforum.org/kb/thread.jspa?threadID=1962021&tstart=0
Do Concrete Examples Actually Hinder Students' Learning of Mathematics?
Posted: Jul 1, 2009)
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The links to reports of teachers using the approach of combining music amd math are:
http://www.yale.edu/ynhti/curriculum/units/2006/4/06.04.02.x.html#a
http://www.childcareexchange.com/resources/view_article.php?article_id=5015846
They refer to the very young and are obviously quite serious, hands on, experienced, approaches, not to be taken lightly.
I did not find much detail on the Ohio State study itself, there is an earlier study from the same source:
http://cogdev.cog.ohio-state.edu/Simple_sybmols-PBR-04-080-revision.pdf
THE ADVANTAGE OF SIMPLE SYMBOLS FOR LEARNING AND TRANSFER Vladimir M. Sloutsky, Jennifer A. Kaminski, and Andrew Heckler The Ohio State University July 14, 2004
"However, it seems likely that facilitative effects of concreteness are limited to cases when concrete representations communicate relevant aspects of the to-be-learned information (see Goldstone & Sakamoto, 2003, for a review).
...Similarly, Dienes blocks (Dienes, 1960) can communicate the idea of the base 10 number system, thus possibly facilitating learning of the system. However, even if this “relevant concreteness” facilitates learning, its effects on transfer are questionable (e.g., Goldstone & Sakamoto, 2003).
...(It)... could be because concrete, perceptually-rich entities are more likely to be considered objects than symbols denoting other entities."
Dienes blocks were looked up naturally:
http://www.arcytech.org/java/b10blocks/description.html
This page is specifically for teachers and describes a complete lesson plan on how to teach base 10 place value as well as basic arithmetic operations using this online version of the base 10 blocks manipulatives.
( It was rather inovative and donated to the internet by Jacobo Bulaevsky, now maintained by his brother Alejandro, in memorium.)
And something about Dienes himself:
http://education.umn.edu/rationalnumberproject/
"... Dienes championed the use of collaborative group work and concrete materials, as well as goals such as democratic access to the process of mathematical thinking..."
I have a particular interest in Dienes Blocks since I was considering designing something similar in Flash, now I don't have to. A bit of an extension perhaps - something along the lines of the Equal Temperament 2 animation, Continued Fractions. (July 8, 2009 - now in development - Dienes Blocks)
I'm taking a closer look at some of the ideas in the study:
| http://social.jrank.org/pages/6/ Abstract-Reasoning.html |
A child who has developed good abstract reasoning skills easily uses symbols instead of concrete objects when learning new information. The beginning learner usually needs concrete aids. |
| http://www.instructables.com/ community/When-do-humans- develop-abstract-reasoning/ |
When do humans develop abstract reasoning? Apparently kids will (almost) always fail this test up to age 4, and almost always pass after age 7 |
| http://www.math.vanderbilt.edu /~schectex/courses/whystudy.html |
mathematics -- or any kind of abstract reasoning -- works by selectively suppressing information |
http://www.challengeconsulting.com.
|
Abstract Reasoning Measures the ability to understand abstract logical problems and use new information outside the range of previous experience, |
| http://www.learning-styles-online.com/overview/ | ...Learning styles group common ways that people learn. Everyone has a mix of learning styles. |
Musemath will generally try to follow an approach that is supplemental to whatever is done in classrooms. However, the website is dependent on visitors having a pre-existing interest sufficient to type in the proper keywords and it is always available to a very diverse, worldwide audience. And it can't threaten you with a failing grade if you don't pass the next test.
A presentation of concrete applications of math and music is expected here. I don't think I will be able to abstract it much beyond the geometrical treatment of a guitar string as a number line continuum. Some of the historical perspective may be useful in any event. The animations seem to me most helpful in providing intuition and insight if they are recreated by whatever animation material is available.
The internet is essentially an unstructured enviornment with overwhelming information. It could probably use all the help it can get from a rigorous, disciplined classroom, however that may be interpreted and finally decided upon by those best qualified.
I suppose it would be too simplistic to say the classroom exists to teach how to use the internet.
The following are some ideas on combining math and music:
July 26 , 2008
There are many similarities between the motion of a guitar string and the motions of an electron trapped in an atom. Both are described by wavefunctions that oscillate in time and space.
Rather than using a metaphor based on light, as Schödinger did, I am going to construct his equation by the analogy to a standing wave such as the vibration of a guitar string.
April 23, 2008
It is sometimes difficult to interest the young in mathematics whereas it is hardly possible to keep them from an interest in music. If they make a connection between the two, the rewards can be great.
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"This study is surely one that a person wanting to learn about physics and mathematics should take up early on, since it includes so much of value, especially as a concrete example of wave motion, and of how to set up and solve differential equations."
Dr. J. B. Calvert
http://www.du.edu/~jcalvert/waves/strings.htm
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Study the guitar and watch the musician who plays it. The guitar touches on a rich set of engineering principles, among them: resonant frequency, period, amplitude, distortion, harmonics, wavelength, stress & strain, elastic limit, am, fm, damping coefficient, Doppler effect, step response, coupled oscillations, fft’s and signal processing. Let’s play around with some of these and see what we can learn from the simple acoustic guitar.
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"Interactive simulation, treated with the proper reverence, does have its role in education. Exercises will more likely be done when they resemble a video game than a homework assignment. Integrating video, speech and text, these programs will adapt to a student's likes and dislikes. For example, an explanation of harmonic waves would be described to a musician via the analogy of the vibration of a guitar's strings."
Stephan Wolfram
http://www.stephenwolfram.com/publications/articles/general/88-academic/5/text.html
Academic Computing in the Year 2000 (1988)
The Effect on Curriculum and Coursework
One of my own arguments:
The tradional western guitar (12 frets/octave - Equal -Temperament) is a popular instrument in the world and has its logarithmic scale clearly marked on face. In its acoustic and electric forms, it has arguably become one of the more observed and experienced of human artifacts. It furnishes a basis for mathematical inquiry and is an analogy to many important physical phenomena. It is a common, well known, generally well liked and desirable empirical experience.
"I think it is a relatively good approximation to truth—which is much too complicated to allow anything but an approximation—that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure."Von Neumann, John - The Mathematician
Re: the nature of intellectual effort in mathematics
quoted in Newman, The World of Mathematics, P2063
In this instance, the geneology is long but not obscure. The details are more myth than math but the Western Mathematical and Scientific Traditions are traced squarely to Pythagoras (Gk - ca 580-500 B.C.) and his experiments with musical strings. Not so much what he did - which apparently had its roots in Mesopotamia thousands of years earlier, but what he did with it, the notion of science.
The relationship between math and music is concisely stated in the following words from Harry Partch, a composer of microtonal music:
"Tone is number , and since a tone in music is always heard in relation to one or several other tones - actually heard or implied - we have at least two numbers to deal with: the number of the tone under consideration and the number of the tone heard or implied in relation to the first tone. Hence, the ratio."
Partch, Harry
Genesis of a Music, Second Edition, P76
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From another mathematician, Decartes we have:
"...the mathematicians, however different their objects, they all agree in considering only the various relations or proportions subsisting among those objects."
quoted in Newman
World of Mathematics, Vol I, P21
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Clearly, the assertion that ratios are at the heart of traditional Western tuning systems is an irrefutable one.
Robert Asmussen
http://www.terraworld.net/c-jasmussen/thesis_asmussen.pdf%20offline 09/26/07
Chapter 8 Conclusions
NOTE: this is the pdf version of a doctoral thesis
More:
The soundwave and harmonic analysis analogize very well to the common scientific understanding.
"The whole domain of periodic phenomena—the motion of the tides, the vibrations of a plucked string, the emission of light waves from an incandescent filament—is governed by the simple trigonometric functions sin x and cos x."
Courant and Robbins
What is Mathematics? P27
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”If one landmark overtops all others in the evolution of science, it is the discovery by Pythagoras of the connection between musical harmonies and numbers.”
Bell, Eric Temple, The Magic of Numbers, P104
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“Arithmetic, like language, begins in legend. But mathematics in our sense, reasoning with numbers… begins here.”
Bronowski, The Ascent of Man, P155
"...musical training is a more potent instrument than any other, because rhythm and harmony find their way into the inward places of the soul, on which they mightily fasten, imparting grace, and making the soul of him who is rightly educated graceful, or of him who is ill-educated, ungraceful; and also because he who has received this true education of the inner being will most shrewdly perceive omissions or faults in art and nature, and with a true taste…"
Plato -Translated by B. Jowett Edited, with introduction by Louis Ropes Loomis,Published for the Classics Club by Walter J. Black, Inc. Roslyn, NY 1942 - P289
The man that hath no music in himself,
Nor is moved with concord of sweet sounds,
Is fit for treasons, stratagems, and spoils.
The motions of his spirit are dull as night,Shakespeare (1564-1619)
The Merchant of Venice V, i, 83 (P610)
The Complete Works
Edited By G.B. Harrison
"The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty. "
D'Arcy Thompson
probably "On Growth and Form" (1915, rev.1941))
One of my own arguments:
The planet has become very small indeed; we have no where left to go, and we will all have to come to terms with ourselves. A knowledge of both math and music is likely to prove helpful both in dealing with the problems and in temporarily escaping from them.
I am not thinking of Musemath as a part of the 'Official Education Community' but I would like to keep abreast of what is being done there- what is needed by way of form and content. (April 6, 2006 I have not done much with these links, they are somewhat distracting, but am not yet ready to delete them)
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http://www.ed.gov/about/offices/list/os/technology/plan/2004/plan_pg8.html
"...The access point for technology use, particularly for older students, is at home, not at school..."
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(offline/moved December 15, 2008)
http://www.asbj.com/specialreports/0905SpecialReports/S1.html
American School Board Journal
The World of
E-Learning
September 2005
"...Many schools have moved to a learning model in which the system detects whether a student is having trouble learning a specific task or skill, such as solving simultaneous equations. The system provides additional content on that topic, while allowing a student who has mastered that skill to move on. Such a model, he says, is "more about achievement and less about putting in your time in class..."
"..."How does this teach? How does the teaching and learning
process change? What’s the administrator’s role? What’s the ongoing need for it?
Teachers and principals change; how do you build in support, evaluation, and
ongoing mentoring and training for all levels, including the
community?..."
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April 23, 2008: Recently came across the following two sites detailing some music/math teaching techniques for the very young.
http://www.yale.edu/ynhti/curriculum/units/2006/4/06.04.02.x.html#a
Sunny Jonas
Yale-New Haven Teachers Institute
Math, Music, and Architecture: Kindergarten Geometry and Aesthetics in Music and Architecture
http://www.childcareexchange.com/resources/view_article.php?article_id=5015846
Karen Sawyers and Janet Hutson-Brandhagen
Music and Math: How Do We Make the Connection for Preschoolers?
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https://edu-gelc.dev.java.net/nonav/index.html
Sun Microsystems is promoting the Global Education Learning Community.
http://www.skoool.com/?iid=aboutinteledu%2Bresources_skool&
This will get you right into the FREE online lessons for schools in England, Ireland and Swedan (as of November 14, 2005)
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http://www.mindinstitute.net/index.html
The "Mozart Effect" and other theoretical relationships between math and music education appear to be coming from this source. I would like to see the theories thoroughly challenged and explored within the professional community.