What is tactile math?

Question

My daughter has atypical Autism and absolutely cannot learn subtraction. She can do any other kind of math with ease, but can't subtract. She is almost 10. Someone suggested tactile math to me for her, because she is sensory. What is this style of math and how do I get my hands on it?

Answer 1

I suppose it is with (soft) forms: "Here are cubes... How many? (5 - or any suitable number). Now bring 2 to to that box there. How many are there left? Great, you have subtracted those 3 (from the 5 that you had)!

I HAVE NO EXPERIENCE WITH AUTISTIC CHILDREN. The part between brackets above may or may not be useful.

General opinion is that autists need a lot of repetitions of the same sequences, but you should decide for yourself. I have been diagnosed with a form of autism (Asperger syndrome) and I like repetitions, but only interesting ones i.e. with slight variations each time. You know your daughter best, decide for yourself.

Okay, let's return to the scene with the cubes. You might continue to do EXACTLY the same subtraction with different shapes: 5 balls, 5 dolls, 5 sweaters, etc. Make sure that your daughter can touch each of the items if she wants;  the calculation should be carried out quite litterally: your girl should be encouraged to choose a place for the subtracted elements. It may be necessary to use a large box or closet or perhaps a doll bed for all the subtracted balls, cubes, etc.

Hold to the same calculation as long as she remains interested. She may want to go on almost indefinitely. In this case you could try to  start the sequence with 6 balls after some days, but NOT with 4.

It might (or might not) be useful to let her add the subtracted elements again after some time. This way she will learn the relation between adding and subtracting. I suppose that it is unwise to tell her about that relation: she needs to find out. You might of course after a week or something try to find out if she discovered such a relation but I think you should be very reluctant to tell her explicitly about it.

I wondered why subtraction poses such difficulties. It may well be that she is disturbed by losing things, which is what happens with subtraction. Autists are sometimes seen as unemotional, but in my opinion the opposite is often true: they are very emotional but unable to handle all these emotions.

If you use a box for the subtracted elements, it may be necessary to leave it open so that she can see that no subtracted element is lost, but only moved to a different place.

I am not absolutely sure about this point: it may as well be the other way around: she could become distracted by subtracted elements in her field of view. so you need to try out what does or does not work. The same is of course true of taking breaks and coping with her attention span.

I will try to find web sites but it will be another day before I find time to do so. An updated answer will be preceded by the word UPDATED in the first line.

You might want to contact me via my profile. Keep in mind however that I have no professional experience with autism and that I do not know much about the US school system, nor about any other educational system, except for the system in the Netherlands, my country.

Answer 2

Tactile Mathematics

Abstract

Visual Computing has led directly to Visual Mathematics -- this by virtue of the ability to directly compile 'natural' mathematical language into machine-executable and graphical form.

Direct 3-D computer printers allow fully concrete, 3-D representations of mathematical systems, directly from the numerical representation.

The author references his work to date in the field of mathematical sculpture.

The author has begun work on integrating text information in Braille into three-dimensional models of mathematical surfaces. Future work, including manipulating computer-specified tactile surface texture in Computer-Aided Design, presents challenges to the technical interfaces in common practise in the Mechanical Prototyping industry. This paper will outline some proposed solutions.

This paper proposes the thesis that a richer, synergistic tactile experience can be afforded by combining the abstract information on a mathematical surface with the surface itself in-the-round in physical form.

I) Background: Visual Computing/Visual Mathematics

The Mathematica [1] system for doing mathematics by computer is a compiler of "natural" mathematical language with integrated graphical methods. Mathematica supports full mathematical typesetting while retaining the ability for the computer to execute typeset statements without modification. Rendering the results of mathematical statements is the direct concretization of a human language, whose evolution predated computers.

The advantage of visual computing is that an image condenses voluminous data to a single representation which can be quickly assimilated. [2] Visual computing affords the scientist a view of the problem which is different from considering raw data or abstract statements. To view an image, the scientist makes broader use of his/her sense of vision.

Visual computing has resulted in unexpected discoveries and breakthroughs in the sciences. In 1975, Benoit Mandelbrot used a computer at IBM to make a graph of a dynamical procedure which was known to be chaotic. [3] Complex dynamics was a "monstrous" topic of inquiry, because there were no known methods for dealing with such systems - least of which was simply graphing the behavior, because so many - billions - of calculations are required. What Benoit found was a graph with symmetry, hierarchical self-similarity and infinite detail, which was only apparent visually. No graph of this kind had ever been seen before. Mandelbrot had to invent a new branch of mathematics - fractal mathematics - to describe what he found.

In 1983 a graduate student in Rio de Janeiro named Celsoe Costa wrote down an equation for what he thought might be a new minimal surface, but the equations were so complex, they obscured the underlying geometry. [4]

David Hoffman at the University of Massachusetts at Amherst enlisted programmer James Hoffman to make computer-generated pictures of Costa's surface. The pictures they made suggested first, that the surface was probably embedded (non-self-intersecting) in three-space - which gave them definite clues as to the approach they should take toward proving this assertion mathematically - and second, that the surface contained straight lines, hence symmetry by reflection through the lines.

The symmetry led Hoffman and William Meeks, III to extrapolate that the surface was radially periodic and that new surfaces of the same class could be achieved by increasing the periodicity. They did so by altering the mathematical description of the surface to be the solution to a boundary-value problem constrained by the behavior of a minimal surface at the periodic lines of symmetry. The result: Hoffman and Meeks proved that Costa's surface was the first example of an infinitely large class of new minimal surfaces which are embedded in three-space.

The technique Hoffman and Meeks used was to make a picture which caused them to modify their mathematical theory and discover something totally unexpected about that theory. They later extended their techniques to find minimal surfaces of more complex geometry and they also created pictures of them. This is a new kind of experimental mathematics and a procedure not far from creative visual art.

II) Computer 3-D Printing -- Concrete Mathematics

Breakthroughs in science and mathematics due to visual computing may now be brought back from the vacuum of the cathode-ray tube by so-called Rapid-Prototyping (R-P) printers which can render a CAD model in physical materials in three dimensions. [5] [6] A growing list of similar Automated Fabrication technologies are appearing, among them: Stereolithography (3D Systems), Selective Laser Sintering (DTM Corp.), Fused Deposition Modeling (Stratasys), Laminated Object Manufacturing (Helisys) and the several licensees of the MIT 3-D Printing Consortium -- Direct Shell Production Casting (Soligen) and Z-Corporation. [7] [8] [9] [10] [11] [12]

These various technologies all build three-dimensional objects via the common principle of dividing the object in software into a sequence of horizontal slices, from bottom to top, which the machine constructs in a physical material and binds together. Thus, these technologies may also be termed various means of Layer-manufacturing [13], each of which uses some kind of three-dimensional computer object slicing software.

Compared to a two-dimensional computer rendering, a physical model in three full dimensions restores a dimension of information which was abstracted away in the perspective rendering of the image [14]. Of course, the actual, 3-D object gives access to a computer model to someone who might not otherwise be able to benefit from computer modeling because of a visual impairment [15]. But, as viewing an image exercises the scientist's visual cortex and integrates this additional processing path into the mental experience of evaluating an abstract hypothesis, might not viewing a sculpture be a more corporeal experience than viewing a two-dimensional image? [16]

I would like to make the claim, to be tested later, that viewing a sculpture is a different sense of apprehension compared to viewing a two-dimensional image. Certainly the tactile experience of an object enhances our understanding of the object. Is the integration of abstract information on an object with the physical object itself a synergistic experience? Is the total effect greater than the sum of the component elements?

III) 4th, 5th, ... Dimensions

There is precedent for integrating additional dimensions of information into three-dimensional representations of computer data. Typically, this is done by mapping a varying color to the surface of the object.

In the work of David Hoffman and James Hoffman in the Hoffman-Meeks extrapolations of Costa's three-ended minimal surface, the computer-generated images show the Gauss Map as a color which varies across the surface according to the point-wise orientation of the normal vector to the surface. [17]



Figure 1): Costa's Three-Ended Minimal Surface, Image by David Hoffman and James Hoffman.

Likewise, the Sullivan/Francis/Levy "Optiverse" shows the orientation of the sphere as a color map during the eversion metamorphosis. [18] The 'inside' of the sphere has a different mapping of the normal vector to color from the 'outside', such that the two sides may be identified during the metamorphosis.



Figure 2): Optiverse Sphere Eversion, image by George Francis, John Sullivan and Stuart Levy.

Hanson et al. have used color to encode 4-D "Depth" and the complex phase of their so-called "Fermat" equations. [19]

In many usual case, a parametric surface maps a two-dimensional domain (u, v) to a three-dimensional range (x, y, z) in a one-to-one fashion. That is, every point (x, y, z) in the parametric surface in three-space corresponds to a unique (u, v).

Computer rendering systems typically employ a two-dimensional parameterization of a three-dimensional surface in order to depict a detailed colored texture on the surface. To achieve photorealism in a computer rendering, the image used for the texture map may be derived from a photograph of a real-world object's surface.

In communicating abstract information on a system represented in three-space in a computer visualization system, I propose mapping text to the surface to convey the connection between information-space and real three-space.

The following figure:



Figure 3): Annotated Hyperbolic Paraboloid -- Computer Rendering by Stewart Dickson

Depicts visually-readable captions on a mathematical 3-D surface in-the-round. [20] In particular, depicted in the captions are the rearranged equations and curves one achieves by holding one of the variables in the equation for the hyperbolic paraboloid to a constant value. I.e., the parabolas in the X = 0 and Y = 0 planes, the hyperbolas parallel to the X-Y plane and the degenerate hyperbola (two straight lines) in the X-Y plane at Z = 0.

I believe this is a form of high-level 3-D Information Integration. It restores abstract information on the surface to the physical representation. Typically, when a mathematical surface is cast into physical form -- concretized -- the abstract information which brought about the three-dimensional object is left behind in the computer. There is typically a good deal of verbal explanation required to supplement the object itself. The object does not stand on its own.

Attaching captions to the surface might be a way of restoring the integrity of the abstract system which the object is intended to represent.

IV) Tactile Fonts, 3-D Textures

There seems to be a synergism inherent in the idea of attaching captions, intended to be "read" by touch -- by a blind person -- to the surface  

Answer 3

Like some others said before, tactile math is just math that uses some sort of hand-held manipulative objects to perform an operation. You can use bears, plastic chips, paper clips, pencils, M & M's (mmm!), or anything. Food is good for some kids...put down 6 candies, take away (eat) 2 of them, count how many are left. If she moves up into multiplication, it would also be good to group sets of objects into equal piles to show how to multiply.

As for solving problems on paper, there is a program called Touch Math. It uses dots on each number that the student memorizes and uses when adding/subtracting/multiplying, etc on paper. You can check out the website at www.touchmath.com . I have used it before with some severely LD students and it works great. I even use it sometimes!

Answer 4

She needs to work with concrete objects.  Lay out ten pennies, have her take two away, and then count to see how many are left.  Break it down for her to the simplest possible step, and walk her through it, slowly building into the math she should be doing.  Start, as I said, using concrete objects, then maybe add  drawing it symbolically while still using concrete objects.  It will help her to transition into working math problems using numerals later.  Find ways to make subtraction hands-on.  Use real-life situations to sneak in math lessons, like when you see birds, and some fly away, or  how many cookies are left when we eat one?  Good luck, and go slow.  If it takes years for her to pick it up, or just weeks, be patient.  Keep trying new things.  If this doesn't work for your daughter, do some research and try something new.