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Osterhaus, S.A. (2002). Susan's math technology corner: Teaching A Blind Student How to Graph on a Coordinate Plane: No Tech, Low Tech, and High Tech Tools. Division on Visual Impairments Quarterly, 47(3), 23-26.

APH Graphic Aid, with diagrams

Susan's Math Technology Corner

Background

Although the use of scientific graphing calculators is now a secondary math classroom mainstay, all students should first understand the concept of graphing on a coordinate plane manually. I REALLY insist that my students be able to physically plot points, graph lines, and find the slope as well. This ability is even more critical for blind students because most math technology is not accessible to them.

The Problem

Recently, I have received an avalanche of requests for help from teachers of  students with visual impairments and math teachers. Question: “How can blind students graph linear equations, inequalities, and systems of inequalities independently and efficiently? Or is this the time the student doesn’t participate because of the visual nature of the task?” Answer: Most academic blind students, even those with spatial orientation problems, are quite capable of graphing, and as one of my students exclaimed, “Not only can we do it, it’s fun!”

No Tech, Low Tech, and High Tech Solution

The Graphic Aid For Mathematics from APH is excellent for graphing algebraic equations but can be used in geometry, trigonometry, etc. It consists of a cork composition board mounted with a rubber mat, which has been embossed with a grid of 1/2-inch squares. My students use two perpendicular rubber bands held down by thumbtacks for the x- and y-axes. Then, points are plotted with pushpins at the appropriate coordinates. Points are connected with rubber bands (for lines), flat spring wires (for conic sections), or string (for polynomial functions). Sighted math teachers can easily interpret the student-made graphs correctly. You can also make your own rubber graph board by affixing a piece of raised line graph paper (also from APH) to a cork board and proceeding as outlined above.

I do mention the use of Wikki Stix and high dots on APH graph paper when the student MUST hand in copies of graphs for homework to insistent math teachers. However, this method can be quite expensive and is very time consuming and is more of a test of artistic ability. I REALLY want my students to graph extensively; and they can do so incredibly fast on the APH Graphic Aid for Mathematics. In fact, many of my print students insist on using it as well because it is faster, fun, and allows graphing skills to be learned in one more modality.

At the same time, the students are being exposed to the ORION TI-34 talking scientific calculator from Orbit Research, which allows them to perform any necessary computations to speed up the graphing process.

Orion TI-34 Talking Scientific calculator Audio Graphing Calculator Screen Shot

I introduce the AGC (Accessible Graphing Calculator from ViewPlus Technologies) when we start exploring what is and isn't a linear equation. For example, our textbook presents an exploration problem where the students are to first make an educated guess as to whether the graph of an equation will be a straight line or not. Then, they are to test their hypothesis by graphing it. The book lists about 10 equations. Well, that would take quite a long while if the students did everything manually, especially since most haven't had exposure to quadratic equations and rational functions. However, the equations can be quickly entered into the AGC, and the students can listen to the audiowave and immediately tell the differences among y=3*x+4, y=x^2, and y=3/x+2 (the way you must enter equations on the AGC). Additionally, we have a TIGER Advantage networked to each computer, so my students can also emboss each graph very quickly.

My pride and joy is a braille student who I had in Algebra 1 last year. I introduced him to graphing manually and then showed him the AGC, as indicated above. He is now in Algebra 2 and is proficient at both. I continue to show him how to solve Algebra 2 problems manually and with technology, and he analyzes which method is best for which circumstance. For example, he might graph a quadratic function manually because it was "too easy to bother with the computer." Yet, he will use the AGC to graph an exponential function.

Specifics

1. How do students represent inequalities that require a solid line or a dotted line on the graph?

Again, my students use the APH Graphic Aid for Mathematics (has raised grid lines), rubber bands held down by thumbtacks to form the x- and y-axes, and pushpins to plot points. We connect the points with a rubber band when the boundary line is to be included in the solution (solid line in print), and we leave off the rubber band when the boundary line is not included in the solution (dotted or dashed line).

2. How do they show shaded parts on the graph?

When graphing one inequality in two variables, my students simply place their hand on the shaded side. When graphing a system of two inequalities, the student places one hand on the shaded side of the first inequality. Then they place the other hand on the shaded side of the second inequality. Where the two hands overlap (including the boundary lines where applicable) is the solution. Pretty soon most of my students are able to handle three or more inequalities without multiple overlapping of hands. We even progress to linear programming problems involving four or more inequalities. In these problems, a bounded area with vertices is often found, and it is pretty obvious where the shaded portion (solution) is located.

3. Is there a way for them to do multiple problems on a piece of paper?

I check each graph as my students complete them. For example, during a test, they have me check each graph and write a notation on their paper before they move onto the next problem. I check to see if the boundary lines are drawn correctly (with or without rubber band) and if they place the "shading" in the correct area.

If your student does need to hand in several graphics, here are my suggestions:

When needing to graph on a coordinate plane, the student could use APH raised line graph paper attached to a corkboard. Then, he could plot his points using stick-on high dots, puff paint, etc. He could form the solid lines using Wikki Stix. He could actually use a colored pen, pencil, or crayon to color the shaded area of the solution. Of course, this all takes MUCH longer than our method, but this would be necessary if a student-made, manually produced, paper copy is required. Then again, the student could easily hand in a paper copy of any single function (can't graph multiple functions on the same graph) created on the AGC.

One year I had a student whose math teacher insisted that all graphs needed to be handed in on a two-sided piece of paper containing 9 small coordinate planes on each side. This student graphed each equation on the graph board, and I copied the work onto the “designated” sheet. The student was perturbed because I couldn’t keep up with her and was slowing her down! Nevertheless, she passed with flying colors.

I would rather see students become proficient at using the rubber graph board, as they will learn SO MUCH more with this method, and they can do so independently. As an alternative, you could divide the APH Graphic Aid for Mathematics into 4 to 6 small, separate coordinate planes. If you have a digital camera, you could even e-mail or print a picture of your student’s graphs! Better yet, have the student or his parents take the photo!

Bottom Line

PLEASE be sure your students are allowed to participate in all kinds of graphing and are supplied with the proper tools. This creative exploration should begin in the early grades and be allowed to blossom. Remember, the beauty of a tactile graphic is found in the fingertips of the beholder. And there can be no more beautiful and meaningful a graphic than one created by those very same fingertips.

Sources for No Tech, Low Tech, and High Tech Tools:

Susan A. Osterhaus, M.Ed.
Secondary Mathematics Teacher
Texas School for the Blind and Visually Impaired
1100 West 45th Street
Austin, TX 78756
Phone: (512) 206-9305
Fax: (512) 206-9453
E-mail:
Website: http://www.tsbvi.edu/math

A vision assistant asks: How important is it for our elementary kids to do a subtraction problem in the brailler (example: 3 digit subtraction with cancellation signs) if they are using the abacus? When you do a subtraction problem, on the brailler, do you have them solve the problem from right to left?

Susan replies:

I'm going to answer your question from a secondary math teacher's viewpoint. I believe that elementary students need to be exposed to working addition, subtraction, multiplication, and division problems on the braillewriter in a spatial arrangement - not necessarily using cancellation signs - until they understand the concept. They should solve the problem very similar to the way a print student would - in the case of subtraction from right to left. Although the textbook or worksheet may give examples using correct Nemeth Code including cancellation signs, you want the students' calculation procedures to be easy and quick. Your students can always use the abacus to check their work on the braillewriter. All the while, they should be learning mental math techniques as well. Once the concept is learned, speed, accuracy and flexibility are more important, and we should see the student quickly progressing to the abacus and mental math, basic calculators, and eventually scientific calculators as they begin higher mathematics. If a blind student has never had to work a math problem in a spatial arrangement (on a braillewriter, with TACK-TILES, or other manipulative) and has only used an abacus, they will most likely have difficulty in algebra with the concept of adding, subtracting, multiplying, and dividing polynomials when presented in a spatial arrangement. This is especially true with division. You just can't manipulate variables on an abacus.

For more detailed information on suggested calculation procedures with the braillewriter see pages 41-54 from: Gaylen Kapperman, et al., Strategies for Developing Mathematics Skills in Students Who Use Braille, Research and Development Institute, Inc., August, 1997. This publication can be ordered from:

Association for Education and Rehabilitation of the Blind and Visually Impaired (AER)
4600 Duke Street, Suite 430
P.O. Box 22397
Alexandria, VA 22304
Phone: 877-492-2708 or 703-823-9690
FAX: 703-823-9695
E-mail:
Website: http://www.aerbvi.org

Price: $20.00 for AER Members; $30.00 for Non-Members (Shipping charges: Canadian orders add $2.00/book; overseas orders, add $10.00/book)

From ViewPlus Technologies, Inc.

Osterhaus, S.A. (2010). Susan’s math technology corner: The Audio Graphing Calculator (AGC) From ViewPlus Technologies, Inc. Division on Visual Impairments Quarterly, 55(4), 25-27.

Background

The use of a scientific graphing calculator is now an integral part of advanced mathematics classrooms across the U.S., and they are a requirement for many statewide assessments.

The Audio Graphing Calculator (AGC) is a computer software program available from ViewPlus Technologies, Inc. www.viewplus.com The AGC was first introduced in March 2001 as the Accessible Graphing Calculator. Since that time, ViewPlus released Version 2 in 2006 and most recently Version 3 in 2007 and changed its name to the Audio Graphing Calculator.

As you know from previous evaluations, I am always looking for the "best buy" in any math technology. However, although my definition of best buy includes price affordability - user-friendliness, features, and reliability are equally important.

Affordability

You may download a free 30-day fully functional copy of the AGC from the web at http://downloads.viewplus.com/software/AGC/.  Then, check it out for yourself to make sure that it is the right product for your needs. The cost of a single copy license is $295. You can upgrade AGC 2.0 to 3.0 for $195. 

User-Friendliness
The AGC was developed from the ground up to be of universal design, and it is a Windows computer program that features a graphing calculator capable of displaying graphs both visually and audibly as a tone graph. With technology specifically designed for the blind, we often find that although it may be user-friendly for the blind student, there is an extensive learning curve for their sighted teacher. Since the AGC is truly accessible by both the blind and sighted, teachers, parents, and peers do not need to learn to use a special device in order to teach or assist these students.

In fact, the regular ed math teacher may very well wish to use the AGC with the entire class. In this day and age when math anxiety is rampant, and there is an emphasis on the multi-sensory approach, the math teacher is looking for any tool that will capture the attention of and increase understanding in all her students.

The AGC is a complex program and requires some practice before a student will become proficient and totally independent in its use. However, I have noticed that those of my students already technologically proficient in the use of a screen reader are the quickest to learn how to navigate in the self-voiced AGC.

It is not necessary that the student learn all the features of the AGC at once in order to benefit from its use. I feel that features should be introduced as needed, and the AGC should never replace the student learning how to graph manually. The best student knows how to use all the various tools in his/her toolbox.

The AGC comes with an HTML and a self-voiced user manual, getting started instructions, specific instructions regarding the use of screen readers and magnification, and help both off and on-line. On-line tutorials can be found at: http://downloads.viewplus.com/tutorials/AGC+3.0/

Screen shot of AGC, showing a function plot.  AGC matrix multiplication screen shot

Features

The AGC has a scientific keypad calculator; an expression evaluator with the ability to define or import constants and expressions; two data set screens that permit the user to enter equations to graph, import and edit data tables, and compute a number of standard statistical properties; and the ability to plot either data set, their sum or difference, or their first derivative. The user can now graph two functions on the same coordinate plane, and work with matrices. There are various hot keys that allow the user to find relative maximums and minimums and x-axis intersections for functions. When two functions are graphed, there is a hot key to find points of intersection. There are several display options for tone-graph audio plots, and it is self-voicing for usability by people who are blind or dyslexic or kinesthetic learners.

Navigating the AGC is actually pretty user-friendly. Using the arrow, space bar, tab, and shift-tab keys let one move around fairly quickly, but many items can also be selected by using a hot key shortcut. The speech rate, pitch, and volume controls on the Speech screen can be easily adjusted to the individual user's satisfaction. The AGC screen can also be magnified repeatedly (using F8) and then decreased again (using F7). The domain, range, use of grid lines, tick marks, or none, and thickness of the graph can be adjusted to the user's specific requirements from the Plot screen, allowing for further individualization.

Reliability and Flexibility

The AGC is accessible to all. The on-screen graphics are easily seen by a low vision student, and the graph can be listened to by using the audio wave feature. Print copies can be made using any standard printer using a variety of fonts including braille. The print copies created with a braille font can be copied onto Swell Touch paper (available from the American Thermoform Corp. www.americanthermoform.com or Humanware www.humanware.com) and run through a tactile imaging machine to create a raised line graphic. The Braille Swell Paper font (called Swell Braille) is highly recommended, and it can be downloaded for free from http://www.tsbvi.edu/Education/fonts.html. However, in my opinion, the best way to create a tactile graphic is to emboss directly from the AGC to a TIGER (braille/graphics embosser) from ViewPlus.

Recommendations

I highly recommend this calculator program for students taking high school algebra and beyond. It is especially beneficial at the Algebra II level.

Susan A. Osterhaus, M.Ed., CTVI
Statewide Mathematics Consultant
Outreach Department
Texas School for the Blind and Visually Impaired
Austin, TX 78756
E-mail:
Website: http://www.tsbvi.edu/math

Citation for this page: Osterhaus, S.A. (2003). Susan's math technology corner: Scientific Notebook + DBT WIN = Nemeth Code. Division on Visual Impairments Quarterly, 48(3), 23-28.

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Background

Two things have long presented unique problems in the computerized delivery of accessible materials for mathematics instruction: 1) the graphical nature of mathematical notation and 2) the Nemeth Code (the braille code for mathematics) that is needed to represent complicated formulas.

Scientific Notebook (SNB) is a software program based on an easy-to-use word processor that can create textbook quality print documents that contain text, mathematics, and graphics. In addition, it formats mathematics in a way that is ideal for transcription into Nemeth Code. Moreover, SNB includes a built-in scientific graphing calculator, which has a view screen that can be enlarged to 400%, used with a computer program that enlarges text, or used independently by a student with low vision to complete math assignments.

Because adaptations to SNB would make it possible for students with visual impairments to have large print and braille access to mathematics, a project to adapt SNB with braille conversion was started by volunteer Jack Medd and extended by Chris Weaver, program coordinator, Mathematics Accessible to Visually Impaired Students (MAVIS), New Mexico State University. In the earliest stages of the project, I was asked to beta test the SNB/conversion software. I believe the collaboration resulted in a product that produced excellent Nemeth Code by computer generator standards.

Nevertheless, I chose to bring up the already-translated braille file in Duxbury's DBT for final proofreading and possible corrections and formatting changes. At this point, I asked Neal Kuniansky at Duxbury to speak with Chris Weaver to see if it would be possible to incorporate the MAVIS filter into DBT, which would reduce the number of steps in the translation process to two. The result of this networking was DBT WIN 10.3, which incorporates the MAVIS LaTeX importer. Since that time, SNB Version 4.0 has been released, followed by the release of DBT WIN 10.4 to match (Osterhaus, Weaver, Amerson, & Siller, 2001).

As you know from previous evaluations, I am always looking for the "best buy" in any math technology. However, although my definition of best buy includes price affordability - user-friendliness, features, and reliability are equally important.

Scientific Notebook $129 Duxbury DBT $595

Affordability

You may download a free 30-day copy of Scientific Notebook. Then, check it out for yourself to make sure that it is the right product for your needs. The cost of a single copy license is $129 for academic/government users and $99 for students. The cost of a new single user license for DBT WIN 10.4 is $595; however, upgrades are considerably less.

The packaging of these two pieces of software becomes affordable if you consider the versatility involved. Scientific Notebook can be used by teachers, transcribers, and low vision students to create textbook quality technical documents and as a scientific graphing calculator. DBT WIN 10.4 is not limited to Nemeth translation. Among other features, it is capable of contracted and uncontracted literary braille, American computer code, and foreign language translation.

User-Friendliness

At first glance, Scientific Notebook looks like any other word processor. The SNB standard toolbar looks quite familiar, with the exception of one button: the Math/Text Toggle.

SBN toolbar: buttons for - New, Open, Save, etc.

It is this math/text toggle that allows the user to enter text and mathematics in the same document in a seamless easy fashion without the need to cut and paste.

Text is entered at the position of the insertion point when the Toggle Text/Math button on the Standard toolbar shows. alt

Mathematics is entered when the Toggle Text/Math button on the Standard toolbar shows alt

After creating a Scientific Notebook document, you simply save it. As you attempt to open the SNB file with DBT WIN, it will prompt you that you are using the TeX or LaTeX Import Filter, and you may select standard or textbook format. After selecting "OK," DBT WIN imports the document as a print file with hidden codes. It has not yet been translated, so the next step is to translate. Finally, format, proofread, correct any errors with six-key entry, save, and emboss.

Duxbury 'import file' dialog shoing TeX or LaTeX highlighted Duxbury 'File Menu' showing 'Translate' selected.
Duxbury edit window showing all math information formated as standard paragraph (indented 2 spaces). Duxbury edit window showing same materials after translation to Nemeth. Each problem is outdented to cell 1.

Features

In addition to the standard toolbar, Scientific Notebook has several other toolbars that allow for easy insertion of the most advanced mathematics symbols and notation. The Math Templates and Math Objects toolbars allow you to enter fractions, radicals, superscripts, subscripts, parentheses, square brackets, summations, integrals, unit names, displays, more operators, more brackets, matrices, functions, binomials, labels, and decorations.

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The compute toolbar allows you to evaluate, evaluate numerically, evaluate exactly, simplify, expand, graph 2D, graph 3D, and define variables and functions.

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Beginning with SNB Version 4.0, there are several improvements to productivity and compatibility that directly address our needs.

  • Customize the Symbol Cache Toolbar by removing the symbols you don't need and adding any symbols that you do want from the expanded symbol panels. This is quite a time-saver.
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  • Change the style more easily. Use the greatly expanded Tag Appearance dialog box to change the face, style, size, and color of the font to design the perfect on-screen and print document for your low vision student. Previously, we had to rely on Chris Weaver to provide us with an LP or LP2 style file.
  • Work faster with enhanced keyboard shortcuts for standard document operations.
  • Open .rtf documents with greater accuracy. The new rtf2latex2e converter converts many more documents than the previous converter. This means scanning certain documents (science and math documents with a considerable amount of text) is now a more time-efficient option.

If you have an old version of DBT WIN (10.2 or lower), it does not contain the LaTeX Import Filter, and you will need to upgrade to use Scientific Notebook.

A free copy of QikTac is included with DBT WIN 10.4, if you wish to use it to create and produce tactile graphics and combine your math and graphs together on the same page.

Reliability and Flexibility

Scientific Notebook does have its limitations. It is not geared toward the spatial mathematics found in elementary mathematics. In fact, it does not even have a symbol for the "goes into" division symbol. Nevertheless, you can write arithmetic problems horizontally, and it will compute the answers. Furthermore, DBT WIN 10.4 can easily translate the arithmetic problems to correct Nemeth Code.

The original MAVIS filter was able to translate such spatial arrangements as matrices, data tables, and charts correctly into Nemeth Code, but DBT WIN 10.4 cannot. This is probably related to the fact that the MAVIS filter converted the LaTeX directly to braille dots, whereas the DBT WIN import filter automatically puts in the DBT math codes before translating to see the "dots." Nevertheless, DBT WIN does allow an easy method to enter spatial math problems manually with six-key entry. I use this same method for creating my standardized braille number lines (Osterhaus, 2003).

Each time Scientific Notebook upgrades its product, changes need to be made in the LaTeX importer. Previously, Chris Weaver made the various "grammar" updates for the MAVIS filter, but MAVIS is now closed, and Chris continues to do so only on a volunteer basis. Therefore, I believe the MacKichan/Duxbury partnership will be more readily able to upgrade in unison in a timely manner.

About the same time that I began testing the MAVIS filter, math professor Henry Gray from Metroplex Voice Computing approached me regarding the efficacy of applying voice recognition to blind users' needs. The result of this networking was MathTalk/Scientific Notebook .

Recommendations

If you are creating mathematics materials for low vision and blind students, I believe that you will want to use Scientific Notebook 4.1 and take advantage of the LaTeX importer found in DBT WIN 10.4. The combination does not yet produce perfect Nemeth Code, but it does a credible job. Furthermore, your efforts result in a quality print and braille copy.

It continues to be my hope that we can convince more regular ed math teachers to use Scientific Notebook in the creation of their math materials. Then, all students will have textbook quality math materials delivered in a timely fashion. The December 2000 issue of the Mathematics Teacher contained a review of SNB 3.0 (Clark, 2000). The reviewer found the ease of toggling back and forth from text mode to mathematical-expression mode to be more convenient than Equation Editor in Microsoft Word; however, he criticized SNB's font sizing and font type and style. These criticisms have been addressed in SNB 4.1.

WARNING: Always format, proofread, and correct any errors with six-key entry. Your students deserve the best!

References

Clark, T.B. (2000). Reviews: technology-based materials: Scientific Notebook. Mathematics Teacher, 93(9), 794-795.

Osterhaus, S.A., Weaver, C., Amerson, M., & Siller, M.A. (2001). More accessibility for math students: AFB Solutions Forum stakeholders and their pursuit of braille conversion software. Journal of Visual Impairment & Blindness, 95(3), 184-188.

Osterhaus, S.A. (2003). Susan's math technology corner: Standardized braille number lines. Division on Visual Impairments Quarterly, 48(2), 9-11.

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A student learns to use Scientific Notebook.

Susan A. Osterhaus, M.Ed.
Secondary Mathematics Teacher
Texas School for the Blind and Visually Impaired
1100 West 45th Street
Austin, TX 78756
Phone: (512) 206-9305
Fax: (512) 206-9453
E-mail:



 

Osterhaus, S.A. (1998) Braille/Print Protractor teacher's guide. Louisville, KY: American Printing House.

by Susan A. Osterhaus

It is often necessary to measure and draw angles in the geometry classroom as well as every day situations. The most common device for measuring angles is a protractor, usually in the shape of a semicircle. The semicircular edge of the protractor is marked with evenly spaced divisions from 0o to 180o. Simply adding dots to the commercially available product doesn't facilitate its use by a blind user. However, another instrument for measuring angles exists, and it is called a goniometer. Geologists use a goniometer for measuring crystal angles, and physical therapists use one for measuring movement at a joint. A goniometer (protractor with a wand) when adapted provides a very user-friendly angle measuring and drawing device for both blind and sighted users.

The Braille/print adapted protractor has several features that make it easier to use than most adapted protractors currently available. Bold large print numbers and two Braille dots are used to mark degrees at 10o increments. A single dot represents those increments ending in the numeral 5 (with the exception of the 45o, 90o, and 135o which have three dots for quick reference). Probably the most important feature however, is the wand which is attached to the "center" of the semicircle's diameter. When the pointed end of the wand is aligned with a degree measurement, the corresponding angle is formed by the extended straightedge of the wand and the edge (diameter) of the protractor. An additional bonus is that the desired angle's supplement is also formed, allowing a significant teachable moment. Other attractive features are its size and plastic flexibility; it can easily be used to measure most angles in textbooks - even ones close to the binding. Furthermore, this protractor can be used independently to both draw and measure angles.

The recommended procedure for using this protractor consists of the following steps:

Measurement:

  1. Examine the drawn angle to determine roughly whether it is acute, obtuse, or possibly a right angle and as a check after the final measurement.
  2. Loosen the screw positioning the wand, and with both hands form an angle and tentatively align the edge of the protractor and the edge of the wand (making sure that the pointed end is located on top of the protractor) with the respective rays of the provided angle - being sure that the two vertices are aligned as well. Tighten the screw somewhat; fine-tune; and then tighten completely.
  3. The pointed end of the wand will be aligned with the angle's measure. Read the measurement directly in print or begin counting from either end of the protractor's edge (diameter) until the tip of the wand is reached. At this point, check yourself to be sure that you have the angle's measurement and not its supplement. If you completed step 1, you will know. Practice also improves speed and allows short-cuts.

Drawing:

  1. If the plan is to copy a drawn angle using a protractor, then go through Steps 1 - 3 listed above. If the task is simply to draw an angle with a designated measure, carefully align the pointed end of the wand with that measurement and tighten the screw completely.
  2. Place the protractor on the desired drawing surface - plain unlined paper for a visual result or Braille paper on a hard rubberized surface (The Sewell Raised-Line Drawing Board is well-suited for the task.) for a raised-line drawing. Firmly hold the protractor in place with the fingers of one hand - making sure the thumb is holding down the wand securely.
  3. Draw the angle formed by the edge of the protractor and the edge of the wand (being sure to define the vertex) using a print writing instrument for a visual result or an appropriate raised-line tool for a tactile graphic: The Sewell drawing stylus is very easy to hold and manipulate. Line tools 3 and 6 from the APH Tactile Graphics Kit also work well. However, an ordinary pencil or ball point pen may be the best and most easily accessible tool to use.

Osterhaus, S.A. (2001). Susan's math technology corner: The ORION TI-34 talking scientific calculator from Orbit Research. Division on Visual Impairments Quarterly, 46(3), 37-41.
Susan with student using Orion Calculator

Background

Approximately two years ago I decided to evaluate all the available accessible standalone talking scientific calculators to facilitate the purchase of several for high school students' needs. After the evaluation period, I constructed a chart comparing the various features of these calculators and listing possible vendors and suggested prices; I then published this chart on my website. I found that no one calculator had everything for all people. Therefore, I still maintain that each individual should assess their own needs and/or that of their student and make a choice based on their unique situation. Maxi-Aids, Independent Living Aids, Inc., and Orbit Research graciously loaned me their products for this evaluation and should be commended for their support.

The last such calculator I evaluated was the ORION TI-34. All of my students and I were in agreement that the ORION was the calculator best suited for our needs. In my case, I am always looking for the "best buy" in any math technology. However, although my definition of best buy includes price affordability - user-friendliness, features, and reliability are equally important.

Affordability

The  flyers http://www.orbitresearch.com/ from Orbit Research all proclaim the ORION to be the "world's first affordable talking scientific calculator." At $249 per unit, this is truthful advertising. In previous years, we could not afford to pay up to three times this amount for the convenience of one standalone calculator, especially when we needed a classroom set. Instead, we chose to use four function talking calculators in combination with talking scientific calculator software located on various desktop and laptop computers and notetakers - which didn't make for a very unified approach to instruction. We can now afford multiple classroom sets and an individual can now afford her own personal ORION.

User-Friendliness

The ORION is small (5.8 by 2.9 by 1.5 inches) and light in weight (11 oz), so a classroom set is easily stored. For individuals, an ORION is very portable and can easily fit into a pocket, purse, or backpack. The top .5 inch layer of the ORION (including the keypad and sliding, impact resistant plastic cover) begins with the popular TI-34 scientific calculator from Texas Instruments, so the ORION user automatically receives the benefits of TI's extensive experience in calculator design. The remaining 1-inch depth contains the heart of the ORION and allows it to voice every key on the keypad in natural speech. In addition, this extra casing provides the volume control (thumb-wheel), ear outlet, adapter (battery charger) input port, learn/speech mode button, and repeat/off button.

In our experience, high school students in academic and remedial math classes became familiar with the layout of the basic keys quite quickly. They found the learning mode button extremely helpful. Initially, it allowed them to identify the various keys and learn which keys would be of benefit to them at their respective math levels. As they began to perform operations, they liked the convenience of being able to exit a computation problem, enter the learning mode to find the location of a specific key, exit the learning mode, and re-enter the computation operation without disturbing their problem in progress. The students also liked the clear speech output.

With technology specifically designed for the blind, we often find that although it may be user-friendly for the blind student, there is an extensive learning curve for their sighted teacher. Since the ORION is truly accessible by both the blind and sighted, teachers, parents, and peers do not need to learn to use a special device in order to teach or assist these students.

In fact, if the regular ed math teacher has selected the TI-34 as the calculator of choice for the classroom, since both the TI-34 and the ORION have identical functionality, the entire class can use the same calculator. An algebra teacher visited me recently. His district had asked that he learn everything there was to know about teaching algebra to a blind student in one afternoon, as a blind student would be enrolled in his class the n year. I proceeded to do the best I could in the limited amount of time and showed him all my various tools and technology - all of which were new to him. Imagine the smile of recognition, when I showed him the ORION - the only piece of technology with which he was familiar. He breathed a sigh of relief as he told me that he had used the TI-34 extensively with his previous classes and teaching a blind student how to use the ORION would be a snap.

The ORION user manual comes in large print, cassette, and floppy disk, and an optional teacher's manual is available for the TI-34. A high quality earphone for quiet operation is included, as well as the AC adapter.

Orion calculator, manual, charger, ear piece, audio cassett, software, cheat sheet.

Features    

In addition to the user-friendly features, the ORION offers over 95 scientific functions including statistics and trigonometry. This should not be thought of as intimidating, but instead as versatile. After learning the basic keys, students can slowly learn each new function key as their math skills increase.

Although the students in remedial mathematics could perform the vast majority of their computations on a regular 4 function talking calculator, they all fell in love with the fraction key, which is not available on any other standalone talking calculator. The steps necessary to solve fraction problems were learned quickly and easily. Not only were the students able to add, subtract, multiply, and divide fractions and mixed numbers, they were able to reduce their answers to lowest terms and convert them to improper fractions or decimal equivalents. Another popular key was the backspace key found on computer keyboards. The only other standalone talking calculator that has this feature is the Audiocalc from Canada. This function allows the user to backspace during construction, and erase any digit in a constituted number, starting with the last digit entered. The ability to quickly fix a mistake without having to start all over again while performing a complicated calculation makes this key extremely valuable.

Pre-algebra and algebra students, who encounter a great many fractions and are just as prone to making entry errors, were equally enamored with the fraction and backspace keys. They quickly added the +/- key to their repertoire, enabling them to manipulate negative numbers as well. The x! (factorial) key also proved handy. Order of operations is preserved including the use of parentheses; therefore little need exists for extra memory positions. The square and square root functions and scientific notation quickly followed. As students advance into higher algebra, they begin to use the universal power and root functions. Geometry students will want to add the use of the pi constant.  

Reliability

We have found the ORION TI-34 calculations to be accurate and reliable. It is capable of displaying and speaking entries and results up to 10 digit+2 digit (-99 to +99) exponents and has an internal accuracy of 12 digits.

It is extremely important to initially charge the battery for at least 16 hours before using it. Once fully charged, it should be capable of over 6 hours of continuous operation, which translates to several days or even weeks of average intermittent use. The calculator has the added feature of shutting itself down after being idle (for a time pre-selected by the user) to preserve the battery charge. Then it is best to charge the unit overnight for about 14-16 hours, shortly after it starts announcing "low battery". It is of course important that the battery be discharged as much as possible before being recharged, to avoid the "memory effect" that is present in every rechargeable battery. The battery should last between 500-1000 charging cycles.

The ORION is so sophisticated that one should think of it more as a miniature computer; and it should be handled with care. Similar to a computer, the earliest models of the ORION TI-34 had certain software bugs and hardware problems. The ORION has been expressly designed to be software upgradable, in order to fix any bug that may arise. Specific circuit changes have now removed the buzzing sound that was present on some of the earlier units. Orbit Research assures me that other bugs have been fully investigated, understood, and fixed. They have a complete database, which tracks all problems that occur in the field so that they can fully address these efficiently. These issues have been resolved by enhanced circuitry in the newer units and by updating circuitry on older units. The newer models also feature a new molded case. Orbit Research backs up the product with a 1-year warranty which covers manufacturing defects. I have always found them to be most cooperative, supportive, and greatly appreciative of all input from their customers.

Recommendations

I definitely recommend this calculator for students taking high school algebra, geometry, and pre-calculus. Furthermore, it can easily be used by middle school students in pre-algebra and by any student requiring assistance in working with fractions, after thorough instruction in fraction concept development. WARNING: After exposure to the ORION, 99% of students are addicted and want one of their own.

Susan A. Osterhaus, M.Ed.
Secondary Mathematics Teacher
Texas School for the Blind and Visually Impaired
1100 West 45th Street
Austin, TX 78756
Phone: (512) 206-9305
Fax: (512) 206-9453
E-mail: or

Website

Osterhaus, S.A. (2004). Susan's math technology corner: Secondary Mathematics Education: The Years of Growth and Challenge. Division on Visual Impairments Quarterly, 50(1), 17-20.

How I wish that someone would ask me the question: How do you teach secondary mathematics to sighted students? Then I could reply: The same way I teach blind students! I do not mean that each individual student should be treated exactly the same. Each student is unique, but all students need access each year they are in school to a coherent, challenging mathematics curriculum that is taught by competent and well-supported mathematics teachers. (NCTM, 2000)  I strive to appeal to as many senses as possible, so I encourage all of my students to read, speak, listen to and look at, touch and feel, sing, and sometimes even smell and eat mathematics - basically completely immerse themselves in the problem at hand. In my experience the more in-roads math concepts have to access the brain, the more likely your student will be able to out-put a correct solution to a problem and transfer that knowledge when learning a new concept. Too many students: especially students who are poor, not native speakers of English, disabled, female, or members of minority groups - are victims of low expectations in mathematics. (NCTM, 2000)  Unfortunately this has often been the case with the majority of blind and visually impaired students, who fall under this umbrella - frequently in several categories. Classroom mathematics teachers must provide high expectations for all their students, and they should be strongly supported by staff trained in the special needs of students with visual impairments. 

A Few Secondary Math Education Links to Get You Started

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Sources for quality manipulatives and other math materials:

What major challenges are encountered when teaching math concepts to blind and visually impaired students?

One of the most difficult challenges for me has been teaching concepts involving three-dimensional objects. When I first did my student teaching (over 35 years ago), I taught geometry in a regular education classroom. My nickname was The Tinker Toy Lady because I was always coming to class with some kind of physical 3-D model to illustrate the day's lesson. 3-D problems are found in all levels of mathematics. They are often difficult for students with vision to understand, especially when trying to create 3-D objects in a two-dimensional drawing. Hey! They are difficult for me!! Such a drawing, even when tactually raised, makes little sense without sighted perspective. Yet, the textbooks continue to draw these 3-D raised line drawings that seem to contradict what the math teacher has just taught the student. For example, a teacher may have just explained to a student that a cylinder has two bases, which consist of two congruent circles and their interiors, and let them examine several real cylinders. Then, when the homework is assigned or the test is administered, they are given a two-dimensional drawing that would seem to indicate that a cylinder only has one base, which consists of an ellipse and its interior. Sometimes my students would be better off without the picture. Whereas, it may help the sighted student, it often causes confusion for the blind student. In addition, the blind student has to learn what the 3-D object really feels like, and then what it feels like as a sighted person would see it. Talk about extra work!

The next most immediate challenge is keeping up with the advancement in math technology tools for the sighted. The scientific graphing calculator is now becoming a requirement for coursework and even standardized tests. There is not yet an accessible equivalent for the very popular TI-83 for example. The blind student can work the majority of these problems without a scientific graphing calculator, but the point is that they are at a disadvantage if they must do everything manually. Nevertheless, the Accessible Graphing Calculator combined with the ORION TI-34 talking scientific calculator allow these students to at least approach a level playing field. (Osterhaus, 2003; 2002; 2001) My latest dilemma is finding an accessible equivalent to the Geometer's Sketchpad.

What advice would you give to a general education teacher who has a student with a visual impairment?

These are my collaborative/inclusive strategies:

  • Adapted educational aids are a necessary component of any mathematics class. They are especially needed to supplement textbooks that have omitted tactile graphics or contain poor quality ones. However, they are also needed to help in interpreting mathematical concepts - just as their sighted peers benefit from various manipulatives. It is very beneficial to the entire class when the braille student's aid is a fun and useful tool for the sighted students and teacher as well.
  • Math teachers need to verbalize everything they write on an overhead, blackboard, or whiteboard and be precise with their language. If the braille learner still has difficulty keeping up, the math teacher should be encouraged to give the student/vi teacher a copy of their overhead transparencies prior to class if pre-prepared or immediately after. Another alternative might be for a classmate to make a copy of their notes to share. The use of whiteboard technology, which allows transmission of the board contents to a low vision student's laptop works very well.
  • Math teachers need to give worksheets, tests, etc. to VI teachers to transcribe into Nemeth far enough in advance, so that the braille student can participate with their fellow students in class - not later alone. Print copies should be legible as well. One way to insure this is for the math teacher to prepare their print materials using Scientific Notebook; then all students can receive high quality materials in a timely fashion in regular and large print and Nemeth code.
  • Relate various mathematical applications to student activities enjoyed by blind students as well as the sighted students -

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  • Put various mathematical concepts to song or at least teach similar to an athletic cheer.
  • The FOIL method for multiplying binomials F - O - I - L: First, Outside, Inside, Last!!!!
  • Quadratic formula sung to the tune of Pop Goes the Weasel
  • Be sure to include athletic experiences that a blind student can relate to; include the parabolic curve of a diver, as well as the football quarterback's pass.
  • Math teachers need to realize that it is their job to teach the mathematical concepts to their students. This is not the job of the VI teacher. The VI teacher can be very helpful by insuring that all materials are in proper Nemeth code and all graphics are of good quality if the math teacher is able to supply these in print in a timely manner. However, any math teacher will tell you that there is always that teachable moment that you cannot anticipate. This is when it is imperative that the math teacher has some tools at his/her disposal. It is the responsibility of the VI teacher to expose the math teacher to the various tools and aids available to him/her. Math teachers can be quite creative, as many VI teachers have discovered. See Assistive math tools for a list of suggested tools and technology.
  • Blind students should not be excused from learning a math concept because they are blind: "Blind students can't graph." "Blind students can't do geometric constructions." Not only can they graph and draw geometric constructions, with the right tools, they can often do so better than their sighted peers. Consideration should be taken into account however with regard to number of problems assigned. It is permissible to shorten the assignment, as long as the student can demonstrate competence in the content area.
  • It is very important for all students (and especially for the VI student) to use as many senses as possible when learning a new math concept. They need to read a new math problem, write it, listen to it, tactually explore it through manipulatives, and when possible move their body and/or manipulative through space. If it's a fractional problem involving food for example, they can even taste and eat the problem.
  • There is an ongoing need for four-way communication among the math teacher, the VI teacher, the family, and the student. Braille textbooks, materials, and aids need to be ordered early. The source of a problem needs to be discerned as quickly as possible - is it the math concept, the braille, or the quality of the tactile graphic? Vocabulary in itself can be a problem. Fractions have numerators and denominators in print and braille; however, they have "tops" and "bottoms" in print and "lefts" and "rights" in braille.
  • For classroom test taking, the student should be given the test in their reading medium (with an option for partial oral administration; for example, in the case of students with learning disabilities who need word problems read) and supplied with appropriate enlarged/tactile graphics, aids, abacus, and/or a talking/large display scientific/graphing calculator. Blind students should be given at least twice the time to complete tests. At times, it may be desirable for the blind student to take the test separate from the group due to the needed extra time, use of aids (especially those involving speech), and/or partial oral administration.

References

NCTM (2000). Principles and Standards for School Mathematics [On-line] Available: http://www.nctm.org/standards/

Osterhaus, S.A. (2003). Susan's math technology corner: Back-2-School: What's new and what's improved. Division on Visual Impairments Quarterly, 49(1), 5-8.

Osterhaus, S.A. (2002). Susan's math technology corner: The Accessible Graphing Calculator (AGC) from ViewPlus Software. Division on Visual Impairments Quarterly, 47(2), 55-58.

Osterhaus, S.A. (2001). Susan's math technology corner: The ORION TI-34 talking scientific calculator from Orbit Research. Division on Visual Impairments Quarterly, 46(3), 37-41.

image009 altSusan A. Osterhaus, M.Ed.
Secondary Mathematics Teacher
1100 West 45th Street
Austin, TX 78756
Phone: (512) 206-9305
Fax: (512) 206-9453
E-mail:

Citation for this article: Osterhaus, S.A. (2004). Susan's math technology corner: Early Childhood: Where Learning Mathematics Begins. Division on Visual Impairments Quarterly, 49(3), 41-47.

Early childhood is when and where children begin to discover that math is all around them. They use, enjoy, and think about math and don't even realize it. Math activities are embedded in real-life activities and "help children develop language as they ask questions, develop fine motor skills as they touch and move objects, and improve social skills as they work" with parents and others on a problem (Fromboluti & Rinck, 1999). All children develop at different rates. Visually impaired students may need extra time to develop and learn, so it is extremely important that they are given opportunities to participate and learn at an early age. There is a great wealth of information on teaching mathematics in early childhood that is not specific to visual impairment. Nevertheless, many of these activities either need no adaptation or can be easily adapted. Just make sure that the adaptations are appropriate.

A Few Early Childhood Mathematics Education Links to Get You Started

What sort of activities might reinforce the NCTM principles and standards at the early childhood level?

  • Read children's books that rhyme or repeat, contain numbers and tactile pictures, and use multiple senses.
  • Sort objects looking for similarities in either color, shape, or size.
  • Sort objects looking for differences, like which box is bigger.
  • Touch and manipulate containers, blocks, shape sorters, and puzzles.
  • Count and become familiar with numbers.
  • Share food, chores, or treats by dividing into equal portions.
  • Place stickers in columns on a piece of braille graph paper to make a chart of how many of each type of food were consumed during the week.

To implement these activities, parents and teachers can use many items easily found in the home or environment. There are also several sources for early childhood math manipulatives that need no adaptation or are easily adapted (Osterhaus, 2002). However, there are also a few vendors who are making a real effort to invent/locate/adapt math tools specifically for the young visually impaired child. These vendors include (but are not limited to):

American Printing House for the Blind

On the Way to Literacy Storybooks Illustrated with Thermoforms

The Gumdrop Tree

A tree which grows from a gumdrop! The raised-line drawings depict the growth of the tree. The gumdrops are shown in a row, circle, square, etc. Includes scented stickers.

Jellybean Jungle

A counting rhyme about a magical jungle filled with jellybeans. From one to ten, the jellybeans appear in rows for easier counting. Scented stickers included.

Puzzle Form Board Kit

Aid for teaching basic concepts such as shape, texture, color, size, and matching. This kit includes five different puzzle shapes (3 variations each) with puzzle frames: circle, diamond, square, rectangle, and triangle.

Peg Kit

Contains nine pegs of varying sizes, colors, and textures; a manipulation/storage container; and six inset frames with holes of varying number and size. Frames fit into the top of the container and provide for manipulating the pegs.

Tactile Treasures

Teaches more than 90 concepts related to shape, size, comparison of two or more objects, amount, position, and page orientation. 79 thermoformed sheets featuring tactile pictures created from real objects that illustrate math and language concepts.

Feel n Peel Stickers

Multi-use tactile/visual stickers.

  • Assorted
  • Point Symbols
  • Reward Statements
  • Faces
  • Alphabet

Exceptional Teaching Aids

Hands-On Soft Geometric Solids

Contains 12 geometric solids: cone, sphere, hemisphere, cube, pyramid, cylinders, prisms, and rectangular solids.

Geometric Shapes Jumbo Knob Puzzle

These super-sized solid wood puzzles are the perfect first puzzle.

Shape Sorting Pounder

Sort geometric shapes and develop fine motor skills at the same time.

Deluxe Magnet Wipe-Off Board and Holding Magnets

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This dual-purpose coated metal board is perfect for use with strong colorful plastic coated magnets. The set includes 40 round magnets and 40 rectangular magnets housed in sturdy storage containers. Four dry erase markers are also included. The magnets are excellent for developing math concepts.

Peel & Stick WonderFoam®

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WonderFoam®has peel and stick backing. It is available in a 720-piece bag of assorted colors, shapes, and sizes and a package containing 20 sheets of assorted bright colors. The sheets are 9" by 12".

Independent Living Aids

Head Start Set of Tactile Images with Braille Booklets

Three booklets introduce the young child to the braille alphabet, numbers, and shapes. Using a polymeric material on heavyweight paper, results in braille and graphics that are easy to feel and durable.

Braille Math Blocks

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Sixteen colorful wooden blocks with numbers (0 - 9) and plus, minus, and equals signs that are embossed and brailled in Nemeth code. In addition, the corresponding number of recessed circles have been embossed next to the numbers.

Variety Texture, Color and Size Round Stick-Ons

Included are 2 sheets each of 23 brown felt, 28 green velour and 18 cork round stick ons. The large stick-ons measure ½" and the small ones ¼" in diameter.

Feel and Find

Each of the 10 geometric and 10 object shapes fits into its corresponding cut out that has a textured base. Children play by dealing the tiles and then reaching into the bag to feel for the matching shape. Great tactile exercise for all kids.

Geometric Sorting Games

Young children learn size progression, shapes and quantities when playing with this colorful wooden toy. It has 20 geometrically shaped wooden pieces in size progression.

Talking Ship Ahoy Sorter

Three fun games teach shape identification, sorting, color recognition, animals and their sounds, and matching. The child places the shapes into their matching holes to hear the names of the shapes. When the four differently shaped objects are placed into the ship, it plays "Row, Row, Row Your Boat". Includes volume control & automatic shut-off.

Brailled Count and Learn The Shapes

This fun and educational puzzle teaches numbers and basic shapes. Made of high-density foam, each of 10 numbers fit into the form with the equivalent number of shapes. Each number section has been brailled and each of the shapes on each section has been brailled with the same number, allowing progressively more difficult math concepts to be learned.

References

Susan A. Osterhaus
Texas School for the Blind and Visually Impaired
Phone: 512-206-9305
E-mail:

Osterhaus, S.A. (2001). Susan's math technology corner: Back-2-school shopping list for algebra and geometry. Division on Visual Impairments Quarterly, 47(1), 23-25.
Susan watching a group of students using math software on a computer

Math Materials

Braille Reader

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Large Print Reader

  • High Quality Large Print Textbook
  • Enlarged Materials
  • Regular Print with Magnification
  • Be Alert for Color-Keyed Graphics

Basic Tools or Technology

Braille Reader

  • Braillewriter
  • Refreshable Braille
  • Braille Paper
  • Braille Eraser
  • Abacus

Large Print Reader

  • Black Line Paper
  • Proper Writing Implement
  • Desktop Computer
  • Laptop
  • Low-Vision Software

Calculators

For All

Large Display

Drawing Tools

  • Adaptive Math Tools.htm
  • Drawing Board
  • Howe Press Compass
  • Straightedge
  • Tracing Wheel
  • APH Braille/Print Protractor
  • Stylus and/or Pen

Measuring Tools

Braille compass and drawing pad. Braille prodractors.

Student-Generated Graphics

Geometric Manipulatives

Math Graphics Made by Others

Computer-Generated Graphics

Susan A. Osterhaus
Texas School for the Blind and Visually Impaired
Phone: 512-206-9305
E-mail:

Osterhaus, S.A. (2003). Teaching a blind student how to graph on a coordinate plane: No tech, low tech, and high tech tools. SEE/HEAR, 8(1),19-21.

Susan Osterhaus, Secondary Mathematics Teacher, TSBVI

Editor's Note: In the author's words, "Although graphing calculators are mainstays of most secondary math classrooms, it is important for all students to understand the concept of graphing on a coordinate plane before they move to the graphing calculator." This is especially important for visually impaired students, and Susan Osterhaus, math teacher at TSBVI, ensures that her students learn to do so manually _ they must physically plot points, graph lines, and find slope. Below are her answers to questions about how to teach this skill, and her suggestions for students, teachers, and parents.

1. How can blind students graph linear equations, inequalities, and systems of inequalities independently and efficiently? Or is this the time when the VI student doesn't participate because of the visual nature of the task?

Most academic blind students, even those with spatial orientation problems, are quite capable of graphing, and as one of my students exclaimed, "Not only can we do it, it's fun!" There are several tools they can use to do so:

The Graphic Aid for Mathematics, from the American Printing House for the Blind (APH), is excellent for graphing algebraic equations. It can also be used in geometry, trigonometry, etc. It consists of a cork composition board mounted with a rubber mat, which has been embossed with a grid of 1/2-inch squares. Two perpendicular rubber bands, held down by thumbtacks, can create the x- and y-axes. Points are plotted with pushpins at the appropriate coordinates. Points are connected with rubber bands (for lines), flat spring wires (for conic sections), or string (for polynomial functions). I like for my students to graph extensively, and they can do so incredibly fast on the APH Graphic Aid. In fact, many print students also like using it because it is fast, fun, and allows them to learn graphing skills in another modality. You can make your own graph board by affixing a piece of raised line graph paper (also available from APH) to a cork board and proceeding as described for the Graphic Aid.

If a student needs to turn in copies of graphs for homework, he can use Wikki Stix and High Dots on APH graph paper. This method can be quite expensive, however, and is very time consuming. It also tends to be more of a test of artistic ability than a demonstration of understanding of graphing concepts.

The ORION TI-34 talking scientific calculator (from Orbit Research) and the Accessible Graphing Calculator (from ViewPlus Technologies) are examples of more high tech solutions for graphing activities. I described them in a previous See/Hear article (Winter, 2002), but strongly recommend that students be able to graph manually as well. It is important for visually impaired students to be able to use a variety of tools, and know when to use each of them. For example, a former student decided to graph a quadratic function manually because it was "too easy to bother with the computer." Yet, he will use the AGC to graph an exponential function.

2. How do students represent inequalities that require a solid line or a dotted line on the graph?

The APH Graphic Aid described above works well. Plot the points with pushpins and connect them with a rubber band when the boundary line is to be included in the solution (a solid line in print). Leave off the rubber band when the boundary line is not included in the solution (dotted or dashed line in print).

3. How can VI students show shaded parts on a graph?

When graphing one inequality in two variables, I simply have my students place their hand on the shaded side. I check each graph as my students complete them. When graphing a system of two inequalities, the student places one hand on the shaded side of the first inequality. Then they place the other hand on the shaded side of the second inequality. Where the two hands overlap (including the boundary lines where applicable) is the solution. Pretty soon most of my students are able to handle three or more inequalities without multiple overlapping of hands. We even progress to linear programming problems involving four or more inequalities. In these problems, a bounded area with vertices is often found, and it is pretty obvious where the shaded portion (the solution) is located.

4. Is there a way for them to do multiple problems on a piece of paper? What if they need to be turned in to another teacher, or you can't check each graph as it's completed?

If a student-made, manually produced paper copy is required, the student could use APH graph paper attached to a corkboard. She could plot her points using stick-on high dots, puff paint, etc. and could form solid lines using Wikki Stix. She could actually use a colored pen, pencil, or crayon to color the shaded area of the solution. This would take much longer, however, and would be very labor intensive. It will be important to know the purpose of the assignment and the concept(s) being taught. A paper copy of a single function can be created on the AGC (it can't graph multiple functions on the same graph). Often, it is possible for a sighted person (teacher, peer, parent, teaching assistant) to make a print copy of the student's graph _ the visually impaired student graphs on the Graphic Aid and someone copies it exactly onto a piece of paper to turn in. You can also divide the Graphic Aid into 4 to 6 small, separate coordinate planes for multiple problems. If you have a digital camera, you could even e-mail or print a picture of the student's graphs. Better yet, have the student take her own photos!

Please be sure that visually impaired students are allowed to participate in all kinds of graphing activities and that they are supplied with the proper tools. I would rather see them become proficient using a rubber graph board because they will learn so much more with this method, and they can do so independently. Creative exploration should begin in the early grades and allowed to blossom. Remember, the beauty of a tactile graphic is found in the fingertips of the beholder. And there can be no more beautiful and meaningful graphic than one created by those very same fingertips.

Sources for materials described in this article:

APH Graphic Aid for Mathematics and APH Graph Paper: www.aph.org

Wikki Stix: www.wikkistix.com

High Dots: www.exceptionalteaching.com

ORION TI-34 Talking Scientific Calculator: www.orbitresearch.com

Accessible Graphing Calculator (AGC): www.ViewPlusSoft.com